Comprehensive Multi-Scale Progressive Failure Analysis for Damage Arresting Advanced Aerospace Hybrid Structures

Brandon Horton

Dissertation submitted to the Faculty of Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of

Doctor of Philosophy In Mechanical Engineering

Javid Bayandor, Chair Francine Battaglia Fayette S. Collier Dawn C. Jegley Walter F. O’Brien Wayne A. Scales

July 18th, 2017 Blacksburg, Virginia

Keywords: PRSEUS, stitched composites, finite element, composite failure, progressive damage

Copyright 2017, Brandon Horton

Comprehensive Multi-Scale Progressive Failure Analysis for Damage Arresting Advanced Aerospace Hybrid Structures

Brandon Horton

ACADEMIC ABSTRACT

In recent years, the prevalence and application of composite materials has exploded. Due to the demands of commercial transportation, the aviation industry has taken a leading role in the integration of composite structures. Among the leading concepts to develop lighter, more fuel- efficient commercial transport is the Pultruded Rod Stitched Efficient Unitized Structure (PRSEUS) concept. The highly integrated structure of PRSEUS allows pressurized, non-circular fuselage designs to be implemented, enabling the feasibility of Hybrid Wing Body (HWB) aircraft. In addition to its unique fabrication process, the through-thickness stitching utilized by PRSEUS overcomes the low post-damage strength present in typical composites. Although many proof-of- concept tests have been performed that demonstrate the potential for PRSEUS, efficient computational tools must be developed before the concept can be commercially certified and implemented. In an attempt to address this need, a comprehensive modeling approach is developed that investigates PRSEUS at multiple scales. The majority of available experiments for comparison have been conducted at the coupon level. Therefore, a computational methodology is progressively developed based on physically realistic concepts without the use of tuning parameters. A thorough verification study is performed to identify the most effective approach to model PRSEUS, including the effect of element type, boundary conditions, bonding properties, and model fidelity. Using the results of this baseline study, a high fidelity stringer model is created at the component scale and validated against the existing experiments. Finally, the validated model is extended to larger scales to compare PRSEUS to the current state-of-the-art. Throughout the current work, the developed methodology is demonstrated to make accurate predictions that are well beyond the capability of existing predictive models. While using commercially available predictive tools, the methodology developed herein can accurately predict local behavior up to and beyond failure for stitched structures such as PRSEUS for the first time. Additionally, by extending the methodology to a large scale fuselage section drop scenario, the dynamic behavior of PRSEUS was investigated for the first time. With the predictive capabilities and unique insight provided, the work herein may serve to benefit future iteration of PRSEUS as well as certification by analysis efforts for future airframe development.

Comprehensive Multi-Scale Progressive Failure Analysis for Damage Arresting Advanced Aerospace Hybrid Structures

Brandon Horton

GENERAL AUDIENCE ABSTRACT

In recent years, the prevalence and application of composite materials has exploded. Due to the demands of commercial transportation, the aviation industry has taken a leading role in the integration of composite structures. Among the leading concepts to develop lighter, more fuel- efficient commercial transport is the Pultruded Rod Stitched Efficient Unitized Structure (PRSEUS) concept. The highly integrated structure of PRSEUS allows a new type of fuselage design to be implemented, known as Hybrid Wing Body (HWB) aircraft. PRSEUS unique fabrication process, the through-thickness stitching utilized by PRSEUS overcomes the low post-damage strength present in typical composites. Although many proof-of-concept tests have been performed that demonstrate the potential for PRSEUS, efficient computational tools must be developed before the concept can be commercially certified and implemented. In an attempt to address this need, a comprehensive modeling approach is developed that investigates PRSEUS at multiple scales. The majority of available experiments for comparison have been conducted for small specimens. Therefore, a computational predictive methodology is developed to accurately model the response of PRSEUS. A thorough analysis is performed to identify what needs to be considered in the model to predict an accurate result while remaining computationally efficient. From the baseline analysis, realistic models of the PRSEUS structure are created numerically and validated against the existing experiments. Finally, the validated approach is extended to panel and a fuselage section to compare PRSEUS to the current state-of-the-art. Throughout the current work, the developed methodology is shown to make accurate predictions that are well beyond the capability of existing predictive models. While using commercially available softwares, the methodology developed herein can accurately predict local behavior up to and beyond failure for structures such as PRSEUS for the first time. Additionally, by applying the methodology to a fuselage section drop scenario, the dynamic behavior of PRSEUS was investigated for the first time. With the predictive capabilities and unique insight provided, the work herein may serve to benefit future iteration of PRSEUS as well as certification by analysis efforts for future airframe development.

Acknowledgements

I wish to acknowledge my gratitude to my advisor, Dr. Javid Bayandor for his guidance and the many opportunities he has provided throughout my graduate studies. I would also like to acknowledge my Ph.D committee members, Dr. Francine Battaglia, Dr. Fay Collier (Program Manager), Ms. Dawn Jegley (Program Manager), Dr. Walter O’Brien, and Dr. Wayne Scales. Their involvement in my thesis work and the critiques they have provided has improved my dissertation. This research was supported by the “Multiscale Modeling of Advanced Aerospace PRSEUS Structures” grant from the National Aeronautics and Space Administration (NASA), for which I am very grateful. It has provided me with the opportunity to collaborate with many technical experts and exposed me to much of the cutting edge research performed at NASA. In particular, the guidance and insights of the Program Manager, Ms. Dawn Jegley, has had a significant impact on the success of this work. I also greatly appreciate the collaboration with the Marion branch of General Dynamics to conduct non-destructive imaging for this project, particularly with Joseph Butler and Kurt Ruoff. I wish to thank my colleagues and friends in the CRASH Lab for their endless advice and support throughout the many long nights of research. In particular, the mentoring and collaboration with the recent Dr. Yangkun Song has made a huge impact on my technical abilities in finite element analysis. I wish to thank my parents and my sister for their love and emotional support that has carried me through my graduate work. Without them, I wouldn’t be where I am today. Finally, I wish to thank Joyci for her endless patience, care, and understanding, especially during the final stages of my dissertation.

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Table of Contents

1 Introduction ...... 1 2 Element Formulation Theory ...... 6 2.1 Beam Element Formulation ...... 7 2.2 Shell Element Formulation ...... 8 2.3 Hourglass Control ...... 9 2.4 Solid Element Formulation ...... 13 2.5 Continuum (Thick) Shell Element Formulation ...... 15 2.6 Contact in Finite Element Analysis...... 16 3 Material Theories ...... 19 3.1 Composite Material Model ...... 19 3.2 Cohesive Zone Modeling ...... 23 3.3 Metal Material Model ...... 25 3.3.1 Isotropic Model ...... 25 3.3.2 Johnson-Cook Model ...... 27 3.3.3 Equation of State ...... 29 4 Preliminary Element Study ...... 31 4.1 Cantilever Beam Analysis ...... 31 4.2 Buckling Analysis ...... 42 5 Component Level Analysis ...... 50 5.1 Stringer Development ...... 50 5.1.1 Stringer Characteristics ...... 51 5.1.2 Stringer Model Development ...... 52 5.1.3 Boundary Condition Setup ...... 57 5.1.4 Epoxy Property Identification ...... 59 5.1.5 Validation of the PRSEUS Stringer ...... 64 5.1.6 Reduced Expense Model...... 73 5.1.7 Uncertainty Quantification ...... 76 5.2 Frame Development ...... 81 5.2.1 Frame Characteristics ...... 82 5.2.2 Frame-Stringer Model Development ...... 84 5.2.3 Validation of the PRSEUS foam-core and tapered L-frames ...... 87 5.3 Chapter Summary and Conclusions ...... 96

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6 Large-Scale Analysis ...... 98 6.1 Panel Analysis ...... 98 6.1.1 PRSEUS Panel Characteristics ...... 98 6.1.2 Computational Analysis of PRSEUS Panel ...... 102 6.2 Reduced Fuselage Analysis ...... 115 6.2.1 Conventional Metal Alloy Fuselage Section ...... 116 6.2.2 Reduced Fuselage Section Setup ...... 117 6.2.3 Comparison of the PRSEUS and Metal Fuselage ...... 122 6.2.4 Comparison of the Stitched and Un-Stitched PRSEUS Fuselage ...... 130 7 Conclusions ...... 137 7.1 Future Work and Recommendations ...... 138 Appendix A : Stringer Mesh Study ...... 140 Appendix B : Delamination Validation ...... 145 B.1 Element Formulation Parametric Study ...... 148 B.2 Mesh refinement study ...... 149 B.3 Parametric Study of Composite Damage Models and Validation...... 152 Appendix C : Metal Fuselage Validation ...... 155 Bibliography ...... 163

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List of Tables

Table 4.1 Material properties for Alu 2024-T3...... 32 Table 5.1 Summary of material properties reported from experimental testing and the initial numerical input...... 54 Table 5.2 Summary of updated material properties based on stiffness knockdown factors ...... 67 Table 5.3 Summary of observed parameter influences on the computational model and their relative impact ...... 72 Table 5.4 Material inputs for manufacturing tolerance in the Class 72 Type 1 material stack ...... 78 Table 6.1 Input parameters for Johnson-Cook material of Alu 7075-T6 [136] and Alu 2024-T3 [91] with damage properties ...... 117 Table 6.2 Mass distribution for the metal and PRSEUS fuselage models ...... 122 Table A.1 Summary of results for the GCI analysis ...... 142 Table B.1 Continuum parameters based on literature [82,104,147]...... 145 Table B.2 Normalized computation expense for each element formulation...... 149 Table B.3 Summary of the results for the GCI analysis ...... 151 Table B.4 Summary of backface displacement, total delamination area at time of bullet arrest, and number of failed plies for each of the selected material failure models ...... 153 Table C.1 Mesh characteristics for B737 model and corresponding computation expense ...... 156

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List of Figures

Figure 1.1 Image of the N3-X BWB concept [27] ...... 1 Figure 1.2 Schematic of loads present in a non-circular pressurized fuselage such as the HWB [45] ...... 2 Figure 1.3 Exploded view of the PRSEUS concept ...... 3 Figure 1.4 Illustration of the material stack breakdown for the Class 72 Type 1 PRSEUS material with the 0° fiber direction aligned with the pultruded rod ...... 4 Figure 1.5 Illustration of the chain stitching architecture utilized by PRSEUS ...... 5 Figure 2.1 Schematic view of a 2-D quadrilateral element (4-node) with a single Gauss point ...... 8 Figure 2.2 Schematic diagram of a fully integrated shell element with four Gauss points ...... 9 Figure 2.3 Zero-energy (hourglass) deformation of an under-integrated quadrilateral element .... 10 Figure 2.4 Simulation results for no hourglass control (top) and viscous hourglass control (mid) with under-integrated shells and a fully integrated shell model ...... 12 Figure 2.5 Reduced integration (left) and fully integrated (right) solid elements ...... 14 Figure 2.6 Single Gauss point (left) and four Gauss point (right) options for continuum shell elements ...... 15 Figure 2.7 Schematic of penalty contact by shifting penetrated nodes to the contact surface ...... 16 Figure 2.8 Contact surfaces for lower dimension elements such as shells (left) and beams (right) ...... 17 Figure 3.1 Schematic of unidirectional composite laminate with representative failure modes (adapted from [64])...... 19 Figure 3.2 Schematic of three primary separation modes (adapted from [83]) ...... 23 Figure 3.3 Schematic of the bilinear mixed-mode stress-displacement model ...... 24 Figure 3.4 Engineering stress-strain comparison between a representative experimental test (black) and linear elastic-plastic hardening model (dashed red) ...... 27 Figure 3.5 Representative illustration of change in Alu 2024-T3 and Alu 7075-T6 behavior at different strain rates ...... 28 Figure 4.1 Simulation setup of square cantilever beam with a single load at the beam tip for the solid (top), vertical shell (left) and horizontal shell (right). All dimensions are in meters .... 31 Figure 4.2 Comparison of load vs. beam tip displacement and energy for under-integrated elements with no hourglass control (left) and with viscous hourglass control (right) ...... 34 Figure 4.3 Example of stress bands produced by shear locking for the solid element ...... 35

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Figure 4.4 Load vs. beam tip displacement for implicit analysis with a single through-thickness element and isotropic material ...... 36 Figure 4.5 Load vs. beam tip displacement for implicit analysis with two through-thickness elements and isotropic material ...... 38 Figure 4.6 Load vs. beam tip displacement for explicit analysis when the displacement is applied over 1 s (left) and 100 s (right) ...... 39 Figure 4.7 Load vs. beam tip displacement with two through-thickness elements and composite material [4 Gauss point thick shell (left) and selectively reduced thick shell (right)] ...... 40 Figure 4.8 Final deformation for both solid and 2x2 quadrature assumed strain thick shell models ...... 41 Figure 4.9 Simulation setup for square beam buckling with a symmetric boundary condition and the fixed-fixed loading condition. All dimensions are in meters...... 42 Figure 4.10 Load vs. beam tip displacement plot (left) and energy distribution (right) for the under-integrated element formulations with isotropic material ...... 43 Figure 4.11 Load vs. beam tip displacement for higher integration methods with a single through- thickness element (left) and two through-thickness elements (right) with isotropic material 44 Figure 4.12 Load vs. beam tip displacement for a full-length specimen (left) and error magnitude compared to the symmetric model (right) ...... 45 Figure 4.13 Load vs. beam tip displacement for explicit analysis when the displacement is applied over 1 s (left) and 100 s (right) ...... 45 Figure 4.14 Load vs. beam tip displacement for two through-thickness elements with composite material ...... 47 Figure 5.1 Test article for the PRSEUS stringer compression experiment (reprinted from [23]).. 51 Figure 5.2 Schematic of a PRSEUS stringer ...... 52 Figure 5.3 Schematic view of the mesoscale modeling methodology ...... 53 Figure 5.4 PRSEUS stringer for tiebreak model (top) and beam-stitch model (bottom) ...... 55 Figure 5.5 Illustration of local material angles along the overwrap for the pultruded rod ...... 56 Figure 5.6 Illustration of the three tested boundary condition ranging from the highest fidelity (left) to the simplest (right) ...... 57 Figure 5.7 Load vs. compression displacement for each of the applied boundary conditions ...... 59 Figure 5.8 Load vs. opening displacement (left) and crack length vs. opening displacement (right)

for a DCB test of [0/90]s preforms ...... 60 Figure 5.9 Simulation setup for DCB test with pre-cracked sections colored red ...... 61 Figure 5.10 Load vs. opening displacement for a range of applied loading rates ...... 61

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Figure 5.11 Bilinear traction-displacement rule for different peak tractions ...... 62 Figure 5.12 Y-stress contours of the DCB simulation at the highest mesh density ...... 63 Figure 5.13 Load vs. opening displacement for each peak traction overlaid with experimental data ...... 64 Figure 5.14 Load vs. displacement for both the tiebreak-stitch and beam-stitch stringer models . 66 Figure 5.15 Loading vs. displacement response for progressive development of the PRSEUS stringer ...... 69 Figure 5.16 von-Mises stress contours for both stitched and un-stitched stringer configurations (the near surface is at the symmetric boundary) ...... 71 Figure 5.17 Illustrations of shell-beam (left) and solid-beam (right) low reduced computational expense models ...... 74 Figure 5.18 Loading vs. displacement for the beam and shell reduced fidelity models ...... 75 Figure 5.19 Average load-displacement response for the beam-solid model with error and standard deviation bars ...... 77 Figure 5.20 Plots of load vs. displacement for positive and negative changes in axial (top) and lateral (bottom) moduli with zoomed area in buckling regime ...... 79 Figure 5.21 Load vs. displacement response for two different initial perturbations in the geometry ...... 80 Figure 5.22 Test article for the PRSEUS frame compression experiment (reprinted from [23]) .. 82 Figure 5.23 Schematic of a PRSEUS foam-core frame (left) and tapered L-frame (right) ...... 83 Figure 5.24 Exploded view of tapered L-frame at the frame-stringer intersection (no stitching shown) ...... 84 Figure 5.25 Illustration of a low fidelity approach for the tapered L-frame ...... 85 Figure 5.26 Load vs. displacement response for the low fidelity tapered L-frame model ...... 86 Figure 5.27 Illustration of a model with and without stack dropdowns ...... 86 Figure 5.28 Illustration of high fidelity foam-core frame (left) and tapered L-frame (right) ...... 87 Figure 5.29 Load vs. displacement for the high fidelity foam-core frame model (left) and tapered L-frame model (right) [*the data was clipped beyond this point due to the edge of the plot]88 Figure 5.30 Stress contours (left) and bonding contours (right) for the foam-core frame model .. 90 Figure 5.31 Stress contours (left) and bonding contours (right) for the tapered L-frame model ... 92 Figure 5.32 Sequentially labeled locations of local element failure for the high fidelity foam-core frame (top) and tapered L-frame (bottom) ...... 95 Figure 6.1 PRSEUS panel harvested from the MBB (left) [130]...... 99

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Figure 6.2 Cross-sectional views of the stringer (left) and frame (right) from the PRSEUS panel ...... 100 Figure 6.3 Example oscilloscope reading for a fully bonded sample (top left) and a debonded sample (top right) with delamination locations shown on the PRSEUS panel (bottom) ..... 101 Figure 6.4 Schematic of the desired test panel based on the non-delaminated area...... 102 Figure 6.5 High fidelity models of the foam-core frame panel (top) and tapered L-frame panel (bottom) with simulation setup and designations for each frame and stringer ...... 104 Figure 6.6 Load vs. displacement for both the foam-core panel (left) and tapered L-frame panel (right) with comparison to component predictions and ultimate failure estimates...... 105 Figure 6.7 Sequential contours of resultant displacement for the foam-core frame panel with the symmetric boundary at the near surface ...... 107 Figure 6.8 Sequential contours of von-Mises stress for the foam-core frame panel with the symmetric boundary at the near surface ...... 108 Figure 6.9 Sequential contours of resultant displacement for the tapered L-frame panel with the symmetric boundary at the near surface ...... 112 Figure 6.10 Sequential contours of von-Mises stress for the tapered L-frame panel with the symmetric boundary at the near surface ...... 113 Figure 6.11 Correct delamination contours (left) and incorrect delamination contours (right) ... 115 Figure 6.12 Illustration of the conventional metal frame and stringer substructure (approximated rivet locations are shown by the blue dots) ...... 116 Figure 6.13 Illustration of the entire reduced section model with the conventional metal design (top) and PRSEUS tapered L-frame (bottom) with a close-up of a single substructure ...... 120 Figure 6.14 Schematic of boundary condition and setup for the reduced section drop test ...... 121 Figure 6.15 Energy distribution for both the metal and PRSEUS fuselage during the initial impact and bounce ...... 123 Figure 6.16 Progression of y-velocity contours for both the metal (left) and PRSEUS (right) fuselage models. The view is cut just beyond the centerline to illustrate the deformation at the centerline ...... 125 Figure 6.17 Normalized vertical acceleration response for the metal and PRSEUS fuselage section drop analysis during the initial impact and bounce ...... 126 Figure 6.18 Sequential contours of von-Mises stress for both the metal (left) and PRSEUS (right) fuselage models at the bottom center of the frame ...... 127 Figure 6.19 Contours of plastic strain for the metal fuselage at the areas subjected to the greatest stress at t = 0.04 s ...... 129

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Figure 6.20 Contours of delamination for the un-stitched (left) and stitched (right) models of the PRSEUS fuselage at the bottom center of the frame ...... 130 Figure 6.21 Contours of delamination for the un-stitched (top) and stitched (bottom) PRSEUS structures 30-50˚ from the bottom center of the frame ...... 132 Figure 6.22 Contours for von-Mises stress (top) and delamination (bottom) at the passenger floor attachment, t = 16.75 ms ...... 134 Figure 6.23 Contours for von-Mises stress (top) and delamination (bottom) at the passenger floor attachment, t = 25.00 ms ...... 135 Figure 6.24 Contours for von-Mises stress (top) and delamination (bottom) at the passenger floor attachment, t = 30.00 ms ...... 136 Figure A.1 Load-displacement plots at three mesh densities (left) and the response of the medium mesh density with error bars (right) ...... 143 Figure B.1 : Illustration of mesoscale model setup and methodology based on experimental testing by Gower et al...... 146 Figure B.2 Delamination energy plots for each element formulation ...... 149 Figure B.3 Percent delamination area and computation time versus number of elements ...... 150 Figure B.4 Delamination shapes at three different mesh densities ...... 151 Figure B.5 Projectile velocity with and without a mechanism for delamination ...... 152 Figure B.6 Qualitative comparison of delamination area from the simulation (left) with comparison against an analytical prediction [161] (middle) and experimental observation [162] (right) ...... 154 Figure C.1 B737-200 forward fuselage section model for dynamic crash test ...... 156 Figure C.2 G-loading time history of the left and right sides of the passenger floor from the section drop test at each mesh density (top) and sequential von-Mises stress contours for the metal fuselage at the medium grid refinement corresponding to the local acceleration peaks (bottom) ...... 158 Figure C.3 Comparison of acceleration history between the experimental measurement and numerical prediction at the medium mesh density for each side of the passenger floor...... 160 Figure C.4 Comparison between section drop experiment conducted by NASA [133] (left) and metal alloy computational result (right) ...... 161 Figure C.5 Qualitative comparison of post-drop frame shape for metal (left) and composite (right) with the fuel tank made transparent ...... 162

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Preface

Over the past several decades, the prevalence of composite materials in every application has exploded. The aviation industry in particular has made a concerted effort to replace as many metal components as is feasible with a composite counterpart in an effort to reduce long term costs by reducing weight [1]. Composites have been so widely accepted due to their superior moduli and strength, tailorability through various ply orientations, and low weight compared to metal alloys

[2–5]. Before the 1980s, composites were virtually unheard of in commercial aircraft structural design [5], but now composites account for 50% or more of the structural weight in many aircraft.

Although composites have many advantages over traditional metal alloys, some of these advantages lead to critical limitations. The combination of matrix material and stiff high-strength fibers is a major part of what makes fiber-reinforced polymer (FRP) structures so versatile, but at the same time, the disparity in material response makes composites susceptible to interlaminar separation known as delamination [6,7]. If delamination occurs, then the composite becomes incapable of supporting any load in the affected region, often propagating the delamination until full failure has occurred. Additionally, the combination of materials with diverse mechanical properties make composites inherently anisotropic, resulting in an orientation dependent material response. Due to their complex nature, predicting failure in composites is an arduous task and is still a major topic of research today [8–12].

In an effort to develop more damage-tolerant aerospace structures, researchers at Boeing and

NASA collaborated to develop the Pultruded Rod Stitched Efficient Unitized Structure (PRSEUS)

[13–15]. This advanced structural concept utilizes through-thickness stitching to arrest delamination propagation, therefore mitigating the damaged area [16–18]. Unlike other state-of- the-art composite aircraft structures, PRSEUS uses its stitching to replace almost all of the welds, rivets, and bolts necessary to attach different components [19]. In addition to making PRSEUS a highly damage-tolerant design, the stitching also plays a major role in the manufacturing process.

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Sewing dry preforms of the skin, frames, stringer, tear straps, and T-caps together into a single self- supporting assembly allows the entire integrated preform to be co-cured together [20]. By simplifying the fabrication process, the proposed method significantly reduces the total manufacturing cost and eliminates the need for any complex mold tooling. In addition to lowering costs, the unique architecture of PRSEUS enables the possibility for a new generation of hybrid wing body (HWB) aircraft [21]. These aircraft would be able to move away from the traditional tube-wing aircraft and implement non-circular fuselage designs, allowing more efficiency and diversity than currently exists in the aviation industry.

Although many proof-of-concept tests have been performed that demonstrate the potential for

PRSEUS, efficient computational tools must be developed before the concept can be commercially certified and implemented. To address this goal, a comprehensive modeling approach must be developed by investigating PRSEUS at multiple scales. The majority of available experiments for comparison have been conducted at the component level. Therefore, using this scale as a baseline, a physics based approach is developed that can be used at a wide range of scales. Most analyses use either a mesoscale approach at the small-scale or a smeared system property model at the large- scale. However, such a constrained approach requires the fundamental modeling assumptions to change with the scale of the problem. Therefore, the proposed methodology combines the mesoscale and smeared property approaches into a single, versatile model. For example, unlike most other proposed models, the current methodology explicitly represents each stitch with a beam element, overcoming several of the disadvantages present in the more widely used tiebreak model without being computationally infeasible.

Due to the complex construction of PRSEUS, the finite element model must accurately represent the material, adhesive layers, and stitching. At the component level, this is achieved by implementing a unique combination of solid elements, beams, and cohesive zones, allowing high accuracy simulations at minimal computational expense. On the other end of the spectrum, the analysis of individual components is coupled together to effectively simulate the response of an

xiv | P a g e entire panel. Eventually, a full fuselage can be developed, allowing the possibility for optimization or certification tests to be virtually performed without incurring any of the experimental costs. With the proposed computational methodology, PRSEUS can be further developed and implemented into the aviation industry.

While PRSEUS has been experimentally proven to be an effective damage mitigation structure, relatively little research has been performed to develop corresponding high fidelity computational models. Current computational models for PRSEUS and other stitched composites use simplified approaches to determine a global, macroscopic response or do not consider failure under extreme loading [22–24]. Before PRSEUS can be implemented into a full-scale aircraft, it must be certified for a wide variety of loading conditions and tests. Without an accurate computational model, the certification process would require an incredible number of physical tests, making it prohibitively expensive. The goal of this study is to develop a comprehensive high fidelity modeling methodology for PRSEUS that accurately incorporates failure. The main objectives of this research are:

1. Develop a methodology to accurately capture the response of PRSEUS at multiple

scales and correlate with experimental data, specifically at the stringer and frame level.

2. Investigate the failure and damage development of stitched structures at higher scales,

specifically at the panel and fuselage level.

With additional contributions of:

a. Provide comparison to the existing metal alloy architectures for conventional tube-wing

aircraft design.

b. Investigate the uncertainty associated with numerical error and manufacturing tolerance.

c. Aid in future certification by analysis efforts for future airframe development.

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Publications

Accepted Journal Papers [1] Horton, B., Song, Y., Feaster, J., and Bayandor, J., 2017, “Benchmarking of Computational Fluid Methodologies in Resolving Shear Driven Flow Fields,” J. Fluids Eng.

[2] Song, Y., Horton, B., Perino, S., Thurber, A., and Bayandor, J., 2017, “A Contribution to Full-scale High Fidelity Aircraft Progressive Dynamic Damage Modeling for Certification by Analysis,” Int. J. Crashworthiness

In-Progress Journal Papers [3] Song, Y., Horton, B., Schroeder, K., and Bayandor, J., 2017, “Ingestion of Commercial Unmanned Aircraft Systems into High Bypass Propulsions Systems,” J. Aircraft

[4] Horton, B., Song, Y., and Bayandor, J., 2017, “High-fidelity Computational Modeling of Stitched Composites for Aerospace Structures,” J. Aircraft

[5] Horton, B., Song, Y., and Bayandor, J., 2017, “Numerical and Experimental Investigation of Through-thickness Stitched Panels for Hybrid Wing Body Aircraft,” J. Aircraft

Published Conference Papers [6] Song, Y., Horton, B., and Bayandor, J., 2017, “Investigation of UAS Ingestion into High- Bypass Engines, Part 1: Bird vs. Drone,” 58th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, American Institute of Aeronautics and Astronautics, Reston, Virginia.

[7] Schroeder, K., Song, Y., Horton, B., and Bayandor, J., 2017, “Investigation of UAS Ingestion into High-Bypass Engines, Part 2: Parametric Drone Study,” 58th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, American Institute of Aeronautics and Astronautics, Reston, Virginia.

[8] Horton, B., and Bayandor, J., 2016, “Numerical Investigation of Fan-Blade Out using Meso- scale Composite Modeling,” Proceedings of the 30th Congress of the International Council of the Aeronautical Sciences, ICAS, Daejeon.

[9] Song, Y., Schroeder, K., Horton, B., and Bayandor, J., 2016, “Advanced Propulsion Collision Damage due to Unmanned Aerial System Ingestion,” Proceedings of the 30th Congress of the International Council of the Aeronautical Sciences, ICAS, Daejeon.

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[10] Horton, B., and Bayandor, J., 2016, “Quantifying the Fluid Modeling Capability of SPH and CLE Through the Study of the Lid-Driven Cavity Problem,” ASME Proceedings of the Symposium on Applications in CFD, ASME, Washington DC.

[11] Song, Y., Horton, B., and Bayandor, J., 2015, “A Contribution to Full-Scale High Fidelity Aircraft Progressive Dynamic Damage Modeling for Certification by Analysis,” Aerospace Structural Impact Dynamics International Conference (ASIDIC), Seville, Spain.

[12] Horton, B., Matta, A., Battaglia, F., Müller, R., and Bayandor, J., 2015, “Test Setup for Determination of Bat Flight Kinematics,” ASME Proceedings of the 6th Symposium on Bio- Inspired Fluid Mechanics, ASME.

[13] Asbury, P., Nichols, R., Gadell, G., Elsheikh, M., Galbraith, B., Horton, B., Marino, J., Nesaw, C., Kossa, M., Collie, Z., Amaya, J., Feaster, J., Bender, M., Matta, A., Bayandor, J., Kurdila, A., Battaglia, F., and Mueller, R., 2014, “Unsteady Flow Analysis Strategies for Flapping Flight,” ASME Proceedings of the 5th Symposium on Bio-Inspired Fluid Mechanics, ASME, Chicago.

In-Progress Conference Papers [14] Horton, B., Song, Y., Jegley, D., and Bayandor, J., 2017, “High-fidelity Computational Modeling of Stitched Composites for Aerospace Structures,” Proceedings of the 20th International Conference on Composite Structures, ICCS, Paris.

[15] Horton, B., Song, Y., and Bayandor, J., 2017, “Full-scale Aircraft Ditching Prediction using Compressible Fluid-solid Interactive Strategy,” Aerospace Structural Impact Dynamics International Conference (ASIDIC), Wichita.

[16] Horton, B., Song, Y., and Bayandor, J., 2018, “Numerical Investigation of Stringer-Frame Intersections for Stitched Aerospace Structures,” 59th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, American Institute of Aeronautics and Astronautics, Kissimmee.

[17] Horton, B., Song, Y., Jegley, D., Collier, F., and Bayandor, J., 2018, “Predictive Analysis of Stitched Aerospace Structures for Advanced Aircraft,” Proceedings of the 31st Congress of the International Council of the Aeronautical Sciences, ICAS, Belo Horizonte.

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[18] Song, Y., Horton, B., and Bayandor, J., 2018, “Fuselage Section Drop Simulation Using Advanced Finite Element Method,” Proceedings of the 30th Congress of the International Council of the Aeronautical Sciences, ICAS, Belo Horizonte.

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Dissertation Map

2 Element Formulation Theory

Chapter 2 presents a comprehensive discussion of the element types used in the current research and their respective theories. Considering that the primary focus of the work is computational model development, understanding of the underlying theory is highly important. An example of hourglass energy is presented with a discussion of its effect on simulation quality.

Specific details of contact algorithms that have direct application to the current work are also discussed.

3 Material Theories

Chapter 3 discusses the various material model formulations for both anisotropic and isotropic materials. Underlying considerations that a composite material model must take into account are presented, with focus on the Chang-Chang composite theory. The fundamental of the bilinear traction-displacement law for cohesive zones is discussed. Both the simplified linear elastic-plastic and phenomenological Johnson-Cook material models for representing metals are presented. A brief discussion of the Mie-Gruneisen equation of state concludes the chapter.

4 Preliminary Element Study

Chapter 4 conducts a large parametric element study to determine the actual working capabilities of each element type. Simulations were performed at loading scenarios deemed to be representative of existing experimental coupon tests as well as expected flight loads: cantilever beam bending and column buckling. The elements considered for the analysis were solids, shells

(in both horizontal and vertical orientations), continuum shells, and beams. The parametric analysis also included the formulation of each element type, time integration schemes, and assigned material model. By the conclusion of the chapter, element types are selected that are applied throughout the rest of the entire modeling methodology.

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5 Component Level Analysis

Chapter 5 is split into two primary parts. The first major section introduces the PRSEUS stringer and thoroughly investigates the necessary modeling conditions required to obtain accurate results. Multiple analyses are conducted to determine the necessary input conditions before a stringer model can even be simulated including boundary studies and interlaminar property identification. With all of the required inputs, two different modeling approaches are presented and attempted. Once a single approach is chosen for further analysis the stringer is validated against an experimental baseline. A brief investigation of numerical uncertainty and the effect of manufacturing tolerance is performed.

The second major section applies the methodology developed for the stringer to two different

PRSEUS frame designs. Using the same modeling approach, the two frame designs are validated against existing experiments. In both of these sections, attempts are made to mitigate computational expense without jeopardizing the accuracy of the numerical predictions.

6 Large-Scale Analysis

Chapter 6 applies the methodology that was previously validated for the PRSEUS stringer and frame components to large-scale structures such as a panel and fuselage section. The characteristics of a PRSEUS panel provided for future experimental testing and numerical validation are presented.

Based on these characteristics, two PRSEUS panel configurations are created and simulated. A thorough discussion of the failure mechanisms and estimated ultimate failure is provided.

The final discussion investigates the largest scale, the reduced fuselage section. A fuselage section drop test is chosen to analyze the dynamic performance of a PRSEUS fuselage for a scenario relevant to aircraft certification. A conventional metal alloy fuselage is also developed based on the design of a B737 to serve as a baseline for comparison. Both stitched and un-stitched configurations of the PRSEUS fuselage are also created to determine the effect of stitching in such a large-scale dynamic scenario.

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1 Introduction The hybrid wing body (HWB), also known as Blended Wing Body (BWB), is one of the leading innovative concepts to replace the conventional tube-wing aircraft design. In preceding publications, the conceptual idea for the wing-body structure was initially discussed by Eriksson in

1982 [25]. Later in 1994, a team of researchers with multidisciplinary backgrounds investigated the HWB concept to initiate a revolution in aircraft design and development [26]. In conventional tube-wing aircraft, the primary lift production is only generated by the wing. Alternatively, the

HWB concept utilizes the unique design of its fuselage to create lift across the entire body as shown in Figure 1.1.

Figure 1.1 Image of the N3-X BWB concept [27]

The unique contouring of the aircraft body creates the possibility for many advantages of current aircraft. The reduced drag and increased lift production of the fuselage can reduce fuel consumption, while also increasing the payload capacity by allowing storage over a much greater surface area. Due to these potential advantages, the HWB concept promises to be a revolutionary concept in high efficiency subsonic transport, maybe one day replacing the current commercial

1 | P a g e aircraft. However, the unique fuselage geometry also introduces many complexities that have to be addressed. Many studies have focused on optimizing the HWB concept by consolidating the main parameters such as wingspan, mass, and flight speed to develop an optimized shape [26,28,29].

The concept has provoked further detailed investigations by many different disciplines including fluid dynamics [30–32], propulsion systems [33–36], structural development [17,37–40], and environmental impact [41–43].

One of the primary concerns from the structural perspective is the complex pressure distribution developed on the interior of a HWB. For conventional tube-wing fuselage design, calculating a stress caused by the internal pressure is well-assessed using hoop stress calculations.

The hoop stress is directly proportional to the internal pressure and radius of the tubular structure but is inverse to the skin thickness. In a HWB, however, the hoop stress calculation is no longer valid. Due to the unique geometry, physical experiments have been conducted to test the structural integrity as a function of the pressure load for new structural technologies designed for the HWB

[44]. In addition to the uneven pressure distribution, the flight loads present on the wing are directly transferred to the cabin section of the HWB as shown in Figure 1.2. Therefore, the structure of a

HWB requires a new state-of-the-art structural concept to be developed in order to sustain the combination of out-of-plane and aerodynamic loads on the aircraft.

Figure 1.2 Schematic of loads present in a non-circular pressurized fuselage such as the HWB [45]

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One such future technology is the pultruded rod stitched efficient unitized structure (PRSEUS) developed by a collaboration between NASA and Boeing, as shown in Figure 1.3. The technology uses a carbon fiber pultruded rod in conjunction with uni-directional carbon/epoxy laminates that are stitched together to overcome the most prominent problem in composite failure, delamination.

Carbon fiber-reinforced polymer (CFRP) composites are well-known for their outstanding strength-to-weight ratio relative to aerospace grade metal alloys. Among other significant improvement, the ability of PRSEUS to arrest delamination propagation gives it a huge advantage over typical composites. During manufacturing, fibers are stitched through the entire lamina and hold the PRSEUS components into a single part. Then this part can be co-cured all at once using the state-of-the-art Controlled Atmosphere Pressure Resin Infusion (CAPRI) process [46], producing a high strength bond. Unlike metallic structures, this advanced stitched composite structure does not require rivets or other external connections, reducing the weight of the airframe significantly.

Figure 1.3 Exploded view of the PRSEUS concept

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As previously mentioned, PRSEUS utilizes uni-directional CFRP laminates throughout the entire construction. A wide variety of fiber and epoxy matrix combination have been used throughout the development of the PRSEUS concept, but the for the current research, the Class 72

Type 1 material is implemented throughout. Each “stack” of Class 72 material is composed of seven layers, which are each individually composed of AS4 fibers and VRM-34 epoxy as shown in Figure 1.4. Multiple orientations of the individual layers are used to create a material stack with effective properties that are strong in all directions, including under shear loading. A symmetric stacking sequence is used to prevent any out-of-plane bending or twisting under in-plane axial loads.

When assembling a PRSEUS structure, multiple layers of these stacks are used in which the orientation and stacking sequence is always the same. Throughout the dissertation, the 0° fiber orientation is referred to as the primary fiber direction of the stack.

Figure 1.4 Illustration of the material stack breakdown for the Class 72 Type 1 PRSEUS material with the 0° fiber direction aligned with the pultruded rod

Each of the material stacks are stitched together using a 3-D chain stitch, as shown in Figure

1.5. Instead of a conventional two-sided stitching approach that would require access to both sides of the structure, PRSEUS is fabricated using a one-sided stitching technology. Two needles are operated from the same side of the laminate, with one inserting the thread and the other catching the loop and pulling it back through the other side of the laminate. The approach can be used for

4 | P a g e large structures with a six degree of freedom robotic arm that allows up to fifteen stacks to be stitched together at once.

Figure 1.5 Illustration of the chain stitching architecture utilized by PRSEUS

The presented work aims to assess the possibility of designing tube-wing airplane structures using the PRSEUS technology as an intermediary step before completely shifting to the HWB concept. By adopting the advantages of the PRSEUS technology, the existing tubular fuselage design may be further improved to be both lighter and more damage resistant than current state-of- the-art in aircraft development. As a precursor to assist in the development and certification of future aircraft implementing the PRSEUS concept, extensive computational analyses have been conducted to demonstrate the possibilities of this technology without incurring the extreme costs of experimental testing. By developing a high fidelity, physics based modeling methodology, the current work performs investigations at scales ranging from the component level to a reduced fuselage section. Eventually, the methodology may be applied to design and optimize PRSEUS for future HWB structures.

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2 Element Formulation Theory To conduct the high fidelity computational analysis of PRSEUS, the commercial finite element software LS-Dyna is used. As the goal of this work is to develop an effective methodology that can be applied toward future PRSEUS certification by analysis efforts, commercially available code is a much more practical option than a personal research code. While this does introduce some limitations with regards to the available element and material models, it is more important to make the results of this work accessible to future investigators.

Although LS-Dyna is primarily known for its time explicit finite element analysis, both implicit and explicit methods are used intensively throughout the presented analysis. Implicit methods are typically used when element deformation is relatively low and dynamic effects are negligible. By simultaneously solving the equations describing each node, implicit schemes are more computationally expensive per iteration, but are unconditionally stable and allow much larger time steps to occur [47]. In the current work, implicit methods are used whenever possible for quasi-static loading cases. For highly dynamic systems where there is large element deformation or failure, then explicit methods are used. Unlike the implicit scheme, explicit calculates the solution at each node based on the information from the previous time step [48]. Once every node has been updated at a given iteration, the simulation advances a time step and recalculates the solution. This forward time-marching approach requires stability criteria based on the material speed of sound in order to remain stable [49]. While this versatile approach allows failure and damage progression to be modeled, its necessarily small time step is unsuited for long duration scenarios such as quasi-static loading.

The inherent complexity of the PRSEUS architecture requires multiple types of elements to be used in order to effectively characterize the response. At the component level, solid elements are used to define the composite stacks and are initially used for the pultruded rod. Shell and continuum shell elements are investigated at the component level, but are generally used for higher scales such as the fuselage section-drop test. Beams are used to represent the stitching that is critical

6 | P a g e to the PRSEUS concept and are also used to model the pultruded stringer rod. The limitations of triangular and tetrahedral elements are well documented [50] and are not implemented for the proposed research. Each of the discussed elements is thoroughly investigated under relevant loading conditions in Chapter 5.

2.1 Beam Element Formulation

Beams are among the simplest element formulations, generally composed of only two nodes with a third virtual node that acts as a reference point. Due to their inherent simplicity, beams are a good option when computational speed is more important than model fidelity. The lowest order beams can be used as truss or cable elements, where only axial displacement and force are required.

Higher order forms incorporating Hughes-Liu [51] or Timoshenko [52] beam theory can be used to quickly calculate estimates for bending intensive problems.

Because beam elements are only one-dimensional, various integration schemes are used to approximate the cross-sectional area of different shapes. To achieve this, a four Gauss point quadrature rule is typically used to calculate stress distribution and approximate the moment of inertia. The virtual third node is necessary to calculate the moment of inertia because it provides a reference to the initial, un-deformed configuration of the beam. Because a beam has only one node at each end, applying realistic boundary conditions can be problematic. Within the current analysis, the primary application of beam elements is to represent the stitch. In this capacity, the beam acts as a cable, which only provides tensile axial resistance. For the purpose of reducing computational expense, higher order beam elements are considered to approximate the pultruded rod in the

PRSEUS stringer.

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2.2 Shell Element Formulation

The simplest quadrilateral element is the four node two-dimensional element. Because of the relatively low computational expense, the quadrilateral element is widely used for much research in aerospace applications because the structure is thin enough to warrant using shell elements [53–

56]. Shell elements operate using plane stress assumption and are therefore incapable of carrying a through-thickness stress. 2 shows a coordinate system of the quadrilateral element with a single

Gauss point and the corresponding boundaries. The shape function N1 to define node 1 does not influence nodes 2, 3, and 4.

Figure 2.1 Schematic view of a 2-D quadrilateral element (4-node) with a single Gauss point

To numerically eliminate the influence of the node 1 shape function on the neighboring nodes, the expression shown in Eqn. (2-1) is defined. Similar expressions are derived at the remaining nodes, and all are divided by four to normalize the magnitude of the shape function:

(1 + 푠)(1 + 푡) N (푠, 푡) = (2-1) 1 4 (1 − 푠)(1 + 푡) N (푠, 푡) = (2-2) 2 4 (1 − 푠)(1 − 푡) N (푠, 푡) = (2-3) 3 4 (1 + 푠)(1 − 푡) N (푠, 푡) = (2-4) 4 4

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While the under-integrated quadrilateral element requires little computational effort, its single

Gauss point makes it prone to accuracy problems. For scenarios where a higher degree of accuracy is required, the number of Gauss points can be increased as shown in Figure 2.2. By allocating four

Gauss points, any strain will be detected by the integration rule. Therefore, no “zero-energy” deformation can occur, which negates the possibility of hourglass energy (discussed in Section 2.3).

As expected, the enhanced strain formulation provided by the fully integrated shell element requires significantly greater computational expense than the under-integrated shell (approximately 2.5 times as much [57]). More details of the shell element formulations can be found in Cook et al. [58] and other publications [59,60].

Figure 2.2 Schematic diagram of a fully integrated shell element with four Gauss points

2.3 Hourglass Control

Even though the under-integrated quadrilateral element point can deliver a solution with low computational expense, it is prone to accuracy problems because of the overly simplified element calculations. A common occurrence for any reduced integration formulation is the “zero-energy” mode deformation, also known as hourglass mode deformation, as illustrated in Figure 2.3.

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Figure 2.3 Zero-energy (hourglass) deformation of an under-integrated quadrilateral element

The reason that hourglass modes occur in the under-integrated shell element is because of the insufficient number of Gauss points. Deformation shapes shown in Figure 2.3 are subject to bending loads which do not register any strain when only a single Gauss point is used. Because the Gauss point is located in the center of the element, such a deformation does not cause any change in the integration rule. Consequently, the element is incapable of providing any resistance to the applied load, leading to spurious energy and a reduction in accuracy.

Many computational models require supplemental corrections because of the accuracy issues associated with the hourglass deformations. Hourglass control was developed as a remedy for under-integrated elements, allowing the user to avoid computationally intensive elements with full integration, but requires careful use in order to be implemented effectively. The hourglass control models tend to make the continuum artificially stiffer, so an adequate hourglass control setting is required. If used correctly, proposed control strategies are able to reduce the total amount of hourglass deformations without a considerable increase in computational cost.

The hourglass control models are developed based on either the viscous or stiffness forms. The viscous method uses a nodal velocity which contributes to hourglass deformation by generating hourglass forces. Because of the viscous-based formulation, this type of hourglass control is recommended for simulations associated with high velocity or high strain rate problems such as high speed impacts, ballistics, or even explosions. On the other hand, the stiffness-based

10 | P a g e formulation uses a nodal displacement that contributes to hourglass modes. Hence the hourglass algorithm generates hourglass forces proportion to the nodal displacement, which is more suitable for relatively low speed scenarios such as quasi-static testing and low speed crash.

A simple study examining a specimen modeled with under-integrated shells with no hourglass control, under-integrated shells with viscous hourglass control, and fully integrated shells is shown in Figure 2.4. The geometry is subjected to quasi-static tensile loading across the top edge, with the bottom edge fixed in both translation and rotation. A prescribed load condition is applied to the top edge instead of a prescribed displacement to clearly show the characteristic wedge or “half hourglass” shape previously shown in Figure 2.3. When compared to the fully integrated counterpart shown on the bottom left in Figure 2.4, the under-integrated model with no hourglass control also predicts elevated stress contours. Although the stress contours look similar, the degree of hourglass deformation is somewhat less severe in the model with the viscous hourglass control.

The influence of the hourglass mode on the solution quality can be further observed in the energy-time plots shown at the bottom of Figure 2.4. For a physically accurate system, the total energy should be equal to the summation of the kinetic and the internal energy over the entire simulation domain. Since the specimen is quasi-statically loaded and no major failures occur, the kinetic energy for the system approaches zero, making the total energy and internal energy equal.

However, the top right plots of Figure 2.4 indicate that a the total energy of the models using the under-integrated shell is always greater than the internal energy due to spurious hourglass energy.

The viscous hourglass control reduces the magnitude of hourglass energy by approximately 23% compared to the case with no hourglass control, but the hourglass energy is still a significant percentage of the total system energy. As previously discussed, this energy is a result of nodes shifting without any resistance and is shown to exponentially increase in the plot. Alternatively, the fully-integrated simulation correctly predicts a perfect match between internal energy and total

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Figure 2.4 Simulation results for no hourglass control (top) and viscous hourglass control (mid) with under-integrated shells and a fully integrated shell model

12 | P a g e energy, with a 22% lower total energy magnitude at the termination of the simulation compared to the no hourglass control model.

If under-integrated quadrilateral elements must be used, then the suggested hourglass energy is less than 10% of the peak internal energy [61,62]. However, this is still a significant percentage when the analysis is considering the onset of failure. In a quasi-static test, nearly all of the energy applied on a specimen is dissipated through internal energy in the form of deformation. Since the applied load is directly proportion to the internal energy, a 10% deviation in energy can be interpreted as a 10% difference in load when failure occurs, hardly an insignificant error. In short, while the reduced integration quadrilateral element is computationally inexpensive, it is generally avoided in favor of a fully integrated shell unless hourglass control is applied.

2.4 Solid Element Formulation

Instead of using the plane stress theory required for 2-D element formulations, the 3-D element formulations can accurately predict the through-thickness stress. Because of their versatility, solid elements are often an ideal choice for many structural applications. With respect to PRSEUS, the

3-D element can be used to calculate detailed damage mechanics such as through-thickness stress, ply failure, and delamination. However, the utility of the solid element comes at the expense of increased computational effort. The simplest solid brick element must use eight nodes, which makes it approximately double the computational expense of a standard shell element.

An under-integrated, eight node hexahedral solid element is depicted in the left side of Figure

2.5. This element formulation is the 3-D counterpart to the quadrilateral element with a single Gauss point in terms of relative computational expense and accuracy. Because the element has only one

Gauss point, this type of solid element is also known as a constant stress solid. Although it can account for through-thickness effects, this element type is also prone to hourglass deformation and energy. As a result, the under-integrated solid is not ideal in regions of large deformation.

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Figure 2.5 Reduced integration (left) and fully integrated (right) solid elements

In the same manner as the fully integrated shell, the hexahedral solid element may also be adjusted to include eight Gauss points inside to become a fully integrated solid element. Figure 2.5

(right) shows an eight-node fully integrated hexahedral element. By implementing all eight Gauss points, the element becomes more suitable to capture large deformations, such as those present in crash and impact simulations, but it is approximately four times more expensive. To alleviate the computational expense, LS-Dyna also offers a selectively reduced integration solid element, which utilizes both the reduced and fully integrated formulations to save computation time while preventing hourglass modes.

For anisotropic and orthotropic materials such as composites, the material coordinate system is tied to the element coordinate system, making the element connectivity very important. Typically, the strong direction of an orthogonal material is defined as the vector from node 1 to node 2.

However, a local material coordinate system can be assigned to each element, allowing the material and element coordinate systems to be decoupled. This capability is crucial for correctly modeling composite materials.

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2.5 Continuum (Thick) Shell Element Formulation

The continuum shell, also called the thick shell, is a combination of both shell and solid elements. Like the solid element, continuum shells are composed of eight nodes, eliminating any geometric assumptions and allowing for more realistic boundary conditions. However, the continuum shell has the option to either use the plane stress assumption like a typical shell or the fully 3-D constitutive laws used by solid elements. This versatility allows the thick shell to be used for a wide variety of applications. As expected, the continuum shell is between standard shell and solid element in terms of computational expense. Unlike either the shell or solid element, no fully integrated scheme exists for the continuum shell available in LS-Dyna. Instead, the continuum shells can utilize either one or four Gauss points depending on whether the plane stress assumption or full 3-D constitutive laws are desired as shown in Figure 2.6.

Figure 2.6 Single Gauss point (left) and four Gauss point (right) options for continuum shell elements

In both continuum shell formulations, the Gauss points are located at a virtual mid-surface.

With the single Gauss point, the response of a continuum shell is nearly identical to that of the reduced integration shell but is somewhat less prone to hourglass energy due to a selectively reduced integration. The four Gauss point scheme is capable of capturing Poisson effects as well as stress distribution in the through-thickness direction, which is not possible for the standard shell formulation. Despite its advantages, this scheme requires hourglass control in order to mitigate spurious energy generation and correctly calculate the stress distribution.

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2.6 Contact in Finite Element Analysis

The fundamental goal of contact algorithms is to allow multiple Lagrangian parts to interact with one another without overlapping volumes. Initially, the contact (or impact) algorithms were proposed by Hughes et al. [63] and the methodology has since been applied to modern finite element framework. Contact algorithms are a broad field that encompass many physically-based interactions that are observed in the real world such as impact, friction, and adhesion. Additionally, contact models have been extended to include two-dimensional elements such as shells and one- dimensional elements such as beams, which only have a theoretical thickness.

Among the many contact algorithms, the penalty-based contact algorithm is the most common.

When a surface penetration is detected, as shown in Figure 2.7, a virtual spring, is generated between all of the penetrating nodes on the master part and the contact nodes on slave part. The virtual spring is elastic and is activated only for compression scenarios. The penetrating nodes are shifted back to the contact surface and a spring force is generated normal to the master surface to prevent further penetration of the nodes on the slave part through the surface of master elements.

The artificial spring constants are continuously updated based on the contact conditions and are directly added to the stiffness matrix at each time step to calculate the contact force and corresponding displacement [49].

Figure 2.7 Schematic of penalty contact by shifting penetrated nodes to the contact surface

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For solid elements, the definition of the contact surface is fairly intuitive, as all of the surfaces that make up the element are direct connections between constituent nodes. However, the same rules do not apply to lower dimension elements such as shells and beams. As shown in Figure 2.8 shells use a theoretical thickness to define the contact surface, which is usually assumed to be symmetric about the mid-plane with normal projection vectors. Thickness changes in the shell can be accounted for in some instances, though the maximum thickness of any shell connected to the node is assumed to be the nodal thickness. The reference surface for projection, whether it be the mid-surface, top, or bottom can also be defined for different scenarios.

Figure 2.8 Contact surfaces for lower dimension elements such as shells (left) and beams (right)

Unlike either solid or shell elements, the contact surface of the beam element may not match the assigned cross-section. Individual shell and solid elements can only represent a hexahedral body, as they are already comprised of at least two dimensions. Alternatively, one dimensional beams can be defined to have any cross-section, such as the I-beam shown in Figure 2.8. The discrepancy creates a contact surface that can be different from the designated cross-section. To calculate contact, the contact surface of the beam is always treated as a circle, where the diameter is equal to the square root of the area of the smallest rectangle that can contain the defined section. In this way, contact with beams must be treated with care, as only beams with a circular cross-section will have a physically accurate contact surface.

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Due to the highly integrated nature of PRSEUS, multiple contact algorithms are required. The primary contact throughout the simulation is the interlaminar contact between different stack layers.

However, due to the epoxy binding each layer together, the interlaminar contact must act as adhesion before delamination occurs. In that case, the identical stiffness between each stack does not create any problems for contact. However, when materials with diverse stiffness come into contact, such as the fuselage skin and rigid ground, the contact algorithm requires artificial stiffness to be added to the softer material to prevent penetration from occurring. This additional stiffness is typically introduced based on the nodal mass of the slave material and the current time step.

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3 Material Theories For any structural finite element analysis, each part must be assigned a material model. The choice of material model has a huge impact on the response of the system and therefore must be selected with care. Even isotropic materials have dozens of different formulations that can significantly affect the quality of the solution. PRSEUS is almost entirely constructed from different types of carbon fiber-reinforced polymer (CFRP). The only non-composite structures in the assembly are the stitch and foam core of the foam-filled frame. For the large-scale fuselage section drop test discussed in Chapter 7, the traditional fuselage section is constructed from isotropic metal alloys. This chapter discusses the fundamental theory of both composite and isotropic materials, as well as the cohesive zones used for interlaminar bonding.

3.1 Composite Material Model

CFRP is widely used in aviation applications due to its high strength to weight ratio, and is the primary constituent of PRSEUS. Composed of unidirectional layers, CFRP is often stacked at different orientations to produce an overall effective stiffness that is resistant to multiple loading directions. For PRSEUS, several stack architectures are used depending on the application of the component and its target location within an aircraft.

Figure 3.1 Schematic of unidirectional composite laminate with representative failure modes (adapted from [64])

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Due to the layering of diverse mechanical properties, composites are prone to a variety of damage and failure mechanics. As shown in Figure 3.1, these failure mechanisms include fiber- matrix debonding, matrix rupture, fiber rupture, fiber pull out, and fiber bridging. At the mesoscale, these internal failures can lead to initiation and propagation of delamination. With so many types of failure, accurately predicting when the composite will fail at the macro-scale is extremely difficult and has forced manufacturers to over-design components in order to ensure safety.

For the damage prediction of composite materials, many failure criteria were developed throughout the mid to late 1900's, such as the widely known Tsai-Wu model in 1970 [65] and

Chang-Chang model in 1986 [66], as well as more recent models such as Daimler-Pinho model

(2006) [67,68]. While many theories have been developed, the Tsai-Wu and Chang-Chang composite damage models are among those that have been commonly implemented into finite element analysis (FEA) software. These damage models do not perfectly represent all of the represented failure mechanisms, but they are widely considered to provide reasonable estimates of the highly non-linear dynamic damage mechanics of composite laminates. For this study, both Tsai-

Wu and Chang-Chang are considered, as well as a generalized 3-D anisotropic strain-based damage model.

The Chang-Chang composite damage model is an updated form of Hashin’s comprehensive composite damage model [69], in which a mode mixity term 훽 is added to allow interaction between the shear and normal failure modes. The model assumes that the unidirectional fiber reinforced composites have a transversely isotropic (x2=x3) behavior along the fiber direction (x1).

The analytical composite model allows the calculation of stress and damage progression of the fiber and the matrix separately to determine the overall condition of an individual ply. The Chang-Chang failure model of the matrix is described by Eqns. (3-1) and (3-2) [66,69]:

2 2 2 𝜎푏푏 𝜎푎푏 Tension: 휀푚푡 = ( ) + ( ) (3-1) 푌푡 푆푐

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2 2 2 2 2 𝜎푏푏 푌푐 𝜎푏푏 𝜎푎푏 Compression: 휀푚푐 = ( ) + [( ) − 1] ( ) + ( ) (3-2) 2푆푐 2푆푐 푌푐 푆푐 where subscripts t and c are the tension and compression correspondingly. The fiber direction is a, while b and c are the matrix direction and through-thickness direction, respectively. 휀푚 is a failure parameter that determines the occurrence of matrix cracking, 𝜎푎푎 is the longitudinal stress of each layer, 𝜎푎푏 is the shear stress of each layer, 푌푡 and 푌푐 are the transverse tensile and compressive strength of matrix, and 푆푐 is the shear strength of the matrix in the xy-plane. The fiber behavior is described by Eqns. (3-3) and (3-4) [66,69]:

2 2 2 𝜎푎푎 𝜎푎푏 Tension: 휀푓푡 = ( ) + 훽 ( ) (3-3) 푋푡 푆푐

2 2 𝜎푎푎 Compression: 휀푓푐 = ( ) (3-4) 푋푐 where 푋푡 and 푋푐 are longitudinal tensile and compressive strength of fiber material and 휀푓 is the failure parameter for the fiber. Damage occurs when the damage parameter (휀) is equal to or greater than unity.

Unlike the Chang-Chang failure model, the Tsai-Wu failure criterion evaluates the damage based on the stresses of the entire laminate in each orthotropic orientation, which is similar to the von-Mises stress criterion. The Tsai-Wu failure criterion is defined as [49,65]:

2 2 2 퐹1𝜎1 + 퐹2𝜎2 + 퐹11𝜎1 + 퐹22𝜎2 + 퐹66휏12 − √퐹11퐹22𝜎1𝜎2 < 1 (3-5) where 퐹푖 is the tension and compression strength in both the longitudinal and transverse directions, and 퐹푖푗 is the maximum shear strength of anisotropic material. When the combination of these stresses is equal to one, then failure is predicted to occur. As with the Chang-Chang criterion, the individual stress components mutually interact with each other, also accounting for failure as a result of complex loading. However, by lumping all of the stress interaction into a single equation, it is difficult to identify the primary failure mode for Tsai-Wu. Based on the authors experience and

21 | P a g e demonstrated in Appendix B, the Tsai-Wu model is considered to be more conservative (predicts failure sooner) than the Chang-Chang model.

In addition to the established Chang-Chang and Tsai-Wu models, a generalized 3-D anisotropic damage model was selected. Unlike either of the established models, the 3-D anisotropic model is not based on phenomenological observations. However, it does not require the simplifying plane stress theory assumption. This model purely uses strain to calculate the accumulation of damage. The damaged compliance matrix is given by Eqn. (3-6), where E is

Young's modulus, 휈푖푗 is Poisson's ratio, 퐺푖푗 is shear modulus, and 푑푖푗 is the damage parameter

[70,71].

1 −휈 −휈 21 31 0 0 0 퐸1(1 − 푑11 ) 퐸2 퐸3 −휈12 1 −휈32 0 0 0 퐸2 퐸2(1 − 푑22) 퐸3

−휈13 −휈23 1 0 0 0 푑푎푚 퐸1 퐸2 퐸3(1 − 푑33) [푆푖푗] = (3-6) 0 0 0 1 0 0 퐺23(1 − 푑23) 1 0 0 0 0 0 ( ) 퐺13 1 − 푑13 0 0 0 0 0 1 [ 퐺12 (1 − 푑12)] The damage parameter itself is calculated based on user-defined damage and failure thresholds and is constrained so that damage can never decrease:

휀 − 휀푡ℎ 푐 푖푗 푖푗 푑푖푗 = 푚푎푥 [푑푖푗, 푑푖푗 [ 푐 푡ℎ] ] (3-7) 휀푖푗 − 휀푖푗 where, 휀푖푗 is the strain, superscript c represents the critical value, and superscript th represents the damage threshold value. This assumption is reasonable for both the quasi-static testing and dynamic scenarios because the load is continuously applied during experimental coupon compression tests and the small time window of interest in dynamic simulations makes any unloading negligible. As with both the Chang-Chang and Tsai-Wu models, the 3-D anisotropic model does not account for any plasticity effects. Many experiments have been performed that demonstrate the brittle response of composites, particularly CFRP [72–74], which makes any plastic response negligible.

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3.2 Cohesive Zone Modeling

Delamination is the separation of two plies within a laminate as a result of the difference in stiffness between neighboring plies. To simulate the delamination between composite plies, the cohesive zone model (CZM) is used to define the adhesion between ply layers. CZM is a finite element extension of classical fracture mechanics. A. A. Griffith initially proposed the fracture mechanics by using energy methods to consider existing micro cracks in solid continuum [75,76].

The work was later revisited by G. R. Irwin [77,78] in the late 1940s to include external loadings.

These works proposed the definition of the now widely used strain energy release rate (G), fracture toughness (Kc), and stress concentration factors (K), as well as their mutual relationships, to analytically predict the material response of a continuum with a preexisting crack. Since then, these parameters have been applied to crack propagation stemming from diverse applications. Fatigue analysis in particular has adopted Irwin’s work, with many studies devoted to characterizing the fracture toughness of various materials [79,80]. Many computational composite simulations rely on these parameters to characterize delamination from extreme loading or impact and have been shown to match experimental observations [81,82].

Figure 3.2 Schematic of three primary separation modes (adapted from [83])

As illustrated by Figure 3.2, the fracture modes can be categorized into Mode I (crack opening damage), Mode II (in-plane shear damage) and Mode III (out-of-plane shear damage). In the

23 | P a g e presented computational work, only Mode I and Mode II are implemented to predict damage. Mode

I fracture is typically a result of through-thickness tensile loading, which can occur between the tear strap and neighboring components for a pressurized vessel. Mode II often occurs during bending, when the outermost plies are subjected to much higher strain than the interior layers. The resulting in-plane shear stress can cause the epoxy to fail and debonding to occur. For the purposes of the current study, fracture Mode III is assumed to be nearly identical to Mode II separation, with transverse displacement instead of a lateral displacement.

Figure 3.3 shows a schematic diagram of the bilinear stress-displacement model that is typically used for CZM as well as a mixed mode to represent the interaction of modes I and II. In this model, energy release rate is the area under a curve defined by peak normal and shear tractions and the crack opening displacement at which the bond has completely failed. The energy release rate of Mode I is defined to be completely independent from the energy release rate of Mode II, but the mixed mode capability combines the energy release rate from each. This capability is necessary, because delamination rarely propagates based on a single fracture mode, except for in closely controlled experimental setups.

Figure 3.3 Schematic of the bilinear mixed-mode stress-displacement model

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As an alternative to the traditional bilinear stress-displacement model, the Dycoss Discrete

Crack Model [84] is also used to approximate the interlaminar bonding shown in Fig. 5.3. This bilinear model works in both tension and compression, with shear (𝜎푖푗) and normal (𝜎푖푖) failure criteria that govern interface strength as shown in Eqn. (3-8).

2 푚푎푥(𝜎 , 0) 𝜎 2 [ 푛 ] + [ 푠 ] = 1 (3-8) 𝜎푁 𝜎푆 − 푠푖푛(휃) ∙ 푚푖푛(0, 𝜎푛) where 𝜎푁 is the normal failure stress and 𝜎푆 is the shear failure stress of interlaminar bonding. This model allows separation to be calculated on a continuous scale from zero to one, where zero represents full adhesion and one represents full separation. In addition, an energy release rate is assigned for both Mode I and Mode II separation, allowing the delamination to propagate from the initiation zone. Unlike the traditional bilinear cohesive model, the Dycoss model does not require an additional layer of elements to be defined at the interface. Instead, a segment based approach is used, in which the element faces of the bonded plies are initially co-linear. Because the model does not require a separate material model, the crack opening displacement at failure is approximated based on the penalty contact stiffness.

3.3 Metal Material Model

3.3.1 Isotropic Model

Although composite material is the focus the current work, isotropic materials are used to conduct the preliminary element study and are required to model the conventional metal alloy fuselage. Isotropic materials are defined by a single set of mechanical properties that apply for all loading directions, making them simpler to characterize within the elastic regime. However, unlike most composites, isotropic materials such as metal alloys are much more susceptible to plasticity effects. Metal alloy failure is characterized by both ductile and brittle fracture, where the strain rate plays a major role in determining the type of material response [76].

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When possible, the isotropic materials implemented for the current work were represented using a linear elastic-plastic hardening damage model. The simplified approach of the material model can predict reasonable results for relatively low strain-rate scenarios in which the stress- strain curve remains approximately constant. Due to the low computational expense, the material model has been used to accurately describe the behavior of many metals used in aircraft manufacturing [55,56,85]. The material model formulation is described by:

퐸 ∙ 휀 (𝜎 ≤ 𝜎 ) 𝜎 = { 1 푒 표 (3-9) 퐸2 ∙ 휀푝 + 𝜎표 (𝜎 > 𝜎표) where 퐸1 is the elastic modulus, 퐸2 is the simplified plastic hardening modulus, 𝜎표 is the yield strength, 휀푒 and 휀푝 are the elastic and the plastic strain respectively.

An example of the difference between the results of the bilinear computational model and a typical experimental test is illustrated in Figure 3.4. Although the simplified computational model does not match the non-linear path of the representative experiment, the onset of plasticity (𝜎표) and ultimate strain (휀푓 ) are identical. Depending on the application, the parameters of the bilinear material model can be adjusted to either match the ultimate stress observed in the experiment (𝜎푢), the stress at fracture (𝜎푓), or the total internal energy dissipated through deformation. While other, more accurate material formulations exist, these formulations are much more computationally expense and require the use of an equation of state [86–89]. The author has effectively used this simplification in other works [55,56].

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Figure 3.4 Engineering stress-strain comparison between a representative experimental test (black) and linear elastic-plastic hardening model (dashed red)

3.3.2 Johnson-Cook Model Unlike the validation studies conducted at the component level, the fuselage section drop test is subjected to significant strain rates. Under such dynamic loading, the high strain rate deformations resulting from impact can induce a critical shock pressure at the interface. Such rapid energy interaction causes the material to response non-linearly, which is not considered in the linear elastic-plastic hardening model. The two primary metal alloys considered for the current study are

Alu 2024-T3 and Alu 7075-T6, which are both commonly used in aircraft fuselage manufacturing

[90]. Both types of aluminum have been extensively studied and shown to exhibit a non-linear response under high strain rate loading [91–94]. A representative example of the change in behavior for these materials is shown in Figure 3.5. In general, Alu 7075-T6 appears to exhibit relatively little strain hardening, but increases in failure strain significantly at high strain rates. Alternatively, the failure strain of Alu 2024-T3 is relatively insensitive to strain rate, but shows a more significant strain hardening before yield occurs.

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Figure 3.5 Representative illustration of change in Alu 2024-T3 and Alu 7075-T6 behavior at different strain rates

To overcome this shortage, a more comprehensive material failure model is explored. One of the commonly used constitutive material failure models that accounts for the effects of strain rate

(휀̇) is the Johnson-Cook model. Johnson-Cook is derived by a purely empirical approach that considers large deformation, strain rate hardening, and thermal softening [95]. The deformation can be calculated by employing an equation of state (discussed in Section 3.3.3) in conjunction with the Johnson-Cook model.

The Johnson-Cook description of the von-Mises flow stress is split into three separate expressions as shown in Eq. (3-10):

푚 푛 ε̇ 푇 − 푇푅표표푚 σ = [A + B(εp) ] [1 + 퐶 ∗ 푙푛 ( )] [1 − ( ) ] (3-10) ε0 푇푚푒푙푡 − 푇푅표표푚 where 휀푝 is the equivalent plastic strain, 휀̇ is the plastic strain rate, and 휀0 is the experimental strain rate. The five material constants A, B, n, C, m, and 푇 are evaluated experimentally. A is the yield stress, B and n are the strain hardening constant and exponent, C is the strain rate hardening constant, and m is the temperature softening exponent. The first bracketed expression calculates stress with strain hardening, but without accounting for strain rate or thermal effects. These effects are included in the second and third bracketed expressions respectively. Because of the numerical contribution of strain rate and temperature in addition to strain, the model is much more versatile than the linear elastic-plastic hardening model.

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The Johnson-Cook material model must be supplemented with additional parameters to characterize the effect of damage on the local material behavior. The failure strain is calculated based on the relationship of damage parameters shown in Eqn. (3-11).

f 푃 ε̇ 푇 − 푇푟표표푚 ε = [퐷1 + 퐷2 exp(퐷3 ∗ )] [1 + 퐷4푙푛 ( )] [1 + 퐷5 ∗ ( )] (3-11) 𝜎̅ ε0 푇푚푒푙푡 − 푇푅표표푚 where D1-5 are the damage parameters, 𝜎푚 is the average of the three normal stress, and 𝜎̅ is the von Mises equivalent stress. D1, D2 and D3 are the damage parameters respecting strain, D4 is the damage parameter respecting strain rate, and D5 is the damage parameter respecting temperature contribution. The overall damage is represented by D, which is defined in Eqn. (3-12) [96].

∆휀̅푝 퐷 = ∑ (3-12) εf where ∆휀̅푝 is the incremental effective plastic strain. When the total effective plastic strain is greater than the local failure strain, (i.e. 퐷 > 1), fracture occurs. As with the description of stress, the bracketed expression each represent an individual damage source. The first term describes the inverse relationship between hydrostatic tension and failure strain. The failure strain calculated by the first time is directly scaled with the relative contributions of strain rate and temperature.

Depending on the sign of the assigned damage parameters, the damage model can be used to define both increases and decreases in ductility in response to the loading scenario.

3.3.3 Equation of State In order to accurately calculate the material response with the Johnson-Cook model, the state variables (pressure, volume and temperature) and their relative influence on each other must be evaluated. The relationship between each of these variables is known as an equation of state. The linear elastic-plastic hardening model is purely a function of strain, and therefore does not require an equation of state. For highly dynamic scenarios such as impact or crash, the equation of state must also account for the presence of shock waves in the material. The shockwave is caused by the energy wave moving faster than the speed of sound in the continuum, whether it be in gas, liquid

29 | P a g e or solid phase. Unlike some equilibrium conditions, which obey the mass, momentum and energy conservation law, an interaction with a shock is not a linear phenomenon and consequently does not necessarily obey the momentum and energy conservations [97,98]. When it occurs, the shock increases pressure, density and temperature within infinitesimal regime.

Mie, in 1903, initially derived a mathematical correlation of high temperatures with solids

[99]. Later, Gruneisen adopted Mie’s equation to numerically predict a comprehensive response to the density and internal energy of a homogeneous solid under hydrostatic pressure [100]. Unlike a typical equation of state, the Mie-Gruneisen form assumes that the state changes occur so rapidly that no significant heat transfer occurs[101]. Instead, the internal energy is related to pressure and volume. The Mie-Gruneisen equation of state for a compressed material is defined as [102]:

훾 푎 퐶2𝜌 휇 [1 + (1 − 0) 휇 − 휇2] 0 2 2 푝 = + (훾 + 푎휇)퐸 휇2 휇3 0 (3-13) [1 − (푆 − 1)휇 − 푆 − 푆 ] 1 2 휇 + 1 3 (휇 + 1)2 where 휇 is the relative density ratio (휇 = 𝜌⁄𝜌0 − 1), C is the speed of sound of the material, S1-3 are the empirical Hugoniot slope constants for the pressure-volume relationship, 훾0 is the

Gruneisen gamma, a is the first order dimensionless empirical volume correction factor, and E is the internal energy. Under expansion, the equation of state can be expressed as:

2 푝 = 𝜌0퐶 휇 + (훾0 + 푎휇)퐸 (3-14)

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4 Preliminary Element Study Before investigating PRSEUS, a preliminary element study was conducted to determine the element configuration that would predict accurate results while remaining computationally feasible.

Simulations were performed at loading scenarios deemed to be representative of the flight loads that are expected in a HWB aircraft. The two primary loading conditions used for the initial element characterization were cantilever beam bending and beam buckling, both using rectangular cross- sections. The elements considered for the analysis were solids, shells (in both horizontal and vertical orientations), continuum shells, and beams. The formulation of each element type was also parametrically evaluated to determine the most effective methodology for further analysis.

Additionally, both implicit and explicit methods were used to perform the analysis. For every simulation, the corresponding force-displacement history and energy dissipation is recorded.

4.1 Cantilever Beam Analysis

The cantilever beam is a classic engineering benchmark problem for which a simple analytical solution exists. The boundary conditions and loading of the cantilever beam with a single point load at the beam tip are easily reproduced in the computational domain. Figure 4.1 illustrates the simulation setup for the solid element and both orientations of the shell element.

Figure 4.1 Simulation setup of square cantilever beam with a single load at the beam tip for the solid (top), vertical shell (left) and horizontal shell (right). All dimensions are in meters

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Both horizontal and vertical orientations of the shell are separately considered due to the plane stress assumption inherent to shell theory. The boundary condition at the fixed wall of each virtual beam model fully constrains the nodes in both translation and rotational degrees of freedom. A prescribed displacement curve is used to apply the load at the free end of the cantilever beam, where the corresponding total force is equally distributed across the nodes. Finally, transverse displacement on one side of the beam was fixed to eliminate drift while still allowing for Poisson effects. The first investigation was performed using implicit methods with only ten under-integrated elements across the length of the beam and a single element through the thickness. A simple elastic, isotropic material based on the properties of Alu 2024-T3, shown in Table 4.1, was specified for each computational model. The results of each element type are compared against an analytical baseline using the cantilever beam equation with a single load applied at the tip.

Table 4.1 Material properties for Alu 2024-T3 Material Property Value 퐸 (GPa) 73.1 푣 0.33

𝜎0 (MPa) 345

휀푓 0.18

Figure 4.2 compares the response of each element type for their respective under-integrated formulations. While just a single Gauss point was used for the under-integrated solid and shell elements, the thick shell required the 2x2 quadrature due to limitations within LS-Dyna. Both viscous and stiffness forms of the hourglass control were investigated, but only the viscous form was observed to have an appreciable effect. With no hourglass control, the only element type to match the analytical result is the beam. Alternatively, the response predicted by the solid, thick shell, and vertical shell are all far below the analytical solution. The significantly lower stiffness for each of these elements indicates the presence of hourglass modes, which often result in a softer response. This hypothesis is confirmed by the corresponding energy plot, which depicts internal

32 | P a g e and hourglass energy for the solid, horizontal shell, and vertical shell elements. Due to the quasi- static loading conditions, the internal energy and hourglass energy are the only energy components in the system. The beam element is not shown because it was not prone to hourglassing while the energy distribution of the thick shell corresponded to the solid element.

For both the solid and horizontal shell elements, the hourglass energy far exceeds the internal energy, but the total energy (sum of the internal and hourglass energy) is different between each case. This suggests that the under-integrated vertical shell is inherently less stiff in the given loading direction when compared to the solid. Both the solid and vertical shell predict far less energy than the horizontal shell, which reaches a total energy of approximately 8 MJ at the final beam tip displacement and closely matches the energy response of the beam element. The horizontal shell does actually accumulate some hourglass energy near the final displacement, but the magnitude of the energy is less than 0.1% of the internal energy. Due to the under-integrated element formulations used in this case, the simulation also error terminated before the final displacement could be achieved.

No Hourglass Control Viscous Hourglass Control

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Figure 4.2 Comparison of load vs. beam tip displacement and energy for under-integrated elements with no hourglass control (left) and with viscous hourglass control (right) When the hourglass control is applied as shown on the right of Figure 4.2, only three primary differences are observed. First, the simulation was now able to run to completion instead of error terminating. However, the second major difference is the loss of accuracy for the horizontal shell solution. The sudden drop in load is caused by a sudden and extreme rise in hourglass energy.

While the exact reason for the sudden rise in hourglass energy is unknown, this affect was only observed for the viscous hourglass control model. Finally, both the internal energy and hourglass energy of the vertical shell dropped by several orders of magnitude. The relative proportion between the hourglass and internal energy remained similar, but the total energy was significantly reduced. From this preliminary study, the hourglass control appeared to have little to no improvement on the quality of the solution, possibly even exacerbating the spurious numerical effects.

Another observation from this case study is that all of the elements that under-predicted the loading were subjected to shear locking. Because all of the elements in the current study are linear, the curvature imposed by bending is interpreted as an applied shear [103]. Due to the applied displacement boundary condition, the final displacement at the tip is the same for each element type, but the shear-locked elements predict a distinctive pattern of high stress that alternates in

34 | P a g e tension and compression as shown in Figure 4.3. The effect of shear locking becomes more apparent as the length of the beam is increased.

Figure 4.3 Example of stress bands produced by shear locking for the solid element

Given the lack of improvement by the hourglass control, the same single element through- thickness analysis was conducted using a more robust integration scheme. Figure 4.4 illustrates the response of each model for the cantilever beam loading. The solid model uses selectively reduced integration, both the horizontal and vertical orientation shells are fully integrated, the thick shell uses selectively reduced integration, and the beam implements a 2x2 Gauss quadrature. Overall, the load-displacement predictions match the analytical result with significantly greater accuracy.

As with the previous case, the beam matches the analytical solution exactly, with a perfectly linear load vs. displacement curve. Due to the plane stress theory utilized by shells, the two shell orientations produced different results.

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Figure 4.4 Load vs. beam tip displacement for implicit analysis with a single through-thickness element and isotropic material

Due to the fewer degrees of freedom, the horizontal shell model predicts a response that is co- linear with the beam and analytical prediction whereas the vertical shell demonstrates a non-linear response that has a significantly greater peak load. Both the solid and continuum shell also produce a non-linear response, but vary greatly in magnitude. The reason for the discrepancy in slope between the analytical solution and solid, continuum shell, and vertical shell models is the inclusion of Poisson effects. Despite this, the solid is unusual in that it initially predicts a lower slope than the analytical baseline. Since the element formulations used in this simulation set do not allow hourglass energy, this is likely the result of an unrefined mesh and shear locking. While the continuum shell accounts for the Poisson effects, it has an overly stiff response, producing a peak load that is approximately 50% greater than the analytical prediction. From the investigation of models with just a single through-thickness element, the vertical shell appears to have the most physically accurate result, though the beam, solid, and horizontal shell models all closely match the analytical prediction.

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Although the loading response of the models with the higher integration scheme demonstrated clear improvement over the point integration schemes, the predicted peak loading is not sufficiently accurate. Therefore, a second set of models using two elements through the thickness was simulated.

In order to keep the element aspect ratio constant, a global mesh size refinement factor of two was applied, causing each solid element to split into eight, each shell element to split into four, and each beam element to split into two. For the horizontal shell and beam models, the elements cannot be refined in the plane of the applied displacement, so the only effective change was the number of elements along the beam length. The same fully integrated and reduced integration schemes were applied to the refined model.

Compared to Figure 4.4, the responses shown in Figure 4.5 are much more tightly grouped.

As before, both the beam and horizontal shell predictions align with the analytical solution, but the responses of the solid, continuum shell are noticeably shifted. Now, the solid model does not exhibit a non-physical drop in stiffness, but instead produces a trend that closely matches the response of the vertical shell. Also, the thick shell has significantly decreased its previously over-stiff prediction, with an overall trend that approximates the solid and vertical shell. The shift in response suggests that the solid element converges from below (initially under-predicts response) whereas the thick shell appears to converge from above (initially over-predicts response). In both cases, a substantial benefit is apparent when the number of through-thickness elements is increased. Further analysis will formalize the mesh dependency for the cantilever beam bending response of an isotropic material.

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Figure 4.5 Load vs. beam tip displacement for implicit analysis with two through-thickness elements and isotropic material

Because PRSEUS must be evaluated for a wide variety of different loading conditions and rates, both implicit and explicit methods must be used. For the previously shown element study, all of the simulations were conducted using an implicit scheme in which the displacement was applied over a 100 second interval. However, because explicit methods purely rely on the input from the previous time step to advance the solution, they must have a small time step. This requirement makes quasi-static analysis or other long-duration loadings computationally intensive. However, for strain-rate independent material formulations such as the linear elastic model used here, the rate of loading is theoretically independent from the material response, allowing the displacement to be applied much more quickly.

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Figure 4.6 Load vs. beam tip displacement for explicit analysis when the displacement is applied over 1 s (left) and 100 s (right)

Figure 4.6 illustrates the response of the cantilever beam using an explicit scheme in which the displacement is applied over 1 s interval (left) and a 100 s interval (right). It is immediately apparent that the explicit scheme results in a much noisier response than the implicit case. Despite the computational noise, the prediction resulting from the 100 s interval to apply the displacement is in close agreement with the implicit results. Alternatively, the prediction from the 1 s displacement interval is extremely jagged, making it difficult to distinguish the response of each element type. The beam element has by far the most noise, with load oscillations that are an order of magnitude larger in amplitude than observed for the other element types. While filtering can alleviate some of the oscillations, the primary reason for the apparent computational noise is stress wave fluctuation within each cantilever beam model.

The computational effort required for both explicit models exceeds the requirements of the implicit method. While the implicit model only ran for 45 seconds with a single central processing unit (CPU), the fast and slow-loading explicit models ran for 108 and 13,452 seconds respectively when using 12 CPUs! This vast difference in computational expense requires that explicit methods be used only when necessary for long-duration loading. When explicit schemes are required for a long-duration response, then the loading rate must be carefully adjusted to provide a balance between a high-quality response and computational expense.

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In addition to the simple isotropic elastic material, simulations were conducted using the

Chang-Chang composite damage model described in Chapter 3.1. The composite properties are based on the reported properties of the Class 72 Type 1 stack used for PRSEUS [13], which are described in Section 5.1.2 and summarized in Table 5.1. Beam elements are incapable of representing the anisotropic response required for composite materials, therefore the linear elastic- plastic model was applied to act as a 1-D pseudo-composite. A hardening modulus was applied to approximate the softening caused by damage accumulation and the ultimate strain was calculated from the experimentally determined composite strength assuming a purely elastic response. The

Chang-Chang damage model is incompatible with the previously used continuum shell formulation, so both forms of the assumed strain integration were used instead. Figure 4.7 shows the implicitly calculated loading response of each element type with two elements through the thickness. An analytical cantilever beam solution assuming the primary fiber-direction modulus as isotropic is also provided.

Figure 4.7 Load vs. beam tip displacement with two through-thickness elements and composite material [4 Gauss point thick shell (left) and selectively reduced thick shell (right)]

Surprisingly, the analytical and beam solutions are within reasonable bounds for the composite bending response, indicating that the out-of-plane properties have relatively little effect on the primary fiber-direction modulus. This has important implications for future, large-scale analysis where computational expense becomes a limiting factor. Both of the shell orientations and solid

40 | P a g e models produce similar loading predictions, where a reduction in stiffness is clearly visible as the displacement is increased. There is a noticeable difference in the slope of each elements prediction, which could be the result of the different degrees of freedom. The failure response is also significantly different for each element type; both the solid and vertical shell models exhibit significant damage within the applied displacement range whereas the horizontal shell and beam models do not experience a significant drop in stiffness.

Figure 4.8 Final deformation for both solid and 2x2 quadrature assumed strain thick shell models

The two thick shell formulations each predict vastly different loading trends. The assumed strain 2x2 Gauss quadrature response shown on the left-side predicts far more damage and element failure early in the applied displacement than the solid model as shown in Figure 4.8. Alternatively, the reduced integration formulation never observes a major reduction in stiffness, but rather increases linearly throughout the analysis. Although this continuum shell formulation is reduced integration, the amount of hourglass energy is nearly equal to the total internal energy, indicating an inaccurate result. A number of different hourglass models were used in an attempt to mitigate the hourglass energy, but none reduced the hourglass energy below the 10% recommended minimum [61,62]. Additionally, the response of the continuum shell was nearly identical for both the viscous and stiffness forms of the hourglass control.

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4.2 Buckling Analysis

While cantilever bending is useful for an initial element characterization study and does have some application to the expected loads during flight, buckling is another primary form of critical loading on most load-bearing structures. Additionally, the majority of available experiments for validation were conducted under compressive loading. Therefore, the elements investigated in the cantilever beam analysis are extended to buckling characterization. The geometry from the simple cantilever beam test was extended to facilitate buckling in the Euler regime and a symmetric boundary condition was introduced as shown in Figure 4.9. Although the symmetric boundary is not necessary to reduce computational effort in the preliminary study, the symmetric assumption is made for many of the PRSEUS models and therefore requires validation. Due to the constrained nodal translation and rotation at the loaded end, the simulation represents a fixed-fixed buckling condition under quasi-static loading. An additional boundary condition was applied to keep the buckling for each model in the same plane. As with the cantilever beam analysis, the isotropic linear elastic Alu 2024-T3 material model and Class 72 Type 1 stack material properties are applied.

The analytical baseline is calculated based on the Euler buckling equation for a square beam under the described loading conditions.

Figure 4.9 Simulation setup for square beam buckling with a symmetric boundary condition and the fixed-fixed loading condition. All dimensions are in meters. Initial simulations were conducted for the under-integrated element formulations with and without hourglass control. Both viscous and stiffness forms of the hourglass control, but once again was not observed to result in any significant improvement. As shown in Figure 4.10, all of the

42 | P a g e under-integrated elements produced an incorrect response. The horizontal shell and beam, which are collinear, did not buckle, but instead only compressed uniaxially. Additionally, the simulation lost stability at approximately half of the total applied displacement, causing error termination.

Because each element type is run simultaneously, it is difficult to determine which formulation is responsible for the global termination of the simulation.

Figure 4.10 Load vs. beam tip displacement plot (left) and energy distribution (right) for the under- integrated element formulations with isotropic material As expected, the softened response of the elements that did exhibit buckling (solid, thick shell, and vertical shell), was a result of excessive hourglass energy. For each of these element types, the hourglass energy was greater than the internal energy (the thick shell response is collinear with the solid element response). Additionally, the total energy magnitude was significantly lower than the expected energy magnitude of 9.5 MJ at 0.25 m tip displacement. Despite the lack of buckling observed in the horizontal shell and beam models, the energy trend and magnitude closely matches the correct solution, though this is likely due to the early simulation termination.

The higher integration element formulation was able to produce significantly more accurate results as shown in Figure 4.11. The fully integrated beam and vertical shell, in particular, achieved a near exact agreement with the analytical prediction of 267 MN. Despite the more diverse response of the solid and continuum shell elements, all of the investigated models predicted almost the exact same stiffness up until buckling. Due to stability problems, the element formulation for the

43 | P a g e continuum shell was required to be the 2x2 Gauss quadrature with selectively reduced integration, but without the assumed strain model. The lack of the enhanced strain formulation is likely why the thick shell predicts an overly stiff response for both the one and two through-thickness cases.

Although a coarse mesh typically results in a stiffer response, the solid element predicts a lower peak load than any of the other element types. However, when the mesh density is increased, the solid element response closely matches both the beam and vertical shell buckling loads. The increased mesh density also reduced the magnitude of the initial over-prediction of the peak loading during the buckling transition for all of the element formulations except the thick shell. Although the thick shell response improved in accuracy compared to the single through-thickness simulation, it is still more deviated from the analytical prediction than any of the other element formulations.

Figure 4.11 Load vs. beam tip displacement for higher integration methods with a single through- thickness element (left) and two through-thickness elements (right) with isotropic material Considering that the symmetric boundary condition is a valuable method to reduce computational time by approximately 50%, a preliminary validation was conducted to ensure that the symmetric and full models produce the same result. The results of the analysis are plotted in

Figure 4.12 with the corresponding error compared to the fully integrated model with a single through-thickness element. In general, the error is less than 1% other than at the location of the buckling peaks. While the spikes in error appear significant, they are a result of a slight shift in the location of the peak. Due to the steep drop in load at the buckling point, the error near these points

44 | P a g e is artificially inflated. Therefore, the symmetric and full length specimens produce an acceptably similar results, with only slight deviations in the location of the peak load.

Figure 4.12 Load vs. beam tip displacement for a full-length specimen (left) and error magnitude compared to the symmetric model (right) Unlike the cantilever beam loading, buckling is an inherently unstable response. While implicit schemes achieve an equilibrium at each individual time step, explicit methods have no such way to avoid numerical drift. Therefore, modeling unstable systems with explicit methods can produce incredibly diverse results. As with the cantilever beam, a series of explicit analyses were conducted for the buckling case at a wide range of loading rates. The results of the analysis with a displacement applied over 1 s and 100 s is shown in Figure 4.13.

Figure 4.13 Load vs. beam tip displacement for explicit analysis when the displacement is applied over 1 s (left) and 100 s (right)

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The effect of loading rate on the explicit buckling model is far greater than was observed for the cantilever beam model. While the faster loading rate primarily generated noise in the cantilever beam solution due to the stress wave fluctuation, in this case the faster loading rate results in significantly higher buckling loads for all of the element formulations considered. The loading rate does not appear to have any effect on the stiffness, which matches the implicit solution exactly for both loading rates. The 100 s loading rate case closely matches the implicit predictions for buckling load and post-buckled response of the solid, thick shell, and beam element models. Although the post-buckled response is somewhat noisy, the averaged response has a slope of zero and matches the analytical buckling load calculation. The explicit solution does not appear to predict the same over-predicted buckling point observed for the thick shell, instead transitioning into the constant load regime relatively smoothly.

Both of the explicitly calculated shell models exhibited a significantly different response than their implicit counterpart. Although the horizontal shell failed to buckle in the explicit simulation as well as in the implicit simulation, the predicted response became bilinear, with a reduction in stiffness occurring at less than 0.1 m of compression. The vertical shell followed the same general trend as the implicit solution, but the buckling load is 10% lower and the post-buckling response is highly oscillatory, with a higher noise level than average superimposed on a lower frequency oscillation. The significant difference in the shell response suggests than either a slower loading rate must be used or that shells are not appropriate for explicitly modeling buckling.

Finally, the Class 72 Type 1 stack material was applied to the model with the 0° fibers in line with the beam length. An analytical estimation of the buckling load was made using the primary fiber-direction stiffness of the composite stack. The same linear elastic-plastic surrogate composite material used in the cantilever beam test was applied to the beam model. The higher integration thick shell element was incompatible with the composite material model, so the reduced integration formulation with enhanced strain formulation and viscous hourglass control was applied instead.

No element failure was observed for any of the element formulations, but significant non-visible

46 | P a g e damage developed in each. In this case, the non-visible damage represents the micro-scale damage accounted for by the Chang-Chang composite failure model. The results for the implicit calculation with two through-thickness elements are shown in Figure 4.14.

Figure 4.14 Load vs. beam tip displacement for two through-thickness elements with composite material As expected, the beam and vertical shell models predicted a peak load that coincides with the analytical approximation. Due to the lack of discrete thickness and enhanced strain formulation, these methodologies are unable to account for the Poisson effects, though both exhibit post buckling softening. The solid predicts a buckling point that is 16% higher than the analytical solution, but quickly softens to approximately the same magnitude as the beam and vertical shell. The increase in predicted buckling load is a result of including Poisson effects. The horizontal shell did not buckle, but did begin to show damage near the simulation termination, with the slope abruptly dropping to approximately zero as a result of accumulated compressive damage. Due to the change in element formulation to a reduced integration scheme, the thick shell model predicted the lowest peak. In this case, the hourglass energy did not exceed 6% of the internal energy, but it is unclear

47 | P a g e if this energy is the sole reason for the under-predicted peak or if the thick shell formulation does not perform well under buckling conditions.

In summary, the analyses conducted in the current chapter highlight the capabilities of each element type. Considering that the current study requires elements that are capable of performing accurately for bending and buckling scenarios, the only acceptable elements are the solid and beam element with higher integration schemes. Although the solid element is the most computationally expensive of those considered in the study, it was the only one able to accurately account for

Poisson effects in addition to approximating the analytical solutions. The beam also performed remarkably well despite its simplicity, though it is unable to capture Poisson effects or model an anisotropic material.

Of the two shell models, the vertical shell performed much better, providing reasonable approximations of the analytical solution in both bending and buckling case studies with the implicit time integration. However, during the explicit analysis of buckling, the vertical shell under- predicted the estimated buckling load by 10% and exhibited a highly oscillatory post-buckling response. Although the horizontal shell was able to match the analytical prediction for cantilever bending, it failed to buckle in any of the buckling simulations. This weakness is due to the Gauss points lying in a single plane that was perpendicular to the buckling direction. Because there is no vertical separation of the Guass points, no through-thickness stress gradient could be developed to result in buckling. This issue could have been resolved by perturbing the geometry with an artificial bias in the buckling direction, but that is not possible for more complex simulations where the mode of failure is unknown.

Finally, the thick shell performed surprisingly poorly in both of the chosen case studies, especially when modeling the composite material. Although the underlying theory of the continuum shell suggested that it would perform better than the two-dimensional shell models, the results do not support this. Another significant issue is that the current implementation of the thick shell is limited in the number of compatible materials and higher integration schemes. In general, both the

48 | P a g e viscous and stiffness forms of the hourglass control failed to significantly improve the simulation quality. Therefore, while it is possible to use the thick shell formulation with a composite material model when an hourglass control model is applied, a numerically robust result is not guaranteed.

49 | P a g e

5 Component Level Analysis

The element study is an important step in the computational modeling process, but the true validation of a virtual methodology is how well its predictions compare to experimental testing.

This chapter discusses the development, numerical testing, and validation of a finite element representation of the PRSEUS component compression tests conducted by Leone et al. [16,23].

The experimental investigation conducted by Leone et al. utilized multiple stringer and frame configurations that were harvested from the alternate center keel panel of the large multi-bay box test. For the research conducted in this work, only the structures using Class 72 Type 1 material were modeled. Additionally, while many different permutations of the PRSEUS stringers and frames have been designed, the current work is focused on replicating the baseline configurations.

In the case of the stringer, the Class 72 overwrap with no additional adhesive is investigated. For the frame model, both the foam-core and tapered L-frame are investigated as they are significantly different in construction and application. A major goal of this research is to determine how well

PRSEUS can be applied to tube-wing fuselage, so an emphasis is placed on creating models that are computationally feasible at higher scales. The chapter is organized into two primary sections:

Class 72 stringer and stringer-frame intersection. Each of these sections are further sub-divided into model development and validation against the experimental baseline.

5.1 Stringer Development

In traditional aircraft, the stringer is typically some form of extruded aluminum that primarily provides shape to the aircraft and acts as an attachment point for the skin. While is this sufficient for tube-wing designs, the non-circular shape of a HWB aircraft requires far more reinforcement.

Also, the possibility of delamination in such a large-scale composite structure must be addressed so that a reasonable service life can be achieved. To overcome these challenges, the PRSEUS stringer shown in Figure 4.9 utilizes a unique architecture and widespread through-thickness stitching. Multiple PRSEUS stringer configurations have been experimentally tested, but all of the

50 | P a g e stringer designs have the same baseline architecture. The development and validation of the modeling methodology detailed in this section is focused on the work by Leone et al [17], specifically the Class 72 Type 1 stringer without extra adhesive in the overwrap. All of the experimental stringer compression tests were conducted using 17.5 in. long specimens that were potted in an epoxy compound within a rectangular steel frame. For each stringer type, only two specimens were tested, so the observed results may not represent the true average response.

Figure 5.1 Test article for the PRSEUS stringer compression experiment (reprinted from [23])

5.1.1 Stringer Characteristics The PRSEUS stringer is constituted of three primary components: the tear strap, pultruded rod, and overwrap. The tear strap acts as the attachment point between the load-carrying structure of the stringer and the skin. For the current study it is one stack thick, with the 0° fiber direction in the same axis as the pultruded rod, which is perpendicular to the primary fiber direction of the skin.

The pultruded rod increases the local strength and stability of the stringer while simultaneously increasing the moment of inertia and shifting the neutral axis away from the skin surface. The cross- section of the pultruded rod is actually a teardrop shape to minimize the high matrix concentration deposits near web-rod intersection. A number of different candidate materials have been considered for the pultruded rod, but the current study uses Grafil 34-700WD carbon fibers with PUL6 epoxy resin. All of the stitching is performed using 1600 denier Vectran thread at a spacing of 5.08 stitches per inch. A schematic of the Class 72 PRSEUS stringer is reprinted in Figure 5.2.

51 | P a g e

Figure 5.2 Schematic of a PRSEUS stringer

The pultruded rod is attached to the stringer by an overwrap stack that is bonded to the tear strap. In the presented study, one stack is used for the overwrap, producing a web that is two stacks thick. PRSEUS utilizes a unique stitching structure to bind the stringer geometry together, with six stitch paths in total. One stitch path is applied through the top of the web to fix the rod location while a second stitch path is applied just above the point where the web stack is folded out to become the flange. The final stitch paths are located in the flange; near the web is the 45˚ stitch sewed through the overwrap, tear strap, and skin toward the stringer centerline. Possibly the most critical stitch location is at the overwrap and tear strap edge to prevent any initial delamination from propagating toward the stringer center.

5.1.2 Stringer Model Development Composites are typically modeled using one of two scales. The first is a mesoscale approach in which each ply is individually modeled and the bonding between each layer is often directly modeled using cohesive zone models. This modeling approach allows delamination to be directly calculated and observed, making it ideal for identifying damage at small scales. The methodology has been successfully implemented by multiple groups [82,104–106] and is also validated in the ballistic impact on composite analysis shown in Appendix B. As shown in Figure 5.3 the mesoscale

52 | P a g e approach models each individual ply as a continuum with interlaminar cohesive models between each layer.

Figure 5.3 Schematic view of the mesoscale modeling methodology

The second type of modeling approach utilizes smeared (also called lumped) system properties.

Although the smeared model does not explicitly account for individual response of each ply and delamination, it is much less computationally expensive, making it desirable for large-scale simulations. The smeared system approach has been utilized extensively by design oriented works

[107] and has been shown to provide reasonably accurate global trends when no major failures occur [37,108]. However, the removal of delamination mechanisms from the model causes the prediction of failure to deviate significantly from experimental observations [37,108,109]. The present work combines the smeared system approach with the delamination modeling capability of the mesoscale analysis to create a modeling methodology that can accurately simulate PRSEUS at a wide range of scales.

Based on a dimensioned schematic provided by NASA and the hybrid modeling methodology, a high fidelity model of the PRSEUS stringer geometry was reproduced in the computational domain as shown in Figure 5.4. No initial finite element models were provided for this work, so all of the models, inputs, and setup were developed by the author. Each stack of material uses the smeared system approach to combine the seven individual plies into a single representative material.

The material properties of the stack in both tension and compression are reported in previous experimental investigations [13], but the composite material model in LS-Dyna only allows a single value for moduli and Poisson ratio. Therefore, the tensile and compressive properties were averaged

53 | P a g e as shown in Table 5.1 and used for the initial model inputs. These properties, as well as the properties of the Grafil 34-700 used for the pultruded rod [110], are discussed further in Section

5.1.5.

Table 5.1 Summary of material properties reported from experimental testing and the initial numerical input

Material Property Tension Compression Numerical Input

퐸푥 (GPa) 70.7 63.6 67.2

퐸푦 (GPa) 35.0 32.1 33.6

퐺푥푦 (GPa) 17.1 15.6 16.4

푣푥푦 0.403 0.397 0.400

Two fundamentally different model types were created for the current analysis, as shown in

Figure 5.4. The tiebreak model utilized a tiebreak approach to simulate both the stitching and epoxy.

Although computationally expensive, tiebreak models are commonly used to represent stitch bonding [108]. However, this approach is highly mesh-dependent and can only approximate some aspects of the unique stitching configuration required for PREUS (i.e. the 45˚ stitches through the web, tear strap, and skin). Additionally, while the tiebreak allows stress concentrations to form at the stitch interface between layers, it ignores the through-thickness stress concentrations that would be present in the actual specimen.

54 | P a g e

Figure 5.4 PRSEUS stringer for tiebreak model (top) and beam-stitch model (bottom)

Therefore, a secondary approach with an arguably higher fidelity model was created that individually models each through-thickness stitch. Instead of modeling each stitch as an actual penetration in the stack, the nodes on either side of the stitch location are connected with a beam.

Modeling each stitch as a beam allows the through thickness tensioning effect to be captured under deformation. However, because there is no direct connection to the interlaminar surface, the beam approach must rely on the local stress concentration and friction to simulate the effect of embedded stitch. For both of these modeling strategies, the location of the nodes is of critical importance to accurately represent the placement of the stitching. Another strength of the beam stitching approach is that it is theoretically less mesh dependent as long as the maximum element size is no greater than the minimum distance between each stitch.

For the initial model, the entire geometry (excluding the stitch) is modeled using selectively reduced integration solid elements. While this is the most computationally expensive element type, it also does not require any geometric assumptions and performed the best in the preliminary element study. Stitch material properties were assumed to be linear elastic and were based on the

Vectran manufacturers specifications [111]. The overwrap, tear strap, and skin were all applied the same material properties, but the primary fiber orientation of the skin material was rotated 90˚ to be perpendicular to the rod axis in accordance with the stringer layup [16].

Based on preliminary analysis, the local material orientation of the composite stack has a huge impact on the numerical results. With typical solid elements, the user must rely on either the

55 | P a g e element connectivity or a globally defined material direction. In the case of the stringer, a global material direction cannot be used because of the curvature of the overwrap stack around the pultruded rod. Alternatively, the user has very little direct control over the element connectivity.

From past experience, the element connectivity often does not remain consistent throughout the mesh, especially when the mesh is imported from a program other than LS-Dyna. To avoid these issues, the current study manually defined the local material coordinate system at each element as shown in Figure 5.5.

Figure 5.5 Illustration of local material angles along the overwrap for the pultruded rod

Although no additional adhesive is modeled between the overwrap and pultruded rod, the tear strap, skin, and overwrap are still bonded together with VRM-34 epoxy from the co-curing process.

Between each stack, a theoretical interlaminar epoxy layer is modeled using the Dycoss Discrete

Crack model [84] previously described in Chapter 3. The property determination to define the required peak tractions is described in Section 5.1.4. Static and dynamic friction coefficients for

56 | P a g e stack interaction were defined to be 0.65 and 0.40 respectively based on existing literature

[112,113].

5.1.3 Boundary Condition Setup In the experimental setup, the final inch of each specimen is potted on either side in a metal frame filled with Hysol EA 9394 paste adhesive. However, in the computational domain, directly including the potting and adhesive material can be difficult due to the complex geometry of the specimens and increased computational time. Therefore, a brief investigation was conducted to determine what fidelity of boundary condition is required to accurately represent the test conditions.

Due to the complex geometry of the PRSEUS stringer, a simpler model was used for the boundary condition study. The simplified model was created such that the cross-sectional area was approximately the same as the PRSEUS stringer, with the Class 72 Type 1 stack material was applied. As shown in Figure 5.6, three levels of boundary fidelity were applied to the simplified geometry.

Figure 5.6 Illustration of the three tested boundary condition ranging from the highest fidelity (left) to the simplest (right) The most complex and realistic boundary is to actually model the potting with the adhesive surrounding the specimen. The Hysol EA 9394 was approximated as an elastic material, with properties based on the manufacturer’s datasheet [114]. For simplicity, it was assumed that the adhesive was perfectly bonded to the specimen. The steel frame surrounding the adhesive was not

57 | P a g e directly modeled, but a fully fixed boundary condition was applied to the outer perimeter and loading surface of the adhesive. A symmetric boundary condition was applied at the mid-plane of each model.

The next highest fidelity boundary did not directly model the epoxy, but instead approximated its effect. For the one inch of potted material, a secondary boundary condition was applied to the outer perimeter that restricted transverse translation, but allowed the potted material to compress in the loading direction. Due to the fixed boundary at the potting perimeter, the Poisson’s effect of the material increases the resistance to axial loading without completely restricting displacement.

As with the previous model, a fully fixed boundary condition was applied to the loading surface.

The final model utilized the lowest fidelity boundary, but also the most computationally efficient. Both the high and medium fidelity models require the material in the potted area to be included, even though they are outside of the effective length of a buckling beam. Alternatively, the low fidelity model reduces the length of the specimen by the pot width and has a fixed boundary condition applied at the new loading surface. This approach prevents any additional axial displacement from the potted material, suggesting a stiffer response.

The load-displacement results of the boundary condition study are plotted in Figure 5.7. A clear difference in stiffness is visible between the fixed boundary model and the higher fidelity approaches. As expected, the stiffness of the fixed boundary model is greater, in this case by 15%.

Alternatively, the model with the pseudo potted boundary conditions matches the response of the model with the highest fidelity almost exactly. In terms of computational effort, the distribution was as expected. The epoxy potting simulation ran nearly twice as long as the lowest fidelity model due to having approximately double the number of elements, and the pseudo potting model fell between the two. Due to the combination of high accuracy and low computational expense (only

13% greater than the lowest fidelity model), the pseudo potting boundary conditions were applied to all of the following compression tests.

58 | P a g e

Figure 5.7 Load vs. compression displacement for each of the applied boundary conditions

5.1.4 Epoxy Property Identification Because epoxy is used in addition to the stitches to bind each stringer component together, the energy release rate and peak tractions must be identified. Unfortunately, the material properties of the VRM-34 epoxy used for the PRSEUS stacks is not publicly available, so an estimate was made by comparing against available experiments. The process to determine Mode I bonding parameters is typically performed using the double cantilever beam (DCB) test, which is governed by ASTM

Standard D5528-13 [115]. Grenoble et al. [116] conducted a series of DCB tests for [0/90]s preforms of the Class 72 Type 1 material as part of an effort to identify material properties for

PRSEUS that could later be used for model development. Figure 5.8 plots the collective data from five experimental DCB tests for load vs. opening displacement and crack length vs. opening displacement (obtained through a personal correspondence). Grenoble provides the Mode I energy release rate (390.53 J/m2), but doesn’t directly report either peak traction or maximum crack opening displacement when full debonding occurs. Therefore, an iterative approach is used in which an initial estimate of peak traction is made and then the corresponding load vs. opening

59 | P a g e displacement curve is plotted. This process is repeated until a close agreement is achieved between the experimental result and computational prediction.

Figure 5.8 Load vs. opening displacement (left) and crack length vs. opening displacement (right) for a DCB test of [0/90]s preforms

The simulation setup for the numerical DCB test is shown in Figure 5.9, where the red section represents the initial pre-crack. Because the experimental testing procedure is not specified, the numerical model applies a prescribed displacement at the ends of the specimen assuming that piano hinges were used in the experiment, allowing the material at the location of the applied displacement to rotate freely. The opposite face of the specimen is constrained using a fixed boundary condition. The Dycoss Discrete Crack model was used to represent the interlaminar bonding of the DCB specimen, but due to an unknown numerical issue within LS-Dyna, the implicit method cannot be used to in conjunction with virtual CZM representations. Therefore, the DCB test was conducted using explicit analysis.

60 | P a g e

Figure 5.9 Simulation setup for DCB test with pre-cracked sections colored red

As previously discussed, the computational expense of the explicit method for long-duration loading problems is extreme. Therefore, an accelerated loading rate must be used to achieve the prescribed displacement many times faster than for the experiment. Figure 5.10 illustrates the load vs. opening displacement of a DCB test for a range of loading rates. While some slight deviation in the phase of the predicted response can be observed for higher loading rates, the trend and peak load are nearly identical across all loading rates. Due to the relatively low computation time required for the 0.05s loading rate (displacement applied in 0.05 s), this loading speed was selected for the parametric analysis.

Figure 5.10 Load vs. opening displacement for a range of applied loading rates

61 | P a g e

As previously described in Chapter 3, the Dycoss Discrete Crack model is a bilinear traction- displacement law that uses the penalty contact stiffness as the initial slope of the bilinear curve.

Due to this, the slope remains unchanged for a given mesh and material definition. Prescribing different peak normal tractions, as shown in Figure 5.11 has the effect of changing the crack opening displacement to failure. This effectively represents a range of epoxy brittleness for a given energy release rate. For the parametric analysis, peak tractions ranging from 10-50 MPa were prescribed.

Figure 5.11 Bilinear traction-displacement rule for different peak tractions

A stress contour plot from the numerical DCB analysis is shown in Figure 5.12, in which the delamination has progressed to steady propagation. The distinctive tension-compression region at the crack tip is clearly visible. Initially the crack growth is unstable, matching the right-side plot for Figure 5.8. In addition to different peak tractions, a mesh study was conducted. While lower resolution grids were unable to qualitatively resolve the crack tip contour distribution with the same level of precision, the quantitative results remained relatively constant.

62 | P a g e

Figure 5.12 Y-stress contours of the DCB simulation at the highest mesh density

The results of the DCB investigation are plotted in Figure 5.13 for each of the simulated peak tractions. As the peak normal traction increases, a significant increase in oscillation is observed. In several cases (peak tractions of 40 and 50 MPa), the results had to be filtered using a Butterworth filter at a peak frequency identified using the Fast Fourier Transform (FFT). The increased oscillation is a result of modeling a more brittle epoxy. When the brittle epoxy fails at each node, the springback causes a stress wave to propagate through the specimen. While this effect can be alleviated by using a higher resolution mesh, the overall trend remains unchanged. The initial slope of each response closely aligns with experimental data, but whereas the experiment observed a linear decrease in stiffness, all of the computational models predicted a nonlinear response. Despite this deviation, the initial slope, peak load, and final load at the maximum displacement were well approximated by the 30 MPa peak traction. Therefore, this peak traction value is used for all instances where components are bonded with VRM-34 epoxy.

63 | P a g e

Figure 5.13 Load vs. opening displacement for each peak traction overlaid with experimental data

The DCB test provides parameters for the Mode I failure, but is completely independent of

Mode II. To determine the peak shear traction, an in-plane shear test is typically performed.

Although in-plane shear tests were performed by Grenoble et al. [116], no information beyond the reported energy release rate of 369.52 J/m2 could be obtained. Therefore, an initial value of 7 MPa was used based on other epoxies typically used for carbon fiber laminates [70]. The peak shear traction was eventually updated to 11 MPa. Due to the recursive nature of updating simulation inputs across multiple scales, all results shown in the following sections use the 11 MPa peak shear traction.

5.1.5 Validation of the PRSEUS Stringer Finally, with both normal and shear peak tractions defined for interlaminar bonding, nearly all inputs required to run the virtual PRSEUS stringer are defined. A mesh refinement study was conducted using the Grid Convergence Index procedure outlined by [117,118], which is thoroughly discussed in Appendix A. As a final step, an adequate loading rate of 0.00125 m displacement applied over 0.01 s for the model was determined using the same procedure outlined in both the

64 | P a g e element preliminaries and DCB study. The first major investigation was to compare the response of the tiebreak and beam-stitch models to determine which should be used throughout the research.

The load-displacement for each modeling approach is shown in Figure 5.14 in addition to the experimental baseline and the existing numerical model. Compared to the experimental baseline, both of the predicted curves are overly stiff, though not to the same extent as the model used by

Leone et al. Despite the increased stiffness, both the tiebreak and beam-stitch correlate well with the buckling load from the experiment, each with less than 1% error compared to the peak magnitude. Alternatively, only a change in stiffness is observed for the existing numerical model, with no reduction in the load that would indicate buckling. The post-buckling response of both the tiebreak-stitch and beam-stitch model approximately match the trends from the experiment, though both display much greater oscillation. The oscillations could be a result of the fast loading rate required for the explicit analysis, though there is no way to verify this hypothesis. In general, the two models performed similarly, achieving the same exact stiffness up until buckling and then both peaking at approximately 142 kN. Additionally, the computational effort required by each approach was nearly the same. The beam-stitch model was able to reach simulation termination just before the tiebreak-stitch model, but it is unclear if the slight difference is a result of the extra contact algorithm required for the tiebreak stitches or if it was due to interference from other computational processes.

65 | P a g e

Figure 5.14 Load vs. displacement for both the tiebreak-stitch and beam-stitch stringer models

Ultimately, the decision was made to continue with the beam-stitch model. Although the results of the models were comparable, the beam-stich approach could be implemented much more easily. Using the tiebreak-stitch approach requires each individual segment face to be manually selected, greatly increasing the model creation time. Additionally, the lower mesh dependency of the beam-stitch makes it a more attractive option at large-scales.

The high stiffness predicted by both the tiebreak-stitch and beam-stitch models was an unexpected response that prompted a more thorough investigation of the material property inputs.

Upon further investigation, the material parameters for both the Class 72 stack material and Grafil

34-700WD rod were updated. An experimental investigation of stitched vs. un-stitched material by

Grenoble et al. [116] discovered that the stitched material incurs a loss in both axial moduli and strength as a result of the fiber misalignment and imperfections introduced by the stitch. Although the difference was only compared for specimens loaded in tension, it is expected that a similar trend would be observed in compression and transverse directions. Based on the results of the study, both

66 | P a g e the axial and lateral stiffness inputs were reduced by approximately 5%, which is reflected in Table

5.2. No significant difference in Poisson’s ratio was observed between the two configurations.

Table 5.2 Summary of updated material properties based on stiffness knockdown factors

Material Property Initial Input Updated Input

Ex (GPa) 67.2 65.0

Ey (GPa) 33.6 32.0 Class 72 Type 1 Gxy (GPa) 16.4 15.6

vxy 0.400 0.400

Grafil 34-700WD Ex (GPa) 137.0 109.6

The pultruded rod had no stitching, but the initial estimate was based on the manufacturer datasheet [110]. The reported parameters were based on a 60% fiber volume fraction combination of Grafil 34-700 fibers and Mitsubishi Rayon #340 resin instead of the PUL6 epoxy used in the

PRSEUS stringer. Unfortunately, no information about PUL6 or the fiber volume fraction of the pultruded rod could be found on the public domain, so the properties of the pultruded rod were scaled based on the previous candidate material for the stringer rod. In earlier investigations, Toray

T800 fibers were used with a 3900-2B resin, with properties reported in the NRA Phase I report

[13]. Using these properties in conjunction with the Toray manufacturer datasheet [119], the relative reduction in fiber stiffness was used to scale the stiffness of the Grafil 34-700WD. The updated simulation input is also shown in Table 5.2.

With the updated material parameters, the PRSEUS stringer was simulated at the progressive stages of model development shown in Figure 5.15. Initially, just the stringer model with no interlaminar bonding or stitching is simulated, providing resistance from geometric effects alone.

As expected, the structure separates early in the loading history and loses structural integrity. When the effect of the epoxy is added to the model, a huge gain in strength is observed, more than doubling the initial peak load. The interlaminar bonding does not appreciably affect the stiffness of the specimen, matching the stiffness of the un-bonded model. Once buckling occurs, the load-

67 | P a g e bearing capacity of the structure quickly drops to near zero, which is characteristic of typical composites. The final model with both stitching and epoxy shows that stitching primarily improves the post-buckling response of the structure. A small increase in peak load is observed, but this difference falls within the range of numerical uncertainty (discussed in Section 5.1.7). The increased load-bearing capacity in the post-buckled regime clearly demonstrates the delamination arrestment due to stitching.

Compared to initial prediction with the original material inputs, the predicted response of the

PRSEUS stringer is much closer to the experimental baseline. In the early stages of loading, the error is negligible. However, at an applied displacement of approximately 0.8 mm, a reduction in stiffness is observed for both of the structures with interlaminar bonding. The softening effect was also present in the initial models and occurred at the same displacement level. The change in stiffness at this point coincides with the formation of low amplitude waves in the outer edges of the stringer skin. These waves were not observed in the experimental investigation and it is believed that this local buckling is an artifact of the fast loading rate. In early models, attempts to use implicit time integration did not predict a change in slope until the buckling point was reached. However, these past simulations do not include updated properties or any tiebreak conditions, so it is difficult to draw a clear conclusion. If not for the slight change in stiffness, the entire load-displacement prediction up to the buckling point would be collinear with the experimental result.

The loading rate is also responsible for the highly oscillatory post-buckling response. As previously discussed in the preliminary element study of buckling (Section 4.2), it was observed that the faster loading rates correlated with higher amplitude load fluctuations. Because the loading rate has such a significant impact on the response, many efforts were made to use implicit and hybrid methods to achieve the experimental loading rate. However, in every case, the implicit time integration was found to be incompatible with the tiebreak conditions. Therefore, the loading rate was selected based on fluctuations in the peak load instead of the post-buckling response so that a feasible analysis could be conducted with explicit methods.

68 | P a g e

Figure 5.15 Loading vs. displacement response for progressive development of the PRSEUS stringer

While the load-displacement plots show the effect of the stitching on the post-buckling load capacity, the difference between a stitched and un-stitched structure is shown more clearly in Figure

5.16. Contour plots for each of these representations are shown at three points in the loading history.

At an applied displacement of 1.50 mm, the stress contours for both models were very similar, with no asymmetry in contours that would suggest imminent buckling. At this point, the waves on the stringer skin have already formed, though the increase in amplitude is very gradual.

As the buckling point is approached at an applied compression of 2.09 mm, differences begin to appear between the two models. The un-stitched model has started to develop clear signs of delamination in the stringer flange, which is indicated by the blue band of low stress where it was previously carrying load. Alternatively, the stitched model has arrested the delamination at the edge of flange, with just a small area of separation beyond the outer stitch line. Both models exhibit signs of buckling, with the development of stress contours on the side of the overwrap and a stress gradient across the pultruded rod. A stress concentration is also visible in both models at the base

69 | P a g e of the web, though the magnitude is greater for the stitched model due to the restricted movement imposed by the stitches.

In the final set of contour plots at 2.25 mm of compression, the un-stitched stringer has completely delaminated. The overwrap is no longer attached to the skin and tear strap, as evidenced by the loss of stress, and has also separated along the web. The separated components display individual buckling modes, with the skin and tear strap developing a half wave in the vertical plane and the rod and overwrap buckling in the horizontal plane. Alternatively, the stitched model shows no visible signs of major delamination, though the previously delaminated band has extended toward the center of the stringer (at the symmetric boundary) and grown in toward the stitch line.

The magnitude of the stress concentration at the base of the web has also increased, suggesting that the stitch line is preventing a delamination near the web base. The stitch locations appear to have a lower stress than the surrounding material, but this is just a result of the contour nodal averaging and lower stress in the transverse directions. Although no element failure is observed in the simulation, the development of damage closely matches the observations by Leone et al. [23]. In the experiment, failure was consistently observed to originate at either the edge of the flange or at the base of the web before propagating to the rod overwrap.

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Figure 5.16 von-Mises stress contours for both stitched and un-stitched stringer configurations (the near surface is at the symmetric boundary)

Using the validated PRSEUS stringer model, a parametric study investigating the effect of different parameters on the final load-displacement response was conducted. The parametric study investigated the effect of changes in the geometry, stitch density, epoxy strength, and stack properties. For this analysis, the geometry and stack properties were only varied slightly, while stitch density and epoxy strength were changed up to a factor of two. The relative influence of each of these parameters, in addition to the previously investigated effect of model type and loading rate,

71 | P a g e are summarized in Table 5.3. Out of all the parameters, geometry and stack properties were observed to have the largest impact on the simulation result. Due to this, these variables are recommended to be thoroughly checked for accuracy.

Another important variable to consider is the strength of the epoxy. Without the DCB test to use as a baseline, the peak normal traction of the interlaminar bond would have probably devolved into a tuning parameter because of its strong influence on peak buckling load. Unlike the peak normal traction, changes in the peak shear traction did not result in any significant changes in the response of the stringer. Other variables such as stitch density did not significantly affect the overall performance of the modeling methodology, but can provide valuable insight for future optimization.

At very high stitch densities, a drop in post-buckling performance was actually observed, likely as a result of over-constraining the surrounding material.

Table 5.3 Summary of observed parameter influences on the computational model and their relative impact

Variable Effective Stiffness Peak Load Post-Buckling Model Type None Low Low Geometry High Medium None Loading Rate None High Medium Stitch Density None None Medium Peak Normal Traction None High None Peak Shear Traction None None Low Stack Properties High High None

In summary, the developed modeling methodology was successfully validated against experimental tests and predicted failure development that coincided with experimental observations.

Despite slight deviations in the slope and post-buckling response as a result of the fast loading rate, the full model with interlaminar bonding and stitching provided highly accurate predictions. The error in peak load is less than 1% between computational results and experimental data and though under-predicted, the post-buckling response is substantially improved over that of a typical

72 | P a g e composite. Additionally, the developed methodology is a substantial improvement over the existing computational model for PRSEUS [16], which could not predict a drop in load following buckling and overestimated the stiffness by 19%. The difference in model performance can be attributed to the greater fidelity of the current methodology and efforts to include the effects of physically based observations.

5.1.6 Reduced Expense Model With the methodology for the PRSEUS stringer model validated, attempts were made to reduce the computational expense of the model for application at higher scales. Based on the results of the preliminary element investigation, the stringer was developed using solid elements with selectively reduced integration. While the results produced by the solid element model are quite accurate, the computational time required is not feasible at higher scales. Each simulation required approximately 6500 seconds to reach the termination time on a system with 12 CPUs. Considering that the PRSEUS stringer model is only 8.75 in. in length with a symmetric boundary condition, applying the same approach at the panel or fuselage level would be computationally infeasible.

Therefore, reduced expense models were developed in an attempt to reduce computational effort while preserving the prediction accuracy.

Due to the difficulty in meshing a curved surface with typical finite elements, the pultruded rod contained 37% of the total number of solid elements in the model. Therefore, the first approach simply replaced the pultruded rod with beam elements as shown in Figure 5.17. Because of the way that beams calculate contact, described in Section 2.6, the switch to beam elements is not a perfect geometric match due to the teardrop cross-section of the pultruded rod. Therefore, the nodes in the surrounding overwrap were adjusted to remove the void. Based on the preliminary element study from Chapter 3, the beam element is capable of predicting accurate responses in both bending and buckling, though it cannot account for Poisson effects.

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Figure 5.17 Illustrations of shell-beam (left) and solid-beam (right) low reduced computational expense models

The second reduced fidelity model takes a more aggressive approach to reduce computational effort by replacing all stack layers with shells in addition to replacing the pultruded rod. Due to the reduction to two-dimensional elements, the number of nodes is reduced to only 43% of the validated solid element model. Although theoretically efficient, this model requires more user effort to create because the entire geometry is based on assumed thicknesses. The fully integrated shell element did not perform as well as the solid or beam elements in the preliminary element study, but the anticipated reduction in computation time was so great that an attempt was made regardless.

The results of applying these two model reduction approaches to the stringer compression test is shown in Figure 5.18.

Of the two reduction approaches, the solid-beam model performs significantly better.

Compared to the validated solid element model, the overall trend is very similar, but the peak buckling load is 10% lower. The reduction in the peak loading is due to the combination of the adjusted cross-section and inability to account for Poisson effects. Despite this, the predicted stiffness aligns perfectly with the solid element response up until buckling. The post-buckling behavior also remains consistent, but with a phase shift due to the reduced peak load.

Alternatively, the shell-beam model is lacking in both peak strength and stiffness. Unlike any of the previously observed models, the shell-beam stringer predicts a continuous nonlinear

74 | P a g e softening as the model is compressed. Also, the peak loading is only 48% of the experimentally observed load, resulting in a load history that resembles the previously discussed case with no interlaminar bonding or stitching. The fully integrated element formulation prevents any hourglass energy from accumulating, which would also explain the softened response. Upon further inspection, several unusual characteristics are observed about the energy distribution of the shell- beam model. The primary issue is a negative sliding energy that is a result of the tiebreak model.

As a result of having only one surface, the tiebreak condition for the tear strap is unstable, often resulting in small penetrations. The negative energy acts similarly to hourglass energy in that it often produces a softer material response than expected. The combination of negative sliding energy and limitations of the shell element formulation are why the shell-beam model performed so poorly. It is possible that the thick shell element would be able to solve the issue of negative sliding energy since it has separate top and bottom surfaces, but because of the hourglass problems encountered in the preliminary element analysis, this approach was not attempted.

Figure 5.18 Loading vs. displacement for the beam and shell reduced fidelity models

The beam-solid model was able to reach simulation termination in only about 1200 seconds for the same 12 CPU system, resulting in a computational time reduction of more than 80%. Such a

75 | P a g e huge reduction in computation time was not expected based on the percentage change in the number of solid elements. However, the elements near the center of the solid element pultruded rod had a poor aspect ratio, which decreased the time step size. When the rod was replaced with beam elements, it not only reduced the number of solid elements by 40%, but also increased the time step by nearly an order of magnitude. Considering the significantly lower computational effort and reasonable approximation that the beam-solid approach was able to produce, it was selected as the primary model for further analysis. When accuracy takes precedence over all else, the solid element model may be used, but for most applications the beam-solid model is sufficient.

5.1.7 Uncertainty Quantification A common expectation with most applications in computational analysis is that for a given set of inputs, the user will always get the same output. However, with an inherently unstable behavior such as buckling, this is not necessarily true [120–122]. Due to discretization error associated with explicit numerical analysis, the accrual of error will cause an initially perfect rod to buckle. In the current study, this error was found to have an effect on the predicted response of the PRSEUS stringer, even when the calculation timestep was fixed well below the minimum stability requirement. To quantify the numerical uncertainty associated with the stringer compression test, the beam-solid model was run multiple times for the same input conditions. The results of the analysis are shown in Figure 5.19.

As expected, differences in the response only become significant as the model approaches buckling. Up until an applied displacement of 1.8 mm, all of the runs effectively predict the same load. However, beyond that point, the uncertainty increases rapidly. Because the buckling load varied between the individual simulations, error is reported with respect to the average response.

From the figure, two primary areas of uncertainty can be identified: peak load and post-buckling.

In the regime of the peak load, the error fluctuates from 1.2% at 1.8 mm to 5.3% at 2.0 mm. During the initial drop in load, the error increases to a maximum of 8.4% before suddenly dropping to only

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2.3%. The sudden drop in error at 2.25 mm suggests that the inflection point of the post-buckled response is a common occurrence. Past the inflection point, the error increase continuously until simulation termination, peaking at 11.4%. Despite the error, the general trend of the response stays consistent with the experimental observations. Additionally, no major changes in damage propagation or the failure mechanisms are observed.

Figure 5.19 Average load-displacement response for the beam-solid model with error and standard deviation bars

One of the major advantages of having a reliable, physics based computational methodology is that it can be used to guide design without requiring a significant number of experimental tests.

One area in particular that can benefit from computational models is uncertainty quantification.

Typically, the only way to quantify the uncertainty in the response of a material or structure is run a large number of tests to define the mean response and standard deviation. However, with a computational model, many simulations can be run without incurring the same costs as with experimental testing.

Considering that PRSEUS is a cutting edge technology, many of the manufacturing processes used to construct PRSEUS were developed recently. A preliminary study was conducted to

77 | P a g e determine the range of uncertainty in the response of the PRSEUS stringer, in which several of the stack material properties were adjusted based on the tolerance for the Hypersizer trade study inputs

[13,24,123]. At this preliminary stage, only the axial and transverse moduli were chosen for the study, but future studies could include other factors such as geometry or bond quality. Each of these variables was varied individually to their respective maximum and minimum values (as shown in

Table 5.4) to determine the individual effect of each variable on the load response. Each dataset consisted of five simulations for a given set of input conditions so that fluctuations due to numerical discretization would be averaged out.

Table 5.4 Material inputs for manufacturing tolerance in the Class 72 Type 1 material stack

Material Minimum Value Initial Input Maximum Value Axial Stiffness (GPa) 64.53 65.00 65.75 Lateral Stiffness (GPa) 31.61 32.00 32.58

The results for the variability in axial and lateral stiffness are shown in Figure 5.20. Due to the fluctuation of the response as a result of numerical discretization, the averaged response of each case is compared to the averaged response of the baseline inputs. The standard deviation of the baseline model is used as the metric to determine if the difference in response is significant. If the average response of the varied property case lies at least one standard deviation away from the baseline case, then that section of the response is considered to be statistically significant.

As an example, the load history predicted by the minimum input value for the axial stiffness does not fall outside of the baseline standard deviation at any point. Therefore, the entire response is considered to be statistically the same as the baseline input case. Alternatively, when axial stiffness is set to its maximum value, the resulting 2.4% increase in the peak load is statistically significant over the range of applied displacement from 1.8 mm to 2.25 mm. Past the post-buckling inflection point, the predicted response falls back within the standard deviation range of the baseline input case. Overall, the model response seems to be much more sensitive to increases in the axial stiffness than decreases.

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Figure 5.20 Plots of load vs. displacement for positive and negative changes in axial (top) and lateral (bottom) moduli with zoomed area in buckling regime

When the lateral stiffness is varied, the opposite trend is observed. The maximum value of the lateral stiffness results in little statistically significant difference from the baseline case. The only significant difference is for a brief range of applied displacement at the inflection point of the post- buckling curve. At the minimum lateral stiffness, the load magnitude becomes statistically greater for applied displacements greater than 2.0 mm but less than 2.3 mm. This result in particular is surprising, though it is possible that the model requires more simulations to be run to better characterize the average response. While the results from the property analysis indicate response changes at the minimum and maximum acceptable stiffness, this does not necessarily represent the outer bound scenarios. Typical composites already have highly non-linear behavior that make

79 | P a g e sensitivity analysis difficult [124], and the integrated nature of PRSEUS makes these interactions even more complex.

In the present uncertainty analysis, multiple trials have been used to produce an average response curve and corresponding standard deviation. However, an alternative method can be used to characterize a single prediction for an unstable system such as buckling, eliminating the need to run multiple simulations for each set of inputs. As previously discussed, the inconsistent response is a result of numerical discretization error causing an initially perfect geometry to gradually become imperfect and eventually buckle. This issue can be resolved by introducing a perturbation into the initial geometry that makes it imperfect [37,120,121]. The overall effect essentially mimics experimental testing in that no physical specimen is truly perfect, allowing the simulation to consistently converge to a single prediction. However, the selection of the perturbation must be performed carefully so that reliable results may be achieved.

Figure 5.21 Load vs. displacement response for two different initial perturbations in the geometry

Figure 5.21 illustrates the load-displacement response for the initial material inputs when two different perturbations are applied. In both cases the perturbation was a shift in node position on the order of 1 x 10-5 m, but the location of the shift was different in each model. The first perturbed system, shown on the left of Figure 5.21, demonstrates clear improvement in the overall magnitude of error and standard deviation. As opposed to the initial baseline, the maximum error is negligible

80 | P a g e until well past the peak buckling load. The error only becomes significant beyond the post-buckling inflection point and eventually reaches a maximum of 4.6% just before the final displacement is achieved. Overall, the general trend of the first perturbed system matches the initial baseline, though the predicted peak load is 5.7% lower and the increase in load past the post-buckling inflection point is much steeper than previously observed.

The second perturbation case shows a much more similar response to the initial baseline. The average response as well as the location and magnitude of error are all very similar to the initial baseline. Unlike the first perturbed system, the second perturbed model does not appear to have converged to a single solution. Although a perturbed system may allow the model to converge to a single, repeatable response, a thorough analysis is still required to select a proper perturbation that requires many simulations to be run to determine the overall effect on the system. Future work will investigate how this method may be applied toward conducting a sensitivity analysis of the

PRSEUS stringer. As previously mentioned, other factors such as geometry or bond quality could be included to develop a more holistic view of manufacturing uncertainty.

5.2 Frame Development

While the primary job of the stringer in semimonocoque airframe construction is to provide shape to the to the traditional aircraft fuselage and act as an attachment point for the skin, the frame is the primary load bearing structure for the internal pressure. The frames must also support the stringers and longerons, which transmit significant loads to the frame during flight. In passenger aircraft, the internal cabin pressure must be cycled continuously, causing structural fatigue. For these reasons, the frames must be highly damage tolerant to maintain a lasting service life. In a

HWB aircraft, the frames must be even more robust due to the out-of-plane and bi-axial loading caused by the non-cylindrical fuselage body. To overcome these challenges, the PRSEUS frame utilizes a unique architecture that firmly bonds the stringer to the frame and widespread through- thickness stitching. As with the experimental baseline and validation for the numerical PRSEUS

81 | P a g e stringer model, the frame tests and experimental baselines are detailed by Leone et al [17]. The same general test setup was used for the frame compression tests, as shown in Figure 5.22, though the specimens were only 16 inches long and anti-buckling guides were used. Additionally, the load- displacement response of each frame type varied much more between the two tested specimens than it did during the stringer compression tests. In each of the compression frame specimens, the two stringers that pass through the frame are different configurations. However, because the fiber direction of the stringers is perpendicular to the frame direction, any disparity was assumed to be negligible.

Figure 5.22 Test article for the PRSEUS frame compression experiment (reprinted from [23])

5.2.1 Frame Characteristics Unlike the PRSEUS stringer, the PRSEUS frames vary significantly in their geometry and construction. In the current analysis, two different frame designs are investigated: the foam-core frame and tapered L-frame as shown in Figure 5.22. The foam-core frame is the baseline frame configuration that was developed for use in a HWB. The core is a Rohacell 110 WF closed cell foam which primarily acts as a mechanism to shift the Class 72 overwrap away from the central axis, thereby increasing the moment of inertia of the frame. Additionally, the foam prevents the surrounding material from buckling inward when under a compressive load. The Class 72 stack material used for the stringer is also used for the frame construction, though the orientation of the fibers in the frame tear strap and overwrap are collinear with the primary fiber direction of the skin.

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Figure 5.23 Schematic of a PRSEUS foam-core frame (left) and tapered L-frame (right)

The tapered L-frame is a relatively recently developed configuration that is designed to be comparable to the frames used in conventional tube-wing fuselage. Because the typical cylindrical fuselage is not prone to the same out-of-plane and bi-axial loads, the tapered L-frame uses significantly less material. A flange at the top of the frame adds overall stiffness to the design and assists in resisting buckling modes. The tapered L-frame has a far more complex construction than either the foam-core frame or stringer and requires approximately twice as much stitching. The frame, not including the tear strap and skin, is comprised of six individual Class 72 stacks as opposed to the two required for the foam-core frame. Additionally, the design uses far more stitch lines, which are present in both the bottom flange and web of the tapered frame.

At the intersection of the frame and stringer, the construction becomes even more complex because of the layering of the two structures. An exploded view example of the stringer-intersection for the tapered L-frame is shown Figure 5.24. One of the primary advantages of PRSEUS over typical metal and composite structures is that it maintains a continuous load path without the use of rivets or other attachments. At the frame-stringer intersection, the structure is layered in the order of skin, frame tear strap, stringer tear strap, stringer, and finally the frame. Then, this highly integrated structure is co-cured and stitched, producing a final, fully attached assembly.

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Figure 5.24 Exploded view of tapered L-frame at the frame-stringer intersection (no stitching shown)

The tapered L-frame adds an additional layer of complexity to the construction by introducing a change in cross-sectional area neat the stringer keyhole. In this region, two layers of the Class 72 stacks are extended beyond the top of the stringer keyhole and the tapered regime around the keyhole varies from two to three stacks thick. Due to the extreme complexity of the design, the next step in the methodology development focuses on accurately characterizing the frame-stringer intersection.

5.2.2 Frame-Stringer Model Development From the PRSEUS stringer analysis, a thorough, physically based modeling approach was developed. Because no tuning parameters or other scenario-dependent variables are used, the same methodology can be extended to modeling the frame compression tests. However, due to the significant complexity at the frame-stringer intersection, an investigation was performed to see what level of fidelity the frame model requires to give an accurate prediction. Figure 5.25 illustrates the low fidelity model for the tapered L-frame. The low fidelity model neglects the change in cross- sectional area in the frame web at the stringer keyhole and also ignores the stack reduction at the

84 | P a g e base. Due to the constant, single stack thickness in the tapered section of the frame web, no additional stitching was applied. While these changes are significant, the user effort required to create the model is very low and could easily be parameterized. In the experimental test, anti- buckling guides were used to avoid premature buckling of the skin in a global buckling mode. A thorough description of the guides is not available, so the same translational constraints used by

Leone et al. were implemented. These constraints are applied on both sides of the frame, covering a 1 inch area from the free edge and leaving space around both the stringer and loading surface as shown in Figure 5.25.

Figure 5.25 Illustration of a low fidelity approach for the tapered L-frame

The results of the low fidelity tapered L-frame are shown in Figure 5.26. The low fidelity model initially predicts a stiffness that is 10% less than the experimentally observed stiffness, but at 0.001 m of displacement the frame web begins to bow. The bowing in the tapered regime causes the stiffness to drop further, increasing the error to 30.5%. While a slight change in stiffness is observed in the first experimental load curve, the change is not nearly as severe. The actual buckling of the top flange occurs at a displacement of 0.00185 m, which correlates well with the first experiment, but the buckling load is under-predicted by 22.5%. Due to these shortcomings, the low fidelity model was not implemented in further analysis.

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Figure 5.26 Load vs. displacement response for the low fidelity tapered L-frame model

Based on the underperformance of the low fidelity model, a high fidelity model was developed for both the baseline foam-core frame and tapered L-frame. In both cases, the individual components were progressively added to accurately represent the layering and stack drop-offs present around the intersection of the stringer and frame. The stack dropdown must be included in the high fidelity model to accurately represent the physical specimen because the dropdown allows the material stack to remain continuous as shown in Figure 5.27. If no dropdown is present, then the structure is much more prone to damage and delamination.

Figure 5.27 Illustration of a model with and without stack dropdowns

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5.2.3 Validation of the PRSEUS foam-core and tapered L-frames After determining that the low fidelity model could not provide predictions with sufficient accuracy, the validation for both of the frame designs proceeded with the high fidelity model. It is important to note that the high fidelity models shown in Figure 5.28 are not based off of a properly dimensioned schematic. Unlike the stringer model, no dimensioned sketches could be provided for the compression frame specimens. Therefore, the models were developed based on images of section views and known outer dimensions of each structure. The known dimensions of the stack thickness and stringer were also used to ensure that the model was as accurate as possible. Based on the stringer validation, solid elements with selectively reduced integration were used to represent both structures with the same mesh density used in the stringer validation. The beam-solid approach for the stringer was used to mitigate computational cost, especially since the stringer is not expected to have a significant contribution to the frame compression. The Rohacell 110 WF foam in the foam-core frame model is represented with the linear elastic-plastic model with properties based on the manufacturer datasheet [125].

Figure 5.28 Illustration of high fidelity foam-core frame (left) and tapered L-frame (right)

The results for both the foam-core frame and tapered L-frame are shown in Figure 5.29. Both predictions closely match the experimentally observed stiffness as well as the peak buckling load for one of the specimens. As previously stated, the lack of experimental trials makes it impossible

87 | P a g e to determine what a typical response of the structure is. However, the fact that the numerical prediction closely represents one of the experimental trials indicates that methodology is capturing many of the same primary damage mechanics.

The foam-core model matches the experimentally observed stiffness closely, over-predicting it by only 2.2%. Additionally, it follows the same trend of gradual softening as the displacement is increased. The buckling peak location of the experiment and numerical prediction coincide at a displacement of 0.00261 in. at which point the simulation peak load of 540.7 kN is 3.3% higher than the experimentally observed buckling load. When compared to the existing numerical model, the prediction by the high fidelity foam-frame model shows significant improvement. The methodology is able to accurately predict the location and magnitude of the buckling load, while the existing model can only provide a general loading trend that does not indicate the location of failure. An additional benefit in the case of the foam-core model is that it is able to capture the point at which catastrophic failure occurs, resulting in the near instantaneous drop in load. This capability was not observed for the stringer model, though local element deletion indicated the impending failure.

Figure 5.29 Load vs. displacement for the high fidelity foam-core frame model (left) and tapered L- frame model (right) [*the data was clipped beyond this point due to the edge of the plot]

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The tapered L-frame model also correlates extremely well to the experimental results, in which trends similar to the foam-core frame were observed. The predicted stiffness had a maximum error of 4.6% during the initial loading and just before buckling, but maintained less than 1% error during the majority of the softening regime. As with the foam-core model, the tapered L-frame model closely matched the stronger of the two replicates tested. The difference in the location of the buckling load is only 0.028 mm, while the error in the predicted buckling load is just 3.6%. Near the buckling point, a brief drop in stiffness is observed when a section of frame web at the symmetric boundary debonds. However, unlike with the low fidelity model that did not include the stack buildup around the stringer keyhole, the reduction in stiffness is only temporary. The brief decrease in stiffness due to the sudden debonding in the web may also be an artifact of the fast loading rate introducing dynamic effects. The post-buckled response is unable to capture the catastrophic failure point, but as with the stringer, local element deletion indicates the location of the impending failure.

The damage progression at both the structural and interlaminar level allows for greater insight into the failure mechanics of the frame. These effects are visualized for the foam-core frame in

Figure 5.30 with contours of von-Mises stress and delamination percentage at three points in the loading history. Well before buckling, at a displacement of 2.45 mm, clear stress concentrations are visible above the stringer keyhole and at the frame base stack dropdown at the stringer-frame intersection. As expected, neither the foam nor the stringer are significant contributors to supporting the load, though they do provide support to delay buckling. While in the case of the foam, the lack of stress is likely due to a relatively low modulus, the lack of contribution from the stringer is primarily the result of the stack drop-offs. Due to the slight gap introduced by the dropdown, stress can only be transferred to the stringer via shear. At the extreme loads supported by the foam-core frame, the relative difference in stiffness between the frame tear strap and frame base near the dropdown causes delamination to slowly spread. This slow extension of delamination contributes

89 | P a g e to the slight non-linearity in the load history, which becomes gradually less stiff as the buckling point is approached.

Figure 5.30 Stress contours (left) and bonding contours (right) for the foam-core frame model

At an applied displacement of 2.625 mm, the foam-core frame is supporting its peak load. The previous stress concentrations on the foam frame are increased in severity, with the stress

90 | P a g e concentration around the stringer keyhole developing a distinctive butterfly shape that strongly resembles the contours at the head of a crack. However, the foam-core frame does not exhibit any clear signs of buckling such as asymmetric stress contours or displacement in the frame web. The previously described delamination extending from the frame base dropdowns has propagated, though primarily on the side closest to the loading surface. The delamination between the two stacks of the frame base has been largely arrested by the stitching.

Finally, the foam-core frame has failed at an applied displacement of 2.750 mm. The development of buckling was extremely rapid for the foam-core frame, transitioning from a state with no signs of buckling to a state where the frame web has delaminated and the supported load has dropped by over 50%. The web of the frame slightly buckled, immediately causing the stacks in the web to separate from one another. Due to the lack of stitching in the foam-core frame web, the delamination immediately propagates across the length of the frame. The spread of delamination to the stringer intersection coincides with the formation of a crack at the top of the keyhole. This crack is discussed in more detail in the following discussions. No buckling of the foam-core web was observed in the experimental observations, but due to the extremely rapid development of buckling, it is possible that it could be overlooked. Unfortunately, the angle of the high speed cameras used in the baseline experimental study makes buckling difficult to discern.

A similar progression of contour plots is shown for the tapered L-frame in Figure 5.31. At a displacement of 1.425 mm, the frame web has slightly bowed away from the top flange and clear stress concentrations are visible at each of the corners of the web dropdown on either side of the stringer keyhole. Due to the increased stack thickness around the stringer keyhole, the distinctive stress concentration visible for the foam-core frame is no longer present. A significant disparity in stress magnitude is visible between the sections of the frame web that have one stack through the thickness and the neighboring buildups, particularly at the corners of the web dropdown and in the web above the stringer keyhole. In the area above the keyhole, the construction of stacks and fiber orientation in this regime only allows forces be transmitted to the stack buildup though shear forces.

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As a result, signs of developing delamination are visible at the edges of the stacks above the stringer keyhole and at the dropdowns in the frame web. More significant delamination is present at the dropdowns in the frame base, though its progression has been largely arrested by the stitching through the frame base. Some slight delamination is present at the intersection between the stringer and frame as a result of the slight bowing in the frame web. As expected, the stringer does not carry significant load since it has no direct connection to the loading surface, though it has started to delaminate between the stringer tear strap and skin due to the disparity in load.

Figure 5.31 Stress contours (left) and bonding contours (right) for the tapered L-frame model

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Just before reaching an applied displacement of 1.625 mm, the stresses at the corner of the web dropdown on the loading side of the stringer exceeded the material strength, causing local element failure. This sudden failure caused the already developing delamination to quickly spread from the corner, but was quickly arrested from spreading toward the stringer keyhole and frame base by the stitching. Almost simultaneously, the bowing of the single stack regime in the frame web caused it to debond from the top and bottom frame buildups on the symmetric plane side of the stringer. The sudden spread of delamination and increase in frame web bowing at this point are the reason for the drop in stiffness observed in the load-displacement plot. However, the delamination was once again quickly arrested from propagating toward the stringer keyhole and pre-existing delamination at the frame base dropdown. Although not clearly visible in the figure, the delamination extending into the top flange is arrested by the stitch line at the base of the flange.

Finally, at an applied displacement of 1.775 mm, the tapered L-frame has buckled and is in the post-buckled regime. The buckling point coincides with the deformation of the flange at the top of the frame. Compared to the von-Mises stress contour at 1.625 mm of displacement, the stresses have been significantly redistributed. The buckled flange now displays relatively low stress where it previously was one of the most highly stressed components in the model. Consequently, the skin and frame tear strap now are bearing the majority of the load and have high stress contours. The buckling in the flange coincides with bond failure between the stacks that make up the flange. The delamination in the flange immediately propagates extensively across the top of the flange and also joins the delamination at the top of the frame web that had previously been arrested by the stitch line. Despite the sudden deformation and spread in delamination, the stitching around the stringer keyhole and at the frame base dropdowns continues to arrest further delamination.

While the developed methodology has the capability to accurately predict the buckling point, it does not necessarily reproduce the ultimate failure observed in the experimental testing. For the foam frame, the point of ultimate failure is clear from the instantaneous drop in load following buckling. Alternatively, the tapered L-frame model did not display such a drastic reduction in load

93 | P a g e and was predicted to maintain a significant load until simulation termination. Therefore, the estimated ultimate failure point of the tapered L-frame is chosen based on a sudden increase in element deletion during the post-buckling response.

Figure 5.32 depicts the local element deletion at the estimated point of ultimate failure with labels in order of occurrence. As the foam-core frame approached its maximum load, the material at the frame base dropdown closer to the loading surface began to erode, effectively simulating the development of a crack parallel to the stringer tear strap. Then, immediately after buckling, a small crack developed above the stringer keyhole. These failure characteristics closely match the observations of the experiment, which also found the crack to originate at the root of the frame near the stringer-frame intersection and extend up though the stringer keyhole.

For the tapered L-frame, the initial crack development was very similar. During loading, a small amount of element erosion was observed underneath the frame base and along the stringer tear strap, but now on the side closer to the symmetric boundary. As the structure continued to compress, the stack dropdown at the buildup around the stringer keyhole began to erode. After the frame entered the post-buckling regime, the final two cracks appeared almost simultaneously. The first was the crack at the top of the web, extending toward the top flange and the second appeared almost directly below at the stack dropdown. Due to the buildup around the stringer keyhole, no failure is observed to occur in that region. Once again, these characteristics are representative of the experimental test, which observed that the crack could propagate on either side of the stringer.

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Figure 5.32 Sequentially labeled locations of local element failure for the high fidelity foam-core frame (top) and tapered L-frame (bottom)

Although the methodology may over-predict the duration of the post-buckling response, this apparent strength is an artificial numerical response due to the lack of crack extension in the frame web. With discrete elements, an extreme level of mesh refinement or an extended finite element

95 | P a g e method is often required to accurately model crack propagation. In such a model, it is possible that the crack would propagate through the entire web, but such a simulation would require significantly greater computational time for a relatively small improvement in modeling prediction. The methodology that has been developed in this work is able to demonstrate the primary locations of failure while still remaining computationally feasible.

5.3 Chapter Summary and Conclusions

Overall, the methodology developed in this chapter has been validated against multiple experimental baselines for a variety of different structures. This was done using an approach entirely developed based on physical observations instead of using numerical tuning parameters. A boundary condition study was first performed to accurately represent the experimental testing conditions. After selecting an approach that was both accurate and computationally efficient, interlaminar bonding properties were characterized using the results from a DCB test. Finally, the material inputs available in the literature were scaled based on experimental works to more accurately represent the effect of stitching on the structure.

With all of the input parameters defined, a high fidelity stringer model was validated against experimental data, predicting a response with less than 1% error in peak load. The accuracy of the model is significantly greater than the existing computational model and adds the capability of including the post-buckling response. With a mindset for large-scale applications, a reduced expense model was developed that was able to approximate the high fidelity model response for only 20% of the initial models computational effort. With the reduced expense model, a preliminary study was performed to characterize the uncertainty that manufacturing tolerance might introduce, which will be continued in future studies.

With the stringer validated, the next higher scale to validate was the frame level, particularly the intersection of the stringer and frame. Two different types of PRSEUS frames were chosen for further analysis: the foam-core frame and tapered L-frame. The two configurations are designed for different applications, with the foam-core frame as the baseline concept for HWB aircraft and

96 | P a g e the tapered L-frame as a potential option for conventional tube-wing fuselage. Each structure was computationally modeled with the same overarching methodology used for the stringer (same boundary conditions, material inputs, bonding parameters) and validated against their experimental counterparts. The model predictions were highly accurate, closely matching the stiffness and buckling point for each frame type and once again outperforming the existing computational model, despite the significant differences in construction. Additionally, the methodology predicted a damage progression that agrees with the observations from the experimental analysis. While the final crack that spanned through the entire specimens could not directly be captured by simulations, the locations and development of the ultimate failure crack path was accurately predicted. It is also noteworthy that the variation in response observed with the stringer model due to discretization error did not appear to be present at the frame scale, though an exhaustive study has not yet been completed. With a methodology that was validated for both the frame and stringer, the primary structural components of an airframe, the panels and fuselage of a PRSEUS aircraft may be investigated.

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6 Large-Scale Analysis

Following the component level validations, one of the goals of the proposed research is to apply the developed numerical methods to large-scale simulations of a PRSEUS aircraft component, such as a panel or fuselage section. Any technology or design proposed to be used in an aircraft must be extensively certified for a wide variety of scenarios. So far, PRSEUS has only been tested under quasi-static conditions. If PRSEUS is to be applied to commercial aircraft, then one potential validation of its performance as a fuselage is the section drop test. In the future, if full aircraft simulations [55] are desired, then the section drop test can act as an intermediary validation. This chapter applies the previously developed and validated methodology to first a panel compression test, and then to a reduced fuselage section drop test. While no validation is currently available for either of these scenarios, the analysis conducted here may provide valuable insight into unobserved damage mechanics and the design of future configurations.

6.1 Panel Analysis

6.1.1 PRSEUS Panel Characteristics Previous analyses of PRSEUS panels have included pressure testing [126,127], tension and compression loading across the stringers [128,129], and one frame loaded, compressive panel experiment [109]. These tests are extremely costly and require intense preparation, which is why they are conducted so infrequently. However, the designs for various PRSEUS components have continued to evolve and change over time. As an alternative to conducting a new experimental test each time the design is updated, high fidelity analysis can be used instead. In the previously mentioned compressive panel test, a numerical model was created, but was unable to accurately predict the onset of failure [109]. With the thoroughly validated methodology discussed in the previous chapters, compressive tests are simulated with the more recent stringer and frame architecture. Although no validation exists at this point, a panel cut from the MBB box was recently provided and is being prepared for testing.

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A picture of the provided panel is shown in Figure 6.1. The panel consists of three frames and twelve stringers and measures approximately 6 x 6 ft. Just a single type of configuration is used for both the frames and stringers, providing an opportunity for multiple tests. No obvious cracks or damage were visible on the panel, though the strain gauges from the MBB experiment were still attached in some places on the structure. No fully updated schematic for the provided panel section could be provided, so initial characterization of the configurations used were made using ruler measurements.

Figure 6.1 PRSEUS panel harvested from the MBB (left) [130]

The cross-sections of both the stringer and frame are shown in Figure 6.2. Based off of ruler measurements, the stringer configuration in the panel matches the Class 72 baseline stringer modeled in Section 5.1. While the frame closely resembles the foam-core frame analyzed in Section

5.2, the overwrap appears to have at least one extra buildup at the top of the frame web. Despite the apparent buildup on the frame web, the stitch locations appear to be the same as the previously investigated specimens.

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Figure 6.2 Cross-sectional views of the stringer (left) and frame (right) from the PRSEUS panel

Because the panel was previously used in the MBB experiment, it was unclear if it had sustained any damage or debonding. Therefore, non-destructive imaging (NDI) was used to locate any pre-existing delamination in the panel. To conduct the NDI, the panel was delivered to the

Marion Operations plant of General Dynamics, as they specialize in composites manufacturing.

The NDI was performed using the ultrasonic C-scan method [131], which uses a transducer that acts as both the emitter and receiver to generate an ultrasonic pulse that propagates through the laminate and returns an echo. If any defects are present in the material, such as debonding, then the pulse is scattered or reflected.

An example of the oscilloscope reading for an acceptably pristine panel and a panel with delamination is shown in the top of Figure 6.3. The signal in the pristine specimen does not encounter any significant defects that would cause the signal to reflect, unlike the debonded panel, which shows a clear reflection as a result of delamination. Additionally, because the ultrasonic C- scan visualizes the echo from the bottom of the laminate, it can indicate approximately where in the through-thickness the delamination has occurred. A disadvantage of this method however, is that it does not perform well around curvature or in regions that are very thick. Therefore, while the skin, the layering at the stringer-frame intersection, and the stringer web could be inspected, the overwrap around the stringer rod and the frame web could not be checked for pre-existing

100 | P a g e delamination. Due to the time consuming process of manual inspection, the panel was inspected with a resolution of 1 in2.

Figure 6.3 Example oscilloscope reading for a fully bonded sample (top left) and a debonded sample (top right) with delamination locations shown on the PRSEUS panel (bottom)

The bottom of Figure 6.3 shows some of the results from the NDI on the panel. Pre-existing delamination was found on one side of the panel, at the stringer-intersection locations of all three frames and five of the stringers. The delamination was not widespread, but was consistently located at the base of the frame web, approximately 40% of the way down the through-thickness. This suggests that the delamination occurred between the frame base and stringer base, which agrees with previous experimental observations and simulation results for initial damage development.

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Given the existence of debonding in many of the stringer-frame intersections for the panel, the panel was cut down to yield a smaller test specimen with pristine bonding. A schematic for the cut panel is shown in Figure 6.4. Due to the pre-existing delamination, the reduced panel contains just six stringers and two frames, with a total length and width of 36 in. and 30 in. respectively. Due to the relatively low aspect ratio of the panel, it is uncertain whether global buckling can be achieved, but the specimen can certainly exhibit local buckling and will still provide a reasonable large-scale validation of the numerical methodology developed in this work. The ends will be potted in the same manner as the single-frame specimens. The panel will be loaded in the frame direction and the outer edge will likely be unsupported. The primary reason for leaving the outer edge unsupported is that due to the relatively short length of the specimen, the load required to bring the specimen to ultimate failure is expected to exceed 700 kN (157 kips). Preparation for the experimental test is currently underway and future works will report both the experimental results and numerical comparison.

Figure 6.4 Schematic of the desired test panel based on the non-delaminated area

6.1.2 Computational Analysis of PRSEUS Panel Based on the reduced test panel that was described in Section 6.1.1, high fidelity models using both the foam-frame and tapered L-frame configurations were created. Although only the foam- frame model could be validated by experimentally testing previously discussed physical panel, the

102 | P a g e tapered L-frame is also simulated to determine if any changes in the damage progression is observed at the higher scale. Because no schematics are available for the test panel, the simulation of the foam-core frame panel was conducted using the same geometry described previously. Once the frames in the panel architecture can be accurately characterized, the numerical geometry will be adjusted to perform a validation.

The high fidelity models for both the foam-core frame and tapered L-frame panels are shown in Figure 6.5. The view and designation for each stringer and frame remains constant throughout this section. Each are developed by repeating the previously developed frame and stringer models using a 6 x 6 in. base for the frames and a 24 x 6 in. base for the stringers. Special care it taken to ensure that the stacks are correctly connected at the interface between each repeat and that the interlaminar bonding is maintained. The same mesh density from previous cases was used, so the foam-core frame panel requires 858,522 elements while the tapered L-frame panel only needs

592,800 elements. The only change from the models at the previous scales is that now the applied boundary condition is the simplified fully-fixed condition since there is no experimental baseline.

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Figure 6.5 High fidelity models of the foam-core frame panel (top) and tapered L-frame panel (bottom) with simulation setup and designations for each frame and stringer

The load-displacement results of the panel compression test for both the foam-core and tapered

L-frame are shown in Figure 6.6. The response of the panel is plotted with the response from the single substructure (which have been clipped at the expected ultimate failure point) for reference.

In both frame architectures, significantly greater change in stiffness is observed for both of the panels than for the component-level test. Although at different magnitudes, both panels are initially stiffer than their component counterparts. However, the initial stiffness is immediately observed to

104 | P a g e soften, becoming equal to the component stiffness at approximately 100 kN of load and continuing to drop. In both cases, the softening is due to the relatively early onset of local buckling of the skin in between all of the stringers for the area between the two frames. A brief, but significant loss in stiffness is observed in both panels at approximately 1mm of applied displacement that corresponds to the point at which all of the sections of skin between stringers (including at the symmetric boundary) have exhibited local buckling. However, because the drop is stiffness is relatively brief, it is likely that this is a dynamic effect in response to the loading rate. During a quasi-static test, it is expected that this drop in stiffness would not be observed, though the overall softening would still be present. The relative increase in peak stress between the panel and component tests is similar in both the foam-core frame and tapered L-frame, at 46.6% and 45.1% increase respectively.

Figure 6.6 Load vs. displacement for both the foam-core panel (left) and tapered L-frame panel (right) with comparison to component predictions and ultimate failure estimates

Beyond the temporary drop in stiffness at the 1 mm displacement, the load continues to increase and the out-of-plane deformation of the skin in between the stringers slowly increases in magnitude. Sequential contour plots of the resultant displacement and von-Mises stress contours for the foam-core frame panel are shown in Figure 6.7 and Figure 6.8 respectively. At a displacement of 2.763 mm, the skin on the outer side of both frames begins to locally buckle in between stringers S2 and S3. This bucking is apparent in the stress contour plot by the significantly

105 | P a g e lower stress at the buckled skin and coincides with a second drop in stiffness as shown in loading history.

Initially, the buckling in between stringers S2 and S3 instead of S3 and the symmetric boundary was believed to be a computational artifact, however, this is not the case. The reason that the outer skin buckling does not occur at the symmetric boundary, even though this has the greatest magnitude of deflection, is the direction of the initial skin buckling. At the symmetric boundary the skin buckles down toward the un-stiffened side, but buckles upward to the stiffener between stringers S2 and S3. The downward buckling of the skin at the symmetric boundary applies a moment to the frames to tilt them inward toward the frame panel center, while the skin buckling between S2 and S3 applies a moment to tilt the frames away from the panel center. The combination of moments causes the outer skin between S2 and S3 to be loaded in greater compression than the sections with a negative buckling displacement of the center skin. The direction of the center skin buckling can be determined from location of the keyhole concentrations on the inner surface of frame F-1.

The state of panel just before the inflection point at the peak load occurs at an applied compression of 5.038 mm. At this point, the skin at the unsupported edge between S1 and S2 has also buckled, though the unsupported edge skin at the symmetric boundary is still straight. The downward buckling of the skin between the stringers at the symmetric boundary is so significant at an applied displacement of 9 mm that the applied moment to tilt the frames inward delays the onset of buckling along the supported edge skin. The primary stress concentration at the frame has shifted back to being directly above the stringer keyholes, though asymmetric bands of stress are observed to extend toward the centerline between stringers S2 and S3. Slight indications of imminent buckling can be observed from the shape of the displacement contours.

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Figure 6.7 Sequential contours of resultant displacement for the foam-core frame panel with the symmetric boundary at the near surface

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Figure 6.8 Sequential contours of von-Mises stress for the foam-core frame panel with the symmetric boundary at the near surface

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Finally, the foam-frame panel buckles and is expected to completely fail at a displacement of

5.200 mm. As with the test at the component level, the onset of total failure is extremely rapid. The combination of opposing moments created by the upward skin buckling between S2 and S3 and the downward skin buckling at the symmetric boundary caused the frame to develop a slight kink, which immediately resulted in buckling and total failure. Due to these applied moments, the buckling of both frames is away from the panel center. As previously discussed in Section 5.2.3, the methodology developed in this work cannot fully propagate the final crack that extends through the entire specimen. However, the local element failure and stress development can allow reasonable predictions to be made.

Unlike the component-level failure, the final failure crack is not necessarily expected to pass through the stringer keyhole. Two primary failure paths are estimated to be likely to occur. In both cases, the failure would originate at the frame base, near the frame-stringer intersection with S2.

The crack would propagate near the frame base dropdown in between S2 and S3 before either extending straight up to the kink at the top of the frame or deviating toward the stringer keyhole and then up toward the kink. Between the two scenarios, the first case of the crack extending through the frame kink is considered to be more likely, considering the sudden onset of buckling extreme loads applied. In either case, the crack would also extend from the origin point back across the panel, curving toward the centerline between S2 and S3 and extending across the width of the panel. The failure path described is similar to the failure characteristics observed from the previous panel compression by Yovanof and Jegley [109], which also experienced local buckling long before the final fracture. One major difference between the current analysis and the experimental investigation of the previous frame loaded panel compression is the presence of the edge supports.

In the experiment, significant edge stresses were observed, which were not present in the numerical prediction due to the unsupported outer skin edge. The edge stresses contributed to the development of the final crack, which propagated from the edge toward the frame web.

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A key observation is that the panel is not predicted to fail between the symmetric boundary and S3 due to the direction of the initial center skin buckling. It is quite possible that if the center skin had initially buckled upward, the resulting applied moments would cause the panel to develop a kink in the frame near the panel center. However, even in such a scenario, the expected failure mechanisms and final crack would remain the same. Additionally, the permitted translation of the free edge on the outside of the frames contributed to the development of buckling. Future studies will also numerically investigate the response of a panel with an edge support, as with the experiment by Yovanof and Jegley, and compare the results to the current unsupported edge configuration. The results of this study many be used to guide the future testing of the panel described in Section 6.1.1.

The sequential plots of resultant displacement and von-Mises stress for the tapered L-frame panel are shown in Figure 6.9 and Figure 6.10 respectively. The first state illustrated for the L- frame panel occurs at a compression of 1.813 mm, at the onset of outer-skin buckling between stringers S2 and S3. The buckling of the outer-skin causes the stiffness of the panel to drop visibly due to the relatively low stiffness of the tapered L-frame members to compare with the foam-core frame. The overall pattern and displacement magnitude of center skin buckling is similar to the observations from the foam-core panel, with the center skin at the symmetric boundary buckling downward and the center skin between S2 and S3 buckling upward. At this point in the loading, the stress concentrations at the corners of the frame web dropdowns are visible but not yet significant in magnitude.

At the peak load for the tapered L-frame, the applied displacement is 2.925 mm. Due to the significantly lower overall strength of the tapered L-frame, the outer skin between S1 and S2 has not completely buckled. The primary difference between the state at the peak load and previous state is the stress magnitude on the panel. Now, stress concentrations are clearly visible at each of the stack dropdown locations within the frame web. Of particular interest is the asymmetry of the contours on the frame web between S2 and S3. The contours displayed on F-1 closely approximate

110 | P a g e the stress distribution observed at the component level whereas the contours on F-2 flare from the web dropdown at each corner toward the dropdowns at the frame base. The connection of these two critical regions suggests a possible fracture zone. Additionally, the lower stress at the top of the F-2 flange in between S2 and S3 and at the top of the F-1 flange in between the symmetric boundary and S3 is indicative of developing buckling.

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Figure 6.9 Sequential contours of resultant displacement for the tapered L-frame panel with the symmetric boundary at the near surface

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Figure 6.10 Sequential contours of von-Mises stress for the tapered L-frame panel with the symmetric boundary at the near surface

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Unlike the buckling of the foam-core panel, the frames do not exhibit buckling simultaneously.

The point at which each side of the frame buckles is different because of the asymmetry of the panel. Because the top flange is pointing away from the unsupported edge, F-1 buckles at the symmetric boundary first, causing the initial drop in load shown in Figure 6.6. The top flange of F-

2 between S2 and S3 quickly follows, resulting in the secondary load drop. The difference in buckling location along the top flange is due to the moments applied by the buckling of the center skin. The point of ultimate failure is estimated to occur at 3.163 mm based on the development of local element erosion in the frame base near the frame-stringer intersection and in the frame web dropdown.

Based on the stress development and buckling locations, the tapered L-frame panel also has two primary crack development paths at complete failure. In both cases, the crack is expected to have the same origin point and initial propagation. The crack could initiate at either the F-1 frame base dropdown or the corner of the F-1 frame web dropdown and would propagate up through the

F-1 frame web near the buckling point in the flange. In the other direction, the crack would propagate across the skin along the symmetric boundary (at the centerline between the two center stringers in a full specimen). From here the crack could either continue up through the frame web at the symmetric boundary, or travel laterally along the frame base before extending upward through the frame web dropdown and top flange between S2 and S3. Due to the higher stress at the secondary bucking point between S2 and S3, it is believed that the second scenario is more likely, despite requiring the crack to travel through the frame-stringer intersection.

No plots of delamination are shown throughout this analysis, even though they would undoubtedly provide greater insight into the failure mechanics in the frame. The reason no delamination plots are shown is during the analysis, the visualization of delamination produced an unusual checkered pattern that does not make any physical sense. An example of a correct and incorrect delamination contour is shown in Figure 6.11. While the plot on the left shows a clear delamination that coincides with the buckling of the outer skin, the second plot depicts a seemingly

114 | P a g e random pattern that has no relevance to the underlying physics or theory. Additionally, this problem was seen to occur even when running the exact same input file on the same system. Sometimes the delamination would be visualized correctly, but otherwise the unusual checkered pattern would appear. The odd visualization is not believed to have an impact on the load-displacement response of the structure, though a more in-depth analysis is required. Considering that the issue is not a weakness of the methodology, but instead one of the code, the author has contacted LSTC, the developer of LS-Dyna, in an attempt to resolve the issue for further works.

Figure 6.11 Correct delamination contours (left) and incorrect delamination contours (right)

6.2 Reduced Fuselage Analysis

In the final stage of this research, the modeling methodology developed herein is applied at a large scale to simulate the response of a reduced fuselage section drop test. Not only is this a real world scenario that can be used for certification [132], but also an opportunity to test investigate the dynamic behavior of PRSEUS for the first time. Additionally, such a test also allows the

PRSEUS concept to be compared against conventional aircraft design. The following sections discuss the development and crashworthiness of both a conventional metal alloy fuselage based on the B737 and a PRSEUS fuselage utilizing the tapered L-frame concept. Due to the extreme computational expense required, only the passenger floor and undercarriage are modeled for a single frame section. The work discussed in this section is an extension of a previous work to accurately model the section drop test performed by Jackson et al. [133] as further described in

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Appendix C. The same experiment was used to guide the input conditions of the current investigation.

6.2.1 Conventional Metal Alloy Fuselage Section Many different iterations of the frame and stringer geometry for metal fuselage design have been developed during the course of aviation history. Out of the many possible options, the current work uses the design of the B737 as a benchmark due to its wide implementation in commercial aviation and availability of publicly released research documents. Illustrations of the baseline structure for the conventional metal fuselage model are shown in Figure 6.12.

Figure 6.12 Illustration of the conventional metal frame and stringer substructure (approximated rivet locations are shown by the blue dots)

The stringer and frame geometry is based on the stringer and frame design described for the

B737 [90]. The thicknesses of the stringers and frames are defined from [134,135]. All of the components in the structure are either Alu 7075-T6 or Alu 2024-T3, which are both commonly implemented in aircraft structure. The material response of both of these materials are represented using the Johnson-Cook material model in conjunction with the Mie-Gruneisen equation of state as described in Section 3.3. The input parameters for each of these metals are shown in and are based on [88,91,136]. The skin and tear strap are Alu 2024-T3 while the frame, stringer, and frame- stringer connection are defined to be Alu 7075-T6. Since the tear strap is in the B737 is often referred to as a bonded waffle doubler [137], it was assumed to have the same thickness as the skin

116 | P a g e and the bonding was approximated by merging the interface nodes together since no bonding properties were available.

Table 6.1 Input parameters for Johnson-Cook material of Alu 7075-T6 [136] and Alu 2024-T3 [91] with damage properties A (MPa) B (MPa) n C m 517 405 0.41 0.0075 1.1 Alu 7075-T6 D1 D2 D3 D4 D5 0.015 0.24 -1.5 -0.039 8.0 A (MPa) B (MPa) n C m 369 684 0.73 0.0083 1.7 Alu 2024-T3 D1 D2 D3 D4 D5 0.112 0.123 1.5 0.007 0.0

Unlike PRSEUS, which is fundamentally integrated, the majority of the components for a metal fuselage must be attached by rivets. The spacing of the rivets in the conventional metal fuselage model is approximately 1 in. [137,138], resulting in a total of 23 rivets per unit substructure shown in Figure 6.12. Directly modeling each individual rivet would be excessively computationally expensive, so as an alternative, the rivet connection between components is approximated by merging the nodes between components at only the rivet locations. The size of the merged area is based on the 0.159 in. diameter hole drilled for each rivet [138]. An approximate mass for each rivet was also applied at the rivet locations, assuming the rivets are made of 304 steel and have the same volume as a 0.25 in. x 0.159 in. diameter circular pin. Finally, dynamic and static coefficients of 0.15 and 0.3 respectively are applied to account for rubbing between components during contact [139]. The same global element size used to create the mesh for the previously discussed PRSEUS structures was also used for the metal fuselage.

6.2.2 Reduced Fuselage Section Setup The full model for both the conventional metal fuselage and the PRSEUS tapered L-frame fuselage is shown in Figure 6.13. The total diameter of 3.76 m matches the outer diameter of the

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B737, though the curvature of the model is simplified to a half circle. The majority of each model was created by repeating the substructure multiple times to create a half-circle with stringer located at the centerline. The metal fuselage has a stringer spacing of 9.5 in. [90] while the PRSEUS fuselage has a stringer spacing of 6 in. based on the existing panel designs. In total, the metal fuselage model is created from repeating twenty-three of the previously shown substructures while the PRSEUS structure requires thirty-seven. The tapered L-frame substructure for the PRSEUS fuselage uses the same construction and mesh density as discussed in Section 5.2.2, but now with an applied curvature.

In the absence of any detailed information, the same simple connection was created for both models to attach the passenger floor. In both cases, the passenger floor was defined as a shell with its nodes directly merged to the top surface of the frames. The nodes at the overlapping section of the passenger floor were manually shifted to match the mesh of the frame surface underneath, effectively simulating a perfect bond at the interface of the passenger floor and frame surface.

Additionally, a vertical stiffener was created underneath each floor to provide support. The support was created in each model by removing the top flange and extending a 0.126 m (5 in.) section of the frame web along the bottom of the passenger floor, again ensuring to align the mesh to fully attach the stiffener. The passenger floor is assumed to be a simplified, linear elastic-plastic form of

Alu 2024-T3 to reduce the computational expense. Additionally, because the passenger floor is not near the impact zone, it is not expected to experience significant variations in strain rate. The vertical stiffener matches the same material as the frame, so therefore is composed of Alu 7075-T6 in the metal fuselage and the Class 72 stack material in the PRSEUS fuselage.

In addition to the conventional metal and PRSEUS fuselage models, an un-stitched PRSEUS fuselage was created. The only difference between it and the PRSEUS fuselage is the presence of stitching, all other geometry, input parameters, and contact algorithms remained constant between the two. The un-stitched model is not truly representative of a “typical” composite due to the unique architecture of PRSEUS and the material properties of the stack buildup. However, it does allow

118 | P a g e some insight into the damage mechanisms present in an unstitched composite. Future studies will also investigate current composite fuselage structures and compare them to performance of the metal and PRSEUS models discussed herein.

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Figure 6.13 Illustration of the entire reduced section model with the conventional metal design (top) and PRSEUS tapered L-frame (bottom) with a close-up of a single substructure

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In order to simulate the response of an interior frame member of the fuselage, symmetric boundary conditions are applied on either side of the reduced section as shown in Figure 6.14. This boundary condition includes the edges of the passenger floor, frames, and stringers, but does not include the vertical support at the center of the passenger floor. To minimize the number of calculation steps, the structure was moved close to the impact surface and an initial velocity was prescribed. The initial velocity of the entire structure is 9.1 m/s with gravity applied over the entire domain. The velocity was calculated based on the expected velocity of the experimental fuselage in [133] just before impact. A frictionless, rigid surface is used as the impact surface and is modeled with shells. The simulation is not intended to represent the actual response of a fuselage during a section drop test, but instead serves as a relevant test case to compare the response of conventional metal designs and PRSEUS for a dynamic scenario.

Figure 6.14 Schematic of boundary condition and setup for the reduced section drop test

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6.2.3 Comparison of the PRSEUS and Metal Fuselage Due to the significantly different designs and materials used for the two fuselage structures, the overall mass of each fuselage is different. A breakdown of the mass distribution for each fuselage is summarized by Table 6.2. The PRSEUS fuselage has a 36.5% greater total mass than the metal fuselage, which was unexpected considering the approximately 36% lower density of the

Class 72 stack material compared to aerospace grade aluminum. However, the primary difference for the magnitude of the disparity is due to the far greater number of stringers present in the

PRSEUS fuselage (37 vs. 23). Even at the substructure level though, the 6 in. PRSEUS substructure has a 12.5% greater mass than the 9.5 in. metal substructure.

While the models investigated in the current analysis clearly show that the PRSEUS structure is more massive, it is important to keep in mind that the tapered L-frame PRSEUS design is extremely recent and relatively untested, while the B737 baseline is an optimized structure that has been proven in the commercial market. Additionally, while the architecture of the metal fuselage is based on existing documents on the public domain, the exact component dimensions and rivets locations are not known. Future iterations of PRSEUS that have been designed specifically for tube- wing aircraft will certainly investigate potential weight savings and optimize the structure.

Table 6.2 Mass distribution for the metal and PRSEUS fuselage models

Fuselage Component Metal (kg) PRSEUS (kg) Total Mass 16.71 22.81 Passenger Floor 7.97 7.97 Floor Stiffener 1.01 0.84 Floor Attachment 0.38 0.44 Entire Frame* 7.35 13.2 Single Substructure** 0.32 0.36 *Not including the passenger floor, floor stiffener, or floor attachment **9.5 in. section for the metal fuselage and 6 in. section for the PRSEUS fuselage

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The normalized energy vs. time distribution during the initial impact and rebound for both the metal and PRSEUS fuselage is shown in Figure 6.15. The energy is normalized by the initial total energy and is used because the reduced fuselage drop simulation is only meant to serve as a means of comparing metal and PRSEUS instead of being representative of actual test conditions.

Additionally, the deviation in total mass of the metal and PRSEUS models requires normalization to make an adequate comparison.

Figure 6.15 Energy distribution for both the metal and PRSEUS fuselage during the initial impact and bounce

The metal fuselage was not observed to exhibit significant material failure, instead dissipating the majority of the initial kinetic energy through elastic and plastic deformation of the fuselage.

Alternatively, the PRSEUS fuselage dissipates approximately 10% of the incident energy through element erosion (EE) by the time the fuselage has stopped moving downward. Additionally, the rate of energy dissipation is significantly greater for the PRSEUS fuselage than it is for the metal.

The primary reasons for this greater rate of energy dissipation, besides the material failure, is the higher wave speed through the composite material. Other works in the literature [140–142] have

123 | P a g e observed that a greater wave speed allows more material to become part of the initial impact, allowing more energy to be dissipated.

The deceleration of each fuselage structure is qualitatively illustrated with contours of y- velocity in Figure 6.16. Soon after impact at 5.0 ms, the region of fuselage at 0 m/s is larger for the metal fuselage than it is for the PRSEUS structure. A contributor to the difference in contour level is the formation of a crack above the center stringer of the PRSEUS fuselage, which reduces the stiffness of the structure near the location of the crack. The formation of the crack coincides with the sudden increase in energy dissipation through element erosion as shown in Figure 6.15. The development of the crack is more thoroughly discussed later in the current section as well as in

Section 6.2.4.

As each fuselage continues to decelerate, a similar trend to the contours at 5.0 ms can be observed at 15.0 ms. Once again, the metal fuselage has a larger region of the fuselage that is at approximately 0 m/s, with a linear transition of y-velocity contours across the vertical stiffener for the passenger floor. The majority of the PRSEUS fuselage is still descending and the contours of y-velocity do not transition linearly. On the passenger floor stiffener in particular, the bands of varying velocity regions indicate the presence of greater oscillation, likely as a result of the higher stiffness of the composite material.

Finally, at 25.0 ms, both fuselage structures are almost entirely at a y-velocity of 0 m/s or greater. Although some fluctuation in the y-velocity is still visible along the passenger floor stiffener for the PRSEUS fuselage, the contours of both models are highly similar. The greater change in contour levels between 15.0 ms and 25.0 ms suggests that the PRSEUS fuselage slows faster than the metal fuselage in the later stages of impact. Once again, this could be due to the formation of the center crack, which reduced the structural stiffness in the early stages of impact, but was not observed to grow significantly as the PRSEUS fuselage continued to descend.

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Figure 6.16 Progression of y-velocity contours for both the metal (left) and PRSEUS (right) fuselage models. The view is cut just beyond the centerline to illustrate the deformation at the centerline

The acceleration transmitted to the passengers is often a driving factor in the certification of commercial aircraft [143]. As long as the fuselage does not catastrophically fail, a lower acceleration and shorter duration of acceleration is preferred. The normalized acceleration for both the metal and PRSEUS fuselage are shown in Figure 6.17. The normalization is based on the peak

125 | P a g e acceleration experienced among the two structures. In both cases, the acceleration was measured from the center of the passenger floor, downsampled, and filtered with a Butterworth filter at a cutoff frequency of 182 Hz before being plotted in Figure 6.17. In both cases, a general trend of positive acceleration (in opposition to the initial downward velocity of the fuselage) until approximately 30 ms after the initial impact is visible. Beyond this point, the acceleration becomes negative as the fuselage begins to bounce off of the impact surface. Most importantly, the peak acceleration of the PRSEUS fuselage is only 60% of the peak acceleration of the metal fuselage.

The lower peak loading is even more promising considering the greater mass of the PRSEUS fuselage. The primary reason for the significantly lower peak acceleration of the PRSEUS fuselage is the combination of high wave speed due to the composite material and the development of local material failure during the impact, dissipating more energy overall than the metal fuselage.

Figure 6.17 Normalized vertical acceleration response for the metal and PRSEUS fuselage section drop analysis during the initial impact and bounce

The primary feature that distinguished the response of the metal and PRSEUS fuselage was the development of the crack above the center stringer. The crack facilitated greater energy dissipation in early stages of impact and allowed the PRSEUS fuselage to have an overall lower acceleration than the metal baseline. Figure 6.18 illustrates the development of the crack with

126 | P a g e sequential contours of von-Mises stress and compares it to the stress distribution around the center stringer of the metal fuselage.

Figure 6.18 Sequential contours of von-Mises stress for both the metal (left) and PRSEUS (right) fuselage models at the bottom center of the frame

At only 0.25 ms after impact, the crack in the PRSEUS fuselage has not yet formed, though a clear stress concentration is visible at the top of the stringer keyhole. Stress concentrations are also present at the corner of the stack dropdown in the frame web, but are not as large in magnitude as the stress concentration at the keyhole. A triangular region of low stress is visible as a result of the stitch lines near the top of the frame web. As expected, the stringer itself does not contribute significantly to the loads experienced by the structure. The metal fuselage also has several stress

127 | P a g e concentrations visible at the locations of the rivets. As with the PRSEUS fuselage, the stringer for the metal structure does not carry much of the transverse tensile load. The stress concentration at the connection between the stringer and frame is due to the vertical deflection in the flange.

By 0.50 ms after impact, the crack has been initiated in the PRSEUS fuselage, causing the surround stress distribution to change significantly. After propagating nearly instantaneously through the stack dropdown of the frame web, the crack is arrested by the diagonal stitch lines. The triangular region of relatively low stress above the diagonal stitch lines is still present, though increase in tensile load and proximity of the crack tip have increased the stress magnitude. While the area above the stringer keyhole has generally increased in stress, the formation of the crack has caused the rest of the frame to lose its load carrying capacity. Stress concentrations are no longer present at the corners of the frame dropdown and the immediate area around the stringer keyhole has dropped to zero stress. Alternatively, the metal fuselage does not exhibit any signs of crack formation and has generally increased in stress magnitude. The stress concentrations that were previously located at the connection between the stringer and frame are no longer present because the stringer flange has largely returned to its original position.

Finally, after being arrested for another 1.00 ms, the crack in the PRSEUS fuselage extends through the stitch line at 1.50 ms after impact and passes through the entire frame web. The outer stacks of the web are present throughout the entire frame web, which caused the crack to continue to propagate straight upward from the keyhole. Alternatively, the inner stack that is introduced at the top of the web to reinforce the top flange does not have a pre-existing crack from the initial impact. Instead the crack in the inner stack is observed to develop above the minimum gauge region between the center and right adjacent stringer. The reduced cross-sectional area in this zone results in a higher stress concentration that causes a crack to develop and propagate through the inner stack, severing all connection of the frame above the center keyhole. The loss of structural integrity in this region drops the local load capacity close to zero, reducing the stiffness of the overall structure

128 | P a g e as previously described for Figure 6.16. As expected, the higher ductility of the metal allows it deform without cracking, with the greatest tensile load carried by the top of the frame web.

While no obvious signs of failure are observed for the metal fuselage, damage is still present.

Unlike most composite structures, metals can be subjected to significant plastic deformation before reaching ultimate failure. Contours of plastic deformation at a time of 0.04 s are shown in Figure

6.19 at the areas subjected to the greatest stress. The greatest plastic strain is present near the bottom center of the fuselage, at the cutout in the frame for the stringer pass-through. At this point the frame has visibly crimped, with some element erosion in the areas of greatest plastic strain. Some plastic strain is also visible at the rivet locations connecting the stringer to the frame and the stringer to the skin. However, these are a result of the oscillations caused by the initial impact instead of developing throughout the descent of the fuselage. At approximately 45˚ from the center stringer, several more areas of plastic strain can be observed. Once again, the plastic strain is primarily located near the cutout for the stringer, which has the lowest cross-sectional area. As the fuselage descended, the frame buckled slightly at the top flange and stringer-frame connection, which is indicated by the areas of greatest plastic strain.

Figure 6.19 Contours of plastic strain for the metal fuselage at the areas subjected to the greatest stress at t = 0.04 s

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6.2.4 Comparison of the Stitched and Un-Stitched PRSEUS Fuselage The metal fuselage serves as a reasonable baseline to compare the response of the PRSEUS fuselage to a proven aircraft structure, but the comparison of the un-stitched and stitched PRSEUS structure allows greater insight into the damage arrestment capability introduced by the stitching.

While the un-stitched PRSEUS fuselage is not intended to be representative of a state-of-the-art composite fuselage, it does exhibit many of the same damage mechanics expected in a more typical composite architecture. The current section explores the differences in delamination progression between the un-stitched and stitched response for the three primary areas of observed damage: the center stringer, approximately 45˚ from the center stringer, and the passenger floor attachment.

Figure 6.20 Contours of delamination for the un-stitched (left) and stitched (right) models of the PRSEUS fuselage at the bottom center of the frame

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Delamination contours for the un-stitched and stitched PRSEUS fuselage are depicted in

Figure 6.20. In both models, the crack above the center stringer keyhole is initiated almost immediately after impact and extends just past the stack dropdown in the frame web buildup at only 0.5 ms after impact. The area of delamination around the crack is greater for the un-stitched case and also extends along both interlaminar interfaces present at the frame web buildup.

Alternatively, the stitched case localizes the delamination to the crack head and immediate area around the stack dropdown. In both models, some debonding occurs at the minimum gauge region of the frame web.

For both the un-stitched and stitched models, the crack above the stringer keyhole grows only slightly until suddenly, at a time of 1.25 ms, the crack in the un-stitched PRSEUS fuselage extends through the rest of the of frame web. Delamination immediately extends from the crack surface, completely debonding the single stack buildup above the stringer keyhole and spreading into the top flange. Alternatively, the stitched PRSEUS fuselage is still able to arrest the crack from fully extending to the top of the frame web. Despite withstanding further propagation of the crack, the stitching is unable to arrest delamination from spreading to the single stack buildup above the stringer keyhole.

As the fuselage continues to descend, the stitching becomes unable to prevent the frame from cracking through the top of the web and top flange. By 1.50 ms after the initial impact, the delamination contours of both the un-stitched and stitched PRSEUS fuselage models appear quite similar. In each case, delamination has completely spread throughout every ply interface in the web and top flange. Although not clear in the viewing area shown in Figure 6.20, the stitching near the top flange of the frame has prevented delamination from spreading quite as far as observed in the un-stitched model. Throughout the remainder of the fuselage descent, the delamination surrounding the center stringer maintains the same shape, with relatively little increase in delamination area.

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In addition to the region surrounding the center stringer, another primary source of damage development is located in the frame, approximately 45˚ from the center stringer. As the fuselage continues to decelerate in its descent, the bottom of the fuselage comes to a stop or even begins to bounce off of the impact surface while the top of the fuselage continues to descend. The relative difference in motion applies a significant moment to the frame in-between the center stringer and passenger floor attachment. Because the loading condition is a result of bending in the fuselage, no damage is observed to propagate until after the initial contact has occurred. The delamination contours of the fuselage frame for the both the un-stitched and stitched PRSEUS fuselage models is shown in Figure 6.21.

Figure 6.21 Contours of delamination for the un-stitched (top) and stitched (bottom) PRSEUS structures 30-50˚ from the bottom center of the frame

Surprisingly, the stitched PRSEUS fuselage exhibits sooner delamination development and propagation than observed for the un-stitched case. Initially, such a result appears to be completely at odds with all of the previous observations where stitching was clearly shown to delay the onset of delamination. The reason for this non-intuitive response is a result of the large-scale dynamics of the fuselage model and the constraints imposed by the stitching. Unlike previous cases, the length

132 | P a g e of the frame in the fuselage section and applied loading causes significantly more sliding and relative motion between laminate as the frame bends. In the case of the stitched PRSEUS fuselage, the presence of the stitching restricts the deformation of the material at the stitch lines, causing greater local material failure. At 2.25 ms, the stitched PRSEUS fuselage develops several areas of delamination as a result of local material failure. As the frame continues to bend, more delamination is initiated from local element erosion. At 3.00 ms, the un-stitched PRSEUS fuselage model also exhibits delamination, though the total delamination area in the region is significantly less during the early stages of the impact. Although the onset and development of delamination is faster in the stitched PRSEUS fuselage, both the un-stitched and stitched PRSEUS fuselage have the same overall delamination area in the discussed frame region by the time that the passenger floor has stopped descending.

The final location of damage development, located at the connection between the passenger floor and fuselage frame, is shown for both the un-stitched and stitched PRSEUS fuselage structures at multiple states. As with the previously discussed frame section approximately 45˚ from the center stringer, the damage near the passenger floor attachment is the result of bending. After the relatively rigid fuselage frame has stopped moving downward, the passenger floor continues to drop, despite the presence of the vertical stiffener below the floor. The relative bending causes a stress concentration to form at the transition from the top frame flange to the vertical stiffener as shown in Figure 6.22. At this point, a significant difference in delamination development is already visible between the un-stitched and stitched PRSEUS fuselage. While the vertical stiffener has already debonded from the adjacent frame stacks in the un-stitched model, the stitched model is still able to arrest the delamination from propagating through the entire web. As a result of the debonding of the vertical stiffener from the other frame stacks, the stress magnitude in the un-stitched model is greater overall in the area below the attachment to the vertical stiffener than for the stitched case.

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Figure 6.22 Contours for von-Mises stress (top) and delamination (bottom) at the passenger floor attachment, t = 16.75 ms

At a time of 25.00 ms, the previously discussed stress concentration in the un-stitched model has developed into a crack that extends straight down from the edge of the vertical stiffener. Unlike the crack above the center stringer that formed nearly instantaneously, the crack at the passenger floor attachment propagates relatively slowly. In this region of the fuselage, it is important to consider both the delamination and crack propagation in the laminate independently, since there is only a single stack in the middle of the frame web. The interlaminar bonding from the epoxy has now failed in the stitched PRSEUS fuselage at the top of the frame web, but the presence of the stitching prevents the laminates from completely delaminating. As a result, no crack formation is observed at the corner between the base of the vertical stiffener and the top of the frame web.

Instead, some local element deletion is observed near the top of the vertical stiffener and a crack has developed just under the passenger floor. The slight material failure is located directly on the stitch line, suggesting that the stitching may have over-constrained the deformation in this region, whereas the crack under the passenger floor is due to the stiffness mismatch between the composite stiffener and metal passenger floor.

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Figure 6.23 Contours for von-Mises stress (top) and delamination (bottom) at the passenger floor attachment, t = 25.00 ms

The final state at 30.00 ms approximately corresponds to the point at which the entire fuselage has a velocity equal to or greater than zero. The un-stitched passenger floor attachment has almost completely separated from the rest of the structure, with two cracks now extending near the base of the frame. The crack that originated at the corner between the base of the vertical stiffener and the top of frame web has now extended to the same level as the pultruded rod in the stringer keyhole.

Additionally, a second crack that developed just under the passenger floor as a result of the stiffness disparity between metal and composite has rapidly extended into the base of the frame web. The uncontrolled propagation of the second crack initiates delamination in the base of the frame web, which swiftly extends across the frame towards the stringer keyhole. The stitched PRSEUS fuselage, however, is still able to maintain its structural integrity in this region due to the additional damage tolerance afforded by the stitching. Despite the development of a significant stress concentration around the previously discussed local element erosion, no crack ever propagates through the stitch line at the top flange. As a result, no delamination can form in the base of the

135 | P a g e frame web, allowing to passenger floor attachment to support the loads throughout the remaining descent and subsequent bounce of the fuselage section.

Figure 6.24 Contours for von-Mises stress (top) and delamination (bottom) at the passenger floor attachment, t = 30.00 ms

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7 Conclusions

Due to the demands of commercial transportation, the aviation industry has taken a leading role in the integration of composite structures. Among the leading concepts to develop lighter, more fuel-efficient commercial transport is the Pultruded Rod Stitched Efficient Unitized Structure

(PRSEUS) concept. The highly integrated structure of PRSEUS allows pressurized, non-circular fuselage designs to be implemented, enabling the feasibility of Hybrid Wing Body (HWB) aircraft.

The manufacturing process for PRSEUS allows pre-made stacks of uni-directional CFRP to be assembled into the desired geometry, held together with through-thickness stitching, and co-cured as a single structure using the state-of-the-art Controlled Atmosphere Pressure Resin Infusion

(CAPRI) method. In addition to assisting in the fabrication process, the through-thickness stitching utilized by PRSEUS overcomes the low post-damage strength present in typical composites.

Although many proof-of-concept tests have been performed that demonstrate the potential for

PRSEUS, efficient computational tools must be developed before the concept can be commercially certified and implemented.

The work herein addresses this need, developing a comprehensive modeling approach that investigates PRSEUS at multiple scales. The majority of available experiments for comparison have been conducted at the coupon level. Therefore, a computational methodology was progressively developed based on physically realistic concepts, without the use of any tuning parameters. A thorough element-study was performed to identify that the selectively reduced integration solid and beam elements consistently provided the most accurate results for applicable loading scenarios. Using the results of this baseline study, a high fidelity stringer model was created at the coupon scale and thoroughly developed with a series of investigations to determine the necessary approach to consistently achieve accurate results. The final methodology was then successfully validated against experimental baselines at both the frame and stringer coupon level for a multiple PRSEUS architectures and levels of complexity. Finally, the validated model was

137 | P a g e extended to the component and large-scale further investigate PRSEUS and compare it to the current state-of-the-art.

Throughout the course of this research, the developed methodology was demonstrated to make accurate predictions that are well beyond the capability of existing predictive models. For the first time, the methodology developed herein can accurately predict local behavior up to and beyond failure for stitched structures such as PRSEUS while still using commercially available analysis tools. Additionally, by extending the methodology to the possible fuselage section drop certification scenario, the dynamic behavior of PRSEUS was investigated for the first time. The insight provided by the investigation highlighted the strengths of PRSEUS as well as identified some behavior that had not previously been considered. With the predictive capabilities and unique insight provided, the work herein may serve to benefit future iteration of PRSEUS as well as certification by analysis efforts for future airframe development.

7.1 Future Work and Recommendations

A new modeling methodology to accurately predict the response of stitched composites was developed and validated in the course of this research at the stringer and frame coupon level.

However, validation at the component level is both feasible and valuable to ensure that the developed methodology is capable of providing accurate predictions at higher scales. One immediate area of future work will focus on the experimental testing of the panel described in

Section 6.1.1. If the results of the compression test can be closely matched by the numerical model, then the modeling methodology can be applied to other scenarios with greater confidence. In addition to the validation at the higher scale, performing validation for other types of loading would also be beneficial. Since experimental tests with multiple loading conditions have been performed throughout the development of PRSEUS, the developed methodology could be applied retroactively for validation. However, considering that the constituent materials and assembly geometries have changed significantly over time, many of the preliminary validations presented herein may have to be repeated.

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In addition to other validations, the developed methodology can be applied to investigate other scenarios to provide guidelines for future iterations of PRSEUS. In the current work, the large-scale dynamics of PRSEUS were investigated for the first time, and were conducted for a scenario that is applicable to future certification. However, there are many other certification scenarios that have never been investigated such as bird strike or maximum wing deflection. Additionally, while the

PRSEUS fuselage was compared to a metal baseline, a comparison should also be made against the current state-of-the-art in the composite fuselage design. From these analyses, PRSEUS can be improved and optimized, perhaps eventually being implemented into commercial tube-wing fuselage structures.

In order to effectively conduct an uncertainty analysis for PRSEUS, an in-depth perturbation analysis must be performed. As described in Section 5.1.7, the combination of a theoretically perfect structure and compressive loading causes the predicted response to be somewhat unstable.

By perturbing the structure to more effectively imitate the imperfections present in a physical specimen, an uncertainty analysis can be conducted. Such an uncertainty analysis would be valuable, because it could define a range of expected response for some variation in parameters such as geometric or bonding imperfections, allowing a minimum safety threshold to be determined.

Several improvements in the current capability for commercial finite element tools such as LS-

Dyna would greatly assist future predictions for PRSEUS and bonded structures in general. A major roadblock in efficiently simulating many of the scenarios discussed in the current research was the inability to use implicit-explicit time integration switching in conjunction with interlaminar bonding. Such a capability would drastically reduce the computation time required for many of the validation studies. Instead of running the entire simulation with explicit time integration, the initial loading up until the onset of failure could be simulated using implicit time integration in before switching to explicit analysis during buckling. Another application of the capability would be to predict the steady-state response of large-scale structures following initial impact, such as the section drop test.

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Appendix A: Stringer Mesh Study Before conducting the stringer analysis discussed in Section 5.1.5, a mesh refinement study was conducted to determine an appropriate mesh for further analysis. Considering the focus of the analysis, the load-displacement response was used as the fitness criteria, though the considerations described in Appendix B were also taken into account. Three different mesh densities for the stringer were analyzed, which were adjusted based on a global element size constraint. The fully solid element model was used for the analysis and only the solid elements were refined while keeping the stitch locations between models constant. Due to the inherent uncertainty caused by the discretization error in buckling analyses, as described in Section 5.1.7, each model was simulated five times for the same input conditions and then the average response was calculated.

Only five trials could be effectively completed for each because the most refined model required

6500 seconds per run. A future study may consider increasing the number of trials or using the reduced expense beam-solid model for the mesh refinement study.

The Grid Convergence Index (GCI) procedure outlined in [117,118] was used to determine the numerical error associated with the current mesh refinement when compared an extrapolated “exact” result. It is important to distinguish this error from the uncertainty inherent to the buckling problem.

Since only the averaged response is used for the GCI analysis, the uncertainty caused by discretization error when predicting buckling is theoretically removed. For three meshes of sufficiently different refinement, the GCI procedure uses the relative mesh refinement (r) and simulation solution (휙), to quantify the apparent order of accuracy (푝). When the refinement factor is variable, the order of accuracy must be solved iteratively as shown in equations A-1 to A-3 [118].

1 푝 = |ln|휀32/휀21| + 푞(푝)| A-1 ln(푟21) 푝 푟21 − 푠 푞(푝) = ln ( 푝 ) A-2 푟32 − 푠

푠 = 1 ∙ sgn(휀32/휀21) A-3

140 | P a g e where 휀푖푗 = 휙푖 − 휙푗 and 휙 is the solution for a given level of grid refinement. The highest grid refinement has subscript 1 while the coarsest grid has subscript 3. In the current analysis, these parameters are locally calculated at each point in the response. Once the observed order of accuracy

푖푗 is determined, an extrapolated solution 휙푒푥푡 is calculated using:

푝 푖푗 푟푖푗 휙푗 − 휙푖 휙푒푥푡 = 푝 A-4 푟푖푗 − 1

Finally, three estimates of error may be calculated for the relative error between each set of grid refinements. As shown in equations A-5 to A-7, these error estimates are approximate relative error, extrapolated relative error, and the GCI.

푖푗 휙푗 − 휙푖 푒푎 = | | A-5 휙푗

휙푖푗 − 휙 푖푗 푒푥푡 푗 A-6 푒ext = | 푖푗 | 휙푒푥푡

푖푗 푖푗 1.25 ∙ 푒푎 GCI = 푝 A-7 푟푖푗 − 1

Using the procedure outline above, the results shown in Table A.1 were calculated. A refinement factor greater than 1.3 is recommended by [118] based on past experience, but could not quite be achieved for the stringer model without exceeding an aspect ratio of 3:1 in the stringer rod at the lowest mesh density, or increasing the number of stack through-thickness elements at the highest mesh density.

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Table A.1 Summary of results for the GCI analysis

Property Value

푁1, 푁2, 푁3 75,272; 35,910; 16,560

푟21, 푟32 1.280, 1.294 푝̅ 5.740 ̅̅21̅̅ ̅̅32̅̅ 푒푎 , 푒푎 0.025, 0.019 ̅̅21̅̅̅ ̅̅32̅̅̅ 푒푒푥푡, 푒푒푥푡 0.013, 0.006 ̅̅̅̅̅̅21̅̅̅ ̅̅̅̅̅̅32̅̅̅ 퐺퐶퐼푓푖푛푒, 퐺퐶퐼푓푖푛푒 0.010, 0.007

Using the locally calculated value for the observed order of accuracy results in artificially large error near points where 휀32 = 휀21, so the values reported in Table A.1 are calculated using the globally averaged observed order of accuracy as recommended in [118]. The error estimates are then averaged over the entire domain to calculate the global error. The observed order of accuracy is much greater than expected considering the simulations are performed using second-order elements. The discrepancy could possibly be a result of the 42.3% oscillatory convergence observed, which has been reported to cause discrepancies in standard GCI methods [144–146]. Additionally, before buckling occurs, the loading history of all three mesh densities are extremely similar as shown on the left plot in Figure A.1. At such small error values, the prediction of the order of accuracy can be erroneous [118]. Each of the estimated error values indicate that while the average error between grids is low, the relative error between the medium and fine grid is greater than the error between the coarse and medium grid, which would suggest that the solution either is not converging, or that all of the mesh resolutions used are already in the asymptotic range. The relatively low number of trials to average out the uncertainty due to the prediction of buckling is also certain to have an effect on the error estimations.

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Figure A.1 Load-displacement plots at three mesh densities (left) and the response of the medium mesh density with error bars (right)

The response predicted by each mesh density and the response of the medium density mesh with error bars calculated from GCI are shown in Figure A.1. As previously mentioned, the response by each mesh pre-buckling is highly similar, but starts to diverge at the peak load of the buckling point. While all mesh densities follow the same trend of post-buckling response, the magnitude varies significantly. Generally, the slope of the drop in load following buckling and the magnitude of the load drop is greater at a higher level of refinement. However, the sensitivity of the post-buckling response to the discretization error as shown in Section 5.1.7 makes distinguishing the buckling uncertainty error from the mesh refinement error difficult. As expected, the errors bars predicted by GCI for the medium mesh density case are the greatest in the post- buckled regime.

Another important factor in the selection of a grid refinement level is the computational time required to produce a solution. As previously mentioned, the highest mesh density required 6,500 seconds of wall time on a system using 12 CPU. Alternatively, the medium and coarse mesh refinements require only 2,800 and 1,000 seconds of wall time respectively. Considering the low error at the peak load and similarity of post-buckling trends, the medium mesh density was selected for continued analysis. The mesh refinement study also suggests that the compression loading

143 | P a g e validations are unlikely to be significantly affected by the mesh density, but simulations that extend significantly beyond the buckling point may be much more sensitive to the grid resolution.

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Appendix B: Delamination Validation

The purpose of this section is to develop and validate the baseline mesoscale modeling approach to include delamination and discuss special considerations that must be taken into account when modeling delamination. The investigation was based on the high speed impact of projectiles onto nineteen plies of Kevlar® 29. In the preliminary investigation, all three of the composite damage theories discussed in Section 3.1 (Tsai-Wu, Chang-Chang, and the 3-D orthotropic damage model) were implemented to model the damage. In addition to the composite damage theory, the element type and mesh resolution were also considered to investigate their relative effects on the solution quality. The observations and understanding gained during the analysis were critical to developing the final hybrid methodology used in the modeling of PRSEUS.

The computational investigation is based on the parametric experimental analysis conducted by van Hoof and Gower et al [15, 17]. Both studies used a 2x2 Kevlar 29 basket weave, 1500 denier,

82% fiber volume fraction fabric that was laminated using polyvinyl butyral (PVB)-phenolic. In these analyses, both Gower and van Hoof conducted parametric studies incorporating different projectile shapes, multilayer Kevlar 29 fabrics, and various projectile speeds. A summary of the parameters investigated by each of these works is given in Table B.1.

Table B.1 Continuum parameters based on literature [82,104,147]

Property Kevlar® 29 𝝆 (풌품⁄풎ퟑ) 1230

푬ퟏ (푮푷풂) 18.5

푬ퟐ (푮푷풂) 18.5

푬ퟑ (푮푷풂) 6.0

푮ퟏퟐ (푮푷풂) 0.77

푮ퟐퟑ (푮푷풂) 2.71*

푮ퟏퟑ (푮푷풂) 2.71*

흂ퟏퟐ (푵/푨) 0.25

흂ퟐퟑ (푵/푨) 0.33

흂ퟏퟑ (푵/푨) 0.33

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𝝈풏 (푴푷풂) 34.5

𝝈풔 (푴푷풂) 9.0 * Estimation by van Hoof

In this analysis we are focusing on modeling the 146 m/s impact of a 9mm diameter hemispherical projectile onto 19 plies of Kevlar®-29 as described by Gower et al. This test was chosen because it provides us with an experimental baseline for comparison and was thoroughly discussed in his work. The mesoscale modeling approach implemented is illustrated in Figure B.1, and directly models both the individual Kevlar-29 layers with the interlaminar bond surface. By doing so, the progressive failure of each ply can be individually represented as well as the delamination between each of those layers, allowing a comprehensive perspective of the damage mechanics. To reduce computational time, a quarter-symmetric mode is used, with fixed boundary conditions at the outer edge of the laminate. The interlaminar bond is represented with the Dycoss

Discrete Crack model as discussed in Section 3.2 and [84].

Figure B.1 : Illustration of mesoscale model setup and methodology based on experimental testing by Gower et al.

Due to the highly crystalline structure of Kevlar and the relatively low impact velocity of the projectile, temperature effects can be neglected. The thermal resistance of Kevlar 29 causes it to have a very high decomposition temperature of 482 ̊C [148], far beyond the temperatures expected to arise based on the work of Prosser [149] and Prevorsek [150].

Deformation of the projectile can also be neglected for this analysis due to the negligible energy absorption and resulting plastic strain of the projectile head. Starratt [151] reported relatively low

146 | P a g e energy absorption by a flat-headed projectile (4%) in his experimental analysis of Kevlar 129 ballistic properties. Considering that the impact speed in this analysis is approximately half the lowest impact velocity investigated by Starrat, projectile deformation is expected to be negligible.

In order to verify this, a simulation was conducted that modeled the AISI 4340 steel projectile using the Johnson-Cook material model [95]. This showed that only 0.3% of incident projectile kinetic energy was absorbed by the projectile, despite being nearly 20% more computationally expensive.

Therefore, modeling the projectile as a rigid body in this case is a good assumption that produces accurate results for less computational cost.

Because the Kevlar 29 used in this study is a 2x2 basket weave, the asymmetrical effects of crimp common in plain-woven structures can be approximated as negligible [152]. Unlike a standard plain weave, which often has an asymmetric crimp factor and is relatively stiff [153], the basket weave has symmetric crimp and a higher compliance. Based on work by Lee and Iremonger, this greater ease of fiber movement leads to higher ballistic performance [154,155]. In this model, crimp is assumed to be part of the constitutive properties for the homogenized continuum since crimp would have been present in any tests involving woven Kevlar.

The significance of friction in composite impact has been shown by many researchers [153,156–

159]. Briscoe et al. found that introduction of a lubricant could reduce effective fabric modulus and ballistic capture efficiency by a factor of three, highlighting its importance in performing accurate damage assessment. Because the continuum approach accounts for micro-scale yarn interaction, friction is modeled here as a dissipative energy caused by the macro-scale interaction of projectile and composite as well as interaction between Kevlar layers. Friction is included in the numerical model in the form of both static and dynamic friction coefficients. For simplicity of implementation, these values are assumed to be constant instead of dependent on load or rate.

Several dynamic loading experiments such as that by Shim et al. [160] illustrate the brittle tendency of Kevlar fibers when acting under high strain rates. This results in a highly linear elastic modulus with a large peak stress and low breaking strain. Because of the linearizing effect, constant

147 | P a g e elastic values from Table B.1 were used to represent the Kevlar plies, greatly simplifying the model complexity. These properties, such as the elastic modulus, are based off of high strain-rate measurements (2000 s-1) that are more representative of impact conditions. Damage criteria allow the elastic properties to update throughout the impact event, allowing stiffness reduction as the projectile is arrested.

B.1 Element Formulation Parametric Study

As previously described in Chapter 2, proper element formulation selection is important to achieve a high fidelity investigation. An in-depth element formulation study was conducted for the ballistic impact onto Kevlar, similar to the discussion in Chapter 4. However, the element study for the ballistic impact only considered different solid element formulations: constant stress, constant stress with nodal rotations, selectively reduced integration (SRI), SRI with enhanced strain

(efficient), and SRI with enhanced strain (accurate). The enhanced strain formulations are specifically designed for low aspect ratio elements, which can be used to reduce computational expense. For the reduced integration element formulation, the viscous hourglass control was assigned to suppress the zero-energy mode.

Considering the previously discussed importance of delamination in modeling composite failure, the delamination energy of each Kevlar laminate was used to compare the results of each element formulation as shown in Figure B.2. The trends of fully integrated elements were similar, while the under-integrated (constant stress) element formulation diverged after 0.1 ms. Beyond this point, the hourglass energy increased dramatically, eventually becoming 60% of the initial projectile energy. Hence, the under-integrated solid element was excluded for further investigation.

The SRI without an enhanced strain formulation was also eliminated due to the presence of shear locking, which caused excessive delamination instead of ply damage. Both SRI formulations with enhanced strain (accurate and efficient) were predicted almost identical to each other while predicting a similar delamination energy to the other element formulations.

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Figure B.2 Delamination energy plots for each element formulation

By comparing the computation times of each simulation, summarized in Table B.2, the SRI element formulation with the efficient enhanced strain calculation is the clear choice for further analysis. The efficient formulation required only 51.4% of the computational time required for the accurate formulation, but matched the delamination prediction almost perfectly.

Table B.2 Normalized computation expense for each element formulation

Element Formulation CPU Time (s) Constant Stress 442 Constant Stress w/ Nodal Rotations 1180 Selectively Reduced Integration (SRI) 663 SRI w/ enhanced strain (efficient) 713 SRI w/ enhanced strain (accurate) 1388

B.2 Mesh refinement study

To quantify the computational uncertainty, six different mesh densities were generated to find the mesh dependency of the 3-D composite damage modeling methodology in accordance with the procedure described in Appendix A and [118]. The mesh resolutions were generated using a consecutive refinement as a factor of 1.31. The percentage of delamination area and computational

149 | P a g e time with respect to the number of elements used to model the laminate are illustrated in Figure

B.3. The area of delamination was measured at the point when the projectile reached a velocity of zero. This point was chosen because it represents the contribution of delamination energy that was directly involved in energy dissipation process required to stop the projectile. After this point, the bullet begins to rebound and further delamination is a result of the stress-strain waves induced by the initial impact. As shown in the figure, the area of delamination appears to converge to approximately 7% of the total area, but with an exponential increase in the required computation time.

Figure B.3 Percent delamination area and computation time versus number of elements

Based on the results of the GCI analysis summarized in Table B.3, there is a discretization error of 0.54% between the fine and medium mesh density and 3.29% between the medium and coarse mesh density. The observed order of accuracy was far beyond the second order discretization of the element formulation, suggesting that strong monotonic convergence is present. By comparing the diminishing error to the increasing computation time, the most efficient mesh is shown to be the medium density where the plies are modeled with 153,900 fully integrated solid elements with the efficient enhanced strain formulation.

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Table B.3 Summary of the results for the GCI analysis

Property Value

푁1, 푁2, 푁3 346,275; 153,900; 68,400

푟21, 푟32 1.31, 1.31 푝 6.77 21 32 푒푒푥푡, 푒푒푥푡 0.023, 0.138 ̅̅̅̅̅̅21̅̅̅ ̅̅̅̅̅̅32̅̅̅ 퐺퐶퐼푓푖푛푒, 퐺퐶퐼푓푖푛푒 0.0054, 0.0329

It is important to note that other parameters, such as backface displacement and delamination energy all reached convergence before the delamination zone shape. While this could suggest that delamination shape is not a primary factor for some responses such as backface displacement, it can certainly have an effect on the damage propagation. An example of the sensitivity that delamination shape has with the mesh is shown in Figure B.4. The predicted delamination pattern completely changes as the mesh density increases. At low mesh densities, the majority of the delamination occurs away from the primary fiber directions. Alternatively, as the mesh density increases, the amount of delamination along the fiber directions increases, eventually changing the delamination shape from an approximate quarter circle to distinctly triangular. Beyond the medium mesh refinement (153,900 elements), the shape of the delamination area no longer changes, but just becomes further defined.

Figure B.4 Delamination shapes at three different mesh densities

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B.3 Parametric Study of Composite Damage Models and Validation

Before conducting the parametric study of the selected composite damage models, a brief investigation of the importance of modeling delamination was performed. The velocity response of a laminate with and without a mechanism for delamination is shown in Figure B.5. The laminate without any mechanism for delamination is fully bonded between each layer, effectively simulating a fully continuum approach instead of a mesoscale approach. The inclusion of delamination mechanisms has a huge impact on the simulation result, with the projectile fully penetrating the laminate when no delamination is possible. Therefore, it is recommended to include delamination whenever possible in order to predict an accurate response without resorting to tuning parameters or artificial material constants.

Figure B.5 Projectile velocity with and without a mechanism for delamination

Having defined the element type and appropriate mesh density, a parametric analysis of each composite damage model was conducted. The peak backface displacement, total delamination area, and number of failed plies are summarized in Table B.4. For the Tsai-Wu and Chang-Chang damage theories, both stress and strain based failure was investigated. Tsai-Wu was observed to

152 | P a g e have a significant dependence on the basis of failure criteria, and significantly under-predicted backface displacement when using the stress based failure. Alternatively, the Chang-Chang model predicts a similar response, regardless of the failure method. Of the three methods, the 3-D orthotropic model gives the most accurate prediction of backface displacement, peaking at 11.5 mm while the experimentally observed response peaked at 12.0 mm. However, due to the large number of parameters required to define the 3-D orthotropic model, it may have limited application with many composites.

Table B.4 Summary of backface displacement, total delamination area at time of bullet arrest, and number of failed plies for each of the selected material failure models

Failure Tsai-Wu Chang-Chang 3-D Orthotropic Method Back Face Stress 8.04 13.8 N/A Displacement (mm) Strain 15.6 14.3 11.5 Overall Delamination Stress 0.04074 0.1191 N/A Area (m2) Strain 0.0749 0.0689 0.0522 Stress 4 16 N/A # of Failed Plies Strain 16 15 13

The area of delamination through the thickness direction of composite laminate is shown in

Figure B.6 at the moment of projectile arrest. As expected, the delamination extends primarily in the fiber directions because the stress-strain waves from impact travels most quickly along the fiber directions, resulting in a surface wave that induces ply bending. This bending causes relative sliding between the plies, in this case Mode II failure. The predicted delamination shape agrees well with the analytical prediction by Naik et al. [161] as well as the experimental observations by Wu and

Chang [162]. By comparing the energy dissipated by delamination to the delamination area, the energy contributions of Mode I and Mode II interface failure can be calculated. In the mesh convergence studies, it was found that 90% of the delamination energy at the time of bullet arrest was due to Mode II failure. This agrees with Naik's analysis, where it is stated that the majority of energy required to propagate delamination would be due to Mode II failure.

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Figure B.6 Qualitative comparison of delamination area from the simulation (left) with comparison against an analytical prediction [161] (middle) and experimental observation [162] (right)

Throughout the parametric study using multiple damage model, it was found that both Tsai-

Wu and Chang-Chang damage model overly predicted damage (conservative) to compare with

3-D orthotropic composite damage model. The disparity is likely caused by the plane-stress assumption, which does not include factor in through-thickness effects. However, if all of the parameters required to effectively use the 3-D orthotropic model are unavailable, the Chang-Chang model can still provide a reasonable prediction. The Tsai-Wu model was discovered to be highly sensitive to the failure method, and the predicted response was the furthest from the experimental response. From the analysis, the inclusion and modeling of delamination has a significant impact on the predictive capability. Additionally, special care must be taken to ensure that the delamination shape and area has converged in addition to other parameters such as backface displacement and total energy dissipation.

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Appendix C: Metal Fuselage Validation

In addition to the high-fidelity reduced fuselage section analysis conducted in Section 6.2, a reduced fidelity investigation of the metal fuselage was performed as a validation of the general modeling approach. The reduced fidelity model of the B737 incorporates the entire, 10 ft. fuselage section dropped in the experimental study by Jackson and Fasanella [133], and is validated using qualitative images of the post-drop damage and quantitative plots of the acceleration time history.

In addition to the metal fuselage, an approximated composite fuselage is also investigated.

In accordance with the experimental test, a B737-200 section is modeled and shown in

Figure C.1. The fuselage section has a 4.09 m height, 3.76 m width, and 3.05 m length. Both a single cargo door and fuel tank are also modeled in order to replicate the structural discontinuities.

The overall mass of fuselage section is 3909 kg (8613 lbs) which is close to the reported value

(3977 kg or 8780 lbs) [133]. The fuel tank is 168 kg (371 lbs), suspended from the passenger floor truss through two longitudinal fuel rails. The additional weight of the fluid inside the tank is represented by evenly distributing mass over the entire mesh, creating a total fuel tank mass of

1694 kg (3740 lbs). To account for auxiliary weight; such as dummies, seats and other experimental instruments used in the report, the overall mass of passenger floor is defined to be 1640 kg (3620 lbs). The impact velocity is set at 9.10 m/s, in order to match the measured velocity from the experiment. In addition, to ensure realistic impact kinematics, gravity is applied over the domain.

Instead of attaching each individual component by tied-contact or failure-enabled connections, the nodes are selectively merged, similar to the bonding assumption used for the rivets in Section 6.2.1.

In the current reduced fidelity approach, this assumption is conservative by considering the failure of the structural components before that of the mechanical joints.

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Figure C.1 B737-200 forward fuselage section model for dynamic crash test

Fully integrated shell elements were used to create the forward fuselage section of the B737 to avoid zero energy mode deformation while keeping the computation time reasonable. Based on the impact surface description from the experiment, the ground in the fuselage section drop simulation was modeled as being rigid. As a preliminary modeling verification, a grid resolution study based on the GCI method [118] previously discussed in Appendix A was attempted to quantify the numerical uncertainty in order to determine the appropriate mesh density for further investigations. Three different mesh densities are applied over the structure in order to perform the grid convergence analysis. The model information and corresponding computational expense are tabulated in Table C.1.

Table C.1 Mesh characteristics for B737 model and corresponding computation expense Coarse Mid Fine # of Element 34,967 142,193 572,455 Time (sec) 768 2,885 25,147

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The acceleration time history at two different locations on the passenger floor for each of the three mesh densities are presented in Figure C.2. Each of these plots are filtered using a low-pass filter, where the filter frequency was determined using an FFT. For the analyses reported, a 48 Hz filtration frequency using the Butterworth filter was selected to remove the numerical noise. From the plots of the acceleration predicted by each mesh resolution, no clear convergence trends can be observed, particularly on the left side of the passenger floor. While the initial peak is somewhat similar for each mesh density, the response beyond that peak becomes significantly different for each case. Despite the disparity, three major steps can be identified from the impact sequence: (1) initial contact with ground, (2) fuel tank and fuselage section impact, and (3) fuel tank and passenger floor interaction.

The first local peak in acceleration corresponds to the initial contact of the fuselage with the ground, as shown in the acceleration plots and stress contours of Figure C.2. As expected, the response on the side of the cargo door is stiffer due to the greater reinforcement on the door. Since the door introduces structural discontinuity, the fuselage is manufactured with additional structural support around the door to compensate for this vulnerability. Because of indirect interaction between the passenger floor and impact surface through the fuselage primary structures, there is a time delay before any deceleration is observed at the passenger floor. After peaking from the initial contact, the acceleration drops as the fuselage frame deforms to dissipate the impact energy. The greater drop in acceleration on the cargo door side of the passenger floor suggests that the fuselage tilts slightly during impact because of the greater stiffness of the cargo door. The tilt causes the acceleration on the opposite side of the passenger floor to be sustained while the acceleration near the cargo door briefly drops to approximately zero.

Once the fuel tank makes direct contact with the primary structure at approximately 48 ms after initial contact, the second major G-loading event occurs (2). The reason for the sudden rise in acceleration is the more direct stress transfer to the passenger floor from the fuel rails as opposed to the sides of the fuselage. Because of fully loaded fuel tank, the passenger floor truss deflects

157 | P a g e downward upon impact, whereas the fuselage ribs at the impact region buckle upward. Both components contact one another and generate a large stress wave that results in the secondary peak acceleration on the passenger floor. While the medium and coarse mesh densities predict similar peak magnitudes, a noticeably lower peak acceleration is predicted by the highest mesh density on both sides of the passenger floor. The greater level of refinement results in more element erosion, dissipating energy through material failure and reducing the amount of interaction between the fuel tank and fuselage primary structure.

Figure C.2 G-loading time history of the left and right sides of the passenger floor from the section drop test at each mesh density (top) and sequential von-Mises stress contours for the metal fuselage at the medium grid refinement corresponding to the local acceleration peaks (bottom)

The final G-loading peak (3) occurs after the fuel tank mounting rails buckle, causing the fuel tank and passenger floor to make direct contact. On the cargo door side of the passenger floor, this

158 | P a g e final stage in the impact sequence is consistently predicted to have the greatest G-loading. As previously mentioned, the presence of the cargo door is the primary contributing factor to the greater acceleration. The height of the cargo door is approximately the same as the fuel tank, so after the buckling of the fuel rail, the cargo door and fuel tank are the primary components under load. The opposite side of the passenger floor experiences G-loading at least 16% lower than predicted for cargo door side, and the magnitude of the final peak is highly dependent on the mesh refinement. For the finest grid resolution, the final peak on the non-cargo door side is the greatest due to the relatively lower energy dissipation at the previous impact stage. Alternatively, for the medium mesh density, the final peak acceleration is lower than the second peak acceleration.

Despite the differences in the individual peak magnitudes, the total energy dissipation is approximately the same for each level of mesh refinement.

As expected from observing the lack of qualitative convergence shown in Figure C.2, the results of the GCI analysis based on the acceleration response suggest that higher mesh densities should be explored. The global average of GCI was 10.6%, with an observed order of accuracy (p) of 0.92. Considering second order accuracy for the Lagrangian FEA formulation, the derived p is relatively low. While this result does not necessarily suggest that the numerical results are not satisfactory [118], it certainly suggests that a distinct monotonic convergence was not present.

Additionally, in scenarios such as the current study in which large deformations or domain separations of the continuum are present, GCI may not correctly quantify the computational uncertainty [117]. Therefore, in order to determine the appropriate mesh density for further analysis, the acceleration response of each mesh density was compared to the experimental measurements.

The comparison of the experimental and medium mesh density results are depicted in Figure C.3.

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Figure C.3 Comparison of acceleration history between the experimental measurement and numerical prediction at the medium mesh density for each side of the passenger floor

While the relative magnitude and location of the local peak accelerations do not always match between the numerical prediction and experimental observation, the three previously discussed stages of impact can be observed. It is important to note that the reduced fidelity model of the metal fuselage is not intended to perfectly replicate the geometry of all the individual structures, many of which were unknown, but instead was created to assess the overall trends and peak magnitude of

G-loading. The first peak caused by the initial contact of the fuselage with the ground is approximately twice as a large in the numerical prediction, indicating that the amount of energy dissipation from the initial impact is significantly lower in the physical experiment. However, the magnitude of the second peak corresponding to the contact between the fuel tank and primary structure is predicted more accurately. In the experiment, the peak on the cargo door side of the passenger floor is observed to occur well before the same peak on the opposite side of the passenger floor, indicating that the fuselage was somewhat titled during impact. The magnitude and time of the final peak correlated extremely well for the cargo door side of the passenger floor, but was difficult to observe on the left side of the passenger floor. Despite the disparity at the final peak on the left side of the passenger floor, the overall trends, stages, and magnitude of the acceleration were captured reasonably well with the medium mesh density, which was used for further study.

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A qualitative comparison of the final state post-drop is presented in Figure C.4 with the cargo door on the left side (1) and the fuel tank near the rear of the section. The G-loading shown in the previous plots of Figure C.2 and Figure C.3 were measured at points (3) and (4). Several distinct failure locations may be observed in both the experiment and simulation. The fuselage rib critical buckling point is observed at (2), while contact with the fuel tank is also captured. An additional failure point at the right bottom corner of the fuselage is also evident. In both the computational and experimental tests, the post-drop position of the fuselage indicates a clear tilt, with the passenger floor on the cargo door side higher than the opposite side.

Figure C.4 Comparison between section drop experiment conducted by NASA [133] (left) and metal alloy computational result (right)

Having developed a reduced fidelity model of the full experimental test and determined an adequate mesh density to capture the global response of the metal fuselage, the model was extended to an approximated composite fuselage. To provide a reasonable comparison to the metal fuselage, the thickness of the composite frames and stringers were iteratively adjusted until the maximum G- load predicted during the section drop of the composite fuselage matched the predicted maximum

G-load from the metal fuselage drop, 36 g’s. Overall, the general trends and development of the local acceleration peaks remain the same between the composite and metal fuselage drop, though the primary mode of energy dissipation changes completely. Whereas deformation and plastic

161 | P a g e strain is the primary form of energy dissipation in the relatively ductile metal fuselage, element erosion representing material failure dissipates most of the energy in the composite fuselage.

The qualitative difference between the post-drop characteristics of the metal and composite fuselage is shown in Figure C.5. The total stroke length of the composite fuselage is greater, with passenger floor noticeably closer to the ground than for the metal fuselage. Additionally, the frames and stringers directly underneath the fuel tank have been eroded significantly, contributing to the energy dissipation. The previously discussed center buckling point and failures at the corners of the fuselage are also more prominent in the composite model, once again indicating the greater level of material failure. While not intended to necessarily represent the response of an existing composite fuselage, many of the observations made herein correspond to the conclusions of the comparison between the high fidelity reduced section drop of the metal and PRSEUS fuselage.

Figure C.5 Qualitative comparison of post-drop frame shape for metal (left) and composite (right) with the fuel tank made transparent

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