The Response of a Pultruded Fiberglass Tube to Lateral Loading
Louis Brunet
Department of Mechanical Engineering WGill University, Montréal
A thesis submitted to the Faculty of Graduate Studies and Research in partial fLEl.lment of the requirements of the degree of Master of Engineering.
May 1997
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This is an investigation of the response of a double fixed-ended pultruded fiberglass tube subjected to large lateral displacements, in both quasi-static and dynamic conditions. The experiments yield a two-mode response of the tubing to loading. The kst mode, referred to as the local mode, is dominated by cnishing of the cross-section. The second mode, referred to as the global mode, consists of beam flexure and shear components. An analytical model, based on deformed geometry, has been developed to predict the response of the tubing under quasi-static loading conditions. Mode1 resuits compare weii with experiments and minimal computation time is required. Dynamic tests demonstrate a signifïcant rate dependency, which is more pronounced in the globdy-dominated portion of the behavior. The model has been used to demonstrate that steel tubing, commonly used as beams or energy absorbers, rnay successfully be replaced by fiberglass tubing produced by pultmion or nlament winding. Résumé
Le travail présenté est une recherche dans la réponse de tubes, fabriqué par pultrusion, encastrés et sujets à de grandes déformations sous contraintes quasi-statiques ainsi que dynamiques. Les essais démontrent une réponse à deux modes. Le premier mode, dit local, est caracterisé par la déformation de la section sous l'impacteur. Le deuxième mode, dit global, est la somme des déflections dûent à la flexion et au cisaillement. Un modèle analytique basé sur la géométrie déformée est développé pour simuler la réponse du tube sous contrainte quasi-statique. Les résultats se comparent bien aux essais et le temps de computation est minimal. Les essais dynamiques révèlent une dépendance au taux d'application de la contrainte, qui se prononce d'avantage dans la section dominée par le mode global. Le modèle est utilisé pour démontrer qu'un tube en acier peut être remplacé par un tube en fibre de verre, fabriqué par pultmion ou par enroulement filamentaire, dans des applications utilisant des tubes comme poutres ou comme absorbeurs d'énergie. Premièrement, je veux remercier mes parents, JeamClaude et Cécile, pour leur soutien. 1 wish to thank Professor Jim Nemes, my advisor, for his guidance and friendship. 1 am endebted to d the members of the MCGill Composite Materials Laboratory, and in particular to Hamid Eskandari and Marie-Josée Potvin for their fnendship, support and advice. Thanks aiso to Professor Larry Lessard for several teaching assistantships and discussion regarding this project.
MMFG Co. was a material sponsor for this research. Contents
Abstract 1 .. Résumé ll .. . Acknowledgments Ill Contents iv Symbols vii Figures X .-- Tables Xlu
Chapter 1: Problem Statement and Context 1.1 Objectives 1.2 Motivation 1.3 Historical Background 1.4 Research Context 1.4.1 Laterd Loading on Composite Tubes 1.4.2 Lateral Loading on Steel Tubes 1.4.3 Behavior of Pultruded Composites 1.4.4 Composites Subjected to Axial Impact 1.4.5 Instnimented Impact Testing
Chapter 2: Experimental Configuration 2.1 Tubing 2.1.1 Geometry 2.1.2 Material Properties 2.2 Test Equipment 2.2.1 Hydraulic Test Machine 2.2.2 Drop Weight Impact Tower 2.2-2.1 Tower Calibration 2.3 Hardware 2 -3.1 Fixturing 2.3.2 Tmpactor and Tup 2.3 -3 Adapters 2.4 Test Grid
Chapter 3: Analytical Model 3.1 Local Displacement 3 -2 Global Displacement 3.2.1 BeamFlexure 3.2.2 BeamShear 3.3 Load 3.4 Solution 3 -5 Sensitivity Analysis 3.5.1 Damage Zone Angle: a 3 -5-2 End Moment: 3 -6 Dynamic Analysis
Chapter 4: Experimental and Model Results 4.1 Quasi-Static Testing 4.1.1 Narrow Tup - 8 Inch Tube 4.1.2 Narrow Tup - 16 InchTube 4.1.3 Wide Tup - 8 Inch Tube 4.1.4 Wide Tup - 12 Inch Tube 4.1.5 Wide Tup - 16 Inch Tube 4.1.6 Wide Tup - 20 Inch Tube 4.2 Supported Tiibe Tests 4.3 Dynamic Testing 4.3.1 Dynamic Analysis 4.3.2 Dynamic Test Results 4.4 Model Results 4.5 Model Predictions 4.6 Additional Transverse Reinforcernent 4.6.1 Experimental Results 4.6.2 Model Results 4.6.3 Comparison to Steel Data
Chapter 5: Conclusions and Future Work Test Method FUrhire Performance Effect of Indentor Length Effect of Tube Length Effect of Loading Rate Model Behavior Comparison to Steel Tubing Future Work
References Appendix A: Drawings Appendix B: Cornputer Code Symbols
A -- cross-sectional area Al -- dynamic response amplitude A2 -- dynamic response amplitude B 1 -- dynamic response amplitude B2 -- dynamic response amplitude cl -- dynamic response coefficient cz -- dynamic response coefficient c3 -- dynamic response coefficient cq -- dynamic response coefficient C -- darnage zone angle gradient Cl - damage zone angle parameter C2 - darnage zone angle parameter C3 -- darnage zone angle parameter C4 - damage zone angle parameter CDM -- central difference method D -- differential operator DWIT -- drop weight impact tower E, -- longitudinal modulus 5 -- transverse modulus F -- load on damage zone G -- shear modulus h -- derivative interval used in CDM H -- mid-line height of the deformed cross-section 1 - second moment of inertia Io -- initial, undeformed second moment of inertia k - shear shape factor k, -- global spring constant
vii kI-- local spring constant L -- tube Iength mb -- beam mas M -- moment Mend-- reaction moment at tube end Mb -- equivalent beam mass Mi -- impactor mass P -- applied load r~ - average radius of defomed cross-section R - average tube radius Ri -- inside tube radius R, -- outside tube radius S -- position in damage zone Sm, -- maximum damage zone width So -- damage zone width SDOF -- single degree-of-freedom t -- tube thickness V -- impact velocity x -- longitudinal position xl-- impactor position X.L -- neutral axis position y -- center-span deflection 2DOF -- two degree-of-fieedom a -- damage zone angie cq - initial damage zone angle h, -- damage zone angle at So=S,, p - angle to neutral axis A -- off-axis local displacement AB -- beam flexure displacement AL -- local displacement As -- beam shear displacement AT -- total displacement Am -- cross-head displacement E -- strain in damage zone @ -- remaining angle in deformed cross-section
viii y -- off-axis arc angle h - off-axis darnage zone length L, - maximum damage zone length -- damage zone length 0 - deflection angle ml -- In natural fiequency of tube and impactor system -- 2ndnatural fiequency of tube and irnpactor system Figures
Chapter 1: Problem Statement and Context Fig. 1.1 : Pontiac Fiero Space Frame.
Chapter 2: Experimental Configuration Fig. 2.1 : Experimental Configuration. Fig. 2.2: Continuous Pultrusion Process. Fig. 2.3 : MTS Hydraulic Test Machine and Controller. Fig. 2.4: Drop Weight Impact Tower: a) Full View, b) Loaded with Specimen. Fig. 2.5: Terminal Velocity of Drop Weight Carriage. Fig. 2.6: Average Acceleration of Drop Weight Carriage. Fig. 2.7: End-Clamp Fixturing. Fig 2.8: Front and Side View of Impactor and Tup.
Chapter 3: Analytical Mode1 Fig. 3.1 : Spruig Model. Fig. 3 -2: Deformed Cross-Section of the Tube. Fig. 3 -3: Remaining Angle for an Imposed Local Displacement. Fig. 3.4: Local Damage Zone, a) Evolution, and b) ~etdedView of one Quadrant Fig. 3.5: Damage Zone Arc Angle. Fig. 3.6: Idealized Local Darnage Zone Seen fiom the Side. Fig. 3.7: Reduced Moment Bearing Section. Fig. 3.8: Remaining Angle for a Given Damage Zone Geometry. Fig. 3.9: Solution Algorithm. Fig. 3.10: Sensitivity to Variable Damage Zone Angle: ai= nl8 and L = 0.2m. Fig. 3.1 1: Sensitivity to Variable Damage Zone Angle: ai = d8and L = 0.4m. Fig. 3.12: Sensitivity to Variable Damage Zone Angle: ai = dl 2 and L = 0.2m. Fig. 3.13 : Sensitivity to Variable Damage Zone Angle: ai = dl2and L = 0.4m. Fig. 3.14: Sensitivity to Variable Damage Zone Angle: ai= 7dl6 and L = 0.2m. Fig. 3.15: Sensitivity to Variable Damage Zone Angle: ai = dl6 and L = 0.4m. Fig. 3.16: Sensitivity to End Moment (L = 0.2m): FledDisplacement. - Fig. 3.17: Sensitivity to End Moment (L = 0.2m): Total Transverse Displacement. Fig. 3.18: Sensitivity to End Moment (L = 0.4m): Flemal Displacement. Fig. 3.1 9: Sensitivity to End Moment (L = 0.4m): Total Transverse Displacement. Fig. 3.20: SDOF Spring Model.
Chapter 4: Experimental and Mode1 Results Fig. 4.1 : 8 Inch Tube at a) 10 mm, b) 20 mm and c) 30 mm Transverse Displacement. Fig. 4.2: Load vs. Displacement Curve for 8 Inch Tube Using Narrow Impactor. Fig. 4.3 : Load vs. Displacement Cuve for 16 Inch Tube Using Narrow hpactor. Fig. 4.4: 8 Inch Tube at a) 10 mm, b) 20 mm and c) 30 mm Transverse Displacement. Fig. 4.5: Load vs. Displacement Curve for 8 Inch Tube Using Wide Impactor. Fig. 4.6: Load vs. Displacement Curve for 12 Inch Tube Using Wide Impactor. Fig. 4.7: 16 InchTube at a) 10 mm, b) 20 mm and c) 30 mm Transverse Displacement. Fig. 4.8: Load vs. Displacement Curve for 16 Inch Tube Using Wide Impactor. Fig. 4.9: Load vs. Displacement Curve for 20 Inch Tube Using Wide Impactor. Fig. 4.10: Implicit Solution of Remaining Angle, $. Fig. 4.1 1 : Local Displacement in a Supported Configuration vs. Fixed-Ended Response for 8 inch Tubing. Fig. 4.12: Dynarnic Analysis of Case 1: a) Initial Condition, b) Final Condition. Fig. 4.13: Dynamic Analysis of case II: a) Initial Condition, b) Final Condition. Fig. 4.14: Dynamic Analysis of Case III: a) Initial Condition, b) Final Condition. Fig. 4.15: Dynamic Analysis of Case TV: a) Initid Condition, b) Final Condition. Fig. 4.16: Dynamic Load vs. Displacement: m = 33.1 kg, v = 3.0 m/s. Fig. 4.17: Dynamic Load vs. Displacement: m = 15.1 kg, v = 4.5 m/s. Fig. 4.18: Dynamic Load vs. Displacement: m = 9.89 kg, v = 5.5 ds. Fig. 4.19: Rate Comparison of Load vs. Displacement Response of an 8 Inch Tube. Fig. 4.20: Energy vs. Displacement. Fig. 4.21 : Model vs. Experimentai Behavior for an 8 Inch (0.2032m) Tube. Fig. 4.22: Model vs. Experimental Behavior for a 12 Inch (0.3048m) Tube. Fig. 4.23: Model vs. Experimental Behavior for a 16 Inch (0.4064m) Tube. Fig. 4.24: Model vs. Experimental Behavior for a 20 Inch (0.5080m) Tube. Fig. 4.25: Model vs. Experimental Behavior for a 20 Inch (0.5080m) Tube with Fuily Fixed End Conditions. Fig. 4.26: Tube Response at Various Lengths with D=ln,and t =1/8". Fig. 4.27: Tube Response at Various Lengths with D=2", and t =1/4". Fig. 4.28: Tube Response at Various Lengths with D=3 ", and t =3/8". Fig. 4.29: Tube Response at Various Lengths with D=4", and t =1/2". Fig. 4.30: 8 Inch Standard vs. Reinforced Tubing under Quasi-Static Test Conditions. Fig. 4.3 1 : Experimental vs. Model Results for Reinforced Tubing. Fig. 4.32: Model Behavior for Various Lengths of Reinforced Tubing. Fig. 4.33: Behavior of Fiberglass Contrasted with Steel for a 12 Inch (305 mm) Tube. Fig. 4.34: Behavior of Fiberglass Contrasted with Steel for an 1 8 Inch (457 mm) Tube.
xii Tables
Chapter 2: Experimental Configuration Table 2.1: Test Geometry. Table 2.2: Matenal Properties. Table 2.3 : Oscillocope Settings. Table 2.4: Tower Mass Configurations. Table 2.5: Test Grid-
Chapter 3: Analytical Model Table 3.1 : Sensitivity Analysis Pararneters.
Chapter 4: Experimental and Model Results Table 4.1 : Initial Conditions for Dynamic Analysis. Table 4.2: Final Conditions for Dynamic Analysis. Table 4.3 : Modeled Tubing Geometries. Table 4.4: Steel and Fiberglass Tubing Geometries.
... Xlll Chapter I
Problem Statement and Context
The use of composite materiais is becoming increasingly widespread. New applications are considering composite materials as design solutions as the cost of traditional matenals such as steel, aluminum and wood increases and improved manufacturing methods for composite materials make composites an attractive alternative. Several properties such as non-conductivity, corrosion resistance, and low coefficients of thermal expansion have already made composite materials attractive design options for many structural, marine and aerospace applications.
The applications where composite materiais are considered often require knowledge of their impact response. Impact has been of interest for several years beginning in the Second World War, when the study of plates and shells subjected to impact was undertaken to improve shelter and submarine designs [LI. Much of the early work focused on the response of traditional matenals, but for several years, the response of composite materials to impact has also become a source of interest.
Within the field of research on impact, there exists two distinct types of problems; blast wave and mass impact problems. This investigation wiIl focus on the static and dynamic response of a double fixed-ended pultruded fiberglass tube subjected to a mas impact. A test method and adequaîe fktmhg are developed for both loading conditions. An analyticai mode1 capable of predicting the load vs. displacement response of the tubing is developed. Experiments are perfomed under static and dynamic loading conditions.
Chapter 1: ProbZem Statement and Context Finally, the model is compared to test results, and a cornparison to steel tubing is attempted.
1.1 Objectives
There are two objectives in this investigation. The fUst is to develop static and dynarnic test methods that sustain the conditions of the test configuration, and to perform an array of tests varying geometric and loading conditions. The second objective is to develop a model that predicts the load vs. displacernent response of the tubing. With a functional, experirnentally validated model in hand, it becomes possible to compare the response of pultnided fiberglass tubing to that of steel tubing over the same range, and engineee have a reliable design tool that rnay be used to malyze beam structures.
1.2 Motivation
Composite matenals are king used in an increasing number of applications. As new oppormnities are sought, the timiting factor is often cost. Low cost rnanufacturing methods are key to opening new markets for composite matenals. Traditional applications have been in the aerospace and sporting goods industries, but industry leaders in the pultnision industry have begun to target the construction market [2] and there is increasing interest in exploiting transport applications [3-51. Glass fiber composites are most applicable to the automotive industry due to theu low cost, whiie only incmg a 25% reduction in specific energy absorption when compared to carbon fiber [q.
With new applications of these materials must corne improved design methodologies which engùieers may employ. Finite-element modeling is available, but costly and not within the means of many potential users, thus alternative design tools have potential.
Chpter 1: Problem Staternent and Context 1.3 Historical Background
Pultnision is a manufacturing method whereby wetted out fibers are drawn through a heated curing die to produce hi& performance constant cross-section composites at Iow cost. Compared to short fiber extrusion, using the sarne matrix (nylon 6-6) and the sarne fiber volume hction (40%), a 300% increase in impact strength and a 60% increase in the flexurai modulus is obtained for the pultruded composite while other properties are maintained [7]. Pultruded goods have many potential applications depending on certain design critena. Pultruded composites have been used in Street sweeper bumpers, springs and automotive components, and sports applications where high impact and corrosion resistance are required. They have also been used in biomedical braces, harvester tools, threaded end rods and truss fiames where high reforrnability is required, and sections with tapered profiles or with flexural and torsional variability have been used where variable properties are required [8]. Io the aerospace industry, a mature market for composite rnaterials, the Beechcraft Starship 1's aidiame has achieved 100% composite use [4], pultruded composite sections are cornmonly used as trailing edge profiles for fked wing applications and helicopter rotors [9].
Fig. 1.1: Pontiac Fiero Space Frame.
The automotive industry has taken two different directions where the design of a composite automobile is concemed: the honeycomb laminate structure and the space-
Chapter 1: Problern Statement and Contexi 3 hestructure. Nakada & Haug [3] performed fiont and side impact nnite-element simulation for honeycomb laminate structues, concluding that simulation of automotive stnictures was feasible. Ashley [5] reports on crash tests performed on honeycomb laminate structures. The structures have achieved technical feasibility, although the cost of composite parts is somewhat too high for commercial applicability 151. Shorter cycle times must be achieved to lower their cost [5]. The emerging use of space fiames as seen in the Fiat Uno and Pontiac Fiero, where composite body panels are suspended on the structure, would be ideal for composite substitution [4]. Although present space &mes are made of steel, it is feasible that these could be changed to pultruded or filament wound structures [4]. 1.4 Research Context
Two elements of this investigation are key in its charactenzation: structural response and matenal response. Nurnerous papers detailhg the response of composite materials to impact loading have been written, as have papers detailing the response of structures to impact. Only systems subjected to single impact loading will be examined, although for some structures, repeated impact loading may be expected.
Impact may be characterized in many ways, one of which is by impact velocity. Impact events may be in the low-velocity, medium-velocity or hypervelocity range. Some discrepancy exists conceming the definition of these ranges, and low-velocity impact, which is the concem of this investigation, is no exception. Abrate [IO] defines low- velocity impacts as those occurring at velocities infenor to 100 m/ç while Cantwell & Morton [Il] de fine them as impacts occurring at velocities inferior to 10 m/s.
In Iow-velocity impact, the ability of the fiber to store energy elastically is fundamental to the material's ability to withstand impact [Il]. While the primary role of the polymer matrk in a composite is to protect, stabilize and assure stress transfer among the fibers, the matrix must aiso be tough, resisting delambation and fiacture if the material is to perform well under impact [Il]. In high velocity impact, damage is highly localized and takes the form of surface spalhg and shear plugging, and the fiber sees little Ioading [lO]. Thus, the response of the material changes with impact velocity.
Chapter I : Problem Statement and Context Moreover, the response of the materid to impact, and the ensuing failure mode, is not independent of the specimen geometry and thus, care should be taken if the response of a structure is to be predicted based on results obtained f?om small simple specimens [l 11. In fact, static and dynamic tests should be performed on coupons and structures [Il].
The interpretation of results obtained fiom impact tests is also di"cult, as the response may not always be reducible to a simple spring mass system [l O]. When large impactor masses are used, it is often possible to reduce two degree of fieedom systems (2DOF) to single degree of fkeedom systems (SDOF) [IO]. For example, Abrate quotes an impactor mass superior or equal to 14 times that of the specimen to be used for impact between a spherical impactor and a clarnped circular plate [l O].
Steel structures subjected to impact loading and large deflections have been studied extensively, and where a fked-ended steel tube subjected to lateral loading is concerned, three modes of fdure have been ideniified: I) large inelastic ductile deformation; II) tensile tearing failure at the supports; III) transverse shear failure at the supports [Il. Several methods of analysis have been developed to determine the response of thùi-walled tube structures in simply supported and fked ended configurations including variations on the plastic hinge concept for both static and dynamic analysis, and models based on deformed geometry and plastic work [Il. Analysis of composite beams require some modification as they are subject to fiacture radier than plastic deformation under lateral loading, leading to degradation of *ess [12].
Lateral Loading on Composite Tubes
Laterai impact on composite tubes is a relatively unexplored problem, and work thus far has focused on finite element simulation using constitutive models [1 3 - 151, or experimentd investigations into the onset of damage [16], with few investigators looking at tubes subjected to large deformations [17]. Corbett & Reid [18] took the next practical step, evaluating and comparing the performance of composite pipes and steel pipes.
Constitutive models have been used effectively in finite-element simulations to study the behavior of composite tubes by several authors. Nemes & Bodelle [13] showed, using a finite element simulation Uicorporating a rate-dependent continuum damage mode1
Chupter 1: Problem Statement and Context 5 developed by Randles & Nemes [19]; that substitution of a glas weave / epoxy composite oval for a steel "Wu highway guardrail will result in higher specific energy absorption. Chang & Kutlu [14] applied a progressive damage mode1 to the quasi-static behavior of a filament wound tube to evaluate the maximum cmhing load the tube can sustain. They obtained experimental confirmation of the maximum crushg load, and the finite element results were within 10% of the predicted value. Mustafa et al. [15] investigated and modeled the behavior of a Mysuppoaed filament wound graphite reinforced tube. They reported the formation of a diamond-shaped indentation zone under the spherical indentor. The constitutive model used in their finite-element simulation was a simple orthotropic linear elastic model and gave good results at low levels of indentation for the defomed geometry of the tube and strain distribution within the tube. At large defomations, shelï buckling was observed to be the cause of failure.
Onset of damage in composite tubes is of interest prharily to those investigating the potential of these materials in fluid transport. Pang & Kailasam [16] investigated the energy absorbing capacity of two different glass epoxy / glas pipes subjected to low velocity impact using indentors of various geometries, and their work led to numerous conclusions. Sharp-edged impactors such as cone and wedge tipped impactors were found to induce localized damage tbrough local plastic deformation of the surface beneath the impactor due to the near-zero radius of cwvature at the contact point. Hemispherical impactors develop a damage zone larger than the impact zone, and cracks are initially formed internally due to the reflected tende stress wave at the inner surface. Lower fiber volume fiactions are conducive to higher energy absorption, reflecting the ductile properties of the resin. hcreasing the impact velocity, while the impactor mass is held constant, results in an increase in absorbed energy, and similarly increasing the mas, while the impactor velocity is held constant, results in an increase in absorbed energy.
Wang et al. [17] investigated the behavior of a box section pultruded composite to Suaal and flexural loading. The flexural loads were applied on sections in a double cantilever configuration. In both axial and bending configurations, the weak transverse properties of the material were found to be the cause of failure. Improving the transverse properties of the tubing using additional weaves or using braiding improved the response of the box section tubing and changed the failme mode fiom wall buckling to end compression failure with a 25% load bearing improvement.
Chapter 1: ProbZem Statement and Contat Corbett & Reid [18] investigated the behavior of filament wound epoxy / glas pipes subjected to projectile impact fiom a gas gun. The ïoad was applied using a hemispherically tipped impactor. Damage was seen to occur in sequence through resin cracking, delamination of a rectangular zone around the impact area, and fiber breakage on penetration. The overall behavior is observed to be indentation followed by indentation and bending. Experimentai cornparison of a steel pipe (101.6 mm OD, 3.2 mm wail thickness) and an epoxy/glass pipe (101.6 mm OD, 4.3 mm wall thickness) subjected to perforation demonstrated the greatly superior energy absorption per area density of the epoxy / glas pipe (77 Lrn2/kg vs. 12.8 ~.m~/k~).
Lateral Loading on Steel Tubes
Lateral impact on steel tubes has been of concem for several industries, though primady for those involved in oii exploration. Important experimental work by Reid & Goudie [20] and Jones et al. [21] has provided data to be used in the evaluation of models developed to predict the behavior of thin-walied steel tubes subjected to lateral impact. Several such models have been developed based on deformed geometry [20,22-241.
Reid & Goudie [20] acknowledge that the problem of modeiing thin-walled tubes subjected to local indentation is a difficult theoreticai problem because of the large plastic deformations governed by interactions between bending, stretching and shearing within the shell. deolivena [22] and Reid & Goudie [20] identiS. three phases of collapse for the behavior of a simply supported tube; local denting, denhg and bendhg, and &al collapse. The dent is observed to develop into a diamond-shaped damage zone [20, 22-24], but the growth of that damage zone may cease as the cross-section at the point of loading is cmshed [20], or the rate of growth of the damage zone may change with the cnishing rate of the cross-section at the point of loading [24]. In fact, the angle characterizing the damage zone varies with pipeline size, externd kinetic energy and impact position [24]. deoliveria [22], Soreide & Amdahl [23], Reid & Goudie [20], and Jones & Shen [24] developed subsequent plastic analyses for ked-ended steel tubes subjected to Iateral loading assuming two distinct phases of deformation; local deformation (cross-sectional
Chupter 1: Problern Statement and Context cnishing) and global deformation @am flexure and beam shear). These phases reflect what is physically observed, and the results obtained by considering the two modes of displacernent are superior to analyses considering only global deformation, which greatly overestimate the load for a given displacement.
The fint model developed by deolivena [22] was a simple ring collapse mode1 for a rigid plastic beam subjected to lateral loading with a beam response capable of accounting for non-ideal beam end-nxity. The solution requires data or a function to descnbe the amount of translation and rotation incurred at the ends. The response of the beam is separated into two distinct and sequential phases. Initidly, there is dent formation, or local deformation, and only once local deformation has ceased is it possible for beam flexure to be*
Soreide & Amdahl [23] performed a fite element simulation using a fidl shell anaiysis incorporating strain hardening, and obtained good correlation to experimental data. The same sequence of events is assumed as in the deoliveria model, but this model brought some improvement over a simple ngid plastic analysis.
Reid & Goudie [20] proposed an improved numerical solution and sought to incorporate membrane stretching, ignored in deOlivenals solution. A similar rigid plastic solution as deoliveria's is developed with local deformation ceasing p50r to the onset of global deformation. Incorporating membrane stretching yields better predicted deformed geometxy than that which had been obtained previously.
Following an experimental investigation [21], Jones & Shen [24] developed a model that allows local and global deformation to occur concurrently. Also of interest is the treâtment of the damage zone. They allow the characteristic angle to Vary such that the defiection is maxirnized for a given impact energy. This approach is consistent with upper-bound evaiuation in the theory of plasticity, and it yields very good results.
These subsequent models each yielded improvements over the model proposed by deoliveria in the form of improved prediction of the load vs. displacement cuve or of the deformed geometry of the tube. However, these improvements have been rnodest, and the deoliveria model provides results of reasonable precision with a simpler analysis.
Chapter 1: Problem Statement and Context 1.4.3 Behavior of Pultruded Composites
The main emerging market for pultruded composites is the construction market. As structural pultruded composites penetrate this market, interest in detennuiing the modes by which these materials fail is increasing.
The behavior of commercially available (Creative Pultnisions Inc.) pultruded polyester / E-glass and vinylester / E-glas 1-bearns in three-point and four-point bending configurations has been studied 125-281. Where the polyester / E-glas beams are concemed, two simultaneous modes of failure are observed for this type of bûam: torsional failure due to instability in the flange, and top £lange buckling [25-261. For vinylester / E-glas, the fmst ffaure occurs in the compression flange by buckling, followed by failure through the web at constant load after some time delay [27-281.
Boukhili et al. investigated the rate-dependent response of pultmded polyester / glas composites [29]. They observed increases in the shear and flexural strength with increased loading rate on half-round sections. However, the shear strength increases only to a threshold loading rate after which a decreasing trend was noted. The loading rate effect is attributed to the ductile nature of the polyester matrix. For intermediate span to depth ratios (iength : radius, 4.5-9), higher loading rates change the mode of failure fiom tensile to shear due to dynamic embrittlement of the matrix.
1.4.4 Composites Subjected to Axial Impact
There has been a considerable amount of work done in the study of the response of composite structures to axial loading. This interest has stemmed fiom the desire to use them in energy absorbing structures. In this configuration, it is believed that composite sections may make good energy absorbers with potential in the transport industries. Composite tubes subjected to axial impact achieve and maintain a constant cash load dominated by fracture rather than buckling as seen in steel tubing [6, 30-321. It is during this cmshing that energy is absorbed.
The principal energy absorbing rnechanisms during crushing are: intmwall crack propagation, fiond bending due to delamination, axial splitting between fionds, fled
Chapter 1: Problem Statement and Context 9 damage of plies, fictional resistance to axial sliding between lamime, frictional resistance to penetration of the annular debris wedge and fkictional resistance to fionds sliding across the platen [6, 3 1-331. In fact, progressive crushg requires the formation of a well- defïned cnish zone featuring an annular debris wedge that is forced axially through the tube wall fomllng delaminated strips or fions [3 11.
The materiai structure plays an important role in energy absorption for axially impacted composite structures. Smaller fiber diameter, and Iower fiber volume fiaction reduce energy absorption [W. Dinerent ma& materials available also affect the energy absorption @henolic Equivalently, geometry plays an important role in energy absorption for axially irnpacted composite structures. Square pultruded polyester 1 glas tubes exhibited a reduction of 30% in energy absorption when loaded dynarnically, though round tubes did not seem as rate dependent [6]. A reduction in energy absorption is also observed in square Wiylester/glass and polyester / glass laminated tubes due to splining dong the edges [30]. There is some debate regarding rate-dependence of axially impacted energy absorption of various polyester composite hibes with sorne authors noting increases, and others decreases [30], but for glas-mat / polyester composite tubes, increased crush speed reduces the specific energy absorbed [3 O]. Instrumented Impact Testing Several methods have been developed to perform impact tests. Cantwell & Morton [Il] suggest the use of Charpy and Izod pendula, drop weight impact towers (DWT), and hydraulic test machines for use in low vel ocity impact testing. Charpy pendulum impact tests subject short, simply-supported, notched specimens to destructive impact. Izod pendulum impact tests subject similar specirnens in a cantilevered configuration to destructive impact. While it is possible to instrument the pendulurn with load cells or Chapter 1: Problem Statement and Contexr acceierometers, generaIiy only the energy absorbed by the specimen is obtained by comparing the kinetic energy of the pendulum pnor to and following impact 1351. The problem with these test methods however, is more fundamental; the use of notched specimens results in an evduation of the propagation energy rather than initiation energy [36]. DWIT1s and hydraulic test machines provide the greatest flexibility [Il], allowing tests to study total energy absorption as well as onset of damage energy absorption. DWIT's consist of an instruinented Carnage with an impactor tip. The carnage is dropped fiom a given height to obtain the desired initial conditions. Both D WIT's and hydraulic test machines enable complex structures to be tested, but hydraulic test machines are limited by the velocities they can generate- Lifshitz et al. observed the natural modes of sûiker and crossbar superimposed on the response of the accelerometer housed in the impact carrïage of their DWIT; however, these additional modes are absent fkom the output of strain gages on the beams [37]. This dynamic phenomenon is due in part to the contact sti-ess of the target, and in part to the relative stiffiless of the impactor compared to the test materiai [38]. The beam's response thus becomes that of a 2DOF systern, by which the beam spring-mass is loaded via a contact spring. Oscillations in the response may be the result of ill-conceived experiments, and care must be taken to ensure that the system is balanced such as to avoid exciting unwanted natural fiequemies in the system. If the systern cm be shown to be a SDOF system, the acceleration history can be twice integrated with respect to tirne to detennine the displacement and thus obtain acceleration vs. displacement, or load vs. displacernent [37]. The travel tune between two sensors may be used to find speed to deterrnine the initial velocity of the impactor at the beginning of the impact event [l O]. Chapter I.. Problem Statement und Context Chapter 2 Experimental Configuration The experimental configuration used in this study consists of a double fixed-ended pulû-uded fiberglass tube subjected to a lateral Ioad at midspan applied by a cylinder of the same radius, R (see Fig. 2.1). The experiment is designed to simulate the transverse impact between two cylinders where one cylinder is ngid, and the other cornpliant. It is intended to investigate the rate-dependency of the material and structure, the effect of beam length on the response, and the effect of impactor geometry. Chapter 2: Fxpehental Configuration The material used, commerciaily available fiom Morrison Molded Fiber Glass Co. (MMFG), is an isophtalic polyester / glas fiber composite [39]. More specifically, the Extren Series 500 is used in the tests. It was selected because it is an inexpensive and readily available matenal, thus fitting the motivating objectives of this investigation and the requirements of transport industries. The matenal is manufactured by pultnision (see Fig. 2.2), which is a continuous rnanufacturing process. This process draws rovings and fiber mats through a resin bath to wet the fibers, then through a hot die to form the desired final shape. Finally, the profile passes through a tunnel oven to accelerate curing, and the composite is cut to length. For this product, the fiber volume fiaction is expected to be approximately 30 %, although it is possible to obtain as high as 75 % fiber volume fiaction using this process. CUT-OFF CATERPILLAR- SAW IMPREGNATOR PUUBLOCKS MATERIAL ROVING CREES Fig. 2.2: Condinuuus PuCfrusion Process [39/. 2.1.1 Geometry One size of tubing was provided for testing; therefore, diameter and thickness of the tubing are not varied. The tube span is increased to investigate its effect on the total response of the tubhg as detailed in Table 2.1. Chapter 2: Experimental Configuration Table 2.1: Test Geomeîry. Total Tube Span, L (mm) 203.2 508.0 Inside Radius, Ri (mm) 19.05 Outside Radius, R, (mm) 25 -40 Average Radius, R (mm) 22.23 Wall Thickness, t (mm) 3.18 Material Properties The material properties, used Later in modeling, are those provided by the manufacturer. Tende and compressive moduli are quoted as equal. These properties were not experimentally confirmed due to the difficulty in making samples fiom cylindrical tubing. Table 2.2: Maferr'cslProaertr'es f391. Shear Modulus, G (GPa) 2.2 Test Equipment Loads are applied quasi-statically and dynamically using a standard hydraulic test machine and a drop weight impact tower, respectively. The range of loading rates provided by these two machines is sufncient to investigate rate dependency in the material and structure. The range also spans a significant portion of the operating range of the applications discussed in the objectives and motivations. Chapter 2: Experirnental Configuration 2.2.1 Hydraulic Test Machine The hydraulic test machine, shown in Fig. 2.3, is a standard hydrauiic tension / compression test machine rnanufactured by MTS. Specificaily, a mode1 204.71 test machine is used with a 458.20 MicroConsole controiler. The machine has a maximum load capacity of 250 kN, and a stroke of 100 mm. It is used under displacement control at a rate of 5x 1O-' m/s in order to ensure that no dynamic effects manifest themselves in the system response. The total transverse displacement imposed during the experiments is 40 mm, or approximately 4 average radii, R, and the test loads are on the order of 5 kN. The 25 kN and 50 mm cartridges are used in the controller. Fig. 2.3: MTS Hydraulic Test Machine and Controlier. 2.2.2 Drop Weight Impact Tower The drop weight impact tower, shown in Fig. 2.4, is a mode1 designed and built at MCGill Universiv. The tower houses a PCB 200B05 piezoelectric load cell dnven by a PCB 482A06 power source. The signal acquisition is performed with a Nicolet Pro40 digital oscilloscope, with the oscilloscope settings detailed in Table 2.3, thus obtaining the voltage history of the impact. The voltage is proportional to load and the load cell has been caiibrated to 4236 NN in the range of O kN and 22 W. Experiments are performed with an irnpactor mass rnuch greater than that of the tube, such that the impact response cm be approximated by a single degree-of-freedom (SDOF) system, as shown in Section 4.3. Effectively, this reduces the tube to a massless spring. Chapter 2: ErperUnental Configuration 15 Table 2.3: Osciiiosco e Settïn S. Tri er 200 mV Acquisition Rate 50 us I Points Dynamic tests are performed on samples of one length = 203.2 mm), due to the physicai constraints of the drop tower. The &op weight is varïed using different loading plates on the lower carriage. DBerent mass configurations are listed in Table 2.4. These are total carrïage masses including the weight of the appropriate load cell, impactor, and all required hardware. The &op height wiU be chosen to provide the same input energy for each carriage configuration. a) 6) Fig. 2.4: Drop Weight Impact Tower: a) Full View, 6) Loaded witlr Specimen. Chapter 2: Experimental Configuration 2.2.2.1 Tower Calibration Friction in the linear bearings on the drop caniage resists gravitational acceleration. The tower is not nomaily equipped with instrumentation that determines the velocity of the carnage. Thus, experimental calibration for this purpose is canied out. Drop Height (m) Fig. 2.5: Terminal Velocity of Drop Weight Carriage. Two photodiode gates were designed (see Appendix A) and set at the bottom of the stroke of the impact Carnage to measure its velocity in the foiiowing way. The circuits each generate a boxcar (square) signal, captured on the oscilloscope, and the physical distance separating the two photodiodes is measured; thus, the velocity (see Fig. 2.5) is determined by dividing the distance traveled by the time between the two rising portions of the boxcar signals. From the terminal velocity of the camiage, it is possible to determine the average acceleration during the fdl, dlowing cornparison t O gravitational acceleration (see Fig. 2.6). Chpter 2: Eqehental Configuration O 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Drop Height (m) Fig. 2.6: Average Acceieration of Drop Weight A curve fit through the data displayed in Figs. 2.5-2.6 generates the following relationships that may be used within the range of the tests, although direct use of Fig. 2.5 is more precise. Eqn 2.1 Eqn 2.2 2.3 Hardware In order to perform the tests described, proper fixturing and irnpactors meeting the experimental constraints have been designed. These devices have been designed such that they may be quickly adapted to both the hydraulic test machine and the DWIT. The fixturing is show in shop drawings in Appendix A, and details are discussed below. Steel is used in al1 designs to ensure that al1 components are much stif5er than the material subjected to testing. It is critical that the hardware be considerably stiffer than the material tested in order that the behavior be attributable to the material and not its fixming. Chapter 2: Erperimenfal Configuration The double fixed-end condition is achieved using a newly designed ee.The fkme consists of sockets inserted in the tubing to prevent cmsh damage fiom extending to the ends of the tubing. The socketed tube ends are then clamped to prevent end rotation of the tubing, as show schematically in Fig. 2.7, similar to nxturing developed by Reid & Goudie [20] for steel tubing. Care must be taken when fastening the fixture to the test machines, as tension in the tube may cause the iktwhg to translate. Sorne researchers [20-211 perfomiing experiments on steel tubing have used more elaborate nxtures, incorporating translation limiting devices. This experiment did not warrant such precautions because the tende loads generated in the tube membrane are not expected to be suEcient to draw the nxturing inward. Impactor and Tup Two cyhdrical impactors have been designed for the experiments, one narrow and one wide. Both have the same radius, R, as the average radius of the tubing, and both are machined fiom steel bar stock. Blunt cylindrical impactors have been selected as they have been observed to produce a larger, distrïbuted damage zone [16]. This will ailow large transverse displacements to be imposed on the tubing with little risk of central failme. The narrow impactor is of tength 2&, twice the outside radius of the tubing. The wide impactor is of length greater than TC%, the flattened width of the tubing. This impactor will simulate the intersection of two long cylinders, and depending on the amount of flattening iocurred during testing, may result in different failure than the narrow impactor. Fig. 2.8: Front and Side View of Lwpactor and Tup. The impactors are tapped to be fixed to a tup as shown in Fig. 2.8. The tup is simply a stem onto which the impactor is fastened. This stem is initially designed to fit the load cell housing of the drop tower. Adapters The kture is designed to fit the hydraulic test machine's 3-point-bend test apparatus. Adapters accommodate the DWIT such that it may use the same fimes. Sirnilarly, the tup's design fits the load cell housing of the DWIT. An adapter accommodates the hydraulic test machine's 3 point bend test apparatus. 2.4 Test Grid A variety of tests are possible within the constraints of the machines used for the experiments. To investigate the effects that the tup width, beam length, and loading rate have on the behavior of the tubing, the following tests are perfonned. These isolate a given variable while investigating its ef3ect for a few configurations. In addition, some tubing reinforced with one layer of cross ply glass cloth will be tested to provide a stifier alternative for cornparison to steel hibing, as set out in the objectives. For ail test conditions, a minimum of 3 samples were tested to ensure repeatability of the results. Table 2.5: Test Grid. Test TUP Beam Impact Velocity (ds) Tubing Number Lent@ (mm) and Mass (kg) 1 narrow 203.2 Quasi-static Standard II narrow 406.4 Quasi-static Standard m wide 203.2 Quasi-static Standard N wide 304.8 Quasi-static Standard V wide 406.4 Quasi-static Standard VI wide 508.0 Quasi-static Standard lm wide 203.2 5.5,9.89 Standard Vm wide 203.2 4.5, 15.1 Standard IX wide 203.2 3.0,33.1 Standard X wide 203.2 Quasi-static ~einforced* One layer of cross-pIy epoxy/giass added to the standard pultruded polyester/glas tubing. Chapter 2: Experimental Configuration Chapter 3 Analytical Model An analytical model capable of predicting the load vs. displacement response of the composite tubing being studied is developed in this section. This model must be able to predict the diin-walled cnishing response of the tubing in addition to the global beam response. Models of this type have been developed for traditional materials, notably those of deoliveria [22], Soreide & Amdahl [23], Reid & Goudie [20], and Jones & Shen [24], who have developed plastic models based on the deformed geometry of metal tubes in the same configuration. To the author's knowledge, no models have sought to predict the behavior of orthotropic materials. In addition, the models mentioned above require an iterative solution method, whereas the proposed method solves for load vs. displacement in one iteration. Isotropic metal tubes subjected to lateral loading have been observed to exhibit a two- mode response [20,22-241, and such a response is expected for the pultnided composite tube. The modes correspond to regions of dominance of one or another mode of deformation, and the model must reproduce these modes. The model accounts for three components of displacement. The fxst component, the local displacement, is the deformation of the tube's cross-section beneath the impactor. The second and third cornponents of displacement are the beam flexural displacement and the beam shear displacement, both of which are considered global components of displacement. This system is illustrated by an in-series, 3 spring system in Fig. 3.1. Chapter 3: Analytical Model A damage zone associated with the local displacement is assumed. It is an extension of the cross-sectional deformation, or local displacement, to the area adjacent to the impactor. The extent and geometry of the damage zone is determined directly fiom the cross-sectional deformation. It is similar, though somewhat simplified, to that observed experimentally. The resultant load is based on the strain developed in the damage zone, thus the stif3kess of this response mechanism is a function of the local displacement. The components of global displacement (flexure and shear) are linked to the local displacement, as the stiffness of these components of displacement are functions of the geometry of the tube, which is detemiined fiom the local displacement. More specifically, the flexural stifkess of the tube changes because the second moment of inertia changes with the changing cross-section, and the shear stiffhess of the tube changes because the shape factor changes with the changing cross-section. Fig. 3.1: Spring Mudel. The illustrated mode1 is subject to the following constraints. Eqn 3.1 Eqn 3.2 Eqn 3.3 Eqn 3.4 Eqn 3.5 Chapter 3: Analytical Model where the subscnpt 1 refers to local displacement, 2 to flexural displacement, and 3 to shear displacement. 3.1 Local Displacement Local deformation is defmed as the cnishing of the cross-section beneath the impactor. It extends into a damage zone to the area adjacent to the impactor. The geometry of this damage zone is subsequently used to predict the load generated on the tubing. Fig. 3.2: Deformed CrossSection of the Tube. The cross-section of the tubing deforms under the impactor as shown in Fig. 3.2, as observed experimentally. This deformation is assumed to be inextensionai, such that the circumferential length of the mid-line of the tubing wall is maintained [20, 22-24]. This assumption permits the deformed geometry of the tubing to be detennined fiom a prescribed local displacement, AL, and it also implies that the transverse modulus, E,,, plays no load bearing role in the behavior of the tube. As the circumference of the deformed section and the orïghdl section must rernain the same; 2M = 2rL(#+ sin @) Eqn 3.6 Chapter 3: Analytical Mode1 where R is the average radius of the undefonned cross-section, r~ is the average radius of the deformed cross-section, and @ is the arc angle of the circular segment of the deformed section. With P dehed as the angle to the neutral axis, the following must also hold. nR = 2r,$ Eqn 3.7 From Eqns 3 -6 and 3.7, it follows that, Eqn 3.8 Eqn 3.9 The local displacement may be detemiined fiom the previous angular relationships. Looking at the projections of radius, the displacement is: AL = R+rLcos@-r,cosp. Eqn 3.10 Substituting Eqns 3 -8 and 3 -9 into Eqn 3.10, the local displacement becomes, Eqn 3.11 Fig. 3.3: Remaining Angle for an Imposed Local Displacemen f. (Eqn 3.11). For an imposed local displacement AL, the angle $ may be calculated by solving Eqn 3.1 1, which determines the complete deformed geometry of the system. The solution of Eqn 3.1 1 is illustrated in Fig. 3 -3. The local displacement is limited to: Eqn 3.12 which corresponds to complete flattening of the cross-section. Thus the ratio of local displacement to original radius can not reach unity while the tube has a fbïte waU thickness. Fig. 3.4: Local Damage Zone, a) Evolufion, and 6) Defailed View of one Quadrant. The deformation of the cross-section beneath the impactor is assumed to extend to the adjacent regions of the tubing. The darnage zone is idealized as diamond-shaped as seen in Fig. 3.4%and this damage zone geometry has been observed by Mustafa et al. [15]. The darnage zone is characterized by an angle cc determined fkom the damage zone width, So, and damage zone length, ho, as seen in Fig. 3.4 Eqn 3.13 However, So may also be defhed in terms of the horizontal projection of the deformed radius of the tubing (see Figs. 3.2 and 3.4), such that: S,, = r, sine Eqn 3.14 The geornetric analysis thus far is consistent with that developed for the plastic analyses discussed previously . For metallic tubes, the angle a is known to Vary with the size of the damage zone [24] and the following general relationship relating the darnage zone angle and size is proposed: Eqn 3.15 where the maximum width of the darnage zone is, Eqn 3.16 Taking the relationship to be iinear gives: Eqn 3.17 where C and cci would need to be experimentally detennùied. These terms are dependent on the material properties (E, $, G), as well as the tubingis geometry @, t) [24]. Fig. 3.5: Damage Zone Arc Angle. Off-axis characteristics of the damage zone may be determined based on the existing assumptions. These will characterize the longitudinal elements in the damage zone. Chapter 3: Analytical Model Fig. 3.4b yields the off-axis length of the damage zone, h, based on an off-axis position in the damage zone, S, where S is a segment of So. Eqn 3.18 The off-axis deflection of the damage zone, A, is also based on S, and the associated arc angle y (see Fig. 3.5). The angle y is defined as the arc angle fiom the vertical to S in the undeformed codiguration. Fig. 3 -5 shows the arc angle for S = So. This Ieads to, Eqn 3.19 Eqn 3.20 Fig. 3.6: Idealized Local Damage Zone Seen from the Side. The damage zone surface would be expected to be cuniluiear as seen ftom the side, though as a simplification, the surface is assumed to Vary hearly as shown in Fig. 3.6. The deflection angle, 8, is determined for a given longitudinal element based on the above off-axis quantities. Eqn 3.21 Finally, it shouid be noted that the mode1 is constrained by a minimum tube length. The tube damage zone rnust remain constrained to the tube span and not extend to the tube ends. Therefore, L$.2A, = 2Lx Y Eqn 3.22 tan a- Eqn 3.23 3.2 Global Displacement Global displacement refers to components of deformation dependent on the geometry and configuration of the entire system. Namely, these cornponents include the flexural and shear components of deformation. The global displacement is not uncoupled fiom the local displacement as the state of deformation due to local displacement has an effect on the flexural and shear components of the total displacement (see Eqns 3.1 - 3 S). Beam Flexure Flexural displacement of the tube is determined by solving the beam flexure equation. However, it is necessary to account for the variable moment of inertia and the redistribution of moment due to cross-sectional changes incurred during local deformation. Taking FO to be at the fixed end of the bearn, the basic beam flexure equation is: Eqn 3.24 E, is assumed to be constant throughout the process. There is no degradation of material properties. The moment of inertia must be determined concurrently with the deformation zone in the local analysis. The moment of inertia of the tubular section in the undefomed configuration is known, and is valid in the region outside the damage zone. Chpter 3: Analytical Model 29 Eqn 3.25 Once in the damage zone, the section subjected to moment is reduced to that seen in Fig. 3 -7. The damage zone (dehed in Section 3.1) is removed fkom the moment bearïng section as it is in tension radier than compression, and thus, does not support the flexural moment. Therefore, it is excluded fkom the computation of the second moment of inertia. Fig. 3.7: Reduced Moment Bearing Section. The second moment of inertia of the section becomes, Eqn 3.26 where $ = @(x), and which reduces to Eqn 3.25 in the undefomed configuration (@ = n). It is important to recall that $ is dependent on the angle a,which is in turn dependent on the materiai properties E, Ey,G), thus representing the material's anisotropy, as well as the tubing's geometry @, t) [24]. Using Eqns 3.6 and 3.14, the following relationship arises. Eqn 3.27 which is solved implicitly for $ and shown in Fig. 3.8. Fig. 3.8: Remaining Angle for a Given Domage Zone Geometry (Eqn 3.27). The bending moment dong the tube is subject to a redistribution as the tube is deformed and the ability to bear moment of a given section is changed The basic moment equation is: Eqn 3.28 where the end moment for a fîxed-ended, non-pnsmatic beam is: Eqn 3.29 based on the superposition of bearns for indeterminate configurations Eqns 3.24 - 3.26,3.28 and 3 .B in conjunction with the appropiate boundary conditions are solved numerically using the central difference method (CDM) [41]. Although numerical integration may be considered as a means of solution, dficulties in application of the prescribed boundary conditions makes CDM, which is a finite merence method, more attractive. Also available and similar methods include the forward ciifference method and the backward difference method. CDM is more efficient than either of these other two options as it is a second order approximation rather than a fust order approximation ~411. CDM evaluates first and second derivatives of a function based on points before and after the point of interest. Eqn 330 Eqn 3311 where h is the interval between derivative points, and is taken to be regular. Thus, Eqn 3.24 becomes: Eqn 332 where, i=1,2,3,.-., iV. Eqn 3.33 In order to render this system of equations determinate, the appropriate end conditions are applied. The boundary conditions of a fixed-ended bearn are, Eqn 3.34 Eqn 335 Eqn 336 or, setting the appropnate indices, and developing Eqns 3.34 - 3.36. Y0 =O Eqn 337 Y-, - Yi = 0 Eqn 338 Y N-I - Y N+I = O Eqn 339 Chupter 3: Analytical Mode1 The final system of resulting equations is of the form, Eqn 3.40 where D is the matnx of differentials. The system of equations cmbe solved by inverting the differential matrix, Eqn 3.41 As will be shown subsequently, experimental observations indicate that a tme kedend condition is not maintained during the latter stages of the experiment. As a result, both hedand simply supported boundary conditions, considered lower and upper bounds for the flexural displacement, are solved with the appropriate boundary conditions considered in the differential matrix. The boundary conditions of a simply supported beam are, Eqn 3.42 Eqn 3.43 or, setting the appropriate indices, and developing Eqns 3.42 and 3.43. Eqn 3.44 Eqn 3.45 The end moment, Mend, is zero for this condition (cf Eqn 3.29). Beam Shear The second type of global deformation is shear deformation. It is also dependent on the change in geometry of the beam due to local deformation. However, the dependence on cross-sectional geometry is weak, and thus it is neglected. Moreover, the contribution of shear displacement to the total displacement becomes negligible as the beam becomes very compliant in flexure in response to the changing cross-sectional geometry. The shear term is Uicluded for completeness, but its role is significant ody in the initial portion of the displacement range discussed in this thesis. Again, no degradation of properties is assumed, thus the shear modulus remains unchanged. The shear deformation for the given beam is: to be evaluated at center-span, or at x = L/2. The shape factor, k, is assumed to be that of a thin circular sheli, and is also assumed constant, and the area of the cross-section of the tube is constant if the assurnption of inextensibility discussed in Sect. 3.1 is maintained. Eqn 3.47 Eqn 3.48 3.3 Load The load resulting fiom the deformation described in Sec. 3.1 is based on the strain in a longitudinal elernent in the damage zone of the tubing, in conaast to the plastic moment based analysis developed by authors analyzing steel tubes [20, 22-24]. This analysis is purely elastic, with no explicit assumption of dmage or matend degradation. Refdgto Fig. 3 -4, Eqns 3.18 and 3 -20, the strain in a given linear element is expressed as, Eqn 3.49 The resultant vertical force, exploiting symrnetry, is, Sn F = 4 ~,t1E(S) sin B(s)~s. Eqn 3.50 O Once the local displacement, AL, has reached its maximm (see Eqn 3-12}, any additional load must arise fkom another mechaaism. The global stfiess is evaluated for this deformed geometry, and increments in load now results in an increment in the global components of displacement ody. It is noteworthy that the shear stiffhess does not change throughout this andysis, as it is assumed independent of the tubing's deformed geometry . 3.4 Solution The mode1 is solved using MatLab as the programming environment. It is not necessary to solve the system of equations by iteration, but once the local displacement reaches its maximum the final state of the tube must be known in order to evduate the shear and fiexural slopes such that fuaher displacement may be computed. The computation time for the solution of a given case is minimal, requiring approximately 5 minutes on a desktop Macintosh to generate 100 points of data to give the local, flexural, shear and total displacements and the generated load. The solution (see Fig. 3.9) begins in the program beamode1.m with an assumed local displacement, AL, the incremented quantity. beamode1.m calls the subroutine 1ocal.m with AL, which cds f1.m with AL. fl.m solves for the remainuig angle 9, and returns it to local.m, which uses it to solve for the deformed geometry of the tube. 1ocal.m then cds f0rce.m with AL and the details of the deformed geomew, and f0rce.m returns the load, P, defined as F in Eqn 3.50. The load, P, and some additional geomehical information, Iro, are retunied to beamode1.m. beamode1.m then cas the subroutine beam.m with the deformed geometry and load information. beam.m cails the subroutine moment.m, which returns the redistributed moment, M(x), as dehed in Eqns 3.28 and 3.29. rn0ment.m requires that inertiam determine the moment of inertia of the section at a position dong the beam with the remaining angle $, for which f2.m solved. bearn.m then solves the CDM system of equations (Eqn 3.4 1) with calls inertia-m and thus f2.m, and beam.m Chapter 3: Analytical Model retums the center span fleddisplacement, AB, to beamode1.m. beamodelm then cds the subroutine shear.m with the load, P, and shearm solves Eqn 3.46 to retum the shear component of displacement, As. The displacements are summed by beamode1.m to obtain the total displacement, AT, and the three components of displacement, AU As and As¶ the total displacement, AT, and the load are output to a text file. Fig. 3.9: Solution Algorithm Pre-existing subroutines have been exploited in solving the mode1 when appropriate. Three such routines have been used in the solution code, namely; 'fzero', 'quad8', and 'trapz'. The solution of Eqns 3.1 1 and 3.27 for the rernaining angle, 41, is obtained using 'fzero' [4 11. 'fkero' uses a combination of bisection, secant and inverse quadratic interpolation methods to detennine the angle at which the hction is nil. The load, Eqn 3.50, is integrated using 'quad8' [41]. 'quad8' uses a Newton-Cotes 8 panel deto numencaily integrate the function. The end moment for a fixed-ended beam, Eqn 3.29, is integrated using 'trapz' [41]. 'trapz' uses a trapezoidal approximation to integrate a given function. 3.5 Sensitivity Analysis Of the parameters entering into the model, the only one that is not known explicitly is a, which in tum is dependent on two parameters, aiand C, as seen in Eqn 3.17. The angle and its evolution are known to be dependent on the properties of the material, and the geometry of the tube. An anaiytical relationship seekùig to determine this angle has not been developed for isotropie materials, and no attempt is made here to develop a relationship for orthotropic materials. Rather, the effect of the angle on a given configuration of tube will be investigated using bounding cases. A second parameter arïsing fkom experirnent is also considered: the end moment of the tubing. The effect of modfiing the end condition on the results of the model are signifïcant, and will also be investigated. Experimentally, the end condition is not maintained as the tube is irnpinged upon. Some pullout and rotation is observed as the transverse displacement becomes large. Damage Zone Angle: a The sensitivity of the model to the damage zone angle will be investigated by varying ai and C in Eqn 3.1 7, as well as the length of the tubing as shown in Table 3.1. Table 3. I: Serzsitivity Analysis Parameters. ai (rad) C L (ml d8 O O -2 - 8 0.61 0.92 0.b3 0.b4 o. Transverse Dispiacement (m) Fig. 3.10: Sensitiviq to Variable Damage Zone Angle: ai= d8and L = 0.2nz. -8 0.61 0i2 oh3 oh4 O.& Transverse Displacement (m) Fi& 3.11: Sensitivity to Variable Damuge Zone Angle: ai= d8and L = 0.4itz. Chapter 3: Annlytical Mode1 20000- : O 0.01 0.02 0.03 0.04 O. Transverse Displacement (rn) Fi'. 3.12: Sensitivity tu Variable Durnage Zone Angie: ac, = dl2 arzd L = 0.2m. 7000 6000 n SOOO 5 4000 O 3000 2000 1000 n - I O 0.81 0.à2 0.83 oh4 o. Transverse Displacement (m) Fig. 3.13: Sensitivity tu Variable Damage Zone Angle: ai = dl2 and L = 0.4nz. Transverse Displacernent (m) Fi'. 3.14: Sensitivity to Variable Durnage Zone Angle: = n/I 6 aiid L = 0.2m. O 0.01 0.02 0.03 0.64 O Transverse Displacernent (m) Fig. 3-15: Sensitivity to Variable Damage Zone Angle: a;. = dl6 and L = O.4rn. The model is run assuming fixed-ended conditions for the beam, and results are shown in Figs. 3.10 - 3.12. The model behaves weil in al1 cases, not generating any discontinuities or asymptotes, which are not expected in the response of the tubing. The value of the initial damage zone angle, cci, influences greatly the amplitude of the load response of the tubing. This is because a wider angle results in greater strahs seen in the damage zone angle, although the Chnpter 3: Annlytical Mode1 damage zone itself will be smaller (see Eqns 3.13, 3.18 - 3.20, and 3 -49 - 3 -50). By the same mechanisms, the damage zone evolution factor, C, has a great influence on the load response of the tubing, and as the factor becomes Iarger, reducing the damage zone angle, the response of the tubing is softened. In fact, the mode1 exhibits little softening, which results in the bilinear or two-mode response expected fiom the tubing, dess the damage zone angle, a,evolves. This evolution may be seen as analogous to the generation of damage in the tube. Retaining a Linearly varying oc, Eqn 3.17, results in a model that responds stiffly at fust, and with considerable softening as the tube is Merirnpinged upon. This is the qualittative response desired fiom the model. End Moment: Il&,d The end moment is known, fiom experimental observation of the end condition, to decrease fiom that of a fully hedend condition due to translation, rotation and damage production. The redistribution of moment due to the evolution of the damage zone, assuming hedends, will overestimate the moment borne at the fked end of the beam. This component of the response has been isolated in Figs. 3.16 and 3.1 8. Model results for both the fixed-ended configuration and the simply-supported con£iguration are shown in Figs 3.1 7 and 3.1 9, and in addition a mixed average solution is shown. The following damage zone angle has been chosen for this analysis: Eqn 3.51 Chapter 3: Analyticul Model Load (N) Fig. 3.16: Sensirivil)>fo End Moment (L = 0.2m): FIaural Daplacemenf. Simply Supported 0.000 0.010 0.020 0.030 0.040 O. Transverse Displacement (m) Fi'. 3.1 7: Sensifivity to End Moment (L = 0.2w Totai Transverse Displacement, Chapter 3: Analytical Mode1 - I O 580 1080 1580 2àoo 2580 3c Load (N) Fig. 3.18: Sensitiviry fo End Moment (Z = O.4m): Flexural Dkpl~cement. 1 Fixed Ended Transverse Displacement (m) Fig. 3-19: Sensitivi@ fo ~nd~oment (Z = O.4m): Total Transverse Displacement. Isolating the flexural displacement shows that the tnie effect of changing the end condition is in the change brought to the flexural stifThess of the tube when the cross-section is fully crushed (Figs. 3.16 and 3.1 8). This results in markedly different responses in the second mode of the beam's response as seen in Figs. 3.17 and 3.19, particularly at the shoaer length where local deformation wili be exhausted more quickly as a mode of deformation. The approximation used for the tube's end conditions codd be improved using a two- spring model, one translational and one rotational, and a fiction criterion for the fixture. Chapter 3: Analytical Model 3.6 Dynamic Analysis A dynamic model is developed in order to evaluate whether the tests detailed in Chapter 2 may be approximated by a SDOF system as illustrated in Fig. 3.1. This implies that the response of the structure is independent of the beam mas. Complete dynamic modeling would require the incorporation of the contact stiffhess and material rate dependence into the existing model. A simplified sp~g-massmodel that neglects damping, similar to those presented in Abrate [10], is deemed appropriate for this analysis and is illustrated in Fig. 3.20. It represents a two degree of fieedom systern, where kg and kl represent the global and local stifiesses respectively, Mb and Mi represent the equivalent beam mass and the impactor mass respectively, and xi and x2 are the respective motions of the two masses. V is the impact velocity. The global stiffness, kg, consists of flexure and shear stiffriesses, whereas the local stSess, kl, is the cross-sectional deformation stifiess. Fig. 3.20: 2DOF Spring Model. Generally, spring-mass models, such as those presented in Abrate [IO], adopt a load vs. displacement fùnction of the following form: F = k(xl -x2)" Eqn 3.52 For the purposes of this analysis, a linear approximation, as seen in Eqns 3.53-54, will be used to bound rather than accurately rnodel the response of the tube to a dynamic Ioad. Thus, The equations of motion for the system are: Eqn 3.53 Eqn 3.54 These equations may be rewritten in the following form: Chupter 3: Amlyticul Mode1 Eqn 3.55 Eqn 3.56 where D is the dzerential operator. A solution is assumed of the foilowing form: COS(U~~)+ c2sin(m1t)) + c3cos(m2t) + c4 sin(~~f)).Eqn 3.57 [::] = [41( Subject to the initial conditions: xi (O) = x,(O) = O Eqn 3.58 x,(O) = O,X,(O) = v Eqn 3.59 Yields the foilowing parameters, Eqn 3.60 Eqn 3.61 Eqn 3.62 Eqn 3.63 In order to determine the equivalent beam mas, Mb, it is necessary to look at the center span deflection of a point loaded fked-ended beam and a continuously loaded hed-ended beam. The distributed mass of the beam, m&, will yield the same center-span deflection as the equivalent mass, Mb, applied as a point load. Eqn 3.64 Eqn 3.65 Comparing Eqns 3 -63 and 3 -64, the equivalent mass of the beam becomes, M, = 5. Eqn 3.66 2 The results of this analysis will be studied in Chapter 4, foilowing a quasi-static experimental investigation of the tubing's response. The stiffriesses, kg and kl, wiIl be deterrnined from the static model, and fiom static experimental observation as detailed in Section 4.3 and in Tables 4.1-2. The masses, Mb and Mi, and velocities, V, will be chosen to simulate experimental conditions and provide bounding approximations to the dynarnic response of the tubing, while maintainhg a constant input kinetic energy. Chapter 3: Analytica l Model Chapter 4 Experimental and Model Results Experiments have been performed as detded in the test grid at the end of Chapter 2. These tests were concerned with the effect of several parameters including the effect of impactor geometry, the effect of beam length, the effect of loading rate in equal energy impacts, and finally, the effect of additional transverse reinforcement to the tubing. The model is evaluated using the data acquired during the experiments. The parameters discussed in Chapter 3 are estimated, to evaluate the performance of the model versus the experimental data. The results are good, and confirm the validity of the model and the associated anaIysis. 4.1 Quasi-Static Testing Al1 quasi-static tests were performed using the MTS hydraulic test machine detailed in Chapter 2. The test procedure and test conditions are dso detailed in Chapter 2. The parameters varied include the tup width and tube length for quasi-static tests. The f~stor narrow impactor is of length 2%, twice the outside radius of the tubing. The second impactor is of length superior to the flattened width of the tubing, or half the outside circumference, rrk. The length of the hibing is varied fiom 8 inches (203.2 mm) to 20 inches (508.0 mm). Chapter 4: Experimental and Model Results Narrow Tup - 8 Inch Tube Fig. 4.1: 8 Inch Tube at a) 10 mm, b) 20 mm and c) 30 mm Transverse Displacement. Quasi-static tests performed using the narrow tup demonstrate a failure sequence associated with punchue of the tube surface. This is similar to that observed by Pang & Kailasam [16] for sharp edge impactors, in that the zero radius of curvature edge of the impactor punctures the target surface. Fig. 4.1 shows that at 20 mm bansverse displacement, the surface of the tube has been punctured by the tup. Also of note on this senes of photographs is the significant amount of end rotation of the tube in Fig. 4.k at 30 mm transverse displacement. This is the result of slip and failure in this region. Fig. 4.2 in tushows the load vs. displacement result, which is the average of 3 tests. Quick initial loading is observed, followed by a constant load segment broken by sharp unloading sequences. These unloadings are thought to correspond to the puncture of the Chapter 4: Experimental and Mode1 ~esults top surface of the tubing occ~gbetween 13 and 22 mm, and puncture of the bottom surface, which occurs at approximately 35 mm. Transverse Displacernant (m) Fig. 4.2: Load vs. Displacement Curve for 8 Inch Tube Using Narrow Impactor. Narrow Tup - 16 Inch Tube OLnCUi~u,~* 0800~O 0- - "NO8 Transverse Displacement (rn) Fig. 4.3: Load vs. Displacement Cuwe for 16 Inch Tube Using Narrow Impactor. Chapter 4: Experimental and Mode1 Results The behavior of the 16 inch (406.4 mm)tube is similar to that of the behavior of the 8 inch (203.2 mm) tube. Quick initial loading is observed, followed by a 1ow gradient gradua! loading. The puncture failure of the top surface of the tubing occurs much Iater, at approximately 37 mm as seen in Fig. 4.3. This is due to the more gradual accumulation of local displacement in the longer tubes, which is in turn due to the lower flexural stiffness of the beam associated with its increased length. Wide Tup - 8 Inch Tube Fig. 4.4: 8 Inch Tube at a) 10 mm, b) 20 mm and c) 30 mm Transverse Displacement. Quasi-static tests performed on 8 inch (203.2 mm) hibing using the wide tup do not induce puncture type fadures as had been seen in narrow tup tests. The height of the cross-section is significantly reduced beneath and adjacent to the impactor, as can be seen in the series of photos in Fig. 4.4, and this is what is referred to as the local displacement Chapter 4: Experimental Md Mode1 Results of the impactor. Fig. 4.4 also shows signincant end rotation of the tubing at 30 mm transverse displacement, due again to slip and failure in the region of the clamp, indicating that the moment beariog ability of the tubing is less than that of a true fked-ended beam. Fig. 4.5 shows the average of 3 tests pei.formed under these conditions. Quick initial loading is observed and the behavior to up to 10 mm transverse displacement is the same as that observed for the narrow tup in Figs. 4.1 and 4.2, but the subsequent region is characterized by a low gradient loading, which differs signincantly from the narrow tup. The wide tup has effectively eliminated the puncture mode of failure fiom the tests. This test will serve as the baseline for the rernainder of the tests. The tubing has undergone a transition fiom locally dominated to globally dominated behavior. The loads at the transition point wiil serve as cornparison points in future tests. The transition load for this case is 3 kN and the peak load at iïua.1 displacement is 5.4 W. Transverse Displacement (m) Fig. 4.5: Load vs. Displacement Curve for 8 Inch Tube Using Wide Impactor. 4.1.4 Wide Tup - 12 Inch Tube Quasi-static tests perforrned on 12 inch (304.8 mm) tubing using the wide tup demonstrate a behavior similar to that observed for 8 inch (203.2 mm) tubing. Fig. 4.6 shows that there has been significant reduction of the load for both segments of the load Chapter 4: ExperUnental and Mudel Results displacement curve. The reduction in the transition load, where the tube goes fiom the locally-dominated to globdy domuiated regime, occurring at 10 mm transverse displacement, is of approximately 25%, and the reduction of the peak load at 40 mm transverse displacement is also of approximately 25% as compared to the 8 inch (203.2 mm) tubing. Transverse Displacement (m) Fig. 4.6: Load vs. Displacement Cuwe for 12 Inch Tube Using Wide Impactor. Wide Tup - 16 Inch Tube Quasi-static tests performed on 16 inch (406.4 mm) tubing agaùi demonstrate sirnilar behavior. Fig. 4.7 shows end rotation of the tubing at large transverse displacement, due again to slip and fdure in the region of the clamp, thus the moment bearing ability of the tubing is sti1l comprornised at this length. Fig. 4.8 shows the average of 3 tests. There has been signincant reduction of the load for both segments of the load displacement cuve when compared to the results for 8 inch (203.2 mm) tubing. The reduction in the transition load is of 35%, and the reduction in the peak load is of 45%. Chapter 4: Experimental and Mode1 Resulrs , Fig. 4.7: 16 Inch Tube at a) 10 mm, b) 20 mm and c) 38 mm Transverse Displacement. O Transverse Displacernent (m) Fig. 4.8: Load vs. Displacernent Curve for 16 Inch Tube Using Wide Impactor. 4.1.6 Wide Tup - 20 Inch Tube The basic behavior for quasi-static tests performed on 20 inch (508.0 mm) tubing remains unchanged. Fig. 4.9 shows the average of 3 tests, with a reduction of approximately 45% Chapter 4: Ejcperirnental and Model Resulrs in both the transition and peak loads when compared to the results for 8 inch (203 -2 mm) tubing. Transverse Displacement (m) Fig. 4.9: Load vs. Displacement Curve for 20 Inch Tube Using Wide Impactor. 4.2 Supported Tube Tests The quasi-static tests performed in Section 4.1 dernonstrated in all cases a two-mode behavior for the tube. The analysis developed in Chapter 3 assumed that this behavior was the result of a locdy-dominated initial mode and a globaily-dominated second mode in the response of the tube. This assumption was the same as had been made by authors considering steel tubhg in similar loading conditions. As a means of verification of the regions which are dorninated by the local or global modes of displacement, tests in which the tube is supported continuously dong the bottom surface are perfomed on the tubing. These tests were performed on 8 inch specimens supported on a 1 inch thick steel plate and loaded by the wide tup. Only displacement of the top and bottom surface of the tube, with respect to the neutrai axis of the tube, can occur during these tests. The local displacement must be isolated from the displacement of the bottom surface, and theri these results may be compared to the previous. In order to isolate the local displacement kom the total displacement of the hydraulic test machine's cross-head, the deformed cross-sectional geometry of the tubing developed in Chapter 4: ExperVnental and Model Resul fs 54 Chapter 3 is reexamined (see Fig. 3 -2). Defîning the movement of the hydraulic test machine's cross-head as AXH, which includes the local displacement, Ar, and the movement of the bottom surface, and H as the total height of the deformed section at its mid-line, Eqn 4.1 Where R is the radius of the undefonned cross section, and 2R is the midine height of the undefonned cross-section. Using the angular relationships developed n Chapter 3, the remaining angle 9, as expressed in Eqn 4.2, may be solved irnplicitly, as were Eqns 3.1 1 and 3.27, with the results shown in Fig. 4.10: Eqn 4.2 Then the local displacement is known as defined in Eqn 3.1 1. Eqn 4.3 Fi,.4.10: Inrplicit Solution of Remaining Angle, # (See Eqrz 4.2). Chapter 4: Ejcperirnental and Mode1 Results The tests are compared to the test results of 8 inch (203.2 mm) tubing, as this is the condition which sees the most severe crushing of the cross-section under the tup. The cornparison reveals that the local mode of displacement is dominant at the onset of loading with very good superposition of the supported and fixed-ended tests in the initial 8 mm of displacement (see Fig. 4.1 1). Thus, it can be concluded that the behavior of the beam is a two-mode behavior that is initidly locdy dominated. Transverse Displacernent (m) Fig. 4.11: Locai Displacement in a Supported Configuration vs. Fixed-Ended Response for 8 inch Tubing. 4.3 Dynamic Testing Ai1 dynamic tests were performed using the DWIT as detailed in Chapter 2. The test procedures and test conditions are also detaired in Chapter 2. The input energy to the system was maintained at approximately 150 1 by varyiog the mass of the chage and adjusting the impact velocity accordingly. All testing was perforrned using the wide impactor, and only tubing of 8 inches (203.2 mm) in length was used due to equipment cons~aints. Chapter 4: Experimental and Mode1 Results Dynamic Analysis A dynamic analysis is performed as developed in Chapter 3 in order to cobthat the response of the tubing may be approximated as a SDOF system. Rather than perform a non-linear analysis, the solution is bounded using hear approximations for the initial sp~gconstants, using an experimentally O btained local stiaess, kl, fiom the locally- dorninated portion of the response (see Fig. 4.9, and an analytical solution for global sti&ess, k. The final conditions assume a very high local stiaess, ki, which corresponds to the fully crushed tube, and a global stiffhess, kg, obtained &om the globdy dominated portion of the response (see Fig. 4.5). Other quantities required are the equivalent beam mas, Mb, the impactor mass, Mi, and the impact velocity, V, ail of which are already known. Al1 impacts analyzed have the same initial kinetic energy of 150 J. Table 4.1: Initial Conditions for Dvnamic Analvsis. Case kn Wm) kl CNW Mb(kg) Mi (kg) V 1 3.0x106 1.2x106 0.070/2 0.07012 92.5 II 3.0x106 1.2x106 0.07012 10 5.5 III 3.0~1o6 1.2x106 0.07012 15 4.5 IV 3.0x106 1.2x106 0.07012 33 3 .O Table 4.2: Final Conditionsfor Dynamic Anaiysis. Case ke (N/m) kl (NW Mb Ocg) Mi Org) V WS) 1 87.5x103 lxlo9 0 .O7012 0.070/2 92.5 11 87.5x103 lxlo9 0 .O7012 10 5.5 III 87.5x103 lxlo9 0.07012 15 4.5 IV 87.5~1o3 lxlo9 0.070/2 33 3 .O Prior to consideration of the cases II, III and IV,which represent actual experiments to be performed, Tables 4.1 & 4.2 present a hypothetical case, case 1, in which the effective beam mass and the impactor mass are the same with an initial impact velocity of 92.5 m/s. The results of this analysis, shown in Fig. 4.12a, clearly demonstrate that the response of the beam and the response of the impactor are out of phase, and thus would require that the problem be treated as a 2DOF system. Chapter 4: Experhental and Mode1 Results Figs. 4.12b-15b show two superimposed curves for the final conditions of the tube. This implies that there is iittle motion of the surface toward the neutral axis. This result is reasonable as the cross-section has been assumed completely crushed. Time (s) Time (s) a) b) Fig. 4.12: Dynamic Anuiysis of Case 1.- a) Initial Condition, 6) Final Condition. CU 6 s O O Time (s) Time (s) Fig. 4.13: Dynamic Andysis of Case II: a) initiai Condition, 6) Final Condition. Figs. 4.13-15, corresponding to cases &IV, show that for the masses used in the dynamic tests performed, sorne high fiequency oscillation will occur in the response of both the Chapter 4: Experimental and Mode2 Results neutral axis and tube surface. However, the amplitude of these secondary vibrations is small. Thus the tests under consideration can be adequately modeled as SDOF systems, and the displacement history of the impacts may be obtained by double integmtion of the acceleration history, or Ioad history, of the impacts, and the load vs. displacement response of the tubes subjected to impact may be obtained. 1 - Neutral Axis 1 1 - Neutral Gis 1 Tirne (s) Time (s) Fig. 4-14: Dynamic Analysis of Case III: a) Initial Condition, 6) Final Condition. 1 - Neutral Axis 1 0.060 A E 0.050 zc a 5 0.040 O Q I 8 0.030 -P a 2 0.020 >aJ E 0.010 E' I- 0.000 9OorM8szze 0 agaz~g Time (s) Time (s) a) 6) Fig. 4.15: Dynamic Analysk of Case W: a) Initial Condition, 6) Final Condition. Chapter 4: Etperintental und Mode1 Results 4.3.2 Dynamic Test Results The parameters varied include the impact velocity and mas, and these are adjusted so as to maintah a constant impact energy. The impact velocities are 3.0,4.5 and 5.5 m/s with respective impact masses of 33.1, 15.1 and 9.9 kg. The voltage history cuves have been integrated and the load vs. displacement response of the tube is shown in Figs. 4.16-1 8. 8000 7000 6000 sA 5000 4000 O gJ 3000 rnS3.l kg, v=3.0m/s 2000 - Transverse Displacernent (m) Fig. 4.16: Dynamic Load vs. Displacement: m = 33.1 kg, v = 3.0 m/s. Transverse Displacement (m) Fig. 4.17: Dynamic Load vs. Displacement: m = 15.1 kg, v = 4.5 mfs. Chapter 4: Experimental mid Model Results Fig. 4.16 shows the average of 3 dynamic tests performed with a carriage mass of 3 3.1 kg and an initial impact velocity of 3.0 m/s. The locally-dominated initial portion of the cuve has remained nearly unchanged, but the globally-dominated second portion of the curve shows signiticant increase in the generated load. The peak load has increased by approxirnately 35%. This increase is Likely to be due to a dynamic hcrease in strength of the mat& modfiing the geometry of the damage zone angle and its evolution. Fig. 4.17 shows the average of 3 dynamic tests performed with a carriage mass of 15.1 kg and an initial impact velocity of 4.5 ds. The locally-dominated initiai portion of the curve is again essentially unchanged, but the giobally-dominated second portion displays dependence on the rate of loading, with a peak load increase of 20%. The increase is kely to be due to the same dynamic increase in matrix strength suggested for the previous test, but the peak load increase would be expected to increase Merif this were the only parameter in play. Transverse Displacement (m) Fig. 4.1 8: Dynamic Load vs. Displacement: m = 9.89 kg, v = 5.5 m/s. Fig. 4.18 shows the average of 5 dynarnic tests performed with a caniage mass of 9.9 kg and an initial impact velocity of 5.5 ds. Again, the locally-domuiated initial portion of the curve is unchanged, but this time, the globdy-dominated second portion of the curve is also unchanged. This may indicate that the response is rate independent, but the preceding tests would refute that conclusion; therefore, there must be a change in the failure mode of the tubing as loading rate is increased. Chapter 4: Expeninental and Model Results 61 8000 7000 6000 -z 5000 Y = 4000 5 3000 2000 - m=IS.l kg, v=4.5m/s 1000 , , , - m=33.l kg, v=3.0m/s Transverse Displacernent (m) Fig. 4.19: Rate Cornparison of Load vs. Displacement Response of an 8 Inch Tube. Comparing the response of the tubing subjected to the 3 dynamic loading conditions in Fig. 4.19, one must conclude that there is no change in the locally-dominated response of the tubing. However, the globally-dominated mode is responding inversely to the loading rate, which seems initialIy to be counter-intuitive. Dynamic effects have been eliminated as the cause of such a response; thus, the conclusion is that there is a new mode of failure comùig into play. This would appear to be a transition from a tensile failure to a shear failure as tests by Boukhili et al. suggest [29]. They observed two dflerent rate effects in the response of pultruded polyester / glas. The longitudinal strength of pultruded polyester / glass increases with increasing loading rate, while the shear strength increases up to a loading rate of approximately 0.1 mk, beyond which there is a decreasing trend. Visual inspection of the samples used in testing reveaied no clear dues as to the change in failure mode that may be taking place. However, this does not indicate that the mode of failure is not shifting to shear as suggested. Chapter 4: fiperimental and Mode1 Results 1 - m=lS.lkg, ~=4.51n/s - m=33.l ka. v=3.0rn/s ooc Transverse Displacement (m) Fig. 4.20: Energy vs. Displacement. As confirmation that the energy input into the system is the same for each loading condition, and that the calibration of the drop tower is accurate, the curves in Fig. 4.19 have been integrated. This is a numericd integration performed as follows: E=~F&=~FAX Eqn 4.4 Fig. 4.20 demonstrates that the energy input into the system is indeed the sarne for each loading condition. 4.4 Model Results Experimental results demonstrate that it is difficult to maintain a Mly fixed-ended condition when the tubing is subjected to large transverse displacements at its rnidspan. Lifshitz et al. [37] experîenced sunilar difficulty using fuced ended beams due to slip at the ends. Since small end rotations result in large deviations in displacement, they reverted to using simply supported beams. Jones &Shen [24] noting sirnilar difnculties when testing steel tubing suggested that a fiction criterion for the behavior of the tubing within the mecould be developed that would lead to an appropriate end moment function, but they have made no attempt to develop such a relationship. For the purposes of this Chapter 4: Experimental and Model Results analysis, the model is solved using the numerical average of the fully fixed-ended condition and the simply supported condition, as there is no suitable expression for the behavior of the end moment bction of the beam- As shown and discussed in the sensitivity analysis of the model to the end moment in Chapter 3, using the average end condition affects only the flexure cornponent of the global displacement, and this effect, whiIe significant, is moderate. The model may be evaluated against the test data for the various lengths tested in Section 4.1, in order to evaluate the degree of precision obtained and the sensitivity to beam length. Figs. 4.21 to 4.25 show these results. The model is run using the following damage zone evolution equation: Eqn 4.5 Transverse Displacement (m) Fig. 4.21: Model vs. Esperimental Behaviorfor an 8 inch (0.2032m) Tube. Chapter 4: Experimentd and Model Results Transverse Displacement (m) Fi&. 4.22: Modei vs. Ekperimentui Behavior for a 12 Inch (0.3048m) Tube. Transverse Displacement (m) Fig. 4.23: Model vs. Ewperimeirtal Behavior for a 16 Inch (0.4064m) Tube. Chapter 4: Experimental Md Model Results Transverse Displacement (m) Fig. 4.24: Model vs. Experimerital Behavior for a 20 inch (0.5080rn) Tube. 0 000 0 0 0 00- Transverse Displacement (m) Fig. 4.25: Model vs. Experimental Behuvior for a 20 Inch (0.5080m) Tube with Fdly Fked End Conditions. The mode1 reproduces the behavior of the beam quite well over the full range of displacement, but behaves particularly weil for the shorter tubing lengths. Figs. 4.2 1 & 4.22 demonstrate a good reproduction of the initial locally-dominated portion of the curve and of the globally-dominated second portion of the curve. Fig. 4.23 displays a widening discrepancy in the locally-dominated mode of the tube response, and Fig. 4.24 displays a Chapter 4: Experimentul and Model Resulrs significant discrepancy in both the locaily and the globally dominated portions of the curve. This is attributable to the end condition assumed in the solution. As the beam lengthens, the end rotation observed for a given transverse displacement decreases (contrast Figs. 4.4 and 4.7) resultïng in a moment distnbution along the beam more closely approximating that of a fked-ended beam. To illustrate thïs, the model was run again using fked end conditions and Fig. 4.25 shows much better agreement for the second, globally-dominated portion of the load vs. displacement curve for a 20 inch beam. 4.5 Model Predictions The previous section demonstrated that the mode1 is capable of reproducing the load vs. displacement response of the tubing with good accuracy. The damage zone angle, a, is known to depend on geometry of the tubing. Assuming that a's sensitivity to geometry is limited to the diameter to thickness ratio, the model may be used again to generate predictive curves for commercidy available tubing of the same material with the same diameter to thickness ratio. In this case, using the average diarneter of the tubing, the ratio to be respected is 7: 1. Table 4.3 lists comrnercially available tubing, including that which was tested, and their diameter to thickness ratios. TabIe 4.3: Modeled Ming Geometrr'es. Outside Diameter (in.) Wail Thickness (in.) a,& 1.O . 1/8 7: 1 2.0 114 7: 1 3 .O 3/8 7: 1 The model has been run for the sections listed in Table 4.3 for various lengths of the hibing using the damage zone evolution equation as expressed in Eqn 4.5. Chapter 4: Ejrperimental and Model Results Transverse Displacement (m) Fig. 4.26: Tube Response at Various Lengths with D=1'', and t =I/8"- Transverse Displacement (m) Fig. 4.2 7: Tube Response at Various Lengths with 0=-2 ': and t =I/4': Chapter 4: Experimental and Model Results Transverse Displacement (m) Fig. 4-28: Tube Respom at Various Lengths witlt 0=3': and t =3/8". Transverse Displacement (m) Fig. 4.29: Tube Respome at Various Lengfhs with D=4'', and t =1/2". The responses of the four selected sections are quite similar. The response displays the two-modes of displacement, as required, with a sofiening transition fiom locally to globally dominated behavior as the beam lengthens. Chapter 4: Experimental und Mode1 Results 4.6 Additional Transverse Reinforcement The objectives of this study, stated in chapter 1, inctuded an experimental and andyticai investigation of the behavior of a laterally-loaded pultruded fiberglass tube, and the evaluation of the performance of such tubing in comparison to traditional materials. The results of the experiments and model clearly indicate that energy absorption of the tubing is iimited by its low local s~ess,which is a consequence of the materiai's low transverse strength; therefore, additionai reinforcement, in the form of one Iayer of woven fiberglass, is added to the tubes in order to improve their resistance to crushing whde having liale effect on the tubing's linear density. Results fkom this additional experimental data and the fitted model are extended for a comparison against experirnental steel tube data obtained by Reid & Goudie [20]. The comparison will be made on an equivalent linear density basis. The additional reinforcement used was a single layer of [0°/90']epoxy / glass weave. It was cut to size and applied to the outer surface of the tubing. The tubing was then vacuum bagged and cured at a temperature of 175'F (80°C)for 7 hours. The thickness added to the tube with the addition of the woven layer is of approximately 0.25 mm. 4.6.1 Experimental Results Transverse Displacement (m) Fig. 4.30: 8 Inch Standard vs. Reinforced Tubing under Quasi-Stalic Test Conditions. Chapter 4: fiperimental and Model Results 70 Fig. 4.30 demonstrates that additional transverse reinforcement results in significant &enhg of the beam response in the locally dominated region, increasing the transition load, at 10 mm transverse displacement, fiom 3 kN to 4.4 kN. This is to be expected as this response mode is dependent on the-transverse properties of the material. The fial peak Ioad is only modestly increased, but the overd energy absorbed is greatly increased. 4.6.2 Mode1 Results A new relationship for the damage zone angle is adopted for the additionally reinforced tubing. The aspect ratio (width to length) of the damage zone will be greater with the parameters in Eqn 4.6 than in Eqn 4.5, thus the damage zone will not have the same extent as that of the previous analysis. Eqn 4.6 Transverse Bisplacement (m) Fig. 4.31: Evperimental vs. Mode2 ResuCtr for Reinforced Tubing. The model is compared to experimental data in Fig. 4.31. Cornparison of the model against the test case demonstrates good agreement for both the locally and the globally dominated portions of the beam response. The model is run agah for various lengths, and Chapter 4: Experimental and Mode1 Results 71 results are illustrated in Fig. 4.32- The behavior is similar to that seen in Figs. 4.26-29, demomtrating the same stability in response. Transverse Displacement (m) Fig. 4-32: ModeC Behaviar for Various Lenghs of Reitzforced Tubing. 4.6.3 Cornparison to Steel Data Steel and fiberglass tubes are compared on the basis of equivalent linear density. Using available experirnental data fiom tests performed on fked-ended steel tubes [20], the following tubing geometries are compared. Table 4.4: Steel and Fiberpfass Tubina Geometries. Steel .- - -- - . .. -. - .- -- .. . Outside Diameter, D (m) 0.0508 0.0508 1 WaU Thickness, t (m) 1 0.00 16 0.008 1 1 Lengths, L (m) 1 0,305, 0.457 1 0.305, 0.457 1 Chapter 4: Experhental and Model Results Transverse Oisplacement (m) Fig. Beh avior of Fiberghss Contrasted with Steel for a 12 Inch (305 mm) Tube. Results in Figs. 4.33 and 4.34 indicate that on a weight basis, fiberglass is a viable alternative to steel. At both lengths, the fiberglass tube has approximately the same response as the steel tubing with only one layer of woven fiberglass added to the pultruded tubing. Tubing with additional transverse reinforcement, as can be produced with pultrusion or filament winding, should provide significant increases in absorbed energy. Transverse Displacernent (m) Fig. 4.34: Behavior of Fibergiass Contrasted with Steel for a 18 Inch (457 mm) Tube. Chapter 4: Experimental and Mode1 Results Chapter 5 Conclusions and Future Work This investigation's objectives consisted of developing static and dynamic test methods, and an andytical mode1 capable of accurately generating the load vs. displacement response of a pultnided fiberglass tube in a fked-ended, laterally-loaded configuration. Once the mode1 was experimentally validated and determined to perform adequately, it was used for a technical feasibility study of the substitution of composite tubing for steel tubing in a simple energy absorbing structure. 5.1 Test Method The test method required the development of nxtmiog capable of restricting translation and rotation of the tubing consistent with a My-fixed configuration. The fixturing designed resembles that which had been used by other authors in the testing of steel tubing in the same configuration. The nxhiring consists of end clamps with fitted end plugs, meant to iimit the translation, rotation and the propagation of damage to the gage region of the tubing. Various test parameters are investigated to detennine their role in the response of the tubing. This is achieved by varying one or more of these in a series of tests. Parameters varied during the tests include the length of the cylindrical indentor, the length of the tubing and the velocity of impact. In addition, filly-supported tests are added to help Chapter 5: Conclmions and Future Work 74 isolate the nature of the displacements observed in the response of the tubing. Quasi- static tests were performed using a MTS hydrauiic test machine at 5x10" ds, whereas dynarnic tests were performed on the DWIT at various velocities. Tests performed on fully-supported tubing cobed that the initial portion of the response curve was attributable to the local mode of displacement. These results were very conclusive. Fixture Performance Diaculties in lirniting rotation at the fixed end of the tubing has been experienced by many authors as the transverse displacements become large. In these experiments, significant rotation was aiso observed at the tubing's ends during quasi-static testing,. This was made obvious in the photographic record in Chapter 4. Translation was essentially eliminated as a degree of fieedom for the tubkg, though some is observed in the £inai stages of displacement and is found to coincide with the rotation of the tubing within the socket. The propagation of the damage zone is effectively stopped at or before the hedend of the hibing, depending on the span length, but some shear damage and cracking is observed at the ends of the tube. Where dynamic testing is concemed, no photographic record was obtained due to equipment limitations. The behavior is expected to be similar to that observed in quasi- static testing with significant rotation at large transverse displacements. Observations of the final state of the fixture and specimen, reveal a minimuni amount of translation in the fixture, and there appears to be more damage in the irnpacted specimens than in the quasi- statically loaded specimens. Effect of Indentor Length Two indentor lengths were used. The fist indentor was of length equal to the outside diameter of the tubing, 2&, and the second indentor was of length superior to the flattened width of the tubing, or half the circwnference, x&. Chapter 5: Conclusions and Future Work The indentor geometry plays a signincant role in the mode of failure generated in the tubing. The narrower tup induces a puncture type failure in the tube regardless of tube length once a threshold load of approximately 3500 N has been reached. This failure is attributable to the state of stress at the edge of the impactor once the fidl width of the impactor is in contact with the flattened tube section. The zero radius-of-curvature edge of the impactor punctures the tube in tum at the top surface and subsequently at the bottom surface. 5.1.3 Effect of Tube Length The length of the tube was varied from 8 inches to 20 inches (203 -2 mm to 508 .O mm) in 4 inch (101 -6 mm) incrernents. As the tube length is increased the total response of the tubing is observed to sofien. The two-mode response of the tubing is maintained with an initial locally-dominated response followed by a globaily-dominated response. The transition fkom local to global occurs at approximately 10 mm of transverse displacement in all cases, but the threshold load is seen to decrease fiom approximately 3000 N to 1600 N at the shortest and longest lengths respectively. The mechanism responsible for the reduction in this threshold transition load is decreased flexural and shear stiffriess due to increased length, which plays on both the flexurai and shear components of the global mode of displacement, coupled to a significant local damage zone capable of reducing the beam's ability to support moment. Effect of Loading Rate Dynamic tests were performed on the DWIT at velocities of 3.0, 4.5 and 5.5 m/s while varying the impact carriage mass to maintain a constant initial kinetic energy. These tests revealed that the tubing's response is rate-dependent for equivalent energy impacts with respect to quasi-static impacts. Fwther, this rate-dependence does not manifest itself in both the locally-do~ninatedand globdy-dominated modes of displacement, but only in the globally-dominated mode of displacement. At the Iowest impact velocity, 3.0 ds, the tests revealed a signincant increase in peak load (35%) over the quasi-static results. Tests at 4.5 m/s also demonstrated an increase in peak load, although this time, the increase was of only 20%, and the last tests at 5.5 mis demonstrated no appreciable peak Chapter 5 Conclusions and Future Work 76 load increase. The tests were designed to minimize dynamic interaction, as cobedb y a dynamic analysis of the tubing thus the system can be confidentiy approximated as a SDOF system, so dynamic effects are not the source of this apparent contradiction. There must be concurrent rate-dependent mechanisms. In fact, the initial increase in the load is atûibuted to a strength increase in the transverse, matrix dominated, direction of the material modi@ing the geometry of the damage zone, and the subsequent decrease is attributable to a decrease in shear strength of such composite materials at higher loading rates. These cornpethg effects are thought to be causing the initial increase in peak load with the subsequent decrease observed. 5.2 Mode1 Behavior The model developed has been tested against several geometries, and found to behave quite weli in predicting the load vs. displacement response of the hibing. Once the damage zone angle and the end conditions have been weli established, the model is capable of generating the load vs. displacement cwe quickly and accurately and to very large transverse displacements. Difficulty in determinhg the true end conditions of the tubing does impede the behavior of the model, although the approximations made have yielded good results. Brunet & Nemes [43] compared this analyticai solution to a finite-element simulation incorporating a rate-dependent continuum damage model 1191. The analytical solution required approximately 5 minutes on a desktop Macintosh where the finite-elernent sindation required 10 hours of CPU time on an Hewlett Packard 720 machine, all while providing superior results in the quasi-static loading condition. 5.3 Cornparison to Steel Tubing The model was subsequently modified to generate the load vs. displacement response of a tube incorporating one layer of weaved cross-ply as additional transverse reinforcement. This additionai transverse reinforcement resists local displacement by increasing the tube's resistance to cross-sectional cnishing, and delays the associated beam softening due to the reduced moment of inertia of the crushed section. The threshold transition load Chapter 5: Conclusions and Future Work 77 fiorn locally-dominated to globaUy dominated behavior is increased fÎom 3000 N to 4400 N, and the model regenerates the response with good precision. The model is then used to compare the response of the tubing to that of steel tubes as observed by Reid & Goudie [20] on an equivalent linear density basis. The addition of transverse reinforcement to the fiberglass tubing would indeed make this material a cornpetitive alternative to steel in terms of mechanical performance and energy absorption. Production of similar tubes using either pultrusion or marnent windîug should provide an adequate design alternative for applications requiring energy absorption in a transverse loading condition. 5.4 Future Work Further research on this problem shouid focus on the integration of the analytical model into a dynamic model. This would require accounting for the rate-dependency of the material, particularly in the transverse direction. This could be integrated relatively easily in the existing routine, and would manifest itself by modifying the geometry of the damage zone. To this end, there would need to be more research in the factors ùifluencing the geometry of the damage zone. Additional research in the parameters governing the damage zone would have to investigate the role of the material properties, as stated above, and also the role of the tube section properties. More specifically, the two ratios thought to define the damage zone angle are the ratio of moduli, material based, and the diameter to thiclmess ratio, geometry based. Work could also be done to develop an analytical solution for the end moment incorporating a fiction criterion for the fixture. A two spring model, one translational and one rotational, may be adequate to accomplish this task. As it stands, the model must be used judiciously for geometries and materials similar tu those experimentally investigated. Further experimental investigation of both of these parameters will dow the model to be used in a predictive manner for a variety of composite materials in a variety of geometries. Chapter 5: Conclusions und Future Work References Jones, N., "Structural Impact", Science Progress, Vol. 78, 1995, pp. 89-1 18. Magill, W.C., "Expanding the Niche: The Next Step for the North American Pulûxsion Industry", GZass Indusîry, Jan. 1995, pp. 22-23. Nakada, I., Haug, E., "Numerical Simulation of Crash Behavior of Composite Structures for Automotive Applications", Matériaux er Techniques, 1992, pp. 3 3- 38. Margolis, J.M., "Advanced Composites for Airfi.arnes and Car Bodies", Chernical Engineering Progress, Dec. 1987, pp. 3 0-43. Ashley, S., "Composite Car Structures Pass the Crash Test", Mechanical Engineering, Dec. 1996, pp. 59-63. Thornton, P.H. and Jeryan, R.A., "Crash Energy Management in Composite Automotive Structures", International Journal of Impacr Engineering, Vol. 7, 1988, pp. 167-180. O'Brien, K.?'. and Crincoli, J.F., "Long Strand Fiber Reinforced Engineering Thermoplastics: Applications in Automotive Marketstt, RETEC, 1987, pp. 33 1- 335. Taylor, S.R., "IndustriaVCommercial Applications for Pultmded Thermoplastics Compositesr1,24'h International SAMPE Technical Confireence, Oct. 1992, p p. T80-T88. References 9. Anonynous, "Pultruded Composites: The Alternative Structurai Mate rial " , Engineering Materials and Design, Vol. 3 1, 1987, pp. 54-57. 1o. Abrate, S., "Impact on Laminated Composite Materialsu, AppZied Mechanics Review, Vol. 44, 1991, pp. 155-190. Il. Cantwell, W.J. and Morton, J., "The Impact Resistance of Composite Materials - A Review", Composites, Vol. 22, 1991, pp. 347-362. 12. Rotem, A., "Residual FIexurai Strength of FRP Composite Specimens Subjected to Transverse Impact Loading", SAMPE Journal, Mar JApr. 198 8, pp. 19-25. 13. Nemes, J.A. and Bodelle, G., "Simulation of Vehicle Impact on Steel and Composite Highway Guardrail Structures", Symposium on Crashworthiness and Occupant Protection in Transportation Systems, San Francisco CA, Nov. 1995, Eds. J.C. Reid, et al., pp. 179-190. 14. Chang, F.K. and Kutlu, Z., "Mechanical Behavior of Cylindrical Composite Tubes Under Transverse Compressive Loads", 32" International SAMPE Symposium, Apr. 1987, pp. 698-703. 15. Mustafa, B., Li, S., Soden, P.D., Reid, S.R., Leech, C.M., Hhton, ML, "Lateral Indentation of Filament Wound GRP Tubes", International Journal of Mechanical Science, Vol. 34, 1992, pp. 443-457. 16. Pang, S.S. and Kailasam, A.A., "A Study of Impact Response of Composite Pipe", Transactions ofthe ASME - Journal of Energy Resources Technology, Vol. 113, 1991, pp. 182-188. 17. Wang, Y., Du, S., Zhao, D., Ramasamy, A., "A Study of Composites fkom Weaving, Braiding, and Pultnision Processes", 7* Technical Conference - Proceedings of the American Society for Composites, Oct. 1992, pp. 3- 1 1. 18. Corbett, G.G. and Reid, S.R., "Failure of Composite Pipes Under Local Loading with a Hemisphencally-Tipped indenter", International Journal of Impact Engineering, Vol. 15, 1994, pp. 465-490. References Randles, P.W. and Nemes, LA., "A Continuum Damage Mode1 for Thick Composite Materials Subjected to High-Rate Dynamic Loading", Mechanics of Materials, Vol. 13, 1992, pp. 1-13. Reid, S.R and Goudie, K., "Denthg and Bending of Tubular Beams Under Locd Loads", Sîrucfural Failure, 1989, Eds. T. Wiezbicki, N. Jones, pp. 331-364. Jones, N., Birch, S.E., Birch, ES., Zhu, L., Brown, M., "AnExperimental Study on the Lated Impact of Fully Clamped Mild Steel Pipes", Proceedings of the Imtifurion of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering, Vol. 206, 1992, pp. 1 11-127. deoliveria, J.G., "The Behavior of Steel Offshore Structure Under Accidental Collisions", 13" Annual Ofshore Technology Confrence, May 1981, pp. 187- 198. Soreide, T.H. and Amdahl, J., "Deformation Characteristics of Tubular Members with Reference to Impact Loads From Collision and Dropped Objects", Nonvegian Maritime Research, Vol. 10, 1982, pp. 3-12. Jones, N. and Shen, W.Q., "A Theoretical Study of the Lateral Impact of Fully Clam ped Pipelines", Proceedings of the Institution of Mechanical Engineers, Parr Er Journal ofProcess Mechanical Engineering, Vol. 206, 1992, pp. 129-146. Bank, L.C., Nadipelli, M., Gentry, T.R., "Local Buckling and Failure of Pultruded Fiber-Reinforced Plastic B eams", Use of Plastics and Plarric Composites: kiaterials and Mechanics Issues - ASME, Vol. 46, 1993, pp. 499-5 19. Bank, L.C., Gentry, T.R., Nadipelli, M., "Local Buckiing of Pultnided FRP Beams - Analysis and Design", Journal of Reinforced Plastics and Composites, Vol. 15, 1996, pp. 283-294. Barbero, E. and Fu, S.H., "Local Buckling as Failure Initiation on Pultruded Composite Beams", Impact and Buckling of Structures, Nov. 1990, Eds. D. Hui, 1. Elishakoff, pp. 41-45. References 8 1 Barbero, E.J., Fu, S.H., Raftoyiannis, I., "Ultimate Bending Strength of Composite Beams", Journal of Materials in Civil Engineering, Vol. 3, 1991, pp. 292-3 06. Boukhili, R., Hubert, P., Gauvin, R., "Loading Rate Effect as a Function of the Span-to-Depth Ratio in Three-Point Bend Testing of Unidirectional Puitnided Composites", Vol. 22, 1991, pp. 39-45. Mamaiis, A-G., Yuan, Y.B., Viegelban, G.L., "Collapse of Thin-Wall Composite Sections Subjected to High S peed Axial Loading", International Journal of Vehicle Design, Vol. 13, 1992, pp. 564-579. FaW, A.H. and Hull, D., "Energy Absorption of Polymer Matrix Composite Structures: Frictional Effecîs", International Symposium on Structural Failure, Massachusetts Institute of Technology, June 1988, pp. 255-279. Farley, G.L. and Jones, R.M., "Prediction of the Energy-Absorption Capability of Composite Tubes", Journal of Composite Materials, Vol. 26, 1992, pp. 3 88-404- Mamalis, A.G., Manolakos, DE., Demosthenous, G.A., Ioannidis, M.B., "Analysis of Failure Mechanisms Observed in Axial Collapse of Thin-Walled Ckcular Fibreglass Composite Tubes", Thin- Walled Structures, Vol. 24, 1996, p p. 33 5-3 52. Schmueser, D.W. and wckliffe, L.E., "Impact Energy Absorption of Continuous Fiber Composite Tubes", Jownal of Engineering Materials and Technology: Transactions of the ASME, Vol. 109, 1987, pp. 72-77. KakaraIa, S.N. and Roche, J.L., "Experimental Cornparison of Several Impact Test Methods" , Instrumenteci Impact Testing of Plastics and Composite Materials, Mar. 1985, Eds. S.L. Kessler, et al., pp. 144-162. Crawford, R-J., Plastics Engineering, 2ndEdition, 1987. References 37. Lifshitz, J.M., Gov, E., Gandelsman, M., "Instnunented Low-Velocity Impact of CFRP Beams", International Journal of Impact Engineering, Vol. 16, 1995, p p. 202-215. 38. Williams, S.G. and Adams, G.C., "The Analysis of Instrumented Impact Tests Using a Mass-Spring Model", International Journal of Fracture, Vol. 33, 1987, pp. 209-222. 39. MMFG Co. Design Manual. 40. Timoshenko, S ., Strength of Materiah, 2ndEdition, 193 0. 41. Gerald, CF. and Wheatley, P.O., Applied Numerical Analysis, 4" Edition, 1989. 42. MatLab Version 4, User's Guide, 1995. 43. Brunet, L.J. and Nernes, J.A., "The Dynamic Response of a Pultruded Fiberglass Tube Subjected to Lateral Loading", Proceedings of the II" International Conference on Composite Materials - ICCM-I 1, 1997. References Drawings Appendix A: Drawings Photodiode r------3 I 1 1 1 i I 1 I I I 1 I I 1 I 1 I I 1 I I 1 I 1 1 1 I 1 I I 1 a Oscilloscope 1 I 1 I 1 O0 -- - Photodiode Circuit 1 9 96/04/02 Louis Brunet Composite Materials Laboratory Appendix A: Drawings Front View 1/4" dia - 20 tapped- 3/8" dia - Side View 16 tapped Bottom Block 1996/01/16 Material: AlSI 1040 Note 1: Fine Iines denote hidden detail Note 2: Tolerance on semi-circular Scale: 100% cutouts, (0, t.005) Louis Brunet 1 Composite Materials Laboratory 1 Front View 1 1/4"dia 7 1/2" 1"dia Side View Material: AIS11 040 Note 1: Fine lines denote hidden detail Note 2: Tolerance on semi-circular Scale: 100% cutouts, (0, t.005) 1 Louis Brunet 1 Composite Materials Laboratory 1 Side View 514' dia End Plug 1996/01/16 Note: Tolerance on diameters, (-.005;0) Material: AIS11040 Scale: 100% Louis Brunet Composite Materials Laboratory Appendix A: Drawings 1 ' dia 13/32' dia J114' slot 114' slot Tor, View \ 1 * i Y'3' 6 ' 1 I 1 1 I 1 112- I Edae View II I 1 I 1 1 I 1 1 1 I Note 1: Dotted lines denote hidden detail DWlT Mounting Plate 1 9 96/O3/l 1 Material: AIS11 040 Scafe: 80% Louis Brunet Composite Materials Laboratory Side View Narrow Wide lmpactor Impactor 114' dia - off '2' End View 718' dia. Note: Dotted Iines denote hidden detail Impactors 1996/01/O3 Material: AISI1 040 Scale: 100% Louis Brunet Composite Materials Laboratory 112' dia 1 112' dia End View Side View TV 1996/01/03 Material: AIS11 040 1 Louis Brunet 1 Composite Materials Laboraroiy AppendixA: Drawings Front View 17/32' dia C) - Side View Note 1: Dotted lines denote hidden detail 1 MTs Adapter 6f /O3 1 Note 2: Tolerance on circular cutouts, Material: AIS11 040 (+.005,+.O1 5) 1 Scale: 100% I 1 Louis Brunet 1 Composite Materials Laboratory 1 Computer Code AppendU: B: Computer Code kamiel .m Fiouis Brunet - 97/02 -tes load-displacerrient data for a tube of given dirIiensions, and of a aven rriateria.1. The code accounts for local displac-t in addition to beam flexure and shear displacement. Loads are initially cmpted based on a longitudinal derrient stifkess in the tube surface danage zone. Once the tube bas ben crushd at center-spn, a kam flexure and shear stif bess is extrcipolated to predict the rest of the tube's behaviaur. Irqnit required: -fulltukespan - outside tube radius - tube Wall thic3aiess - final total displacerrient - output file nam - end condition mte: - present material is isophtalic polyester /glass fiber cmposite- %-*+,:::::::::,::::*+++::::t:::::+::::::::: %: : : : : : : : : : :-f++++ff+++i-i4+++++.f+ffs.+f+++w-t4+ % % LIST OF FOR MAIN EwXRAM AND E'mRXIC%JS % % A = CTOSS-sectionalarea % aJ-Pha = damage zone angle % beni = function dl to barn-m % bearrPl = local function m Appendix B: Compter Code = angle to neutxal axis = beam flsaire slope = beam £1-e displac-t = barn £1-e diçplacerrent under fixed end conditions = beam £1-e displacarient mdar simply siq?ported conditions = off-axk local displac-t = local displac-t = flexurai displacenent of beam tmder fdend conditions = flexmal displacment of bearn under simply-sugprted end conditions = local displacmt = maximum local ~~~~~~t = beam shear displacerrient = local displacerrrent in=-t = damge zone width Ulcrezlbent = totai displacecrient = mximm total displac-t = 10ng-itudinal position incremmt = differential matrices for fixed end condition = differential rriatrices £or simply slq-ported end condition = beam end conditions = longitudinal m&ius = transverse &US = function call to force - integrable load function = local function MIE = phi0 at a given longitudinal position = function call to f1.m - solved for phi0 = local function name = function cal1 to £2 .m - solved for phi0 = local hction name Appendix B: Cornputer Code = danage zone arc angle in undeforrried configuraticai = shear I-rdulus = heaviside function = cauntuiçT indtrx = counting * = function cal1 to inda-m = local function = second rmnznt of inertia ratio = initial second 111~k[bent of inertia = local second mtof inertia under fixed end codtions = local second ~~=arientof inertia under simply support& end conditions = shear shape factor = off-axis Mgezone leng-th = Wgezone length local = function cal1 to 1d.m- determines load = local, hction name =beamspm = mment distribution dong &£O& beam = end nmmmt on bearn = function cal1 to mment.rn = local function narric3 = second dif ferential of position (d2y/W)under fWend conditions = second differential of position (d2y/W)under simply s~q~~ortedend conditions = mtdistrihtion under fixed end conditions = mtdistyhtion mdcx simply wrted end conditions = output file (1- displac-t, flexurai displac-t , shear displac-t , total displacenient, l&) = output file narrie = beam f lmeinc~enxtt (pst-local) Appendix B: Computer Code = beam shear increm€zlt (pst-local) = renir3ining cross-sectian angle = load = load increrruint = pre-estkg BtLab functim - integrates = local radius = asmxge &us = outside radius = hction cal1 to &ear.m = local function narre size = nmka of longihrdinal points on hibe stretch = strain in a longitudinal el-t of the damge zone = point along the damge zone width = damge zone width at longitudinal psition = damage zone width at longitudi nal position = mximm dazMge zone width - fully flattened = damage zone width at cent= span = tube thickness = inclination angle O£ a longitudinal elment = pre-acisting MXLzib hction - integrates = longitudinal position = longitudinal position under fixed end conditions = longitudinal position under simply wrted end conditions %-t-+t+++4tttt+++f++++++tt+ : : : **- clear %1:::1.1.*::::::::::::~ % DMZARATIrn OF GLa3AL - %: : : :-+ff+ff+t:: : : : : : ++++++-t-i-~W+f++++++f+++++f+++ global alpha; global db; gIoW deltal; global dl global ds ; global dt; global EbdConCtn; global global 'Ey; global G; global L; global global P; global R; global S; global SfiO Appendix B: Cornputer Code global SO; global t %**::::::::+:::::::-:::::::*::::::::::::* % P!mmnL mm %*-::::"::::::::::::*::::::::::::+::::::::::H E?x = 2500000*6895; EY = 800000*6895; G = 425000*6895; %::::::+::::::::::::::::-::::+:::::+::::::*:::::::: % ?UBEGEmElxY %++f+-:::::::::::::::ff:::::::H:::+++::::i-H::::- L = input('Full Tube Çpan is (m)?:'); Raut = irq-t('ûutside Tube Radius is (rn)?:'); t = bput('Tube WI thiclaiess is (m)?:'); dtrriax = input ( ' Final Total Displacerrient is (m) ? : ' ) ; R = Rout-t/2; %++tt+++ttt+tt+:: : : +++++tt-t+: : : : : : : : :++++ % 'IUBE END alNDITION %:::::::::::::+:::::::::::::::+ff:::::::::::t-:::::::::::::w &sp(' '1 diS.I? ( ' Possible E2d Conditions : ' ) disD (' 1-Fixed-End&' ) disp (' 2-Simply-Sqported ' ) diSP (' 3 Average ' ) E~~COII&I =input-- ('EndConditionis?:'); %+: :: t :::-l-+i-H:: : :+++++++++"'.-"+"+ % ~OUrPCTT~ %~+++++++cs-t-c-ti-i-+*~*+++~~~ out£ile = input ('Nanii= of Output Fi1e?:','s1); %:::::::::::*::::::::::::::ff+ff-(::::::::::::::::::::::::::+ % lNlTmLzZATICBN AND SIIEPPra %-+ll.~w**** baCkst€p = RlSOO; deltal = t/2; ast~ = W50; almax = R-deltal-backstep; dt = 0,000; Appendix B: Compter Code idx = 1; %t:::::++++++--::::-(::::tf++-é++tf:::::::~+f+-::::::+ffS. % lxxzxLY l33ammm BEHAVIQR %::::::::::.+.ttttfS*:::t-++t::::::+::::::::+:::::-t+f+ff-ttf-+++ &le (deltal~~& dt-cdtx-ax) ; P = local(deltal); out(indx,l) = Mtal; out(*,2) = kEEn; out(indx,3) = shear; out(hdx,4) = out(~,l)+out(~,2)+out(~,3); out(hdx,5) = P; dt = out (*, 4) ; delta1 = deltal+dstep; idx = *+l; %:::::::::::w::::::::::;I::~:::::.++fc+3-f-t-e-H+++++++C+u % GrJmxLY DOM3NATED BEHASAOR % NO ADDITIm m DISPLAClEMENT %~~~++u-+u++-i-f-u~ % DEimmmE BEAM P;ND SHEAR L5rnmmS %++++++++-++++++i: : : : :u: : : : : : : : : :*++++tf-t+f++++++ Wtal = R-t/2-bahtep; P = local (MW); out(in&, 1) = am; out(~,2)= beam; out(hdx,3) = sheaz; out(bdx,4) = out(indx,l)+out(ir1dx~2)+out(~,3); out(hdx,5) = P; m = *+l; = deftal - backstep; = local(deltal); = (out(*-1,s) -beam)/ (out(in&-1,s) -P); = (0ut(~-1,3)-~h~)/(0~t(~-1,5)-~); pst- Appertdix B: Cornputer Code aut2uic = bslope*Pstep; out3k = s~lOpe*PStep; %u;::1.11~~::::;:::~ % mBAL DISPLACEMENT FOR REmINmR OF PRESCRIBED mm %*::::::::::::::ttf::::::::::::::::::::::::::::::::::::::+f+ Wle (out(in&-l, 4) cd- & bslape>O) ; out (in&, 1) = out (in&-1,l) ; out(indx,2) = out(*-lt2)+out2rnc; out(*,3) = out(*-1,3)+out3inc; out (*, 4) = out (m,1) +out (kn.dxt2) +out (hdx, 3 ) ; 0ut(*,5) = out(hd~-1,5)+Pstep?; indx = *+l; m; %+:::H::::r-t-t-tt+++tt+-t*:::::::::::iP* % OUrPUT %++++f-tsi.:::::::::::::::::::::::::::::+::::H-::::::::::+++f++ £id = fapen(outfile,'w'); @rintf(£idt1%10.5e%10.5e %10.5e %10.5e %10.5e\nf,out'); fclose (£id); %~+i-+++++++++i-+++++++f++++f++f++f+4:; : : : : : : *++ %~:::::::::It:::;:;::::::::::::::::::::+::: Appendii B: Cornputer Code %f+f++++++++-+$-tf::::::::-:::::::::::::::::::tf:::::::k % local .m % Lmis Brunet - 97/02 % % Ccanputes the load borne by the tube due to an asçurried % assumed de£orrried gecmetry- %+ffff+f+f++++-::::::::::::;:::+::::::::::::::::::::::::::* function localn=local (ddtal) %3-1-~-c.++-t-::::+::::::::::-+-t+:::::::::~i:::-t+++:::::: % D-m OF mBAL %fff+f+fttt:::::::::k::::$-ttt+f++::::::::::+::::::::::::::*- global. alpha; global delta; global Ek global lamiho; global R; global SO; global t %::::::::++:::::::~::::::::::::::::::f-t:::::::t++ff+-t-++ % DEFOHMED czmEma %f+++~-t+++++-t~-3.$-)-+-~: : : : : : : -f+++++++ff++f+++f++- phiio = fzero('£lt,3*pi/4) ; &ta = (phiO+sin(phi0) ) /2; 50 = pi*R* (1-phiO/(S*beta) ) ; 3-m~ = pi*R/2; %alpha = pi/9* (1-4/7*SO/%~x); %for reinforcd tubhg a?$= = pi/lO* (1-6/11*SO/Smax);%for standard tubing larda0 = SO/tan(al@a); ds = SO/lO; %ffff+*+++++*~ffff++++-++H+++-f+++ : : : :+-: : : : : : : : : : +f-t+++f+* % m Appendix B: Cornputer Code % FuI'JcI?CIN EsDxmrIm %++++++++$::: :: : : : : : """-""""""'-"::: 1IIUI.::::: : :*+ flri =-deltal+R* (l+pi*( (cos(phiO) -cos ( (phiO+sin(phiO) ) /2 ) ) / . . - (phiO+sin(phiO)) 1 1 ; %:: :::: ::: :*::: :--+++++:: :4: :::: ::: :*~u-i- Appendix B.- Cornputer Code Appendix B: Cornputer Code % barn-m % Louis Brunet - 97/02 % % Determines the barn flexure displacemnt at center-span % based on a central difference rriethcd qprQximation. %+-::::::+:::::~-::::::::::::::::::::: ::::: function beamn=beam(P) %~~::::::::::;::-+f++++C:::::::::::::-++ft::::k++ % DECiSWTCN OF GLDBAL WRDBIXS %u~,,,,.,,,,,::::::::::+-::::::::::::::-tf++++~ glo;bal ~ondn;global Ex; global 1-0; global L; global P; global R; global size; global SO; global t; Appendix B: Compter Code = P/2*k2; = inertia (S) ; = M2 (size); = ï2 (size); = P/2*x2; = inertia(A2); = M2 (size); = ï2 (size); Appendix B: Cornputer Code = (W2/BC)*(M2,/I2) ; = O; = MbyEa (size); = inv(D2) -' ; = -disp2 (siz-1) ; = (W2/EX)*(rn./I2) ; = 0; = MbyEIS (size); = h~(D2)-'; = -disp2 (siz-1) ; dr> = (dbl+db2) /2 ; erd ~-ttt++t~++++-H-+w:: : : : : *tl-H+++ % DI- %*++++H-HH-+*-- lEElmn = db; %f+f+f+f+f++ftt+f+ft:::: : :f+f+-i::::: :-+tf-i-+ff+++ttt++f++fffff+ %++f++::::::::::+++f++t::::::;:::::w+f++ff++++++f+ Appendix B: Computer Code %::::-:::::::*:::::::::::::::::::)++* %:::::::f++ff+f++f:::+:::::::::::::::::::::::::::::::::* % I~YCPTY?~~,m % Louis Brunet - 97/02 % % C-tes the redistributed end rru=ariicrt due to the % changing cross-section. %::::-+tSt-::::::::k:::::::::::::::::::::::::+::::::::+:::::+++ function ~tn~t %:::::::::::::+-Hi-++:::- % DElXARXTON OF GLOBAL WWABLES %::::::::::::::::::::+::::::-:::::--::::::* global larndao; global aiph; global L; global P global R; global SfiO; global size; global SO; global t %:::::::::::::-+:::::::::::::::::::::::::::*:::::::: % lCMENT AND 5XCND MXBYT OF DEKiIA DI-a %:::::::::::*ff-:::::::m:::+::::::::::t++-::::tttttt* x = O :L/ (2*size):L/2; i0 = pi* ( (R+t/2)"4- (R-t/2)"4) /4; ir = iO./inertia(x); rra = P/4.*ir.jcx; 1fy=klys1tn = trapz (x,ma)- /trapz (x,iY) ; %-+L-::: %-+L-::: :: :: :-* %----- %----- Appendix B: Cornputer Code %-,. %-,...... %:::::+::::::::::::-:::::-:::::::::::- % uierti2.m % Liouis Brunet - 97/02 % % Carputes the 11~3a~3ntof inertia of the tube at a given % longitudinal position. %::::::::::+$.:::::::::::+:::::::::::::::t++ft::::::::::+:::++ function inertian=in&ia (x); %:::::*:::::::::::::::f-l::::Llll::::::::m % DECLARATI:rn OF GLcBAL, laRmmEs %:::::::-+++f::::::::+:::::::::$-tt+++f++::::::::::::t-tf+f+++++ global alpha; global 1-0; gloM L; global R global SfiO; global SO; global t %-r:::::::;M:::-e-r-c+++:::-::P* % LlxxTUDTNAL CHARACTERfSI?CS OF THE DEFORMATIa ZQlm %f++++ff++f+++++f+++++tt++f++++++ff:: : f-i: :: : : : : : : : f-fff+f+++++ i0 = pi* ( (R+t/2)"4- (R-t/2) "4) /4; heavy = ( (x-(L/2-iaIrK%lO) ) >O) ; Çfi = (x-(L/2-landaO) ) - *tan (alpha). *heavy; * = 1; *le index <= lengt-li(Sfi) SfiO = Sfi(h&x) ; fiO(k&x) = fzero('f2',3*pi/4); indsr =*+1; end %f+++++++tt+++++++++++++$.~~+~$-t-3-)-$-l-fS-t-t++++ % rnDIFIED l!a4ENr OF lxEEmm %::::::::::::::::::::::*:::::::::::::::::*::::::::::::::: beta = (fiO+sin(fi0) ) ,/2 ; fi = pi*R. / (2*beta); = iO* (l-hea.)+ ( ( (rL+t/2) ."4- (rL,-t/2 ) .^4) . /4. * - - . (fio. * (1+2* (cos (fi0/2) ) ."2) +... sin(fi0). * (cos (fi0)-4*cos (W2)) ) ) .*&y; % : : : : : : ++-: : : : ~-f--!-~~~+-i4+ %f~uu-f-~~-::: :+: :::::: :-+ Appendix B: Computer Code %-:::-:::::+f++S-:::""'-'.F"::I::t::::H %f+f::::::::::::::*:::::-:::::::::::::::::t-f-t-t-t-+-t-fSff+ % £2 .m % Louis Brunet - 97/02 % % Determines the regcross-section angle coinciding % with a prescribed darrage zone width, %-:::::::::::::::::f+ftt+f::::::i:::i:::- function f2n=f2 (phiO) %+*:::::::::::::::++++::::::::::::::::::+::::::::H:::* % DElXmmIrn OF GLmAL l7zmxmm %~:::::::::::::~ global R; global SfiO %*::::::::::::::*::::::::::::::::::::i:::::- % cm!Em %+++++-:::+++:::)-H-Hf+f++:::::::::::::::::;::::::::.f+ f2n = SfiOvhiO + (SfiO-pi*R) *siri (&O) ; %tttt-t-++++++++-tf+f++++++++f++t-t-:::: -:::: ;:: ;+ %::::::i:::::~+H+::::::::;f+s-)-)-t-e++++t.tttf~ Appendix B: Cornputer Code Appendix B: Cornputer Code IMAGE EVALUATION TEST TARGET (QA-3) APPLlEO -1 IMAGE. lnc --- 1653 East Main Street ----. - Rochester. NY 14609 USA ------Phone: 7161482-0300 ---- Fa: 71 6/288-5989 O 1993. Applied Image. Inc. AI1 Righiç Reserved