DOCUMENT NO: EFRC – ABS02 (1) FAA PROJECT NO: CURRENT REVISION: IR INITIAL DATE: 07/07/11
Status Report 2
EVALUATION OF RESIDUAL STRENGTH OF BEECHCRAFT BONANZA SPAR CARRY‐THROUGH WITH FATIGUE CRACKS
Conducted for the
AMERICAN BONANZA SOCIETY
DEPARTMENT: Embry-Riddle Aeronautical University - Eagle Flight Research Center
SECTION: Technical Analysis
PREPARED BY: TECHNICAL APPROVAL: Snorri Gudmundsson PREPARED BY: MANAGER APPROVAL Richard P. Anderson
REVISION APPROVAL REV REVISED BY APPROVED BY DATE REV REVISED BY APPROVED BY DATE IR SG RPA 07/07/11 1 2 3
REVISION HISTORY Revision DESCRIPTION OF CHANGE
A Initial release
Snorri Gudmundsson Principal Investigator: Assistant Professor of Aerospace Engineering Embry‐Riddle Aeronautical University
Prepared by: Snorri Gudmundsson
Research team supervisor: Dr. Eric Hill
Research Team (alphabetical order by last name): Christopher Foti, Ning Leung, Zachary Sager, Michael Scheppa, Isadora Thisted, Joseph Tabarracci, Dr. Jean‐Michel Dhainaut.
Contents
1. INTRODUCTION ...... 7 2. PROJECT TASKS ...... 7 3. LOAD TESTING ...... 7
GENERAL ...... 7 MOUNTING INTERFACE ...... 8 LOAD APPLICATION GIZMO ...... 8 APPLICATION OF LOADS ...... 9 STRAIN GAGES AND DATA ACQUISITION ...... 9 EXPERIMENTAL PROCEDURE ...... 11 FIDELITY OF STRAIN GAGES DURING TEST ...... 11 4. FINITE ELEMENT MODEL ...... 13
GENERAL ...... 13 MODEL CONSTRAINTS ...... 16 APPLICATION OF LOADS ...... 16 LIMITATIONS OF THE FE MODEL ...... 19 5. COMPARISON OF EXPERIMENT TO PREDICTION ...... 20
COMPARISON OF STRAINS ...... 20 COMPARISON TO CLASSICAL STRESS ANALYSIS ...... 21 6. LOAD CASES ...... 23
LOAD CASE 1: SYMMETRICAL LOAD (12600 LBF) ON EACH WING ...... 23 LOAD CASE 2: ASYMMETRICAL LOAD CASE 100/60 ...... 23 LOAD CASE 3: SYMMETRICAL TOUCH‐DOWN ON MAIN GEAR ONLY...... 24 LOAD CASE 4: ASYMMETRICAL TOUCH‐DOWN ON MAIN GEAR ONLY...... 24 7. STRESS FIELDS AT LIMIT LOAD ...... 25
LOAD CASE 1: SYMMETRICAL LOAD (8400 LBF) ON EACH WING – LIMIT LOAD...... 26 LOAD CASE 2: ASYMMETRICAL LOAD CASE 100/60 – LIMIT LOAD ...... 27 8. STRESS FIELDS AT ULTIMATE LOAD – BASELINE STRUCTURE ...... 28
LOAD CASE 1: SYMMETRICAL LOAD (12600 LBF) ON EACH WING – ULTIMATE LOAD ...... 29 LOAD CASE 2: ASYMMETRICAL LOAD CASE 100/60 ...... 30 LOAD CASE 3: SYMMETRICAL TOUCH‐DOWN ON MAIN GEAR ONLY...... 31 LOAD CASE 4: ASYMMETRICAL TOUCH‐DOWN ON MAIN GEAR ONLY...... 32 9. STRESS FIELDS AT ULTIMATE LOAD – 3.5” LONG CRACK ...... 33
LOAD CASE 1: SYMMETRICAL LOAD (12600 LBF) ON EACH WING – ULTIMATE LOAD ...... 34 LOAD CASE 2: ASYMMETRICAL LOAD CASE 100/60 ...... 35 LOAD CASE 3: SYMMETRICAL TOUCH‐DOWN ON MAIN GEAR ONLY...... 36 LOAD CASE 4: ASYMMETRICAL TOUCH‐DOWN ON MAIN GEAR ONLY...... 37 10. STRESS FIELDS AT ULTIMATE LOAD – CRACK THROUGH THREE FASTENERS ...... 38
LOAD CASE 1: SYMMETRICAL LOAD (12600 LBF) ON EACH WING – ULTIMATE LOAD ...... 39 LOAD CASE 2: ASYMMETRICAL LOAD CASE 100/60 ...... 40 LOAD CASE 3: SYMMETRICAL TOUCH‐DOWN ON MAIN GEAR ONLY...... 41 LOAD CASE 4: ASYMMETRICAL TOUCH‐DOWN ON MAIN GEAR ONLY...... 42
11. STRESS FIELDS IN SPARS AT ULTIMATE LOAD ...... 43 12. CONCLUSION ...... 47 APPENDIX A: BEECH BONANZA DATA ...... 50 APPENDIX B: VON MISES YIELD CRITERION ...... 52
EVALUATION OF RESIDUAL STRENGTH OF BEECHCRAFT BONANZA SPAR CARRY‐THROUGH WITH FATIGUE CRACKS
1. INTRODUCTION This status report is considered the final deliverable in an investigation conducted by Embry‐Riddle Aeronautical University (ERAU) research faculty on behalf of the American Bonanza Society (ABS). The investigation is performed in accordance with ERAU Research Project 13776 (signed on 9/22/2010) and an agreement between ERAU and ABS presented in a Statement of Work (SOW).
The precursor to this project and initial work is detailed in Reference 1, a report titled EFRC – ABS01, Evaluation of Residual Strength of Beechcraft Bonanza Spar Carry‐Through with Fatigue Cracks. This report details the tasks accomplished in the second phase of the project and its conclusion.
2. PROJECT TASKS The effort in the second phase consisted of the following tasks:
1. The spar carry‐through was load tested to obtain data for validation of a Finite Element (FE) model. The load testing is discussed in Section 3 of this report. 2. A FE model of the spar carry‐through was created using the solid modeler CATIA, and then meshed using the pre‐ and post‐processor FEMAP. This model is discussed in Section 4 of the report. 3. The FE solver NASTRAN was used to predict strains in the FE model using the test loads. A comparison of measured and predicted strains was conducted to evaluate the quality of the model and to better understand the nature of load paths. This validation is discussed in Section 5 of the report. 4. Four load cases to be applied to the FE model are discussed in Section 6 of this report. The load cases, which are detailed in the Reference 1 represent symmetric and asymmetric flight load cases and landing load cases. 5. The application of the four load cases to the FE model and evaluation of stresses in the region of the huckbolts, near the lower wing attachment mount is discussed in Sections 7 through 11 of the report. 6. Introduction of a simulated crack in the region, similar to what has been experienced in the fleet of Bonanza aircraft, and repeat solution using the four load cases is performed and discussed in Sections 7 through 11 as well.
3. LOAD TESTING General The spar carry‐through was load tested in order to obtain experimental strains for comparison to strains predicted by applying the same loads to the FE model. The general setup of the test is best described by viewing the schematic in Figure 1. The spar carry‐through is mounted to a stiff C‐channel beam fixture using the left side wing attachment mounts. Then a vertical shear load is applied to the right side mounts using a special gizmo, especially designed and fabricated for this purpose. It allows the test load to be applied to the carry‐through mounts without introducing moments at the interface of the hydraulic jack and attachment mounts. This way, only an axial couple (RAX) is applied to the upper and lower mounts, and a shear force (RAY) is applied to the lower mount only (see Figure 1).
Figure 1: A schematic showing the spar carry‐through mounted to the test fixture.
Mounting Interface As stated earlier, the spar carry‐through was mounted to a C‐channel beam fixture using the left hand wing attachment mounts (see Figure 3) using ¾” fasteners. Although stout, the maximum test load of 4000 lbf acting through an arm of approximately 46”, twisted the beam and it was estimated the far end of the carry‐through deflected about 1.5” or about 1.9°.
Load Application Gizmo As can be seen in Figure 1, the gizmo is only attached to the lower spar cap mount on the right hand side of the spar carry‐through. Its purpose is to apply pure shear to the structure. Only a modest amount of axial load is applied in the process. Although not exactly representative of flight loads, this approach causes the applied load to be reacted as shear in the shear web and bending loads in the spar caps, without additional moments being introduced. When the primary structural members are loaded this way, all eight strain gages are simultaneously loaded as well, which is very convenient for validation purposes.
The gizmo transfers the load from the hydraulic jack (P) via offset, but this ensures it will rest solidly against the surfaces of the upper wing mount. The offset for the load is small, about 2 inches, so the horizontal reaction force RAX is small compared to RAY. Note that the actual device (see Figure 2) looks different from the one in Figure 1 although it works using the same principle.
Figure 2: CATIA and up‐close view of the load application gizmo.
Application of Loads The loads were applied using an MTS machine, which allows the magnitude of load to be applied to the structure to be precisely controlled. A time history of the load application is shown in Figure 9.
Spar carry‐ through
Spar is mounted to this steel C‐channel, using the upper and lower wing mounts.
Load application Gizmo
Figure 3: Spar carry‐through mounted to the test fixture.
Strain Gages and Data Acquisition A total of eight 350Ω strain gages were bonded to the structure at six carefully chosen locations. The locations were chosen based on the local stiffness of the structure to ensure plausible load paths were picked up by the strain gages. With this in mind, two strain gages were mounted on the shear web as shown in Figure 4 to allow multi‐directional strains to be measured. One strain gage was mounted on the upper spar (Figure 5) to measure the bending and axial load in the upper cap. Another one was mounted to the lower C‐channel spar cap (Figure 6), to the vertical tie plates (Figure 7), and the bottom side of the upper C‐channel (Figure 8).
Figure 4: Strain gage location on the shear web. Figure 5: Strain gage location on the outer upper spar.
Figure 6: Strain gage location on the lower spar cap C‐ Figure 7: Strain gage location on the vertical tie plate. Channel.
Figure 8: Strain gage location on the lower side of the upper C‐Channel.
The strain gages were each connected to a circuit that served as a strain gage amplifier and the remaining three legs of the quarter bridge configuration. The output of this circuit was routed to a portable Data Acquisition (DAQ) system (NI USB 6008), which was then connected to a computer. Through the use of a program called LabVIEW, the computer converted the changes in voltage to strains.
Experimental Procedure
The test was implemented by applying a set of four static loads of 1000, 2000, 3000, 4000 lbf to the spar carry‐ through while the resulting strains were recorded using the apparatus described above. The 4000 lbf load was applied to the finite element model1 and the predicted strains recorded (see Figure 19). With this knowledge, the predicted strains can be compared to the measured strains, allowing model fidelity to be evaluated. Each strain gage was checked for functionality before the test by measuring resistivity. They were considered functional before the test, although two of them returned data that is now considered unreliable.
The test was performed by slowly increasing the load applied by the MTS machine. A time history of the load application can be seen in Figure 9. The horizontal axis shows time in “time steps” but each corresponds to 0.04 seconds (25 Hz collection frequency). This way 16000 time steps amounts to 640 seconds or 10.67 minutes. The load was maintained for approximately 30 seconds to a 1 minute at each load step, to allow an average strain to be collected. The vertical axis shows the strains indicated by the strain gages.
The figure shows unexpected behavior of one of the strain gages mounted to the shear web. This strain gage is identified as SG5 in Figure 10. Its signal rises at first, but then returns back to zero. This behavior renders its signal unreliable, and, thus, it is omitted from further consideration. Also, the strain gage mounted to the tie plate (SG1) showed very limited responsiveness, which can mean one or a combination of the following: (1) practically no load went through it, (2) it was poorly bonded to the structure, or (3) it was not functional during the test.
Fidelity of Strain Gages during Test It is very important to monitor the fidelity of the strain gages during load testing and this was done by maintaining the applied loads for several seconds as explained before (and shown in Figure 9) to allow an average value to be collected. This data is then plotted to evaluate the linearity of the strain gage signals with load. If the load applied to the structure results in stress levels below yield strength and the strain gages are “healthy”, such a graph should be linear.
Results of this nature are presented in Figure 10. It shows that Strain Gages 1 and 5 give unreliable results, or are mounted in locations that are not a primary load path for that particular load. Strain Gage 1 was mounted on the tie plate (see Figure 7) and SG5 is one of the four located on the shear web (see Figure 4). The data from these two gages is ignored.
The remaining strain gages all demonstrate linear response and are thus thought to indicate they are indeed properly mounted to definite load paths and functional.
1 The FE model is linear, so the strains at the other loads can be obtained by proportions.
Figure 9: Time history of load applied to the spar carry‐through during the test.
Figure 10: The fidelity of the strain gages can be evaluated by plotting their linearity.
4. FINITE ELEMENT MODEL
A Finite Element model of the structure was created using the geometry previously digitized in the solid modeler CATIA. Inspection of the original spar carry‐ through reveals that shimming was extensively used for better fit of parts. As an example, the digitization revealed the forward and aft shear webs are not perfectly parallel, but bend away from the underlying structure (see simplified side view representation of the carry‐through in Figure 11). This deficiency was resolved by the manufacturer by riveting tapered shims between the shear web and underlying support structure. These imperfections were copied into the original CATIA model, which, in turn, led to considerable difficulties in the FE modeling, as the highly accurate digitized part did not fit well together. Eventually these required modification to be made to the CATIA model. The modifications consisted primarily of redefined geometry such the forward and Figure 11: A schematic showing changes made to the aft shear webs were aligned parallel to the underlying CATIA model to allow for easier construction of the FE structure, as shown in the right image of Figure 11, model. allowing it to be joined without irregular shims. This change is not thought to affect the results.
General The model features three kinds of elements; plate elements, solid parametric elements, and rigid elements. The entire structure is modeled using plate elements, excluding the four mounts, which are modeled using solid elements, and rigid members to which the flight and landing loads were applied. The two shear webs were represented with plate elements and featured all rivet and huckbolt holes, as did the underlying structure. These were matched up with corresponding holes in the underlying structure and joined via node consolidation. The purpose with this approach was to try to get an idea of this effect of stress concentration on the stress field in the structure and simulate with more realism how the primary load paths behave. A full view of the model is shown in Figure 12 and a close‐up of the attachment mounts Figure 13.
To further understand the impact of stress concentrations on the stress field in the shear web, two crack shapes were generated. The first is a 3.5 inch long crack generated by the request of the FAA. A 4” crack was actually requested, but could not be accommodated easily due to its length. For this reason it was shortened by about 0.5”. The crack was modeled by removing elements locally and it features sharp ends. It can be seen in Figure 14. Modeling cracks is not simple and the ones introduced to the model at best return average stresses.
The second crack was prepared by removing elements between the three outboard huckbolts. This was done to simulate the crack in the spar carry‐through sent to ERAU. It can be seen in Figure 15, which shows both the crack in the actual part and how it was modeled. The discrepancy effectively takes 3 fasteners out of commission and this triggered the curiosity of the Principal Investigator.
Figure 12: Full view of the FE model, showing plate and solid elements (mounts).
Figure 13: The model primarily consists of plate elements. The attachment mounts are solid elements.
Figure 14: A close up view of the 3.5” long crack.
Figure 15: A close up view of a crack between the three outboard huckbolts. The insert shows the crack in the actual spar carry‐through.
Model Constraints The model was constrained in two places on the upper wing attachment mounts. The right mount featured a simply supported constraint and a roller type constraint on the left. The rollers were constrained from translating in the x (forward) and z (upward) directions. These constraints were applied to the front and rear faces of the attachment mounts and were identified as Constraint Set 1. Another type was evaluated in combination to, but this added simple supports along the flanges of the shear web to simulate the riveting to the fuselage side. This was identified as Constraint Set 2.
Application of Loads Loads were applied to the model in a fashion similar to what was shown in Reference 1. However, the flight loads must be represented in three‐dimensional space. The ultimate load is calculated by multiplying the maneuvering load, as the airplane stalls while (theoretically) it pulls 6.0 gs, by a factor of 1.5. The airspeed at which this occurs (and is referred to as the maneuvering speed, VA) is calculated below:
2nlimW VA (1) SCLmax
Where; CLmax = Maximum lift coefficient = 9.0 nlim = Limit load factor = 6.0g S = Wing area = 181 ft² WA = Gross weight for aerobatic operation = 2800 lbf = Density of air = 0.002378 slugs/ft3
Of the above variables, the maximum lift coefficient remains to be determined. This can be done by using the 2 published stalling speed with flaps retracted which is 60 KCAS at a gross weight of W = 2900 lbf.
2W 22900 CL max 2 2 1.314 VS S 0.002378 601.688 181
Inserting this into Equation (1) yields:
2nlimW 26.02800 VA 244 ft/s 144 KCAS SCLmax 0.002378 181 1.314
2 Source: F33A Pilot’s Operating Handbook.
The Bonanza features a NACA 23016.5 airfoil at the root and NACA 23012 at the tip. Per NACA R‐824 these airfoils stall approximately at 18° considering the Reynolds Number of 8.3 million for the MGC airfoil at 142 KCAS. An excerpt from NACA R‐824 for the 23012 airfoil, showing the lift curve, can be seen in Figure 16. This represents an upper extreme for the 3D aircraft. A Vortex‐Lattice estimation of the AOA required for trim at that condition yielded approximately 14° AOA (see Results in Appendix A). However, since VLM is linear, the stall will occur at a higher AOA; this represents a lower extreme. In the absence of actual flight test data the average of the two is a good approximation for this project, so a stall AOA of 16° is used. As consequence, for the application of loads to the FE model, it will be assumed the aircraft is approximately at an AOA of 16° (see Figure 17). For this reason, it is prudent to break the total lift force into a component normal to the wing plane and a component parallel to the chordline. These are denoted as FN and FC, respectively, in Figure 17. At stall, this component would force the wing forward, a tendency prevented by the wing’s aft attachment, which is put into tension.
This scenario was simulated in the application of loads to the FE model. Figure 18 shows how the loads are applied via loading truss that react FN and FC. A sample calculation of these components is shown below.
Figure 16: Wind tunnel test data for the NACA 23012 airfoil from NACA R‐824.
Figure 17: Forces acting on the Bonanza wing just before stall.
The aft shear web/fuselage attachment point was represented as a simple‐ support in the FE model.
This rigid member represents the main landing gear leg, used with load cases 3 and 4.
The flight load is applied here (and on the opposite side), but this is the location of the 25% chord of the MAC. The force points forward as it combines the forward and upward components.
Figure 18: A zoomed‐out view of the FE model, showing loads applied via rigid members.
The normal and chordwise load components can be calculated per each wing using the following expressions. Note that although the aforementioned discussion involves the limit load, the load applied to the FE model is ultimate load:
L n W Normal force per wing: F cos ult cos N 2 2
L nultW Chordwise force per wing: FC sin sin 2 2
Where; nult = Ultimate load factor = 9.0 W = Gross weight = 2800 lbf = 16°
Inserting these quantities yields the following normal and chordwise forces
n W 9.0 2800 Normal force per wing: F ult cos cos16 12112 lb N 2 2 f
nultW 9.0 2800 Chordwise force per wing: FC sin sin16 3473 lbf 2 2
This method allows the wing and landing gear loads to be applied in a straight‐forward fashion that is less likely to cause confusion. A total of 4 load cases were applied and these are discussed in greater detail in Section 6. The above component break‐down is only applied to the load cases that involve the maneuvering loads (Load Case 1 and 2). A lift load of 3600 lbf (1 g flight load) is used for the landing cases and that load was not corrected for AOA.
Limitations of the FE Model The following limitations of the FE model must be kept in mind when reviewing these results:
1. Aeroelastic effects are not accounted for in this analysis. Since the FE analysis is done at the highest loads the aircraft is designed for, this would be accompanied by significant bending of the wings. The true wing‐tip deflection at limit load is not known, but it is not unlikely it would be in the 3‐4 ft range. That sort of deflection would likely move the center of lift inboard before the aircraft stalls, perhaps some 2‐3 ft, in turn, reducing the wing bending moment.
2. The contribution of fuselage and horizontal tail to lift is not accounted for. As can be seen in Appendix A, a Vortex‐Lattice analysis was performed on the aircraft and it predicts that 7% of the total lift is generated by the fuselage and HT and 93% by the wing. The loads in this analysis do not account for this difference.
3. Non‐linear effects are not accounted for in this analysis. Once the yield strength of the material used is reached, its elasticity becomes non‐linear and linear predictions are therefore no longer valid. This is particularly important in all highly stressed areas, as well as around fastener holes. Once fastener bearing stresses are reached, the material will yield which changes the geometry of the hole and the stress field in its neighborhood.
4. As discussed earlier, the model is constrained using one simple support and one roller support. Proper model constraints is a hard detail to model accurately because the spar carry‐through is riveted to the fuselage skin, but this really behaves as an elastic foundation. However, defining such a foundation would require an accurate model of the fuselage.
5. Mesh sensitivity. Since the subcomponents of the structure are meshed automatically using the pre‐processor FEMAP, the meshing near holes and cutouts often yields highly skewed elements (i.e. less than “perfect” elements). The skewness of an element affects the predicted strains and consequently high strains (and thus stresses) may appear in such details. Such effects must be kept in mind when evaluating results.
6. Joining of parts. Simulation of rivets and fasteners is accomplished in the FE model by consolidating nodes of adjacent parts. This means that the nodes along the circumference of co‐centric holes of two parts are joined. Generally, a fastener reacts shear force in bearing such that only a half of the hole’s circumference forms a bearing contact with the fastener. In the extreme case, the hole would retain a circular shape on the contact side, but the “contact‐free” side would stretch, making it oblong. In the FE model, the fastener acts as if the “back” side is also in contact with the fastener and this extends the strain field forward and aft of the fastener’s displacement direction. Strictly speaking, this phenomenon could be simulated in the FE model by “pre‐analysis” to determine the fastener direction, and then a “post‐analysis” after the back‐side node had been separated. Such a task is tremendously time consuming, precluding this approach from being used the analysis.
5. COMPARISON OF EXPERIMENT TO PREDICTION Comparison of Strains As has been stated before, the fidelity of the FE model is evaluated by comparing the measured strains to the predicted strains in the same location. These results are shown in the table in Figure 19. In the interest of clarity, it should be stated that these load cases should not be confused with the flight load cases of Section 6. The ones discussed here are the test load cases.
The location of the strain gages was modeled as a region on the FE model. Each region contained a number of elements. The resulting strain of each element varied inside some of these regions, so the average for the region was calculated and this is presented as FE in Figure 19.
With respect to Figure 19, while the magnitude of the strain gage data is off by an order of magnitude for some of the strain gages (for instance, the ratio of predicted to experimental strain, FE/EXP, for Strain Gage 2 is about 5.2) the ratio of predicted to experimental strain, FE/EXP, remains constant for each of the load cases (1000‐2000‐ 3000‐4000 lbf). This means that when the load is doubled, both FE and EXP double, and this indicates there is indeed correlation between the two. A highly accurate match of strains is not always possible in complicated structure.
In this particular project, Strain Gages 2, 3, 4, 6, 7, and 8 all demonstrate such correlation. For instance, consider
Strain Gage 2. The ratio FE/EXP remains uniform throughout the application of loads (5.866, 5.591, 5.490, and 5.490 for lowest to highest) and this constitutes acceptable correlation. The strain gages that most closely matched measured values were SG3 (with 1.852, 1.789, 1.761, and 1.718) and SG6 (with 0.565, 0.512, 0.493, and 0.495).
It can be seen that all the strain gages used to detect shear strains (SG6, SG7, and SG8) display values less than predicted (that’s why the ratio is >1). All the others (SG2, SG3, and SG4) display strains larger than predicted (that’s why the ratio is < 1).
Figure 19: Comparison of measured and predicted strains for the spar carry‐through.
Comparison to Classical Stress Analysis Another way to evaluate if the FE model returns reasonable stresses is to compare it to a simple classical stress analysis, performed in a region of the structure where stresses are relatively uniform. This is only intended to “give an idea” as to what magnitudes of stresses to expect in this region. This was done in the middle of the spar caps (see Figure 20) for Load Case 1 by calculating the stress in accordance with the classical expression:
M h 2 bending I X
Where; M = Bending moment at location = 952295 in∙lbf (see moment diagram in Reference 1). IX = Area moment of inertia (estimated below). h = Structural depth (height of spar carry‐through) = 11.326 in (from the digitized geometry)
The first step is to determine a representative moment of inertia and for this check it will be done using the parallel axis theorem. Based on the digitized geometry, the cross‐sectional area of the spar caps, A, is approximately 1.502 in4. Therefore:
2 2 2 h Ah 1.50211.326 4 I X bending 2A 96.34 in 2 2 2
The resulting stress at the center of an idealized circular spar cap would thus be:
M h 2 95229511.326 2 56000 psi bending I 96.34 X
x ~ 76400 psi
Classical analysis:
x ~ 56000 psi
x ~ 45700 psi
x ~ 51900 psi
x ~ 73900 psi
Figure 20: Values in text boxes show general magnitudes of normal stresses in the upper and lower spar caps.
As can be seen in Figure 20, the stress on the upper surface of the upper spar cap was found to be around 76400 psi, and 45700 psi on the lower surface. The average of the two is around 61000 psi, and similar analysis of the lower spar cap yields an average value of 62900 psi. Both are in the “ballpark” of the value returned by the classical analysis.
6. LOAD CASES The load cases of interest are detailed in Reference 1 and are presented below for convenience. The spanwise location of the Mean Aerodynamic Chord (MAC), yMAC, was determined by calculating the Center of Pressure of the wing using the Vortex‐Lattice Method (VLM). This returned the value 8.22 ft. Therefore:
yMAC 8.2212 75.58 in
For clarity, conventional Mean Geometric Chord analysis, often used as an approximation for the MAC leads to (note that = Taper Ratio = 0.5):
b 1 2 33.5 1 20.5 yMGC 7.444 ft 89.33 in 6 1 6 1 0.5
The VLM value is more realistic as it accounts for the shape of the planform, including the leading edge extension; It is an aerodynamic estimation that accounts for airfoil geometry and location, wing washout, dihedral, wingtip geometry, to name a few, not to mention is predicts lift distribution quite accurately when compared to experiment. The MGC value is solely based on the planform geometry and should only be considered a “ballpark” value. Noting the width of the spar carry‐through, the spanwise distance from the wing attachment mounts to the MAC can be determined as follows:
46.12 yMAC 8.2212 75.58 in 2
Load Case 1: Symmetrical Load (12600 lbf) on each Wing
This represents the airloads at VA or VD, during which the airplane reacts 9g ultimate load. The resulting loads can be seen in Figure 21. This load is applied to two nodes, each at the Mean Geometric Chord (MGC) of each wing, with a forward component of 3473 lbf (6946 lbf total) and 12112 lbf upward (24224 lbf total). The resultant force on each not is 12600 lbf and this is shown in the figure.
Figure 21: Result for a symmetrical load at 9gs.
Load Case 2: Asymmetrical Load Case 100/60
This represents the airloads at VA during which the airplane reacts 9g ultimate load in a yawed condition. This assumes one wing to react the full load and the other 60% of that. This is shown in Figure 22. This load is applied to the two node at the MGC, with a forward component of 3473 lbf on the one and 2084 lbf at the opposite one (5557
lbf total) and an upward forces of 12112 lbf and 12112 lbf, respectively (19379 lbf total). The resultant forces are 12600 lbf and 7560 lbf and this is shown in the figure.
Figure 22: Result for 100‐60 asymmetrical load at 9gs.
Load Case 3: Symmetrical Touch‐Down on Main Gear only. This load case represents a hard landing on both main gears with applied forces consisting of a total wing lift of 3600 lbf (1g lift load) and vertical impact force of 5400 lbf per CAR 3.243. The magnitude of this force is based on an assumed 3g landing impact (so 3 x 3600 = 10800 lbf; divide by 2 to get 5400 lbf). The resultant force on each wing MGC node is 1800 lbf and this is shown in the figure. Note that the landing gear drag force is omitted as this is reacted by the aft wing attachment.
Figure 23: A simple FE model to evaluate the impact of a hard landing on the carry‐through.
Load Case 4: Asymmetrical Touch‐Down on Main Gear only. This severe case assumes the pilot makes a 3g landing on one main gear only. Other than that, it is identical to Load Case 3. The applied loads are as shown in Figure 24.
Figure 24: The simple FE model loaded to represent a hard touch‐down on one main gear only.
7. STRESS FIELDS AT LIMIT LOAD
In the presentation that follows, colors tell the whole story. Per MIL‐HDBK‐5J there is a range of yield strengths for the various plate thicknesses of 2024 alloy used in the carry‐through assembly. Generally, the yellow color represents a band of stresses where the structure should be expected to yield. The red represents where the structure should be expected to fail. Additionally, the abbreviation SC will be used for a Stress Concentration and of which SCs is the plural form. Also, SCR stands for Stress Concentration Region and refers to the general area in which the stress concentration can be found.
This presentation attempts to take advantage of the format in which this document is presented – a PDF. The images are all equally sized to allow the reader to quickly flip from one page to the next, as this will be helpful in understanding the results.
Even though not specifically requested, the analysis would be incomplete without the inclusion of the limit load cases. The two cases presented here are the symmetric and asymmetric 6g flight load the F33C is designed to react. Both of these cases have free edges (i.e. the flanges of the shear web are not riveted to the fuselage).
Load Case 1 Figure 25 shows the shear web is relatively lightly loaded, but the spar regions are highly loaded. SCs near the two bottom cutouts appear near or post‐yielding. These would not necessarily yield in the real airplane if the shear web was riveted to the fuselage, as this would prevent the large deformation present here due to bending (the bottom of the shear web is “bowing out”). This would actually reduce these SCs, although the attachment region would probably see a slight rise in stresses in the process.
A region near and below the lower wing attachment, in the corner of the shear web flange is also highly stressed. The right side of this area (which is actually the left side when installed in the aircraft) is enlarged for clarity. Note that the stress field in the spar carry‐through is slightly asymmetric because of the way the model was constrained (one side is fixed; the other is on a roller).
Load Case 2 Figure 26 shows the asymmetric load significantly loads up the shear web and contradicting the findings of the simplified 2D analysis performed in Reference 1. High SCs at the end of the corrugations on the shear web are visible. The size and shape of the stress concentration below the left wing attachment (the right side of the figure) appears similar as in Load Case 1.
Load Case 1: Symmetrical Load (8400 lbf) on each Wing – LIMIT LOAD
Figure 25: Load Case 1 + Constraint 1 (free edges).
Load Case 2: Asymmetrical Load Case 100/60 – LIMIT LOAD
Figure 26: Load Case 2 + Constraint 1 (free edges).
8. STRESS FIELDS AT ULTIMATE LOAD – BASELINE STRUCTURE
All four load cases are presented in this section and all of these feature free edges. In the airplane, the flanges are riveted to an elastic foundation and this will affect the stress field in some ways. The following sets of images show the front and aft faces of the spar carry‐through and reveal the stress field.
On the request of the FAA the analysis included subjecting the FE model to ultimate loads. Unfortunately, the FE method predicts failure regardless of presence of cracks in the shear web. However, this allows one to observe the location and nature of major SCs.
Load Case 1 Figure 27 shows a lightly loaded shear web, but the spar regions that are stressed beyond failure. Again, the region near and below the lower wing attachment in the corner of the shear web flange is highly stressed.
Load Case 2 Figure 28 shows the asymmetric load significantly loads up the shear web, contradicting the findings of the simplified 2D analysis performed in Reference 1. High SCs at the end of the corrugations on the shear web are visible. The general shape of the SC below the left wing attachment (the right side of the figure) appears similar as in Figure 27, although it is larger.
Load Case 3 Figure 29 indicates that a hard touch‐down on both main landing gear results in considerably less stress build‐up than the flight loads.
Load Case 4 Figure 30, on the other hand, shows a hard touch‐down on one main landing gear results in large stresses in the shear web, as well as the area of the lower wing attachment.
Load Case 1: Symmetrical Load (12600 lbf) on each Wing – ULTIMATE LOAD FRONT
BACK
Figure 27: Load Case 1 + Constraint 1 (free edges).
Load Case 2: Asymmetrical Load Case 100/60 FRONT
BACK
Figure 28: Load Case 2 + Constraint 1 (free edges).
Load Case 3: Symmetrical Touch‐Down on Main Gear only. FRONT
BACK
Figure 29: Load Case 3 + Constraint 1 (free edges).
Load Case 4: Asymmetrical Touch‐Down on Main Gear only. FRONT
BACK
Figure 30: Load Case 4 + Constraint 1 (free edges).
9. STRESS FIELDS AT ULTIMATE LOAD – 3.5” LONG CRACK
In this section the baseline structure is compared to one that features a 3.5” long crack below the fastener pattern of the lower wing attachment. The crack extends through the highly stressed region as can be seen in the following sets of figures. For convenience, the baseline is presented on top and the cracked structure below. The underlying structure does not feature a crack – it is only in the shear web. Again, the constraints are free edges, with Load Case 1 featuring the additional fixed‐edge constraint.
Load Case 1 Figure 31 compares the stress field of the baseline model to the one with the crack in it. The shape of the stress field in the upper row of huckbolts mostly remains the same, but the lower row is affected and some redistribution is evident. If this effect represents reality, this would cause the fatigue sensitive area to move to a different location – for this particular load case, farther toward the bottom of the fuselage and would promote ongoing crack growth. However, this also reveals an important clue. The flight load is transferred to the shear web, first through the huckbolts into the spar caps, whose deformation then transfers it via the fasteners to the shear web.
Load Case 2 Figure 32 shows how the asymmetric loading affects the stress field with and without the crack. Stresses are redistributed around the edges of the crack, which may promote crack growth.
Load Case 3 Figure 33 shows moderate stresses for the baseline, but a SC immediately at the ends of the crack.
Load Case 4 Figure 34 also shows SCs at the ends of the crack.
Load Case 1: Symmetrical Load (12600 lbf) on each Wing – ULTIMATE LOAD ORIGINAL – ULT
3.5” CRACK – ULT
Figure 31: Load Case 1 + Constraint 1 (free edges).
Load Case 2: Asymmetrical Load Case 100/60 ORIGINAL – ULT
3.5” CRACK – ULT
Figure 32: Load Case 2 + Constraint 1 (free edges).
Load Case 3: Symmetrical Touch‐Down on Main Gear only. ORIGINAL – ULT
3.5” CRACK – ULT
Figure 33: Load Case 3 + Constraint 1 (free edges).
Load Case 4: Asymmetrical Touch‐Down on Main Gear only. ORIGINAL – ULT
3.5” CRACK – ULT
Figure 34: Load Case 4 + Constraint 1 (free edges).
10. STRESS FIELDS AT ULTIMATE LOAD – CRACK THROUGH THREE FASTENERS
In this section the baseline is compared to the shear web in which a crack that runs through three huckbolts has been modeled. This is intended to represent the crack in the spar carry‐through that was sent to ERAU for analysis. The constraints are free edges.
Load Case 1 Figure 35 shows that the stress drops between the affected fasteners. The overall shape of the stress field does change with the presence of the crack. The reader should be reminded that the huckbolts transfer the wing loads into the spar caps and there is no crack present in them. The load is ultimately transferred to the shear web by all of the fasteners, although in different magnitudes.
Load Case 2 Figure 36 reveals similar comments can be made as for Load Case 1.
Load Case 3 Figure 37 reveals similar comments can be made as for Load Case 1.
Load Case 4 Figure 38 reveals similar comments can be made as for Load Case 1.
Load Case 1: Symmetrical Load (12600 lbf) on each Wing – ULTIMATE LOAD ORIGINAL – ULT
FASTERNER CRACK – ULT
Figure 35: Load Case 1 + Constraint 1 (free edges).
Load Case 2: Asymmetrical Load Case 100/60 ORIGINAL – ULT
FASTERNER CRACK – ULT
Figure 36: Load Case 2 + Constraint 1 (free edges).
Load Case 3: Symmetrical Touch‐Down on Main Gear only. ORIGINAL – ULT
FASTERNER CRACK – ULT
Figure 37: Load Case 3 + Constraint 1 (free edges).
Load Case 4: Asymmetrical Touch‐Down on Main Gear only. ORIGINAL – ULT
FASTERNER CRACK – ULT
Figure 38: Load Case 4 + Constraint 1 (free edges).
11. STRESS FIELDS IN SPARS AT ULTIMATE LOAD In this section the spar caps of the baseline are compared to the two spar caps whose shear webs feature cracks. The three images reveal a very important clue about the nature of the load paths in the spar carry‐through that are discussed in more detail in Section 12.
ORIGINAL – ULT
3.5” CRACK – ULT
FASTERNER CRACK – ULT
12. CONCLUSION
There are several observations that can be made when inspecting the results from the FE analysis.
(1) Figure 25 and Figure 26 of Section 7 show that, generally, the stresses in the structure are in the elastic region. The local areas where stresses reach yield strength can most likely be attributed to the limit loads being at least 7% higher than what one would expect the airplane to experience when taking fuselage and HT lift3 into account. Of course, Federal Regulations requires the deformation of the structure to be elastic up to and including the limit load and this appears complied with. However, there is a high SC4 in the shear web near and below the lower wing attachment, probably caused by the local stiffness of the shear web flange and the joggle that accommodates the wing attachment mounts. While it is not being claimed this has exceeded the ultimate strength of the shear web (since the contour color is red), it is indicative of a high SC even at moderate flight loads. Based on this model, one would expect an increased likelihood of crack initiation and growth in this region.
As an example, consider an airplane consistently operated at low g‐loads. The “average” von Mises stresses in the shear web might be of the order of say 2500 psi at a 1g flight condition (at gross weight). However, in the region of the SC the prevailing stress level may be five to six–times higher, or between 12500 and 15000 psi. While this is well below the yield strength of the 2024‐T3 material, it may expedite fatigue in this area when compared to other regions of the structure, which would be manifested as the appearance of cracks.
(2) The predictions show that, at ultimate load, the FE structure fails in a number of locations by exceeding the ultimate strength of the aluminum, regardless of whether or not cracks are present. This does not include other failure modes (buckling, crippling, etc.) that might occur first, or load redistribution due to local yielding (which makes the structure behave non‐linearly, not captured by the linear analysis).
This begs the question: Why then perform a linear analysis in the first place? The answer is; (1) Non‐linear analysis requires more time and effort. (2) Linear analysis can reveal the nature of load paths, which is imperative for the structural engineer. And (3) the linear analysis will reveal whether or not a non‐linear analysis is needed. Inevitably the answer will be a yes or no, and here it is yes, something not known prior to its implementation.
Unfortunately, this renders it impossible for a linear analysis to answer the question: “can an airplane with a 4” long crack react ultimate loads?” According to the FE analysis the answer is no. But then again, neither will the intact airplane. Both fail per the FE analysis. The answer to the question, at this stage, calls for alternative means and this is discussed in bullet (4).
(3) The predictions show that in the presence of cracks, the high SCRs are relocated to the ends of the cracks. This means conditions for crack growth may persist and monitoring its length is the correct approach (as is currently stipulated by AD 95‐04‐03).
(4) The FE model reveals the nature of the major load paths and this must be clearly understood before proceeding. When considering the stress fields presented in Sections 9 and 10, it becomes apparent there is not a significant change in the shape or magnitudes of the SCRs. While the structure “accommodate” the cracks by retaining the high SC at each end, stress areas farther away pretty much remain identical. This reveals a very important aspect of the structural load path, further supported by considering the stress distribution in the spar caps in Section 11. They are all identical in shape and magnitude. It is almost as if the cracks are not present at all.
3 No, the HT is not generating “negative” lift at stall, even though the elevator is deflected trailing edge up. It is only generating a whole lot less lift in the upward direction than it would if the elevator were not deflected. The HT only generates downward lift a higher airspeeds (lower AOA). At lower airspeeds (high AOA) the opposite holds true. 4 SC = Stress Concentration, SCR = Stress Concentration Region.
It is easy to overlook that the huckbolts are not intended to transfer the flight loads directly to the shear web, but to the interface to the spar caps. From a certain point of view, the shear web is there for the ride, only to contribute by reacting asymmetric loads when needed. The major load path can be understood as follows: The flight loads are transferred to the wing mounts, which transfer them via the huckbolts to the spar caps, whose deformation transfers it via the multitude of fasteners to the shear web.
If the load paths of the FE model resemble those of the real airplane, one can conclude that there probably is not an imminent risk to aircraft that have developed cracks of the magnitude stipulated in AD 95‐04‐03, because the spar deformation will be transferred to the shear web by functional fasteners elsewhere in the structure. In other words; as long as the huckbolt‐to‐spar interface and spar caps remain intact, it is unlikely the structure will fail. In fact, based on the FE model, it is not unlikely that even larger cracks could be tolerated; although determining the geometry of such cracks was not within the scope of the research project.
Of course this is not intended to downplay the importance of a healthy shear web. Its role is absolutely imperative. However, using the FE model it can be argued that the spar carry‐through assembly appears more tolerant of cracks in the wing attachment area than what one might think at first glance. The high SCR in the lower shear web is of concern, and will likely cause continued crack growth in the area, slow or not, eventually requiring repairs.
REFERENCES
1. EFRC – ABS01, Evaluation of Residual Strength of Beechcraft Bonanza Spar Carry‐Through with Fatigue Cracks, S. Gudmundsson, et al., 2011. 2. MIL‐HDBK‐5J, METALLIC MATERIALS AND ELEMENTS FOR AEROSPACE VEHICLE STRUCTURES. DoD, 2003. 3. AD 95‐04‐03, FAA. 4. F33A Pilot’s Operating Handbook, Beechcraft.
APPENDIX A: Beech Bonanza Data
A three‐view drawing of the Beech F33C Bonanza.
A VL prediction of lift distribution of the Beech F33C Bonanza at stall.
CL of entire aircraft: CL = 1.35869 CL of selected surfaces: CL = 1.27093
The wing generates 1.27093/1.35869 = 0.9347 or some 93.47% of the total lift of the aircraft at stall.
Table of aircraft properties from Section 12 of Reference 1.
Parameter Symbol A36 Bonanza F33C Bonanza Wing area S 181 ft² 181 ft² Wing span B 33.5 ft 33.5 ft Spanwise location of the Mean Geometric Chord yMGC 8.22 ft 8.22 ft Design gross weight W0 3600 2800 Limit load factor n 4.4g 6.0g Ultimate load factor nult 6.6g 9.0g Inertia load at ultimate load factor nult ∙W0 23760 lbf 25200 lbf 5 Load per wing Fwing 11880 lbf 12600 lbf
Landing load factor nldg 3.0g 3.0g Inertia load at ultimate load factor nldg ∙W0 10800 lbf 8400 lbf Load per main landing gear leg 0.5∙nldg ∙W0 5600 lbf 4200 lbf
5 Here incorrectly assuming each wing carries the entire half load (i.e. fuselage and horizontal tail contributions are ignored).
APPENDIX B: Von Mises Yield Criterion
In materials science and engineering the von Mises yield criterion can be also formulated in terms of the von Mises stress or equivalent tensile stress, σv, a scalar stress value that can be computed from the stress tensor. In this case, a material is said to start yielding when its von Mises stress reaches a critical value known as the yield strength, σy. The von Mises stress is used to predict yielding of materials under any loading condition from results of simple uniaxial tensile tests. The von Mises stress is calculated using the following expression:
2 2 2 1 2 2 3 1 3 v 2
Where: i, i = 1,2,3 are the stresses along three principal orthogonal axes.