A novel optimization strategy for composite beam type landing gear for light aircraft Edwin Spencer * United Technologies Aerospace Systems 850 Lagoon Drive Chula Vista Ca. 91910 Abstract Composite beam type landing gear have applications in small light aircraft. A novel optimization strategy using Nastran finite element models and sol 200 (optimization solution sequence) is presented in this paper. The analysis criteria are to meet stiffness strength and weight requirements for multiple landing load cases. The optimization analysis strategy is a novel two stage approach. In the first stage a shape optimization is carried out on a tapered rectangular hollow cross section beam finite element model. Each beam element had 8 design variables: height, width, cap thickness and web thickness at each end of the beam. The design constraints for the optimization analysis is a specified vertical deflection at the wheel attachment to meet landing energy absorption criteria and stresses below allowable for all elements for all load cases. The optimum cross-sections from the beam analysis are used to create a 3D composite shell finite element model with multiple ply lay-up property regions for the second stage of the optimization analysis. In the second stage a ply thickness optimization analysis was carried out on the 3D composite shell finite element model. The design variables are the thicknesses of zero degree, + / - 45 degree and 90 degree plies for the various property regions. The objective function for both shape optimization and ply thickness optimization was to minimize weight. *Currently employed at UTAS, work for this paper was carried out while the author was formerly employed at MSC Software.
Introduction Landing gear for small light aircraft need to satisfy minimum weight criteria, strength criteria with adequate margin of safety against failure and a precisely tuned stiffness or deflection on landing to absorb the landing energy. Laminate composite structures are ideal for this application because the material system can be tailored to meet the specific requirements in a very predictable manner. To design the landing gear structure to meet these requirements manually by trial and error will be very labor intensive and extremely time consuming. Finite element laminate composite analysis coupled with optimization techniques offer a robust and efficient methodology to design a beam type landing gear. Optimization Concept A novel two stage optimization strategy using Nastran finite element models and Sol 200 optimization solution sequence has been devised which can be used to design any generic beam type landing gear for multiple landing load cases. In the first stage the landing gear structure is modeled with tapered rectangular hollow cross section beam elements (see Figure 1). Each beam has 8 design variables : height, width, cap thickness and web thickness at each end of the beam which are initially set to nominal values and a shape optimization is carried out whereby the cross sectional shape can change span-wise as shown on Figure 2 which is a 3D rendering of the optimized beam structure. The optimum cross-sections from the beam analysis (Figure 2) are used to create a 3D composite shell finite element model with multiple ply lay-up property regions (Figure 3 and Figure 4) for the second stage of the optimization analysis. In the second stage a ply thickness optimization analysis was carried out on the 3D composite shell finite element model. The design variables are the thicknesses of zero degree, + / - 45 degree and 90 degree plies for the various property regions. The objective function for both shape optimization and ply thickness optimization was to minimize weight.
Figure 1 Tapered Rectangular Hollow Cross Section Shape Optimization Beam Model
Figure 2 3D Rendering of Shape Optimized Beam Model
Figure 3 3D Composite Shell Finite Element Model Lofted from Optimized Shape Beam Model
Figure 4 Ply Lay-Up Property Regions for Thickness Optimization
Shape optimization Tapered Box Beam Finite Element Model: This is a full model consisting of 32 tapered box section beam elements and 33 nodes. The MSC Nastran solver solution sequence 200 was used. This model uses equivalent isotropic material properties of a typical ply lay-up. This is a shape optimization analysis model the purpose of which is to obtain an optimum shape while meeting the stiffness constraint of 14.5 in at load application point for the 3g ult. vertical load case and also stress constraints of tensile, compressive and shear stresses below the material allowable for the four load cases considered: 1. 3g. Ultimate Vertical load (Figure 5) 2. Max. Drag Load (Figure 6) 3. Max. Side Load (Figure 7) 4. Max. Vertical Load (Figure 8) The loads and boundary conditions for each of the above four cases are shown on Figure 5 to Figure 8. Although the model is symmetric about the aircraft center line a full model has to be used because the loading is un-symmetric for the side load case (case 3). Each beam cross section has 8 design variables: width, height, cap & web thickness at ends A & B as shown on Figure 1. The cross section of end B of each beam is constrained to be same as End A of adjacent beam so that a smooth shape transition is created. Figure 2 shows a 3d plot of the optimized shape beam model. The cross sectional dimensions of this beam model is used to create the 3d lofted surfaces of the composite shell model which is optimized for ply thickness in a further analysis step.
Figure 5-3g Ultimate Vertical Load (lbf)
Figure 6 - Max. Drag Load lbf (ultimate)
Figure 7 - Max. Side Load lbf (ultimate)
Figure 8 - Max. Vertical Load lbf (ultimate)
Ply Thickness Optimization 3d Composite Shell Finite Element Model: Figure 3 shows the 3d composite shell finite element model lofted from the cross sections of the optimized shape beam model shown in Figure 2. The model consists of 72136 predominantly quadrilateral shell elements and 72610 nodes. The MSC Nastran solver solution sequence 200 was used, version 2008 R3 (or higher) of the software is required in order to use the composite failure indices in the optimization routine. The main purpose of this model is to calculate the optimum ply thicknesses while meeting the stiffness constraint of 14.5 in at load application point for the 3g ult. vertical load case and a stress constraint of failure index below 0.95 for all elements for the four load cases considered: 1. 3g. Ultimate Vertical load (Figure 5) 2. Max. Drag Load (Figure 6) 3. Max. Side Load (Figure 7) 4. Max. Vertical Load (Figure 8) The loads and boundary conditions for each of the above four cases are identical to that used for the beam model as shown on Figure 5 to Figure 8. The model was broken up into spanwise regions and each span-wise region was further broken into cap region, corner region and web region (Figure 4). There are 30 such property regions in the finite element model and they are symmetrical about the aircraft centerline. Each of the thirty property regions was assigned a nominal thickness 6 ply symmetric lay-up, [0/ + - 45 /90 ]s. The design variables are the thicknesses of 0 s, 45 s and 90 s plies, the optimization analysis calculates the required thicknesses of 0 s,45 s and 90 s from which the number of plies of each are backed out. Having obtained the required number of plies in each orientation a reasonable stacking sequence and ply drop off details have to be then determined.
Nastran Sol 200 Cards For Design Response, Design Constraint, Design Variable and Design Variable to Property Relation The model has 30 property regions and 6 plies in each region. For each of 6 plies in each of 30 property regions we need to set up 2 design responses, one for failure index based on normal stresses and the other failure index based on interlaminar shear stress. Therefore there are 30*6*2 = 360 response items to set up with Dresp1 cards. DRESP1,1,FP, DRESP1,360,FP, cfailure,pcomp,,5,1,1 cfailure,pcomp,,7,6,30 All of these response items ( failure indices) are limited to max value 0f 0.95 with as many DCONSTR cards ( 360 of them): DCONSTR,200,1,,0.95 DCONSTR,200,3606,,0.95 Z Displacement of a certain node ( 248123) is set up with another Dresp1 card and constrained to have max and min value set by another DCONSTR card. DRESP1,999,ZDISP,DISP,,,3,,248123 DCONSTR,100,999,14.24,14.26 The weight is set up as response number 1000 : DRESP1,1000, WEIGHT, WEIGHT, This is a symmetric layup [ 0/ + - 45 /90]s and there are 30 regions, so there are 30*3 =90 design variables to set up : DESVAR,1,TPLY,0.006,0.006,1.0 DESVAR,90,TPLY,0.006,0.006,1.0 There are 90 DVPREL1 cards needed to relate each design variable to a property card ( Pcomp) field DVPREL1,1,PCOMP,1,13,,,,,+ +,1,1.0 DVPREL1,90,PCOMP,30,23,,,,,+ +,90,1.0
Figure 9 Vertical Displacement
Figure 10 Max. Failure Index 3G Ultimate Vertical Load Case
Figure 11 Max. Bond Failure Index 3G Ultimate Vertical Load Case
Figure 12 Max. Failure Index Max. Drag Ultimate Load Case
Figure 13 - Max. Bond Failure Index Max. Drag Ultimate Load Case
Figure 14 - Max. Failure Index Max. Side Load Ultimate Load Case
Figure 15- Max. Bond Failure Index Max. Side Load Ultimate Load Case
Figure 16 - Max. Failure Index Max. Vertical Load Ultimate Load Case
Figure 17- Max. Failure Index Max. Vertical Load Ultimate Load Case
Figure 18 Eigen Value buckling Load Factor Max. Side Load Ultimate Load Case
FEA Results Versus Test Strain Correlation Strain Gauge Location Test Result Micro Strain FEA Result Microstrain SG 13 Bot. Cap, Y = 0.0 2500 2420 SG 11 Top Cap, Y= 0.0-2230 -2050 SG 23 Bot. Cap, Y = 30.43 2818 3270 SG21 Top Cap. Y=31.1-2500 -2490 SG 33 Bot. Cap, y= 41.99 3068 3840 SG 31 Top Cap, Y=43.08-2700 -3260 SG53 Bot. Cap, Y= 60.3 1750 1830 Conclusions A robust and easy to implement shape/composite ply thickness optimization strategy has been developed for beam type landing gear to meet stiffness, strength and weight criteria for multiple landing load cases. The predicted strains from the finite element analysis compared favorably with strain gauge measurements on a test article. Eigen value buckling analysis was carried out on the optimized structure independent of the optimization solution. For future development, the buckling check could be also be included in the optimization routine.