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, cfailure,PCOMP,,5,1,1

DRESP1,360,FP, 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.