REDUCED BASIS TECHNIQUE FOR EVALUATING THE SENSITIVITY COEFFICIENTS OF THE NONLINEAR

TIRE RESPONSE

Ahmed K. N o d . John A. Tanner** and Jeanne M. peterst NASA Langley Research Center

Harnpton, Virginia

Abstract

An efficient reduced basis technique is presented for calculating the sensitivity of nonlinear tire response to variations in the design variables. The tire is discretized by using three-field mixed finite element models. The vector of structural response and its first-order and second- order sensitivity coefficients (derivatives with respect to design variables) are each expressed as a linear combination of a small number of basis (or global approximation) vectors. The Bubnov-Galerkin technique is then used to approximate each of the finite element equations governing the response, and the sensitivity coefficients, by a small number of algebraic equations in the amplitudes of these vectors. The path derivatives (derivatives of the response vector with respect to path parameters, e.g., load parameters) are used as basis vectors for approximating the response. A combination of the path derivatives and their derivatives with respect to the design variables is used for approximating the sensitivity coefficients. The potential of the proposed technique is discussed and its effectiveness is demonstrated by means of a numerical example of the Space Shuttle nose-gear tire subjected to uniform inflation pressure. The design variables are selected to be the material properties of the cord and rubber as well as the cord diameters, end counts, and angles.

Nomenclature

bo to b3 parameters used in defining the cord

- - end counts (epi) bl, b2 geometric characteristics of the Space

Shuttle nose-gear tire cross section, see Fig. 2

4, d2 cord diameters EL. ET elastic moduli of the individual layers

in the direction of the cords and normal to that direction, respectively

ETO value of ET at s=C=O

*Ferman W. Perry Professor of Aerospace Structures and Applied Mechanics, and Director, Center for Computational Structures Technology, University of Virginia; Fellow AIAA **Head, Landing and Impact Dynamics Branch, NASA Langley Research Center

t ~ e n i o r Programmer Analyst, University of Virginia

epi GLT, GTI

Young's moduli of the rubber and cord, respectively subvector of strain parameters for the shell model cord end count (ends per inch) shear moduli in plane of cords and normal to it shear moduli of the rubber and cord, respectively

vectors of nonlinear terms of the re-

duced systems, see equations (7)-(9),

(11) and (21) vector of nonlinear terms, see equation (1) subvector of stress-resultant para- meters for the shell model total thickness of the tire cross section thickness of the tire cross section at C=0 = hho global linear structure matrix which includes linear stiffness and strain- displacement matrices; see equations (1) and (A.2), Appendix A linear stiffness matrix, see equations (A.2), Appendix A

linear matrices of the reduced systems, see equations (7)-(lo), (13) and (17) bending (and twisting) stress resultants subvectors of nonlinear terms, see equations (A.3), Appendix A extensional stress resultants subvector of normalized applied loads, see equations A.4, Appendix A intensity of normal pressure loading normalized load vector, see equation

(1)

load vectors of the reduced system, see equations (7)-(9), (12), (16) and (22) load parameter linear matrix containing products of integrals of shape functions, see equations A.2, Appendix A

number of basis vectors used in evaluating the response linear strain-displacement mamix, see equations A.2, Appendix A meridional coordinate of the tire cross section total strain energy of the shell model displacement components of the reference surface of the shell in the meridional, circumferential and normal directions subvector of nodal displacements of the shell model coordinate normal to the shell reference surface response vector of the shell model, which includes the subvectors of strain parameters, stress-resultant parameters and nodal displacements

matrices of basis vectors, see equ- ations (4)-(6). (23)-(25) circumferential (hoop) coordinate of the shell model cord angle with the s axis, see equations (29) - parameters used in defining 9 material and geometric parameters of the tire constituents (material properties of the cord and rubber as well as the cord diameters, end counts and angles) major Poisson's ratio of the individual layers dimensionless coordinate along the meridian of the tire cross section rotation components of the middle surface of the shell

vectors of undetermined coefficients of the reduced equations (amplitudes of the global approximation vectors), see eqs.(4)-(6)

t denotes matrix transposition s' denotes order of path derivatives used to construct

basis vectors (see Appendix B)

Ranees of Indices I, J, L : 1 to the total number of degrees of freedom

(nodal displacements, strain parameters and stress resultant parameters) 1 to the total number of reduced degrees of i , j . freedom used in evaluating the sensitivity coefficients

! : 1 to the total number of material and geometric parameters of the tire constituents

Introduction

Significant advances have been made in the development of effective computational models and computational strategies for the numerical simulation of the nonlinear tire response (see, for example, Refs. 1- 5). However, the use of nonlinear analysis in automated optimum design of tires requires the availability of efficient techniques for calculating the sensitivity of the nonlinear tire response to variations in the design variables. The sensitivity coefficients (derivatives of the response vector with respect to design variables) are used to:

a) determine a search direction in the direct application of nonlinear mathematical programming algorithms. When approximation concepts are used for optimum design of tires, sensitivity derivatives are used to construct explicit approximations for the critical and potentially critical, behavior constraints.

b) generate an approximation for the response of a modified tire (in conjunction with a reanalysis technique);

c) assess the effects of uncertainties, in the material and geometric parameters of the computational model, on the tire response; and

d) predict the changes in the tire response due to changes in the parameters.

Two general procedures are currently being used for calculating the sensitivity coefficients of the nonlinear structural response. The two approaches are (see, for example, Refs. 6-14): direct differentiation method and adjoint variable method. The first procedure is based on the implicit differentiation of the equations that describe the nonlinear structural response with respect to the desired parameters and the solution of the resulting sensitivity equations. In the adjoint variable method an adjoint physical system is introduced whose solution permits rapid evaluation of the desired sensitivity coefficients. Both procedures can be applied to either the governing discrete equations or to the functional of the variational formulation of the structure (with a consequent change in the order of discretization and implicit differen tiation).

The discrete models for an aircraft tire typically have large number of degrees of freedom, and the calculation of the sensitivity coefficients of the nonlinear tire response can become quite expensive. Although efficient reduced basis techniques have been developed for substantially reducing the number of degrees of freedom in the initial discretization and the computational effort involved in the nonlinear analysis of tires (see, for

example, Refs. 1 ,2 and 15), no attempt has been made to adapt these techniques to the calculation of sensitivity coefficients. The present study is an attempt to fill this void. Specifically, the objective of this paper is to present a reduced basis technique and a computational procedure for the efficient calculation of the first-order and second-order sensitivity coefficients of the nonlinear tire response (derivatives of the tire response with respect to variations in the material and geometric parameters of the tire). The crux of the technique is the effective choice of basis vectors for approximating the sensitivity coefficients. The path derivatives (derivatives of the tire response vector with respect to a path parameter, e.g., load parameter) are used as basis vectors for approximating the response. A combination of the path derivatives and their derivatives with respect to the material and geometric characteristics of the tire constituents (namely, the tire cord and rubber) is used for approximating the sensitivity coefficients. Numerical results are presented for the case of uniform inflation pressure on the Space Shuttle orbiter nose-gear tire. Both the first-order and second-order coefficients for the generalized displacements and stress resultants are given. The design variables are selected to be the material properties of the cord and rubber as well as the cord diameters, end counts and angles.

Mathematical Formulation

Governing Finite Element Eauations

The tire is modeled by using a two-dimensional, moderate rotation, laminated anisotropic shell theory with the effects of variation in material and geometric parameters included5. A total Lagrangian formulation is used for describing the deformation, and the fundamental unknowns consist of generalized displacements, strain components, and stress resultants of the tire. The governing finite element equations are obtained through the application of the three-field Hu-Washizu mixed variational principle. The governing finite element equations for the response vector and its first-order and second-order sensitivity coefficients can be written in the following compact form:

and

where [K] is the global linear structure matrix which includes "generalized" stiffness matrices of the tire; (Z] is the response vector which includes strain parameters, stress resultant parameters, and nodal displacements; (G(z)) is the vector of nonlinear terms; q is a load

parameter; (QJ is the normalized applied load vector; h refers to a design variable, selected to be any of the material properties of the cord and rubber, cord diameters, end counts and angles. The subscript 1 in equations (2) and (3) signifies the dependence of the vectors and matrices on the particular choice of the design variable. For convenience, the subscript 1 is not, and will not be, used with the individual h's in the equations. However, in succeeding discussions he is used to denote a single design variable. The form of the arrays [K], (G(z)) and (Q} is described in Appendix A. Note that in equations (2) and (3), (~ (z )} is assumed to be independent of he, which is a unique feature of the mixed formulation of the problem. Equations (1) are nonlinear in (Z), but equations (2) and

(3) are linear in - and - , respectively. EL I;:\, Basis Reduction and Reduced System of Eauations

The response vector, {Z}, and its first-order and second-order derivatives with respect to he,

, are each expressed as a linear

combination of a few preselected basis vectors. The approximations can be expressed by the following transformations:

The columns of the matrices [TI. [F](and[:]( in equations (4)-(6) are the basis vectors; and the elements of

the vectors (w}, (@}( and are the amplitudes of the approximation vectors which are, as yet, unknowns. Note that the number of basis vectors in equations (4)-(6) is considerably smaller than the total number of degrees of freedom (components of the vectors ( Z ) , [:It and e ).

A Bubnov-Galerkin technique is now used to replace the governing equations for the tire response and its first-order and second-order sensitivity coefficients,

equations (1)-(3), by the following reduced equations in

the unknowns ( y ), (W)L and {GIt:

expression in terms of [w), equations (4). The vector ( ~ ( y , v ) ) ~ in equations (21) is , obtained - by replacing (Z)

in the vector az, a an an . .-

expressions in terms of { y ) and {v)l, equations (4) and

(5). [[a + [%I ] i wlL = - [gIe { V) + ($1 (8)

Selection and Generation of Basis Vectors [ I + - [ah] j (= + - [a"] - ( w ) - 2 [aE] W j ah2 e ax L

The effectiveness of the proposed technique for calculating the sensitivity coefficients depends, to a great extent, on the proper choice of the basis vectors (the

columns of the matrices [TI, [?If and [=I r 1). An effective choice for the basis vectors used in approximating the response vector (Z), equations (4), was found to be the various-order path derivatives (derivatives with respect to the control parameter q); i.e., the matrix [r] used in approximating (Z), over a range of values of q, includes the response vector corresponding to a particular value of q (viz. qO) and its various-order derivatives with respect to q, evaluated at the same value of qO, or

where

az a% a3z [r] = (z) - - - [ L ) [ a q 2 1 [ a q 3 1 j q o (23)

The basis vectors for approximating the

sensitivity coefficients - and - were selected to (3 (::It be a combination of:

a) the various-order path derivatives, i.e., the columns of the matrix [r]; and

b) the fist derivatives of [T] with respect to A,,

i.e., [$I,: and in the case of - I f 1, c) the second derivatives of [T] with respect to

1 , i . . , ] merefore:

The number of basis vectors in [& and [$, for Note that equations (7) are no&ear in the unknowns {w) each I , is two and three times, respectively, the number

of basis vectors in [r]. The rationale for the particular choice of

and equations (8) and (9) are linear in (W)! and (@If, respectively. The vector { ~ ( y ) ) in equations (1 1) is

obtained by replacing {Z) in the vector ( ~ ( 2 ) ) by its [flf and [qf is based on the following two facts:

a) Differentiating equations (4) with respect to h leads to:

i.e, the expression for includes both [r] and

and the expression for - i n c l u d e s I;:l, [r], - and - ; and [::I, [21

b) the use of free parameters . . ($1 and . a 0 1 ,

instead of the fixed amplitudes (y), - and 7 i n I , [Zb equations (26) and (27), is expected to improve the

accuracy of the approximations for - and - , over I El, . . a wide range of q. The free parameters {vI1 and (?If are

obtained by applying the Bubnov-Galerkin technique to equations (2) and (3), resulting in equations (8) and (9).

The path derivatives (columns of the matrix [r]) are obtained by successive differentiation of the governing finite element equations of the tire, equations (I), with respect to the parameter q. The recursion relations for evaluating the path derivatives are given in Ref. 16. The first three recursion relations are summarized in Appendix B. Note that only one matrix factorization is needed for generating all the path derivatives.

The derivatives of [TI with respect to hi,

recursion relations for evaluating the path derivatives with respect to kt. The resulting equations have the same left- hand sides as those of the original recursion relations, and therefore, no .- addition$ .. ma@x factorizations are needed for

generating Fl d I$], . (see Appendix B). ah f

The computational procedure for generating the nonlinear response vector, {Z), and its sensitivity

coefficients, k), and I"/ , can be conveniently ah2 1

divided into two distinct phases, namely: a) evaluation of the basis vectors, at a particular

value of q (viz, qO) and generation of the reduced equations; a d

b) marching with the reduced equations in the solution space and generating the response and sensitivity coefficients at different values of q.

For each value of q, the vector of reduced unknowns, (w) is obtained by solving the reduced nonlinear equations, equations (7). Then the vectors

($1 and {& associated with the same value of ( w ), are evaluated by solving the reduced linear equations, equations (8) and (9). The response vector and its

sensitivity coefficients, P I , ) are

obtained by using equations (4)-(6). The process is repeated for different values of q.

Numerical Studies

To test and evaluate the effectiveness of the proposed reduced basis technique, the sensitivity coefficients of the nonlinear response of the Space Shuttle nose-gear tire was generated by this technique. Comparisons were made with the sensitivity coefficients obtained by using the full system of equations of the finite element model. Typical results are presented in Figs. 4-12 for the case of uniform pressure, po, on the Space Shuttle nose-gear tire. The geometric and material characteristics of the tire are given in Table 1 and Fig. 2. Because of symmetry, only half the tire cross section was modeled. The tire cross section was divided into seven segments as shown in Fig. 3. Each segment contained a different number of layers, different material properties corresponding to different cord content in the composite, and varying cord orientations (see Refs. 5 and 17). Spline interpolation was used to smooth the experimental data that was used to define the geometric and material characteristics of the two-dimensional shell model (see Refs. 5 and 17). The outer surface of the tire was chosen to be the reference surface for the shell model.

The numerical studies were performed using three-field mixed finite-element models for the discretization of the tire in the meridional direction. Linear interpolation functions are used for approximating each of the stress resultants and strain components, and quadratic Lagrangian interpolation functions are used for approximating each of the generalized displacements. The integrals in the governing equations are evaluated using a two-point Gauss-Legendre numerical quadrature formula. Sixty finite elements were used in modeling half the tire

cross section for a total of 960 strain parameters, 960 = 1 to 15) and their first-order derivatives with stress-resultant parameters, and 560 nonzero generalized displacements. The three displacement components in the respect to hf . For the second-order sensitivity region 0.45 1 E, 4 0.5 are totally restrained, and in coefficients, the forty-five basis vectors consist of the addition, the rotation components are restrained at 6 = 0.5. thirty basis vectors used in approximating the first-order - - -

sensitivity coefficients, and the second derivatives of The cord end counts, epi, and cord angles with -

the meridional direction, 0, were approximated by the (3) with respect to Lf. following formulas:

An indication of the accuracy of the meridional

epi = bo + blS + b2k2 + b3k3 distributions of the displacements and stress resultants, u,

(28) v, w, N,, Ne, Nse, Ms, Me, h& obtained by the reduced 0 = 00 - 016 - 025z (29) basis technique is given in Fig. 10. The displacements

and stress resultants obtained by using fifteen basis where 00 = 54.38, 01 = 3.884, 02 = 148.96; and the vectors are almost indistinguishable from those obtained numerical values of bo, bl, b2 and b3 are given in Table 2. by solving the full system of finite element equations.

Figure 4 shows plots of the loading versus the maximum normal displacement, w,, and the total strain energy, U, as well as their first-order and second-order sensitivity coefficients with respect to the cord diameters dl, d2; and the material parameters E,, &, G,, G,; where subscripts r and c refer to rubber and cord, respectively. As can be seen from Fig. 4, the nonlinear tire response is more sensitive to variations in d2 and E, than to the other cord diameter and material properties.

Figures 5 and 6 show plots for the loading versus the first-order and second-order sensitivity coefficients with respect to the parameters 00,0 1, 02, and bo, bl, b2, b3 for region 2. As can be seen from Figs. 5 and 6, the nonlinear tire response is considerably more sensitive to variations in bo and o0 than to all the other b's and 0's.

The basis vectors for both the response and the sensitivity coefficients were generated for the unloaded tire, po=O, and were thus obtained by solving a linear system of finite-element equations. Then the reduced equations for evaluating the response and the sensitivity coefficients were generated. The basis vectors were not updated throughout the range of loading considered. An indication of the accuracy of w, and U, as well as first- order and second-order sensitivity coefficients obtained by the reduced basis technique is given in Figs. 7-9.

For the range of loading considered, w, and U obtained by using fifteen basis vectors are almost indistinguishable from those obtained by using the full system of finite element equations. The first-order and second-order sensitivity coefficients (first-order and second- order derivatives of w, and U with respect to d2, E,, E,, O0 and bo of region 2), obtained by using thirty and forty-five basis vectors, are as accurate as w, and U obtained by using 15 basis vectors (r=15). For the first-order sensitivity coefficients, the thirty basis vectors consist of the first fifteen derivatives of (Z) with respect to p,,

An indication of the accuracy of the meridional distributions of the first-order and second-order derivatives of U, V, W; with respect to dl, d2, E,, E,, 00,02 obtained by the reduced basis technique is given in Figs. 11 and 12. The high accuracy of the first-order and second-order sensitivity coefficients predicted by the reduced basis technique with r=15 is clearly demonstrated in Figs. 11 and 12.

The computational time associated with the foregoing technique is considerably less than that associated with the direct application of equations (2) and (3). This is particularly true when the sensitivity coefficients are needed at several different values of po. This is because the decomposed full-structure matrix on the left-hand-sides of equations (2) and (3) is needed for each value of po at which the sensitivity coefficients are required. Unless the response vector, (Z), is obtained by solving the full system of nonlinear equations, equations (I), using the Newton-Raphson technique the decomposed left-hand-side matrix of equations (2) and (3) is not readily available (which is the case when the response vector is obtained by using either the quasi-Newton method or the reduced basis technique). By contrast, in the foregoing technique the most time-consuming operations are those associated with the generation of the basis vectors and reduced equations. The computational time expended in the solution of the reduced equations for the response and the sensitivity coefficients is very small. Since the basis vectors are evaluated at p,=O, their generation requires only the decomposition of the linear matrix of the tire model, [K]; evaluation of the right-hand-sides of the recursion relations for the basis vectors, and forward reduction-back substitution. These operations are computationally less expensive than the decompositions of the left-hand-side matrices of equations (2) and (3).

Potential of the Foregoing Reduced Basis Technique

The foregoing reduced basis technique appears to have high potential for use in automated analysis and design of tires. The numerical studies conducted clearly demonstrate the accuracy and effectiveness of the technique. In particular, the following comments seem to be in order:

1. The particular choice of the basis vectors used herein for approximating the first-order and second-order sensitivity coefficients (columns of the matrices

and I? f) allows the accurate prediction of these F l c [=I derivatives for a wide range of values of the control parameters.

2. The computational procedure developed for predicting the nonlinear response using the reduced basis technique (see, for example, Ref. 16) can now be extended to the prediction of the sensitivity derivatives as well. When a new set of basis vectors for approximating the response is generated, the derivatives of each of these vectors with respect to the design variables (columns of

the mamx - ) are generated as well. [El 3. As pointed out in the preceding section on

numerical studies, the computational time associated with the foregoing technique is considerably less than the direct application of equations (2) and (3). This is particularly true when the reduced basis technique is used in generating the nonlinear response, and when the sensitivity coefficients are needed at several different values of the load parameter.

4. The foregoing technique can be easily extended to the evaluation of the cross second-order derivatives, as well as to structural reanalysis of tires (analysis of modified tires). For the cross second-order derivatives the matrix of basis vectors, equations (25), includes the derivatives with respect to more than one design variable. However, in order to keep the number of basis vectors small only a few of the cross derivatives should be used at any one time. In structural reanalysis

the set of basis vectors included in [& and [T are used for predicting the nonlinear response of the modified structure.

Concludine Remarks

An efficient reduced basis technique is presented for calculating the sensitivity of nonlinear tire response to variations in the design variables. The tire is modeled by using a two-dimensional, moderate rotation, laminated anisotropic shell theory with the effects of variation in material and geometric parameters included. A total Lagrangian formulation is used for describing the deformation and the fundamental unknowns consist of generalized displacements, strain components and stress

resultants of the tire. The governing finite element equations are obtained through the application of the three- field Hu-Washizu mixed variational principle.

The vector of structural response and its first- order and second-order sensitivity coefficients (derivatives with respect to design variables) are each expressed as a linear combination of a small number of basis (or global approximation) vectors. The Bubnov-Galerkin technique is then used to approximate each of the finite element equations governing the response, and the sensitivity coefficients, by a small number of algebraic equations in the amplitudes of these vectors. The path derivatives (derivatives of the response vector with respect to a path parameter) are used as basis vectors for approximating the response. A combination of the path derivatives and their derivatives with respect to the design variables is used for approximating the sensitivity coefficients.

The potential of the proposed technique is discussed and the effectiveness of the basis vectors used in approximating the sensitivity coefficients is demonstrated by means of a numerical example of the Space Shuttle nose-gear tire subjected to uniform inflation pressure. The design variables are selected to be the material properties of the cord and rubber as well as the cord diameters, end counts and angles.

&-pendix A - Form of the Arravs in the Governing Discrete Eauations of the Tire

The governing discrete equations of the tire, equations (I), consist of: the relations between the stress- resultants and the strain components of the tire; the strain- displacement relations, and the equilibrium equations of the tire. The response vector (Z], can be partitioned into the subvectors of strain parameters, (E), stress-resultant parameters, (H) , and nodal displacements, (X) , as follows:

The different arrays in equations (1) can be partitioned as follows:

Table B. 1

S' Ry)

where [KO] is a linear stiffness matrix; [R] is a linear matrix containing integrals of products of shape functions; [S] is the linear strain-displacement matrix; (M (x)) and (N (H,x)) are the subvectors of nonlinear terms; (P) is the subvector of normalized applied loads; a 0 refers to a null matrix or vector; and superscript t denotes transposition. Note that the strain components and stress resultants are allowed to be discontinuous at interelement boundaries, and therefore, can be eliminated on the element level.

Ap~endix B - Evaluation of Path Derivatives and

Their First Derivatives with Respect to hl

The path derivatives in equations (23) are obtained by successive differentiation of the governing finite element equations, equations (I), with respect to the parameter q and solving the resulting system of linear algebraic equations. The recursion relations for the path derivatives can be written in the following compact form:

where 1,J range from 1 to the total number of degrees of freedom in the model. The explicit forms of the

components of R!", for s1=1,2 and 3, are given in Table B.1.

In Table B.l a repeated uppercase Latin index denotes summation over its full range and GI are bilinear (or quadratic) functions of Zj.

The derivatives of the path derivatives are obtained by differentiating equations (B. 1) with respect to the design variables hi. The resulting equations can be written in the following compact form:

where h refers to any of the design variables and s' > 0. For the case when sl=O, the sensitivity derivatives are given in equations (2). For convenience, the subscript i of hi has been dropped in equations (B.2).

The work of the first and third authors was partially supported by NASA Cooperative Agreement NCCW-0011 and by NASA Grant NAG-1-1 197. The numerical studies were performed on the CRAY-YMP computer at the San Diego Supercomputer Center.

References

Noor, A. K. and Tanner, J. A., "Advances and Trends in the Development of Computational Models for Tires," Computers and Structures, Vol. 20, NOS. 1-3, 1985, pp. 517-533.

Noor, A. K., Andersen, C. M. and Tanner, J. A., "Exploiting Symmetries in the Modeling and Analysis of Tires," Computer Methods in Applied Mechanics and Engineering, Vol. 63, 1987, pp. 37- 81. Noor, A. K. and Tanner, J. A., "Advances in Contact Algorithms and Their Application to Tires," NASA TP-2781, April 1988. Grigolyuk, E. I. and Kulikov, G. M., Multilayered Reinforced Shells. Analysis of Pneumatic Tires, Mashinostroenie, Moscow, 1988 (in Russian). Noor, A. K., Kim, K. 0. and Tanner, J. A., "Analysis of Aircraft Tires via Semianalytic Finite Elements," Finite Elements in Analysis and Design, Vol. 6, 1990, pp. 217-233. Haftka, R. T. and Mroz, Z., "First- and Second-Order Sensitivity Analysis of Linear and Nonlinear Structures," AIAA Journal, Vol. 24, No. 7, July 1986, pp. 1187-1 192. Szefer, G. , Mroz, Z. and Demkowicz, L., "Variational Approach to Sensitivity Analysis in Nonlinear Elasticity ," Arch. Mech., Vol. 39, No. 3, Warsaw 1987, pp. 247-259.

8. Wu, C. C. and Arora, J. S., "Design Sensitivity Analysis and Optimization of Nonlinear Structural Response Using Incremental Procedure," AIAA Journal, Vol. 25, No. 8, Aug. 1987, pp. 11 18-1 125.

9. Cardoso, J. B. and Arora, J. S., "Variational Method for Design Sensitivity Analysis in Nonlinear Structural Mechanics," AIAA Journal, Vol. 26, No. 5, May 1988, pp. 595-603.

10. Choi, K. K. and Santos, J. L. T., "Design Sensitivity Analysis of Nonlinear Structural Systems, Part I: Theory, and Part 11: Numerical Method." International Journal for Numerical Methods in Engineering. Part I - Vol. 24, 1987, pp. 2039-2055; Part 11: Vol. 26, 1988, pp. 2097-2114.

11. Haftka, R. T. and Adelman, H. M., "Recent Developments in Structural Sensitivity Analysis," Structural Optimization, Vol. 1, 1989, pp. 137- 15 1.

12. Tortorelli, D. A., Haber, R. B. and Lu, S. C.-Y., "Adjoint Sensitivity Analysis for Nonlinear Dynamic Thermoelastic Systems," AIAA Journal, Vol. 29, No. 2, Feb. 1991, pp. 253-263.

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14. Noor, A. K., Tanner, J. A. and Peters, J. M., "Sensitivity of Tire Response to Variations in Material and Geomemc Parameters," Finite Elements in Design and Analysis (to appear).

15. Noor, A. K., Andersen, C. M. and Tanner, J. A., "Mixed Models and Reduction Techniques for Large- Rotation Nonlinear Analysis af Shells of Revolution with Application to Tires," NASA TP- 2343, 1984.

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Table 1 - Values of Elastic Constants of Tire Constituents Used in Present Study

Tire Young's Shear Poisson's Constituent Modulus, Modulus, Ratio

E(P4 G(P4

Rubber 3.10x106 1.04x106 0.49

Nylon 2.41x109 4.83x106 0.66 cord

"Since the deformations are small in the bead area, it is reasonable to assume that the bead wires are isotropic.

Table 2 - Numerical values of the coefficients used in approximating the end count for the Space Shuttle nose-

gear tire used in the present study (see equation (28)).

Region bo bl b2 b3

Note: Regions 1 and 3 refer t the top and bottom two layers of the shell model (which have cords); and Region 2

refers to all the other layers with cords.

89 Pe

External loading

Oeneralized displacements Stress resultants

Figure 1 - Two-dimensional shell model of the tire and sign convention for the external loading, generalized displacemen~ and stress resultants.

- b - 38.54 cm -1 - I b, = 10.465 cm reference surface 1 I

at a s=E,=O s = 24.79 ern h, =_ 1 .go8 crn E, = 0.5 r = b 1

I

I I

C-------- 6, ---i

boundary conditions:

for 0.45 16 s 0.5 a t k=O u = v = w = o symmetry condition:

at 5 = 0.5 u = v = $ - h

s-4e=O I 4s = Qe = 0

Figure 2 - Geometric characteristics of the Space Shuttle orbiter nose-gear tire used in the present study.

/-- tread

C inner lini

---- .3535

Wire beads

''. .4483 - h = 2.185 6=.5

Figure 3 - Tire cross section and thickness variation for Space Shuttle orbiter nose-gear tire used in the present study (T; = hlh,; h, = 1.908 cm).

Figure 4 - Nonlinear response and sensitivity coefficients with respect to cord diameters and material properties. Space Shuttle orbiter nose-gear tire subjected to uniform inflation pressure. p, = 2.068 MPa, qo = 8.0 MPa, b, = 1.908 cm. Point c corresponds to 5 = 0.

Figure 5 - First-order and second-order sensitivity coefficients of the tire response with respect to parameters of cord angle (see equations (29)). Space Shuttle orbiter nose-gear tire subjected to uniform inflation pressure. po = 2.068 MPa, E, = 8.0 MPa, ho = 1.908 cm. Point c corresponds to 5 = 0.

0

Figure 6 - First-order and second-order sensitivity coefficients of the tire response with respect to parameters of cord end count in region 2 (see equations (28) and Table 2). Space Shuttle orbiter nose-gear tire subjected to uniform inflation pressure. po = 2.068 MPa, %o = 8.0 MPa, ho = 1.908 cm. Point c corresponds to 6 = 0.

Figure 7 - Accuracy of tire response obtained by the reduced basis technique. Space Shuttle orbiter nose-gear tire subjected to uniform inflation pressure. p, = 2.068 MPa. qo = 8.0 MPa, ho = 1.908 cm. Point c corresponds to 6 = 0.

959

- full system

x r = 12 reduced system

+ r=15 O 1 I

Figure 8 - Accuracy of first-order sensitivity coefficients of tire response obtained by the reduced basis technique. Space Shuttle orbiter nose-gear tire subjected to uniform inflation pressure. po = 2.068 MPa, ETo = 8.0 MPa, h, = 1.908 cm. Point c corresponds to 5 = 0.

960

- full system

+ r=15

Figure 9 - Accuracy of second-order sensitivity coefficients of tire response obtained by the reduced basis technique. Spxe Shuttle orbiter nose-gear tire subjected to uniform inflation pressure. po = 2.068 MPa qo = 8.0 MPa, h,, = 1.908 cm. Point c corresponds to 5 = 0.

N,, -.lo

PO^,)-.15

boundary condition location (typical)

Figure 10 - Accuracy of the meridional variations of the displacements and stress resultants obtained by the reduced basis technique. Space Shuttle orbiter nose-gear tire subjected to uniform inflation pressure. po = 2.068 MPa, E, = 8.0 MPa, ho = 1 .go8 cm. 0

boundary condition location (typical)

Figure 11 - Accuracy of the meridional variations of the first-order sensitivity coefficients of the tire displacements obtained by the reduced basis technique. Space Shuttle orbiter nose-gear tire subjected to uniform inflation pressure. po = 2.068 MPa, KO = 8.0 MPa, h,, = 1.908 cm.

1800

6 "'

boundary condition location

600 7 P.. k.. (typical)

0 '

Figure 12 - Accuracy of the meridional variations of the second-order sensitivity coefficients of the tire displacements obtained by the reduced basis technique. Space Shuttle orbiter nose-gear tire subjected to uniform inflation pressure. p, = 2.068 MPa, ho = 8.0 MPa, h, = 1.908 cm.

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