Design sensitivity analysis of critical load factor for nonlinear structural systems

Compuners di Strucrures Vol. 36. No. 5. pp. 8234738. 1990 Printed in Great Britain.

0045-7949/90 s3.w + 0.00 p 1990 Pergamon Press pie

DESIGN SENSITIVITY ANALYSIS OF CRITICAL LOAD FACTOR FOR NONLINEAR STRUCTURAL SYSTEMS

J. S. PARK and K. K. CHOI

Department of Mechanical Engineering and Center for Simulation and Design Optimization of Mechanical Systems, College of Engineering, University of Iowa, Iowa City, IA 52242. U.S.A.

(Received 24 April 1989)

Absh&-A ~ntinuum fo~ulation of sizing design ~nsitivity analysis of the critical load factor is developed for nonlinear structural systems that are subject to conservative static loadings. Both geometric and material nonlinear effects are considered. Sizing design variables, such as thickness and cross-sectional areas of structural components, are treated. The total L.agrangian formulation for an incremental equilibrium equation and one- and two-point linearized eigenvalue problems are utilized to estimate the critical load factor. A distributed parameter approach and the adjoint variable method are used to obtain design sensitivity expressions in continuum mechanics formulation. A numerical method is presented to evaluate design sensitivity expressions using established finite element codes. The proposed method is tested using severai examples. The numerical results show that the proposed method for design sensitivity of the critical load factor is accurate. It is found that, as the applied load approaches the critical load, the design sensitivity of the critical load factor does not converge to the design sensitivity of the actual critical load. even though the estimated critical load converges to the actual critical load and the design sensitivity ok the critic2 load factor is correct.

1. INTRODU~ION

The process of structural design optimization tends to lead to a design that exhibits buckling characteristics under the service loading [I]. Moreover, with demand for efficient and economical structural systems and growing sophistication of materials and methods of manufacturing, structural components have become increasingly slender and thin, and thereby more prone to nonlinear behavior and buckling. The term ‘buckling’ denotes the loss of stability of a structure due to snap-through or bifurcation, and ‘critical load’ denotes the load level at which buckling occurs. The initial theoretical investigations of the critical load of structures were based on many simplifying assump tions of the linear structural theory. However, experi- ence and experiments show that in many cases the stability limit is reached at a much smaller load level than that predicted by the linear stability analysis. A better estimation of the critical load can be obtained using the nonlinear structural theory rather than the Iinear structural theory.

The design of nonlinear structures shows one feature that is different from the design of linear structures. During the design optimization process of a nonlinear structure, the structure can coIlapse at a load that is below the given external load. In this case, the design optimization process cannot be continued without improvement of the design to bring it inside the feasible region of the design space. Therefore, design sensitivity analysis of the critical load is in- evitable for the design of nonlinear structural systems. In the design optimization process, if e-active strategy is used, design sensitivity analysis of the estimated critical load has to be carried out. In other

words, design sensitivity analysis of the critical load factor has to be carried out at a final equilibrium that is not the critical limit point.

This paper presents a unified formulation of continuum design sensitivity analysis of the critical load factor for nonlinear structural systems using the total Lagrangian formulation. As in the design sensitivity analysis of nonlinear structural responses, the method presented can readily be extended to use the updated Lagrangian formulation [Z]. An interesting feature for the design sensitivity analysis of the critical load factor for nonlinear structures, unlike that for linear structures, is that an adjoint variable method is necessary to handle the design derivatives of displacements. Effects of large displacements, large rotations, large strains, and material nonlinearities are included in the analytical design sensitivity expressions using appropriate kinematic and constitutive equations. Numerical procedures for evaluating the design sensitivity analysis results of the critical load are developed using postprocessing data of established finite element analysis codes. To demon- strate the feasibility and accuracy of the proposed theory, numerical results are presented for truss and beam structures.

Various linear and nonlinear eigenvalue problems are suggested for the stability analysis of nonlinear structures. Mallet and Marcal [3] presented a nonlinear eigenvalue problem using the fact that the second variation of the total potential energy vanishes at the critical limit point for elastic structural systems. Brendel and Ramm (41, Rammerstofer [5], and Borri and Hufendiek [6] formulated linear eigenvalue problems with info~ation at one loading step. This formulation is called a one-point linear eigenvalue

824 J. S. PARK and K. K. CHOI

problem. Marcal [7] and Bathe and Dvorkin [8] utilized information at two sequential loading steps for a linear eigenvalue problem, which is called a two- point linear eigenvalue problem. The linear eigenvalue analyses, which are performed at the initial configuration for the linear systems, are repeated at several prebuckling configurations on the nonlinear equilibrium path to obtain the critical load factor.

In the past, optimal structural designs with stability constraints have been studied extensively. Kiusalaas [9], Khot ef al. [lo], and Khot [l l] presented redesign equations for the minimum weight of linear structures with stability requirements using optimality criteria. They solved linear eigenvalue problems to obtain the critical load and obtained derivatives of the critical load by differentiating the discretized matrix eigenvalue equation with respect to design variables. The assumption that the stiffness matrix is linearly proportional to design variables whereas the geometric stiffness matrix does not depend on design variables was used. Later, Khot and Kamat [ 121 extended the optimality criteria method to geometrically nonlinear structures, using the fact that the strain energy density in all members is equal at the optimum design. They evaluated derivatives of the total potential energy using the fact that the strain energy is linear with respect to design variables that are the cross-sectional areas of trusses.

Kamat and Ruangsilasingha [ 131 developed a method to obtain the design sensitivity of the critical load parameter through implicit differentiation of the nonlinear equilibrium equation and the stability crite- rion for truss structures. This method is applicable at any point along the equilibrium path except at the critical limit point where the determinant of the Hessian matrix of the potential energy vanishes. Kamat et al. [14] derived an expression of the critical load explicitly in terms of geometric and material properties of two- and four-bar shallow trusses using a minimum potential energy approach. Using their method, the analytical expressions for the design sensitivity of the critical load can be obtained. How- ever, the expression for a general structural system becomes too complicated to obtain in a closed form solution.

Wu and Arora [15] presented design sensitivity analysis of the critical load for nonlinear structural systems by taking derivatives of discretized matrix equations with respect to design variables. This method needs derivatives of the stiffness matrix with respect to design and state variables, which requires many tedious computations for each type of structural component. They introduced three approaches using analysis information at the critical limit point. However, the first two approaches are abandoned because of some difficulties in numerical implementation. In the third approach, design sensitivity formu- las are derived without using derivatives of displacements at the critical limit point. Wu and Arora deleted the terms that include derivatives of

displacements using an argument that multiplication of the left side of the equation of eigenvalue problems (Ky = 0) and derivatives of displacement at the critical limit point, which they assumed to be finite, vanishes. The third approach started from the force equilibrium equation to yield design sensitivity expressions without derivatives of critical displacement or eigenvectors. Wu and Arora implemented their method using only truss structural components since a lot of tedious computations are required for the other types of structural components. For bending- type structural components, they used the semi-analytical method to compute derivatives of stress vectors and strain-displacement matrices with respect to design variables, which were needed for evaluation of design sensitivities [ 161.

Wu and Arora’s third approach has a restriction: it works only at the critical limit point. In the optimal design process of nonlinear systems, an a-active strategy may be used for the buckling constraint. In this case, design sensitivity analysis of the critical load should be performed before the critical limit point is reached. Therefore, we need more general critical load design sensitivity expressions that work at any prebuckling configuration. The purpose of the present paper is to obtain a unified structural design sensitivity expression of the critical load factor of nonlinear structural systems with sizing design variables such as thickness and cross-sectional area of structural components.

The papers cited in [g-16] presented methods of discrete design sensitivity analysis for the critical load factor in matrix equations by taking the derivative of discretized matrix equations with respect to sizing design variables. In contrast to the discrete approach, a continuum approach was used for analytical expressions of design sensitivity of the critical load factor for linear structural systems in [ 17, 181. Reference [ 191 addressed optimization problems of linear structural systems in which repeated eigenvalues may occur. Optimality conditions including directional derivatives of repeated eigenvalues were derived using the continuum approach. It has been shown that the critical load for a nonconservative loading system may not be continuous with respect to design [S]. In the present paper, only the simple eigenvalues of conservative load systems are considered.

2. STABILITY EQUATIONS FOR NONLINEAR STRUCTURAL SYSTEMS

Incremental equilibrium equations

Using the principle of virtual work, the equilibrium equation for a body of domain ’ + ‘92 and boundary ‘+ “r in the equilibrium configuration at time t + At can be expressed as (20-221

r+Ar~,j ,+A,c0 l+A’dR = r+AIR,

for all P E Z, (1)

Design sensitivity analysis of critical load factor 825

where ‘+A’ul/ is the Cartesian component of the Cauchy stress tensor at time r + At, ,+A&, is the component of the infinitesimal strain tensor referred to the configuration at time t + At, t is a kinemati- tally admissible virtual displacment, Z is the space of kinematically admissible virtual displacement, and an overbar indicates the variation of the quantity under the bar. On the right side of eqn (1), ‘cA’R is the virtual work done on the body of externally applied forces through kinematically admissible virtual displacement Z

l+A’R _

s

l+Al 0,

I + A’dR

r+dm

+ c ‘+A’,,,, ‘+AldZ-, (2) J,+~tr

where z represents the incremental displacement, and ‘+Alf; and ‘+A’7’, are the components of externally applied body and surface force vectors, respectively. In the following development, forces ‘+ Ax and ’ + “T, are assumed to be deformation-independent. The notation convention of [22] is used for superscripts and subscripts. Equation (1) cannot be solved di- rectly, since the configuration at time t + At is un- known. A solution can be obtained by referring all variables to a previously known equilibrium configuration, and linearizing the resulting incremental equation. There are two alternative formulations: the updated and total Lagrangian formulations accord- ing to the reference domain.

In the total Lagrangian formulation, all static and kinematic variables are referred to the initial configuration at time 0 and eqn (1) is transformed to [2]

=

s

t+Ar &j OdQ + t + “; Tiii Odr

on I or

= I,(t), for all I l Z, (3)

where ’ + A’z is the total displacement at time t + At, fi is the xi direction component of virtual displacement 5, ‘+A&SU are the Cartesian components of the second Piola-Kirchhoff stress tensor at the configuration time t + At, and h+A’ .sii are the Cartesian components of the Green-Lagrange strain tensor at time t + At, all of which are measured with respect to the configuration at time 0. In eqn (3), ~~(~~~5, i) and I.(r) are energy and load forms, respectively.

After a process of decomposition of stress and strain and linearization of nonlinear incremental strains, the linearized incremental (from t to t + At) equilibrium equation becomes

= l”(I) - s hSU o?U OdQ, for all ,%Z, (4) on

where oCU,s is the incremental material property tensor, OeU is the linear incremental strain tensor, o~ii is the nonlinear incremental strain tensor, and &is the second Piola-Kirchhoff stress tensor at time t. Define energy forms A,, and (-D,) as the first and second terms of the left side of eqn (4), respectively. Also define energy forms a.’ as [2] - _

af (‘z; z, i) = A,(‘z; z, f) - D,(‘z; z, 2).

Then

(5)

A, (‘z; z, i) I 5

0 clJr~(Oz,, 04 + izk./ Ozw Ozk., w

+ ~Zk.rOzk,sOZI./+~~k.r~Zl.iOZk,,O I.] 2 )OdQ

and

(6)

+ 0’zm.r lZm,s) ozk,i 0fk.i ‘dn, (7)

where oz is the incremental displacement measured with respect to the configuration at time 0. The fact that oC,, and material property tensor ;Ciirs are symmetric [22] with respect to their indices has been used in eqns (6) and (7). Thus, the energy bilinear form a,‘(‘~; ., a) of eqn (5) is symmetric in its argument. In eqn (7), the constitutive law iSi, = ;C,, ,& and the relation between the Green-Lagrange strain and displacement [22] are used. In eqns (6) and (7), ,,z,~ denotes the partial derivative of ozi with respect to the coordinate xi.

Stability equations

Based on the incremental equilibrium equations, the equations of a linear eigenvalue problem for the stability analysis of nonlinear structural systems can be presented in variational form. Various linear and nonlinear eigenvalue problems have been suggested to evaluate the stability status for nonlinear structural systems [3-81. They differ in the assumption introduced for the relationship between the critical load factor and the estimated critical load. The critical load factors of nonlinear structural systems can be evaluated by solving a linear eigenvalue problem at any prebuckling equilibrium configuration. Formula- tion of equations of stability analysis may include the effect of large displacements, large rotations, large strains, and material nonlinearities with appropriate kinematic and constitutive description.

The critical load can be found using the fact that at least two adjacent configurations exist at the critical load [23]. The mathematical basis for this interrogation follows immediately from the incremental equilibrium equations, i.e. the left sides of eqn (4) vanish at the critical limit point. With the total displacement “z at the critical limit point t = cr, the


stability equation becomes increment with the same ratio between two configurations at time t - At and t, the energy form (A, - D,) in eqn (8) can be written as

for all j E Z, (8) U’z;~,j9-4(“z;y,.i9

where the incremental displacement ,,z in eqns (6) and (7) is replaced with y to distinguish an eigenfunction from a real incremental displacement Oz. The exis- tence of nontrivial incremental displacement solutions y to this nonlinear relation serves to identify a point of instability. That is, if the final equilibrium configuration is not the crucial limit point, the solution y of eqn (8) must be trivial.

=B,(‘-A’z;y,~)+‘[E,(‘-A’z,‘z;y,3), (10)

where the energy forms B, and E,, are defined as

BU(L-A’~;y,~)=A,(‘-A’z;y,j)-D,(‘-A’z;y,j)

(11)

To estimate the critical load, it is necessary to evaluate the left side of eqn (8) at the critical limit point using the available information at a final prebuckling equilibrium configuration at time t. Note that, unlike in the case of incremental equilibrium equations, t is the final equilibrium configuration time here. Linear extrapolation can be used to approximate the left side of eqn (8) to form an eigenvalue problem. By solving the eigenvalue problem, the lowest eigenvalue is the critical load factor ‘<. Assum- ing proportional conservative static loading, the estimated critical load vector ‘P, can be expressed with the given load vector ‘p and the critical load factor ‘4. Two commonly used approaches, one- and two-point linear eigenvalue problems, are formulated in variational forms, and expressions of the corresponding estimated the critical load are presented.

E,(‘-Ar~,‘~;y,$) = B,(‘z;y,y) - BJ’-A’z;y,y)

and ‘t; is the critical load factor at time t. Then, eqn (8) becomes an eigenvalue problem, which will he called a two-point linear eigenvalue problem

Bu(‘-A’z; y, j) + ‘[Eu(‘-A’z, ‘z; y, jj) = 0,

Solving the eigenvalue problem in eqn (13) at the given load level ‘p that is lower than the true critical

load pE, leads to an estimated critical load

‘PC, = I- Alp + p (‘p _ ’ - AIM).

In eqns (9) and (13), ‘c is the smallest eigenvalue

One-point approaches. Utilizing the information at the equilibrium configuration time t, eqn (8) can be rewritten as an eigenvalue problem. By linearizing the nonlinear relationship between the energy form D, and an additional load increment, D, at critical limit point is approximately

with the critical load factor ‘l at time f. Also, by neglecting the variations of the energy form A, due to the loading change, we can write

and y is the corresponding eigenvector. Note that if the present final equilibrium configuration time t is at the critical limit point, ‘[ = 1 in eqns (9) and (13), and eqns (9) and (13) become eqn (8). The stability analysis of eqn (9) or (13) can be applied at any prebuckling configuration, and the estimated critical load becomes more and more accurate when the considered final equilibrium configuration approaches the critical limit point [5]. The estimated critical loads for both approaches are an unconserva- tive approximation, i.e. larger than the true critical load ‘p,, 2 pc,. The equation for the stability analysis of linear structural systems can be obtained as a special case of the stability equation of nonlinear structural systems with the assumptions of linearly elastic material and small displacement.

Then, eqn (8) becomes an eigenvalue problem, which will be called a one-point linear eigenvalue problem

A,(‘z; y, 1) - ‘iDu(Iz; y, y) = 0, for all p E Z. (9)

3. DESIGN SENSITIVITY ANALYSIS OF CRITICAL LOAD FACTORS

Solving the eigenvalue problem in eqn (9) at the given load level ‘p that is lower than the true critical load pc, leads to an estimated critical load ‘pc, = ‘(‘p.

Two-point approach. Utilizing the information at two configurations at time t - At and t, where t is the final equilibrium configuration time, eqn (8) can be rewritten as an eigenvalue problem. With the assumption that from time r - At onwards the energy form (A, - D,) changes linearly to an additional load

The theory of design differentiability of eigenvalues for the eigenvalue problem of nonlinear structural systems has not been developed in the literature, whereas a reasonably complete theory on differentiability of eigenvalues of linear structural systems is given in [ 181. Under the assumption of differentiability of eigenvalues of nonlinear systems, a method of design sensitivity analysis of the estimated critical load is developed for the conservative system. The adjoint variable method presented in [2] for nonlinear

(12)

for all y E Z. (13)

Design sensitivity analy sis of critical load factor 827

structural systems is used to obtain design sensitivity expressions of the critical load factor in terms of design perturbations. Equations of design sensitivity analysis consider the effect of large displacements, large rotations, large strains and material nonlinearities with appropriate kinematic and constitutive description. The stability equations in eqns (9) and (13), which are nonlinear with respect to design variables, are linearized to obtain the first variation with respect to design variables.

Differentiability of static response and corresponding energy forms

Consider a structural system in the equilibrium configuration at time t (instead of t + At in Sec. 2) with a given design II, and the new equilibrium configuration at time 1+ At with a perturbed design u + r6w. Using eqn (3), equilibrium equations can be written as

a,&, .F) = l,(Z), for all P e 2 (14)

and

a, + du ( ‘+A’z, i) = lu+& (9), for all i E Z. (15)

The smaller the design perturbation TSU, the smaller the change between the equilb~um states of the original and perturbed designs. Hence, as the design perturbation vanishes, so does the difference in state, i.e. At +O as t-*0. Define the first variations of the nonlinear energy and load forms in eqn (15) with respect to their explicit dependence on the design variable u as [2]

d (16)

(17)

where ‘i denotes the state ‘z with dependence on t suppressed, and P is independent of T. Define the first variations of the solutions of the equilibrium equations at time t and t - At with respect to design u as

= lim ‘+“‘z(u + r&) - ‘z(u)

t-0 T (18)

and

Using the above definition, the order of taking the first variation and the partial derivative of the response can be interchanged [2].

Noting that the energy form a, in eqn (14) is nonlinear in ‘+Ar~, the chain rule of differentiation can be used to obtain [2]

${au+r6urtAtz(u + r&u), ~l~lr~O

= a;,f’z, 2) + a:(iz; z’, 5). (20)

For the second term on the right of eqn (20), the linearization process from eqn (3) to eqn (5) has been used. By taking the first variations of both sides of eqn (15) and using eqns (17) and (20)

a,*@; z’, 5) = I&(.z?) - a;,(‘z, 5). (21)

Dt@erentiability of eigenvalue and corresponding energy forms

One -point linearized eigenvahe analysis. For one- point linear eigenvalue problems, the stability equations at the ~uilib~um co~gurations at time t with design u and at time t + At with design u + r& can be written, using eqn (9) as

A,f’z;y,j9 =‘W,(‘Z;Y,J%

for all je2 (22)

A II+& (r+A’~; y,y) = r+A’[DU+r&+A’~; y,j),

for all 9 E 2. (23)

Define the first variation of the nonlinear energy forms in eqn (23), with respect to its explicit dependence on the design variable u, as

~L(‘z;Y,B)E~A”+-~“(‘~;B,~)I~=~ (24)

where ‘i and 9 denote the states ‘z and y, respectively, with dependence on r suppressed, and y is independent of r. With the assumption of differentiability of the eigenvalue and eigenfunction for the linear eigenvalue problem in eqn (23) their first variation with respect to design variable u can be defined as

= iim ‘+AY(t( + r&u) - ‘[(a)

(26) r-0 t


Using the above definition, it can be noted that the order of taking the first variation and the partial derivative of the eigenfunction can be interchanged

PI. In eqn (22), the energy forms A, and D, are

nonlinear in ‘z and linear in y. Using these properties, the chain rule of differentiation can be used to yield

& {A,+,,,r+*‘z(u + 7~Io;Y@ + 7W,A1l,_O

=A;,(lz;Y,R

+A:(‘z;z’,y,~)+A,(‘z;y’,~) (28)

and

& {D”+16”[r+A’r(~ +7du);y(u +7wA~],~O

=o;,(‘z;Y,y)

+D:(‘z;z’,Y,~)+D,(‘z;Y’,~), (29)

where A,+ and D,* are the first variations of A,, and D, with respect to the design variable implicitly through the response ‘z, respectively, and they are defined as

= I Ocii,(Oz~.iOY~sO~kk,j+ OZ~,rOYk,rO~i.j on

+ 0zL.r iZ,,i 0Yk.s OY/,j + 6Zk.r 0'i.i 0Yk.s 0yl.j) ‘a C30)

and

D:(‘z;z’,y,jj) = - I

6c,(Oz:,3 OYLjO9Li on

+ izk,r Oz;, oY/,j 0Yl.i) ‘da. (31)

Taking first variations of both sides of eqn (23), replacing j with y E Z since eqn (23) holds for all j E Z, and using symmetry of the energy forms

A,@; ., .) and D,(‘z; , .) in their arguments

D,(‘z;y,y)l3’=[A:(Iz;z’,y,y)-‘rD:(’z;z’,y,y)l

+[A,(‘z;y,y’)-‘CD,(‘z;y,y’)l

+[A~,(‘z;Y,Y)-‘~D;,(‘z;Y,Y)~.

(32)

Noting that y’ E Z, it can be seen that the term in the second brackets on the right side of eqn (32) is zero using eqn (22). Thus

A,*(‘z;z’,y,y)-‘rD:(‘z;z’,y,y)

‘(f = +A;,(‘z;Y,Y)-‘IDL(lz;y,y)

D,(‘z;Y,Y) . (33)

Two-point linearized eigenvalue analysis. For two- point linear eigenvalue problems, the stability equa-

tions at the equilibrium configurations at time t with design u and at time t + At with design u + r&4 can be written, using eqn (13), as

B”(‘-*‘z; y, j) + ‘@“(‘_A’Z, ‘z; y,Y) = 0,

for all j o Z (34)

B “+r6”(l~;Y~~)+‘+*‘S~“+~d”(‘~,‘+*’~;Y,p)=o,

for all 9 E Z. (35)

Define the first variation of the nonlinear energy form B, in eqn (35) with respect to its explicit dependence on the design variable u as

B;.(‘-*‘z;y,B)-&B.,,,.(‘-P’i;B,);)I,_,. (36)

With the assumption of differentiability of the eigenvalue and eigenfunction for the linear eigenvalue problem in eqn (35), their first variations with respect to explicit dependence on design variable u can be defined as in eqns (26) and (27) respectively.

In eqn (34), the energy forms B, and E,, are nonlinear in ‘z and ‘-*‘z and linear in y. Using these properties, the chain rule of differentiation can be used to yield

& {B,+,,,M +7bu);y(u +7WA~I,=o

=B;,(‘-*‘z;y,jJ)+B:(‘-*‘z;zI,y,j)

+ B,(‘-A’z;y’,Y) (37)

and

& {E,,+ldu[lz(u +7&4), ‘+A’z(u + z&4);

Y(U +7~4~11~J,=o

= B;,(lz;y,jj) + B:(‘z; z’,y,j) + B,(‘z;Y’,~)

-B;,(‘-“‘z;y,j)-B:(‘-*‘z;z!,y,jj)

- B,(‘-*‘z;y’,y) (38)

using eqn (12). In eqns (37) and (38), B: is the first variation of B, with respect to the design variable implicitly through the responses ‘z

B,*(‘z;z’,y,~)=A:(‘z;z’,y,~)-D~(’z;z’,Y,~).

(39)

Taking first variations of both sides of eqn (35), replacing jj with y E Z since eqn (35) holds for all j E Z, and using symmetry of the energy form


f&(‘z, 9, *) in their arguments

Noting that y’ E Z, it can be seen that the term in the second brackets on the left side of eqn (40) is zero using eqn (34). Thus

(1 -'6)[B,*('-A'z;z',y,Y)+B;,('-A'z;Y,Y)l

14’ = +‘r[B:(‘z;z’,Y,Y)+B~,(‘z;Y,Y)l

~“(‘-A’z;Y,Y)--“(‘z;Y,Y) .

(41)

Adjoint variable design sensitivity analysis

Recall that the first variations z’ and z1 depend on the direction of a design change 6~. The objective here is to obtain an explicit expression for ‘[’ in terms of a design change Su. The adjoint variable method [2, 181 can be used to obtain the terms that include z’ and zl in the numerators of the right sides of eqns (33) and (41) explicitly in terms of 6~. Adjoint equations corresponding to z’ and zi are introduced by replacing z’ and z’ by virtual displacements 1 and A_, respectively, and equating terms involving 1 and I_ to the energy form a,*.

One -point linearized eigenvalue analysis. The adjoint equation corresponding to z’ is

a:(‘z;1,z)=A,*(‘z;X,y,y)-‘TD,*(‘z;&y,y),

for all XE Z, (42)

where a solution 1 E Z is desired. Following the same procedure of the adjoint variable method of [2], the terms that include z’ in eqn (33) can be rewritten in terms of design variations, using eqn (21), as

A:(‘z;z’,y,y)-‘CDf(‘z;z’,y,y)

= MA) -a@, A), (43)

where the right side is linear in 6u and can be evaluated once the state ‘z and adjoint solution 1 are determined. Substituting this result into eqn (33), the explicit design sensitivity of ‘c is

,r,= L@) -4,(‘z,~) +A&~z;Y,Y) -'WbCz;y,y)

D,('z;Y,Y) (44

Two -point linearized eigenvalue analysis. The two adjoint equations corresponding to z’ and ZI are

a,‘(‘z;A,X)=B,*(‘z;X,y,y), for all XEZ (45)

and

af(L-A’z; k,X) = B:(‘-A’z;X,y,y),

for all I_ E Z, (46)

respectively, where solutions 1 and L_ are desired. Following the same procedures of the adjoint variable method of [2], the terms that include z’ and ZI in eqn (41) can be rewritten in terms of design variations, using eqn (21), as

B:(‘z; z', y, y) = W) - &(rz, 1) (47)

and

B:(‘-A’z; z!,Y,y) = /;(A_) - a;,(‘-A’z, A-), (48)

where the right sides of eqns (47) and (48) are linear in 6u and can be evaluated once the states ‘z and r-A’~ and the adjoint solutions 1 and 1_ are determined. Substituting these results into eqn (41), the explicit design sensitivity of ‘c for the two-point approach is

(1 -‘r)[l;,(l_)-a;,(‘-A’z,h)+B;,(‘-A’z;y,y)] + ‘i&(n) - ak(‘z, 1) + &(‘z;~,y)l

&(‘-A’r;Y,Y)-~“(‘r;Y,Y) (4%

In eqns (44) and (49), the variations of energy forms on the right sides depend on the structural component. Equations (44) and (49) will serve as the principal tool for sizing design sensitivity of the estimated critical load of nonlinear structural systems with large displacement, large rotation, small strain, and nonlinear elastic material behavior under conservative static loading. The design sensitivity analysis result of eqn (44) is applicable to linear structural systems by simply dropping f;(n) and a;,(‘z, J.).

4. EXAMPLES

The general results presented in Sec. 3 are applied to beam/truss and plate/membrane structural components to derive analytical expressions of design sensitivity including geometric and material nonlinearities. For each component, the energy forms related to the equilibrium and stability analyses are presented. The adjoint system equations of eqn (42) for the one-point approach and eqns (45) and (46) for the two-point approach can be defined using energy forms A: and 0,‘. The eigenvalue design sensitivity expression in eqn (44) for the one-point approach and eqn (49) for the two-point approach can be found using first

. variations a&, ii,, A;,, and 02,.

Beam /truss

Consider a beam/truss component as shown in Fig. 1, with the cross-sectional area and second moment


and OL

Fig. 1. Beam/truss component.

of inertia as design variables. The energy form

P4 251

l

associated with the equilibrium equation in eqn (3) is

OL

a,('z,Z)= iwiOA[hl,, +t;Zk,l liZk.,l

0

where ‘z, , ‘z2, ‘zj, and :E are an axial displacement, two orthogonal lateral displacements, and the elastic modulus at the configuration time t, respectively. Also, ‘I*, ‘Z3, ‘A, and ‘L are two moments of inertia, cross-sectional area, and length of the centroid axis of the beam/truss component in the original unde- formed configuration, respectively. In eqn (50), summation notation is used with k = 1,2,3. The load form of the beam/truss component is

I

OL

f,(f)= ;J 4 d’x,, (51) 0

where ; f, , d fi, and if, are axial load and two orthogonal lateral loads, respectively, as shown in Fig. 2. If there are point loads, a Dirac measure can be used in eqn (51) 1181.

s QL

D,('z,y,P)= - 6E"ACz,., +fd~m.,kn.,l 0

x OY,,, 0Jkk.1 d"x, 3 (53)

where y, , y,, and y, are an axial eigenmode and two orthogonal lateral eigenmodes, respectively, and

The first variations of the energy and load forms in

y = [y,, y,, y,]? In eqns (52), ,E is the incremental

eqns (50)-(53), are from eqns (16), (17), (24), and (25)

(or tangential) modulus at time t with reference configuration at time 0.

x LOG., + OZmI ofm,,l~"A

+ dZ2.,, OF,,,, so4

+ rh, oh, 6'121 d"x,

l;,(f) = s

OL ;fi,oA & 6’A d”x, 0

s

OL

A;~.(‘z;Y,~) = OEUOY,., + hf., OYkJl 0

and

x [oP,., + hn,, oL,I~~A

+ OY2,ll oJ72.1, go4

+ 0~3~1, oh.,, 8’121 d”x,

W’z;y,J)= - s

oL~EEz~.,+i~z,,,d~m,,l 0

X 0yk.lOjk.l hoA d'x,.

(54)

(55)

(56)

(57)

The energy forms associated with the stability eqn (9) for the beam/truss component are

The first variations of A, and D, with respect to the design variable implicitly through the response ‘z are,

OL from eqns (30) and (31) A,(‘z;y,j)= oE{‘A[oy,,, + izk., OYk,,l

Jo

I "L

x IOY,,, + ch,, O~“.,l + Oh oY2.11 0~2.11 A:(‘z;z’,y,j)= oEOAIOY,,, oz:., ov,.~

0

+ o120~3~,, o_C,,,> d”x, + or;., OY,,,(O.?,., + 2&., elm,, )I d’x, (58)

and

I

QL

D:(‘z;z’,y,jj)= - ;E”A[oz;,, +hn,, 0~6~1 Xl 0

b

x Oyk,lO~k.,doX,~ (59)

Plate/membrane

Fig. 2. External forces for beam/truss component. Consider a plate/membrane component as shown

in Fig. 3, with thickness as a design variable. The


x3,/w

Fig. 3. Plate/membrane component.

Xl -

energy form of the plate/membrane component is

124,251

where Oh is the thickness of the component in the original configuration. In eqn (60), summation notation is used with i, j = 1,2. The first term on the right of eqn (60) is associated with stretching of middle surface and the second term with curvature of the strained middle surface. The load form of the component is

where if = C f, , ,j fi, ,j f3]' and i T = [i T, , (I TJT are the body force vector and traction force vector, respectively, as shown in Fig. 4. If there are point loads, a Dirac measure can be used in eqn (61) [18].

The energy forms associated with the stability eqn (9) for the plate/membrane component are

JJ ,E”h = 7 Ioe,, 061 + oe22 o&

ml-ov

-I- IS oh’oE 2 [OY3,1, Oh, + 0Y3.22 Oh,22

on 1w -0v 1

+Ov(O~3.11 073.22 +OY3.22Oh,)

+2(1 - 0v)OY3.12 0~3.121 doa (62)

and

o.(‘z;Y,~)= - JJ ;oh cSijl.r,=o on X (0Yk.i 0Yk.j + Ok.1 d/u) doa, (63)

where y,, y2, and y, are two orthogonal inplane eigenmodes and a lateral eigenmode, respectively. In eqn (71), o E and ov are the incremental modulus and Poisson’s ratio at time t with reference configuration at time 0.

The first variations of the energy and load forms in eqns (60)-(63) are, from eqns (16), (17), (24), and (25)

u;,(‘z, P) = JJ rc&,hj,~x,=, on + !G sij o fv>lx, = o/,/2 IS Oh do0 (W

r;,(5) = JJ (;f;;0&)6~h do% i = 1,2, 3 (65) on

A;,(‘z;y,y) = JJ OE -[e 2 o ~~~~~~~~~~~~~~~ onu-ov 1 + ov(oe22 06, + Oell oz22)

+ 2(1 - ov)oe,2 ot,2]60h doa

+ JJ oh20E 2

onw-ov 1 [ oY3.11 0?3,,,

and

+ OY3.22 0733.22

+ Ov(O~3,11 Oq33.22 + OY3.22 Oh.,,)

+ 20 - 0v)OY3.,2 oY33.,2P”h d”Q (66)

&,(‘z;Yrjj)= - JJ fLsijlx,-O(OYk.iOYkl.j an

+ oYk,j o.Fk,iPOh don. (67)

Fig. 4. External loads for plate/membrane component.


The first variations of A, and D, with respect to the design variable implicitly through the response ‘z are, from eqns (30) and (31)

A,+(‘z;z’,y,y)= ss $$$jjKoeLl +ovoe22)

x 04,10Yk,~ + (0622 + 0~ oelJ

’ 0zi,2 0.h.Z + t1 - 0v)Oe12(Ozh 0jkk,2

+ 042 0ykk.l 11 don (68)

and

D,+(‘z;z’,.Y,~~

0Yk.l 0jk.l + iv OYk,2 O~k,Z)b-;I

+ (Oyk,, Oyk,2 + iv OYk.1 O?kk,l%2

(69)

5. NUMERICAL METHOD OF DESIGN SENSITIWTY ANALYSIS

In this section, a numerical method is presented to evaluate the design sensitivity expressions derived in Sec. 4, using structural analysis results of an established finite element code. The computational efforts involved in the design sensitivity analysis of the critical load factor for nonlinear structural systems are analyses of nonlinear original and linear adjoint systems, linear eigenvalue analysis, calculation of adjoint loads, and evaluation of design sensitivity

Stability Analysis

I I I I I I I

I I

(Ei$-) (+)(-) + c 4 +

Design Sensitivity Analysis

n : Computational Module

( :Data

Fig. 5. Flow chart ofdesign sensitivity calculation procedure.

expressions. Unlike design sensitivity analysis of the critical load for linear structural systems, design sensitivity analysis of the critical load factor for nonlinear structural systems requires an adjoint analysis.

For numerical implementation with an established finite element analysis code, the computational procedure given in Fig. 5 is used. The structural system is defined by analysis data that include geometric and material properties, finite element modeling data, kinematic boundary conditions, and applied loads. Design data such as specification of design variables and performance measures define the design problem. In this paper, the critical load factor is the only performance measure. Using an established finite element analysis code, the structural system is ana- lyzed to obtain the original response ‘z, and the linear eigenvalue problem is solved at the final equilibrium configuration to obtain the lowest eigenvalue and the corresponding eigenfunction.

For the adjoint variable method of design sensitivity analysis, adjoint loads must be computed. Using the original response ‘z, the critical load factor ‘l, and eigenfunction y, the adjoint loads on the right sides of adjoint systems in eqn (42) for the one-point approach and eqns (45) and (46) for the two-point approach are calculated outside the finite element analysis code. An interesting fact is that, unlike in the case of design sensitivity analyses of displacement and stress, the adjoint loads are distributed all over the structure. The solutions of the adjoint system equations can be obtained efficiently by recognizing the fact that the adjoint system equations are linear, even though the original governing equation is nonlinear, and the stiffness matrix of the adjoint system equations in eqns (42) and (45) is the tangent stiffness matrix at the final equilibrium configuration at time t, and the stiffness matrix of eqn (46) is the tangent stiffness matrix at the configuration time f - A.t. Therefore, once the adjoint loads are obtained, the restart option of the finite element analysis code can be used to obtain the adjoint response i for the computation of design sensitivity expressions derived in Sec. 3.

For numerical implementation of the design sensitivity analysis results, consistent methods of incremental equilibrium analysis (total or updated Lagrangian formulation) and the stability analysis (one- or two-point approach) must be used. Estab- lished finite element analysis codes have various different capabilities. The nonlinear codes, ABAQUS, MARC, and ADINA have capabilities for incremental analysis using both the updated and the total Lagrangian formulations for nonlinear static analysis. They utilize the two-point linear eigenvalue analysis for stability analysis, while ANSYS, NAS- TRAN, and EAL use the one-point linear eigenvalue analysis. Recently, ANSYS has added a capability of incremental analysis using the total Lagrangian formulation for nonlinear structural systems in its Ver-


Fig. 6. Two-bar truss.

sion 4.3B. In this paper, numerical implementation is carried out using ANSYS [26].

6. NUMERICAL EXAMPLES

In this section, the numerical method presented in Sec. 5 is used to evaluate the design sensitivity coefficients derived in Sets 3 and 4, using structural analysis results from ANSYS. Geometrically nonlinear truss and beam structures with linear elastic material are considered for the numerical examples. For all examples in this section, the total Lagrangian formulation is used for equilibrium and the one-point linearized eigenvalue analysis is used for stability.

data, Kamat et al.3 actual critical load is 60.14 lb and the design sensitivity coefficient of the actual critical load with respect to design variable A, is 60.14, which is the characteristic value of the given structure independent of the magnitude of applied loading. Wu and Arora [ 161 also obtained the same design sensitivity coefficients of the actual critical load as Kamat et al. They [14, 161 addressed the critical load, not the critical load factor.

Two -bar truss

Consider a shallow two-bar truss that is subjected to a vertical concentrated load p at node 2, as shown in Fig. 6, with two truss elements, three nodal points, and two degrees of freedom. For analysis data, constant cross-sectional area A = 0.8 in2 and Young’s modulus E = 1.0 x 10’ psi are used. The design variables are two cross-sectional areas A, and A, of elements 1 and 2, respectively. Kamat et al. [14] developed an explicit expression of the actual critical load for the two-bar truss. With the given analysis

Structural analysis is performed using a two-dimensional truss element STIF 1 of ANSYS. Using the incremental analysis method of ANSYS, the actual critical load is found to be around 60.060911 lb. At several load levels from p = 1 to 60 lb, the critical load factors are calculated, and design sensitivity analyses are performed for two design variables, A, and A,. For linear structural systems, the design sensitivity coefficients of the linear critical load factor are evaluated using the eigenvalue design sensitivity formula of eqn (44) without a;, and liti, and are shown in Table 1. The design sensitivity coefficients of the nonlinear critical load factor are evaluated using eqns (44) and (54)-(57), and are presented in Table 2. At each load level, the estimated critical load can be evaluated by multiplying the applied load in the first column of Table 1 or 2 by the

Table 1. Design sensitivity of linear critical load factor for two-bar truss

Load (lb) ‘C

1.00 0.311960 + 03 10.00 0.311960 fO2 20.00 0.155980 + 02 30.00 0.103990 + 02 40.00 0.779910 + 01 50.00 0.623930 + 01

aw4

311.5 (100%) 30.71 (100%) 15.11 (100%) 9.907 (100%) 7.308 (100%) 5.750 (100%)

awA,

75.93 (100%) 5.922 (100%) 2.154 (100%) 0.984 (100%) 0.462 (100%) 0.201 (100%)

60.00 0.519940 + 01 4.712 (100%) 0.069 (100%)

Table 2. Design sensitivity of nonlinear critical load factor ‘C for two-bar truss

Load (lb) ‘C axiaA, axiaA,

1.00 0.30897D + 03 312.01 (100%) 77.895 (101%) 10.00 0.281460 + 02 31.257 (100%) 7.8086 (101%) 20.00 0.124850 + 02 15.740 (100%) 3.9303 (101%) 30.00 0.720150 + 01 10.683 (100%) 2.6709 (101%) 40.00 0.44886D + 01 8.3460 (100%) 2.0874 (101%) 50.00 0.275100 + 01 7.4445 (101%) 1.8577 (101%) 60.00 0.10808D + 01 41.161 (101%) 10.316 (101%)


Table 3. Design sensitivity of the displacement .z2 at node 2 of two-bar truss

Load (lb) az,iaA, az,ia,4,

10.0 0.08887 0.022289 20.0 0.20183 0.050595 30.0 0.35625 0.089307 40.0 0.54597 0.14922 50.0 I .0852 0.27137 52.0 1.2745 0.31924 54.0 1.5425 0.38632 56.0 1.9791 0.49549 58.0 2.929 1 0.73341 60.0 18.313 4.5837

critical load factor ‘r in the second column for the one-point linear eigenvalue analysis of nonlinear systems.

For the case of the linear structural system in Table 1, the estimated critical load for all applied load levels is 312 lb, which is more than five times the actual critical load of 60.060911 lb. The third and fourth columns of Table 1 represent design sensitivity coefficients for the two cross-sectional areas, and the numbers inside parentheses are agreements in percentage with sensitivity coefficients calculated using the finite difference method. Agreements are almost lOO%, with 1% design perturbation, which indicates that good design sensitivity results are obtained from the proposed numerical method of eigenvalue design sensitivity analysis for linear structural systems.

For the case of the nonlinear system in Table 2, the estimated critical load changes from 309.0 lb for the applied load p = 1 to 64.8 lb, which is 1.07 times the actual critical load (60.060911 lb), for the applied load p = 60 lb. If the eigenvalue problem is invoked at the critical limit point, the eigenvalue will be 1 and the critical load is just the applied load. The third and fourth columns of Table 2 represent design sensitivity coefficients for the first and second design variables, respectively. The ratio of design sensitivity coefficient for A, to design sensitivity coefficient for A, is independent of the applied load level and is almost equal to 4. This means that the influence of cross-sectional area A, is four times greater than that of A, on the critical load factor. The numbers inside parentheses of the third and fourth columns of Table 2 are agreements in percentage with sensitivity coefficients calculated using the finite difference method. The agreements are almost lOl.O%, which indicates good design sensitivity results from the proposed numerical

method of design sensitivity analysis of the critical load factor for nonlinear structural systems. For the finite difference method, 1% design perturbation is used for the loads up to 50 lb while 0.001% perturbation is used for the load level of 60 lb. The magnitude of design sensitivity coefficient for design A, decreases to 7.4445 as load increases to p = 50 lb. However, at the load level of 601b, design sensitivity coefficients grow rapidly to 41.16, and good agreement requires very small design perturbation for the finite difference; for example, agreement of 101% requires 0.001% design perturbation.

Results for the design sensitivity analysis of displacement in the x,-direction at node 2 using the method developed by Choi and Santos [2,25] are presented in Table 3. The design sensitivity coefficients in the second and third columns increase as load increases. At the load level 60 lb, they increase rapidly. The agreements of the sensitivity coefficients with that of the finite difference results are good with 1% design perturbation for all load cases except at the load level of 60 lb, where good agreement requires a very small design perturbation of 0.01%. These numerical results indicate that the design sensitivity of displacement may increase without bounds as the load approaches the critical load. Since the proposed method for the critical load factor design sensitivity analysis is based on the design derivative of displacement, the increase of the derivative of displacement is the reason for the rapid growth of design sensitivity of the critical load factor at the load level of 60 lb in Table 2.

Design sensitivity analyses of the nonlinear critical load factor are performed as the applied load approaches very close to the critical load of 60.060911 lb, and the results of four more load levels are shown in Table 4. The design sensitivity coefficient in the fourth column increases drastically from 41.161 at load p = 60.0lb to 1231.8 at load p = 60.0609 lb. Good agreements with finite difference results are obtained for all load levels with extremely small step size. For example, a design perturbation of 10e6% is required to obtain agreement of 105.2% for the load of 60.0609 lb. The critical load factor is very sensitive and highly nonlinear with design variations near the critical limit point. Even though the sensitivity of the critical load factor is correct, it did not approach to the sensitivity of the critical load of 60.14. This indicates that the derivative of the estimated critical load does not converge to the derivative of the actual critical load, even

Table 4. Limit study of design sensitivity of two-bar truss

Load (lb) ‘i Estimated critical load (lb) ax/aA,

60.0 0.108080 + 01 64.84800 41.161 60.05 0.103500 + 01 62.15175 87.883 60.06 0.100980 + 01 60.64859 306.86 60.0608 0.100490 + 01 60.35510 550.08 60.0609 0.100210 + 01 60.18703 1231.8


Fig. 7. Plan of dome.

though the estimated critical load converges to the actual critical load.

Dome

Consider a three-dimensional shallow dome with 30 truss elements, 19 nodal points, and 21 degrees of freedom, as shown in Fig. 7 [I l-13, 15,251. Nodal point coordinates are given in Table 5 and Young’s modulus E is equal to 1.0 x 1O’psi. At the initial design, cross-sectional areas of all elements are

Table 5. Coordinates of nodal points of dome

Node x, (in.) x, (in.) x, (in.)

1 0.0 0.000 85.912 2 180.0 311.769 64.662 7 360.0 0.000 64.662 8 360.0 623.538 0.000 9 0.0 623.538 21.709

18 540.0 311.769 21.709 19 720.0 0.000 0.000

Table 6. Design variable linking of dome

Design Elements linked Number of elements

Al l-6 6 A? 7-12 6 -43 13-18 6 4, 19-30 12

1.528 in2. A concentrated load is applied at node 1 in the vertical direction. Structural analysis is performed using the three-dimensional truss element STIF 8 of ANSYS. Using the incremental analysis method of ANSYS, the actual critical load is found to be 1999.89172 lb.

At several load levels from p = 1500 to 1999.89 lb, the critical load factors are calculated, and design sensitivity analyses are performed with four design variables, A,, A,, A,, and Ad. Element groups for each design variable are shown in Table 6. Design sensitivity coefficients of the critical load factor are presented in Table 7 for seven loading cases. Using the applied load and the critical load factor in the first and second columns, respectively, the estimated critical loads are found to change from 5047.35 lb for applied load p = 1500 to 2004.29 lb, which is 1.002 times the actual critical load (1999.89172 lb) for applied load p = 1999.89 lb. The next four columns of Table 7 represent design sensitivity coefficients for four design variables. The sensitivity results are ver- ified to be accurate using the finite difference method. To show the result of the proposed method, design sensitivity coefficients with uniform design at the applied load level of p = 1999.89 lb are compared with the result of finite difference method for several design perturbations in Table 8. In this table, A’[ is

Table 7. Design sensitivity of nonlinear critical load factor ‘[ for dome

Load ‘p (lb) ‘I aYl% wa.4, ww awA,

1500.00 3.3649 2.51226 2.08182 0.050649 0.0091477 1750.00 2.2912 2.51124 2.02482 0.053239 0.0044556 1875.00 1.7856 1.79502 164736 0.041905 0.0036836 1938.00 I .5024 3.46200 2.73462 0.076044 0.0013737 1969.00 1.3337 4.40610 3.46842 0.097326 O.OtJO7623 1999.00 1.0505 21.0024 16.4736 0.466344 -0.0009459 1999.89 1.0022 440.724 345.624 9.786600 -0.0226056


Table 8. Verification of design sensitivity of critical load factor ‘C using finite difference method for dome (applied load =I 1999.89 lb)

Perturbation (%) Area (in2) ‘C A’C S’C Agreement (%)

0 1.528 1 1.54328 0.1 1.529528 0.01 1.5281528 0.001 1.52801528 0.0001 1.52800153 O.OOOOO98 1.52800015 0.O9oOO13 1.52800002

1.0022290 - - 1.2595373 0.2573083 12.16459 4727.6 1.0765525 0.0743235 1.216459 1636.7 I .0237544 0.0215254 0.1216459 565.1 1.0077491 0.0055201 0.01216459 220.4 1.0032063 0.0009773 0.00121646 124.5 1 II02340 1 0.0001111 0.00011942 107.5 1.0022440 0.0000150 0.00001592 106.1

the finite difference and S’[ is the predicted change of the critical load factor by design sensitivity analysis with the design perturbation. The agreement between S’l and A’[ in the last column approaches 106.1% with 1.3 x 10e6% design perturbation from 4727.6% with 1% design perturbation. This convergence study indicates that the result of finite difference method converges to the result of this work, i.e. the proposed design sensitivity analysis is correct. As in the case of the two-bar truss, near the critical point, the critical load factor becomes very sensitive and highly nonlinear with respect to design.

Shallow folded frame

Apart from truss problems, the proposed method is applied to a shallow 20-member folded frame. Geometric configuration and the location and direction of applied load are the same as for the two-bar truss, as shown in Fig. 6, except that the left bar is composed of 16 two-dimensional beam elements and the right bar of four beam elements. For boundary conditions, both ends are hinged. This frame structure has 20 beam elements, 21 nodal points, and 59 degrees of freedom. A solid rectangular cross-section with the initial design of height 1.0 in. and width 0.8 in. and a linear elastic material is used with

Young’s modulus E = 1 .O x 10’. Design variables are heights h, of the left 16 elements and h2 of the right four elements. Structural analysis is performed using the two-dimensional beam element STIF 3 of ANSYS. The critical load is found to be around 12.1093 lb using the incremental analysis of ANSYS, and the design sensitivity of critical load with respect to the design variable hl is 29.0 using the finite difference method.

At several load levels between p = 2.5 and 12.109 lb, design sensitivity coefficients of the nonlinear critical load factor are evaluated for the design variable h, using design sensitivity expressions in eqn (44), and are presented in Table 9. In Table 9, the estimated critical load level changes from 17.927 lb for the applied load level p = 2.5 lb to 12.145 lb for p = 12.109 lb, by multiplying the applied load in the first column and the critical load factor in the second column. The third column represents design sensitivity coefficients for h,, and the next three columns represent agreements of design sensitivity coefficients with results of the finite difference method. With 1% design perturbation, agreements are good (between 98.3 and 101.4%) up to the load level of 11 lb, 136.7% for the load level of 121b and 2123.4% for the load level of 12.109 lb. As the design perturbation

Load (lb)

Table 9. Design sensitivity of nonlinear critical load factor for shallow folded frame

Agreement (%) with finite difference ‘I ax/ah, 1% perturbation 0.1% perturbation 0.01% perturbation

2.5 7.1706 0.19080 + 2 99.2 99.8 99.8 5.0 3.4898 0.9515D + 1 98.3 99.6 99.7 7.5 2.2389 0.64550 + 1 99.0 99.5 99.5

10.0 I.5699 0.53800 + 1 99.7 99.5 99.5 11.0 1.3554 0.57300 + 1 101.4 99.3 99.3 12.0 1.0910 0.13lOD + 2 136.7 105.8 101.2 12.109 1.0030 0.32480 -I- 3 2123.4 791.0 411.1

Table 10. Verification of design sensitivity of critical load factor ‘C using finite difference method for shallow folded frame (applied load = 12.109 lb)

Perturbation (%) -

0 1 0.1 0.01 0.001 0.0001 0.00001

Height (in.) ‘I A’C

1.0 1.0029500 - 1.01 1.1559056 0.1529556 1.001 1.0440113 0.0410613 1.0001 1.0108512 0.0979012 1.00001 1.0040997 0.0011497 1.OOOOOl 1.0031321 0.0001821 1.OOOOOOl 1.0029750 0.0090250

J’Y Agreement (%) - -

3.24790 + 0 2123.4 3.24790 - 1 791.0 3.24790 - 2 411.1 3.24790 - 3 282.5 3.24790 - 4 178.4 3.24790 - 5 130.0

O.OOOOOl l.OOOOOOO1 1.0029530 0.0009030 3.24790 - 6 108.3


decreases to O.Ol%, the agreement improves to factor at any load level. This raises a question. When 101.2% for the load level of 12 lb and to 411.1% for does an accurate design sensitivity of an approximate the load level of 12.109 lb. This indicates that the solution converge to an accurate design sensitivity of critical load factor at the load level of 12.109 lb the true solution if the approximate solution con- becomes more nonlinear with respect to design vari- verges to the true solution? That is, under what ation than at the load levels below 12.109 lb. condition can we say accurate design sensitivities of

The sensitivity coefficients in the third column finite element analysis results converge to accurate decrease to 5.38 at the load level of lOlb, and then design sensitivities of true solutions if the finite increases rapidly to 13.10 at the load level of 12 lb and element results converge to the true solutions? Fur- to 324.79 at the load level of 12.109 lb. TO see which ther research is required and is being carried out for term contributes to the rapid increase of design continuum design sensitivity analysis of the critical sensitivity, (&, - a&)/D, and (A;, - ‘lD;,YD, in the load. sensitivity expression in eqn (44) are evaluated sepa- rately at load levels of 10 and 12.109lb. The term REFERENCES

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To show the accuracy of the proposed method for design sensitivity analysis of the critical load factor near the critical limit point, design sensitivity results at the load level of p = 12.109 lb are compared with the finite differences for several design perturbations in Table 10. The agreement between S’[ and A’( in the last column approaches 108.3% with 10b6% design perturbation from 2123.4% with 1% design perturbation.

7. CONCLUSIONS

In the process of design optimization, a design of the nonlinear structural system at a given design iteration may not buckle and yet be close to the critical limit point. To prevent buckling of the nonlinear structural system at the next iteration, design sensitivity information of the critical load is necessary. However, the present design may not buckle and only the estimated critical load is available at the final equilibrium. In this case, design sensitivity information of the estimated critical load is necessary. For nonlinear structural systems with design independent loads, the design sensitivity of the critical load factor can be used to obtain the design sensitivity of the estimated critical load. The proposed method works at any prebuckling equilibrium configuration and yields accurate design sensitivity information of the critical load factor. However, it is found that the design sensitivity of the estimated critical load does not approach the design sensitivity of the actual critical load even though the estimated critical load converges to the actual critical load. This is the case even though the proposed method yields very accurate design sensitivity information of the critical load

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Design sensitivity analysis of critical load factor for nonlinear structural systems

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