optimization for nonlinear stability

Computers % Structures Vol. 30, No. 3, pp. 529-535, 1988 004%7949/88 $3.00 t 0.00 Printed in Great Britain. 0 1988 Civil-Comp Ltd and Pergamon Press plc

OPTIMIZATION FOR NONLINEAR STABILITY

ROBERT LEVY~ and HUEI-SHIANG PERNG$

tFaculty of Civil Engineering, Technion-Israel Institute of Technology, Haifa, Israel SDepartment of Civil Engineering, Rutgers University, New Brunswick, New Jersey, U.S.A.

A&tract--This paper is concerned with the optimal design of trusses to withstand nonlinear stability requirements. While basically a geometrically nonlinear problem, nonlinear stability accounts for large rotations and equilibrium in the deformed state in contrast to linear stability which results in a generalized eigenvalue problem and handles small rotations and ~uilib~um in the initial state.

A two-phase iterative procedure of analysis and redesign is proposed for the nonlinear stability optimization problem. Phase one utilizes an incremental technique of analysis until the point of instability and phase two utilizes a recurrence relation based on optimality criteria for redesign. Examples are presented to illustrate the technique and weight savings involved.

I~KODU~ON

Overall nonlinear structural stability as a constraint on the minimum weight design of trusses is the concern of this paper. The term nonlinear stability is used here in conjunction with overall loss of stiffness

‘: in cases involving large rotations and equilibrium in the deformed state whereas linear stability is used in conjunction with overall loss of stiffness in problems restricted to small rotations and equilibrium in the initial state.

Optimization for linear stability is considered by Khot et al. in [l] and by Levy in [2]. In both cases ‘linear stability is regarded as a generalized eigenvalue problem. Optimality criteria based iterative schemes are presented and used for design. Now, resulting designs may have coinciding critical buckling modes and are sensitive to geometric imperfections. Khot [3]

.‘overcomes simultaneous mode designs by assigning specific interval separations to critical eigenvalues. Linear stability problems are fully analogous to frequency constrained problems [2,4] to the extent that the same schemes may be used for design. An interesting relation between stability and frequency is discussed in [5] in relation to the follower force. A reflection on trusses might be the significance of complex valued frequencies and their relation to stability.

Primarily, interest lies in an allowable stress algorithm as a solver for the nonlinear problem. Allow- able stress algorithms are very popular and stem from the simplest optimization case of designing trusses to withstand allowable stress criteria [6]. An outstanding work in utilizing such algorithms is that of Venkayya [7].

Stability in its nonlinear form is formulated by Kamat et al. [8]. They resort to solving two special cases. A two-bar shallow truss and a four-bar shallow space truss. They obtain explicit relations for the critical load in terms of design variables and then set out to maximize the critical load subject to a given volume.

Khot [9] performs a nonlinear stability analysis on trusses that were optimized for linear stability only to find a 50 per cent load capacity reduction and suggests the need to incorporate nonlinear analysis in the optimization algorithm.

The philosophy in this paper is indeed to utilize an allowable stress recurrence relation based on optimality criteria for the linear stability problem and to redesign using an appropriate nonlinear analysis procedure.

As far as analysis is concerned Hangai and Kawamata [lo] obtained the nonlinear equilibrium equations from the derivatives of the total potential. Strains of up to third order were considered, resulting in a generalized stiffness matrix composed of three individual matrices: the linear elastic stiffness matrix and another two incorporating second and third order nonlinear terms of strain respectively. In- crementally, they followed the equilib~um path beyond the critical point into the postbuckiing range. Paradiso et al. [l l] use succesive linear approxi- mations and an iterative solution to carry them along the equilibrium path into the post buckling range. In fact for the same examples that were solved in [lo, 1 l] similar critical loads were attained.

Spillers [ 12,131 derives the nonlinear effects from equations of equilibrium that are usually used for linear structures without sliding into higher order theories. By introducing prestress he shows, elegantly, that only a first order Taylor approxi- mation of the deformed length is needed to describe nonlinear phenomena for small rotations and to derive the geometric stiffness matrix. In a consistent manner, if second order terms are permitted then length changes due to rotations will have been included. He further shows that the geometric stiffness matrix can be conveniently obtained by taking the gradient of the equilib~um equation with respect to the joint coordinates. The above formu- lation forms a smooth transition from zero order theory (linear elastic) to first order theory (geo-

529

530 ROBERT LEVY and HUEI-WANG PERNG

metrically nonlinear restricted to small rotations) and renders itself suitable for including stability and utilization in optimization. Other examples for the analysis of space trusses for stability may be found in the work of Rothart et al. [14] and Jagannathan et al. [15].

In relation to geometric imperfections Thomson and Hunt [16] are concerned with safety with optimization. Rosen and Schmit [17-191 present an analysis and an optimization procedure for structures having local and system imperfections in an attempt to answer some of the safety questions that arise in optimized structures in relation to simultaneous failure. Basically they use a sequence of unconstrained minimization techniques (SUMT) for optimization in conjunction with an approximate analysis to obtain their optimum designs. Concern over safety of optimized structures is raised by Patnaik and Srivastava [20]. They argue that since several modes of failure (e.g. stability, stress, displacement) are active at the optimum some structures maybe dangerously unsafe. They introduce imperfections and suddenly applied loads to the cylinder and show an increase in weight of 230% compared to that of a perfect cylinder. Interaction of stability and frequency causes an additional weight increase of 21%.

Following this introduction the optimization strategy is discussed, optimality criteria are derived, an algorithm is presented and examples are solved.

OPTlMlZATlON STRATEGY

It is aimed to use a technique which is somewhat close to a designers heart-that of analysis and redesign.

Analysis

The Appendix provides a general background on linear analysis, geometrically nonlinear analysis and stability.

Analysis is performed using a modified version of the program described in [13]. The modification is such that the existing nonlinear analysis program can take incremental proportional loading until the point of instability. The program updates the nodal coordinates based on equilibrium in the deformed configuration and stores the resultant forces as prestress ready for the next increment. Coding details can be found in [21].

Remembering that the optimization problem re- quires design for a given load, scaling and reanalysis will yield a buckling load equal to the given design load for a given ratio of member stiffness. The real question of course is what is the optimal ratio of member stiffnesses or what the redesign step (reshuffling of stiffnesses) should be before the next analysis is performed.

Redesign is performed via an optimality criterion which is henceforth derived. The derivation is made

for the linear stability problem and its validity for every analysis step is assumed.

Optimality conditions

Linear stability can be cast as

Minimize 1 &(A:)*.

Subject to 1 = 1. (1)

Here 15 is a specified buckling load factor and the objective function is proportional to weight since

1 Ki(Ay)’ = (0:/E) 1 A,L, N volume, (2) I

where

K, = (A,E)/L,,

A; = (a. LJE

and 0, is the allowable stress. Following eqn (A9) the lowest buckling load factor

can be written as

Optimality conditions are derived via the Lagrangian

L = CK~(A;)~ + p(n - I). (4)

Differentiation yields

The comma in eqn (5) stands for a/aK,. From eqn (3) we obtain

A.( = - A;

Z$ (7)

where A, is the member displacement associated with t&e eigenvector S. For normalized eigenvectors 6K,6 = 1. Substituting eqn (7) into eqn (5) yields

1 A. 2 -_= 1 . P 0 Ay

At the optimum, therefore, the ratio of the member length change associated with the lowest buckling mode to the allowable member length change is a constant.

Optimization for nonlinear stability 531

This property suggests an immediate use of member displacements instead of normalized eigenvalues together with the fundamental allowable stress algorithm for redesign. Before exiting the redesign step the stiffnesses may be scaled using eqn (3) to arrive at 1 = 1.0.

While eigenvalue realizability may not be satisfied optimality conditions are. Hence the motivation. The redesign step is therefore

where p is the convergence factor. To avoid unreal- istic stiffnesses the member displacements are scaled according to

A?) = A?) ,/(min{pi}). (10)

Note that while the different approach used in [2] where an eigenvector based recurrence relation is derived may be more proper, the heuristic approach of eqn (9) is rather simple, very easily implemented and avoids the use of eigenvalue calculations.

A TWO PHASE ALGORITHM

The two phases are the analysis and redesign which have already been discussed in the previous sections.

Phase 1

Pick arbitrary starting values for the stiffness with K, # 0. Analyze for nonlinear stability and determine

Pclitical . Assume Pcritical < Pdesign. Otherwise scale design variables as

Determine deformed equilibrium configuration and calculate weight.

Phase 2

1. Evaluate the Langrange multipliers (pi) using eqn (8). Let p = [min{pi}]“2.

2. Scale Ai-‘Ai*p. 3. Redesign using eqn (9), i.e.

Phase 1 is returned to after resetting the problem with the original status except for member sizes.

The stopping criteria is set (1) by the number of iterations, (2) by limiting CPU time and (3) by the percentage difference in the weight of the structure for successive iterations.

EXAMPLES

A computer program coded in FORTRAN was used to solve the examples in this section. The analysis routine of [13] was modified to suit the needs of the algorithm. The basic changes include: proportional incremental loading until buckling, an added routine for redesign and making the whole program iterate. For more details the reader is referred to [21].

The 132-bar truss

Figure 1 shows three views of the 132-bar dome with dimensions in inches. All the circumferential nodes are fully fixed and each internal node is loaded with 1000 lb vertically downwards. Young’s modulus is taken as 10' psi, the density as 0.1 lb/in3 and the allowable stress as 75,000 psi. This structure was optimally designed for linear stability in [ 1,2] and here for nonlinear stability using initial cross- sectional areas of 0.216 in2 and a convergence factor of p = 100.

Figure 2 shows the iteration history for two cases. Curve L portrays the linear stability design for starting values of 0.216in2. The optimum of 76.34Ib is reached in an oscillating manner and then drifting away from the optimum is evidenced. This is because first and second modes coincide at the optimum and the derivative of the buckling load factor with respect to the design variables is undefined, making the optimum numerically unstable and sensitive to starting values. Oscillations are due to modes inter- changing. Curve N portrays the nonlinear stability design. Oscillations are no longer observed and it appears that deformed equilibrium stabilizes the curve but somewhat slows convergence for this case of a problematic example. An optimal design of 92.43 lb was attained after 40 iterations. For additional accuracy one may want to increase p in the redesign phase and fine tune the analysis phase. The nonlinear stability optimization factor of 92.43/76.34 = 1.21 is rather small. Table 1 lists the optimal design. The deformed shape of the optimal truss is such that all nodes displace downwards.

The 24-bar truss

Figure 3 shows the 24-bar truss with dimensions in centimeters. Here too all the circumferential nodes are fully fixed and the central node is subjected to a load of 100,000 g acting vertically downwards. Young’s modulus is taken as 21.093 x lo6 g/mm2, the density as 0.01 g/mm3 and the allowable stress as 30,000 g/mm2. This truss was analyzed in [lo] for nonlinear stability. It is interesting to note that for initial cross-sectional areas of 10 mm2 [lo] documents a nonlinear buckling load of 67,500 g whereas the analysis program which is used here produces a very close buckling load of 65,261 g (-3% difference). The iteration history is shown in Fig. 4 and Table 2 lists the optimal design. In the deformed configuration the central node displaced downwards

532 ROBERT LEVY and HUEI-SHUNG PERNG

Fig. 1. The 132-bar truss.

10 times as much as the rest (N 8.5 mm). An optimum of 529.4 g was attained after 20 iterations and did not improve further.

The four-bar truss

Figure 5 shows the four-bar truss with dimensions in inches. Peripheral nodes are fixed and a load of 173.8341b acts on the free node vertically downwards. Young’s modulus is taken at 75,000psi. the density as 0.1 lb/n?, and the allowable stress as

751 , , , I I ITERtTlONS ’

Fig. 2. Weight history of the 132-bar truss.

75,000psi. This truss was optimized for nonlinear stability in [8]. Actually the dual problem was solved in the cited reference since the buckling load was maximized for a given volume (weight) of material with results of a buckling load of 173.834lb and equal cross-sectional areas of 1.73472 in* for a weight of 1001b.

Figure 6 shows the iteration history and Table 3 the design history for individual members. Initial analysis revealed that a load of only 159.942 lb could be supported. Scaling increased the weight to

Table 1. Optimal design of the 132-bar truss

Members Area

192 0.229 1 0.228

13,16 0.230 14,15 0.228 31,32 0.219 43,48 0.233 44,41 0.220 45,46 0.230 13,15 0.230

74 0.221 91,98 0.244 92,91 0.222 93,96 0.230 94,95 0.224

t Weight = 92.4 lb (six-fold symmetry).


n 15 6

/-pw..+$ c /I

L 25 x_

x ‘x3 z L3.2 n L

Fig. 3. The 24-bar truss.

Fig. 6.

Fig. 4. Weight history of the 24-bar truss.

108.705 lb and an optimum of 103.358 lb was finally attained. Convergence is very fast. In the deformed position the central node displaced 1.01 in. downwards.

This example was also solved for linear stability in [2] resulting in an optimal weight of 19.24 lb and a nonlinear stability optimization factor of 103.358119.24 = 5.37.

Table 2. Optimal design of the Mbar truss

Members

2,:‘: 6 8) i

7,9,10, 12 13, 18,19,24 14, 17,20,23 15. 16.21.22

Area

16.256 16.206 13.767 13.803 2.081 2.116 2.045

t Weight = 529.4 g (four-fold

-2 I I_ lu ~60 IC

1 1 1

Fig. 5. The four-bar truss.

‘O+Yiix7 Weight history of the four-bar truss.

Table 3. Design history of the four-bar truss

Iteration Member areas No. 1 2 3 4

0 1.7347 1.7347 1.7347 1.7347 : 1.8858 1.9422 2.5996 1.8859 I 1.8857 .5520 1.4900 1.8857

3 1.9421 2.5994 1.5520 1.4900 4 1.9421 2.5994 1.5520 1.4900

t Weight = 103.4 lb.

CONCLUSION

The simplest of allowable stress algorithms has demonstrated its rise in a rather complex optimal design problem of optimization for nonlinear stability in conjunction with a powerful nonlinear analysis program to form an analysis/redesign scheme for practical design.

The need for nonlinear type solutions was demonstrated in solved examples due to the large nonlinear stability optimization factors that may arise (1.21, 5.37).

Beyond this it is hoped that optimization for nonlinear stability is now closer to the designers desk.

Acknowledgements-The authors gratefully acknowledge the help of the staff of the CCIS at Rutgers and the technical

534 ROBERT LEVY and HUEI-SHIANG PERNG

assistance of the Structural Engineering Staff at the APPENDIX: THE NODE METHOD, GEOMETRIC Technion. NONLINEARflIES AND STABILITY

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N. S. Khot, V. B. Venkayya and L. Berke, Optimum structural design with stability constraints. Inc. J. Nu- mer. Method. Engng. 10, 1097-l 114 (1976). R. Levy, Optimization for Overall Stability. In Recent Deuelopments in Structural Optimization. (Edited by F. Y. Cheng), pp. 86-100. American Society of Civil Engineers, New York (1986). N. S. Khot, Optimal design of a structure for system stability for a specified eigenvalue distribution. Inter- national Symposium on Optimum Structural Design (Edited by E. Atrek and R. Gallagher). University of Arizona, Tucson, AZ (1981). W. R. Spillers, S. Singh and R. Levy, Optimization with frequency constraint. J. Struct. Div., ASCE 107, 2337-2347 (1981). S. P. Timoshenko and J. M. Gere, Theory of Elastic Sfabihty, 2nd Edn. McGraw-Hill, New York (1961). W. R. Spillers, Iterutice Structural Design. North Holland Elsevier, New York (1975). V. B. Venkayya, Structural optimization: a review and some recommendations. Inc. J. Numer. Meth. Engng. 13, 203-228 (1978). M. P. Kamat, N. S. Khot and V. B. Venkayya, Opti- mization of shallow trusses against limit point instability. AIAA J. 22, 4033408 (1984). N. S. Khot, Nonlinear analysis of optimized structure with constraints on system stability. AIAA J. 21, 1181-I 186 (1983). Y. Hangai and S. Kawamata, Nonlinear analysis of space frames and snap-through buckling of reticulated shell structures. Proc. 1971 IASS Pacific Syposium, Part II on Tension Structures and Space Frames, Tokyo and Kyoto, Architectural Institute of Japan, Paper No. 94, 803-816 (1972). H. Paradiso, E. Reale and G. Tempesta, Nonlinear Post-buckling analysis of reticulated dome structures. Proceedings of the IASS World Congress on Shells and Spatial Structures, Vol. 1, pp. 67-80 (1979). W. R. Spillers, On the relationship between buckling and optimal structural design. J. Franklin Inst. 229, 463466 (1975). W. R. Snillers. R. B. Testa and N. Stubbs, Analysis of fabric structures. J. Franklin Inst. 306, 341-353 (1978). H. Rothert, T. Dickel and D. Renner, Snap-through buckling of reticulated space trusses. J. Struct. Div. ASCE 107, No. STl, 1299143 (1981). S. Jagannathan, H. I. Epstein and P. Christiano, Non- linear analysis of reticulated space trusses. J. Struct. Div., ASCE 101, 264-2658 (1975). J. M. T. Thompson and G. W. Hunt, A General Theory of Elastic Stability. John Wiley, New York (1973). A. Rosen and L. A. Schmit, Jr, Design oriented analysis of imperfect truss structures. Part I: Accurate Analysis. Inr. J. Numer. Meth. Enang 104. 1309-1321 (1979). A. Rosen and L. A. Schmit:Jr, Design oriented analysis of imperfect truss structures. Part II: approximate analysis. Int. J. Numer. Mech. Engng IS, 483494 (1980). A. Rosen and L. A. Schmit, Jr, Optimization of truss structures having local and system geometric imperfections. AIAA J. 19, 626-633 (1981). S. N. Patniak and N. K. Srivastava, Safety of optimality designed structures like cylinders and plates. Comput. Struct. 11, 363-367 (1980). H.-S. Pemg, Structural weight optimization with overall stability constraint in reticulated space truss. M.Sc. Thesis, Dept. of Civil Engineering, Rutgers University, New Brunswick, NJ (1987).

me. W. R. Spillers, Automated Structural Analysis: An In- troduction. Pergamon Press, New York (1972).

REFERENCES The node method

The node method [22] of analysis serves as the basis for all derivations in this paper. Its equations of structures are

#F = P (nodal equilibrium) (Al)

F = F’ + KA (constitutive equations) (A2)

N6 = A (member/nodal displacement eqns). (43)

In these equations F = member force matrix; P = nodal load matrix; Fc’ = initial member force matrix (prestress); K = primitive stiffness matrix; A = member displacement matrix; d = nodal displacement matrix; and N = generalized incidence matrix. The primitive stiffness matrix is diagonal and positive definite with the ith diagonal element K, = A,E/L,, where A, is the cross-sectional area of member i, E is Young’s modulus and L, is the length of member i.

Analysis is performed by combining eqns (AlHA3) and solving for the nodal displacements from

(flKN)b = P - &+. (A4)

Geometric nonlinearities

Following 112, 131 the total derivative of the equilibrium equations yields

(dfi)F + RdF = dP. (A5)

With the nodal displacements regarded as variations of nodal coordinates eqn (A5) becomes

[#(VF) + (VR)Fjd = dP. (A6)

The term #(VF) is simply NKN 3 Ks, the linear elastic stiffness matrix, whereas the term (Vm)F simply constitutes a definition for the geometric stiffness matrix, K,(F). Equation (A6) may, therefore, be written as

[Ks + K,(F)]6 = dP. (A7)

Explicitly the contribution of member i whose incidence nodes are A and C is given by

r (I - n,fi,) -(I - nir$)l F . . Row A

WG), = -(I-&i,)... (I-&i,) 2 . Row C.

Col. A Cal. c

Here n, is the unit vector associated with member i.

Linear stability

For proportional loading AdP, eqn (A7) may be Written as

[KE + IK,(F,)]6 = P. (A@

Here F0 = KN(flKN)-’ dP, and P = ,ldP,. In the unstable state where 6 is unbounded eqn (A8) holds as

[KE + nK,(F,)]c5(‘) = P

and

[Ks + nKc(F,)]b(z’ = P.

Subtraction yields

IKE + nKo(F,,)]b = 0. (A9)

Equation (A9) constitutes a generalized eigenvalue problem.


The buckling load factor of interest is

I,, = sgn(&) min 1 Ail, (A101

where n is the position in the array of Al’s of the arith- metically smallest Li (first or last).

Still in the realm of analysis restricted to small rotations eqn (A7) may be enhanced by iterating as follows

[KS + KG(F”)]G”+’ = dP.

The use of eqn (Al 1) produces a realizable solution. More- over if instability is regarded as a general loss of stiffness an alternative to eqn (A9) can be. achieved through eqn (Al 1). The load is proportionally increased by small increments and each time an analysis is performed. The resultant internal forces Pk of the kth load, dP, are used as prestress for dP,+, in the following manner

[K,+ K,(Pn,)J6”+’ = dP,+, - mPK. (AW

This procedure is continued until

det[K, + KG@‘,)] = 0. (A13)

Of the two techniques for attaining buckling load factors the second is more appealing from an engineering stand- point since realizable stress resultants and displacements are available at each increment.

Nonlinear stability

Here small rotations are no longer tolerated and equilibrium is required in the deformed configuration. As far as buckling is concerned eqn (A13) may still be. used in conjunction with the incremental approach. The only difference in analysis now is that within each increment the nodal coordinates are corrected for each equilibrium trial and this in turn triggers the varying of KE and KG because of the unit vectors and an additional change in KG because of a change in internal forces (prestress) due to member length changes. Convergence toward a deformed equilibrium configuration is controlled via an error norm which dictates the acceptance of nodal displacements for a certain equilibrium trial or halving of the same. For further details of large rotations analysis the reader is referred to [I 9.

optimization for nonlinear stability

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