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Optimization for Maximum Buckling Load of a LatticeSpace Frame With Nonlinear Sensitivity Analysis
byPei-Shan Chen and Mamoru Kawaguchi

Reprinted from

INTERNATIONAL JOURNAL OFSPACE STRUCTURES
Volume 21 · Number 2 · 2006

MULTI-SCIENCE PUBLISHING CO. LTD.
5 Wates Way, Brentwood, Essex CM15 9TB, United Kingdom

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Optimization for Maximum
Buckling Load of a LatticeSpace Frame With Nonlinear
Sensitivity Analysis
Pei-Shan Chen1 and Mamoru Kawaguchi21Dr. -Eng., Chief Engineer, Structure Design Division, Maeda Corporation
2Dr. -Eng., Professor Emeritus, Department of Architecture, Hosei University

Received February, 13, 2006 – Accepted June, 22, 2006

ABSTRACT: The paper presents an optimization analysis method to find theoptimum shape of a lattice space frame with maximum buckling load. The
buckling load of the lattice space frame is determined by a geometricallynonlinear analysis. The mathematic analysis of the optimization is based on
Generalised Reduced Gradient method (GRG) [1], in which the slope of theobjective function (the load parameter at buckling point) with respect to thedesign parameters is required. In order to determine the slope, the authors
introduced a so-called Stiffness Sensitivity Tensor in the nonlinear structural
analysis, such that the sensitivity analysis can be carried out simultaneouslywith the incremental iteration of the geometrically nonlinear structuralanalysis. A single layer lattice dome is taken as a numerical example, and theresults show that the buckling load parameter for full area loading caseincreases 32.75% compared to that for the initial shape.
Key Words: Shape optimization, maximum buckling load, sensitivity analysis,
Stiffness Sensitivity Tensor, nonlinear analysis, lattice space frame, lattice dome.

1. INTRODUCTION
Stability discussion plays an important role in structuraldesign of large span spatial structures, especially for the
single layer lattice dome structures. Ramm andReilinger [2, 3] have presented optimization methods
for shells. However, there are few researches focusing
on the optimum shape of a lattice space frame withmaximum buckling load. In the past years, the authors
have researched such a theme and issued parts of theirworks [4, 5]. In the present paper, the authors will
present an optimization method for maximum bucklingload of a lattice space frame.
For most optimization methods with mathematictreatments, sensitivity of the objective function withrespect to the design parameters is required. But theInternational Journal of Space Structures Vol. 21 No. 2 2006

sensitivity analysis based on a nonlinear structural
analysis is known as a difficult problem, especiallywhen material nonlinear behaviors have to be taken into
account. In the present paper, the authors define a socalled Stiffness Sensitivity Tensor, by which thesensitivity of the objective function with respect to thedesign parameters can be determined simultaneouslywith the iteration of geometrically nonlinear structuralanalysis. Therefore, the derivative of buckling load
parameter with respect to design parameters can beobtained easily. GRG is adopted for the optimization in
the present paper, in which the incremental direction of
the objective function is determined by the derivative of
the objective function with respect to the design
parameters. The objective function in the present111

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Optimization for Maximum Buckling

optimization is the buckling load parameter of a lattice
space frame, which is determined by a geometricallynonlinear structural analysis.

2. FORMULATION
2.1. The basic theory
In general, a nonlinear structural analysis is based on
the incremental equation shown below.
Kd = λ f(1)

where K is the tangential stiffness matrix, d thevector of changes in displacements, f the load mode
vector, and λ the change in load parameter. Many
numerical techniques, such as Riks-Method, loadincremental and displacement incremental methods
are developed to find the equilibrium path (loaddisplacement curve) [6–8], and the buckling point isobtained as a limit point and/or a bifurcation point in
the load-displacement curve.
For a common analysis procedure, the load mode
vector keeps constant, and its coefficient, the loadparameter, is increased step by step by the iteration.
For the convenience of the mathematic analysis, we
make the load parameter as an explicitly objectivefunction. As the first step, we define a vector b by thefollowing equations.

Maximize Λ = b T T K u d u∑ u=1
Subject to :h j (X, U ) = 0 ; (j=1,..., s)
g k (X,, U ) ≤ 0 , (k =1,..., t)
(5)
(6)
(7)

where X is the joint coordinate vector with n elements
and U the other design parameters vector with melements. Equation (6) indicates the equality
constraints, Equation (7) the inequality constraints,
and these conditions make a constraint space S.
X ⊂ ℜn
S= m
U ⊂ ℜh ( X, U ) = 0 j
i = 1, 2,... t, j = 1, 2,... s < n + m g ( X, U ) ≤ 0,i(8)Then, the process of the optimization is to search a
point of the design parameters (X, U), X∈S andU∈S, at which the total buckling load parameter Λreaches its maximum.

2.2. Formulation for multi-loading modes
Tb f =1(2a)0, ∀f i = 0
bi =  1 −1
 f i , ∀f i ≠ 0
N(2b)

where f = {fi} and N is the number of nonzero
elements of f. Then, an explicit function of the load
parameter at incremental step u can be obtained as thefollowing equation.

In fact, a structure should be able to support multiloading cases. On the other hands, the optimum shapefound by the present method depends on the loading
mode f. Therefore, the authors present an optimization
for multi-loading modes by introducing the weighting
function method.
For every loading mode if (i=1,2,…,l), we candefine a vector ib (i=1,2,…,l) by Equations (2a, 2b)
and the load parameter by Equation (3). Then, the totalload parameters iΛ (i=1,2,…,l) can be obtained byT

λ =b K duTuu(3)If the buckling point is found at step T of thenonlinear structural analysis, the total buckling load
parameter Λ can be obtained as:TΛ = b T∑ K u d u

(4)

u =1

Then, the optimization for maximum buckling loadof a lattice space frame can be formulated as below.

112

Λ = i bT ∑ i K u i d u ;iu =1(i = 1, 2,..., l )(9)Then the objective function is in formlF = ∑ wi i Λ

(10)

i =1

where w 1 ,…,w l , ∑ w i = 1

are weighting factorsiwhich are determined by the designers. Then theoptimization for multi-loading cases can be formulated
as:International Journal of Space Structures Vol. 21 No. 2 2006

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Optimization for Maximum Buckling

8design parameters is more efficient than that withgeometrical shape-parameters only.
The numerical example demonstrated that an
optimum shape with maximum buckling load may not
be sensitive to its shape imperfection, especially whenmember section dimensions are taken as designparameters. Nevertheless, a mathematic theory ofimperfection-sensitivity analysis for the present
optimization is remained. Instead of the imperfection
sensitivity analysis, the authors promoted a method ofcomparing the nonlinear behaviors between theimperfection shapes of the optimum and initial ones.

Λ = 6.905
d = 9.462

76Load parameter

4_CHENΛ = 5.846
d = 6.719

543Optimum shape2

Initial shape1002

46810121416Displacement (cm)Figure 15. The load-displacement curves of point aShape imperfection for case 2; Full area loading case;
Point a is shown in Fig. 3(b).

maximum buckling load with member section
dimensions as design parameters is more efficient, andcan provide more information for the structuralworking design. Hence, an optimum shape withmaximum buckling load may not be more sensitive toits shape imperfection.

5. CONCLUSION
By mathematical analysis and numerical examples, theauthors demonstrate that the present method is aneffective method for the shape optimization ofmaximum buckling load. However, this method canfind the optima only in the vicinity of the initial design
parameters, but it may not be efficient to find the global
optima in constraint space. The present optimization
can determine the member sections automatically andout put more information for the structural working
design. Authors deduct that optimization for maximum
buckling load with member section dimensions as118

REFERENCES[1] Minoux, M., “Mathematical Programming (Theory andAlgorithms)”, John Wiley and Sons, 1983.
[2] Reitinger, R. and Ramm, E., “Optimization of geometrically nonlinear buckling sensitive structures”, Optimizationof Structural System, pp. 525–540.
[3] Reitinger, R., Bletzinger, K. -U., and Ramm, E., “Shapeoptimization of buckling sensitive structures”, Computing
System in Engineering, Vol. 5, No.11994, pp. 65–75.
[4] Pei Shan Chen, Mamoru Kawaguchi, and Masaru Abe,
“An application of inverse problem techniques to spatial
structures”, Spatial/Lattice and Tension Structures(Atlanta, Georgia), April 1994, pp. 113–122.
[5] Pei Shan Chen and Mamoru Kawaguchi, “Shape of aSpace Frame With Maximum Buckling Load”, Journalof Structural and Construction Engineering, AIJ, No.
489. November, 1996, pp. 41–46.
[6] Riks, E., “An Incremental Approach To the Solution ofSnapping and Buckling Problems”, Int. J. SolidsStructures, Vol. 15, 1979, pp. 529–551.
[7] Riks, E., Bifurcation and Stability, a Numerical
Approach, “Proceeding of the international conference
on innovative methods for nonlinear problems”, Lui, W.
K., Belytschko, T., and Park, K. C. (eds.), Pineridge Press
International Limited, 1984, pp. 313–344.
[8] Riks, E., “Some computational aspects of the stability
analysis of nonlinear structures”, Computer Methods
in Applied Mechanics and Engineering, 1984, 47,
pp. 219–259.

International Journal of Space Structures Vol. 21 No. 2 2006