1 Maximum loads of imperfect systems corresponding to stable bifurcation
Maximum loads of imperfect systems corresponding to
stable bifurcation
Makoto Ohsaki *
Department of Architecture and Architectural Systems,Kyoto University,Sakyo,Kyoto 606-8501,Japan
Received 15February 2001
Abstract
Asimple and computationally inexpensive approach is presented for obtaining the maximum load factor of an elastic structure considering reduction of load-carrying capacity due to inevitable initial imperfections.The structure has a stable bifurcation point if no initial imperfection exists.An antioptimization problem is formulated for minimizing the maximum loads reduced by the most sensitive imperfection within the convex bounds on the imperfections of nodal locations and nodal loads.The maximum loads may be de?ned by bifurcation points or deformation constraints.A problem of simultaneous analysis and design with energy method is formulated to avoid laborious nonlinear path-following analysis.The stable bifurcation point is located by minimizing the load factor under constraint on the lowest eigenvalue of the stability matrix.It is shown in the examples that a minor imperfection that is usually dismissed is very important in evaluating the maximum load of a ?exible structure.ó2002Elsevier Science Ltd.All rights reserved.Keywords:Buckling analysis;Imperfection sensitivity;Minor imperfection;Antioptimization;Convex model;Simultaneous analysis and design
1.Introduction
The lower bound of the maximum load factor of a geometrically nonlinear structure that exhibits bi-furcation-type instability may be evaluated based on the most critical mode of imperfection that maximizes the reduction of the load carrying capacity under constraint on the norm of the imperfection.There have been several studies for ?nding the most critical mode of imperfections for simple and coincident unstable symmetric bifurcation points (Ho,1974;Ikeda and Murota,1990)based on a perturbation approach (Koiter,1945;Thompson and Hunt,1973).For a symmetric structure subjected to symmetric proportional loads,which is called symmetric system for brevity,an antisymmetric imperfection is classi?ed as ma-jor imperfection or ?rst-order imperfection in the sense that the imperfection has direct e?ect on the de-rivative of the total potential energy in the direction of the buckling mode.Ohsaki et al.(1998)presented an *Tel./fax:+81-75-753-5733.
E-mail address:ohsaki@archi.kyoto-u.ac.jp (M.Ohsaki).
0020-7683/02/$-see front matter ó2002Elsevier Science Ltd.All rights reserved.
PII:S 0020-7683(01)00232-3
928M.Ohsaki/International Journal of Solids and Structures39(2002)927–941
optimization method considering the reduction of maximum load factor due to the most critical mode of major imperfection.
It is laborious,however,to?nd the buckling load factor by a nonlinear path-following analysis and to obtain the most critical imperfection based on a perturbation approach because the formulations are complicated and are di?cult to implement in a?nite element analysis program.In addition to this di?culty, the maximum load factor is estimated at the perfect system using sensitivity informations;i.e.a moderately large imperfection is not considered.
Ohsaki(2000)presented an algorithm for obtaining optimum designs of symmetric systems with coin-cident critical points,and showed that optimization does not always increase imperfection sensitivity.For a symmetric system,a symmetric imperfection is classi?ed as minor imperfection or second-order imper-fection(Roorda,1968)in the sense that the imperfection does not have direct e?ect on the derivative of the total potential energy in the direction of the buckling mode.For a minor imperfection,the critical point of an imperfect system remains to be a bifurcation point,which is contrary to the fact that the critical point of an imperfect system corresponding to a major imperfection turns out to be a limit point.Although the imperfection sensitivity of a bifurcation load factor corresponding to a minor imperfection is bounded, Ohsaki(2000)showed that a minor imperfection is sometimes more critical than major imperfections if moderately large imperfection is allowed.
One approach for avoiding di?culty in numerical implementation for obtaining the most critical im-perfection is to use a stochastic approach.In this case,however,the probability distribution of initial imperfection should be given appropriately,preferably based on experiments.The convex model is very e?ective for the case where stochastic approach cannot be used(Ben-Haim and Elishako?,1990).It con-siders uncertainty within known bounds on the parameters.Elishako?et al.(1994c)applied the convex model to stability analysis of imperfection sensitive columns on elastic foundation,where prebuckling deformation can be neglected in buckling analysis and the buckling load factor is linearized with respect to the imperfection parameters.They compared the results by a stochastic approach and the convex model, and showed that only few modes are necessary for buckling analysis and for modeling imperfections. Elishako?et al.(1994a)presented a method for obtaining the most critical imperfection for elastic static problem using the antioptimization approach to obtain the possible worst case values of the parameters. They considered uncertainty in loads as well as the nodal locations.Uncertainty in the elastic modulus has been considered in Elishako?et al.(1994b).Pantelides(1996a)introduced imperfections in geometry and material properties.Pantelides(1996b)used elliptic bounds for buckling analysis of columns on uncertain elastic foundation.Aconvex model for buckling of bars connected by springs are discussed in Pantelides (1995).
The method called simultaneous analysis and design,which is abbreviated as SAND,is very e?ective for reducing the computational cost for path-following analysis that should be carried out at each step of optimization or antioptimization of geometrically nonlinear structures.It considers the state variables as well as the design variables as independent variables.Haftka(1985)incorporated the equilibrium equations into the objective function by using the interior penalty functions,and presented an e?cient approach for avoiding illconditioning of the Hessian of the Lagrangian or the objective function.His approach has been shown to be applicable to truss topology optimization problems(Sankaranarayanan et al.,1994).The method with direct incorporation of the equilibrium equations as equality constraints has also been pre-sented(Wu and Arora,1987;Orozco and Ghattas,1997).
Contrary to imperfection-sensitive structures such as cylindrical shells and sti?ened plates,the bifur-cation point of a column that has a stable postbuckling path disappears due to a small major imperfection;
e.g.antisymmetric imperfection of a symmetric system.Aquestion then arises how the maximum load factor of such stable structures should be de?ned.One approach is to allow deformation along the bi-furcation path that has the load factor above the bifurcation load(Pietrzak,1996).In this case,the maximum load factor may be determined by the constraints on displacements and/or stresses.The anti-
symmetric component of initial imperfection,however,may happen to be very small,and sudden defor-mation may occur near the bifurcation point.Therefore in some situations it may be unsafe to allow loading along the bifurcation path,and the bifurcation load factor should be used for de?ning the maxi-mum load.
In this paper,a simple and numerically inexpensive approach is presented for determining the maximum load factors of imperfect elastic structures considering imperfections of nodal locations and nodal loads.It is shown in the examples that the reduction of the maximum loads de?ned by displacement constraints is very small if a major imperfection is considered and moderately large displacements are allowed.Therefore, minor imperfections should be considered in de?ning the most critical mode of imperfection.An antiop-timization problem is formulated so as to minimize the bifurcation load factor within the convex bounds on the imperfection parameters.Arelaxed problem is solved based on the SA ND,where the bifurcation load is determined by minimizing the load factor under constraint on the lowest eigenvalue of the stability matrix allowing imperfections of nodal loads.This way,laborious nonlinear path-following analysis is successfully avoided.It is shown in the examples of a20-bar truss that the most critical mode of minor imperfection can be successfully obtained by the proposed approach.
2.Maximum load factor of an imperfect system
Consider a?nite dimensional elastic structure subjected to quasi-static proportional loads P de?ned by the load factor K as P?K p,where p is the speci?ed vector of load pattern.The vector of nodal dis-placements is denoted by U?f U i g.Let n denote an imperfection parameter that represents any type of imperfection including initial dislocation of nodes,distortion of cross-sectional shape of a member,etc.The total potential energy PeU;K;nTis de?ned as
PeU;K;nT?HeU;nTàK p TenTUe1Twhere HeU;nTis the strain energy which is assumed not to depend explicitly on K.This assumption is valid for a proportionally loaded structure modeled by a?nite element formulation using nodal displacements as variables for de?ning deformation.
The equivalent nodal force FeU;nT?f F jeU;nTg is de?ned by
F jeU;nT?o H
o U j
ej?1;2;...;nTe2T
where n is the number of degrees of freedom.The equilibrium equations are written as
F jeU;nT?K p jenTej?1;2;...;nTe3TIn the following,the arguments U,K and n are omitted except the case where dependence on those variables is important.The stability matrix S,which is the tangent sti?ness matrix used for nonlinear analysis,is given as
S?
o2H
o U i o U j
e4T
The r th eigenvalue k r and eigenvector U r of S are obtained from
S U r?k r U r;er?1;2;...;nTe5TThe critical load factor K c corresponds to k1?0,where k1is the lowest eigenvalue.
M.Ohsaki/International Journal of Solids and Structures39(2002)927–941929
De?ne a as
a ?X n i ?1o 2P o n o U i U 1i e6T
where U 1i is the i th component of U 1.The major and minor imperfections are characterized by a ?0and a ?0,respectively (Roorda,1968).For a symmetric system,the prebuckling deformation is symmetric and the bifurcation mode is antisymmetric.In this case,a symmetric and antisymmetric imperfections corre-spond to minor and major imperfections,respectively.Figs.1and 2illustrate the relation between K and a representative antisymmetric displacement component U for the cases of major and minor imperfections,respectively.It is seen from Fig.1that K increases above the bifurcation load factor K c of the perfect system if a major imperfection exists.For the case of minor imperfection,as shown in Fig.2,an imperfect system still has a bifurcation point,and the bifurcation load factor may increase or decrease depending on the sign of the imperfection parameter.
Since antisymmetric components of deformation along the bifurcation path of the stable bifurcation point may be very large,the maximum load should be de?ned in view of the stresses and/or displacements.Aquestion arises whether it is safe to expect loads above the bifurcation load.A lthough a critical point does not exist for a structure with a major imperfection,the imperfection may happen to be extremely small,and the structure may reach a bifurcation point that causes sudden dynamic antisymmetric mode of deformation.Therefore,the maximum load factor of a structure exhibiting stable bifurcation may be de-?ned by either of the following criteria:
C1Bifurcation load factor.
C2Load factor corresponding to the speci?ed limits on stresses and/or displacements.Consider a case where the maximum load factor is de?ned by the upper bound U
of the displacement component U for a system illustrated in Figs.1and 2.It is observed from Fig.1that the magnitude of reduction of the maximum load factor due to a major imperfection is very large for a small range of U ,e.g., U ?U a ,but it decreases as U is increased to,e.g., U ?U b .For a minor imperfection,the magnitude of reduction does not strongly depend on U ,and it is larger than that to a major imperfection if U is mod-erately large,e.g., U ?U b .Therefore,for the case where C2is used,the most critical mode of imperfection will be a major imperfection if U
is su?ciently small,otherwise the maximum load factor should be de?ned by a minor imperfection.If C1is used,the most critical imperfection should be a minor imperfection because the bifurcation point disappears if a major imperfection exists.In this paper,we consider a
?exible
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system allowing a moderately large deformation.In this case,minor imperfection plays a key role in evaluating the maximum load factors of the imperfect systems considering both criteria C1and C2.Agood approximate maximum load factor and the corresponding most critical mode of imperfection will be found by minimizing the bifurcation load factor even for the case of C2.
Summarizing the discussion above,we de?ne the most critical imperfection by reduction of the bifur-cation load factor due to minor imperfections based on the following reasons:
(1)Even for a stable bifurcation,reaching the bifurcation point should be avoided because it leads to a sudden dynamic deformation.Since the bifurcation point disappears if a major imperfection exists,the most critical imperfection for this case is a minor imperfection.
(2)Since we consider a ?exible structure and allow moderately large deformation,the maximum load de?ned by deformation constraints is dramatically reduced by minor imperfections rather than major imperfections,and sensitivity of the maximum load is almost equivalent to that of the bifurcation point.
3.Antioptimization problem
If we consider only symmetric systems,imperfections can easily be divided into major and minor im-perfections based on the symmetry conditions.For a more complicated structure without explicit symmetry properties subjected to nonsymmetric loads,classi?cation of imperfection is not straightforward.In this section,a method is presented for obtaining most critical minor imperfection without carrying out any preprocessing for orthogonalization or classi?cation of imperfection modes.Note that the most critical major imperfection of an unstable symmetric bifurcation point may be found directly by the perturbation approaches (Ho,1974;Ikeda and Murota,1990).There has been no study,however,for ?nding most critical imperfection of a stable symmetric bifurcation point.
Let n i ei ?1;2;...;m Tdenote the vector of i th set of imperfection parameters including any possible type of imperfection such as nodal locations and cross-sectional areas.The norm of n i is denoted by e i en i Twhich is a convex function of n i .Suppose an upper bound e i is given for e i en i Tby an approach similar to that of the convex model (Ben-Haim and Elishako?,1990).In the formal convex model,the objective function is linearized by utilizing the ?rst order sensitivity information,and the optimal or antioptimal solution is uniquely determined.In this paper,however,the nonlinear buckling load factor is directly used as objective function in order to rigorously incorporate the geometrical
nonlinearity.
M.Ohsaki /International Journal of Solids and Structures 39(2002)927–941931
The set of vectors n i is divided into major imperfections n I
i and minor imperfections n II i .The val-ues corresponding to major and minor imperfections are indicated by superscripts eTI and eTII ,respec-tively;e.g.the upper bound for e II i en II i Tis denoted by e II i .Let n II denote the vector that consists of all the
elements of the vectors n II
i ,ei ?1;2;...;m II T.The maximum load of the imperfect system considering re-
duction due to the most critical mode of minor imperfection is de?ned as the solution of the following optimization problem:
P1:minimize
K c en II Te7Tsubject to e II i en II i T6 e II i ei ?1;2;...;m II Te8T
This type of problem for ?nding the minimum load factor is called antioptimization problem (Elishako?
et al.,1994a).The variables in P1are n II
i ,ei ?1;2;...;m II T.
As noted above,the objective function of P1is not linearized with respect to the imperfection para-meters;i.e.the rigorous nonlinear formulation is used for K c .P1may be solved by using an appropriate gradient-based optimization algorithm if sensitivity coe?cients of the critical load factors can be found (Ohsaki,2000).However,K c en II Tcorresponding to the given set of n II
i should be determined by tracing the
fundamental equilibrium path at each iterative step of optimization.Therefore,the formulation of P1is computationally expensive.
If we ?x n II and only consider major imperfections,the region in the eK àU T-space where k 160is satis?ed is as indicated by feasible region in Fig.3,where U is a representative generalized displacement generated due to existence of a major imperfection.For a symmetric system,U represents an antisymmetric component of deformation.The thick curve in Fig.3is the bifurcation path of the perfect system,and thin curves are equilibrium paths of imperfect systems.The dotted curves indicate unstable equilibrium points.Note that the region bounded by the dashed curve ABC is feasible for the constraint k 160.Since we consider the case where the perfect system exhibits stable bifurcation,the feasible region in the vicinity of the bifurcation point is convex with respect to U and K .Hence,the buckling load factor is found by minimizing K with respect to n I
i under constraint of k 160.We further minimize K with respect to n II i to
obtain the most sensitive imperfection.Since both processes correspond to minimization of K ,those can be carried out
simultaneously.
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In order to reduce the computational cost for geometrically nonlinear path-following analysis,nodal displacements are also considered as variables of the optimization problem,and analysis and optimization are simultaneously carried out.If we use the most simple formulations of SAND,the state variables U are iteratively updated to satisfy the equality constraints.The optimization problem with equality constraints, however,are computationally costly.Atwo stage algorithm can be used for reducing the computational cost(Wu and Arora,1987).In this case,a special algorithm should be implemented for optimization.If the equality constraints are included in the objective function as penalty terms(Haftka,1985),the optimization problem can be solved by simply using an optimization package.
In this study,we consider U as variables which are the same level as n II and relax P1allowing imper-fections in nodal loads.Therefore,we do not need the exact equilibrium state corresponding to the perfect nodal loads.The ranges of the nodal loads are given as
K p L
j 6P
j
6K p U
j
ej?1;2;...;nTe9T
where p L
j and p U
j
are the speci?ed lower and upper bounds,respectively.Suppose the case where the upper
bound D P for the error in the load is proportional to K as D P?K D p.Eq.(9)is then written as Kep jàD pT6P j6Kep jtD pTej?1;2;...;nTe10TLet n denote the vector consisting of n i including minor and major imperfections.From Eq.(2),the internal nodal forces F?eU;nT?f F?
j
eU;nTg equivalent to the displacements U of an imperfect system are de?ned by
F?j eU;nT?
o HeU;nT
o U j
ej?1;2;...;nTe11T
whereeT?indicates a function of U and n.F?
j is then calculated for each trial displacement vector during
the optimization process.
The optimization problem to be solved is formulated as follows for?nding the minimum value of K under constraints on the norms of imperfections and the lowest eigenvalue k?
1
eU;nTof the stability matrix: P2:minimize Ke12Tsubject to e ien iT6 e iei?1;2;...;mTe13T
Kep jàD pT6F?
j
eU;nT6Kep jtD pTej?1;2;...;nTe14T
k?
1
eU;nT60e15T
The variables of this problem are U,n and K.Only computation of F?
i eU;nTand k?
1
eU;nTis needed for the
current value of U and n at each iterative step of optimization,and the laborious path-following analysis is not needed.
4.Examples
Consider a column-type20-bar plane truss as shown in Fig.4.The lengths of members in x-and y-directions are100and200cm,respectively.The cross-sectional areas are2.0cm2for all the truss members. The proportional loads in the negative y-direction at nodes7and8are given as K p,where p?98kN.The elastic modulus is205.8GPa.The axial strain is de?ned by the Green’s strain.Optimization is carried out by ID ESIG N Ver.3.5(Arora and Tseng,1987),where the sequential quadratic programming is used,and the gradients of the objective and constraint functions are computed by the?nite di?erence approach.Very strict convergence criteria have been assigned for obtaining rigorous optimal solutions;i.e.the constraint violation is limited to1:0?10à5,and the di?erence of the objective values in two consecutive steps should M.Ohsaki/International Journal of Solids and Structures39(2002)927–941933
be less than 1:0?10à6.The computational cost will be reduced if larger limits are used in practical https://www.sodocs.net/doc/6c7466057.html,putation has been carried out on a personal computer with AMD Athron 1.0GHz.
The vector n of the imperfection parameters consists of the coordinates of all the nodes except two supports.Note that n includes minor and major imperfections.Therefore the size of n is equal to n .The
norm ~e
en Tof the imperfection is de?ned as ~e en T?1n
????????n T n q e16TThe upper bound D p for the error in the nodal loads is 1%of p .Note that the imperfection of loads is assumed to exist in all the displacement components.The number of variables n ,U and K in P2is 33.The extensional sti?ness of each spring attached at nodes 7and 8is denoted by j .We consider two cases with j ?0and j ?102:9kN/m which are referred to as column-type truss and laterally supported truss ,re-spectively.
Let U A and U S denote the lowest antisymmetric and symmetric linear buckling modes,respectively,of the perfect system.Imperfection sensitivity properties are ?rst investigated for imperfections in the direc-tions of U A and U S which correspond to major and minor imperfections,respectively.Imperfection modes may also be de?ned by the eigenmodes U r of S at the critical point.Since the bifurcation mode U 1is antisymmetric and the lowest symmetric mode of S does not have any physical meaning,and since the prebuckling deformation is not very large for a perfect column-type trusses,it is reasonable to de?ne the imperfection by the linear buckling modes.
The upper bounds 100and 300cm are given for the absolute values of the components of n and U ,respectively.Moderately large upper bounds should be given for the components of n and U so as
to
934M.Ohsaki /International Journal of Solids and Structures 39(2002)927–941
exclude deformation above possible local maxima of the bifurcation path in the process of optimization,even if those constraints are inactive at the optimal solution.
4.1.A column-type truss
Consider the column-type truss without springs;i.e.j ?0:The critical load factor of the perfect system is 3.9366,where the buckling mode is antisymmetric with respect to the y -axis and the critical point is a symmetric bifurcation point.
We ?rst investigate imperfection sensitivity of the maximum load factor in the directions of U A and U S ,respectively,which are as shown in Fig.5(a)and (b).Note that the imperfection mode D p of the nodal loads is also considered in the same directions as the nodal imperfections,where D p is scaled so that its maximum absolute value is equal to 1%of p .Fig.6shows the relation between the horizontal displacement d of node 8
and the load factor for three cases of perfect and imperfect systems in the direction of U A with ~e
en T?1:0and 5.0cm.It is observed from Fig.6that K slightly increases along the bifurcation path,and the critical point of the perfect system is a stable symmetric bifurcation point.Fig.7shows the relation between d and K for three cases of perfect and imperfect systems for minor imperfection corresponding to U S with ~e en T?1:0and 5.0cm.Suppose the maximum load factor K M is de?ned by the displacement constraint d 6 d
.It may be ob-served from Figs.6and 7that the reduction of K M due to a major imperfection is larger than that to a minor imperfection if d
is small,but a minor imperfection dominates if d is su?ciently large.For instance,if ~e
en T?5:0cm,reduction in the direction of U S is larger than that of U A in the range d >179cm.The important property observed in Fig.7is that the magnitude of reduction of K M does not strongly
depend M.Ohsaki /International Journal of Solids and Structures 39(2002)927–941935
on the value of d
.Therefore,the most critical mode of minor imperfection may be successfully obtained by solving P2considering only the bifurcation load factor.
The minimum value of K of P2for ~e en T6 e ?5:0cm is 2.6693which is about 68%of K c ?3:9366of the
perfect system.The most critical mode of nodal imperfection n M is symmetric as shown in Fig.8(a),where the initial value for n has been given as U S .The displacements and load factor at buckling of the perfect system have been assigned to the initial values of U and K ,respectively.Most critical imperfections of nodal locations and nodal loads are also listed in Table 1.It is observed from Table 1that all the components of D p are equal to the upper or lower bound.
The number of optimization steps,CPU time and the optimal objective value are as listed in the ?rst row
of Table 2.Note that K c of the imperfect system corresponding to ~e
en T?5:0cm in the direction of U S is 3.4747which is larger than that for n M .Therefore,U S cannot be used as an approximation for n M .Although the number of steps is considerably large,an almost optimal solution has been found within 30steps.The relation between d and K for the most critical case is also plotted in Fig.7.
Since the structure and loading conditions considered here have obvious symmetry properties,it is very easy to divide n and U into symmetric and antisymmetric components.If we consider only symmetric components of n and U ,the maximum load factor of the symmetric system is 2.6693which agrees within the accuracy of ?ve digits with the value obtained by the formulation including asymmetric imperfections and deformations.The mean absolute value of deviation of n M from those of the symmetric system
is
936M.Ohsaki /International Journal of Solids and Structures 39(2002)927–941
4:3335?10à3which is very small compared to the maximum absolute value 13.7810of n M .The compu-tational cost is reduced as shown in the second row of Table 2if we consider only symmetric imperfections and
deformations.
Table 1
Most critical imperfections of nodal locations and nodal loads
Node
Direction Location (cm)Load eD p i =p T1
x 13.7810.01y à0.22715à0.012
x à13.783à0.01y à0.22766à0.013
x 2.94100.01y 0.22275à0.014
x à2.9375à0.01y 0.22211à0.015
x 0.682350.01y 0.013910à0.016
x à0.68769à0.01y 0.011772à0.017
x à0.0496430.01y 0.91936à0.018x
0.049153à0.01y 0.91907à0.01
M.Ohsaki /International Journal of Solids and Structures 39(2002)927–941937
938M.Ohsaki/International Journal of Solids and Structures39(2002)927–941
Table2
Number of iteration steps,CPU time and the objective value for the column-type truss
Number of steps CPU time(s)Objective value
Symmetric initial solution3836.8 2.6693
Symmetric imperfection14 6.7 2.6693
Asymmetric initial solution2725.2 2.6693
Linear strain2926.5 2.6864
Incremental analysis6754.2 2.6708
If we do not exclude major imperfections and the initial values of all the components of nodal imper-fections and nodal displacements are equal to0.1which are not symmetric,the deviation from the sym-metric solution increases to3:4390?10à2,but the deviation is still very small.The number of steps and CPU time for this case are listed in the third row of Table2.Note that the computational cost is smaller than that from the symmetric initial solution.Therefore,symmetricity of initial solution does not always lead to reduction of computational cost,but usually leads to a rigorously symmetric solution.It should be noted that the optimal objective values are same up to?ve digits for several cases we tested from di?erent initial solutions.
The maximum load factor obtained by linearizing the equilibrium equation(11)with respect to U is 2.6864.Therefore,the prebuckling deformation may be neglected for the column-type truss as this example. The convergence property,however,does not improve as the result of neglecting the geometrical nonlin-earity as observed from the fourth row of Table2.In general cases including dome-type structures,pre-buckling deformation cannot be neglected,and the geometrically nonlinear formulation presented in this paper should be used.
Problem P1has been directly solved for comparison purpose,where path-following analysis is to be carried out to evaluate the bifurcation load factor at each step of optimization.Only symmetric imper-fections are considered.If asymmetric imperfection exists,the bifurcation point disappears and the opti-mization process obviously does not converge.Finite di?erence method has been used for sensitivity analysis,and the equilibrium path is traced by the displacement increment method.The computational results are listed in the last row of Table2which should be compared to the second row because only symmetric imperfections are considered here.It is observed from Table2that computational cost for P1is very large compared to that for https://www.sodocs.net/doc/6c7466057.html,putational cost,however,strongly depends on the methods of path-following analysis and optimization.The ratio of CPU time for P1to that of P2will be di?erent if analytical sensitivity analysis is used instead of?nite di?erence approach.However,the cost for evaluating the constraint functions for P2is very small because the equivalent nodal loads are obtained by an algebraic computation and k1is computed by carrying out eigenvalue analysis only once which is not costly com-pared to the path-following analysis.Since imperfections on p are also considered in P1,the number of variables for P1and P2for this case are32and33,respectively,which are almost same.Therefore,it can be concluded that the computational cost for P2is generally smaller than that for P1.
4.2.A laterally supported truss
Consider next a laterally supported truss with j?102:9kN/m.The ratio of the extensional sti?ness of the spring to that of the horizontal truss member is0.005.The buckling load factor of the perfect system is 15.497,where the buckling mode is antisymmetric with respect to y-axis.Therefore the critical point is a symmetric bifurcation point.
Fig.9shows the relation between d and K for three cases of perfect and imperfect systems corresponding to a major imperfection U A with~eenT?1:0and5.0cm.It is seen from Fig.9that the critical point of the perfect system is a stable bifurcation point,and the critical point of imperfect systems are limit points that
are far above the bifurcation point.Note that the reduction of K M is very small for ~e
en T?1:0cm in the moderately large range of displacement.Variation of k 1with respect to K for ~e
en T?1:0cm is as shown in Fig.10.The lowest eigenvalue has a local minimum near the bifurcation point,but increases to positive values before reaching 0at the limit point.
Fig.11shows the relation between d and K for three cases of perfect and imperfect systems for minor
imperfection in the direction of U S with ~e
en T?1:0and 5.0cm.The equilibrium paths are plotted up to the second and ?rst critical point,respectively,for perfect and imperfect systems.Note that the bifurcation load
factor for ~e en T?5:0cm is 14.107,where imperfection in nodal loads are also considered in the direction of U S .
The optimal value of K obtained by solving P2for ~e en T6 e ?5:0cm is 12.483which is about 81%of
K c ?15:497of the perfect system,where the initial value for n has been given as U S .The displacements and load factor at buckling of the perfect system have been assigned to the initial values of U and K ,respec-tively.The relation between d and K for n M which is symmetric as shown in Fig.8(b)is also plotted in Fig.
9.In this case the reduction of K c due to the most critical imperfection is much larger than that to the imperfection with the same norm in the direction of U S .Note from Fig.8(a)and (b)that n M strongly depends on the extensional sti?ness of the spring.
If we consider only symmetric components for n and U ,the maximum load factor of symmetric system is 12.483which agrees within the accuracy of ?ve digits with the value obtained by the formulation
including
M.Ohsaki /International Journal of Solids and Structures 39(2002)927–941939
asymmetric imperfections and deformations.The mean absolute value of deviation of n M from the most critical imperfection mode of the symmetric system is 2:0182?10à2which is very small compared to the maximum absolute value 13.7810of n M .If we start from a nonsymmetric initial values such that all the components of n and U are equal to 0.1,the deviation increases to 6:5855?10à2which is still a small value.The maximum load factor for linear case is 12.863.Also for this case,convergence property did not im-prove as a result of neglecting the e?ect of prebuckling deformation.
5.Conclusions
Asimple and computationally inexpensive approach has been presented for obtaining the maximum load factor of an elastic structure that has a stable bifurcation point if no initial imperfection exists.In the proposed method,an antioptimization problem is ?rst formulated for minimizing the load factor within the convex region of possible imperfections.The problem is then relaxed and reformulated by using the SAND as well as the energy based approach to obtain the most critical minor imperfection and the corresponding bifurcation load factor also considering the imperfection in nodal loads.The bifurcation load factor is located as a minimum value of the load factor with respect to the major imperfections under constraint on the lowest eigenvalue of the stability matrix.The variables of the problem are the imperfection parameters,nodal displacements and the load factor,and laborious nonlinear analysis for tracing equilibrium path is avoided.
The equilibrium paths of perfect and imperfect systems have been investigated for a 20-bar column-type truss with and without lateral springs considering major and minor imperfections of various magnitudes.It has been shown for a ?exible structure allowing moderately large displacements that the antisymmetric linear buckling mode cannot always be the most critical mode of imperfection and that a minor imper-fection is very important for estimating the reduction of the maximum load factor de?ned by the dis-placement constraints.It has also been shown that the reduction of the maximum load factor de?ned by the displacement constraints does not strongly depend on the value of the upper bound of displacement if a minor imperfection is considered.The possibility of reaching the bifurcation point that leads to sudden dynamic antisymmetric deformation should also be avoided in practical situation.Therefore it is reasonable to de?ne the most critical mode of imperfection in the direction of symmetric minor imperfection for a ?exible structure allowing moderately large deformation.
The most critical minor imperfection has been shown to be successfully obtained by solving the proposed antioptimization problem using an appropriate nonlinear programming algorithm.The antioptimal
solu-
940M.Ohsaki /International Journal of Solids and Structures 39(2002)927–941
M.Ohsaki/International Journal of Solids and Structures39(2002)927–941941 tions have been found under several problem settings,and it has been con?rmed that the proposed method has advantages over the method with path-following analysis in view of computational cost and conver-gence property.
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