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An improved method for evolutionary structural optimisation against buckling
Computers and Structures 79 (2001) 253±263
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An improved method for evolutionary structural optimisation against buckling J.H. Rong a, Y.M. Xie b,*, X.Y. Yang b b
a Aircraft Structure Strength Research Institute, AVIC, P.O. Box 86, Xi'an 710065, People's Republic of China Faculty of Engineering and Science, School of the Built Environment, Victoria University of Technology, P.O. Box 14428, Melbourne City MC, VIC. 8001, Australia
Received 7 July 1999; accepted 1 May 2000
Abstract In this paper, an improved method for evolutionary structural optimisation against buckling is proposed for maximising the critical buckling load of a structure of constant weight. First, based on the formulations of derivatives for eigenvalues, the sensitivity numbers of the ®rst eigenvalue or the ®rst multiple eigenvalues (for closely spaced and repeated eigenvalues) are derived by performing a variation operation. In order to eectively increase the buckling load factor, a set of optimum criteria for closely-spaced eigenvalues and repeated eigenvalues are established, based on the sensitivity numbers of the ®rst multiple eigenvalues. Several examples are provided to demonstrate the validity and eectiveness of the proposed method. Ó 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction Recently, a simple new approach called evolutionary structural optimisation (ESO) to structure optimisation has been proposed by Xie and Steven [1,2]. It is based on the concept of slowly removing inecient materials from the structure or gradually shifting materials from the strongest part of the structure to the weakest part until the structure evolves to a desired optimum. Compared to other structural optimisation methods, such as the homogenisation method [3,4] and density function method [5], the ESO method is attractive due to its simplicity and eectiveness. In recent years, the ESO method has been demonstrated to be capable of solving many problems of size, shape and topology optimum designs for static and dynamic problems [2,6,7]. At the same time, Refs. [8±10] applied the ESO method to optimum designs against buckling. Optimum design
*
Corresponding author. Tel.: +61-3-9688-4787; fax: +61-39688-4139. E-mail address: [email protected] (Y.M. Xie).
against buckling can be ®nding the minimum weight design of a structure that satis®es the prescribed buckling load constraint. Alternatively, it can be maximising the critical buckling load of the structure while keeping its weight or volume constant. For the convenience of comparing the eciency of dierent designs, the latter approach is generally used. Despite the extensive research on evolutionary structural optimisation against buckling, problems arise when there exist repeated eigenvalues or closely-spaced eigenvalues. First, the eigenvectors associated with the repeated eigenvalues are not unique. Instead, they can be any linear combination of the eigenvectors associated with the same repeated eigenvalue. Therefore, the eigenvalue sensitivities for the repeated eigenvalues cannot be determined uniquely using the formulas developed by Fox and Kapoor [11]. And even if the derivatives of the repeated eigenvalues corresponding to dierent modes are determined, these derivatives must be considered during the optimisation process in order to increase the buckling load factor. Second, when the ®rst eigenvalue becomes close to the subsequent eigenvalues, there will be serious interference between the ®rst and the subsequent modes. If only the ®rst eigenvalue is considered
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and the eect of the subsequent modes is ignored, the ®rst two or multiple buckling modes may swap with each other as a result of structural modi®cations during the iterations thus good results cannot obtained. In the case of closely spaced eigenvalues, Refs. [2,8] adopted a simple optimum criterion that makes use of an arithmetic mean of all closely-spaced eigenvalues to calculate the sensitivity numbers (here, this method is called the mean method). While the mean method can generally reach convergent optimum results, it restricts the range of optimum domains thus it cannot be ensured that the buckling load factor is increased at each iterative step. In this paper, an improved method for evolutionary structural optimisation against buckling is proposed for maximising the critical buckling load of a structure of constant weight. First, based on the formulations of derivatives for eigenvalues, the sensitivity numbers of the ®rst eigenvalue or the ®rst multiple eigenvalues in the case of closely spaced and repeated eigenvalues are derived by performing a variation operation. Then, in order to eectively raise the buckling load factor, a set of optimum criteria for closely-spaced eigenvalues and repeated eigenvalues are established, based on the sensitivity numbers of the ®rst multiple eigenvalues. Three examples are provided to demonstrate the validity and eectiveness of the proposed method.
2. The problem statement The linear buckling behaviour of a structure is governed by the following general eigenvalue problem: K ki Kg fui g f0g;
1
where K is the global stiness matrix; Kg , the global stress matrix or geometric stiness matrix; ki , the ith eigenvalue and fui g, the corresponding eigenvector. The eigenvalues in Eq. (1) are used to scale the applied loading to give buckling loads. The most critical buckling load is equal to the loading multiplied by the ®rst eigenvalue kl , and kl is referred to as the buckling load factor. The objective of buckling optimisation considered in this paper is to increase the fundamental eigenvalue kl so that the buckling load is maximised, by changing the cross-sectional areas of the structural elements.
3. The basic formulations of eigenderivatives If Eq. (1) has distinct eigenvalues, the eigenvalue sensitivity was derived as follows [10,11]:
oki ÿ od
fui gT
oKg oK fui g ki od od
fui gT Kg fui g
:
2
In the case of repeated eigenvalues, Eq. (2) in general does not produce a unique solution due to the non-unique nature of the eigenvectors associated with repeated eigenvalues. To overcome this problem, the following eigensystem is formed and solved [12]: oK oKg oki U T ki U 3 I fai g f0g; od od od where U consists of the original orthogonal eigenvectors that are associated with the same repeated eigenvalue ki , and I is the identity matrix with a dimension corresponding to the multiplicity (r) of the repeated eigenvalues. Eq. (3) is apparently an eigensystem with the repeated eigenvalue sensitivity oki =od being its eigenvalue and fai g the corresponding eigenvector. If the eigenvalues of this smaller eigensystem are distinct, the unique eigenvector fai g can be used to determine the unique eigenvectors that is associated with the repeated eigenvalues by the following linear combination of eigenvectors: n o i g U ai ; fu 4 i g stands for the unique mode shape that corwhere fu responds to the repeated eigenvalue ki . After determining the unique eigenvectors for the repeated eigenvalues, Eq. (2) can be used again to calculate the eigenvalue sensitivities for those repeated eigenvalues, although the solutions have already been found by solving the eigensystem of Eq. (3). 4. Sensitivity numbers and optimum criteria for bucking load The sensitivity analysis is used to identify the best locations for structural modi®cations. Suppose there is a small change in the cross-sectional area of the eth element and assume that the eigenvector fui g is approximately the same before and after such a small change [8±10], based on the basic formulations of eigenderivatives, the sensitivity numbers of the eigenvalues due to such a change can be derived by performing a variation operation. Generally speaking, there are four cases of eigenderivatives in buckling optimisation, namely, distinct eigenvalues, closely-spaced eigenvalues, repeated eigenvalues and combinations of closely spaced and repeated eigenvalues. According to these four cases, a set of criteria determining elements whose cross-sectional areas are increased and those whose cross-sectional areas are reduced will be established. The interference
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between the ®rst and the subsequent eigenvalues or between repeated eigenvalues or between all these eigenvalues including closely spaced and repeated eigenvalues are re¯ected in those criteria. We start with the simplest case of distinct eigenvalues and establish the sensitivity number calculation and optimum criterion. 4.1. Sensitivity numbers and optimum criterion for distinct and non-closely-spaced eigenvalues The equation for distinct eigenvalues derivative is as follows [2,8]: Dki ÿ
fui gT DK ki DKg fui g fui gT Kg fui g
:
5
The change in the global stiness matrix K is equal to the change in the element stiness matrix of the eth element, which can be easily calculated. However, since Kg depends on the current stress distribution in the structure and the cross-sectional change in the eth element aects the stress in its surrounding elements, one cannot assume that Kg is equal to the change in the element stress matrix of the eth element only. The calculation of Kg is generally much involved. Fortunately, Kg is equal to zero if the axial or membrane stress resultant remains constant before and after the crosssectional change in the elements. Such is the situation of all statically determinate structures as the cross-sectional changes do not aect the axial forces in structural members. For a statically indeterminate structure, Kg can be negligible if at each iteration step the crosssectional modi®cations are so small that they do not cause signi®cant changes in the axial or membrane stress resultants. When Kg is ignored, Eq. (5) is reduced to e T ui DK e uei fui gT DKfui g Dki ÿ ÿ ; 6 fui gT Kg fui g fui gT Kg fui g where uei is the eigenvector of the eth element and K e is the stiness matrix of the eth element. If the eigenvector is normalised with respect to Kg , Eq. (6) is further simpli®ed to T Dki ÿ uei DK e uei : 7 From the above equation, we de®ne the sensitivity number for buckling load as follows: T i 8 ae ÿ uei DK e uei : The calculation of the above sensitivity number only involves small matrices of individual elements. This sensitivity number is used to measure the eect of changing the cross-sectional area of the eth element on the buckling load factor. If the structure is statically determinate and the cross-sectional change at each iteration step is small, the sensitivity number gives a very
255
accurate estimation of the change in the buckling load factor. Even for a statically indeterminate frame this sensitivity is reasonably accurate if the cross-sectional variation in the frame only results in slight changes in the axial stress resultants. For a plate structure, this sensitivity number only works in the case of gradually varying the thickness of plate elements. If an element is removed at once which may cause signi®cant changes in the membrane or axial stress resultants in its surrounding elements, Eqs. (6) and (7) will be invalid and the sensitivity number de®ned in Eq. (8) will be incorrect too. Therefore, Eq. (8) should not be used for buckling optimisation involving element removal. In this paper, the buckling optimisation is carried out by changing the cross-sectional area only. In the case of an increase in the cross-sectional area A, we have DK e DK e K e A DA ÿ K e A;
9
and in the case of a reduction DK e DK e ÿ K e A ÿ DA ÿ K e A:
10
Hence, to estimate the eect of cross-sectional changes on the buckling load factor, two sensitivity numbers need to be calculated for each element, one for area increase T 1 11 ae ÿ ue1 DK e ue1 ; and the other for area reduction T 1 ÿ ae ÿ ue1 DK e ÿ ue1 :
12
From the above de®nition of the sensitivity number, it is clear that to raise the buckling load factor it will be most eective to increase the cross-sectional areas of elements with the highest 1 a e values and reduce those with the highest 1 aÿ e values. 4.2. Sensitivity numbers and optimum criterion for distinct and closely-spaced eigenvalues 4.2.1. Two closely-spaced eigenvalue case During the process of buckling optimisation, it is often observed that while the ®rst eigenvalue is increasing, the subsequent eigenvalues are decreasing and gradually the ®rst two or more eigenvalues become very close to each other. This will cause serious interference between the ®rst and the subsequent buckling modes. Therefore the eect of all the participating eigenvectors needed to be included. If, for example, the distance between k1 and k2 is within a certain limit, say 5%, and the distance between k1 and k3 is greater than that limit, we assume that the structure has become bi-modal. For a bi-modal structure, the ®rst two buckling modes may swap with each other as a result of structural
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modi®cation during the iterations if only the sensitivity numbers of the ®rst eigenvalue are considered. There is no point in trying to increase k1 only to see k2 drop its value below previous k1 in the next step. In Ref. [2], this case is dealt with by a simple strategy called Ômean methodÕ, i.e. to increase the average of k1 and k2 , instead of increasing a single eigenvalue k1 only. However, it is not ensured in the mean method that k2 is kept above the previous k1 in the next step thus restricting the range of optimum domains. Therefore, it is necessary to comprehensively consider the sensitivity numbers of the ®rst two eigenvalues to determine the element modi®cation. First, in order to clearly describe the idea of the proposed method and to unify the formula for various cases of two and multiple closely spaced eigenvalues, some mathematics notations are de®ned as follows:
4.2.2. Multiple closely-spaced eigenvalue case Similar to the bi-modal case, a structure become multi-modal when the distance between the rth eigenvalue kr and the ®rst eigenvalue k1 is within a certain limit, say 5%, and the distance between k r1 and k1 is greater than that limit. In such a case, N2 , N2ÿ , N3 , N3ÿ ,. . .,Nr , and Nrÿ are constructed in the following form for the second, the third,. . ., the rth eigenvalue, respectively. Nj ej j a e2S e > ÿ ej Njÿ ej j aÿ e2S j 2; 3; . . . ; r: 18 e > ÿ ej
S f1; 2; . . . ; mg; N1 S; N1ÿ S;
Then it follows:
13
e 2 Sg; N2 ej 2 a e > ÿe 2 ÿ ÿ N2 ej ae > ÿ e e 2 S ;
k2 ÿ k1 ; N
14
15
where N is the total number of elements in which elemental sectional-areas are changed at current step. It is set up in advance based on the requirement of keeping structural weight constant. Then de®ne N N1 \ N2
N ÿ N1ÿ \ N2ÿ ;
16
where \ represents the intersect set operator of two sets. After determining N and N ÿ from Eqs. (11)±(16), we check up the sum NR of the numbers of elements in N and N ÿ . If NR < N , then de®ne ek bekÿ1
k 1; 2; 3; . . .;
e0j
kj ÿ k1 N
j 2; 3; . . . ; r:
N N1 \ N2 \ \ Nr ; N ÿ N1ÿ \ N2ÿ \ \ Nrÿ :
where m is the number of structural ®nite elements to be 2 ÿ modi®ed, and 2 a e and ae are the sensitivity numbers of the second eigenvalue of eth element for area increasing and decreasing, respectively. N2 and N2ÿ are the sets of the structural elements that are de®ned by 2 a e > ÿe, and 2 ÿ ae > ÿe, respectively. e is a small positive real number which can be changed in each optimum step. An initial value of e at current step is de®ned as e0
Similarly, an initial value of ej is given as follows:
17
where b > 1. The above procedures using Eqs. (13)±(17) will be repeated until NR is equal to or greater than N. Now, it is clear that to increase k1 and keep k2 above k1 in the next step, namely to raise the buckling load factor, it will be most eective to increase the crosssectional areas of elements that belong to N and are of the highest 1 a e values, and reduce the cross-sectional areas of elements that belong to N ÿ and are of the highest 1 aÿ e values.
19
20
After determining N and N ÿ , from Eqs. (18)±(20), the sum of the numbers of elements in N and N ÿ are checked up and similar equation to Eq. (17) can be de®ned where applicable: ekj bekÿ1 j
k 1; 2; 3; . . .:
21
After N and N ÿ are determined, it is clear that to increase k1 and keep k2 ; k3 ; . . . ; kr above the previous k1 , it will be most eective to increase the cross-sectional areas of elements that belong to N and are of the highest 1 a e values, and reduce the cross-sectional areas of elements that belong to N ÿ and are of the highest 1 aÿ e values. 4.3. Sensitivity numbers and optimum criterion for repeated eigenvalues It is de®ned that U fu1 g; fu2 g; . . . ; fur g: When the cross-sectional area of an element is changed in the current structure, assuming that DKg is ignored, the following equation can be derived from Eq. (3): oki A 22 I fai g f0g od in which 2 k11 6 k21 6 A 6 .. 4 . kr1
k12 k22 .. . kr2
... ... .. . ...
3 k1r k2r 7 7 .. 7; . 5 krr
23
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n o T i 1; 2; . . . ; r; j 1; 2; . . . ; r; kij uei DK e uej 24 where I is a r r identity matrix. To solve Eq. (22), the matrix fai g i 1; 2; . . . ; r can be obtained, then the ei can be calculated from unique element eigenvector u the following equation: n o ei ue1 ; ue2 ; . . . ; uer fai g i 1; 2; . . . ; r: u 25 ei to Eqs. (11) and (12), the sensitivity Substituting u 1 ÿ 2 2 ÿ r ÿ numbers (1 a ; ae ; ae ; ae ; . . . ; r a e e and ae ) for the repeated eigenvalues can be obtained easily.
De®ne 0
N1 S;
0
N1ÿ S;
e2S ; Nj ej j a e P0 0 ÿ e2S Nj ej j aÿ e P0
26
0
0
N 0 N1 \ 0 N2 \ \ 0 Nr ;
0
N ÿ 0 N1ÿ \ 0 N2ÿ \ \ 0 Nrÿ ;
0
0
0
ÿ
N N [ N ;
e2S ; Nj ej j a e > ÿe e ÿ e2S Nj ej j aÿ e > ÿe
j 2; 3; . . . ; r;
27 28 29
e
e
N 0 N1 \ e N2 \ \ e Nr ;
e
N ÿ 0 N1ÿ \ e N2ÿ \ \ e Nrÿ ;
e
N eN [ eN ÿ;
j 2; 3; . . . ; r; 30 31 32
where [ represents the sum set operator of two sets. In order to have the ®rst eigenvalue increased, the sectional areas of elements whose values of 2 a e ; 3 2 ÿ 3 ÿ r ÿ ae ; . . . ; and r a ; or a ; a ; . . . ; and a all are e e e e greater than zero, and are of the highest 1 a e values or the highest 1 aÿ e values, should be changed. In the general case, it cannot be ensured that 0 N and 0 N ÿ all are not an empty set. When one of 0 N and 0 N ÿ is an empty set, we can have e N and e N ÿ to be a non-empty set by setting up a small positive parameter e (referring to Section 4.2), then change the sectional areas of elements belonging to 0 N or e N , and being of the highest 1 a e e 0 values, or the highest 1 aÿ e values. When N instead of N is used in the optimisation process, the reduction in the ®rst eigenvalue can be the smallest even if an increase cannot be ensured. Moreover, for many engineering structural optimisations, the occurrence of repeated eigenvalues is much fewer compared to the total iterations
257
and thus a temporary eigenvalue decrease at one or two steps will not aect the solution process as a whole. 4.4. Optimum criterion for the combination of closely spaced and repeated eigenvalues When the repeated eigenvalues are not the ®rst eigenvalue but are very close to the ®rst eigenvalue, we can refer to Sections 4.2 and 4.3 and their sensitivity numbers are calculated by using Eqs. (11), (12) and (22)±(25). For the determination of modi®ed elements, each repeated eigenvalue is respectively treated as a closely spaced eigenvalue in relation to the ®rst eigenvalue. In doing so, it may happen that one or two eigenvalues become repeated with respect to the ®rst eigenvalue. This case is solved by combining the optimum criteria presented in Sections 4.2 and 4.3. N~ and N~ ÿ can be de®ned and obtained in such a combination. They serve as the same purpose as N and N ÿ in the case of closely spaced eigenvalues, and 0 N and 0 N ÿ (or e N and e N ÿ ) in the case of repeated eigenvalues. 5. Evolutionary procedure for buckling optimisation Similar to Ref. [2], the evolution procedure for buckling optimisation of columns, frames and plates is outlined as follows: Step 1: Discretise the structure using a ®ne ®niteelement mesh. Step 2: Solve the eigenvalue problem (1). Step 3: Calculate the sensitivity numbers according to each case: 1 ÿ Case 1: distinct eigenvalues: calculate 1 a e and ae for each element. Case 2: distinct and closely-spaced eigenvalues: i ÿ calculate i a ae i 1; 2; e i 1; 2; . . . ; r and . . . ; r for each element. Case 3: repeated eigenvalues: calculate i a e i 1; 2; . . . ; r and i aÿ e i 1; 2; . . . ; r for each element. Case 4: combination of closely-spaced and repeated eigenvalues: calculate the sensitivity number of closely spaced and repeated eigenvalues for each element at the same time. Step 4: According to each case, determine the elements whose cross-sectional areas are to be increased, and those whose cross-sectional areas are to be reduced. The requirements on structural symmetry and loading conditions are considered in the element modi®cation. Case 1: Increase the cross-sectional areas of the elements that have the highest 1 a e values, and decrease the cross-sectional areas of the same number of elements that have the highest 1 aÿ e values.
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Case 2: Increase the cross-sectional areas of the elements that belong to N (Eqs. (16) and (20)) and have the highest 1 a e values, and decrease the cross-sectional areas of the same number of elements that belong to N ÿ and have the highest 1 ÿ ae values. Case 3: Increase the cross-sectional areas of the elements that belong to 0 N or e N (Eq. (28) or Eq. (31)) and have the highest 1 a e values, and decrease the cross-sectional areas of the same number of elements that belong to 0 N ÿ or e N ÿ and have the highest 1 aÿ e values. Case 4: Increase the cross-sectional areas of the elements that belong to N~ and have the highest 1 a e values, and decrease the cross-sectional areas of the same number of elements that belong to N~ ÿ and have the highest 1 aÿ e values. Step 5: Repeat steps 2 to 4 until the increase in the ®rst eigenvalue becomes very slight for a consecutive, say four or ®ve iterations. During the evolution, the cross-sectional area is allowed to vary in small steps. For beam elements of rectangular cross-sections, either the breadth or the depth can be changed. For plate elements, the thickness can be changed. The change in the element stiness matrix DK e can be easily calculated for the above ®niteelement types. Meanwhile, information on eigenvalues and eigenvectors required for the sensitivity number calculation is readily available from the ®nite element analysis. In the above procedures, the number of elements subjected to cross-sectional changes and the step size of the change at each iteration need to be prescribed.
6. Examples In order to demonstrate the validity and eectiveness of the proposed method, two simple frames and a box structure displaying closely spaced or repeated eigenvalues during buckling optimisation are considered. In these three examples, the initial design is of uniform cross-section, and the YoungÕs modulus E 200 GPa, PoissonÕs ratio m 0:3 and mass density q 2700 kg/m3 are assumed.
Fig. 1. Structural model and loading case of the three-member portal frame.
vary to the maximum 40 mm2 and to the minimum 5 mm2 in steps of 1 mm2 . Each member is divided into 10 elements of equal length. A modifying ratio 24% and closely-spaced eigenvalue parameter ec 4% are used. For the frame with a uniform cross-section, the ®rst buckling mode is anti-symmetric with sway and the second buckling mode is symmetric with closely-spaced eigenvalues from the outset. The optimum radius square r2 for each element is shown in the column chart Fig. 2. The buckling load is 1.2514 times that of the uniform frame, in comparison to 1.125 times by only considering the single mode, and 1.2474 by using the mean method [8]. The evolutionary histories of the ®rst two eigenvalues using the mean method and the proposed method are given in Figs. 3 and 4, respectively. Fig. 5 compares the evolutionary histories of the ®rst eigenvalue using these two methods. It is seen that although the optimum factors obtained by the two methods dier only by 0.32%, fewer iterations are involved in the proposed method. The above problem is analysed with dierent values of closely-spaced model parameter ec 2%, 4% and 4.5%, as well as dierent modifying ratios c 10%, 15% and 24%. While the iteration histories of eigenvalues vary slightly in intermediate designs, no dierence is observed in the ®nal design.
6.1. Three-member portal frame
6.2. Three-member space frame example
A three-member pin based frame, which was analysed in Refs. [2,13] is considered for closely-spaced buckling eigenvalues. The frame structural model and the loading are shown in Fig. 1. All the members are of circular cross-sections and of equal length of 1 m. Initial uniform radius square r2 is 20 mm2 , and it is allowed to
A space frame with three beams pinned at the base and clamped at the apex is considered for the optimisation of structures with repeated eigenvalues against buckling. The frame model and the loading condition are shown in Fig 6. All the members are of circular cross-sections and of equal length of 1 m. Initial uni-
J.H. Rong et al. / Computers and Structures 79 (2001) 253±263
259
Fig. 2. Ratio of radius squares of beam cross-sections at optimum point to corresponding initial uniform values.
Fig. 3. Optimisation histories of the ®rst two eigenvalues for the three-member frame by using the mean method.
Fig. 4. Optimisation histories of the ®rst two eigenvalues for the three-member frame by using the proposed method.
form r2 is 20 mm2 , and it is allowed to vary to the maximum 40 mm2 and to the minimum 5 mm2 in steps of 1 mm2 . Each member is divided into 10 elements of
equal length. c 24% and ec 4% are used. This is a triple symmetric structure and the ®rst three eigenvalues coincide for the uniform design and remain coincided
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J.H. Rong et al. / Computers and Structures 79 (2001) 253±263
Fig. 5. Optimisation histories of the ®rst eigenvalue for the three-member frame by using the proposed method and the mean method.
6.3. Box frame
Fig. 6. Optimum model and loading case of the three-member space frame.
throughout the optimisation process. The optimum beam sectional parameters obtained by using the proposed method are given in Fig. 7. The ratios of the ®nal to initial uniform radius of the cross-section are displayed in this ®gure for one member, as it is identical for all members. The optimum buckling load is 1.275 times that of the uniform frame and it is achieved after 10 iterations. The iteration histories of the ®rst three eigenvalues (namely three repeated eigenvalues) are given in Fig. 8.
In the box frame shown in Fig. 9, all the members are of rectangular cross-sections with a constant breadth b 40 mm and an initial uniform depth d 40 mm. The horizontal members at the top and bottom are divided into 12 elements of equal length, and diagonal and vertical members in the middle are divided into three elements of equal length. Numbering of beam elements is shown in the model. Buckling optimisation of this frame was considered to be one of the most dicult examples in the literature. It was studied in Ref. [10] using the mean method of ESO. In applying the method proposed in this paper, the initial value of the design variable, beam depth d is allowed to change without upper limit and to the minimum depth of 1 mm in steps of 1 mm. c 24% and ec 4% are used. The optimum depth ratio is shown Fig. 10. The evolutionary histories of the ®rst ®ve eigenvalues using the proposed method and the mean method are shown in Figs. 11 and 12, respectively. It is observed that while all cases of closely spaced and repeated eigenvalues occur during optimisation process, the ®rst eigenvalue is kept increasing at non-repeated eigenvalue points by using the proposed method. It also takes fewer iterations than the mean method. Fig. 13 shows the results of the ®rst eigenvalues in these two methods. The buckling load factor is 1.9214, compared to 1.8678 with a dierence of 2.9%. It is seen that members of the optimum design have segmented cross sections which are not manufacturally appealing. One solution to this can be some smoothing techniques using interpolation functions so that the structural member displays a smoother outer shape. This point for ESO method is presented in detail in Ref. [2].
J.H. Rong et al. / Computers and Structures 79 (2001) 253±263
Fig. 7. Optimum result of section radius squares of beam elements for the three-member space frame.
Fig. 8. Optimisation histories of the ®rst eigenvalue for the three-member space frame by using the proposed method.
Fig. 9. Finite-element model and loading case of the box frame (allowable minimum depth 1 mm).
261
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J.H. Rong et al. / Computers and Structures 79 (2001) 253±263
Fig. 10. Optimum result of beam section depths for the box frame.
Fig. 11. Optimisation method histories of the ®rst ®ve eigenvalues for the box frame by using the mean method.
Fig. 12. Optimisation histories of the ®rst ®ve eigenvalues for the box frame by using the proposed method.
J.H. Rong et al. / Computers and Structures 79 (2001) 253±263
263
Fig. 13. Optimisation histories of the ®rst eigenvalue for the box frame by using the mean method and the proposed method.
7. Conclusion An improved approach to optimising the structures against buckling is proposed and illustrated with examples. The results demonstrate that the proposed method is valid and eective and is suitable for various complex cases of practical structures. The method can obtain better optimum design for structures against buckling than the mean method. It can be readily implemented in any of the existing ®nite-element codes. References [1] Xie YM, Steven GP. A simple evolutionary procedure for structural optimisation. Comput Struct 1993;49:885±96. [2] Xie YM, Steven GP. Evolutionary structural optimisation. Berlin, Germany: Springer; 1997. [3] Bendsùe MP, Kikuch N. Generating optimal topologies in structural design using a homogenisation method. Comput Meth Appl Mech Engng 1998;71:197±224. [4] Ma ZD, Kikuchi N, Cheng HC. Topology design for vibrating structures. Comput Meth Appl Mech Engng 1995;121:259±80.
[5] Yang RJ, Chuang CH. Optimal topology design using linear programming. Comput Struct 1994;52:265±75. [6] Xie YM, Steven GP. Evolutionary structural optimisation for dynamic problems. Comput Struct 1996;58:1067±73. [7] Querin OM. Evolutionary structural optimisation: stress based formulation and implementation. PhD Thesis, University of Sydney, Sydney, Australia, 1997. [8] Manickarajah D, Xie YM, Steven GP. A simple method for the optimisation of columns, frames and plates against bucking. In: Structural stability and design. Balkema, Rotterdam, 1995. p. 175±80. [9] Manickarajah D, Xie YM, Steven GP. An evolutionary method for optimisation of plate buckling resistance. Finite Element Anal Design 1998;29:205±30. [10] Manickarajah D. Optimum design of structures with stability constraints using the evolutionary optimisation method. PhD Thesis, Victoria University of Technology, Melbourne, Australia, 1998. [11] Fox RL, Kapoor MD. Rates of change of eigenvalues and eigenvectors. AIAA J 1968;6:2426±9. [12] Chen TY. Design sensitivity analysis for repeated eigenvalues in structural design. AIAA J 1993;31:2347±50. [13] Szyszkowski W, Watson LG. Optimisation of the buckling load of columns and frames. Engng Struct 1988;10: 249±56.
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An improved method for evolutionary structural optimisation against buckling J.H. Rong a, Y.M. Xie b,*, X.Y. Yang b b
a Aircraft Structure Strength Research Institute, AVIC, P.O. Box 86, Xi'an 710065, People's Republic of China Faculty of Engineering and Science, School of the Built Environment, Victoria University of Technology, P.O. Box 14428, Melbourne City MC, VIC. 8001, Australia
Received 7 July 1999; accepted 1 May 2000
Abstract In this paper, an improved method for evolutionary structural optimisation against buckling is proposed for maximising the critical buckling load of a structure of constant weight. First, based on the formulations of derivatives for eigenvalues, the sensitivity numbers of the ®rst eigenvalue or the ®rst multiple eigenvalues (for closely spaced and repeated eigenvalues) are derived by performing a variation operation. In order to eectively increase the buckling load factor, a set of optimum criteria for closely-spaced eigenvalues and repeated eigenvalues are established, based on the sensitivity numbers of the ®rst multiple eigenvalues. Several examples are provided to demonstrate the validity and eectiveness of the proposed method. Ó 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction Recently, a simple new approach called evolutionary structural optimisation (ESO) to structure optimisation has been proposed by Xie and Steven [1,2]. It is based on the concept of slowly removing inecient materials from the structure or gradually shifting materials from the strongest part of the structure to the weakest part until the structure evolves to a desired optimum. Compared to other structural optimisation methods, such as the homogenisation method [3,4] and density function method [5], the ESO method is attractive due to its simplicity and eectiveness. In recent years, the ESO method has been demonstrated to be capable of solving many problems of size, shape and topology optimum designs for static and dynamic problems [2,6,7]. At the same time, Refs. [8±10] applied the ESO method to optimum designs against buckling. Optimum design
*
Corresponding author. Tel.: +61-3-9688-4787; fax: +61-39688-4139. E-mail address: [email protected] (Y.M. Xie).
against buckling can be ®nding the minimum weight design of a structure that satis®es the prescribed buckling load constraint. Alternatively, it can be maximising the critical buckling load of the structure while keeping its weight or volume constant. For the convenience of comparing the eciency of dierent designs, the latter approach is generally used. Despite the extensive research on evolutionary structural optimisation against buckling, problems arise when there exist repeated eigenvalues or closely-spaced eigenvalues. First, the eigenvectors associated with the repeated eigenvalues are not unique. Instead, they can be any linear combination of the eigenvectors associated with the same repeated eigenvalue. Therefore, the eigenvalue sensitivities for the repeated eigenvalues cannot be determined uniquely using the formulas developed by Fox and Kapoor [11]. And even if the derivatives of the repeated eigenvalues corresponding to dierent modes are determined, these derivatives must be considered during the optimisation process in order to increase the buckling load factor. Second, when the ®rst eigenvalue becomes close to the subsequent eigenvalues, there will be serious interference between the ®rst and the subsequent modes. If only the ®rst eigenvalue is considered
0045-7949/01/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 0 0 ) 0 0 1 4 5 - 0
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and the eect of the subsequent modes is ignored, the ®rst two or multiple buckling modes may swap with each other as a result of structural modi®cations during the iterations thus good results cannot obtained. In the case of closely spaced eigenvalues, Refs. [2,8] adopted a simple optimum criterion that makes use of an arithmetic mean of all closely-spaced eigenvalues to calculate the sensitivity numbers (here, this method is called the mean method). While the mean method can generally reach convergent optimum results, it restricts the range of optimum domains thus it cannot be ensured that the buckling load factor is increased at each iterative step. In this paper, an improved method for evolutionary structural optimisation against buckling is proposed for maximising the critical buckling load of a structure of constant weight. First, based on the formulations of derivatives for eigenvalues, the sensitivity numbers of the ®rst eigenvalue or the ®rst multiple eigenvalues in the case of closely spaced and repeated eigenvalues are derived by performing a variation operation. Then, in order to eectively raise the buckling load factor, a set of optimum criteria for closely-spaced eigenvalues and repeated eigenvalues are established, based on the sensitivity numbers of the ®rst multiple eigenvalues. Three examples are provided to demonstrate the validity and eectiveness of the proposed method.
2. The problem statement The linear buckling behaviour of a structure is governed by the following general eigenvalue problem: K ki Kg fui g f0g;
1
where K is the global stiness matrix; Kg , the global stress matrix or geometric stiness matrix; ki , the ith eigenvalue and fui g, the corresponding eigenvector. The eigenvalues in Eq. (1) are used to scale the applied loading to give buckling loads. The most critical buckling load is equal to the loading multiplied by the ®rst eigenvalue kl , and kl is referred to as the buckling load factor. The objective of buckling optimisation considered in this paper is to increase the fundamental eigenvalue kl so that the buckling load is maximised, by changing the cross-sectional areas of the structural elements.
3. The basic formulations of eigenderivatives If Eq. (1) has distinct eigenvalues, the eigenvalue sensitivity was derived as follows [10,11]:
oki ÿ od
fui gT
oKg oK fui g ki od od
fui gT Kg fui g
:
2
In the case of repeated eigenvalues, Eq. (2) in general does not produce a unique solution due to the non-unique nature of the eigenvectors associated with repeated eigenvalues. To overcome this problem, the following eigensystem is formed and solved [12]: oK oKg oki U T ki U 3 I fai g f0g; od od od where U consists of the original orthogonal eigenvectors that are associated with the same repeated eigenvalue ki , and I is the identity matrix with a dimension corresponding to the multiplicity (r) of the repeated eigenvalues. Eq. (3) is apparently an eigensystem with the repeated eigenvalue sensitivity oki =od being its eigenvalue and fai g the corresponding eigenvector. If the eigenvalues of this smaller eigensystem are distinct, the unique eigenvector fai g can be used to determine the unique eigenvectors that is associated with the repeated eigenvalues by the following linear combination of eigenvectors: n o i g U ai ; fu 4 i g stands for the unique mode shape that corwhere fu responds to the repeated eigenvalue ki . After determining the unique eigenvectors for the repeated eigenvalues, Eq. (2) can be used again to calculate the eigenvalue sensitivities for those repeated eigenvalues, although the solutions have already been found by solving the eigensystem of Eq. (3). 4. Sensitivity numbers and optimum criteria for bucking load The sensitivity analysis is used to identify the best locations for structural modi®cations. Suppose there is a small change in the cross-sectional area of the eth element and assume that the eigenvector fui g is approximately the same before and after such a small change [8±10], based on the basic formulations of eigenderivatives, the sensitivity numbers of the eigenvalues due to such a change can be derived by performing a variation operation. Generally speaking, there are four cases of eigenderivatives in buckling optimisation, namely, distinct eigenvalues, closely-spaced eigenvalues, repeated eigenvalues and combinations of closely spaced and repeated eigenvalues. According to these four cases, a set of criteria determining elements whose cross-sectional areas are increased and those whose cross-sectional areas are reduced will be established. The interference
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between the ®rst and the subsequent eigenvalues or between repeated eigenvalues or between all these eigenvalues including closely spaced and repeated eigenvalues are re¯ected in those criteria. We start with the simplest case of distinct eigenvalues and establish the sensitivity number calculation and optimum criterion. 4.1. Sensitivity numbers and optimum criterion for distinct and non-closely-spaced eigenvalues The equation for distinct eigenvalues derivative is as follows [2,8]: Dki ÿ
fui gT DK ki DKg fui g fui gT Kg fui g
:
5
The change in the global stiness matrix K is equal to the change in the element stiness matrix of the eth element, which can be easily calculated. However, since Kg depends on the current stress distribution in the structure and the cross-sectional change in the eth element aects the stress in its surrounding elements, one cannot assume that Kg is equal to the change in the element stress matrix of the eth element only. The calculation of Kg is generally much involved. Fortunately, Kg is equal to zero if the axial or membrane stress resultant remains constant before and after the crosssectional change in the elements. Such is the situation of all statically determinate structures as the cross-sectional changes do not aect the axial forces in structural members. For a statically indeterminate structure, Kg can be negligible if at each iteration step the crosssectional modi®cations are so small that they do not cause signi®cant changes in the axial or membrane stress resultants. When Kg is ignored, Eq. (5) is reduced to e T ui DK e uei fui gT DKfui g Dki ÿ ÿ ; 6 fui gT Kg fui g fui gT Kg fui g where uei is the eigenvector of the eth element and K e is the stiness matrix of the eth element. If the eigenvector is normalised with respect to Kg , Eq. (6) is further simpli®ed to T Dki ÿ uei DK e uei : 7 From the above equation, we de®ne the sensitivity number for buckling load as follows: T i 8 ae ÿ uei DK e uei : The calculation of the above sensitivity number only involves small matrices of individual elements. This sensitivity number is used to measure the eect of changing the cross-sectional area of the eth element on the buckling load factor. If the structure is statically determinate and the cross-sectional change at each iteration step is small, the sensitivity number gives a very
255
accurate estimation of the change in the buckling load factor. Even for a statically indeterminate frame this sensitivity is reasonably accurate if the cross-sectional variation in the frame only results in slight changes in the axial stress resultants. For a plate structure, this sensitivity number only works in the case of gradually varying the thickness of plate elements. If an element is removed at once which may cause signi®cant changes in the membrane or axial stress resultants in its surrounding elements, Eqs. (6) and (7) will be invalid and the sensitivity number de®ned in Eq. (8) will be incorrect too. Therefore, Eq. (8) should not be used for buckling optimisation involving element removal. In this paper, the buckling optimisation is carried out by changing the cross-sectional area only. In the case of an increase in the cross-sectional area A, we have DK e DK e K e A DA ÿ K e A;
9
and in the case of a reduction DK e DK e ÿ K e A ÿ DA ÿ K e A:
10
Hence, to estimate the eect of cross-sectional changes on the buckling load factor, two sensitivity numbers need to be calculated for each element, one for area increase T 1 11 ae ÿ ue1 DK e ue1 ; and the other for area reduction T 1 ÿ ae ÿ ue1 DK e ÿ ue1 :
12
From the above de®nition of the sensitivity number, it is clear that to raise the buckling load factor it will be most eective to increase the cross-sectional areas of elements with the highest 1 a e values and reduce those with the highest 1 aÿ e values. 4.2. Sensitivity numbers and optimum criterion for distinct and closely-spaced eigenvalues 4.2.1. Two closely-spaced eigenvalue case During the process of buckling optimisation, it is often observed that while the ®rst eigenvalue is increasing, the subsequent eigenvalues are decreasing and gradually the ®rst two or more eigenvalues become very close to each other. This will cause serious interference between the ®rst and the subsequent buckling modes. Therefore the eect of all the participating eigenvectors needed to be included. If, for example, the distance between k1 and k2 is within a certain limit, say 5%, and the distance between k1 and k3 is greater than that limit, we assume that the structure has become bi-modal. For a bi-modal structure, the ®rst two buckling modes may swap with each other as a result of structural
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modi®cation during the iterations if only the sensitivity numbers of the ®rst eigenvalue are considered. There is no point in trying to increase k1 only to see k2 drop its value below previous k1 in the next step. In Ref. [2], this case is dealt with by a simple strategy called Ômean methodÕ, i.e. to increase the average of k1 and k2 , instead of increasing a single eigenvalue k1 only. However, it is not ensured in the mean method that k2 is kept above the previous k1 in the next step thus restricting the range of optimum domains. Therefore, it is necessary to comprehensively consider the sensitivity numbers of the ®rst two eigenvalues to determine the element modi®cation. First, in order to clearly describe the idea of the proposed method and to unify the formula for various cases of two and multiple closely spaced eigenvalues, some mathematics notations are de®ned as follows:
4.2.2. Multiple closely-spaced eigenvalue case Similar to the bi-modal case, a structure become multi-modal when the distance between the rth eigenvalue kr and the ®rst eigenvalue k1 is within a certain limit, say 5%, and the distance between k r1 and k1 is greater than that limit. In such a case, N2 , N2ÿ , N3 , N3ÿ ,. . .,Nr , and Nrÿ are constructed in the following form for the second, the third,. . ., the rth eigenvalue, respectively. Nj ej j a e2S e > ÿ ej Njÿ ej j aÿ e2S j 2; 3; . . . ; r: 18 e > ÿ ej
S f1; 2; . . . ; mg; N1 S; N1ÿ S;
Then it follows:
13
e 2 Sg; N2 ej 2 a e > ÿe 2 ÿ ÿ N2 ej ae > ÿ e e 2 S ;
k2 ÿ k1 ; N
14
15
where N is the total number of elements in which elemental sectional-areas are changed at current step. It is set up in advance based on the requirement of keeping structural weight constant. Then de®ne N N1 \ N2
N ÿ N1ÿ \ N2ÿ ;
16
where \ represents the intersect set operator of two sets. After determining N and N ÿ from Eqs. (11)±(16), we check up the sum NR of the numbers of elements in N and N ÿ . If NR < N , then de®ne ek bekÿ1
k 1; 2; 3; . . .;
e0j
kj ÿ k1 N
j 2; 3; . . . ; r:
N N1 \ N2 \ \ Nr ; N ÿ N1ÿ \ N2ÿ \ \ Nrÿ :
where m is the number of structural ®nite elements to be 2 ÿ modi®ed, and 2 a e and ae are the sensitivity numbers of the second eigenvalue of eth element for area increasing and decreasing, respectively. N2 and N2ÿ are the sets of the structural elements that are de®ned by 2 a e > ÿe, and 2 ÿ ae > ÿe, respectively. e is a small positive real number which can be changed in each optimum step. An initial value of e at current step is de®ned as e0
Similarly, an initial value of ej is given as follows:
17
where b > 1. The above procedures using Eqs. (13)±(17) will be repeated until NR is equal to or greater than N. Now, it is clear that to increase k1 and keep k2 above k1 in the next step, namely to raise the buckling load factor, it will be most eective to increase the crosssectional areas of elements that belong to N and are of the highest 1 a e values, and reduce the cross-sectional areas of elements that belong to N ÿ and are of the highest 1 aÿ e values.
19
20
After determining N and N ÿ , from Eqs. (18)±(20), the sum of the numbers of elements in N and N ÿ are checked up and similar equation to Eq. (17) can be de®ned where applicable: ekj bekÿ1 j
k 1; 2; 3; . . .:
21
After N and N ÿ are determined, it is clear that to increase k1 and keep k2 ; k3 ; . . . ; kr above the previous k1 , it will be most eective to increase the cross-sectional areas of elements that belong to N and are of the highest 1 a e values, and reduce the cross-sectional areas of elements that belong to N ÿ and are of the highest 1 aÿ e values. 4.3. Sensitivity numbers and optimum criterion for repeated eigenvalues It is de®ned that U fu1 g; fu2 g; . . . ; fur g: When the cross-sectional area of an element is changed in the current structure, assuming that DKg is ignored, the following equation can be derived from Eq. (3): oki A 22 I fai g f0g od in which 2 k11 6 k21 6 A 6 .. 4 . kr1
k12 k22 .. . kr2
... ... .. . ...
3 k1r k2r 7 7 .. 7; . 5 krr
23
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n o T i 1; 2; . . . ; r; j 1; 2; . . . ; r; kij uei DK e uej 24 where I is a r r identity matrix. To solve Eq. (22), the matrix fai g i 1; 2; . . . ; r can be obtained, then the ei can be calculated from unique element eigenvector u the following equation: n o ei ue1 ; ue2 ; . . . ; uer fai g i 1; 2; . . . ; r: u 25 ei to Eqs. (11) and (12), the sensitivity Substituting u 1 ÿ 2 2 ÿ r ÿ numbers (1 a ; ae ; ae ; ae ; . . . ; r a e e and ae ) for the repeated eigenvalues can be obtained easily.
De®ne 0
N1 S;
0
N1ÿ S;
e2S ; Nj ej j a e P0 0 ÿ e2S Nj ej j aÿ e P0
26
0
0
N 0 N1 \ 0 N2 \ \ 0 Nr ;
0
N ÿ 0 N1ÿ \ 0 N2ÿ \ \ 0 Nrÿ ;
0
0
0
ÿ
N N [ N ;
e2S ; Nj ej j a e > ÿe e ÿ e2S Nj ej j aÿ e > ÿe
j 2; 3; . . . ; r;
27 28 29
e
e
N 0 N1 \ e N2 \ \ e Nr ;
e
N ÿ 0 N1ÿ \ e N2ÿ \ \ e Nrÿ ;
e
N eN [ eN ÿ;
j 2; 3; . . . ; r; 30 31 32
where [ represents the sum set operator of two sets. In order to have the ®rst eigenvalue increased, the sectional areas of elements whose values of 2 a e ; 3 2 ÿ 3 ÿ r ÿ ae ; . . . ; and r a ; or a ; a ; . . . ; and a all are e e e e greater than zero, and are of the highest 1 a e values or the highest 1 aÿ e values, should be changed. In the general case, it cannot be ensured that 0 N and 0 N ÿ all are not an empty set. When one of 0 N and 0 N ÿ is an empty set, we can have e N and e N ÿ to be a non-empty set by setting up a small positive parameter e (referring to Section 4.2), then change the sectional areas of elements belonging to 0 N or e N , and being of the highest 1 a e e 0 values, or the highest 1 aÿ e values. When N instead of N is used in the optimisation process, the reduction in the ®rst eigenvalue can be the smallest even if an increase cannot be ensured. Moreover, for many engineering structural optimisations, the occurrence of repeated eigenvalues is much fewer compared to the total iterations
257
and thus a temporary eigenvalue decrease at one or two steps will not aect the solution process as a whole. 4.4. Optimum criterion for the combination of closely spaced and repeated eigenvalues When the repeated eigenvalues are not the ®rst eigenvalue but are very close to the ®rst eigenvalue, we can refer to Sections 4.2 and 4.3 and their sensitivity numbers are calculated by using Eqs. (11), (12) and (22)±(25). For the determination of modi®ed elements, each repeated eigenvalue is respectively treated as a closely spaced eigenvalue in relation to the ®rst eigenvalue. In doing so, it may happen that one or two eigenvalues become repeated with respect to the ®rst eigenvalue. This case is solved by combining the optimum criteria presented in Sections 4.2 and 4.3. N~ and N~ ÿ can be de®ned and obtained in such a combination. They serve as the same purpose as N and N ÿ in the case of closely spaced eigenvalues, and 0 N and 0 N ÿ (or e N and e N ÿ ) in the case of repeated eigenvalues. 5. Evolutionary procedure for buckling optimisation Similar to Ref. [2], the evolution procedure for buckling optimisation of columns, frames and plates is outlined as follows: Step 1: Discretise the structure using a ®ne ®niteelement mesh. Step 2: Solve the eigenvalue problem (1). Step 3: Calculate the sensitivity numbers according to each case: 1 ÿ Case 1: distinct eigenvalues: calculate 1 a e and ae for each element. Case 2: distinct and closely-spaced eigenvalues: i ÿ calculate i a ae i 1; 2; e i 1; 2; . . . ; r and . . . ; r for each element. Case 3: repeated eigenvalues: calculate i a e i 1; 2; . . . ; r and i aÿ e i 1; 2; . . . ; r for each element. Case 4: combination of closely-spaced and repeated eigenvalues: calculate the sensitivity number of closely spaced and repeated eigenvalues for each element at the same time. Step 4: According to each case, determine the elements whose cross-sectional areas are to be increased, and those whose cross-sectional areas are to be reduced. The requirements on structural symmetry and loading conditions are considered in the element modi®cation. Case 1: Increase the cross-sectional areas of the elements that have the highest 1 a e values, and decrease the cross-sectional areas of the same number of elements that have the highest 1 aÿ e values.
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Case 2: Increase the cross-sectional areas of the elements that belong to N (Eqs. (16) and (20)) and have the highest 1 a e values, and decrease the cross-sectional areas of the same number of elements that belong to N ÿ and have the highest 1 ÿ ae values. Case 3: Increase the cross-sectional areas of the elements that belong to 0 N or e N (Eq. (28) or Eq. (31)) and have the highest 1 a e values, and decrease the cross-sectional areas of the same number of elements that belong to 0 N ÿ or e N ÿ and have the highest 1 aÿ e values. Case 4: Increase the cross-sectional areas of the elements that belong to N~ and have the highest 1 a e values, and decrease the cross-sectional areas of the same number of elements that belong to N~ ÿ and have the highest 1 aÿ e values. Step 5: Repeat steps 2 to 4 until the increase in the ®rst eigenvalue becomes very slight for a consecutive, say four or ®ve iterations. During the evolution, the cross-sectional area is allowed to vary in small steps. For beam elements of rectangular cross-sections, either the breadth or the depth can be changed. For plate elements, the thickness can be changed. The change in the element stiness matrix DK e can be easily calculated for the above ®niteelement types. Meanwhile, information on eigenvalues and eigenvectors required for the sensitivity number calculation is readily available from the ®nite element analysis. In the above procedures, the number of elements subjected to cross-sectional changes and the step size of the change at each iteration need to be prescribed.
6. Examples In order to demonstrate the validity and eectiveness of the proposed method, two simple frames and a box structure displaying closely spaced or repeated eigenvalues during buckling optimisation are considered. In these three examples, the initial design is of uniform cross-section, and the YoungÕs modulus E 200 GPa, PoissonÕs ratio m 0:3 and mass density q 2700 kg/m3 are assumed.
Fig. 1. Structural model and loading case of the three-member portal frame.
vary to the maximum 40 mm2 and to the minimum 5 mm2 in steps of 1 mm2 . Each member is divided into 10 elements of equal length. A modifying ratio 24% and closely-spaced eigenvalue parameter ec 4% are used. For the frame with a uniform cross-section, the ®rst buckling mode is anti-symmetric with sway and the second buckling mode is symmetric with closely-spaced eigenvalues from the outset. The optimum radius square r2 for each element is shown in the column chart Fig. 2. The buckling load is 1.2514 times that of the uniform frame, in comparison to 1.125 times by only considering the single mode, and 1.2474 by using the mean method [8]. The evolutionary histories of the ®rst two eigenvalues using the mean method and the proposed method are given in Figs. 3 and 4, respectively. Fig. 5 compares the evolutionary histories of the ®rst eigenvalue using these two methods. It is seen that although the optimum factors obtained by the two methods dier only by 0.32%, fewer iterations are involved in the proposed method. The above problem is analysed with dierent values of closely-spaced model parameter ec 2%, 4% and 4.5%, as well as dierent modifying ratios c 10%, 15% and 24%. While the iteration histories of eigenvalues vary slightly in intermediate designs, no dierence is observed in the ®nal design.
6.1. Three-member portal frame
6.2. Three-member space frame example
A three-member pin based frame, which was analysed in Refs. [2,13] is considered for closely-spaced buckling eigenvalues. The frame structural model and the loading are shown in Fig. 1. All the members are of circular cross-sections and of equal length of 1 m. Initial uniform radius square r2 is 20 mm2 , and it is allowed to
A space frame with three beams pinned at the base and clamped at the apex is considered for the optimisation of structures with repeated eigenvalues against buckling. The frame model and the loading condition are shown in Fig 6. All the members are of circular cross-sections and of equal length of 1 m. Initial uni-
J.H. Rong et al. / Computers and Structures 79 (2001) 253±263
259
Fig. 2. Ratio of radius squares of beam cross-sections at optimum point to corresponding initial uniform values.
Fig. 3. Optimisation histories of the ®rst two eigenvalues for the three-member frame by using the mean method.
Fig. 4. Optimisation histories of the ®rst two eigenvalues for the three-member frame by using the proposed method.
form r2 is 20 mm2 , and it is allowed to vary to the maximum 40 mm2 and to the minimum 5 mm2 in steps of 1 mm2 . Each member is divided into 10 elements of
equal length. c 24% and ec 4% are used. This is a triple symmetric structure and the ®rst three eigenvalues coincide for the uniform design and remain coincided
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J.H. Rong et al. / Computers and Structures 79 (2001) 253±263
Fig. 5. Optimisation histories of the ®rst eigenvalue for the three-member frame by using the proposed method and the mean method.
6.3. Box frame
Fig. 6. Optimum model and loading case of the three-member space frame.
throughout the optimisation process. The optimum beam sectional parameters obtained by using the proposed method are given in Fig. 7. The ratios of the ®nal to initial uniform radius of the cross-section are displayed in this ®gure for one member, as it is identical for all members. The optimum buckling load is 1.275 times that of the uniform frame and it is achieved after 10 iterations. The iteration histories of the ®rst three eigenvalues (namely three repeated eigenvalues) are given in Fig. 8.
In the box frame shown in Fig. 9, all the members are of rectangular cross-sections with a constant breadth b 40 mm and an initial uniform depth d 40 mm. The horizontal members at the top and bottom are divided into 12 elements of equal length, and diagonal and vertical members in the middle are divided into three elements of equal length. Numbering of beam elements is shown in the model. Buckling optimisation of this frame was considered to be one of the most dicult examples in the literature. It was studied in Ref. [10] using the mean method of ESO. In applying the method proposed in this paper, the initial value of the design variable, beam depth d is allowed to change without upper limit and to the minimum depth of 1 mm in steps of 1 mm. c 24% and ec 4% are used. The optimum depth ratio is shown Fig. 10. The evolutionary histories of the ®rst ®ve eigenvalues using the proposed method and the mean method are shown in Figs. 11 and 12, respectively. It is observed that while all cases of closely spaced and repeated eigenvalues occur during optimisation process, the ®rst eigenvalue is kept increasing at non-repeated eigenvalue points by using the proposed method. It also takes fewer iterations than the mean method. Fig. 13 shows the results of the ®rst eigenvalues in these two methods. The buckling load factor is 1.9214, compared to 1.8678 with a dierence of 2.9%. It is seen that members of the optimum design have segmented cross sections which are not manufacturally appealing. One solution to this can be some smoothing techniques using interpolation functions so that the structural member displays a smoother outer shape. This point for ESO method is presented in detail in Ref. [2].
J.H. Rong et al. / Computers and Structures 79 (2001) 253±263
Fig. 7. Optimum result of section radius squares of beam elements for the three-member space frame.
Fig. 8. Optimisation histories of the ®rst eigenvalue for the three-member space frame by using the proposed method.
Fig. 9. Finite-element model and loading case of the box frame (allowable minimum depth 1 mm).
261
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J.H. Rong et al. / Computers and Structures 79 (2001) 253±263
Fig. 10. Optimum result of beam section depths for the box frame.
Fig. 11. Optimisation method histories of the ®rst ®ve eigenvalues for the box frame by using the mean method.
Fig. 12. Optimisation histories of the ®rst ®ve eigenvalues for the box frame by using the proposed method.
J.H. Rong et al. / Computers and Structures 79 (2001) 253±263
263
Fig. 13. Optimisation histories of the ®rst eigenvalue for the box frame by using the mean method and the proposed method.
7. Conclusion An improved approach to optimising the structures against buckling is proposed and illustrated with examples. The results demonstrate that the proposed method is valid and eective and is suitable for various complex cases of practical structures. The method can obtain better optimum design for structures against buckling than the mean method. It can be readily implemented in any of the existing ®nite-element codes. References [1] Xie YM, Steven GP. A simple evolutionary procedure for structural optimisation. Comput Struct 1993;49:885±96. [2] Xie YM, Steven GP. Evolutionary structural optimisation. Berlin, Germany: Springer; 1997. [3] Bendsùe MP, Kikuch N. Generating optimal topologies in structural design using a homogenisation method. Comput Meth Appl Mech Engng 1998;71:197±224. [4] Ma ZD, Kikuchi N, Cheng HC. Topology design for vibrating structures. Comput Meth Appl Mech Engng 1995;121:259±80.
[5] Yang RJ, Chuang CH. Optimal topology design using linear programming. Comput Struct 1994;52:265±75. [6] Xie YM, Steven GP. Evolutionary structural optimisation for dynamic problems. Comput Struct 1996;58:1067±73. [7] Querin OM. Evolutionary structural optimisation: stress based formulation and implementation. PhD Thesis, University of Sydney, Sydney, Australia, 1997. [8] Manickarajah D, Xie YM, Steven GP. A simple method for the optimisation of columns, frames and plates against bucking. In: Structural stability and design. Balkema, Rotterdam, 1995. p. 175±80. [9] Manickarajah D, Xie YM, Steven GP. An evolutionary method for optimisation of plate buckling resistance. Finite Element Anal Design 1998;29:205±30. [10] Manickarajah D. Optimum design of structures with stability constraints using the evolutionary optimisation method. PhD Thesis, Victoria University of Technology, Melbourne, Australia, 1998. [11] Fox RL, Kapoor MD. Rates of change of eigenvalues and eigenvectors. AIAA J 1968;6:2426±9. [12] Chen TY. Design sensitivity analysis for repeated eigenvalues in structural design. AIAA J 1993;31:2347±50. [13] Szyszkowski W, Watson LG. Optimisation of the buckling load of columns and frames. Engng Struct 1988;10: 249±56.