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Available online at www.sciencedirect.com

International Journal of Mechanical Sciences 45 (2003) 13691389

Optimum design of structures with stress and displacementconstraints using the force method

R. Sedaghatia ;;1, E. Esmailzadehb;2

aDepartment of Mechanical and Industrial Engineering, Concordia University, 1455 de Maisonneuve Blvd. West,

Montreal, Quebec, Canada H3G 1M8bUniversity of Ontario Institute of Technology, School of Manufacturing Engineering, Oshawa, Ontario,

Canada L1H 7K4

Received 13 August 2002; received in revised form 18 September 2003; accepted 27 September 2003

Abstract

A new structural analysis and optimization algorithm is developed to determine the minimum weight of

structures with the truss and beam-type members under displacement and stress constraints. The algorithm

combines the mathematical programming based on the sequential quadratic programming (SQP) technique

and the nite element technique based on the integrated force method. The equilibrium matrix is generated

automatically through the nite element analysis while the compatibility matrix is obtained directly usingthe displacementdeformation relations and the single value decomposition (SVD) technique. By combining

the equilibrium and compatibility matrices with the forcedisplacement relations, the equations of equilibrium

with the element forces as variables are obtained. The proposed method is extremely ecient to analyze and

optimize the truss and beam structures under stress and displacement constraints. The computational eort

required by the force method is found to be signicantly lower than that of the displacement method. The

eect of the geometric nonlinearity in the structural optimization problems under the stress and displacement

constraints were also investigated and it is illustrated that the geometric nonlinearity is not an important issue

in these types of problems and hence, it does not aect the nal optimum solution signicantly. Four examples

illustrate the procedure and allow the results to be compared with those reported in the literatures.

? 2003 Elsevier Ltd. All rights reserved.

Keywords: Optimum structural design; Stress constraints; Finite element force method; Size optimization; Displacementconstraints

Corresponding author. Fax: +1-514-848-8635.

E-mail addresses: [email protected] (R. Sedaghati), [email protected] (E. Esmailzadeh).1 Mem. CSME, Mem. ASME.2 Fellow ASME, Fellow I.Mech.E., Mem. CSME, Mem. SAE.

0020-7403/$ - see front matter? 2003 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijmecsci.2003.10.001
mailto:[email protected]:[email protected]:[email protected]:[email protected]

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1370 R. Sedaghati, E. Esmailzadeh / International Journal of Mechanical Sciences 45 (2003) 1369 1389

1. Introduction

The concept of equilibrium of forces and compatibility of deformations are fundamental to the

analysis methods in structural mechanics. The underlying principle behind the equilibrium equationsis the force balance. In general, equilibrium equations are not adequate enough in solving the struc-

tural analysis problems and need to be augmented by the compatibility conditions. In other words,

equilibrium equations are indeterminate by nature, and determinacy is hence achieved by inclusion of

the compatibility conditions. Two analytical approaches, namely the displacement and force methods,

have been developed to analyze the determinate or indeterminate structures.

Structural analysis and optimization algorithms developed in recent years have generally been based

on the displacement method [16]. Although, this method is an ecient approach for the stress

displacement type analysis, but it presents some disadvantages in the optimization problems when the

number of stress constraints are larger than the displacement constraints. On the other hand, the force

method has not yet been very popular among researchers for the structural optimization problems.This is due to the fact that the redundancy analysis required in the force method has not been easily

amenable to the computer automation. However, for the analysis of a not highly redundant structure

in which, the number of redundant elements is lower than the displacement degrees of freedom, or

for a determinate structure the force method is computationally more ecient than the displacement

method.

In the classical form of the force method, it is quite dicult to generate the compatibility con-

ditions. Splitting up the given structure into a determinate based structure and redundant members

would generate the compatibility in the classical force method and the compatibility conditions are

stated by establishing the continuity of deformations between the redundant members and the basis

structure. Navier [7] originally developed this procedure for the analysis of indeterminate trusses.

Prior to the 1960s, the basis structure and redundant members were generated manually. In thepost-1960s, several schemes have been devised to automatically generate the redundant members

and the basis determinate structure [8,9], but with a limited success. In the integrated force method,

developed by Patnaik [10,11], both the equilibrium equations and the compatibility conditions are

considered simultaneously. The generation of the compatibility equations is based on extending the

St. Venants theory of elasticity strain formulation to the discrete structural mechanics by elimination

of the displacement terms in the deformationdisplacement relationship [12].

In the present study, the linear analysis based on the integrated force method has been used to

analyze and optimize the truss and beam structures under both stress and displacement constraints and

the results were compared with those obtained from the geometrically nonlinear analysis. Moreover,

it is intended to investigate the eciency of the force method in the structural optimization of thetruss and beam structures, under displacements and stress constraints, by solving the equilibrium

and compatibility equations simultaneously. A direct method has been developed to generate the

compatibility matrix for indeterminate structures, which, is based on the displacementdeformation

relationship and the singular value decomposition (SVD) technique without the need to select the

consistent redundant members. The equilibrium matrix is also generated automatically through the

nite element analysis.

In most recent works, reported in literatures, the optimization algorithms were mainly based on

the optimality criterion technique because of its computational eciency. For example the optimality

criteria method has been employed to minimize the weight of the truss and beam structures under

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R. Sedaghati, E. Esmailzadeh / International Journal of Mechanical Sciences 45 (2003) 1369 1389 1371

the stress and displacement constraints [1316], stability constraint [17], or frequency constraint [18].

A number of the optimality criterion algorithms are based on satisfying one major critical constraint

in order to avoid the scaling and calculating a large number of Lagrange multipliers. Modern opti-

mality criterion algorithms would involve the case of satisfying the multiple constraints (scaling) andKarushKuhnTuker (KKT) condition (resizing) alternatively. However when the cross-sectional area

and principal moment of inertia are nonlinearly related (frame structures), the scaling procedures,

normally used in the optimality criterion methods, are approximate in nature and the scaling itself

needs an iteration procedure. Considering these facts the powerful nonlinear mathematical program-

ming method, based on the sequential quadratic programming (SQP) technique, has been utilized as

the optimization algorithm in this study to nd the true optimum solution.

The application and eciency of the proposed methodology is illustrated by minimizing the weight

of the truss and beam structures. It is shown that by using either the force or displacement method, as

an analyzer does not aect the nal optimum solutions of the problems with stress and displacement

constraints. However, the force method is more computationally ecient than the displacement one.It is also illustrated that in the practical optimization problems, with stress and displacement con-

straints, the nonlinear analysis does not aect the nal optimum solution signicantly. Moreover, it

is found that using the SQP method as the optimizer could lead into a lighter design when compared

with the conventional optimality criterion technique mostly reported in literatures.

In the following sections, a short description of the structural analysis using the force method and

the geometrically nonlinear nite element analysis is presented and the size optimization algorithm

is fully explained. Finally, the application of the algorithm is illustrated by optimizing four dierent

truss and beam-type structures under the stress and displacement constraints.

2. Structural analysis using the force method

A discrete nite element structure can be designated as structure (d; f), where d and f are the

displacement and force degrees of freedom, respectively. The structure (d; f) has d equilibrium

equations and r= (fd) compatibility conditions. In static problems the equilibrium equations inthe displacement formulation can be written as

KU=P; (1)

where K is the system stiness matrix of the structure (obtained by assembling the stiness matrices

of the individual elements), U is the nodal displacement vector and P is the external applied load

vector. The compatibility conditions have been satised implicitly during the generation of Eq. ( 1).

The equivalent form of Eq. (1) in the integrated force formulation can be written as [10,11]:

SF=P; (2)

where F is the element force vector. The matrix S and vector P can be obtained through the

combination of the equilibrium matrix as

QF=P (3)

and the compatibility equations can be written as

C=0; (4)

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where the element deformation vector can be related to the element force vector F in

accordance to

= GF; (5)

thus

S=

Q

C G

; P =

P

0

; (6)

where Q, C and G are the (d f) equilibrium matrix, (r f) compatibility matrix and the(f f) exibility matrix, respectively. One should note that the matrices Q, C and G are bandedand they have full-row ranks of d, r and f, respectively. The matrices Q and C also depend

on the geometry of the structure and therefore, are independent of the material properties. For anite element idealization, the generation of the equilibrium matrix Q and the exibility matrix

G is straightforward and can be obtained automatically. However the automatic generation of the

compatibility matrix C is a laborious task in the standard force method. Moreover, the generation

of C in the integrated force method is based on the elimination of the d displacement degrees of

freedom from the f elemental deformations. In this study, an ecient method is proposed to derive

the compatibility matrix directly. The method is based on the displacementdeformation relations

and the SVD.

The displacementdeformation relationship for the discrete structures can be obtained by equating

the internal strain energy and the external work as

12

PTU= 12

FT: (7)

By substituting P from Eq. (3) into Eq. (7), we can obtain

12

FTQT U= 12

FT or FT(QTU) =0: (8)

Since the element force vector F is not a null space, we are able to determine the following

relationship between the member deformation vector and the nodal displacement vector as

= QTU: (9)

Eq. (9) relates the f deformations to the d nodal displacement degrees of freedom and hence,

the r= (f d) compatibility equations can be arrived through the elimination of the d nodaldisplacements from the f deformations. To obtain the compatibility matrix, one may express the

nodal displacements in terms of the member deformations by using Eq. (9) as

U= (Q QT)1Q = (QT)pinv; (10)

where the matrix (QT)pinv denotes the MoorePenrose pseudo-inverse of QT. Considering Eq. (9)

and (10), we may have

[I QT(QT)pinv]=0 (11)

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or in short A=0; (12)

where A = [I

QT(QT)pinv] (13)

Eq. (12) is similar to the compatibility equations given by Eq. (4). However, the matrix A is a

(f f) matrix with rank of r. It illustrates that the rows of matrix A are dependent on each other.In order to extract the (rf) compatibility matrix C form the matrix A, i.e. to reduce the matrixA to matrix C, the singular value decomposition (SVD) method is used [19]. By applying the SVD

method to A, we obtain

A = RTT; (14)

where R and T are the (ff) orthogonal matrices and

= 0

0 0

(ff)

(15)

with = diag{1 2 r}, and 12 r 0. It follows that

A = R

C

0

: (16)

Therefore the (rf) compatibility matrix C can be represented by

C=[T1 T2 Ti Tr]T; (17)

where the vector Ti denotes the ith column of the matrix T.

Although Eq. (9) is quite adequate to determine the element deformations using the nodal dis-

placements, but it is not sucient to obtain the nodal displacements using the element deformations

or forces since the redundant structures are represented by the rectangular equilibrium matrix Q

with no inverse. This implies that the compatibility equations should be merged with the equilibrium

equations. For this reason, using S instead of Q in Eq. (9) and solving for the nodal displacements

U, we obtain

U= J or U= J G F; (18)

where

J=d rows of ST

: (19)

3. Nonlinear analysis

It is important that the eect of the geometrical nonlinearity on the nal optimum solution must

be fully investigated. In most cases, although the values of strain are quite small and the material

behaves linearly, the response of the structure, as a result of nite rotations and displacements, would

become nonlinear. It is therefore, necessary to express the joint equilibrium in terms of the nal

geometry of the structure. In the case of large displacements, the strain-displacement relationships

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contain the nonlinear terms. Consideration of these terms and by using the principle of virtual work,

the system stiness matrix can be evaluated in the form of

K= KE+ KG; (20)where K represents the system tangent stiness matrix, KEis the system linear elastic stiness ma-

trix and KG being the system geometric stiness matrix. It is worth noticing that the matrix KGis

associated with the changes in the geometry of the structure. These matrices are obtained by the

assemblage of the element linear elastic and the geometric stiness matrices in the global coordi-

nates. Linear and geometry stiness matrices for the truss and beam elements are well documented

in most nite element books [20,21].

To derive the incremental nite element equations, it is assumed that the equilibrium conguration

at a load level is known and the conguration at a slightly higher load level is to be determined.

Using the NewtonRaphson method and updated Lagrangian formulation [21], these equations may

be written as

t+ tK(k1)U(k) = P(k1) = t+ t(k)Preft+ tP

(k1)

t+ tU(k) = t+ tU(k1) + U(k) (21)

where k is the iteration number, t+ tK is the tangent stiness matrix at time step t+ t, t+ tP(k1)

is the vector of the nodal resultant member forces at time step t+ t, and Pref is a given reference

load. Furthermore, t+ t is a load factor parameter to denote the external load at time step t+ t,

P is the out-of-balance force, U is the vector of increments in the nodal displacements, andt+ tU is the vector of the nodal displacement at time step t+ t. The out-of-balance load vector

P corresponds to a load vector that is not yet balanced by the element forces, and hence an

increment in the nodal displacements is required. This updating of the nodal displacements in the

iteration will be continued until the out-of-balance loads and the incremental displacements are small.

In order to guarantee that both the out-of-balance loads and the incremental displacements are small,

the energy convergence criteria [21](being a product of the out-of-balance force and the incremental

displacement), has been employed. For the purpose of the present analysis an energy convergence

tolerance E= 106 has been selected.

It is noted that any increment in either the load or displacement is conventionally represented as

an evolution in time t. One should note that the problem is pure static and t simply denotes the

incremental steps in the solution.

4. Optimization algorithm

The optimization problem can be dened mathematically as minimizing the structural mass:

Min

M(A) =

ni=1

iLiAi

(22)

subject to the (n+m) stress and the displacement constraints (behavior constraints)

gj(A) =|i=i| 16 0; j= 1; : : : ; n ;

gj(A) =|Uj= Uj| 16 0; j= 1; : : : ; m (23)

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and the n side constraints on the design variables

Ai Ai6 0; i= 1; 2; 3; : : : ; n ; (24)

where i, Ai and Li are the density, the cross-sectional area and the length of the ith element,respectively. M is the total mass of the structure and Ai is the lower limit on the ith design variable.

Furthermore, j ; j are the stresses in the jth and its allowable limit value, respectively, and Uj; Ujare the respective jth constrained displacement and its allowable limit value. It should be pointed

out that the analysis and optimization are two separate modules. The equilibrium and compatibility

equations in the force method, Eq. (2), and the iterative equilibrium equations, Eq. (21) are satised

directly during the nite element analysis (analysis module) and then the results are passed to the

optimization module. Thus it is not necessary to take into account Eq. (2)or Eq. (21) as the equality

constraints inside the optimization algorithm.

In this study, the sequential quadratic programming (SQP) method has been applied to solve the

optimization problem discussed in above. The implementation of the SQP method was performed inMATLAB [22]. Based on the work done by Powell [23], the method allows one to closely mimic the

Newtons method for the constraint optimization just as it is done for the unconstraint optimization.

SQP is indirectly based on the solution of the KKT conditions.

It must be noted that the stress and displacement gradient functions, in Eq. (23), are not both

smooth and convex functions, thus the local optimum result may be achieved using the gradient-based

algorithms such as the SQP algorithm. In this study, several randomly generated initial points have

been used for the SQP algorithm in order to make sure that the optimal solution is either a global

solution or very close to the global solution.

5. Illustrative examples

Four examples on the analysis of the space truss and frame structures, presented in this section,

illustrate the proposed procedure and allow the results to be compared with those reported in lit-

eratures. It is intended to show that in the structural optimization problems, with the stress and

displacement constraints, the linear analysis procedure (either the force method or the displacement

one) does not aect the nal optimum design. Furthermore, it is to establish the fact that the design

optimization procedure based on the force method is more ecient than the displacement method

and that the geometrical nonlinearity is not an important issue in the truss- and beam-type structures.

5.1. The 25-bar space truss

The 25-bar space truss structure, shown in Fig. 1, has previously been investigated by Saka using

the linear analysis based on the displacement method and the geometrical nonlinear analysis with

the optimality criterion approach [13,14].

The structure has identical symmetries about the XZ and YZ planes, so that the design vari-

able linking is used to impose symmetry on the structure and hence, only eight design vari-

ables are identied here. The material selected is steel with the Youngs modulus of elasticity of

E=207 kN=mm2 (30; 000; 000 psi) and the mass density of = 7830 kg=m3 (0:283 lbm=in3) also the

structure is subjected to single load case as shown in Table1. The allowable tensile stress for all the

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Fig. 1. The 25-bar space truss structure.

Table 1

Nodal load components (N) for the 25-bar space truss structure

Node Coordinate directions

X Y Z

1 80,000 120,000 30,000

2 60,000 100,000 30,000

3 30,000 0 0

6 30,000 0 0

elements is set as at= 240 MPa (34809 psi) while the allowable compressive stress is determined

in accordance to the AISC codes [24]. It indicates that ac= 2E=S2R is for the case of SR Cc and

the value ofac= at(1 0:5S2

R=C2c ) is forSR Cc. The slender ratio of each member is SR=L=RG,

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Table 2

Final design solutions for the cross-sectional areas (mm2) for the 25-bar space truss structure

Design variables Member Linear analysis Nonlinear analysis

DM FM

1 1 232.7 232.7 233.0

2 25 1150.6 1150.6 1149.8

3 6 9 895.1 895.1 894.0

4 10,11 230.4 230.4 235.2

5 12,13 223.3 223.3 227.1

6 14 17 1018.4 1018.4 1023.8

7 1821 950.2 950.2 954.4

8 2225 1443.5 1443.5 1440.4

Mass (kg) 649.7 649.7 650.9

No. of iterations 316 424 380No. of A.C.a 8 8 8

CPU time (s) 50.64 16.31 209.77

aNumber of active constraints.

where L is the length, RG is the radius of gyration for each member and Cc=

22E=at. Thus, the

value of the allowable compressive stress varies during the optimization process. Stress constraints

have been imposed on all the elements and the displacement constraints of 10 mm have beenimposed on the nodes 1 and 2 in the X- and Y-directions. The minimum cross-sectional area for all

the elements was set at 200 mm2. All the members have the pipe-type cross-sections with SR= aAb,

where A is the area in square centimeter and the constants a and b are selected as 0.4993 and0.6777, respectively. The number of degrees of freedom for the displacement is m = 18, and that of

the force is n = 25. Therefore, the number of the redundancy is found to be r= 7. Without linking

the design variables, the number of the design variables is 25 and, the number of the constraints is

54. On the other hand, by linking the design-variables into eight groups, the number of the design

variables reduces to 8, and the number of the constraints would change to 20.

For the linear analysis, a minimum value of the mass of 649:7 kg (1432:3 lbm) was obtained

using both the displacement method (DM) and the force method (FM). However, when the nonlinear

analysis is carried out the minimum value of the mass was obtained as 650 :9 kg (1435 lbm). The

nal results for both the linear and the nonlinear analysis are presented in Table 2 with their

iteration histories illustrated in Fig. 2. The initial cross-sectional area for all the elements is chosenas 2000 mm2 (3:1 in2). The CPU time required for the FM is signicantly lower than that of the DM

indicating its superior eciency. For the linear and the nonlinear analyses, the compressive stress

constraint in the elements 1, 2, 6, 10, 13, 16, 18 and 24 (one member from each group was selected)

are found to be active. According to Table 2 there is a slight but insignicant dierence between

the optimum results obtained from the linear and the geometrical nonlinear analyses. Nevertheless,

to conrm the feasibility of the linear results, the structure was also analyzed using the nonlinear

analysis based on the optimum cross-sectional areas. It is found that very little stress violations were

occurred in the elements 10, 13, 16 and 18, and since these members are under compression the

optimum results obtained from a linear analysis could result into structural failure.

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0 50 100 150 200 250 300 350 400 450600

700

800

900

1000

1100

1200

1300

1400

Iteration

Mass(Kg)

Linear analysis (DM)

Linear analysis (FM)Nonlinear analysis

Fig. 2. Iteration histories for the 25-bar space truss for the linear and nonlinear analysis solutions.

0 200 400 600 800 1000200

400

600

800

1000

1200

1400

Iteartion

Mass(kg)

Initial areas=2000 mm2; Feasible guess

Initial areas=500 mm2; Infeasible guess

Fig. 3. Iteration histories for dierent initial areas for the 25-bar space truss.

The problem was solved, using dierent initial values of the cross-sectional areas for all the

elements, and results were found to be exactly the same as for the previous case. The iteration

histories for two dierent initial values of the cross-sectional area, using the force method, is shown

in Fig. 3.

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A minimum value of the weight of, 921 kg (2030:5 lbm) was obtained by Saka, using the linear

analysis, and of 507 kg (1117:7 lbm) for the nonlinear analysis with the optimality criterion based

only on satisfying the displacement constraints [13,14]. The stress constraints were satised through

the stress-ratio technique in the linear analysis. In the nonlinear analysis, the stress constraints weresatised during the steps of nonlinear analysis, so that when a member force exceeds its allowable

limit, then this limit value is used in the computation of the compensating load. In accordance to

the results found by Saka and the outcome of such redistribution, the truss members do not support

a force, which is more than its critical value.

Flurry and Schmit [5] also solved this problem by using the dual method and the approximation

concept and considering only the linear analysis based on the displacement method. This structure

was also analyzed by the authors using the data provided by them for which, identical results were

obtained.

5.2. The 72-bar space truss

The 72-bar space truss structure, shown in Fig. 4, is a relatively large size problem with the

material chosen as aluminum with the Youngs modulus of elasticity E= 69 kN=mm2 (107 psi) and

the mass density = 2768 kg=m3 (0:1 lbm=in3). The allowable stress for all the members is set as

172:37 MPa (25000 psi) and the stress constraints have been imposed on all the elements. Thedisplacement constraints of6:35 mm (0:25 in) are imposed on both the X- and Y-directions forthe nodes 14 and the minimum value of the cross-sectional area for all the elements was set at

64:52 mm2 (0:1 in2).

The structure is subjected to two dierent loading inputs. In the rst case, only the node 1 is

subjected to a pull load of 22:25 kN (5000 lbf ) in the X and Y-directions and a push load of

22:25 kN (5000 lbf) in the Z-direction. However, for the second case, all the nodes of 14 aresubjected to a push load of22:25 kN (5000 lbf) in the Z-direction. The number of displacementdegrees of freedom is m=48, and the number of the force degrees of freedom is n=72, and hence, the

number of the redundancy will be r= 24. Without linking together the design variables, the number

of the design variables is 72 and the number of the constraints is found to be 152. However, by

linking the design variables into 16 groups, the number of the design variables becomes 16, and the

associated number of the constraints reduces to 40.

Using the linear analysis, the minimum value of the mass, for both the DM and the FM, was

obtained as 172:20 kg (379:615 lbm). When the same problem was analyzed, using the nonlinear

analysis, the minimum value of the mass was increased slightly to 172 :400 kg (380:079 lbm). The

nal results for both the linear and the nonlinear analysis are presented in Table 3 and their itera-tion histories are illustrated in Fig. 5. The nonlinear analysis steps, required to obtain the response,

was found to be 2 for the rst load case, and 3 steps for the second load case in all the itera-

tions. The required computational time for the FM is signicantly lower than the DM, illustrating

the eciency of the FM over the DM for the analysis of large size problems. A closer examina-

tion of the results reveals that in both the linear and the nonlinear analysis the nodal displacement

constraints at node 1 in the X- and Y-directions for the rst load case, and the stress constraints

in the elements 1 to 4 for the second load case are active. The cross-sectional areas in groups 7,

8, 11, 12, 15 and 16 reached their minima in both analyses. The linear analysis solution matches

exactly with the solution reported by Flurry and Schmit [5], who solved the problem using the dual

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Fig. 4. The 72-bar space truss structure.

method and the approximation concepts in a linear analysis, which is based on the displacement

method.

It is noted that there is a very slight dierence, although insignicant, between the optimum

results obtained from the linear and the nonlinear analysis. To conrm the validity of the linear

solution, the structure was analyzed considering the geometrical nonlinearities based on the optimum

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Table 3

Final design solutions for the cross-sectional areas (mm2) for the 72-bar space truss structure

Design variable Members Linear analysis Nonlinear analysis

DM FM

1 1 4 100.97 100.97 101.29

2 5 12 352.00 352.00 352.39

3 1316 264.77 264.77 265.42

4 17,18 367.55 367.55 368.58

5 19 22 337.87 337.87 338.45

6 2330 333.61 333.61 334.06

7 3134 64.52 64.516 64.52

8 35,36 64.52 64.516 64.52

9 37 40 818.32 818.32 819.68

10 41 48 330.13 330.13 330.3211 49 52 64.52 64.52 64.52

12 53,54 64.52 64.52 64.52

13 55 58 1216.90 1216.90 1218.70

14 59 66 330.52 330.52 330.71

15 6770 64.52 64.52 64.52

16 71,72 64.52 64.52 64.52

Mass (kg) 172.20 172.20 172.40

No. of iterations 556 557 561

No. of A. C. 9 9 9

CPU time (s) 274.23 107.10 1427.6

0 100 200 300 400 500 60030 0

40 0

50 0

60 0

70 0

80 0

90 0

Iteration

Mass(x0.4

5Kg)

Linear analysis (DM)Linear analysis (FM)Nonlinear analysis

Fig. 5. Iteration histories for the 72-bar space truss for both the linear and nonlinear solutions.

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0 1000 2000 3000 4000 5000200

300

400

500

600

700

800

900

Iteartion

Mass(x0.4

5Kg)

Linear analysis (DM)Linear analysis (FM)

Fig. 6. Iteration histories for the 72-bar space truss with no variables linking.

cross-sectional areas obtained from the linear analysis. Surprisingly enough, both the displacement

constraints at node 1 and the stress constraints in the elements 14 for the second load case were

slightly violated.

Furthermore, the problem was once again solved when the design variables were not linked to-

gether. The results using the FM and the DM were found to be almost identical and the computationaltime for the FM was approximately one half of that required by the DM. The optimum value of

the mass was reduced to 288:8 lbm, demonstrating that when the symmetry is not imposed on the

structure a signicantly lower value of the mass can be obtained for the nal design. In this case

the number of active constraints was 47. The displacement constraints at node 1 in both X- and

Y-directions for the rst load case, and the stress constraints in the members 1, 2, 4 and 19 for the

second load case were found to be active. The cross-sectional areas in the elements 5, 8, 9, 1216,

18, 24, 25, 28, 29, 3136, 38, 40, 41, 4854 and 56 were reached to their minima and the iteration

histories for this case is illustrated in Fig. 6.

In order to better understand the eect of the geometrical nonlinearity, the load in all directions

and for both load cases was increased by a multiple of 100 with an increase in the displacementconstraints of101:6 mm (4 in), simultaneously. The nal results are presented in Table4and theiteration histories are shown in Fig.7.It is worth noting that the eect of the geometrical nonlinearity

is more pronounced at the nal optimum design and the constraint displacements are quite far from

the limiting value, implying that the design is only controlled by the stress constraints. Moreover,

when using the linear analysis, the stress constraints in members 6, 11, 13, 16, 17, 23, 30, 39, 42,

47, 57, 59 and 66, for the rst load case, and in members 14 and 1922, for the second load

case, are found to be active. However, in the case of nonlinear analysis for the rst load case the

situation is similar to that of the linear analysis, but for the second load case, the stress constraints

in the members 3 and 21 are found to be active.

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Table 4

Final design solutions for the cross-sectional areas (mm2) for the 72-bar space truss with increasing load and displacement

constraints

Design variable Members Linear analysis Nonlinear analysis

DM FM

1 1 4 12,727.00 12,727.00 12,787.00

2 5 12 5014.60 5014.60 5068.60

3 1316 4412.20 4412.20 4480.50

4 17,18 5664.90 5664.90 5699.70

5 19 22 12,911.00 12,911.00 13,168.00

6 2330 4575.30 4575.30 4603.90

7 3134 64.52 64.52 64.52

8 35,36 536.84 536.84 624.51

9 37 40 13,865.00 13,865.00 13,998.0010 41 48 4513.50 4513.50 4518.60

11 49 52 64.52 64.52 64.52

12 53,54 64.52 64.52 64.52

13 55 58 19,153.00 19,153.00 19,322.00

14 59 66 4484.20 4484.20 4508.80

15 6770 64.52 64.52 64.52

16 71,72 64.52 64.52 64.52

Mass (kg) 2698.87 2698.87 2723.03


No. of A. C. 15 15 15

CPU time (s) 100.92 46.67 1103.6

0 50 100 150 200 250 3000

1000

2000

3000

4000

5000

6000

7000

Iteration

Mas

s(x0.4

5Kg)

Linear analysis (DM)Linear analysis (FM)Nonlinear analysis

Fig. 7. Iteration histories for the 72-bar space truss with increasing load and displacement constraints.

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Fig. 8. The 10-member frame structure.

Finally, it may be inferred that for the practical truss design problems, which are optimized for

the size under the stress and displacement constraints, the eect of the geometrical nonlinearity is

not signicant and hence, the linear analysis can provide acceptable solutions. Moreover, for the

test cases studied the nonlinear analysis based on the geometrical nonlinearity does not necessarily

produce a better optimal solution (for lighter structures) when compared to that of the linear analysis.

Nevertheless, an optimized structure based on a linear solution may fail when the values of the stress

at some members would reach beyond the allowable design limit, as can be seen in some of the

test cases presented.

5.3. The 10-member frame (two-story and two-bay)

The 10-member frame structure consists of three stories and two bays, illustrated in Fig. 8, is

made of steel with E= 207 kN=mm2 (30; 000; 000 psi) and = 7830 kg=m3 (0:283 lbm=in3). The

stress limit for all the frame members is 165:47 MPa (24000 psi). The horizontal displacements for

all the joints were limited to 0:254 mm (0:01 in), while a minimum cross-sectional area limit of

3225:80 mm2 (5 in2) and a maximum area limit of 64; 516 mm2 (100 in2) were enforced. The initial

cross-sectional area was set as 16; 129 mm2 (25 in2) that being the same for all the elements. The

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following empirical relationships were used for the area A, the section modulus S, and the moment

of inertia I [15].

S= 1:6634 A

1:511

I= 4:592 A2 06A6 15

S= (281:077 A2 + 84100)0:5 290

I= 4:638 A2 15 A6 44

S= 13:761 A103:906

I= 256:229 A230044 A6 100;

where A is the area measured in square inches. The above relationship is stated for a steel section

in accordance to the AISC code.A minimum weight of 3307:23 kg (7291:19 lbm) is obtained with a linear analysis, but with the

use of nonlinear analysis it will increase to 3309:9 kg (7297:08 lbm). The horizontal displacement

constraint at node 4 and the stress constraints on element 6 are identied as active. The horizontal

displacement at node 3 is close to being active and the cross-sectional area of the elements 3, 4 and

10 reached their minimum size. The results were compared with those reported in the literature. As

an example, Khan [15,16] used a displacement based linear analysis with the optimality criterion

technique and has obtained a minimum value of the weight 3391:87 kg (7477:79 lbm) with the

horizontal displacement of the nodes 3 and 4 being active constraints (no active stress constraint).

This problem has also been solved using the CONMIN code and a minimum value of the weight

3969:97 kg (8752:29 lbm) is reported [25].

Table 5

Final design solutions for the cross-sectional areas (mm2) for the 10-member frame structure

Members Linear analysis Nonlinear analysis

DM FM

1 28,387 28,387 28,387

2 23,682 23,682 23,720

3 3226 3226 3226

4 3226 3226 32265 46,255 46,255 46,271

6 10,241 10,241 10,238

7 7236 7236 7251

8 16,433 16,433 16,467

9 16,243 16,243 16,266

10 3226 3226 3226

Mass (kg) 3307.23 3307.23 3309.9


No. of A. C. 5 5 5

CPU time (s) 63.52 20.60 144.39

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Fig. 9. The 25-member frame structure.

The computational time for the force method was signicantly lower than that required by the

displacement method, again pointing out the eciency of the force method when applying the force

method to the frame-type structures. There is a very small discrepancy between the linear and

nonlinear solutions, however insignicant. Nevertheless, to conrm the validity of the linear analysis

results, the structure was simulated by a nonlinear analysis using the optimum cross-sectional areas

obtained via the linear analysis. It was observed that the stress constraint in element 6 is a little over

active, as shown in Table 5, pointing out that the linear analysis have produced acceptable results.

5.4. The 25-member frame (three-story and three-bay)

The 25-member frame structure, shown in Fig. 9, corresponds to a three stories and three bays

structure. The numerical values of the material properties and the stress limit and the relationship

between the cross sectional area, section modulus and the moment of inertia are all the same as

those mentioned in Section 5.3. The values of the horizontal displacements for the nodes 1, 2,

3, 10, 11 and 12 are limited to 0:127 mm (0:005 in) and a minimum value of the area limit of

3225:80 mm2 (5 in2) and a maximum area limit of 64; 516 mm2 (100 in2) have been dened.

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Table 6

Final design results for the cross-sectional areas (mm2) for the 25-member frame structure

Members Linear analysis Nonlinear analysis

DM FM

1 10,007 10,007 9483

2 7146 7147 6916

3 4233 4233 3952

4 16,210 16,210 16,730

5 20,247 20,246 20,307

6 3226 3226 3226

7 51,757 51,749 52,280

8 14,878 14,885 14,155

9 31,429 31,429 31,878

10 53,870 53,862 53,23711 64,516 64,516 64,516

12 20,581 20,581 20,443

13 3226 3226 3226

14 19,581 19,581 19,514

15 3226 3226 3747

16 4826 4832 5067

17 13,153 13,154 12,898

18 3226 3226 3226

19 16,165 16,168 15,072

20 3226 3226 3226

21 18,751 18,754 17,297

22 13,546 13,556 13,765

23 46,754 46,739 47,96724 3226 3226 3226

25 3226 3226 3226

Mass (kg) 9508.32 9508.32 9487.26


No. of A. C. 18 18 17

CPU time (s) 479.93 299.50 1687.90

The minimum value of the weight obtained using the linear and nonlinear analysis is 9508.32(20962:26 lbm) and 9487:26 kg (20915:83 lbm), respectively. The horizontal displacement constraints

at the nodes 2 and 10, as well as the stress constraints on the elements 1, 2, 3, 5, 9, 12, 14 and 17,

are active both in the linear and nonlinear analysis. The cross-sectional areas for the frame elements

6, 11, 13, 15, 18, 20, 24 and 25 reached their minimum size. This problem was also solved by

Khan [15]using the displacement based linear analysis and the optimality criterion technique, having

obtained a minimum weight of 10049:77 kg (22155:95 lbm) with just the horizontal displacement at

nodes 2 and 11 being active (no active stress constraint). Once again results obtained from the nite

element force method, performed in this study, indicate a superior advantage over the ones obtained

from the displacement method, as illustrated in Table 6.

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6. Conclusions

A structural optimization algorithm, using integrated force formulation technique, has been de-

veloped to minimize the weight of truss- and frame-type structures under the stress and displace-ment constraints. The required compatibility matrix in the formulation has been derived directly

by utilizing a displacementdeformation relationship and the single value decomposition technique.

Moreover, the sequential quadratic programming method has been adopted to optimize the truss and

frame structures.

The main objective of this study is to investigate the relative performance of the force and

displacement methods in the design and optimization of dierent space structures with the stress

and displacement constraints. It is found that the optimization technique that is based on the force

method is computationally far more ecient than the displacement one. Moreover, it is shown that

the nonlinear analysis does not signicantly aect the optimum solution of these types of problems.

Last but not least, from the results obtained from four dierent examples, it is demonstrated thatthe sequential quadratic programming method could result into a lighter optimal design of space

structures when compared to the conventionally used optimality criterion techniques. The proposed

methodology has proved to be extremely ecient in the analysis and optimization of the truss- and

frame-type space structures.

References

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