595

European Congress on Computational Methods in Applied Sciences and EngineeringEccomas 2000

Barcelona, 11-14 September 2000 ECCOMAS

ANALYSIS OF STRUCTURES INCLUDINGCOMPRESSION-ONLY AND TENSION-ONLY MEMBERS

Hamid Moharrami* and Masoud Riyazi Mazloumi*** Dr. H. Moharrami, Ph.D., Assistant Prof.,

Civil Engineering Department, Tarbiat Modarres University,Tehran, P.O. Box 14155-4838, Irane-mail: [email protected]

** M. Riyazi Mazloumi, M.Sc., Tarbiat Modarres UniversityNo.22, Emamat22 St., Emamat Blvd., Mashad, Iran.

Key Words: Compression-Only, Tension-Only, Elasto-plastic, Nonlinear Analysis,Nonlinear Behaviour, Mathematical Programming.

ABSTRACT. Compression-only and tension-only members are some special cases of elasto-plastic members that cause nonlinearity in the behaviour of structures. Analysis of suchstructures is traditionally performed in two ways. 1) Reanalysis after eliminating the tension-only members that show compression and/or eliminating compression-only members thatshow tension upon previous analysis. 2) Nonlinear analysis using iterative, gradual loadingtechniques with/without modification of stiffness matrix of the structure. In this paper aninnovative technique is presented that conducts the analysis in one step without the use of anyof these methods. In this method the nonlinearity effects of compression-only and tension-onlymembers, are considered by removing any undesired internal forces in these members. This isdone by adding some unknown compressive and tensile artificial forces to the internal forcesof compression-only and tension-only members that neutralize their primary internal forces.To evaluate the values of these artificial forces, first a direct sensitivity analysis should becarried out using unit load force method. Artificial forces are then determined by establishingand solving a Quadratic Programming problem that is built based on these sensitivitycoefficients. Afterwards internal forces are calculated precisely by employing superpositionprinciple. The method is completely formulated and applied to several structures includingcompression-only and tension-only members. One illustrative example has been provided toexhibit the solution procedure and its robustness. For structures with a few compression-onlyand tension-only members, this method is much more efficient than similar procedures in theliterature, because it concentrates only on a few members, not entire structure.

Hamid Moharrami and Masoud Riyazi Mazloumi

2

1 INTRODUCTIONCompression-only and tension-only members, hereafter called especial nonlinear members,

confine the use of linear analysis for structures including these members. In fact in the linearanalysis of structures, it is implicitly assumed that structural members have infinite capacities.Of course basically this assumption is not correct. However with this assumption nothing willbe violated if stresses in structural members do not exceed their elastic stress limits. If astructure includes some especial nonlinear members, whose stresses do not fit their pre-defined linear elastic stress limits, this assumption is no longer valid because the behaviour ofthe structure will change when some of these members reach their proportionate limits. In thecommon practise, for analysis of structures including tension-only and compression-onlymembers, to consider the actual behaviour of a structure, usually, either the omissiontechnique or a nonlinear analysis is performed. In the omission technique, the structure isanalyzed first, then the internal forces in the especial nonlinear members are checked. If theresult shows tension in a compression-only member or compression in a tension-onlymember, the member is removed from the assembly of the structure and analysis is repeateduntil there is no inconsistency between the result and the behaviour of these especial nonlinearmembers.

Literature is almost mature of nonlinear analysis methods. Crisfield1 and Owen andHinton2 have cited good summaries of classical nonlinear techniques. Basically nonlinearanalysis techniques are combination of the following three basic methods. 1) IncrementalScheme: In this method, displacement is found incrementally by gradually increasing theloading and modifying the stiffness of the structure at the beginning of each load increment.To get reasonable result from this method alone, loading increments should be quite smallotherwise it results in false displacement values. 2) Initial Stiffness method: In this method,the structure is analysed with an initial stiffness. After each analysis internal force ofmember, which can be measured by the previously obtained displacement, is compared toapplied load. Then the unbalanced force is applied again to the structure. The process ofanalysis of structure under unbalanced forces is repeated until external and internal forces andmoments reach equilibrium. 3) Newton-Raphson method: This method is similar to the initialstiffness method except that in Newton Raphson method the modified stiffness of thestructure is used at the beginning of each analysis instead of initial stiffness. In this methodtoo, the process continues until the unbalance load becomes infinitesimally small.

Nonlinear analysis of structures by mathematical programming is an old field of researchin this ground. De Donato3 is perhaps the last one who summarized the previous researchworks and presented fundamentals of this method for both holonomic (path independent) andnonholonomic material behaviours. In this method, it is assumed that displacement of nodesof a nonlinear elasto-plastic structure comprises two parts namely elastic and plastic parts.Then, the problem of finding total displacement vector of a structure is formulated in the formof a quadratic programming (QP) problem with some complementarity yield constraints.These yield constraints state that individual members either are stressed within their elasticlimits and do not accept plastic deformations or they are stressed up to yield limit and, as aresult, undergo some plastic deformations. The output of this sub-problem is linear and


3

nonlinear deformation of structure. Despite its robustness, this method suffers from theconsiderable number of variables that enter in the QP sub-problem and makes the solutionprocess time consuming and tedious. In addition this method is not suitable for analysis ofcompression-only and tension-only members because of the problems that encounter in themodelling of these especial nonlinear members.

There are also some other techniques that enable inelastic analysis of structures based ontheorems of Structural Variation. Structural variation theory studies the effect of change ofproperties, or even removal, of a member on the entire structure. It takes advantage of linearelastic analysis and sensitivity of structure to some self-equilibrating unit loads that areapplied at the end nodes of changing members. This technique has been applied to analysis ofseveral types of inelastic skeletal structures including space trusses (Saka & Celik4), frames(Majid & Celik5) and grids (Saka6), etc. It has been also extended to nonlinear finite elementanalysis (Abu Kassim & Topping7, and Saka8). Although this method takes advantage ofinitial stiffness matrix and does not require change in the stiffness matrix of structure duringthe analysis process, it is a historical and step by step method of analysis in which every stepuses information from the previous step.

The goal of this research work is to present a one step analysis procedure that can bereplaced for iterative techniques in the nonlinear analysis of structures including compression-only and tension-only members. The formulation of the method has been built up on simplestructural behaviour and equilibrium context for truss type elements. This method holdssimplicity of structural variation theorem, advantages of De-Donatos method, robustness ofmathematical programming and precision of the results.

2 FORMULATION OF THE SOLUTION PROCEDUREThe behaviour of compression-only and tension-only members is quite the same but in

opposite sense. For simplicity of presentation of the concept, first the formulation of solutionprocedure for analysis of structures including compression-only members is presented. Then,the formulation of solution of structures with tension-only members is presented. Finally theextension of the method to the analysis of structures including both tension-only andcompression-only members is provided. Since the axial force does not have direct influenceon flexural behaviour of individual members, the solution procedure outlined hereafter isapplicable as well, to frame elements that may carry bending moment beside axial force.

2.1 Compression-only membersConsider the structure in Fig. (1). Suppose that member z is a compression-only member

that, under some special load condition, the result of analysis shows tension force Fz in it. Toreach the correct solution, (i.e. zero tensile force in compression-only member), instead ofomitting the member z for the problem, the effect of presence of this member in the structurewill be eliminated. Adding an artificial compressive internal force Cz to neutralize its internaltensile force may do this action. When member z is in tension, the joints at its two ends, i andj, are forced to approach towards each other by a pair of forces equal to its internal force Fz inits longitudinal direction. Conversely the artificial compressive internal force will push the


4

two joints outward by a pair of self-equilibrating Cz forces as shown in Fig.(1-b). To evaluatethe effect of this artificial internal force, a direct sensitivity analysis may be done byapplying a pair of unit loads in the direction of Cz forces.

Figure 1: A structure with Compression-only and Tension-only members under: (a) Applied loads, (b) Artificialforces for Compression-only members, (c) Unit loads and (d) Artificial forces for Tension-only members.

Let us assume that qzz is the internal force in member z due to the unit loads in fig(1-c). Byapplying the pair of Cz forces to i and j, which is equivalent to adding a compressive Cz forceto members internal force, the internal force in member z, changes to zF

~ as follows: zzzzz qCFF +=

~ (1)The elimination of the effect of presence of member z on joints i and j, necessitates having

the sum of Cz and zF~ equal to 0. i.e.

0~ =+ zz FC (2)Note that compressive and tensile axial forces are assumed negative and positive entities

respectively. Substituting for F z~ from Eqn.(1) into Eqn.(2), results in:

0=++= zzzzzz qCFCR (3)Rz is in fact the resultant force on joints i and j in longitudinal direction of member z that

has been neutralized by the Cz. To apply the pair of Cz forces to the compression-onlymember, it is necessary to be confident a priori that, the member will be in tension otherwise,if member z is in compression, the pair of artificial Cz forces will not be applied to its endsand in this case: 0%zz FR = . Substituting for Rz from Eqn.(3) will give: 0%zzzzzz qCFCR ++= (4)

Combining Equations (3) and (4) will result in the following equation:0++= zzzzzz qCFCR (5)

i

jz

m

nx

z

z

(a) (b)

(c) (d)

i

j

i

j

1

Cz

zC

xn

m

1

x

xT

T


5

Equation (5) states that the value of Rz should be consistent with the expected internalforce in compression-only member z, i.e. a nonpositive force. It should be noted that inEqn.(5) Rz =0. when 0%zC and 0%Rz when 0=zC . This relationship, which indeed is theyield condition for the compression-only member, is equivalent to say: 0.= RC zz or:

0)( =++ zzzzzz qCFCC (6)For structures with more than one compression-only member, the internal force in member

z becomes:

=

+=n

rzrrzz qCFF

1

~ ; z = 1, . . . , n (7)

In this equation which should be written for all compression-only members, n is thenumber of compression-only members, rC is the nonnegative magnitude of artificialcompressive forces at the ends of member r, and qzr is the internal force in member z due tounit loads at the ends of member r. Similar modifications apply to Eqns.(3) to (6).Equations (5) and (6) become:

01

++= =

n

rzrrzzz qCFCR (8)

0)(1

=++ =

n

rzrrzzz qCFCC (9)

If rC values can be found in such a way that Eqns.(8) and (9), representing equilibrium andyield conditions respectively, are satisfied simultaneously, the analysis of the problem iscompleted and Eqns.(8) may be used to obtain internal forces in all members.

One method of solution of this kind of problems which is placed in the category ofcomplementarity programming problems, is to solve the set of n inequalities of Eqns.(8) andcheck the satisfaction of Eqns.(9) in a trial and error process. Another solution method, whichis presented in this research work, is to solve the set of Eqns.(8) and satisfy Eqns.(9)simultaneously. Establishing an optimization problem with these equations can do this.

Comparing equations (8) and (9), it can be concluded that for arbitrary non-negative valuesof rC that satisfy Eqns.(8), left sides of Eqns.(9) become negative values. There are certainvalues of rC that make Eqns.(9) vanish while satisfies Eqns.(8). Therefore if the sum of leftside of Eqns.(9) is taken as an objective function that has to be maximized to zero, whilesatisfying constraints of Eqns.(8) and non-negativity of rC , an optimization problem can beestablished as follows:

.0

)(1

++

=

z

zrr

n

=1rzzz

n

rzrrzzz

n

z=1

C

n ., . . 2, 1,=z ; 0.qC+F+C=R S.T.

qCFCC Maximize

(10)


6

This form of optimization problem is in fact a quadratic programming (QP) problem andhas a straightforward solution. It has been shown that if QP problem does have a solution, itresults in the global optimum and unique values for variables. Therefore since existence ofsolution is confirmed for stable structures, solution of the QP is insured. Solution of aboveproblem gives Cz for all compression-only members, which not only satisfy constraints of theproblem Eqns.(8), but also satisfy complementarity conditions of Eqns.(9) simultaneously.Finally internal forces in members can easily be obtained using Eqns.(8).

2.2 Tension-only MembersIf a structure contains tension-only members, a procedure similar to compression-onlymembers can be followed except that, an artificial tensile force should be added to internalforce of the tension-only member to neutralize the existing compression force. This isequivalent to applying a pair of artificial Tx forces as shown in Fig.(1-d) to the end nodes mand n of member x. In this case, the resultant force for a structure with m tension-onlymembers can be obtained from an equation similar to Eqn(8) as follows.

01

+= =

m

rxrrxxz qTFTR (11)

Note that Since artificial forces are applied in reverse direction of unit loads on thestructure a minus(-) sign appears everywhere in which rT is multiplied by its corresponding q.Note also that by assuming positive entities for tensile forces the resultant internal forces inthe tension-only members should be non-negative. Therefore, the inequality sign is reversedfor this type of members. According to the above explanations, the optimization QP problemfor tension-only members can be written in the following form:

Solution of this problem will reveal the unknown xT values. Then internal forces inmembers are calculated using equation (11).

2.3 Hybrid StructuresTo analyze hybrid structures that contain both compression-only and tension-only

members, it is sufficient to combine equations (10) and (12) to get a unit QP problem thatcontains both objective functions and constraints. To that end, a (1) should be multiplied tothe objective function of Eqn.(12) in order to convert it to a maximization form. Also, the

0.T

m ., . . 2, 1,=x ; qT-FT=R S.T.

) qTFTT inimize

x

xrr

n

=1kxxx

xr

m

rrxxx

m

=1x

+

+

=

0

(M1

(12)


7

effect of unit loads of compression-only members on tension-only members and vice versashould be considered. Applying all these notes, the unit QP problem for solution of hybridstructures becomes as follows:

0.T C

m ., . . 2, 1,=x ; qT-qCFT=R

n ., . . 2, 1,=z ; qTqCFCR S.T.

)qTqCFTTqTqCFCC aximize

xz

xrr

m

=1r

n

sxssxxx

m

rzrr

n

szsszzz

xr

m

rr

n

sxssxxx

m

=1x

n

z

n

sxr

m

rrzsszzz

++

++=

++++

=

==

=== = =

&

0

0

()(M

1

11

111 1 1

(13)

Solution of the quadratic problem of Eqn.(13) can successfully determine the compression-only and tension-only members that contribute in nonlinear behaviour of the structure. In factthose members whose artificial forces possess positive values are the special nonlinearmembers that have been stressed beyond their stress limits and therefore are automaticallyeliminated from the structure. After solution of QP problem, internal forces in any memberm can be obtained from the following equation:

qT-qCFT=R mrrm

=1r

n

smssmmm

=

++1

(14)

To facilitate the establishment of optimization problem (13), it can be written with matrixnotation in the following standard QP form:

0.X BX A

S.T.

X DX21+X C=f(x) ax. TT

M

(15)

In the above equation the vector X represents the vector of unknown artificial forces. Othermatrices and vectors can be obtained by comparing Eqns.(13 & 15). Matrix A, which is thecoefficient matrix of inequality constraints, can be represented by a )()( nmnm ++ matrix.Vector B is the vector of order (m+n) representing constant values of constraints. Similarlyvector C which is the vector of coefficients of linear terms in objective function, is a (m+n)component vector. Matrix D, which is also a )()( nmnm ++ matrix, is the Hessian matrixof the objective function. Its elements are defined as XXf(x)/=d ji2ij . Elements of theabove vectors and matrices may be simply obtained from following equations:


8

[ ] { } BC AA DFIB IIQIA

T=+=

=+= (16)

In the above equation Q is the matrix of direct sensitivities in which its ij element is qij.The matrix I is an especial identity matrix whose n diagonal elements, corresponding to ntension-only members, are equal to (-1).

It is noticed that the values of B and C vectors depend only on initial internal forces ofespecial nonlinear members. Therefore, if external loading do change, these vectors should bealtered. However, A and D matrices depend only on geometry and physical properties of thestructure and evidently for multiple load conditions they remain unchanged. This feature ofthe proposed formulation gives outstanding power to the algorithm in solving nonlinearproblems under multiple load conditions.

3 EXAMPLEFigure (2) shows structure of a statue of a wheel of a trolley. It will be supported at its huband is to be analyzed under its self-weight. In the analytical model the circle rim has beenapproximated with a nine-side polygon. The statue is designated to have concrete rim with adiameter of 4 m. and steel thin spokes. Physical properties of rim and spokes are as follows:

====

=====

Mpa 210000E Cm. 785.0Iyy Ixx Cm. 3.14 A spokesFor Mpa 20000E Cm. 12000 IzzIyy Cm. 20000 Ixx Cm. 400 A rim For the

42

442

Figure 1: A statue of a trolley wheel and its analytical model.

With the above specifications the rim segments are assumed to behave as compression-only members and spokes behave as tension-only members. The result of linear elastic

1 9

8

765

4

32

AB

C

D

EF

G

H

IT8

CB

T8

BC


9

analysis of the wheel under its self-weight is shown in Table 1. It is noticed from this Tablethat sides A, B, C and I of the rim are in tension while they cannot tolerate tension. Alsospokes number 1,2,3, 8 and 9 are in compression while they are tension-only members.

Table 1: Result of linear elastic analysis for the idealized trolley

Spokes No. 1 2 3 4 5 6 7 8 9

InternalForce

-237.75 -209.69 -47.49 -136.96 257.24 257.24 136.96 -47.49 -209.69

Rimmember

A B C D E F G H I

InternalForce

192.49 192.49 102.42 -35.62 -157.03 -204.95 -157.03 -35.62 102.42

Table 2: Results of direct sensitivity analyses, (part 1).

Location of unit loadsMemberI.D. 1 2 3 4 5 6 7 8 9

1 .7087 .0384 -.1421 -.1004 -.0965 -.0965 -.1004 -.1421 .03842 .0384 .7087 .0384 -.1421 -.1004 -.0965 -.0965 -.1004 -.14213 -.1421 .0384 .7087 .0384 -.1421 -.1004 -.0965 -.0965 -.10044 -.1004 -.1421 .0384 .7087 .0384 -.1421 -.1004 -.0965 -.09655 -.0965 -.1004 -.1421 .0384 .7087 .0384 -.1421 -.1004 -.09656 -.0965 -.0965 -.1004 -.1421 .0384 .7087 .0384 -.1421 -.1004-7 -.1004 -.0965 -.0965 -.1004 -.1421 .0384 .7087 .0384 -.14218 -.1421 -.1004 -.0965 -.0965 -.1004 -.1421 .0384 .7087 .03849 .0384 -.1421 -.1004 -.0965 -.0965 -.1004 -.1421 .0384 .7087A .1354 .1586 .1429 .1433 .1440 .1433 .1429 .1586 .1354B .1354 .1354 .1586 .1429 .1433 .1440 .1433 .1429 .1586C .1586 .1354 .1354 .1586 .1429 .1433 .1440 .1433 .1429D .1429 .1586 .1354 .1354 .1586 .1429 .1433 .1440 .1433E .1433 .1429 .1586 .1354 .1354 .1586 .1429 .1433 .1440F .1440 .1433 .1429 .1586 .1354 .1354 .1586 .1429 .1433G .1433 .1440 .1433 .1429 .1586 .1354 .1354 .1586 .1429H .1429 .1433 .1440 .1433 .1429 .1586 .1354 .1354 .1586I .1586 .1429 .1433 .1440 .1433 .1429 .1586 .1354 .1354


10

Employing the traditional omission procedure, that in common engineering practise is thedominant method for analyzing this kind of problems, has two drawbacks. 1) Identification ofcompression-only and/or tension-only members that will not contribute in the actual nonlinearbehaviour of the structure is a difficult task and internal forces upon first trial linear elasticanalysis usually leads to false recognition. 2) Omission procedure is not applicable to somemodels. For example, in the present problem omitting the aforementioned compression-onlymembers, will destroy the loading of the system. Therefore a nonlinear analysis should beperformed. To perform nonlinear analysis via the proposed method, it is necessary to conductdirect sensitivity analysis. This can be done by simply applying a pair of unit loads inlongitudinal and outward direction of all special nonlinear members. Table 2 shows theresults.

Table 2: Results of direct sensitivity analyses, (part 2).

Location of unit loadsMemberI.D A B C D E F G H I1 .00764 .00764 .00894 .00806 .00808 .00812 .00808 .00806 .008942 .00894 .00764 .00764 .00894 .00806 .00808 .00812 .00808 .008063 .00806 .00894 .00764 .00764 .00894 .00806 .00808 .00812 .008084 .00808 .00806 .00894 .00764 .00764 .00894 .00806 .00808 .008125 .00812 .00808 .00806 .00894 .00764 .00764 .00894 .00806 .008086 .00808 .00812 .00808 .00806 .00894 .00764 .00764 .00894 .008067 .00806 .00808 .00812 .00808 .00806 .00894 .00764 .00764 .008948 .00894 .00806 .00808 .00812 .00808 .00806 .00894 .00764 .007649 .00764 .00894 .00806 .00808 .00812 .00808 .00806 .00894 .00764A .9876 -.0121 -.0118 -.0119 -.0119 -.0119 -.0119 -.0118 -.0121B -.0121 .9876 -.0121 -.0118 -.0119 -.0119 -.0119 -.0119 -.0118C -.0118 -.0121 .9876 -.0121 -.0118 -.0119 -.0119 -.0119 -.0119D -.0119 -.0118 -.0121 .9876 -.0121 -.0118 -.0119 -.0119 -.0119E -.0119 -.0119 -.0118 -.0121 .9876 -.0121 -.0118 -.0119 -.0119F -.0119 -.0119 -.0119 -.0118 -.0121 .9876 -.0121 -.0118 -.0119G -.0119 -.0119 -.0119 -.0119 -.0118 -.0121 .9876 -.0121 -.0118H -.0118 -.0119 -.0119 -.0119 -.0119 -.0118 -.0121 .9876 -.0121I -.0121 -.0118 -.0119 -.0119 -.0119 -.0119 -.0118 -.0121 .9876

Now with the help of Eqns.(15) the QP problem of Eqn.(14) can be established. To getmore accuracy in solution of QP problem, more digits of sensitivity coefficients may be used.Note that table 2 is also a tabular presentation of matrix Q. Every column in this table isanalysis result of one unit load condition corresponding to one of the special nonlinearmembers. The result of solution to corresponding problem is presented in the following:

T1 = 1092.791 T2 = T9 =580.702 T3 = T4 = T5 = T6 = T7 = T8 = 0.


11

CA = CB = CC = CD = CE = CF = CG = CH = CI = 0.

It is noticed that although the primary analysis showed compression in members 1, 2, 3, 8and 9, the result of actual nonlinear analysis, shows that in the absence of members 1,2 and 9,the members 3 and 8 take tension internal forces. In addition, the members A, B, C and I thatwere in tension at the primary analysis, in the absence of some tension-only members becomecompressive. i.e. no omission of compression-only members was required.

Now internal forces in all members can be obtained using Eqn(14). The results are reportedin Table 3 and compared to the initial internal forces and the results of analysis in the absenceof spokes No. 1, 2 and 9. As it is shown, the results are almost the same and the very smalldifferences are perhaps due to rounding error.

Table 3: Results of proposed nonlinear analysis

MemberI.D.

Initial ForcesProposedNonlinearAnalysis

Forces inAbsence of

1, 2 & 91 -273.75 0.000 ---2 -209.69 0.000 ---3 -47.49 143.717 143.724 136.96 385.175 385.155 257.24 477.058 477.056 257.24 477.058 477.057 136.96 385.175 385.158 -47.49 143.717 143.729 -209.69 0.000 ---A 192.49 -126.270 -126.29B 192.49 -126.270 -126.29C 102.42 -232.527 -232.50D -35.62 -367.125 -367.09E -157.03 -480.276 -480.27F -204.95 -528.777 -528.78G -157.03 -480.276 -480.27H -35.62 -367.125 -367.09I 102.42 -232.527 -232.50

4 CONCLUSIONAnalysis of structures containing compression-only and tension-only members was

completely formulated. The formulation was built up based on simple structural behaviourand equilibrium context for truss type elements. However, it is possible to extend it to flexuralmembers that can bend in one direction and not the opposite direction. It is shown9 that this


12

concept can also be used in formulation of nonlinear analysis of elasto-plastic structures withmulti-linear stress-strain relationships. With some minor modifications, it can be used as apowerful tool for the assessment of ultimate load carrying capacity of existing structures inreliability analysis.

One example was solved to exhibit the capabilities of the method in the analysis ofstructures that include compression-only and tension-only members. It was shown that, thisnew method conducts analysis in one step, based on its elastic behaviour and initialconfiguration, and it does not require any modification in the structural model or iterativeprocedure. Obviously, this approach is much more efficient compared to iterative techniques.Its efficiency is also higher than classical mathematical programming methods because itconcentrates only on especial nonlinear members, not entire structure, and therefore, forstructures with a few compression-only and/or tension-only members, very much fewernumber of variables enter in the QP sub-problem.

5 REFERENCES[1] Crisfield M.A., Nonlinear finite element analysis of solids and structures, Volume 1-

Essentials. John Wiley & Sons, Chichester, U.K. (1991).[2] Owen, D. R. J. and Hinton, E., Finite elements in plasticity- theory and practise,

Pineridge press Swansea (1980)..[3] De Donato, O. Fundamentals of elastic-plastic analysis, Engineering plasticity by

mathematical programming, M.Z. Cohn and G. Maier, editors. Pergamon Press, NewYork, N.Y. pp. 325-349 (1977).

[4] Saka, M.P., and Celik, T., "Nonlinear analysis of space trusses by the theorems ofstructural variation", Proceeding of the second international conference on Civil andStruc. Eng. Comp. CIVIL-COMP 85, Vol. 2, pp. 153-158 (1985).

[5] Majid, K.I. and Celik, T., "The elastic-plastic analysis of frames by the theorems ofstructural variation", International Journal for Numerical Methods in Engineering, Vol.21 pp. 671-681 (1985).

[6] Saka, M.P., The theorems of structural variation for grillage systems, Innovation inComputer Methods for Civil and Structural Engineering, Edinburgh, Civil Comp Presspp. 101-111(1997).

[7] Abu Kassim, A.M. and Topping, B.H.V. "The theorems of structural variation forlinear and nonlinear finite element analysis", Proceeding of the second internationalconference on Civil and Struc. Eng. Comp. CIVIL-COMP 85, Vol. 2, pp. 159- 168(1985).

[8] Saka, M.P., "Finite element application of the theorems of structural variation", Journalof Computers and Structures, Vol. 41, No.3, pp. 519-530 (1991).

[9] Riyazi Mazloumi, M., " Analysis of structures including truss elements with limitedaxial strengths", M.Sc. thesis, Tarbiat Modarres University, Tehran, IR Iran (1999).

Documents

595