Collapse limit surface generation for multiparametric loading

Collapse limit surface generation for multiparametric loading F. Tin-Loi and Y. F. Lo School

Wales,

of Civil Engineering, Australia

The University

of New South

Wales,

Kensington,

New

South

A systematic computational scheme for the generation of the global collapse load surface of skeletal structures under multiparameter loading is developed. The algorithm is bused on identifying the relevant collapse facets in loadspuce through a hyperplune generution type technique involving solution of u series of linear programming problems. Several examples ure given to illustrate its intuitive simplicity and accuracy. Keywords:

limit analysis, collapse surfaces, linear programming, multiparameter loading

Introduction Limit analysis, namely, the determination of the maximum load amplification or safety factor that can be sustained by a perfectly plastic structure subjected to given loads, is a central problem in structural plasticity. One of the basic assumptions that is usually made in such analyses is that of proportional loading, whereby the system of applied loads, typically assumed to be modelled as concentrated, can be defined by a single monotonically increasing parameter. It is well known’ that the limit analysis problem can be efficiently formulated as dual mathematical programming problems that arise naturally from the static (lower bound) and kinematic (upper bound) theorems of plasticity. Discretization of the structure and piecewise linearization of the nonlinear yield criterion are usually carried out to transform the nonlinear constrained optimization formulation into a linear programming (LP) problem. The use of LP to compute the limit loads of a wide range of structural types is well established; comprehensive surveys are given by Maier and Munro2 and by Maier and Lloyd Smith.3 A more general, albeit more difficult, problem is limit analysis without the restriction of proportional loading. In such cases the individual loads are allowed to vary independently, and a primary objective of the analysis is to determine a failure or limit surface in load space. As mentioned by Nafday et al.,4,s such information can be important for the reliability assessment of structures. Very few methods have been proposed

Address reprint requests to Dr. Tin-Loi gineering,The University of New South

at the School of Civil EnWales, Kensington, NSW

2033, Australia. Received

17 October

1991; accepted

0 1992 Butterworth-Heinemann

10 March

1992

by researchers over the years to obtain collapse limit surfaces for entire structures. Early work by Symonds and Prager” resulted in the generation of simple twoand three-dimensional surfaces in stress space for trusses. Purely geometric arguments were used as the basis of their procedure. A later attempt was made by Lloyd Smith,’ who developed schemes for obtaining inner-bound and outer-bound approximations to the limit load surfaces for frames. The inner-bound approximation was generated from the convex combinations of arbitrary collapse load solutions, obtained through repeated solutions of single-load parameter limit analyses. The outer-bound approximation, on the other hand, was derived from the enumeration of feasible collapse modes. Repeated limit analyses using proportional loading paths were also used fairly recently by Lin and Coroti9 to establish approximate collapse load surfaces in two- and three-dimensional load spaces. The problem of developing a computational method for exact generation of limit hypersutfaces seemed to have been solved only recently by Nafday et al.4*SConsidering framed structures for which a single stress resultant (bending moment) causes plasticity, they formulated the problem as a multiobjective linear programming (MOLP) model and interpreted the obtained weak noninferior’ set of solutions as the global collapse hypersurface. In a brief discussion of this work, Casciati and Faravelli’O mentioned that the proposed approach of Nafday et al.4J appeared to be the same, from an operative point of view, as their”*‘2 parametric linear programming scheme for building the piecewise linear boundary of the safety domain. As we shall demonstrate in this paper, however, a MOLP formulation is not strictly correct in that, for some cases, the collapse surface is not necessarily characterized by the noninferior manifold. The purpose of this paper is to present a simple computational scheme, requiring only a standard LP

Appl. Math. Modelling,

1992, Vol. 16, September

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Collapse limit surface for multiparametric

loading:

F. Tin-Loi and Y. F. Lo

solver, to construct the exact global collapse load hypet-surface of a structure under multiparameter loading. It can be applied to a wide class of discretized structures with piecewise linear (PWL) yield surfaces, often considered by Maier, Munro, Cohn, Lloyd Smith, and many others. I-3 It is also capable of solving those problems for which the MOLP method4,s fails. Interestingly enough, the basic idea of the method was conceived as a result of work on interactive MOLP procedures by Reeves and Franz.13-15 The organization of this paper is as follows. First, we review the static LP approach for the calculation of the collapse limit loads of suitably discretized structures under single-parameter monotonic loading. Various comments regarding the extension of the analysis to the case of multiparameter loading are also made. We then detail the algorithmic procedure of the proposed technique with the help of simple examples. Finally, we present further examples to illustrate application of the method.

The limit analysis problem Single parameter

loading

The basis of a limit analysis rests on application of either the static theorem of plasticity or the kinematic theorem of plasticity.’ In the following, we briefly review application of the static approach for a singleparameter loading. According to the static theorem, the load on a perfectly plastic structure, which corresponds to any arbitrary equilibrium stress field and nowhere violates the yield condition, is a lower bound to the actual plastic collapse load. The best lower bound then clearly requires maximization of the proportionally applied load multiplier under constraints of equilibrium and yield conformity. Discretization of the structure and piecewise linearization of the commonly nonlinear yield conditions are usually implemented to transform the optimization into an LP problem. In particular, the structure is suitably discretized into a number of finite elements that are interconnected by nodes. As usual, the discretization is made on the basis of known or assumed locations at which generalized plastic hinges may occur. All applied loads are transformed to equivalent concentrated nodal forces corresponding in number to the degrees of freedom of the structural model. If P represents the nodal load vector and Q the stress state, also representing conveniently the active stress resultants’,‘h at the nodes, then equilibrium of the whole structure can be expressed as BQ = P

(1)

where B is the structure equilibrium matrix. We can also assume, without undue loss of generality, that the structure is statically indeterminate so that Q can be expressed explicitly as Q = B,,P + BrX

(2)

In structural analysis, (2) represents a familiar step in the flexibility method, where vector X collects the re-

492

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dundants for the structure and matrices B0 and B, are the basic and redundant load matrices, respectively. For simple structures, X can be chosen by inspection, and B0 and BI can be calculated manually; for complex structures, accurate and automatic techniques for performing these are available.i7,‘* The use of (2) instead of (1) is computationally advantageous, since it reduces the number of variables in the LP limit analysis problem (the dimension of X is invariably much less than that of Q). It is also useful to note that the term B,X in (2) represents a self-equilibrated or residual stress field in equilibrium with zero external loads. This is particularly useful for shakedown analyses;’ indeed, the work that we present in this paper can be easily extended to accommodate shakedown loading. As was mentioned earlier, for each node we assume that PWLr9 yield conditions apply. Each of these in effect represents a convex polygonal domain in Q-space for which the stress state at the corresponding node is admissible. A compact and computationally efficient way of mathematically representing the PWL yield condition is in terms of the vertices or corners of the convex yield surface, a scheme that was popularized in the context of structural plasticity by Zavelani-Rossi.20 The set of PWL yield conditions for the whole structure can be written as

Q = Vt

(34

t?-0

(3b)

UT65 1

(3c)

where vectors contain as nonintersecting subvectors the contributions of each node and matrices have a block diagonal structure, each block referring to one node. In particular, matrix V collects the extreme point vertices of the PWL yield polygons of all nodes, 5 is a vector of nonnegative multipliers, U is a Boolean matrix (i.e., with 0 and 1 entries only), 0 is a null vector, 1 is a vector with unit entries, and superscript T denotes a transpose. Equation (3a) simply expresses Q as a convex combination with multiplier 4 of vertices V, and condition (3~) represents the yield conformity requirement. We are now ready to formulate the limit analysis as an LP problem. If we assume that the load P is made up of a fixed dead load component POand a monotonically increasing live load component pPL, then the LP formulation for finding the collapse or limit load factor J_+ can be written as pC = maxp

(4a)

subject to pBoPL + B,X - V& = -BoPD

(4b)

UT5

(4c)

1

where (4b) has been obtained by substitution of (2) into (3a) and, as is conventional, the nonnegativity condition for ,$is not shown. The load multiplier p is usually nonnegative; we shall also adopt this restriction without loss of generality. Problem (4), involving variables

Collapse limit surface for multiparametric

p, X, and 5. can be solved by means of any standard LP package. Extension to multipurameter loading In formulating the problem (4) we made a key assumption that the loading program is defined by a single, monotonically increasing parameter p. Extension to the case in which more than one load multiplier controls the loading is, as we shall see, not straightforward. Before discussing our proposed method, it would be worthwhile to illustrate the idea of a collapse surface in load space through a simple and well-known example and also briefly discuss the MOLP method of Nafday et al.4.5 Consider the portal5 with two-parameter loading shown in Figure I; throughout this paper we shall assume that Pi = p;M,,IL, where the subscript i refers to the ith load parameter, M,, is a reference plastic moment capacity, and L is some reference length. Assuming that plastic hinge formation is governed only by bending moment, a limit hypersurface for the structure can be constructed in two-dimensional load space as in Figure 2. This hypersurface, consisting of hyperplanes AB, BC, and CD, represents the inIinite combinations of (pi, p2) for which plastic collapse of the frame occurs. Clearly, for a fixed ratio of p, to pLzthe problem becomes one of single-parameter loading for which formulation (4) can be applied. When pl and puz are allowed to vary independently, an approximate surface can obviously be obtained through repeated ap2p2 f IMP 1-l

Figure 1.

Portal

frame

(3.333,6) (0,6) A

0

0

PI

(6.667,O) Figure 2.

Limit surface for the portal frame

Appl.

loading:

F. Tin-Loi and Y. F. Lo

plication of (4) for assumed ratios of p, to p2. However, such a scheme is inefficient and limited to simple twoparameter loadings. If we note that the safety domain is a convex polyhedron (OABCD in Figure 2), a viable computational procedure would be to generate as economically as possible the extreme points (A, B, C, and D in our example) of the limit hypersurface from or during which the maximal facets can be systematically identified. This idea in fact forms the basis of the scheme proposed by Nafday et al. 4~sThey formulated the multiparametric limit analysis of frames under a single active stress resultant (bending moment) as a MOLP problem. In the present context the single parameter case then becomes pc. = max p (5a) subject to /_JB,)P~,+ B ,X - V,$ = - BoPn

(5b)

IJTCs 1 (5c) Formulation (5) differs from (4) in that the load multiplier to be maximized is now a vector instead of a scalar. The theory and solution of such multicriteria or vector optimization problems are well developed; refer, for instance, to the excellent books by Steuer,’ Cohen,?’ and Zeleny. 2z In their work, Nafday et al.5 used the multiobjective simplex code ADBASE developed by Steuer9 to obtain the noninferior solutions in load space. We recall that for a MOLP involving vector maximization of n objective functions z(x) and variables x a solution x* is noninferior if there exists no feasible x such that z(x) 2 z(x*) and z;(x) > z,;(x*) foratleastonei = 1,. . . , n. As defined, BC in Figure 2 represents the noninferior solution set. Recognizing that this so-called “strong noninferiority” is unduly restrictive for the limit analysis problem, Nafday et al.4 defined the failure hypersurface as including the “weak noninferior” solutions as well. Mathematically, a solution x’ is weakly noninferior if there exists no feasible x such that z;(x) > zi(x’) for all i. We note that the strong noninferior set is also weakly noninferior, but the converse is not true. This broader definition of noninferiority thus covers the set ABCD of Figure 2. It should also be mentioned that some authors9 prefer to use the term “noninferiority” (Pareto optimality, admissibility, or efficiency) to refer to points in decision or variable space, while the term “dominance” is used for vectors in criterion or objective space. While for most situations the weak noninferior set in load space does define the collapse limit surface, there exist some instances in which this is not correct. We illustrate this briefly by means of the following simple counterexample. Consider a uniform two-span beam under the two-parameter load shown in Figure 3. It is easy to work out manually that the collapse surface is as given in Figure 4. The entire weak noninferior set is obviously BC; the formulation of the structural limit analysis problem as the MOLP (5) is therefore incorrect, since it will not generate solutions AB and CD.

Math.

Modelling,

1992, Vol.

16, September

493

Collapse

limit

surface

for multiparametric

loading:

PI

a

$

-

-

% 2L

*pFigure 3.

Two-span

7

beam

(3,O) Figure 4.

Limit surface for the two-span

beam

Step I (initial phuse)

The proposed method The method that we propose for solving the multipar-. ameter limit analvsis Droblem is based on the work of Reeves and Franzi3-‘~ concerning a simple interactive procedure for MOLP problems. Their basic idea lies in generating systematically a series of hyperplanes to identify, if necessary, all strongly noninferior solutions to the problem. While the primary intention of Reeves and Franz is to implement a scheme whereby the decision maker can reach a compromise solution bv interactive elimination of alternatives, our aim is to [dentify the complete solution set to the limit surface. It is therefore necessarv for us to modifv the method of Franz and Reeves-to capture solutions that may be only weakly noninferior or that may even be inferior points, as demonstrated through the example shown in Figures 3 and 4. We shall first illustrate the key idea of our method by solving the structures shown in Figures I and 3 before we Droceed with a detailed description of the algorithm: Consider again the portal shown in Figure I. The graphical illustration of the solution path shown in Figure 5 will help explain the procedure. We start off by solving a single-parameter problem involving maximization of p1 subject to constraints (5b), (5c), and ~2 = 0. In pl-p2 load space, this gives the solution point 1 with coordinates (6.667, 0). The next run is performed by maximizing p2 subject to (5b), (5c), and pI = 0 to give a second point 2 = (0, 6). We then use vertices 1 and 2 to form a hyperplane, the equation of which gives the objective function for the next maximization under constraints (5b) and (5~) only. We ob-

494

Appl.

Math.

Modelling,

and Y. F. Lo

tain point 3 = (6.667, 4). Two new trial hyperplanes 1-3 and 2-3 are now possible. Selecting hyperplane l-3 as the objective function of the next maximization generates point 1, again indicating that l-3 is also a facet of the collapse polygon. A fifth maximization with 2-3 gives a new point 5 = (3.333, 6). Two further hyperplanes 2-5 and 3-5 are therefore possible. These yield old points 2 and 5, respectively. Since no further trial hyperplane is possible, we stop the computation. A comparison of the results obtained with Figure 2 shows that points 2, 5, 3, and 1 correspond to A, B, C, and D, respectively. Application of the procedure to the beam of Figure 3 results in the solution tree shown in Figure 6, where 1 = (3, 0), 2 = (0, 0.333), 3 = (6, 6), and 5 = (4, 6). Again, these results agree with the limit hypersurface of Figure 4. We now provide a formal description of the algorithm. For the sake of clarity a three-parameter loading is considered. While extension to higher-dimensional problems is straightforward, it should be noted that for practical problems the number of parameters is not likely to be large, since few load multipliers are required to cover all load types. In our description a general phase is assumed to involve points a, h, and c, while we refer to points 1, 2, and 3 when describing the initial phase; each point is a vector of load multipliers.

MP

t_

F. Tin-Loi

1992,

Vol.

16,

September

l

l

Solve a single objective: maximize kl LP problem with the two additional constraints p2 = 0 and p3 = 0 to give optimum point 1. Repeat for: maximize p2 and additional constraints 11, = 0 and p3 = 0 to give point 2.

Figure 5.

‘1 ,,

Geometrical

illustration

,

l-3

of the algorithm

-

I

‘-2-3<2_3_5<2-5-2 3-5

Figure 6.

-

Tree diagram for the solution of the two-span

5 beam

Collapse l

limit surface for multiparametric

Repeat for: maximize p3 and additional constraints pi = 0 and p2 = 0 to give point 3.

Step 2 (start of general phase)

Obtain an objective function by calculating the equation of the current plane a-b-c (use an arbitrarily chosen constant value of 1, if possible). Store the current combination [a, 6, cl in an “already run” combination stack. Solve the single-objective LP problem, using the objective function obtained to give optimal point d. Note that additional constraints of the type p = 0 (used in step 1) are not included. Step 3 l

Check whether point d is one of points a, b, or c. (a) If yes, discard point d, record a-b-c as a limit hyperplane, store points a, h, and c in a “hyperplane” stack, and go to step 4. (b) If no, check whether point d is a new point. If it is new, store as a new point; otherwise, use the old point number. Then check whether the optimal value obtained at step 2 is the same as the value of the objective function used. If yes, record the combination in the “hyperplane” stack and go to step 4. If no, form three different hyperplane combinations (i.e., d-a-b, d-a-c, d-b-c). Check each by comparing with the combinations in the “already run” stack to find out whether it has occurred before; store any combination that is not in the “already run” stack in a “to-be-run” stack; otherwise, discard the combination.

Step 4 l

Pick from the “to-be-run” combination stack the next combination to be used. If the stack is empty, go to step 6; otherwise, delete the combination from the stack and continue.

Step 5 l

Retrieve the coordinates of the three points forming the selected combination and go to step 2.

Step 6 l

l

l l

Retrieve limit hyperplanes and corresponding points from the “hyperplane” stack. Postprocess the results to eliminate any points that are not extreme points by, for instance, forming the convex hull of the points for each hyperplane. Plot the limit hypersurfaces. stop.

loading:

F. Tin-Loi and Y. F. Lo

Steps 2 to 5 constitute the general phase of the scheme and are repeated, with provision for backtracking, until all extreme point solutions have been found. While the number of solutions can theoretically be large, the analyses of practical structures can be, from our computational experience, performed easily, especially if the algorithm is fully automated. Step 2 requires the construction of a plane passing through three points. This involves the solution of the system of equations Ax = b

(6)

where matrix A collects row-wise the coordinates of the three points, x is the required vector of coefficients of p for the plane, and b is an arbitrarily specified vector. We use a vector with unit entries for b so that if A is invertible, the objective function value is initially unity. Three cases, which need to be catered for, are possible: (a) for b = 1 the plane does not pass through the origin; (b) the plane passes through the origin and contains two axes (i.e., pj = 0; i = 1, 2, or 3); and (c) the plane passes through the origin and does not contain two axes (e.g., pi + p2 = 0). During any LP solution the algorithm is designed to move the objective hyperplane away from the origin. It can easily be verified that this is satisfied for cases (a) and (b) described in the preceding remark. For case (c) a two-direction search is necessary; this is simply implemented by carrying out two analyses for objective functions with signs of coefficients reversed. Step 6 reveals a deficiency of the proposed method. The algorithm generates vertices in decision or variable space. These vertices, when mapped onto the objective or load space, do not necessarily stay as extreme points. Hence it is possible and, in fact, is common that “redundant” points are obtained when step 6 is reached. The proposal to form a convex hull of points lying on a particular hyperplane attempts to eliminate such redundancies. An alternative scheme, of course, would be to form the convex hull of all points in load space. In practice, even without postprocessing, once the limit hypersurface has been plotted, it is usually clear which points are redundant. Further examples We have implemented the algorithm described as a pilot code for use on personal computers. The package LP8823 was used as the LP solver. In addition to the two examples introduced earlier, a number of other examples have been solved. We report on two of these in this section.

The following remarks concerning the basic framework of the limit surface generation algorithm will further help explain the procedure.

Example

1. Step 1 initializes the procedure by ensuring that the collapse limits corresponding to each of the load axes are first evaluated.

This is an example used by Nafday et a1.5The structure, subjected to three load multipliers, is shown in Figure 7. Under the assumption that bending governs

1: Two-story portal frame

Appl. Math. Modelling,

1992, Vol. 16, September

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Collapse limit surface for multiparametric

loading:

F. Tin-Loi and Y. F. Lo

3%

Table 1.

1

PI -

1

MP

1 3L

2P, 4 IMP

Figure 7.

3L

2MP

2MP

_

2L

Extreme points for example

1

Extreme point

J_

2L

Example 1: two-story

frame

_I

0.833 0.000 0.000 0.333 0.667 0.500 0.500 0.667 0.833 0.833 0.833 0.000

2 3 4 5 7 8 10 12 13 16 48

7.5P2 b25P,

0.000 0.667 0.000 0.667 0.667 0.333 0.000 0.667 0.333 0.333 0.000 0.667

0.000 0.000 0.667 0.667 0.000 0.667 0.667 0.333 0.333 0.000 0.333 0.667

7.5P2

p2

1

---c

MP v3 A

L

Figure 9.

Figure 8.

Limit surface for example

Example 2: simple frame

1

only the formation of plastic hinges, we have obtained the limit hypersurface shown in Figure 8; the extreme points numbering is as generated by the computer program. Table I lists all of these extreme points. Our results do not agree with those reported by Nafday et al.’ It appears that their solution is incorrect, since a check with p2 = 0 and p3 = 0 gives an optimal value for pI of 0.833 instead of 0.888 as quoted by them.

Figure 10.

Table 2.

Limit surfaces for example

Extreme points for example

Extreme point

Example

2: Single bay portal frame

This example illustrates the influence of including axial force (N) effects in estimating the plastic capacity of the members, assumed to be solid rectangular crosssections with depth to width ratios of 2. Figure 9 shows the structure, and Figure 10 gives the limit polygons for both bending only (M) and axial-bending interaction (M + N). A conventional bisymmetric PWL yield locus with nondimensional (N, M) vertices of (0, l),

496

Appl.

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,

1992, Vol. 16, September

1 2 3 5 1’ 2’ 3’ 4’ 5’ 10’

2

2

CL1

/-Q

3.200 0.000 3.200 1.600 3.182 0.000 3.010 3.165 3.182 1.420

0.000 4.000 2.000 4.000 0.000 3.774 1.979 0.370 0.124 3.761

Collapse limit surface for multiparametric (0.5,0.75),

adopted.

7

and (I, 0) in the positive quadrant has been All extreme points are listed in Tubfr 2.

8

Conclusions

9

A simple method designed to generate the global limit hypersurface of skeletal surfaces under the action of combined stresses is presented in this paper. It has intuitive appeal in view of its easy geometrical interpretation. The numerical examples carried out also lend support to its usefulness and accuracy in solving this type of structural plasticity problem. It suffers from the disadvantage that extreme points detected in variable space are not necessarily extreme points in load space, thus making it necessary to postprocess the results. While it is also well known in mathematical programming theory that problems of this type can have a large number of solutions, most practical structural problems have relatively few load parameters and are therefore amenable to solution by the procedure proposed. However, there is still a need to carry out research into means of improving the efficiency of the algorithm, such as by exploiting the special algebraic features of the LP problems that arise from limit analysis.

10

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I3

I4

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References

I6

Cohn, M. Z. and Maier, G.. eds. Engineering Plasticity by Mathematical Programming. Pergamon Press, New York, 1973 Maier. G. and Munro, .I. Mathematical programming applications to engineering plastic analysis. Appl. Mech. Ret,. 1982. 35, 1631-1643 Maier. G. and Lloyd Smith, D. Update to “Mathematical programming applications to engineering plastic analysis” [AMR 35 (1985): 163 I - 16431. ASME, Applied Mechanics Updaie 1986. 377-383 Nafday. A. M., Corotis. R. B., and Cohon, J. L. Multiparametric limit analysis offrames. I: Model. J. Enjirg. Mech. 1988. 114, 377-386 Nafday, A. M., Corotis. R. B., and Cohon, J. L. Multiparametric limit analysis of frames. II: Computations. J. En~r,q. Mech. 1988. 114, 387-403 Symonds, P. S. and Prager, W. Elastic-plastic analysis of structures subjected to loads varying arbitrarily between prescribed limits. J. Appl. Mrch. 1950, 17, 315-323

I7 I8

I9 20

21 22

23

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Lloyd Smith, D. Plastic limit analysis and synthesis of structures by linear programming. Ph.D. Thesis, Univ. of London, 1974 Lin. T. S. and Corotis, R. B. Reliability of ductile systems with random strengths. J. Structural Engrg. 1985, 111, 1306-1325 Steuer, R. E. Multiple Criteria Optimization: Theory, Computation, and Application. John Wiley and Sons, New York, 1986 Casciati, F. and Faravelli, L. Discussion on paper “Nafday, A. M., Corotis, R. B., and Cohort, J. L. Multiparametric limit analysis of frames: Part I-model, Journal ofEngineering Mechanics, American Society of Civil Engineers 114, 377-386, 1988.” J. Engrg. Mech. 1990, 116, 480-481 Casciati, F. and Faravelli, L. La sicurezza strutturale nei confronti degli stati limite descrivibili come problemi di ottimizzazione vincolata. Costruzioni Metalliche 1977, 26, 153-158 Augusti, G., Baratta, A.. and Casciati, F. Probabilistic Methods in Structural En,qineering. Chapman and Hall, London, 1984 Reeves. G. R. and Franz, L. S. A simplified approach to interactive MOLP. Essays and Surveys on Multiple Criteria Dec,ision Making, ed. P. Hansen. Springer-Verlag, New York, 1983, pp. 310-316 Gonzalez, J. J., Reeves, G. R., and Franz, L. S. An interactive procedure for solving multiple objective integer linear programming problems. Decision Making with Multiple Ohjecfitles. ed. Y. Y. Haimes and V. Chankong. Springer-Verlag, New York, 1985. pp. 250-260 Reeves, G. R. and Franz, L. S. A simplified interactive multiple objective linear programming procedure. Comput. Oper. Res. 1985. 12, 589-601 Grierson. D. E. and Abdel-Baset. S. B. Plastic analysis under combined stresses. J. Engrg. Mech. 1977, 103, 837-854 Domaszewski, M. and Borkowski, A. On automatic selection of redundancies. Comput. Sfructures 1979, 10, 577-582 Borkowski, A. Analysis of’Skeleta1 Structural Systems in the Elasfic and Elastic-Plastic Range. Elsevier Scientific Publishers, Amsterdam, 1988 Maier. G. A matrix theory of piecewise linear elastoplasticity with interacting yield planes. Meccanica 1970, 5, 54-66 Zavelani-Rossi. A. A new linear programming approach to limit analysis. Vuriutional Methods in Engineering, Vol. 2. ed. C. A. Brebbia and H. Tottenham. Southampton University Press, 1973. pp. 8.64-8.79 Cohon, J. L. Multiobjective Programming and Planning. Academic Press, New York, 1978 Zeleny, M. Linear Multiobjective Programming. Lecture Notes in Economics and Mathematical Systems, Vol. 95. SpringerVerlag, New York, 1974 LPSR, Linear programming far the IBM PC. Eastern Software Products. Inc. Virginia, USA, 1987

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