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Optimization of slab reinforcement by linear programming
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 12 (1977) 1-17 0 NORTH-HOLLAND PUBLISHING COMPANY
OPTIMIZATION OF SLAB REINFORCEMENT BY LINEAR PROGRAMMING A. BORKOWSKI Institute of Fundamental
Technological Research, Polish Academy of Sciences,
Warsaw, Poland
Manuscript received October 1976 A numerical procedure is developed to compute the minimum volume design for the reinforcement of concrete slabs. The optimum solution ensures a given safety factor against plastic collapse and meets technological requirements. A linear programming approach to the optimization of rigid-plastic structures is briefly outlined, and a dual discrete model of a slab is then introduced that uses finite elements. A linear function is chosen for the deflection rate, while the bending and twisting moments are kept constant throughout the element. The mesh lines are potential yield lines, and the admissibility of stresses is checked along these lines. As a result a kinematic estimate of the optimum reinforcement is obtained. Several examples illustrate the method.
1. Introduction Considerable experience gained in applying the yield-line theory [ 11 to the design of reinforced concrete slabs has established its efficiency and reliability. Despite its purely kinematical nature, the yield-line approach, when properly employed, gives a conservative estimate of the required reinforcement. This is due to effects neglected in the yield-line theory, for instance arching, steel hardening, and membrane action, each of which is favorable to the safety of the design (compare experimental results in [ 141). It is rather a straightforward task to find an adequate collapse mode for a simply shaped, conventionally loaded plate of uniform reinforcement. This is not the case, however, in optimization problems where the plastic properties of a slab are not known in advance. For the sake of safety it is then desirable to begin with a sufficiently wide set of possible collapse mechanisms and to develop a numerical procedure that recognizes the critical one. A way of doing this by linear programming was indicated by Rzhanitsyn [ 21, who considered the load-carrying capacity of plates and shells. Rzhanitsyn’s method was generalized by Chyras and the author in [ 31 where a physical interpretation of duality was given, and both load-carrying capacity and plastic optimization were considered. The linear and nonlinear programming approaches to the theory of rigidplastic structures were presented in [4], where primary concepts of the middle fifties (e.g. [ 5,6]) were generalized and shaped into a concise dual form. The main results of that paper relating to optimization by linear programming are recalled in section 2. The present paper develops a finite element treatment of the method proposed in [ 31. The author was inspired to do this by the papers of Anderheggen and Knopfel [7,8] which treat loadcarrying capacity by linear programming. Finally, it should be mentioned that analytical solutions of optimum reinforcement are available for some cases (see [9]).
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A. Borkowski, Optimizationof slab reinforcement by linear programming
2. Dual approach to optimization
of rigid-plastic structures
The success of a numerical procedure for structural design depends largely upon the proper choice of a discrete model of the structure. This statement also applies to the finite element method, where a certain freedom exists as regards parametric representation. The main advantage of the dual model proposed below is the completeness of the rigid-plastic solution for a discretized structure. A deviation of such a solution from the analytic one (which is usually not available) may be estimated by considering static or kinematic approximations introduced in a discretized system. Let us consider an arbitrary structure divided into finite elements. The stress-strain state of the eth element is described by the stress vector Q, and the strain vector qe (each having 1, components). Displacements of the element are represented by the J,-dimensional vector u,, and external forces are represented by the vector P, of the same dimension. Components of these vectors are generalized variables in the sense that the scalar product Q: Q, gives the internal virtual work, and the scalar product ui P, equals the external virtual work. Under the assumption of small displacements, the strain-displacement relation is linear,
(le=Ae u,
(1)
Y
and the principle of virtual work yields the dual (adjoint) equation
f’, =A:
Q,
,
which expresses the equilibrium of the element. The entries of the (I, X J,)-matrix A, depend only upon the configuration of the element and the shape functions that are used for the parametrization of stresses and displacements. Eqs. (1) and (2) are independent of the constitutive law and are thus still valid for the rates G,, ic, used in the description of a rigid-plastic behavior, provided small displacements and rotations may be assumed. Aiming at numerical efficiency, we restrict our model to linear programming. This requires a piecewise-linear approximation of the yield surface. After this linearization the constitutive equations for a generic element may be written as follows: G: C, -B:
Q, 20,
(3)
c,>o, 4, =B, ke > 0
(5)
1, ,
(6)
)
k;(G; C, -B;
Q,)= 0
.
(7)
Here C, is the &-vector of plastic moduli, 1, is the N,-vector of plastic multipliers. Entries of the (1, X N,)-matrix B, and the (K, X N,)-matrix G, depend upon the shape of the yield polyhedron. Ineqs. (3), (4) are the stress-admissibility conditions, whereas relations (5), (6), (7) express the
3
A. Borkowski, Optimizationof slab reinforcement by linear programming
associated flow rule. Since we are interested in the optimization of a structure, the notion of cost should be introduced. Let the cost of a generic element be a linear function of the plastic moduli:
v, = A; c,
(8)
(called henceforth the cost function). The components of the &-vector Ae are the assigned cost factors (prices). The smaller the value of V, (cost, weight, volume), the better is the element. Since relations (l)-(7) completely describe the behavior of an element, one can proceed to the assemblage of a structure. Omitting details of this step, we simply state that it results in global vectors Q, 4, P, ti, C, A, 1 and global matrices A, B, G. The behavior of the assembled structure is described by the set of relations (l)-(7) with subscripts e omitted. Boundary conditions usually reduce the number of unknown parameters, and the global dimensions are: I < E 1, for Q and 4, J Q E J, for P and ir, N < EN, for A, where E denotes the number of elements. The dimension K of the global vectors C and A is usually much less than E K, on account of the regularity of the structure. We now are in a position to define a plastic optimization problem. Dealing with the proportional loading P = pOP’, we assume that both the service load P’ and the safety factor p,, are prescribed. Hence the required load-carying capacity P is known. The aim is to find an optimum distribution of plastic moduli C’ that minimizes the cost function V = A’C and ensures that a safety factor against plastic collapse be not smaller than po. It may be proved [4] that such an optimum design corresponds to the solution of the -following pair of dual linear programming problems: Primal problem:
minimize:
V= A’C,
subject to:
A’Q = P ,
(9’) G’C-B’Q>
o,
cao; Dual problem :
maximize:
ci/=P’ir,
subject to:
Air - B1= o ,
(9”)
The primal problem (9’) reflects a static approach: the optimum is selected among statically admissible (safe) designs. The dual problem (9”) expresses a kinematic approach: the external power p is maximized for compatible ir and 1. Note that the prescribed cost factors Ak (k = 1, 2, ... K) appear in the dual problem as upper bounds for plastic multipliers and hence for the strain rates (compare the kinematic optimality criterion in the analytic approach [ 91). If Cl > 0, it follows from the complementary slackness rule that G,,hz = Ak (summed over n = 1, 2, . . . N).
4
A. Borkowski, Optimizationof slab reinforcement by linear programming
Otherwise this constraint will remain passive for the optimum solution. For a realistic design the requirement C Z o is insufficient because it allows plastic moduli to vanish as well as to assume unacceptably large values. It is desirable therefore to bound the values of plastic moduli: C- < C d Cf. The bounds CL, Ci should be adequately chosen to avoid nonexistence of solution (obviously 0 < C, < Cl). Taking into account the above considerations, the following final version of the op~mization problem is obtained: minimized
v= AC,
subject to:
At Q = P ,
(10’) G'C-B"Q>
o,
c-,
maximize:
l$ =p’G + c-‘Z-
subjectto:
Air-B&=0,
)
(10”)
Gk+Z--z+= A> 0,
- C+‘Z+
A,
z-20,
z+>0.
The variables 2; and 2: correspond to the constraints C, 2 Ci and C, Q Ci. If C; < Ci < Ckf, then 2, * = 2:. = 0 and G,,hi = A*. Otherwise, the difference between G,,hi and A., is equal to 2;; * (when Ci = CL) or to Zi* (when Ci = Ck+). Prior to optimization, J linearly dependent components of Q should be eliminated by means of the equilibrium equations. This elimination is performed by the computer and corresponds to the automatic choice of redundancies.
3. Finite element of a reinforced slab A triangular element suitable for plastic optimization of slabs is introduced. Fig. 1 shows its geometric characteristics and local coordinate axes parallel to the directions of reinforcement. The rate of deflection &(x, v) is assumed to be linear, while the moments &&, M,, MX,, are constant over the element. Lines of possible slope discontinuities are separated from the boundary of the element by narrow rigid strips called connectors (fig. 2). This device is necessary because of the possible discontinuity of plastic properties for adjacent elements. The collapse mode shown in fig. 2 has six degrees o_ffreedom: three nodal deflection rates I/i)g and three rates of rotation $, of the connectors (g =: 1,2,3 throughout the paper). Due to the allowed jumps in slope the rates $, are independent of the rates of rotation 8, of the rigid inner triangle. Let
A. Borkowski, Optimizationof slab reinforcement by linear programming
5
(11) where
Thus an external loading has to be represented by the vector p, =
{P,, p,, p,, x,3 x,9 x,1
(13)
(collecting the nodal forces Pg and the couples X8 attached to the unit length of each connector). Any other kind of loading should be reduced to these six resultants by equivalence of virtual work.
Fig. 1. Triangular finite element. 2
-----_ A-A
r----
Fig. 2. Plastic collapse mode for element.
I
6
A. Borkowski, Optimizationof slab reinforcement by linear programming
For example a constant transversal pressure p is equivalent to Xg=O,
p* = PSI39
(14)
where S denotes the area of a triangle. Internal work is done only along the lines of discontinuity of strain rates
of the rate of slope. Let the vector
(15) have components
proportional
to the net rates of rotation in the plastic hinges:
8; = (8, + &)lg .
(16)
The entries of the dual vector
Q, = W,, M,, &I
,
(17)
are then the normal bending moments acting across the lines of possible hinges. These three parameters uniquely describe the stress state of the element because the constant moments M,, M,, M,, are in one-to-one correspondence with them: MS = M, sin2flg + MY cos2& - 2 MXYsin fig cos fig .
(18)
Thus, all four dual vectors P,, ir,, Q, and 4, are defined. Fig. 3 shows positive directions of their components. Expressing the rates of slope 8, in terms of the nodal rates of deflection G.‘sand
I
-MA -
\
Xi%
b) . Fig. 3. Positive directions of: a) nodal forces Ps and displacement rates WP b) external couples Xs and rotation rates $b, normal a)
bending moments MS and strain rates lb.
A. Borkowski, Optimizationof slab reinforcement by linear programming
I
taking ( 16) into account, we obtain -cc2
+ c3>
A, =
c3 -cc1
c3 c2
c2
+c31
Cl
(19)
Cl -(cl
+ c2>
0
0
1
where cg = cot ixg .
(20)
An alternative way is to calculate A’, from the virtual work equations [ 71. Let us now consider the plastic properties of the element. We assume that the cross-sectional areas of the steel rods, f,, f,, (bottom grid) and fi, f; (top grid) (measured per unit length) are constant for the element. Therefore, four principal yield moments completely describe its plastic properties, and
c, = {m,,my,
m:,m;1 .
(21)
When specifying the stress admissibility condition (3), we are restricted by the assumed collapse mode: yield lines may develop only along the sides of the element. Hence it would be inconsistent to use the generalized Johansen’s criterion:
mx-M,>O, my-M,>O,
(%Z-M,Hm,-MJ-M;,>O, m;+M,>O, m;+M,>O, (m:+M,)(m;+M,)-M,Z,>O, because then yielding may occur along any direction. Hence an approximate yield condition is applied: the absolute value of Mg should not exceed the appropriate yield moment mg or rnb calculated for the direction of the gth line. This leads to six linear inequalities for each element: m, sin2flg
+m,, cos2/3,-MS a 0, (23)
m’ sin2/3, + m’ cos2/3, + MS 2 0 . They fit into the matrix form (3) after the following specification: B, = 11,
-1313x6
,
(24)
A. Borkowski, Optimizationof slab reinforcement by linear programming
8
where s = {sin’&,
sin2P2, sin2p,),
c = Icos2pI,
cosZp2, cos2&} .
The last remaining quantity that needs an explanation is the vector A,. Let the cost be proportional to the volume of the reinforcing steel. Each element contributes to this volume by
ve=Xf,
+fy +f; ‘fJ
*
(25)
The cross-sectional areas of the reinforcement formulas
are related to the yield moments by the known
fx= m, /(ox zx) etc.,
(26)
where u denotes the yield stress of steel, and z is the arm of the stress couple in a cross-section. Let the thickness of the slab h and the yield stress u be chosen prior to the optimization. It is common practice to neglect the dependence of z on f,taking e.g. z = const = 0.9 h. Under such circumstances the volume V, depends linearly on the yield moments: V, =a, m, +ay my +a:
rn:+a;
m;,
(27)
where a, = l/(0, zX) etc.
(28)
Comparing this expression with (8) and (.21), we conclude that Ae = Sia,, ay,
a:,a;).
(29)
For a preliminary design the cost factors may be taken proportional to S. Under certain circumstances less than four plastic moduli are subject to optimization. These are the cases of preassigned orthotropy or asymmetry of the top and bottom reinforcement. Let the factors of orthotropy P =
mylmx,
cc'=m;/rnk
(30)
be given. Then the optimum values of m, and rn:may be calculated using
Ae= S{(a,
c; =
+ pa,),
(s + PC)
[
03
62: + da;)) , 03
(s+Er’d
1
6x2
*
(31)
9
A. Borkowski, Optimizationof slab reinforcement by linear programming
If the factors of asymmetry 77, =
m:lm,,
9, =
(32)
m;lmy,
are fixed beforehand (e.g. n, = r), = 0 implies the absence of the top reinforcement),
C, = h,,
then
myI , (33)
GJ =
Finally, in the case that both 1~1 and q are preassigned, the optimization parameter per element; hence
can affect only one
Ce=m,, (34)
G,t= {(s + PC), (q,s + rly P’c)) . 4. Assemblage of elements into a slab In the previous section each element was supposed to have an individual reinforcement. Contrary to this, a rational technology requires the minimum number of reinforcing grids, each of them having a’constant mesh and constant diameters of parallel steel rods. Bearing this in mind, the slab is divided prior to optimization, into areas of constant yield moments. In the following, areas of this kind are called regions. Theoretically, a region may include a single element, but usually it will include many elements. A finite element network is adjusted in such a manner that boundaries of regions do not intersect the elements (fig. 4). Inside each region connectors are cancelled and adjacent elements fit to each other simply by the continuity of deflection rates. Connectors are retained however, along the boundaries of regions where a jump of the yield moment should be covered by a continuous slope. Components of the network are identified by means of the following global labels (fig. 4): elements regions lines nodes connectors
e=l,2 , . . . E, r = 1, 2, .. . R, i = 1, 2, .. . I, j=l,2 ) . . . J’, j = J’ + 1, J’ + 2, . . . J.
Unsupported nodes and potential plastic hinges only are taken into account (i.e. lines coinciding with free or simply supported edges of the slab are excluded).
10
A. Borkowski, Optimizationof slab reinforcement by linear programming
Fig. 4. Finite element network and global labels of elements (e), regions (r), lines (I), nodes (11 and connectors (11.
g Tf-
__------
--T--
I
\
&
i+l
?
i
Fig. 5. Conformity at: a) interelemental
boundaries, b) interregional boundaries.
qi
A. Borkowski, Optimizationof slab reinforcement by linear programming
11
The first J'components of the global vectors P and ic are, respectively, the nodal forces P,. and the nodal deflection rates kj. The remaining components are the couples Xi and the rotation rates 4;. Entries of Q and 4 are the bending moments Mi and the strain rates bi. For the ith line situated inside the region (fig. 5a) Mi =M,:
=Mf
(35)
and
e;=(8; + e;>li
)
(36)
i.e. two adjacent elements contribute to the ith global component. The situation is different at the boundary of the region (fig. 5b) where each element retains its independent component. The global vectors C and A include R subvectors C, and A,, each of which corresponds to a region. The global dimension K depends upon the type of optimization problem: K = 4E for free optimization, K = 2E for prescribed orthotropy or asymmetry, and K = E when P and q are fixed simultaneously. A computer routine generating global matrices and vectors requires information about: a) geometry and topology of the network: coordinates of nodes, correspondence between local and global labels, division into regions; b) loading: transversal pressure, concentrated loads etc.; c) plastic properties: yield stress of reinforcing rods, orthotropy and asymmetry factors etc. The global matrices A, B,G and the vectors P and A are built by subsequent accumulation of the contributions of individual elements (as in the elastic analysis by finite elements [ lo]). Regular nets are preferable because a mesh generator then allows the minimization of input data. A solution of problem (10) may be obtained by the simplex routine that is often included in computer software. The components of Q* are of minor interest because of their multivaluedness in the rigid parts of the slab; C’ gives the required optimum design, and the dual zi’ describes the collapse mechanism.
5. Discussion Despite the completeness of the solution for a discrete model of the slab, such a solution remains a kinematic estimate. It is clearly seen that all kinematic requirements of the rigid-plastic approach are satisfied: the dissipation and the external power are calculated exactly, and the collapse mechanism satisfies the boundary conditions. On the other hand, a piecewise-linear stress field can not be in local (differential) equilibrium with a distributed load. Moreover, the stress admissibility condition (23) has an approximate character. An alternative way would be to try a purely static approach with, say, quadratic shape functions for stresses and the nonlinear condition (22). This leads to a nonlinear programming problem, a solution of which is too cumbersome for practical purposes [ 11 I. A compromise was proposed in [ 7,81: piecewise-constant and piecewise-linear approximations of M,,M,,M,, combined with linear shape functions for G. In order to keep the problem one of linear programming, Wolfensberger’s admissibility criterion was introduced [ 121:
A. Borkowski, Optimizationof slab reinforcement by linear programming
12
mx-M,*M,,,>O, m,-M,-+M,,>O, (37) m:+M,kM,,,>O, m;+M,,kM,,>O. These eight inequalities correspond to the octahedrical domain inscribed into the elliptic cones of Johansen’s yield surface. This kind of approach gives no upper or lower bounds for the solution: a local equilibrium for distributed load is not maintained and the flow rule is obeyed only in the mean. Let us specify (23) for the regular mesh shown in fig. 6. In terms of M,, M,, M,, it reads:
m,-M,>O, (m,+m,)-(M,+M,+2M,,)>O, (38)
m;+M,>O,
The comparison of the yield surfaces corresponding to (22) (37) and (38) is given in fig. 7 in the case of isotropy when m, = m,,= m. The yield surface (38) is circumscribed about Johansen’s.
Fig. 6. Regular finite element network.
A. Borkowski, Optimizationof slab reinforcement by linear programming
13
Fig. 7. Comparison of stress admissibility conditions (22), (37) and (38).
However, the overestimation of strength is not essential. Investigations carried out for other nets showed that acute angles (Yand 0 should be avoided for the sake of accuracy.
6. Numerical examples The above theoretical considerations were implemented in an ALGOL code PROLIN-1 written for a computer ODRA 1204 (core memory - 16 K, speed - 60,000 additions/set). Typical examples solved by this routine are given below. A detailed description of PROLIN-1 may be found in [13]: 6.1. The first example concerns a rectangular slab with three edges simply supported and the<’ fourth free. Fig. 8 shows a solution for isotropic bottom reinforcement (p = 1, r) = 0) which required 70 set of CP time. The yield moment obtained, m = 0.1916 pL2 ,
(3%
will be a reference value for other solutions. Note that the co’mmonly used mechanism - a single positive hinge spreading from each corner towards the free edge - gives [ 14, p. 3201 m=0.1802pL2.
(40)
Therefore, even for such a simple case an intuitive choice of the collapse mechanism may lead to a 6% underestimation of the required reinforcement. The first step towards an optimum design is optimization of the orthotropy factor cc. The
A. Borkowski, Optimizationof slab reinforcement by linear progmmming
14
-
free
supported
-
free
edge
edge
-
positive
hinge
____w_
negative
hinge
Fig. 8. Solution of isotropic rectangular slab with a free edge.
solution is presented in fig. 9 (all optimum solutions were calculated under A, = S,). The optimum values are m; = 0.3000 pLz = 1.5658 m , (41)
m; = 0.0500 pL* = 0.2610 m , M*
=0.1667.
The volume of the reinforcement isotropic case.
(proportional
Fig. 9. Optimum orthotropy
to v) for this design is 8% less compared to the
of rectangular slab with a free edge.
A. Borkowski, Optimization of slab reinforcement by linear programming
I
q5L
_)_
0.5L
15
J
Fig. 10. Division of slab into regions for trial optimizations.
Further decrease of V requires more freedom in the reinforcement pattern. Several possibilities were investigated (fig. 10): 1) two reinforcing grids, regions 1 + 3 and 2 + 4; 2) two reinforcing grids, regions 1 + 4 and 2 + 3; 3) two reinforceing grids, regions 1 and 2 + 3 + 4; 4) two reinforcing grids, regions 1 + 2 + 3 and 4; 5) three reinforcing grids, regions 1, 2 + 3 and 4; 6) four reinforcing grids, regions 1, 2, 3 and 4. A free optimization of the 6th patterm reduced V up to 21%. However, the most suitable from the practical point of view seems to be the first pattern. Under the constraint m min > O.OlOOpL’
= 0.0522
m
it still reduces the volume by 13%. This solution gives (fig. 11):
Fig. 11. Optimum solution for regions 1 + 3 and 2 + 4 under constraint (42).
(42)
16
A. Borkowski, Optimizationof slab reinforcement by linear programming
m :1 =0.3650pL2
= 1.9050m,
rnGI = 0.0100 pL* = 0.0522 m ,
(43) m l2 = 0.2817 pL* = 1.4702 m , rnG2 = O.OlOOpL* = 0.0522 m .
The intuitively promising patterns 3 and 4 resulted in 7% and 10% reduction, respectively. 6.2. The second example shows an influence of boundary conditions on the optimum solution. A simply supported rectangular slab has a cut-out in a corner (fig. 12a). The solution for isotropic bottom reinforcement yields m = 0.0260 pL*
(44)
with the cohapse mechanism shown in fig. 12b. The optimal orthotropy rn:
=
corresponds to (fig. 13):
0.6667 X lo-* pL* = 0.2568 m ,
rn; = 3.8519 x lo-* pL* = 1.4835 m
,
(45)
/J* = 5.7778. This design uses 13% less steel than the isotropic one. Note that, contrary to the case of a continuously supported rectangular plate, the stronger reinforcement is parallel to the shorter side of the slab. Additional calculations rejected the possibility of more economical and still technologically acceptable reinforcement patterns for this slab. a)
-
b)
t .
m =0.0260 pL2
L
0.6L
au
Fig. 12. Solution of isotropic rectangular slab with cutout: a) scheme of slab and yield moment, b) collapse mechanism.
A. Borkowski, Optimizationof slab reinforcement by linear programming a)
17
d
Fig. 13. Optimum orthotropy
of rectangular slab with cutout: a) yield moments, b) collapse mechanism
7, Conclusions A yield-line method combined with computers opens interesting possibilities for the rational design of slabs. The computational effectiveness of the algorithm makes it possible to run several trial optimizations (e.g. for different configurations of regions) in a kind of interactive design. Having performed a preliminary plastic optimization, one may choose a technologically acceptable reinforcement grid that produces yield moments as close as possible to their theoretically optimum values. Then additional requirements, for instance those of sufficient stiffness under service load, crack resistance, should be taken into account - this may require a modification of reinforcement. A final design may be recalculated by the rigid-plastic finite element method in order to establish its actual safety factor against plastic collapse.
References ]l] K.W. Johansen, Yield-line theory (Cement and Concrete Assoc., London, 1962). [Z] A.R. Rhzanitsyn, Kinematic limit analysis of sheHs by linear pro~amming [in Russian], in: Proc. 6th All-Union Conf. on Plates and Shells (Nauka, Moscow, 1968) 656-665. [ 31 A.E. Borkauskas * and A.A. Chyras, On the duality in limit analysis and design of plates, Bull. Acad. Polonaise des Sciences (ser. sci. tech.) 16 (1968) 241-247. [4] A.E. Borkauskas * and A.A. Chyras, Duality in non-linear problems of plastic analysis and design (in Russian], Litovskil Mekhanitscheskir Sbornik (Lithuanian Mechanical Review) no. 3(2) (1968) 55-67. [5] J. Foulkes, Linear programming and structural design, in: Proc. 2nd Symp. (Bureau of Standards, U.S. Dept. of Commerce, Washington, 1955) V. 1, No. 1, pp. 27-29. [6] W. Prager, Linear programming and structural design, RAND Corp. Research Memorial, RM-2021 (1957). [7] E. Anderheggen and H. Knopfel, Finite element limit analysis using linear programming, Int. J. Sols. Structs. 8 (1972) 1413-1431. [8] H. Knijpfel, Berechnung Starr-plastischer Platten mittels fmiter Elemente (Institut ftir Baustatik, ETH, Ziirich, no. 47,1973). [P] W. Prager, Introduction to structural optimization, CISM Courses and Lectures, No. 212, Udine, 1974 (Springer, Vienna, 1974). [lo] J.H. Argyris, Recent advances in matrix methods of structural analysis, in: D. Ktichemann and L.H.G. Sterne (ed.) Progress in Aeronautical Sciences 4 (Pergamon Press, Oxford, 1964). [ll ] P.G. Hodge and T. Belytschko, Numerical methods for the limit analysis of plates, J. Appl. Mech. 35 (1968) 796-802. [12] R. Wolfensberger, Traglast und optimale Bemessung von Platten (Wildegg, Zurich, 1964). [13] A. Borkowski, Optimum design of orthotropic rigid-plastic slab [in Polish], Archiwum Iniyniern Ladowej (Civil Engineering Archives) 21 (1975) 15-28. [14] A. Sawczuk and Th. Jaeger, Grenzt~gf~higkeits-Theorie der Platten (Springer, Berhn, 1963). * Present name: A. Borkowski.
OPTIMIZATION OF SLAB REINFORCEMENT BY LINEAR PROGRAMMING A. BORKOWSKI Institute of Fundamental
Technological Research, Polish Academy of Sciences,
Warsaw, Poland
Manuscript received October 1976 A numerical procedure is developed to compute the minimum volume design for the reinforcement of concrete slabs. The optimum solution ensures a given safety factor against plastic collapse and meets technological requirements. A linear programming approach to the optimization of rigid-plastic structures is briefly outlined, and a dual discrete model of a slab is then introduced that uses finite elements. A linear function is chosen for the deflection rate, while the bending and twisting moments are kept constant throughout the element. The mesh lines are potential yield lines, and the admissibility of stresses is checked along these lines. As a result a kinematic estimate of the optimum reinforcement is obtained. Several examples illustrate the method.
1. Introduction Considerable experience gained in applying the yield-line theory [ 11 to the design of reinforced concrete slabs has established its efficiency and reliability. Despite its purely kinematical nature, the yield-line approach, when properly employed, gives a conservative estimate of the required reinforcement. This is due to effects neglected in the yield-line theory, for instance arching, steel hardening, and membrane action, each of which is favorable to the safety of the design (compare experimental results in [ 141). It is rather a straightforward task to find an adequate collapse mode for a simply shaped, conventionally loaded plate of uniform reinforcement. This is not the case, however, in optimization problems where the plastic properties of a slab are not known in advance. For the sake of safety it is then desirable to begin with a sufficiently wide set of possible collapse mechanisms and to develop a numerical procedure that recognizes the critical one. A way of doing this by linear programming was indicated by Rzhanitsyn [ 21, who considered the load-carrying capacity of plates and shells. Rzhanitsyn’s method was generalized by Chyras and the author in [ 31 where a physical interpretation of duality was given, and both load-carrying capacity and plastic optimization were considered. The linear and nonlinear programming approaches to the theory of rigidplastic structures were presented in [4], where primary concepts of the middle fifties (e.g. [ 5,6]) were generalized and shaped into a concise dual form. The main results of that paper relating to optimization by linear programming are recalled in section 2. The present paper develops a finite element treatment of the method proposed in [ 31. The author was inspired to do this by the papers of Anderheggen and Knopfel [7,8] which treat loadcarrying capacity by linear programming. Finally, it should be mentioned that analytical solutions of optimum reinforcement are available for some cases (see [9]).
2
A. Borkowski, Optimizationof slab reinforcement by linear programming
2. Dual approach to optimization
of rigid-plastic structures
The success of a numerical procedure for structural design depends largely upon the proper choice of a discrete model of the structure. This statement also applies to the finite element method, where a certain freedom exists as regards parametric representation. The main advantage of the dual model proposed below is the completeness of the rigid-plastic solution for a discretized structure. A deviation of such a solution from the analytic one (which is usually not available) may be estimated by considering static or kinematic approximations introduced in a discretized system. Let us consider an arbitrary structure divided into finite elements. The stress-strain state of the eth element is described by the stress vector Q, and the strain vector qe (each having 1, components). Displacements of the element are represented by the J,-dimensional vector u,, and external forces are represented by the vector P, of the same dimension. Components of these vectors are generalized variables in the sense that the scalar product Q: Q, gives the internal virtual work, and the scalar product ui P, equals the external virtual work. Under the assumption of small displacements, the strain-displacement relation is linear,
(le=Ae u,
(1)
Y
and the principle of virtual work yields the dual (adjoint) equation
f’, =A:
Q,
,
which expresses the equilibrium of the element. The entries of the (I, X J,)-matrix A, depend only upon the configuration of the element and the shape functions that are used for the parametrization of stresses and displacements. Eqs. (1) and (2) are independent of the constitutive law and are thus still valid for the rates G,, ic, used in the description of a rigid-plastic behavior, provided small displacements and rotations may be assumed. Aiming at numerical efficiency, we restrict our model to linear programming. This requires a piecewise-linear approximation of the yield surface. After this linearization the constitutive equations for a generic element may be written as follows: G: C, -B:
Q, 20,
(3)
c,>o, 4, =B, ke > 0
(5)
1, ,
(6)
)
k;(G; C, -B;
Q,)= 0
.
(7)
Here C, is the &-vector of plastic moduli, 1, is the N,-vector of plastic multipliers. Entries of the (1, X N,)-matrix B, and the (K, X N,)-matrix G, depend upon the shape of the yield polyhedron. Ineqs. (3), (4) are the stress-admissibility conditions, whereas relations (5), (6), (7) express the
3
A. Borkowski, Optimizationof slab reinforcement by linear programming
associated flow rule. Since we are interested in the optimization of a structure, the notion of cost should be introduced. Let the cost of a generic element be a linear function of the plastic moduli:
v, = A; c,
(8)
(called henceforth the cost function). The components of the &-vector Ae are the assigned cost factors (prices). The smaller the value of V, (cost, weight, volume), the better is the element. Since relations (l)-(7) completely describe the behavior of an element, one can proceed to the assemblage of a structure. Omitting details of this step, we simply state that it results in global vectors Q, 4, P, ti, C, A, 1 and global matrices A, B, G. The behavior of the assembled structure is described by the set of relations (l)-(7) with subscripts e omitted. Boundary conditions usually reduce the number of unknown parameters, and the global dimensions are: I < E 1, for Q and 4, J Q E J, for P and ir, N < EN, for A, where E denotes the number of elements. The dimension K of the global vectors C and A is usually much less than E K, on account of the regularity of the structure. We now are in a position to define a plastic optimization problem. Dealing with the proportional loading P = pOP’, we assume that both the service load P’ and the safety factor p,, are prescribed. Hence the required load-carying capacity P is known. The aim is to find an optimum distribution of plastic moduli C’ that minimizes the cost function V = A’C and ensures that a safety factor against plastic collapse be not smaller than po. It may be proved [4] that such an optimum design corresponds to the solution of the -following pair of dual linear programming problems: Primal problem:
minimize:
V= A’C,
subject to:
A’Q = P ,
(9’) G’C-B’Q>
o,
cao; Dual problem :
maximize:
ci/=P’ir,
subject to:
Air - B1= o ,
(9”)
The primal problem (9’) reflects a static approach: the optimum is selected among statically admissible (safe) designs. The dual problem (9”) expresses a kinematic approach: the external power p is maximized for compatible ir and 1. Note that the prescribed cost factors Ak (k = 1, 2, ... K) appear in the dual problem as upper bounds for plastic multipliers and hence for the strain rates (compare the kinematic optimality criterion in the analytic approach [ 91). If Cl > 0, it follows from the complementary slackness rule that G,,hz = Ak (summed over n = 1, 2, . . . N).
4
A. Borkowski, Optimizationof slab reinforcement by linear programming
Otherwise this constraint will remain passive for the optimum solution. For a realistic design the requirement C Z o is insufficient because it allows plastic moduli to vanish as well as to assume unacceptably large values. It is desirable therefore to bound the values of plastic moduli: C- < C d Cf. The bounds CL, Ci should be adequately chosen to avoid nonexistence of solution (obviously 0 < C, < Cl). Taking into account the above considerations, the following final version of the op~mization problem is obtained: minimized
v= AC,
subject to:
At Q = P ,
(10’) G'C-B"Q>
o,
c-,
maximize:
l$ =p’G + c-‘Z-
subjectto:
Air-B&=0,
)
(10”)
Gk+Z--z+= A> 0,
- C+‘Z+
A,
z-20,
z+>0.
The variables 2; and 2: correspond to the constraints C, 2 Ci and C, Q Ci. If C; < Ci < Ckf, then 2, * = 2:. = 0 and G,,hi = A*. Otherwise, the difference between G,,hi and A., is equal to 2;; * (when Ci = CL) or to Zi* (when Ci = Ck+). Prior to optimization, J linearly dependent components of Q should be eliminated by means of the equilibrium equations. This elimination is performed by the computer and corresponds to the automatic choice of redundancies.
3. Finite element of a reinforced slab A triangular element suitable for plastic optimization of slabs is introduced. Fig. 1 shows its geometric characteristics and local coordinate axes parallel to the directions of reinforcement. The rate of deflection &(x, v) is assumed to be linear, while the moments &&, M,, MX,, are constant over the element. Lines of possible slope discontinuities are separated from the boundary of the element by narrow rigid strips called connectors (fig. 2). This device is necessary because of the possible discontinuity of plastic properties for adjacent elements. The collapse mode shown in fig. 2 has six degrees o_ffreedom: three nodal deflection rates I/i)g and three rates of rotation $, of the connectors (g =: 1,2,3 throughout the paper). Due to the allowed jumps in slope the rates $, are independent of the rates of rotation 8, of the rigid inner triangle. Let
A. Borkowski, Optimizationof slab reinforcement by linear programming
5
(11) where
Thus an external loading has to be represented by the vector p, =
{P,, p,, p,, x,3 x,9 x,1
(13)
(collecting the nodal forces Pg and the couples X8 attached to the unit length of each connector). Any other kind of loading should be reduced to these six resultants by equivalence of virtual work.
Fig. 1. Triangular finite element. 2
-----_ A-A
r----
Fig. 2. Plastic collapse mode for element.
I
6
A. Borkowski, Optimizationof slab reinforcement by linear programming
For example a constant transversal pressure p is equivalent to Xg=O,
p* = PSI39
(14)
where S denotes the area of a triangle. Internal work is done only along the lines of discontinuity of strain rates
of the rate of slope. Let the vector
(15) have components
proportional
to the net rates of rotation in the plastic hinges:
8; = (8, + &)lg .
(16)
The entries of the dual vector
Q, = W,, M,, &I
,
(17)
are then the normal bending moments acting across the lines of possible hinges. These three parameters uniquely describe the stress state of the element because the constant moments M,, M,, M,, are in one-to-one correspondence with them: MS = M, sin2flg + MY cos2& - 2 MXYsin fig cos fig .
(18)
Thus, all four dual vectors P,, ir,, Q, and 4, are defined. Fig. 3 shows positive directions of their components. Expressing the rates of slope 8, in terms of the nodal rates of deflection G.‘sand
I
-MA -
\
Xi%
b) . Fig. 3. Positive directions of: a) nodal forces Ps and displacement rates WP b) external couples Xs and rotation rates $b, normal a)
bending moments MS and strain rates lb.
A. Borkowski, Optimizationof slab reinforcement by linear programming
I
taking ( 16) into account, we obtain -cc2
+ c3>
A, =
c3 -cc1
c3 c2
c2
+c31
Cl
(19)
Cl -(cl
+ c2>
0
0
1
where cg = cot ixg .
(20)
An alternative way is to calculate A’, from the virtual work equations [ 71. Let us now consider the plastic properties of the element. We assume that the cross-sectional areas of the steel rods, f,, f,, (bottom grid) and fi, f; (top grid) (measured per unit length) are constant for the element. Therefore, four principal yield moments completely describe its plastic properties, and
c, = {m,,my,
m:,m;1 .
(21)
When specifying the stress admissibility condition (3), we are restricted by the assumed collapse mode: yield lines may develop only along the sides of the element. Hence it would be inconsistent to use the generalized Johansen’s criterion:
mx-M,>O, my-M,>O,
(%Z-M,Hm,-MJ-M;,>O, m;+M,>O, m;+M,>O, (m:+M,)(m;+M,)-M,Z,>O, because then yielding may occur along any direction. Hence an approximate yield condition is applied: the absolute value of Mg should not exceed the appropriate yield moment mg or rnb calculated for the direction of the gth line. This leads to six linear inequalities for each element: m, sin2flg
+m,, cos2/3,-MS a 0, (23)
m’ sin2/3, + m’ cos2/3, + MS 2 0 . They fit into the matrix form (3) after the following specification: B, = 11,
-1313x6
,
(24)
A. Borkowski, Optimizationof slab reinforcement by linear programming
8
where s = {sin’&,
sin2P2, sin2p,),
c = Icos2pI,
cosZp2, cos2&} .
The last remaining quantity that needs an explanation is the vector A,. Let the cost be proportional to the volume of the reinforcing steel. Each element contributes to this volume by
ve=Xf,
+fy +f; ‘fJ
*
(25)
The cross-sectional areas of the reinforcement formulas
are related to the yield moments by the known
fx= m, /(ox zx) etc.,
(26)
where u denotes the yield stress of steel, and z is the arm of the stress couple in a cross-section. Let the thickness of the slab h and the yield stress u be chosen prior to the optimization. It is common practice to neglect the dependence of z on f,taking e.g. z = const = 0.9 h. Under such circumstances the volume V, depends linearly on the yield moments: V, =a, m, +ay my +a:
rn:+a;
m;,
(27)
where a, = l/(0, zX) etc.
(28)
Comparing this expression with (8) and (.21), we conclude that Ae = Sia,, ay,
a:,a;).
(29)
For a preliminary design the cost factors may be taken proportional to S. Under certain circumstances less than four plastic moduli are subject to optimization. These are the cases of preassigned orthotropy or asymmetry of the top and bottom reinforcement. Let the factors of orthotropy P =
mylmx,
cc'=m;/rnk
(30)
be given. Then the optimum values of m, and rn:may be calculated using
Ae= S{(a,
c; =
+ pa,),
(s + PC)
[
03
62: + da;)) , 03
(s+Er’d
1
6x2
*
(31)
9
A. Borkowski, Optimizationof slab reinforcement by linear programming
If the factors of asymmetry 77, =
m:lm,,
9, =
(32)
m;lmy,
are fixed beforehand (e.g. n, = r), = 0 implies the absence of the top reinforcement),
C, = h,,
then
myI , (33)
GJ =
Finally, in the case that both 1~1 and q are preassigned, the optimization parameter per element; hence
can affect only one
Ce=m,, (34)
G,t= {(s + PC), (q,s + rly P’c)) . 4. Assemblage of elements into a slab In the previous section each element was supposed to have an individual reinforcement. Contrary to this, a rational technology requires the minimum number of reinforcing grids, each of them having a’constant mesh and constant diameters of parallel steel rods. Bearing this in mind, the slab is divided prior to optimization, into areas of constant yield moments. In the following, areas of this kind are called regions. Theoretically, a region may include a single element, but usually it will include many elements. A finite element network is adjusted in such a manner that boundaries of regions do not intersect the elements (fig. 4). Inside each region connectors are cancelled and adjacent elements fit to each other simply by the continuity of deflection rates. Connectors are retained however, along the boundaries of regions where a jump of the yield moment should be covered by a continuous slope. Components of the network are identified by means of the following global labels (fig. 4): elements regions lines nodes connectors
e=l,2 , . . . E, r = 1, 2, .. . R, i = 1, 2, .. . I, j=l,2 ) . . . J’, j = J’ + 1, J’ + 2, . . . J.
Unsupported nodes and potential plastic hinges only are taken into account (i.e. lines coinciding with free or simply supported edges of the slab are excluded).
10
A. Borkowski, Optimizationof slab reinforcement by linear programming
Fig. 4. Finite element network and global labels of elements (e), regions (r), lines (I), nodes (11 and connectors (11.
g Tf-
__------
--T--
I
\
&
i+l
?
i
Fig. 5. Conformity at: a) interelemental
boundaries, b) interregional boundaries.
qi
A. Borkowski, Optimizationof slab reinforcement by linear programming
11
The first J'components of the global vectors P and ic are, respectively, the nodal forces P,. and the nodal deflection rates kj. The remaining components are the couples Xi and the rotation rates 4;. Entries of Q and 4 are the bending moments Mi and the strain rates bi. For the ith line situated inside the region (fig. 5a) Mi =M,:
=Mf
(35)
and
e;=(8; + e;>li
)
(36)
i.e. two adjacent elements contribute to the ith global component. The situation is different at the boundary of the region (fig. 5b) where each element retains its independent component. The global vectors C and A include R subvectors C, and A,, each of which corresponds to a region. The global dimension K depends upon the type of optimization problem: K = 4E for free optimization, K = 2E for prescribed orthotropy or asymmetry, and K = E when P and q are fixed simultaneously. A computer routine generating global matrices and vectors requires information about: a) geometry and topology of the network: coordinates of nodes, correspondence between local and global labels, division into regions; b) loading: transversal pressure, concentrated loads etc.; c) plastic properties: yield stress of reinforcing rods, orthotropy and asymmetry factors etc. The global matrices A, B,G and the vectors P and A are built by subsequent accumulation of the contributions of individual elements (as in the elastic analysis by finite elements [ lo]). Regular nets are preferable because a mesh generator then allows the minimization of input data. A solution of problem (10) may be obtained by the simplex routine that is often included in computer software. The components of Q* are of minor interest because of their multivaluedness in the rigid parts of the slab; C’ gives the required optimum design, and the dual zi’ describes the collapse mechanism.
5. Discussion Despite the completeness of the solution for a discrete model of the slab, such a solution remains a kinematic estimate. It is clearly seen that all kinematic requirements of the rigid-plastic approach are satisfied: the dissipation and the external power are calculated exactly, and the collapse mechanism satisfies the boundary conditions. On the other hand, a piecewise-linear stress field can not be in local (differential) equilibrium with a distributed load. Moreover, the stress admissibility condition (23) has an approximate character. An alternative way would be to try a purely static approach with, say, quadratic shape functions for stresses and the nonlinear condition (22). This leads to a nonlinear programming problem, a solution of which is too cumbersome for practical purposes [ 11 I. A compromise was proposed in [ 7,81: piecewise-constant and piecewise-linear approximations of M,,M,,M,, combined with linear shape functions for G. In order to keep the problem one of linear programming, Wolfensberger’s admissibility criterion was introduced [ 121:
A. Borkowski, Optimizationof slab reinforcement by linear programming
12
mx-M,*M,,,>O, m,-M,-+M,,>O, (37) m:+M,kM,,,>O, m;+M,,kM,,>O. These eight inequalities correspond to the octahedrical domain inscribed into the elliptic cones of Johansen’s yield surface. This kind of approach gives no upper or lower bounds for the solution: a local equilibrium for distributed load is not maintained and the flow rule is obeyed only in the mean. Let us specify (23) for the regular mesh shown in fig. 6. In terms of M,, M,, M,, it reads:
m,-M,>O, (m,+m,)-(M,+M,+2M,,)>O, (38)
m;+M,>O,
The comparison of the yield surfaces corresponding to (22) (37) and (38) is given in fig. 7 in the case of isotropy when m, = m,,= m. The yield surface (38) is circumscribed about Johansen’s.
Fig. 6. Regular finite element network.
A. Borkowski, Optimizationof slab reinforcement by linear programming
13
Fig. 7. Comparison of stress admissibility conditions (22), (37) and (38).
However, the overestimation of strength is not essential. Investigations carried out for other nets showed that acute angles (Yand 0 should be avoided for the sake of accuracy.
6. Numerical examples The above theoretical considerations were implemented in an ALGOL code PROLIN-1 written for a computer ODRA 1204 (core memory - 16 K, speed - 60,000 additions/set). Typical examples solved by this routine are given below. A detailed description of PROLIN-1 may be found in [13]: 6.1. The first example concerns a rectangular slab with three edges simply supported and the<’ fourth free. Fig. 8 shows a solution for isotropic bottom reinforcement (p = 1, r) = 0) which required 70 set of CP time. The yield moment obtained, m = 0.1916 pL2 ,
(3%
will be a reference value for other solutions. Note that the co’mmonly used mechanism - a single positive hinge spreading from each corner towards the free edge - gives [ 14, p. 3201 m=0.1802pL2.
(40)
Therefore, even for such a simple case an intuitive choice of the collapse mechanism may lead to a 6% underestimation of the required reinforcement. The first step towards an optimum design is optimization of the orthotropy factor cc. The
A. Borkowski, Optimizationof slab reinforcement by linear progmmming
14
-
free
supported
-
free
edge
edge
-
positive
hinge
____w_
negative
hinge
Fig. 8. Solution of isotropic rectangular slab with a free edge.
solution is presented in fig. 9 (all optimum solutions were calculated under A, = S,). The optimum values are m; = 0.3000 pLz = 1.5658 m , (41)
m; = 0.0500 pL* = 0.2610 m , M*
=0.1667.
The volume of the reinforcement isotropic case.
(proportional
Fig. 9. Optimum orthotropy
to v) for this design is 8% less compared to the
of rectangular slab with a free edge.
A. Borkowski, Optimization of slab reinforcement by linear programming
I
q5L
_)_
0.5L
15
J
Fig. 10. Division of slab into regions for trial optimizations.
Further decrease of V requires more freedom in the reinforcement pattern. Several possibilities were investigated (fig. 10): 1) two reinforcing grids, regions 1 + 3 and 2 + 4; 2) two reinforcing grids, regions 1 + 4 and 2 + 3; 3) two reinforceing grids, regions 1 and 2 + 3 + 4; 4) two reinforcing grids, regions 1 + 2 + 3 and 4; 5) three reinforcing grids, regions 1, 2 + 3 and 4; 6) four reinforcing grids, regions 1, 2, 3 and 4. A free optimization of the 6th patterm reduced V up to 21%. However, the most suitable from the practical point of view seems to be the first pattern. Under the constraint m min > O.OlOOpL’
= 0.0522
m
it still reduces the volume by 13%. This solution gives (fig. 11):
Fig. 11. Optimum solution for regions 1 + 3 and 2 + 4 under constraint (42).
(42)
16
A. Borkowski, Optimizationof slab reinforcement by linear programming
m :1 =0.3650pL2
= 1.9050m,
rnGI = 0.0100 pL* = 0.0522 m ,
(43) m l2 = 0.2817 pL* = 1.4702 m , rnG2 = O.OlOOpL* = 0.0522 m .
The intuitively promising patterns 3 and 4 resulted in 7% and 10% reduction, respectively. 6.2. The second example shows an influence of boundary conditions on the optimum solution. A simply supported rectangular slab has a cut-out in a corner (fig. 12a). The solution for isotropic bottom reinforcement yields m = 0.0260 pL*
(44)
with the cohapse mechanism shown in fig. 12b. The optimal orthotropy rn:
=
corresponds to (fig. 13):
0.6667 X lo-* pL* = 0.2568 m ,
rn; = 3.8519 x lo-* pL* = 1.4835 m
,
(45)
/J* = 5.7778. This design uses 13% less steel than the isotropic one. Note that, contrary to the case of a continuously supported rectangular plate, the stronger reinforcement is parallel to the shorter side of the slab. Additional calculations rejected the possibility of more economical and still technologically acceptable reinforcement patterns for this slab. a)
-
b)
t .
m =0.0260 pL2
L
0.6L
au
Fig. 12. Solution of isotropic rectangular slab with cutout: a) scheme of slab and yield moment, b) collapse mechanism.
A. Borkowski, Optimizationof slab reinforcement by linear programming a)
17
d
Fig. 13. Optimum orthotropy
of rectangular slab with cutout: a) yield moments, b) collapse mechanism
7, Conclusions A yield-line method combined with computers opens interesting possibilities for the rational design of slabs. The computational effectiveness of the algorithm makes it possible to run several trial optimizations (e.g. for different configurations of regions) in a kind of interactive design. Having performed a preliminary plastic optimization, one may choose a technologically acceptable reinforcement grid that produces yield moments as close as possible to their theoretically optimum values. Then additional requirements, for instance those of sufficient stiffness under service load, crack resistance, should be taken into account - this may require a modification of reinforcement. A final design may be recalculated by the rigid-plastic finite element method in order to establish its actual safety factor against plastic collapse.
References ]l] K.W. Johansen, Yield-line theory (Cement and Concrete Assoc., London, 1962). [Z] A.R. Rhzanitsyn, Kinematic limit analysis of sheHs by linear pro~amming [in Russian], in: Proc. 6th All-Union Conf. on Plates and Shells (Nauka, Moscow, 1968) 656-665. [ 31 A.E. Borkauskas * and A.A. Chyras, On the duality in limit analysis and design of plates, Bull. Acad. Polonaise des Sciences (ser. sci. tech.) 16 (1968) 241-247. [4] A.E. Borkauskas * and A.A. Chyras, Duality in non-linear problems of plastic analysis and design (in Russian], Litovskil Mekhanitscheskir Sbornik (Lithuanian Mechanical Review) no. 3(2) (1968) 55-67. [5] J. Foulkes, Linear programming and structural design, in: Proc. 2nd Symp. (Bureau of Standards, U.S. Dept. of Commerce, Washington, 1955) V. 1, No. 1, pp. 27-29. [6] W. Prager, Linear programming and structural design, RAND Corp. Research Memorial, RM-2021 (1957). [7] E. Anderheggen and H. Knopfel, Finite element limit analysis using linear programming, Int. J. Sols. Structs. 8 (1972) 1413-1431. [8] H. Knijpfel, Berechnung Starr-plastischer Platten mittels fmiter Elemente (Institut ftir Baustatik, ETH, Ziirich, no. 47,1973). [P] W. Prager, Introduction to structural optimization, CISM Courses and Lectures, No. 212, Udine, 1974 (Springer, Vienna, 1974). [lo] J.H. Argyris, Recent advances in matrix methods of structural analysis, in: D. Ktichemann and L.H.G. Sterne (ed.) Progress in Aeronautical Sciences 4 (Pergamon Press, Oxford, 1964). [ll ] P.G. Hodge and T. Belytschko, Numerical methods for the limit analysis of plates, J. Appl. Mech. 35 (1968) 796-802. [12] R. Wolfensberger, Traglast und optimale Bemessung von Platten (Wildegg, Zurich, 1964). [13] A. Borkowski, Optimum design of orthotropic rigid-plastic slab [in Polish], Archiwum Iniyniern Ladowej (Civil Engineering Archives) 21 (1975) 15-28. [14] A. Sawczuk and Th. Jaeger, Grenzt~gf~higkeits-Theorie der Platten (Springer, Berhn, 1963). * Present name: A. Borkowski.