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Application of linear programming to the optimal plastic design of circular plates subject to technological constraints
COMPUTERMETHODS IN APPLIED MECHANICS AND ENGINEERING 0 NORTH-HOLLAND PUBLISHING COMPANY
13 (1978) 233-243
APPLICATION OF LINEAR PROGRAMMING TO THE OPTIMAL PLASTIC DESIGN OF CIRCULAR PLATES SUBJECT TO TECHNOLOGICAL CONSTRAINTS D. LAMBLIN *, C. CINQUINI ** and G. GUERLEMENT
*
Received 13 June 1977 This paper is concerned with plastic minimum-volume design of circular or annular plates under axially symmetric loading. Accordingly, all characteristics of the plate depend only on the radial coordinate r. The plate is divided into rings of constant yield moment. Other technological constraints may be added, for instance bounds on the yield moments of the rings. Linear programming is used to obtain an approximate solution of the problem, and the optimality criteria are derived from the duality relations. The method is illustrated by examples which show that it is quite accurate.
1. Introduction The problem of optimal plastic design of circular sandwich plates with stepwise varying thickness (or yield moment) was intuitively treated in 1955 by Hopkins and Prager [ 11, who obtained analytically the design of a simply supported loaded plate. Since no optimality criterion was used, it could not be proved that the design had minimum volume. The general necessary and sufficient condition of optimality was established in 1969 by Sheu and Prager 121. This condition was extended in [31 and [4] to more general cost functions and loadings. Analytical designs of some circular and annular plates were given in 13 I and 15 I . Despite its advantage of exactness the analytical approach has strong disadvantages. The designer must have some intuition of the solution from the very beginning. Thus the method can only be applied to relatively simple problems. Furthermore, the intrinsic mathematical difficulty of the method increases considerably when technological constraints are imposed such as minimum or maximum values for the yield moments, or when the structural weight must be taken into account. In this paper we develop a numerical approach to the problem that is based on linear programming. Starting from the fundamental static theorem of limit analysis and on the basis of a suitable hypothesis supplemented by the discretization of the conditions of equilibrium and plasticity, we formulate in section 3 the problem of optimal plastic design of circular plates with ringwise varying yield moments in the presence of linear technological constraints. Using the well-known duality concepts of mathematical programming, we obtain the kinematic formulation of the design problem (section 4) from which we derive the optimality conditions already established in [ 2]- [ 41. Several examples are presented in section 5. * **
Faculti Polytechnique de Mons, Institut de Micanique et Construction, Belgium. lstituto di Scienza e Tecnica delle Costruzioni, Universiti di Pavia, Italy.
D. Lamblin, C. Cinquini and G. Guerlement,
234
Applicatiort of linear programming
2. The problem Consider a circular or annular plate divided into n rings, in each of which the yield moment M is constant. Let rO and m.be the radii of the concentric boundaries of the rings. The material of [he plate is assumed to be rigid-perfectly plastic and to obey the Tresca yield condition. We seek a design, that is the values of the yield moments Mpi (i = 1, . . . n) of the rings, which fulfills the following conditions. a) Behavioral constraints. The plate is to be at the verge of plastic collapse under the given load p(r). For brevity, we consider only distributed loads, although the inclusion of ring loads would not cause significant complications. We shall also assume that the plate is supported along only one of the inner or outer edges, but the method described below is readily extended to plates supported along both edges. The usual assumptions of thin-plate theory regarding the smallness of the plate thickness and deflections are adopted. bl Technological consfruints. Each plastic moment Mpi will be subjected to M mitY >Mpi>M
i=
mm ’
l,...n,
(1)
where Mmax and Mmin are given constants which may be assigned different values for each ring. cl Minimizution of cost. The design objective is to minimize the cost of the plate defined by C= 5
i=
1
‘rr(rf - ry_, )Mpi = c
i=
I
For a sandwich plate of constant volume of the sheets.
AiMpi.
core thickness
(2) the expression
(2) is seen to be proportional
to the
3. Static formulation The equilibrium
equation
for the axially symmetric
problem
is
M+rdM/dr-N=rT,
(3)
where T, M and N, respectively, are the shear force and the radial and circumferential ments. We shall replace the differential equation (3) by an equivalent finite difference To this end we divide ring i into s rings of identical width Si (fig. 1) and set x{ = r i_l ‘isi,
j=o,...s,
F; = T(x$), i=
As the finite difference
M/ = M(x:),
bending moequation.
N; = N(xj), (4)
1, . ..n. analog of (3) we adopt the following
equations:
\
(5)
S-l i=
l,..n.
D. Lomblin, C. Cinquini ond G. Guerlement, Application of RING i+i
RING
I
ri
linear
i
RING
,
X{
I-
Fig. 1. Part of a circular
programming
or annular
plate divided
235
i-1
b-1
1
into rings.
Since the plate is to be supported along only one edge, Ti can be calculated from the given loading. If we assume that N/> 0, everywhere, the yield conditions are (fig. 2)
Fig. 2. Tresca hexagon
showing
yield conditions.
236
D. Lamb&,
C. Cinquirli and G. Guerlement,
Application of linear programming
Mpi-ME>0 Mpi-N:2 0
', j=O,...s,i= l,...n.
Mpi+M(-N;>
(8)
0,
To the equilibrium and yield conditions (5)-(7) and (8), we have to add the technological straints (1), the boundary conditions which are expressed in terms of the bending moments, the continuity conditions for the radial moment at yi:
Mtf=My+, fori=l,...nIn matrix min 5
i=
notation
1.
conand
(9)
the optimal plastic design problem
can be formulated
as follows:
AMpi, 1
subject to
RtMi+ , - O’Mi = 0,
i = 1,. . . n _
(10)
1,
C;M,-Ni= Ti
(11)
Mpie-Mi>o
i=
1,.
n,
(12)
Mpie-Ni> o
(13)
Mpie-(Ni-M,)> o
(14)
Mpi ’
Mmin
(15)
-Mpi> -Mmax to which we must add the boundary conditions for the particular problem. Equations (10) correspond to (9), (11) to (5)-(7), (12)-( 14) to (8), and (15) to (1). The (S + 1 )th-order vector e has all components equal to one, the vectors B, D, M, N and T are defined by B’=
{l,o;},
D’ = (05, l} (os is thesth-order
zero vector),
M; = {My,... Ml!},Ni'={N,?,...N,s}, T; = {ri_,T;,.. .qTf), and the (S + 1 )th-order
ci’ =
matrix C,! is (e.g. ifs = 3)
’ - yi_l/Gi
‘i_ll~i
O
0
-xi’ /(26,)
1
xi’ /(26,)
0
0
-XT /(26,)
1
xF/(26,)
0
0
-ri/ai
1 + Yi/$
. I
D. Lamblin,
C. Cinquini and C. Guerlement,
Application
231
of linear programming
4. Dual problem By applying the well-known primal. We thus obtain max Z = C
i=
1
duality
relations
[6], we can construct
the dual of the previous
(T;qi +Mminai - Mmaxfli),
(16)
subject to et (Li + pi + ui) + ai - pi = Ai
BYi-
1
-hi
+
C,q,
-
(Ai - ui) - D,y, =
+ pi + v) <
(17) 0
,
i = 1, . . . n.
(18)
(19)
0
In (18) for i = 1 the term B,y,_ 1 need not be considered; for i = n the term -D,y, has to be deleted. The constraints (19) are inequalities because, as a consequence of the Kuhn-Tucker conditions [ 121 , they are associated with the nonnegative primal variables Nj (see section 3). On the other hand, (17) and (18) are equalities because they are associated with the variables Mpi and Mj, which are not restricted in sign. It should be emphasized that, on account of the technological constraints (1 S), the M i need not be stipulated to be nonnegative. Among &e dual variables, qi and yj are free in sign while the others are nonnegative. The li, ui and ui are associated with the conditions (12)-( 14) and can take positive values only if the corresponding yield condition is satisfied as an equality.
5. Interpretation
of the dual
5.1. General interpretation Denote by w{ the transverse rate of deflection of section j in ring i. These rates are defined within a common positive factor. Interpret $i as increments of the rates, and write qj = K(Wi+’ I -wf’), qp
j=2,...s-
1
(20)
=7r(w’i - wp),
q; = n(wf -
(21)
J
w;-l ),
or, in a more concise
to
(22)
form,
~7 = 6Awi.I From relations is a discretized
(20)-(22) form of
it is easy to see that the first term of the objective
function
of the dual
r
P=27T
n s r T(dw/dr) r0
dr.
(23)
D. Lamblin, C. Cinquini and G. Guerlement,
238
Application
of linear
programming
With our hypotheses (only one supported edge and uniform loading) P is the power of dissipation of the load [41. The objective function Z of the dual problem represents to within a factor (that is only a function of CX[,pi, Mmin and Mmax ) the power of dissipation of the load. 5.2. Interpretation
of A{, pi, 6
When t; is different
from zero, the constraints
(19) will be satisfied as equalities
and become
($ t yj) = -T-$ or, using the appropriate
relation
(20), (2 1) or (22),
.
(24)
(+)ri-l
Q-l;+ ‘-q = -2n(+jri(;L
(6
z)?
(25)
y).
(26)
In the second members of (24)-(26) we recognize the powers of dissipation per unit yield moment for the dashed zones noted O, o and@, respectively, in fig. 1 when the plastic profile (fig. 2) is AB(p{ > 0, r$ = 0), BC# = 0, 4 > 0) or is characterised by point B(< + p{ > 0). We shall write (p{ t 4) = a{&, with ai equal to the mean surface of the pIate element of sectionj in ring i, and xi equal to the discretized curvature in the same element. Let us now consider one of the equalities (18) in the case where i f 0 and i # si (current section). This constraint can be written as xi,-1
xj+l
y-$=26,+‘+?+
iSi
$
‘+
1 )
or A{ - yj =
(27)
In view of (20) it is readily seen that
l-i
A2w i Ar2
i
is the discretized
form of the second derivative
of w or, within a sign, of the radial curvature
K{.
For
D. Lamblin, C. Cinquini and G. Guerlement, Application of linear programming
239
(27) we shall write (28)
Xi - < =a${.
The second member of (28) is the power of dissipation per unit yield moment for the element of the plate surrounding sectionj of ring i when the plastic profile is the side DA (y.> 0, vj = 0) or BC(hS = 0, I,$> 0) of the Tresca hexagon. Let us examine successively the equahties (18) obtained for j = 0 in ring i + 1 and for j = s in ring i. These are functions of Xi. By eliminating Ai between them, one obtains
(29) where the quantities
are the discretized expressions of the first and second derivatives of w, respectively, in sectionj Of ring i. In the second and third terms of the second member of (29) one finds expressions similar to (28). These terms represent the power of dissipation per unit yield moment in elements 0 and 1, respectively of rings i + 1 and i for a plastic profile identical to sides DA or BC of the Tresca hexagon. Denote by H:+’ the first term of the second member of (29) (H:+’ may be different from zero only if there is a relative rotation between rings i and i + 1). First, let us suppose that My =M;+l > 0.
In this case we must have v:t*
= v; = 0.
If Hi” > 0, the last relations imply that at least one of the hy+1 , As must be positive. The KuhnTucker condition implies that one h must be equal to zero if the plastic moments of rings i and i + 1 are different. We may therefore interpret Hit 1 as the power of dissipation per unit yield moment in the hinge circle formed in the ring with the smaller plastic moment. When the radial moment is negative at the boundary of the two rings (X = 0, v + 0), a similar discussion shows that -M; = -M?1+
and that H:” 5.3. Optimality
1 = min
{Mpi>Mpi+lI
is still a power of dissipation in a hinge circle. criteria
According to the previously given interpretation
of the dual variables h{, &, L$,the product
er(Xi+pi+ui)=zi/Mpi
in equation (17) represents the total plastic power dissipated per unit yield moment in ring i.
240
D. Lam blin, C. Cinquini and G. Guerlement,
Application of linear programming
lfMpi = M min, we have (Ye> 0 and pi = 0, and equation
(17) becomes
zi/Mmin < Ai.
(30)
In particular, we may have zi = 0. Similarly, fi MPi= MmQX,we obtain pi B 0, ffi = 0, and (17) becomes zi/Mmax 2 Ai.
(31)
When Mm, < Mpi < Mmax, then CY~ = 0, = 0, and (17) yields ziJMPi = Ai. The conditions
6. Computational
(32) (30)-(32)
are the optimality
criteria derived in different
ways in [21, [4]
remarks
It is well-known [61 that the computer time required for the solution of a linear programming problem is roughly proportional to m3yl, where yy1is the number of constraints and n the number of variables. For ring i subdivided into s elements we have for the primal problem 4s + 7 constraints and 2s + 3 variables, and accordingly m Z 2~2. Since the number of variables of the primal becomes the number of constraints of the dual and vice versa, it has to be expected that the computer time required for solving the dual is only about a quarter of the time required for the primal. Furthermore, to start the solution of a linear programming problem, we need a basic feasible solution. For the dual an obvious feasible solution is obtained by setting all variables equal to zero; this corresponds to the undeformed plate (all n, A, /J, v equal to zero). While on one hand the dual is computationally advantageous, on the other hand the formulation and data input for the primal are easier. The above remarks suggest that it is convenient to use linear programming codes capable of forming and processing the problem dual to that stated in the input and capable of supplying in the output both primal and dual solutions. In the computer program that we have used the variables N/ were extracted from equations (11) and substituted into constraints (12)-( 14). Furthermore, we have taken advantage of equalities (10) to reduce the number of unknows in the vectors Mi and hence the number of constraints in the dual problem. For sandwich plates the structural weight can readily be considered by properly evaluating Ti, which will contain the plastic moments Mpk (k = 1, . . . i) as unknowns. A similar formulation can be used for limit analysis problems, where the Mpi are given, and the scalar multiplier of the loads has to be maximized [5 1.
7. Examples
and comparisons
7.1. Simply supported,
uniformly
loaded
circular plate
Let us consider the plate represented in fig. 3 for which we have no technological constraints of the type (15). The first and second rings are respectively divided into 6 and 4 pieces (s = 6 and s = 4). The analytical solution for this problem has been given in [ l] and [23 . The computed results are
D. Lamblin, C. Cinquini and G. Guerlement,
Fig. 3. Circular
Application of linear programming
241
plate (sec. 7.1).
compared in tables 1 and 2 with the exact analytical values. The relative errors are very small and less than 1 percent for the yield moments and the cost (see equation (2)) and less than 3.4 percent for the radial bending moments with a mean error of 1 percent. The accuracy can be improved by increasing the values of s. With s = 8 for both rings the relative errors are reduced to less than 0.4 percent. In fig. 4 we have plotted the design obtained for a simply supported plate divided into 16 rings of equal width taking s = 2. Here again the computed design is close to the analytical design [ 21. Furthermore, fig. 4 shows that the computed design approximates well the absolute minimumvolume design [7], that is the design without any technological constraint and with continuously variable yield moment. 7.2. Built-in, uniformly loaded circular plate For a plate divided into 2 rings limited by the circles of radii 0, R/2 and R, table 3 collects the results obtained by using s = 5 for both rings. As for the previous example the values furnished by linear programming agree very well with the exact results given in [ 31. The linear programming approach described in section 3 can be used to design a plate of constant thickness (one ring) for a given load or, equivalently, to obtain the limit load of a plate with given constant thickness. For the built-in, uniformly loaded plate we obtained pR* /MP = 1 1.19 for s = 10 and n = 1. The exact value is pR* /Ml, = 11.26.
Radial bending r/R
Yield moments
Table 1 and cost for circular
M;fpR’ exact
moment
Table 2 for circular
plate (sec. 7.1)
fI+pR2 computed
Relative
0.8
(%)
plate (sec. 7.1) 0
0.1907
0.1921
0.2
0.184
0.185
0.65
0.4
0.164
0.1649
0.55
0.6
0.1307
0.1313
0.5
0.8
0.069
0.069
0
0.5
0.9
0.0357
0.0369
3.4
0.6
1
0
0
0
Exact
Computed
Relative
values
values
(%)
M~I/PR’
0.1907
0.1921
0.8
Mpzi~R’
0.1307
0.1313
C/pR4
0.1523
0.1532
error
error
D. Lamblin, C. Cinyuini and G. tiuerlement,
Fig. 4. Radial bending
7.3. Simply supported,
uniformly
moment
for circular
Application of linear programming
plate divided
into 16 rings (sec. 7.1).
loaded annular plate
We finally consider the plate sketched in fig. 5. It is well-known [2], 191 that for a plate of this kind the optimal design is characterized by an “edge effet”, that is by MP2 much larger than MP, . To obtain a realistic design, we have to introduce technological constraints of the form (15). Taking s = 10 and s = 5 for rings 1 and 2, respectively, and setting the constraints i= 1,2,
Mpi < 0.25 pR2, we have obtained
the results given in table 4. The analytical
Yield momentsand
Table 3 cost for circular
Yield moments
plate (sec. 7.2)
of this problem
Table 4 and cost for annular
is found in [5].
plate (sec. 7.3)
Exact values
Computed values
(%)
Mp,fpR2
0.25
0.25
0
Mp2/~R2
0.0833
0.0845
1.4
0.1146 ~____~___.
0.1151
0.4
Exact values
Computed values
Mp,lpR=
0.1184
0.1197
1.1
Mpzl~R2
0.0768
0.0772
0.5
C/pR4
0.08718
0.08777
0.7
C/pR4
____
Relative
solution
error
(%)
___
Relative
error
D. Lamblin, C. Cinquiniand
G. Guerlement,
243
Application of linear programming
Fig. 5. Annular plate (sec. 7.3).
8. Conclusion The problem of optimal plastic design of circular plates with technological constraints can be solved by an efficient linear programming approach. We have shown by means of a few examples that the method is accurate and leads (although not demonstrated) to safe designs.
Acknowledgment This research has been sponsored by NATO and by the Belgian National Research authors express their thanks to Professor W. Prager for having reviewed the text.
Fund. The
References [l] H. Hopkins and W. Prager, Limits of economy of material in plates, J. Appl. Mech. 22 (1955) 372-374. [2] C.Y. Sheu and W. Prager, Optimal plastic design of circular and annular sandwich plates eith piecewise constant cross section, J. Mech. Phys. Solids 17 (1969) 11-16. [3] D. Lamblin and G. Guerlement, Dimensionnement plastique de volume minimal sous contraintes de plaques sandwhich circulaires soumises i des charges fixes ou mobiles, J. Mdc. 15 (1976) no. 1. (41 C. Cinquini, D. Lamblin and G. Guerlement, Variational formulation of the optimal plastic design of circular plates, Comp. Meths. Appl. Mech. Eng. 11 (1977) 19-30. [5 1 D. Lamblin, Analyse et dimensionnement plastique de coCt minimum de plaques circulaires, These de doctorat en Sciences appliqudes, Facultd Polytechnique de Mons, 1975. [6] H.S. Chan, Mathematical programming in optimal plastic design, Int. J. Solids Struts. 4 (1968) 885-895. [7] H. Hopkins and W. Prager, The load carrying capacity of circular plates, J. Mech. Phys. Solids 2 (1953) 372-374. [ 81 M. Save and C. Massonnet, Plastic analysis and design of plates, shells and disks (North-Holland, Amsterdam, 1972). [9] G.J. Megarefs, Method for minimal design of axisymmetric plates, ASCE J. Eng. Mech. Div. 92 (1966) 79-99. [lo] G. Sacchi, A variational formulation of the optimal design problem with linear and convex cost function, HJ TAM Symposium-Warsaw (1973). [ll] M. Save, A general criterion for optimal structural design, J. Opt. Theory Appl. 15 (1975) 119-129. [ 121 G. Hadley, Nonlinear and dynamic programming, (Addison Wesley, Chicago, 1965).
13 (1978) 233-243
APPLICATION OF LINEAR PROGRAMMING TO THE OPTIMAL PLASTIC DESIGN OF CIRCULAR PLATES SUBJECT TO TECHNOLOGICAL CONSTRAINTS D. LAMBLIN *, C. CINQUINI ** and G. GUERLEMENT
*
Received 13 June 1977 This paper is concerned with plastic minimum-volume design of circular or annular plates under axially symmetric loading. Accordingly, all characteristics of the plate depend only on the radial coordinate r. The plate is divided into rings of constant yield moment. Other technological constraints may be added, for instance bounds on the yield moments of the rings. Linear programming is used to obtain an approximate solution of the problem, and the optimality criteria are derived from the duality relations. The method is illustrated by examples which show that it is quite accurate.
1. Introduction The problem of optimal plastic design of circular sandwich plates with stepwise varying thickness (or yield moment) was intuitively treated in 1955 by Hopkins and Prager [ 11, who obtained analytically the design of a simply supported loaded plate. Since no optimality criterion was used, it could not be proved that the design had minimum volume. The general necessary and sufficient condition of optimality was established in 1969 by Sheu and Prager 121. This condition was extended in [31 and [4] to more general cost functions and loadings. Analytical designs of some circular and annular plates were given in 13 I and 15 I . Despite its advantage of exactness the analytical approach has strong disadvantages. The designer must have some intuition of the solution from the very beginning. Thus the method can only be applied to relatively simple problems. Furthermore, the intrinsic mathematical difficulty of the method increases considerably when technological constraints are imposed such as minimum or maximum values for the yield moments, or when the structural weight must be taken into account. In this paper we develop a numerical approach to the problem that is based on linear programming. Starting from the fundamental static theorem of limit analysis and on the basis of a suitable hypothesis supplemented by the discretization of the conditions of equilibrium and plasticity, we formulate in section 3 the problem of optimal plastic design of circular plates with ringwise varying yield moments in the presence of linear technological constraints. Using the well-known duality concepts of mathematical programming, we obtain the kinematic formulation of the design problem (section 4) from which we derive the optimality conditions already established in [ 2]- [ 41. Several examples are presented in section 5. * **
Faculti Polytechnique de Mons, Institut de Micanique et Construction, Belgium. lstituto di Scienza e Tecnica delle Costruzioni, Universiti di Pavia, Italy.
D. Lamblin, C. Cinquini and G. Guerlement,
234
Applicatiort of linear programming
2. The problem Consider a circular or annular plate divided into n rings, in each of which the yield moment M is constant. Let rO and m.be the radii of the concentric boundaries of the rings. The material of [he plate is assumed to be rigid-perfectly plastic and to obey the Tresca yield condition. We seek a design, that is the values of the yield moments Mpi (i = 1, . . . n) of the rings, which fulfills the following conditions. a) Behavioral constraints. The plate is to be at the verge of plastic collapse under the given load p(r). For brevity, we consider only distributed loads, although the inclusion of ring loads would not cause significant complications. We shall also assume that the plate is supported along only one of the inner or outer edges, but the method described below is readily extended to plates supported along both edges. The usual assumptions of thin-plate theory regarding the smallness of the plate thickness and deflections are adopted. bl Technological consfruints. Each plastic moment Mpi will be subjected to M mitY >Mpi>M
i=
mm ’
l,...n,
(1)
where Mmax and Mmin are given constants which may be assigned different values for each ring. cl Minimizution of cost. The design objective is to minimize the cost of the plate defined by C= 5
i=
1
‘rr(rf - ry_, )Mpi = c
i=
I
For a sandwich plate of constant volume of the sheets.
AiMpi.
core thickness
(2) the expression
(2) is seen to be proportional
to the
3. Static formulation The equilibrium
equation
for the axially symmetric
problem
is
M+rdM/dr-N=rT,
(3)
where T, M and N, respectively, are the shear force and the radial and circumferential ments. We shall replace the differential equation (3) by an equivalent finite difference To this end we divide ring i into s rings of identical width Si (fig. 1) and set x{ = r i_l ‘isi,
j=o,...s,
F; = T(x$), i=
As the finite difference
M/ = M(x:),
bending moequation.
N; = N(xj), (4)
1, . ..n. analog of (3) we adopt the following
equations:
\
(5)
S-l i=
l,..n.
D. Lomblin, C. Cinquini ond G. Guerlement, Application of RING i+i
RING
I
ri
linear
i
RING
,
X{
I-
Fig. 1. Part of a circular
programming
or annular
plate divided
235
i-1
b-1
1
into rings.
Since the plate is to be supported along only one edge, Ti can be calculated from the given loading. If we assume that N/> 0, everywhere, the yield conditions are (fig. 2)
Fig. 2. Tresca hexagon
showing
yield conditions.
236
D. Lamb&,
C. Cinquirli and G. Guerlement,
Application of linear programming
Mpi-ME>0 Mpi-N:2 0
', j=O,...s,i= l,...n.
Mpi+M(-N;>
(8)
0,
To the equilibrium and yield conditions (5)-(7) and (8), we have to add the technological straints (1), the boundary conditions which are expressed in terms of the bending moments, the continuity conditions for the radial moment at yi:
Mtf=My+, fori=l,...nIn matrix min 5
i=
notation
1.
conand
(9)
the optimal plastic design problem
can be formulated
as follows:
AMpi, 1
subject to
RtMi+ , - O’Mi = 0,
i = 1,. . . n _
(10)
1,
C;M,-Ni= Ti
(11)
Mpie-Mi>o
i=
1,.
n,
(12)
Mpie-Ni> o
(13)
Mpie-(Ni-M,)> o
(14)
Mpi ’
Mmin
(15)
-Mpi> -Mmax to which we must add the boundary conditions for the particular problem. Equations (10) correspond to (9), (11) to (5)-(7), (12)-( 14) to (8), and (15) to (1). The (S + 1 )th-order vector e has all components equal to one, the vectors B, D, M, N and T are defined by B’=
{l,o;},
D’ = (05, l} (os is thesth-order
zero vector),
M; = {My,... Ml!},Ni'={N,?,...N,s}, T; = {ri_,T;,.. .qTf), and the (S + 1 )th-order
ci’ =
matrix C,! is (e.g. ifs = 3)
’ - yi_l/Gi
‘i_ll~i
O
0
-xi’ /(26,)
1
xi’ /(26,)
0
0
-XT /(26,)
1
xF/(26,)
0
0
-ri/ai
1 + Yi/$
. I
D. Lamblin,
C. Cinquini and C. Guerlement,
Application
231
of linear programming
4. Dual problem By applying the well-known primal. We thus obtain max Z = C
i=
1
duality
relations
[6], we can construct
the dual of the previous
(T;qi +Mminai - Mmaxfli),
(16)
subject to et (Li + pi + ui) + ai - pi = Ai
BYi-
1
-hi
+
C,q,
-
(Ai - ui) - D,y, =
+ pi + v) <
(17) 0
,
i = 1, . . . n.
(18)
(19)
0
In (18) for i = 1 the term B,y,_ 1 need not be considered; for i = n the term -D,y, has to be deleted. The constraints (19) are inequalities because, as a consequence of the Kuhn-Tucker conditions [ 121 , they are associated with the nonnegative primal variables Nj (see section 3). On the other hand, (17) and (18) are equalities because they are associated with the variables Mpi and Mj, which are not restricted in sign. It should be emphasized that, on account of the technological constraints (1 S), the M i need not be stipulated to be nonnegative. Among &e dual variables, qi and yj are free in sign while the others are nonnegative. The li, ui and ui are associated with the conditions (12)-( 14) and can take positive values only if the corresponding yield condition is satisfied as an equality.
5. Interpretation
of the dual
5.1. General interpretation Denote by w{ the transverse rate of deflection of section j in ring i. These rates are defined within a common positive factor. Interpret $i as increments of the rates, and write qj = K(Wi+’ I -wf’), qp
j=2,...s-
1
(20)
=7r(w’i - wp),
q; = n(wf -
(21)
J
w;-l ),
or, in a more concise
to
(22)
form,
~7 = 6Awi.I From relations is a discretized
(20)-(22) form of
it is easy to see that the first term of the objective
function
of the dual
r
P=27T
n s r T(dw/dr) r0
dr.
(23)
D. Lamblin, C. Cinquini and G. Guerlement,
238
Application
of linear
programming
With our hypotheses (only one supported edge and uniform loading) P is the power of dissipation of the load [41. The objective function Z of the dual problem represents to within a factor (that is only a function of CX[,pi, Mmin and Mmax ) the power of dissipation of the load. 5.2. Interpretation
of A{, pi, 6
When t; is different
from zero, the constraints
(19) will be satisfied as equalities
and become
($ t yj) = -T-$ or, using the appropriate
relation
(20), (2 1) or (22),
.
(24)
(+)ri-l
Q-l;+ ‘-q = -2n(+jri(;L
(6
z)?
(25)
y).
(26)
In the second members of (24)-(26) we recognize the powers of dissipation per unit yield moment for the dashed zones noted O, o and@, respectively, in fig. 1 when the plastic profile (fig. 2) is AB(p{ > 0, r$ = 0), BC# = 0, 4 > 0) or is characterised by point B(< + p{ > 0). We shall write (p{ t 4) = a{&, with ai equal to the mean surface of the pIate element of sectionj in ring i, and xi equal to the discretized curvature in the same element. Let us now consider one of the equalities (18) in the case where i f 0 and i # si (current section). This constraint can be written as xi,-1
xj+l
y-$=26,+‘+?+
iSi
$
‘+
1 )
or A{ - yj =
(27)
In view of (20) it is readily seen that
l-i
A2w i Ar2
i
is the discretized
form of the second derivative
of w or, within a sign, of the radial curvature
K{.
For
D. Lamblin, C. Cinquini and G. Guerlement, Application of linear programming
239
(27) we shall write (28)
Xi - < =a${.
The second member of (28) is the power of dissipation per unit yield moment for the element of the plate surrounding sectionj of ring i when the plastic profile is the side DA (y.> 0, vj = 0) or BC(hS = 0, I,$> 0) of the Tresca hexagon. Let us examine successively the equahties (18) obtained for j = 0 in ring i + 1 and for j = s in ring i. These are functions of Xi. By eliminating Ai between them, one obtains
(29) where the quantities
are the discretized expressions of the first and second derivatives of w, respectively, in sectionj Of ring i. In the second and third terms of the second member of (29) one finds expressions similar to (28). These terms represent the power of dissipation per unit yield moment in elements 0 and 1, respectively of rings i + 1 and i for a plastic profile identical to sides DA or BC of the Tresca hexagon. Denote by H:+’ the first term of the second member of (29) (H:+’ may be different from zero only if there is a relative rotation between rings i and i + 1). First, let us suppose that My =M;+l > 0.
In this case we must have v:t*
= v; = 0.
If Hi” > 0, the last relations imply that at least one of the hy+1 , As must be positive. The KuhnTucker condition implies that one h must be equal to zero if the plastic moments of rings i and i + 1 are different. We may therefore interpret Hit 1 as the power of dissipation per unit yield moment in the hinge circle formed in the ring with the smaller plastic moment. When the radial moment is negative at the boundary of the two rings (X = 0, v + 0), a similar discussion shows that -M; = -M?1+
and that H:” 5.3. Optimality
1 = min
{Mpi>Mpi+lI
is still a power of dissipation in a hinge circle. criteria
According to the previously given interpretation
of the dual variables h{, &, L$,the product
er(Xi+pi+ui)=zi/Mpi
in equation (17) represents the total plastic power dissipated per unit yield moment in ring i.
240
D. Lam blin, C. Cinquini and G. Guerlement,
Application of linear programming
lfMpi = M min, we have (Ye> 0 and pi = 0, and equation
(17) becomes
zi/Mmin < Ai.
(30)
In particular, we may have zi = 0. Similarly, fi MPi= MmQX,we obtain pi B 0, ffi = 0, and (17) becomes zi/Mmax 2 Ai.
(31)
When Mm, < Mpi < Mmax, then CY~ = 0, = 0, and (17) yields ziJMPi = Ai. The conditions
6. Computational
(32) (30)-(32)
are the optimality
criteria derived in different
ways in [21, [4]
remarks
It is well-known [61 that the computer time required for the solution of a linear programming problem is roughly proportional to m3yl, where yy1is the number of constraints and n the number of variables. For ring i subdivided into s elements we have for the primal problem 4s + 7 constraints and 2s + 3 variables, and accordingly m Z 2~2. Since the number of variables of the primal becomes the number of constraints of the dual and vice versa, it has to be expected that the computer time required for solving the dual is only about a quarter of the time required for the primal. Furthermore, to start the solution of a linear programming problem, we need a basic feasible solution. For the dual an obvious feasible solution is obtained by setting all variables equal to zero; this corresponds to the undeformed plate (all n, A, /J, v equal to zero). While on one hand the dual is computationally advantageous, on the other hand the formulation and data input for the primal are easier. The above remarks suggest that it is convenient to use linear programming codes capable of forming and processing the problem dual to that stated in the input and capable of supplying in the output both primal and dual solutions. In the computer program that we have used the variables N/ were extracted from equations (11) and substituted into constraints (12)-( 14). Furthermore, we have taken advantage of equalities (10) to reduce the number of unknows in the vectors Mi and hence the number of constraints in the dual problem. For sandwich plates the structural weight can readily be considered by properly evaluating Ti, which will contain the plastic moments Mpk (k = 1, . . . i) as unknowns. A similar formulation can be used for limit analysis problems, where the Mpi are given, and the scalar multiplier of the loads has to be maximized [5 1.
7. Examples
and comparisons
7.1. Simply supported,
uniformly
loaded
circular plate
Let us consider the plate represented in fig. 3 for which we have no technological constraints of the type (15). The first and second rings are respectively divided into 6 and 4 pieces (s = 6 and s = 4). The analytical solution for this problem has been given in [ l] and [23 . The computed results are
D. Lamblin, C. Cinquini and G. Guerlement,
Fig. 3. Circular
Application of linear programming
241
plate (sec. 7.1).
compared in tables 1 and 2 with the exact analytical values. The relative errors are very small and less than 1 percent for the yield moments and the cost (see equation (2)) and less than 3.4 percent for the radial bending moments with a mean error of 1 percent. The accuracy can be improved by increasing the values of s. With s = 8 for both rings the relative errors are reduced to less than 0.4 percent. In fig. 4 we have plotted the design obtained for a simply supported plate divided into 16 rings of equal width taking s = 2. Here again the computed design is close to the analytical design [ 21. Furthermore, fig. 4 shows that the computed design approximates well the absolute minimumvolume design [7], that is the design without any technological constraint and with continuously variable yield moment. 7.2. Built-in, uniformly loaded circular plate For a plate divided into 2 rings limited by the circles of radii 0, R/2 and R, table 3 collects the results obtained by using s = 5 for both rings. As for the previous example the values furnished by linear programming agree very well with the exact results given in [ 31. The linear programming approach described in section 3 can be used to design a plate of constant thickness (one ring) for a given load or, equivalently, to obtain the limit load of a plate with given constant thickness. For the built-in, uniformly loaded plate we obtained pR* /MP = 1 1.19 for s = 10 and n = 1. The exact value is pR* /Ml, = 11.26.
Radial bending r/R
Yield moments
Table 1 and cost for circular
M;fpR’ exact
moment
Table 2 for circular
plate (sec. 7.1)
fI+pR2 computed
Relative
0.8
(%)
plate (sec. 7.1) 0
0.1907
0.1921
0.2
0.184
0.185
0.65
0.4
0.164
0.1649
0.55
0.6
0.1307
0.1313
0.5
0.8
0.069
0.069
0
0.5
0.9
0.0357
0.0369
3.4
0.6
1
0
0
0
Exact
Computed
Relative
values
values
(%)
M~I/PR’
0.1907
0.1921
0.8
Mpzi~R’
0.1307
0.1313
C/pR4
0.1523
0.1532
error
error
D. Lamblin, C. Cinyuini and G. tiuerlement,
Fig. 4. Radial bending
7.3. Simply supported,
uniformly
moment
for circular
Application of linear programming
plate divided
into 16 rings (sec. 7.1).
loaded annular plate
We finally consider the plate sketched in fig. 5. It is well-known [2], 191 that for a plate of this kind the optimal design is characterized by an “edge effet”, that is by MP2 much larger than MP, . To obtain a realistic design, we have to introduce technological constraints of the form (15). Taking s = 10 and s = 5 for rings 1 and 2, respectively, and setting the constraints i= 1,2,
Mpi < 0.25 pR2, we have obtained
the results given in table 4. The analytical
Yield momentsand
Table 3 cost for circular
Yield moments
plate (sec. 7.2)
of this problem
Table 4 and cost for annular
is found in [5].
plate (sec. 7.3)
Exact values
Computed values
(%)
Mp,fpR2
0.25
0.25
0
Mp2/~R2
0.0833
0.0845
1.4
0.1146 ~____~___.
0.1151
0.4
Exact values
Computed values
Mp,lpR=
0.1184
0.1197
1.1
Mpzl~R2
0.0768
0.0772
0.5
C/pR4
0.08718
0.08777
0.7
C/pR4
____
Relative
solution
error
(%)
___
Relative
error
D. Lamblin, C. Cinquiniand
G. Guerlement,
243
Application of linear programming
Fig. 5. Annular plate (sec. 7.3).
8. Conclusion The problem of optimal plastic design of circular plates with technological constraints can be solved by an efficient linear programming approach. We have shown by means of a few examples that the method is accurate and leads (although not demonstrated) to safe designs.
Acknowledgment This research has been sponsored by NATO and by the Belgian National Research authors express their thanks to Professor W. Prager for having reviewed the text.
Fund. The
References [l] H. Hopkins and W. Prager, Limits of economy of material in plates, J. Appl. Mech. 22 (1955) 372-374. [2] C.Y. Sheu and W. Prager, Optimal plastic design of circular and annular sandwich plates eith piecewise constant cross section, J. Mech. Phys. Solids 17 (1969) 11-16. [3] D. Lamblin and G. Guerlement, Dimensionnement plastique de volume minimal sous contraintes de plaques sandwhich circulaires soumises i des charges fixes ou mobiles, J. Mdc. 15 (1976) no. 1. (41 C. Cinquini, D. Lamblin and G. Guerlement, Variational formulation of the optimal plastic design of circular plates, Comp. Meths. Appl. Mech. Eng. 11 (1977) 19-30. [5 1 D. Lamblin, Analyse et dimensionnement plastique de coCt minimum de plaques circulaires, These de doctorat en Sciences appliqudes, Facultd Polytechnique de Mons, 1975. [6] H.S. Chan, Mathematical programming in optimal plastic design, Int. J. Solids Struts. 4 (1968) 885-895. [7] H. Hopkins and W. Prager, The load carrying capacity of circular plates, J. Mech. Phys. Solids 2 (1953) 372-374. [ 81 M. Save and C. Massonnet, Plastic analysis and design of plates, shells and disks (North-Holland, Amsterdam, 1972). [9] G.J. Megarefs, Method for minimal design of axisymmetric plates, ASCE J. Eng. Mech. Div. 92 (1966) 79-99. [lo] G. Sacchi, A variational formulation of the optimal design problem with linear and convex cost function, HJ TAM Symposium-Warsaw (1973). [ll] M. Save, A general criterion for optimal structural design, J. Opt. Theory Appl. 15 (1975) 119-129. [ 121 G. Hadley, Nonlinear and dynamic programming, (Addison Wesley, Chicago, 1965).