- Email: [email protected]
Optimal design of elastic structures for maximum stiffness
hr. J
Sohdr S~rurrurrs.
1968. Vol. 4. pp
OPTIMAL
689 to 700. Pergamon
Press
Prmted
m Great
Brltam
DESIGN OF ELASTIC STRUCTURES FOR MAXIMUM STIFFNESS-f N.
C.
HUANG;
University of California, San Diego. La Jolla. California
Abstract-The purpose of this paper is to establish a general theory of optimal design of elastic structures such that the structure with a given volume would have maximum stiffness. A sufficient condition of optimality is derived from the principle of minimum potential energy. This optimalit) condition IS proven by the variational method to be a necessary one under the condition thar the optimal structure has certain continuity and differentiability properties. Physical interpretations of the optimality condition are discussed for problems of beams. plates and trusses. An application of the theory is illustrated in a problem of optimal design of a simply supported circular plate under uniform pressure. Detailed description of the numerical procedure for the solution of the plate problem is presented.
1. INTRODUCTION
IN the problem of optimal elastic design of structures for maximum stiffness.0 we usually define a design variable and ask how the design variable should vary as a function of position such that the work done by a given load to the structure is maximized while the total weight of the structure remains constant. The choice of the design variable depends entirely on the design purpose. For example. in the>problem of optimal design of solid beams. we may choose the height of the beam as a design variable and keep the width of the beam fixed or vice versa. In the problems of optimal design of plates we would consider the thickness of the plate as a design variable. Equivalent problems of optimal design are : ii). for given deformation and load, find the variation of the design variable such that the total weight of a structure is a minimum: and (ii), for given deformation and total weight of a structure, find the variation of the design variable such that the applied load is a maximum. A general method of treating a variety of problems of optimal design of sandwich structures was given by Prager and Taylor [I]. The problems of optimal elastic design of solid beams to achieve the highest fundamental frequency and of solid columns to reach the largest buckling load or maximum height have been analyzed by the variational method in [J-4]. In this paper, we shall deal with the optimal design of elastic structures for maximum stiffness in a general manner. A necessary and sufficient condition of optimality is derived by means of the calculus of variation and the principle of minimum potential energy. The physical meanings of this optimality condition in the problems of beams, plates and trusses are discussed. An application of the theory is made for the problem of optimal design of a circular
plate with simply supported
edge under a uniform
pressure.
t The results presented in this paper were obtained in the course of research conducted under Contract N00014-67-A-0109-0003. Task NR 064-496 with the Office of Naval Research. Washington. D.C. $ Assistant Professor. Department of Aerospace and Mechanical Engineering Scienc&. $ The term “stiffness” used in this paper is measured by the external work done by the load 6X9
2. BASIC
PRINCIPLE
Consider an elastic structure which may be a beam. a plate. a shell or a truss. Let P, and 8, be the generalized load and the generalized displacement at any point. The external work is FV = 4 1 p,(i, = e C’ dS wI I
Ii)
where dS denotes the line element of a one-dimensional structure or the area element of a two-dimensional structure and L. is the unit strain energy. The potential energy is defined as
Therefore I3 1
f-I=-w. Let s be the stiffness and to the specific strain energy or the strain energy we have
per unit sttffness.
I. = SP
IJI
Denote the design variable by B = B(S) which is a function of position. For an appropriate choice of the design variable, the total voiume of the structure can be written as C’Z‘.
where L’is a positive
constant.
1
- BdS .I The stiffness can usually
r-1 be expressed
as
5 = rB”
16)
where r is a positive constant and II stands for any positive integer. For example. in the problem of optimal design of a beam of rectangular cross section with fixed width h and variable height for maximum bending stiffness. we may take the cross-secttonal area of the beam as the design variable. We then have C‘= 1. n = 3 and x = E. t 12b’I where E is Young’s modulus for the beam material. In general. we have from equations 13). (3). (4) and (6).
For given load. we can consider W as a measure of stiffness. Let us consider two designs specified by B(S) and @St. Suppose would lead to the same stiffness. i.e..
where E and 6, are respectively the specific strain energy corresponding to the design B. By the principle of minimum
that these two designs
and generalized displacement potential energy, we have
Optimal design of elastic structures for maximum stiffness
691
because the displacements of the design B are also kinematically admissible for the design B. From equation (8) and the inequality (9). we have c(B”--B”)dS 2 0. Y If B is infintesimally close to B. we may set B = B+6B
(10)
(11)
with 6B to be a function of position of infinitesimal magnitude. If we substitute equation (11 J into the inequality (10) and neglect the higher order terms of bB we obtain
i
eB”-’ 6BdS 3 0.
(12)
If eB”-l = constani
(13)
then JSB dS 2 0 and the design specified by B is an optimal one in the sense that it does not have a greater structural weight than any neighboring design of the same stiffness. Thus equation (13) is a sufficient condition for optimality in this restricted sense. Equation (13) can also be written as UIB = constant.
(14)
We shall prove that the optimality condition, equation (13). is also a necessary condition of optimal design. For given V, the problem of optimal design for maximum stiffness can be reduced to find B(S) such that the external work W given by equation (1) is maximized under the subsidiary condition of equation (5). If the design variable is continuous and differentiable, this problem can be treated by the calculus of variation. We can set the following variational equation from equations (1J. (4). (5) and (6) : 6
s
(eB”-/LcB)dS
= 0,
(15)
where i is a Lagrange multiplier. The Euler equation of equation (15) would lead to the optimality condition equation (13). Thus equation (13) is also a necessary condition of optimal design. By the condition of optimality and the equation of equilibrium, we can find a solution for our problem corresponding to a local minimum weight of the structure if this solution exists. Whether this local minimum is also the absolute minimum depends on the uniqueness of the solution. In the problems of optimal design of beams, it is evident from equation (14) that a beam design is optimal in the sense specified above if the given loads produce deflections such that the strain energy in any element between adjacent cross sections is proportional to the weight of this element. Equation (14) can also be written as M’/(1,4) = constant
(16)
where M is the bending moment and A and I are the area and the moment of inertia of the cross section respectively. We can also write equation (16) as Mh/(I.4h2?
= constant.
(17)
where h is the height of beams of rectangular cross section or the radius of beams of circular cross section. Since I is proportional to .4/r’. we have Mh:I
IISI
= constant.
Accordingly. the optimal design of beams of rectangular or circular cross section is also the design of uniform strength in which the stresses in the extreme fibres are uniform. By using this property of uniform strength. we can simplify our analysis of optimal design ol beams, particularly when the problem is statically determinate. In the problem of optimal design of plates under bending and without stretching, if the thickness of the plate 11is taken as the design variable. equation i 14) becomes
where M, and 51, are bending ratio of the material. Equation
l19I
2 1 + \~04f,],‘/? = constant
[M,: 4 54: - ZvM,M,+
moments. ,%JI.~, is the twisting f 19) can also be written as
moment
and \I IS the Poisson’<
rrz T ~7: - 2~~7~~7,+ 7( 1 + ~)r& = constant where (T,, gY and TV,,are stress components expression of equation (20) is ‘T,,E,+
CUE,
on the surfaces
-+ 5,,.:‘,,
=
of the plate. An alternative
constant
IllJ
where E,, E, and :lX, are strain components on the surfaces of the plate. Therefore the optimal design of plates under bending and without stretching is the design of constant strain energy density on the surfaces of the plate. This condition of constant strain energy density on the surfaces is of course also applicable to the beam problems. We can also prove by the same method as shown in the beam and plate problems that the optimum truss of given configuration and volume for maximum stiffness is the one m which the stresses per unit area in all members have the same magnitude. In other words, the optimal design of trusses is also the design of uniform strength.
3. OPTIMAL DESIGN OF CIRCULAR PLATES WITH SIMPLY SUPPORTED EDGE UNDER UNIFORM PRESSURE As an illustrative example of the theory developed in the previous section. let us consider the problem of optimal design of an elastic circular plate of radius u under a uniform pressure 4 for maximum stiffness. Again. we shall measure the stiffness of the plate by the external work of pressure y. Set the origin of a polar coordinates system at the center of the plate. We shall assume that the optimum plate is axisymmetric. i.e., the thickness h and deflection w of the plate are functions of radial coordinate r only. The curvatures of the deformed plate are
and I
jc”
zz
--
-_I,.’
r
:23)
Optimal
where (
I’ = d/dr(
design of elastic structures
). The moment-curvature
for maximum
stiffness
693
relations are
M, = DIX, + w,) = - D
(23)
and (25) where Eh3
D=.---.--12(1-P)
(26)
The equation of equilibrium is
M, T+M;-M”+Q
= 0
r
(27)
where Q = qr,/2
is the transverse shear. Substitution leads to
of equations
(28 1 (24). (25) and (28) into equation (27)
(29) where V2 = (d’/dr’)+(l/r)(dldr)
is the Laplace operator. Equation (29) can be written as 1
2&3-(C’,,)’ r
+6&h’-
1 ri
= q
M.”+ :,.’ I
(301
I
where 2 = E(l-?-’ 12. If we choose the thickness of the plate h as the design variable. the condition of optimalitg. equation (14). becomes
where C is a constant. Let i? be the average deflection of the plate. We have 0 M’=--2 wr dr. a’ I o From equation (1). we have (I (1 t+: = f*qEa’ = 2~ Ur dr = 2zC hrdr = CT: s0 i0
= 1(Ga’)
(32)
(33)
After elimination of C from equations (31) and (33). we obtain
iI
1
2ach’ (VW) - 2( 1 - \‘)-W’M”’ r
q
(34)
where ,’ = ‘1 = ‘77
The edge of the plate boundary conditions :
is assumed
,i.“l
-Uhr dr. L,,
to be simply
supported.
We have
the iollolvmg
Il’ (0) = 0.
l:h!
I\‘(U) = 0.
l.J-1
!Cl,(fJI = 0.
/ ;s1
and
Substituting
equation
(24) into equation
(38) and using equation
(33). we have
Since 1~~tr~is bounded. \l”‘(ClI = x From equations
(2-l). I 38) and 140). we have hlrrj = 0.
Let us introduce
the following
dimensionless
quanti
ies
and put (
d , = ci.y(
1 and
V’ = _!!<+ i 2 d.\ - .Ydx
Eyuatlons 1351. (301. (34). (311. ri61. (37). (401 and (41) can be expressed dimensionless quantities : .I rz ds = I. ?0
in terms of these
1431
Optimal
design of elastic structures
for maximum
stiffness
69.5
and f( 1) = 0.
(501
Let % = _I“be the deflection slope. Equations (44). (45). (47). and (49) can be written il terms of %as (51
r
1
l-1
1
1
I--\
and %‘(1) = EC.
(54)
Equations (43) and (5OH541 are the governing equations of our problem. Since the nonlinearity is involved in the coupled differential equations, we shall solve them by an iterative procedure. First of a!l. we shall assume values for B(s). then solve for t(x) from equation (51) and adjust 8(x-) such that equation (43) is satisfied. Next. 8(s) can be solved from equations (52), (53) and (54) and r(s) is solved ffom equations (43) and (51) again. Such iteration will continue until certain convergence criterion is fulfilled. The process involved in this iterative procedure will be described in detail in the following. From equation (54) we can see that close to the edge of the plate there is a region in which 8” $ 8’ b 8 and equations (51) and (52) can be expressed approximately as (55) and 1 p/y
=
_7
1
%(;)d
N
i0
i
(56)
1
where c1 is a constant. Equations (55). (56) and (50) can be solved explicitly. The solutions are % = -4c:
sin-‘s+c2
(57)
and
where c2 is an integration constant. Let 1 -x* be the thickness of the boundary layer region in which the approximate equations (55) and (56) hold. The values of c1 and c2 are determined by matching the values of 8 and H’at x = s*, viz.. - 4c: = %‘(s*)(1 - .Y*2)+
(59)
and c2 = %(x*)+4cf sin-lx_*.
(60)
Let Y = Y, = (i- ! )A. where A is the mesh size. Set 0, = O(.Y,I and t, = ~IY,). In the the regron 1 x I*. we shall use the following central difference equattons to approximate derivatives of O(u) :
Put
Equation
(5 1I can be written
as 1
We shall use the open .\dams
to calculate
method
I, m the tirst iteration
and use the modified
Euler’s method
to evaluate I, in the rest of the iterations. The values of the calculated I(YI depend on the values of 0.~) used in equatrons (61-64). The values of t(.u) thus calculated may not satisfy equation (13). Let .I I Y d.Y = p.
ii,S,
”0 Then the adjusted
values of t(s) and NY) would be
Note that such adjustments
would cause r,,(YI and fln(.ul to satisfy equattons
(431 and I
Optimal
design of elastic structures
for maximum
stiffness
691
After t(x) is determined we shall solve for B(x) from equations (52). (53) and (54). Let us define 6(.x) such that it satisfies the following equations: (71)
B(0)=
0
(72)
and 8’(l) = %.
(73)
For given t(x) we shall solve for 8(x). Let /I(s) = s’/t2.
(74)
s”B” + 21&M + 8’ = fi_
(73
Equation (71) can be written as
Therefore 8’ =
-~i,,n~ia_(I_~‘,8’]‘i.
(76)
Since lim 17= B’(0). x-0 s
(77)
from equation (71), we have 2(1+ \*)[r(0)B’(O)]2
= 1
(78)
(79)
&co, = -[‘(l+v)]-*[t(O)]-‘. If we consider equation (79) as a limiting case of equation (78). we have 8’ = -~I,lB+[ic-(1-i,~)8’]i:.
(80)
Let 8, = &s,). we have 8, =o. 6; = -[?(I
(81) +l’)j-*r;l.
(82)
G2 = -d[‘(l+~)]-‘r;‘.
(83)
At the beginning. we shall trl
then evaluate 0; from equation quadrature formula :
(80) and 8, by iterations
according
to the following
(84)
.After ci, and d2 are evaluated. we shall calculate O,(i > 7) in the region to the following procedure : pi) To use the open Adams method for the first iteration. i.e..
(ii) To use the modified
Euier’s method
for the rest ofthe iterations.
.Y I Y* according
1.e..
We shall choose Y* such that it is a grid point. After g(.x*) IS calculated. we shall ~duatt: to rquatrons (59) and (60) b) replacing H(P) bl 4(.x*) and H’(r*) b! I’ i and (‘2 according (I’(.\-*). For x > x*, d(s) is determined according to equation (57). Let ISi’)
If we set 188)
then
and (4 u) would satisfy equations (52). (53) and (54). This iterative procedure will continue until the following fulfilled :
where (Ip(s) is the v,a!ue of NY) appeared beginning of our calculation. we shall take
in the previous
as the initial tried value of t)(.~). The numerical values of r(.~) and 0(.x) are shown deflection of the optimum plate is found to be
For a plate of the same volume
and uniform
convergence
process
criterion
of Iteration.
15
At the
in Fig. 1 ior 1’ = :. The average
thickness.
the average
deflection
is
Optimal
design of elastic structures
FIG. 1.
Therefore the ratio of the stiffness volume and uniform thickness is
for maximum
stiffness
699
r(x) and H(r) curves for v = 1 :3 of the optimum
plate
to that of a plate of the same
r, = 1.58’.
(94)
4. DISCUSSION The proof that the optimality condition. equation (13). is the necessary condition of optimal design is based on the assumption of continuity and differentiability of the design variation function. Megarefs [5] has shown in a problem of plastic minimum weight design or circular solid plate for maximum safety that the volume of the plate can be reduced if one allows the discontinuity in the variation of plate thickness. In view of the use of piecewise linear yield conditions with an infinity of compatible strain rates at vertices or edges of the yield locus. compatibility is less stringent in plasticity than in elasticity. Certain conditions of continuity and differentiability would be introduced through the compatibility equation in the elastic problems. Since the shear deformations are ignored in our analysis of beam and plate problems. zero cross sections or zero thicknesses are found at the places where the bending moment vanishes. Shear stress would become infinite at the zero cross section. In order to make the design more realistic. shear deformations must be taken into account and the stress at any point must be restricted within the maximum allowable stress. However the analysis involving the shear deformation and the restriction of local stresses would become complicated. Ackno~,le&emenr-The
author
wishes to express
his thanks
to Professor
W. Praper
for his helpful
suggestions.
REFERENCES :I] W. PRAGER and J. E. TAYLOR. Problems of optimal structural desp. J. appl. Mech. 35. 102-106 (1968) [2] F. I. NIORDSON. On the optimal design of a vibrating beam. Q. appl. Marh. 23.37-53 (1965).
[i)
I TADJBAKHSH and J. 9. KELLER. Strongest columns xnd lsoperlmetrlc (19621 [Jj J 9. KELLER and F. I. NIORDSW. The tallest Column. J .Muth. Mwh. [5] G. J. MEGAREFS. Method Etxrr 92. 79-99 ( 1966)
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Sohdr S~rurrurrs.
1968. Vol. 4. pp
OPTIMAL
689 to 700. Pergamon
Press
Prmted
m Great
Brltam
DESIGN OF ELASTIC STRUCTURES FOR MAXIMUM STIFFNESS-f N.
C.
HUANG;
University of California, San Diego. La Jolla. California
Abstract-The purpose of this paper is to establish a general theory of optimal design of elastic structures such that the structure with a given volume would have maximum stiffness. A sufficient condition of optimality is derived from the principle of minimum potential energy. This optimalit) condition IS proven by the variational method to be a necessary one under the condition thar the optimal structure has certain continuity and differentiability properties. Physical interpretations of the optimality condition are discussed for problems of beams. plates and trusses. An application of the theory is illustrated in a problem of optimal design of a simply supported circular plate under uniform pressure. Detailed description of the numerical procedure for the solution of the plate problem is presented.
1. INTRODUCTION
IN the problem of optimal elastic design of structures for maximum stiffness.0 we usually define a design variable and ask how the design variable should vary as a function of position such that the work done by a given load to the structure is maximized while the total weight of the structure remains constant. The choice of the design variable depends entirely on the design purpose. For example. in the>problem of optimal design of solid beams. we may choose the height of the beam as a design variable and keep the width of the beam fixed or vice versa. In the problems of optimal design of plates we would consider the thickness of the plate as a design variable. Equivalent problems of optimal design are : ii). for given deformation and load, find the variation of the design variable such that the total weight of a structure is a minimum: and (ii), for given deformation and total weight of a structure, find the variation of the design variable such that the applied load is a maximum. A general method of treating a variety of problems of optimal design of sandwich structures was given by Prager and Taylor [I]. The problems of optimal elastic design of solid beams to achieve the highest fundamental frequency and of solid columns to reach the largest buckling load or maximum height have been analyzed by the variational method in [J-4]. In this paper, we shall deal with the optimal design of elastic structures for maximum stiffness in a general manner. A necessary and sufficient condition of optimality is derived by means of the calculus of variation and the principle of minimum potential energy. The physical meanings of this optimality condition in the problems of beams, plates and trusses are discussed. An application of the theory is made for the problem of optimal design of a circular
plate with simply supported
edge under a uniform
pressure.
t The results presented in this paper were obtained in the course of research conducted under Contract N00014-67-A-0109-0003. Task NR 064-496 with the Office of Naval Research. Washington. D.C. $ Assistant Professor. Department of Aerospace and Mechanical Engineering Scienc&. $ The term “stiffness” used in this paper is measured by the external work done by the load 6X9
2. BASIC
PRINCIPLE
Consider an elastic structure which may be a beam. a plate. a shell or a truss. Let P, and 8, be the generalized load and the generalized displacement at any point. The external work is FV = 4 1 p,(i, = e C’ dS wI I
Ii)
where dS denotes the line element of a one-dimensional structure or the area element of a two-dimensional structure and L. is the unit strain energy. The potential energy is defined as
Therefore I3 1
f-I=-w. Let s be the stiffness and to the specific strain energy or the strain energy we have
per unit sttffness.
I. = SP
IJI
Denote the design variable by B = B(S) which is a function of position. For an appropriate choice of the design variable, the total voiume of the structure can be written as C’Z‘.
where L’is a positive
constant.
1
- BdS .I The stiffness can usually
r-1 be expressed
as
5 = rB”
16)
where r is a positive constant and II stands for any positive integer. For example. in the problem of optimal design of a beam of rectangular cross section with fixed width h and variable height for maximum bending stiffness. we may take the cross-secttonal area of the beam as the design variable. We then have C‘= 1. n = 3 and x = E. t 12b’I where E is Young’s modulus for the beam material. In general. we have from equations 13). (3). (4) and (6).
For given load. we can consider W as a measure of stiffness. Let us consider two designs specified by B(S) and @St. Suppose would lead to the same stiffness. i.e..
where E and 6, are respectively the specific strain energy corresponding to the design B. By the principle of minimum
that these two designs
and generalized displacement potential energy, we have
Optimal design of elastic structures for maximum stiffness
691
because the displacements of the design B are also kinematically admissible for the design B. From equation (8) and the inequality (9). we have c(B”--B”)dS 2 0. Y If B is infintesimally close to B. we may set B = B+6B
(10)
(11)
with 6B to be a function of position of infinitesimal magnitude. If we substitute equation (11 J into the inequality (10) and neglect the higher order terms of bB we obtain
i
eB”-’ 6BdS 3 0.
(12)
If eB”-l = constani
(13)
then JSB dS 2 0 and the design specified by B is an optimal one in the sense that it does not have a greater structural weight than any neighboring design of the same stiffness. Thus equation (13) is a sufficient condition for optimality in this restricted sense. Equation (13) can also be written as UIB = constant.
(14)
We shall prove that the optimality condition, equation (13). is also a necessary condition of optimal design. For given V, the problem of optimal design for maximum stiffness can be reduced to find B(S) such that the external work W given by equation (1) is maximized under the subsidiary condition of equation (5). If the design variable is continuous and differentiable, this problem can be treated by the calculus of variation. We can set the following variational equation from equations (1J. (4). (5) and (6) : 6
s
(eB”-/LcB)dS
= 0,
(15)
where i is a Lagrange multiplier. The Euler equation of equation (15) would lead to the optimality condition equation (13). Thus equation (13) is also a necessary condition of optimal design. By the condition of optimality and the equation of equilibrium, we can find a solution for our problem corresponding to a local minimum weight of the structure if this solution exists. Whether this local minimum is also the absolute minimum depends on the uniqueness of the solution. In the problems of optimal design of beams, it is evident from equation (14) that a beam design is optimal in the sense specified above if the given loads produce deflections such that the strain energy in any element between adjacent cross sections is proportional to the weight of this element. Equation (14) can also be written as M’/(1,4) = constant
(16)
where M is the bending moment and A and I are the area and the moment of inertia of the cross section respectively. We can also write equation (16) as Mh/(I.4h2?
= constant.
(17)
where h is the height of beams of rectangular cross section or the radius of beams of circular cross section. Since I is proportional to .4/r’. we have Mh:I
IISI
= constant.
Accordingly. the optimal design of beams of rectangular or circular cross section is also the design of uniform strength in which the stresses in the extreme fibres are uniform. By using this property of uniform strength. we can simplify our analysis of optimal design ol beams, particularly when the problem is statically determinate. In the problem of optimal design of plates under bending and without stretching, if the thickness of the plate 11is taken as the design variable. equation i 14) becomes
where M, and 51, are bending ratio of the material. Equation
l19I
2 1 + \~04f,],‘/? = constant
[M,: 4 54: - ZvM,M,+
moments. ,%JI.~, is the twisting f 19) can also be written as
moment
and \I IS the Poisson’<
rrz T ~7: - 2~~7~~7,+ 7( 1 + ~)r& = constant where (T,, gY and TV,,are stress components expression of equation (20) is ‘T,,E,+
CUE,
on the surfaces
-+ 5,,.:‘,,
=
of the plate. An alternative
constant
IllJ
where E,, E, and :lX, are strain components on the surfaces of the plate. Therefore the optimal design of plates under bending and without stretching is the design of constant strain energy density on the surfaces of the plate. This condition of constant strain energy density on the surfaces is of course also applicable to the beam problems. We can also prove by the same method as shown in the beam and plate problems that the optimum truss of given configuration and volume for maximum stiffness is the one m which the stresses per unit area in all members have the same magnitude. In other words, the optimal design of trusses is also the design of uniform strength.
3. OPTIMAL DESIGN OF CIRCULAR PLATES WITH SIMPLY SUPPORTED EDGE UNDER UNIFORM PRESSURE As an illustrative example of the theory developed in the previous section. let us consider the problem of optimal design of an elastic circular plate of radius u under a uniform pressure 4 for maximum stiffness. Again. we shall measure the stiffness of the plate by the external work of pressure y. Set the origin of a polar coordinates system at the center of the plate. We shall assume that the optimum plate is axisymmetric. i.e., the thickness h and deflection w of the plate are functions of radial coordinate r only. The curvatures of the deformed plate are
and I
jc”
zz
--
-_I,.’
r
:23)
Optimal
where (
I’ = d/dr(
design of elastic structures
). The moment-curvature
for maximum
stiffness
693
relations are
M, = DIX, + w,) = - D
(23)
and (25) where Eh3
D=.---.--12(1-P)
(26)
The equation of equilibrium is
M, T+M;-M”+Q
= 0
r
(27)
where Q = qr,/2
is the transverse shear. Substitution leads to
of equations
(28 1 (24). (25) and (28) into equation (27)
(29) where V2 = (d’/dr’)+(l/r)(dldr)
is the Laplace operator. Equation (29) can be written as 1
2&3-(C’,,)’ r
+6&h’-
1 ri
= q
M.”+ :,.’ I
(301
I
where 2 = E(l-?-’ 12. If we choose the thickness of the plate h as the design variable. the condition of optimalitg. equation (14). becomes
where C is a constant. Let i? be the average deflection of the plate. We have 0 M’=--2 wr dr. a’ I o From equation (1). we have (I (1 t+: = f*qEa’ = 2~ Ur dr = 2zC hrdr = CT: s0 i0
= 1(Ga’)
(32)
(33)
After elimination of C from equations (31) and (33). we obtain
iI
1
2ach’ (VW) - 2( 1 - \‘)-W’M”’ r
q
(34)
where ,’ = ‘1 = ‘77
The edge of the plate boundary conditions :
is assumed
,i.“l
-Uhr dr. L,,
to be simply
supported.
We have
the iollolvmg
Il’ (0) = 0.
l:h!
I\‘(U) = 0.
l.J-1
!Cl,(fJI = 0.
/ ;s1
and
Substituting
equation
(24) into equation
(38) and using equation
(33). we have
Since 1~~tr~is bounded. \l”‘(ClI = x From equations
(2-l). I 38) and 140). we have hlrrj = 0.
Let us introduce
the following
dimensionless
quanti
ies
and put (
d , = ci.y(
1 and
V’ = _!!<+ i 2 d.\ - .Ydx
Eyuatlons 1351. (301. (34). (311. ri61. (37). (401 and (41) can be expressed dimensionless quantities : .I rz ds = I. ?0
in terms of these
1431
Optimal
design of elastic structures
for maximum
stiffness
69.5
and f( 1) = 0.
(501
Let % = _I“be the deflection slope. Equations (44). (45). (47). and (49) can be written il terms of %as (51
r
1
l-1
1
1
I--\
and %‘(1) = EC.
(54)
Equations (43) and (5OH541 are the governing equations of our problem. Since the nonlinearity is involved in the coupled differential equations, we shall solve them by an iterative procedure. First of a!l. we shall assume values for B(s). then solve for t(x) from equation (51) and adjust 8(x-) such that equation (43) is satisfied. Next. 8(s) can be solved from equations (52), (53) and (54) and r(s) is solved ffom equations (43) and (51) again. Such iteration will continue until certain convergence criterion is fulfilled. The process involved in this iterative procedure will be described in detail in the following. From equation (54) we can see that close to the edge of the plate there is a region in which 8” $ 8’ b 8 and equations (51) and (52) can be expressed approximately as (55) and 1 p/y
=
_7
1
%(;)d
N
i0
i
(56)
1
where c1 is a constant. Equations (55). (56) and (50) can be solved explicitly. The solutions are % = -4c:
sin-‘s+c2
(57)
and
where c2 is an integration constant. Let 1 -x* be the thickness of the boundary layer region in which the approximate equations (55) and (56) hold. The values of c1 and c2 are determined by matching the values of 8 and H’at x = s*, viz.. - 4c: = %‘(s*)(1 - .Y*2)+
(59)
and c2 = %(x*)+4cf sin-lx_*.
(60)
Let Y = Y, = (i- ! )A. where A is the mesh size. Set 0, = O(.Y,I and t, = ~IY,). In the the regron 1 x I*. we shall use the following central difference equattons to approximate derivatives of O(u) :
Put
Equation
(5 1I can be written
as 1
We shall use the open .\dams
to calculate
method
I, m the tirst iteration
and use the modified
Euler’s method
to evaluate I, in the rest of the iterations. The values of the calculated I(YI depend on the values of 0.~) used in equatrons (61-64). The values of t(.u) thus calculated may not satisfy equation (13). Let .I I Y d.Y = p.
ii,S,
”0 Then the adjusted
values of t(s) and NY) would be
Note that such adjustments
would cause r,,(YI and fln(.ul to satisfy equattons
(431 and I
Optimal
design of elastic structures
for maximum
stiffness
691
After t(x) is determined we shall solve for B(x) from equations (52). (53) and (54). Let us define 6(.x) such that it satisfies the following equations: (71)
B(0)=
0
(72)
and 8’(l) = %.
(73)
For given t(x) we shall solve for 8(x). Let /I(s) = s’/t2.
(74)
s”B” + 21&M + 8’ = fi_
(73
Equation (71) can be written as
Therefore 8’ =
-~i,,n~ia_(I_~‘,8’]‘i.
(76)
Since lim 17= B’(0). x-0 s
(77)
from equation (71), we have 2(1+ \*)[r(0)B’(O)]2
= 1
(78)
(79)
&co, = -[‘(l+v)]-*[t(O)]-‘. If we consider equation (79) as a limiting case of equation (78). we have 8’ = -~I,lB+[ic-(1-i,~)8’]i:.
(80)
Let 8, = &s,). we have 8, =o. 6; = -[?(I
(81) +l’)j-*r;l.
(82)
G2 = -d[‘(l+~)]-‘r;‘.
(83)
At the beginning. we shall trl
then evaluate 0; from equation quadrature formula :
(80) and 8, by iterations
according
to the following
(84)
.After ci, and d2 are evaluated. we shall calculate O,(i > 7) in the region to the following procedure : pi) To use the open Adams method for the first iteration. i.e..
(ii) To use the modified
Euier’s method
for the rest ofthe iterations.
.Y I Y* according
1.e..
We shall choose Y* such that it is a grid point. After g(.x*) IS calculated. we shall ~duatt: to rquatrons (59) and (60) b) replacing H(P) bl 4(.x*) and H’(r*) b! I’ i and (‘2 according (I’(.\-*). For x > x*, d(s) is determined according to equation (57). Let ISi’)
If we set 188)
then
and (4 u) would satisfy equations (52). (53) and (54). This iterative procedure will continue until the following fulfilled :
where (Ip(s) is the v,a!ue of NY) appeared beginning of our calculation. we shall take
in the previous
as the initial tried value of t)(.~). The numerical values of r(.~) and 0(.x) are shown deflection of the optimum plate is found to be
For a plate of the same volume
and uniform
convergence
process
criterion
of Iteration.
15
At the
in Fig. 1 ior 1’ = :. The average
thickness.
the average
deflection
is
Optimal
design of elastic structures
FIG. 1.
Therefore the ratio of the stiffness volume and uniform thickness is
for maximum
stiffness
699
r(x) and H(r) curves for v = 1 :3 of the optimum
plate
to that of a plate of the same
r, = 1.58’.
(94)
4. DISCUSSION The proof that the optimality condition. equation (13). is the necessary condition of optimal design is based on the assumption of continuity and differentiability of the design variation function. Megarefs [5] has shown in a problem of plastic minimum weight design or circular solid plate for maximum safety that the volume of the plate can be reduced if one allows the discontinuity in the variation of plate thickness. In view of the use of piecewise linear yield conditions with an infinity of compatible strain rates at vertices or edges of the yield locus. compatibility is less stringent in plasticity than in elasticity. Certain conditions of continuity and differentiability would be introduced through the compatibility equation in the elastic problems. Since the shear deformations are ignored in our analysis of beam and plate problems. zero cross sections or zero thicknesses are found at the places where the bending moment vanishes. Shear stress would become infinite at the zero cross section. In order to make the design more realistic. shear deformations must be taken into account and the stress at any point must be restricted within the maximum allowable stress. However the analysis involving the shear deformation and the restriction of local stresses would become complicated. Ackno~,le&emenr-The
author
wishes to express
his thanks
to Professor
W. Praper
for his helpful
suggestions.
REFERENCES :I] W. PRAGER and J. E. TAYLOR. Problems of optimal structural desp. J. appl. Mech. 35. 102-106 (1968) [2] F. I. NIORDSON. On the optimal design of a vibrating beam. Q. appl. Marh. 23.37-53 (1965).
[i)
I TADJBAKHSH and J. 9. KELLER. Strongest columns xnd lsoperlmetrlc (19621 [Jj J 9. KELLER and F. I. NIORDSW. The tallest Column. J .Muth. Mwh. [5] G. J. MEGAREFS. Method Etxrr 92. 79-99 ( 1966)
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