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Chapter 10 Optimal Beam Design
Chapter I 0
Optimal Beam Design
1. Introduction
In this and the following chapter we present a unified methodology for the analysis of a class of optimal design problems. The main idea of these two chapters is to treat, within the same conceptual framework, the seemingly related problems of formulating the model, deriving the optimality conditions, and constructing suitable numerical algorithms. In so doing we are naturally enlarging the scope and results of Chapter 8. Here we treat elastic beams and in the next chapter rotating disks. In the first part of this chapter we present a dynamic programming version of the max-min problem for minimum-weight beam design that results from constraining the value of the potential energy. The thorough treatment of this specific problem should be taken more as an example of the use of dynamic programming procedures to a larger class of problems in structural optimization rather than as an isolated case of theoretical importance. Interest in a dynamic programming approach to structural optimization stems from both practical and theoretical considerations. Some of these aspects are already apparent to the reader. 244
2. MAX-MIN OPTIMIZATION PROBLEM
245
The problem is formulated in Section 2, while in Section 3 the basic dynamic programming equations are obtained. In Section 5 the classical EulerLagrange equations for the beam are derived from the fundamental BellmanHamilton-Jacobi equation. It is shown that the conditions of optimality associated with the minimum operation are expressions of the theorem of Castigliano, an expected result repeatedly obtained. An analytical solution for the optimum cantilever laying on elastic foundation is found in Section 4, while a method of successive approximations consisting in a stable, twosweep iterative procedure is developed in Section 6. In order to illustrate the theory, numerical examples are presented in Section 7. In the second part of this chapter, we use the calculus of variations in the fashion of Pontryagin’s minimum principle to derive optimality conditions for a beam under elastic foundation subject to a displacement constraint. Again, this problem is presented as representative of a larger class of problems in structural mechanics. Using ideas of invariant imbedding and the method of successive approximations, the pertinent nonlinear mixed boundary-value problem is reduced to a two-sweep iterative procedure in terms of a system of Riccati differential equations subject to initial values and exhibiting favorable numerical stability properties. Two examples are finally presented to illustrate the application and accuracy of the method. I. DYNAMIC PROGRAMMING 2. Max-Min Optimization Problem
We consider a beam of stiffness a(x) = EZ, where E is Young’s modulus and I is the moment of inertia, and length L, laying on a Winkler foundation k(x), subject to distributed forcesp(x) and a concentrated force P and moment M at the end x = L. The beam is subject to prescribed deformations, deflection w(O), and slope w’(O), at the origin x = 0. We restrict attention to beams of the sandwich type, so that the volume to be minimized is a quantity proportional to the integral jt a dx. The strain energy is given by the functional
where a prime indicates derivative. We define the potential energy V(w, a) = U(w, a) - W(w,a),
where
(2.2)
246
10 OPTIMAL BEAM DESIGN
is the virtual work of the distributed forces p ( x ) and the concentrated forces P and it4 at x = L, over kinematically admissible displacements. By the theorem of minimum potential energy, the function w* that minimizes V in (2.2) and satisfies the boundary conditions is the “true ” displacement function of the beam with a given design a(x). The value of the minimum is min V(w,a) = -+W(w*, a).
(2.4)
W
Clearly, for a given set of forcesp, P,and M , the virtual work W(w*, a) along the true displacement field can be used as a measure of compliance. Therefore, if our aim is to minimize the volume a dx for a fixed value of the compliance W(w*, a), we can formulate the minimization problem:
1
min[JoLa dx - 2p min(U(w, a) - W(w,a)) , a
W
(2.5)
a min-min problem in the deflection w and the design a, where p is a Lagrange multiplier used to incorporate the constraint. Alternatively, if for a given system of forces we wish to determine a design a for which the compliance W(w*,a) is a minimum among all possible designs with the same volume, we formulate the following problem :
or, on account of (2.2) and (2.4),
a noninterchangeable max-min problem in a and w, respectively. We assume to belong to A, the set of all admissible designs. Clearly, the two problems (2.5) and (2.6) are equivalent and L = l/p. In the following we shall use the formulation given by (2.6). 01
EXERCISE Formulate mathematically the following ma& min, problems: (a) maximum buckling load of an axially loaded prismatic bar for a specified volume; (b) largest minimum frequency of a beam for a specified volume. [In both problems assume the volume to be given by V = j!j g(a)dx, whereg is a specified function of the cross-sectionalstiffness a.]
3. Dynamic Programming Instead of a beam of length (0,L), we consider the family of beams of lengths (2,L), where 0 Iz I L, subject to prescribed displacements
w(x)I,=, = 421,
w‘(x)[,=, = 421,
(3.1)
247
3. DYNAMIC PROGRAMMING
at x = z. Recalling Eqs. (2.1), (2.3), and (2.6), we define the optimum value function f(u, u, z ) = max min a.zA
w
[+ (CLW"'+ kw2 L Jz
- 2pw - h)dx
1
- Pw(L) - Mw'(L) , (3.2)
where V ( Z ) = du(z)/dz.
(3.3)
To obtain a differential equation for the optimum value function, we proceed as usual and find the following functional equation
[
f ( u , u, z ) = max min 6 V - 1 5 dE C[z, ) zw(C) +Az]
Jzz+Az + f ( u ( z + Az), u dx
v(z
1
+ Az), z + Az)
,
(3.4)
where
$1
z+Az
6 ~ =
z
(awn2 + kw2 - 2pw) dx
(3.5)
is the potential energy of the slice of beam between z and z + Az. Equation (3.4) together with ( 3 . 9 , which may be easily justified on mechanical grounds, is a local expression of the principle of optimality. Introducing the curvature K
= d2w/dxzl x Z z ,
(3.6)
standard arguments in dynamic programming show that, as Az+O, Eqs. (3.4) and (3.5) yield
fz
= min
max[ - ( u K 2
a~
+ ku2 - 2pu - h ) / 2 -fu
2,
-f,K],
(3.7)
the Bellman-Hamilton-Jacobi equation for the optimum value function, where f,stands for aflaa, a = z, u, u, andf(u, u, z ) is subject to the initial condition f ( u , u, L) = Pu
+ Mu,
(3* 8)
an equation that follows directly from consideration of Eq. (3.2). Minimizing with respect to K in Eq. (3.7) we obtain K =
(3.9)
-fv/u,
that substituted back into Eq. (3.7) yields
fz
= minv;2/2a a
-fuv - ku2/2
+ p u + Au/2].
(3.10)
248
10 OPTIMAL BEAM DESIGN
If the design a is not subject to constraints, we can minimize in (3.10) by taking derivatives, obtaining in this fashion
f,’la’
(3.11)
=I,
an equation that together with (3.9) yields the optimality condition
I
(3.12)
= K2.
Combining Eqs. (3.10) and (3.1 1) we obtain
f, + uf, + ku2/2 - PU
=f:/a,
(3.13)
=A f:,
(3.14)
or, eliminating a using Eq. (3.1 l),
+ ~f.+ ku2/2 - PU)’
(fi
a nonlinear partial differential equation for the optimum value function subject to the initial condition (3.8). EXERCISES 1. It is known that the buckling load P of a prismatic axially loaded bar is given by
P = min w (f fawe2 dx)
/
(4 J:w”
dx) ,
i.e., by the minimum of a Rayleigh quotient. Show that the optimization problem subject to JOLg(a)dx = V,
max P, (I
where V, the volume, is a prescribed quantity and g is a given function of a denoting the cross-sectional area of the bar, may be formulated as max min 4 a
w
:j(awl2- h,w’, -
(4
2h2g(a))dx,
where h, and h, are two constant Lagrange multipliers. 2. In Exercise 1, assume the column clamped at x = 0 and free at x
f(u, u, x)
= max min 0
1
K
where u(x) = w(x), u(x) = u’(x), and differential equation
f
s:
K =
(aK2
-A,
= L.
u2 - 2hg(a)) dx,
Set (4
u”(x), and show that f satisfies the partial
-fx(u, u, x) = max(-(1/2a)fv2 - fhl u2 - A d a ) +f.u), 01
f(u, u, L ) = 0.
(4
249
4. CONSTANT CURVATURE
4. Constant Curvature
If the curvature of the beam does not change the sign, then Eq. (3.14) admits a remarkably simple solution. In fact, assuming K ( Z ) > 0 for 0 Iz < L, Eq. (3.14) reduces to f,
+ Ofu + ku2/2 - p~ = A 1 J 2 f , ,
(4.1)
which admits a solution of the form
f
= rtu2
+ 2r2 uv + r3 v2 + r4 u + r5 v + r 6 ,
(4.2) where r i , i = 1, . . . , 6, are functions of z that satisfy the following initialvalue system :
dr,/dz = -k/2, dr2/dz= - r l ,
r,(L) = 0, r2(L)= 0,
dr4/dz = p + 2A'I2r2, r4(L) = P, dr,/dz = -r4 + 2A'I2r3, r,(L) = M ,
(4.3)
dr,/dz = - 2 r 2 , r3(L)= 0 , dr6/dz = A1l2r5, r 6 ( L ) = O, obtained upon substitution off given by (4.2) into (4.1) and collecting terms in powers of u and v. The initial conditions follow from consideration of Eq. (3.8). Clearly, Eqs. (4.3) admit a solution in quadratures. For example, assume p and k to be constant in 0 I z I L and consider the initial conditions v(0) = 0.
u(0) = 0,
(4.4)
Integration of (4.3) yields
rt = k t / 2 , r2 = kt2/4, r3 = kt3/6, r4 = P - p t - kA'I2t3/6, r5 = M + P t - p t 2 / 2 - kA'I2t4/8, r6 = - M t - P t 2 / 2 p t 3 / 2+ kA'l2t5/40,
(4.5)
+
where t = L - z. On the other hand, integration of (3.12), where by (3.6) subject to the initial conditions (4.4), yields
u(z) = A'/2z2/2, v(z) = A'/2z. The optimal design is given by Eq. (3.1 l), i.e., 01
=fu/A'12
= (2r2u
K
is given (4.6)
+ 2r3 v + r5)/A'J2,
which on account of Eqs. (4.5) and (4.6) yields 01
=
[M
+ PLS - P L ~ ~ ~ / ~+][2( / A1 ' /9)'~ + 85( 1 - 5)/3 - f2][2kL4/8,
(4.7) where [ =
250
10 OPTIMAL BEAM DESIGN
EXERCISES 1 . Consider problem 2 of Section 3. Assumingg(a) = kal/”,where k is a positive constant and n is a positive integer, solve Eq. (e). 2. If n = 1, show that the curvature of the column, in its deformed position, is a constant.
3. Show that when n = 1, the optimum design is proportional to Lz - x z .
5. Euler-Lagrange Equations
We illustrate the procedure outlined in the last chapter, and derive the pertinent necessary conditions of the present optimization problem. Along an optimal design aOpt,Eq. (3.10) holds without the minimum operation, i.e., f z =f,”/2a -fu v - ku2/2+ PU
+ 2~12,
(5.1)
where u satisfies (3.11). On account of (3.11) we can write (5.1) in the form fz
= ACC- f u v - ku2/2
+ PU.
(5.2)
We compute now the total derivatives o f f , and f u with respect to z, i.e., (dldzlfu = f z u +fuuv +fuuv’, (dldzls, = f z u + f u u v + f U U V ’ ,
(5.3)
and also the partial derivatives f , , and fvz from Eq. (5.2), i.e., fuz
= 2% - f u u v - ku
Lz =
-Luv
+ P,
(5.4)
-fu,
which substituted into (5.3) yields (dlWfI4 = P - ku (dldzlf, =
-fu
+
-fufuula,
+ 2%-Lfvul4
(5.5)
where v‘ has been substituted by its equivalent dvldz
=
-fv/U,
(5.6)
obtained recalling (3.3), (3.6), and (3.9). Now, taking partial derivatives with respect to u and v in Eq. (3.11) we have
2% -f v fuula = 0, La” -f u f o v l a = 0, respectively, so that Eqs. (5.5) reduce to (dldz)S, = P - ku, ( 4 W f V
=
-fu
Y
(5.7)
6 . DESIGN CONSTRAINTS. SUCCESSIVE APPROXIMATIONS
25 1
the equations of equilibrium of the beam which together with Eqs. (3.3) and (5.6) are the Euler-Lagrange equations of the original variational problem (2.6). They hold, of course, for any u other than the optimal one given by (5.9) an equation derived directly from (3.11). In Eq. (5.8) werecognizef,, to be the shear force andf, the bending moment of the beam. In fact, the equations (5.10)
f" = -uw" obtained from (3.6) and (3.9), and
f"= -f,' = (olw")',
(5.1 1)
obtained from (5.8) and (5.10), are local expressions of the theorem of Castigliano obtained via dynamic programming. Elimination of fu between (5.8) and (5.1 1) yields (dZ/dx2)(ud2wldxZ)
+ kw = p ,
(5.12)
the classical equation of the beam where u(z) in Eq. (5.8) has been substituted by w ( 4 . EXERCISE Using the method outlined in Chapter 9, discuss the end conditions on the quantities
f. and fu . 6. Design Constraints. Successive Approximations There are several reasons why we wish to solve the problem of optimum beam design using a method of successive approximations. In fact, closedform solutions of the type obtained in Section 4 are the exception rather than the rule. In general we must resort to Eq. (3.14) or to the more complicated ones which are obtained if the hypotheses of beams of sandwich type are relaxed; and although Eq. (3.14) is subject to initial conditions, its numerical solution is not a routine matter in view of the singularities associated with the inflection points of the beam, i.e., where the curvature changes the sign. In addition to this we observe that Eq. (3.14) was derived assuming that there are no constraints on the design, a condition seldom met in practice. It is therefore desirable that the development of successive approximation schemes suitable for the solution of various classes of problems be derived in a unified fashion.
252
10 OPTIMAL BEAM DESIGN
We start by considering Eq. (3.10). For a fixed design oro this equation holds without the minimum operation, i.e.,
f, =fv2/2u0 -f , v
- ku2/2
+ PU + Ia0/2,
(6.1) subject to the initial condition (3.8). Equation (6.1) admits a solution of the form
where si,
f = s1u2 + 2s, uv + s3 v2 + s4 u + s5 v + ss , i = 1, . . . , 6, are functions of z given by ds4/dz= 2s, s5/ao + p (d) ds,/dz= 2s, s5/ao - s4 (e) dS6/dZ= s52/2ao+ h 0 / 2 (f)
dsl/dz= 2szz/u0- k/2 (a) ds2/dz= 2s2s3/a0- s1 (b) ds3/dz= 2s3’/a0 - 2s, (c)
(6.2)
(6.3)
a system of Riccati differential equations subject to the initial conditions S,(L) = S z ( L ) = s3(L) = Sg(L)
s4(L) = P,
= 0,
S,(L) = M ,
(6.4)
obtained from consideration of Eqs. (3.8) and (6.2). Functions u and v can be obtained from Eqs. (3.3) and (5.6), i.e.,
du/dz= V , dvldz = -(2s2 u + 2s3 v
+ s,)/ao,
(6.5)
a coupled system of differential equations subject to the initial conditions
u(0) = uo ,
v(0) = 00,
(6.6)
where uo and vo are prescribed values of the deflection and the slope at z = 0, respectively. Now we are in a position to implement a successive approximation scheme for the computation of the optimum structure. In fact, for a given estimate uo of the design, we integrate Eqs. (6.3a)-(6.3e), and obtain si,i = 1, . . . , 5, which substituted into (6.5)-(6.6) yield functions u and u by forward integration. Using those quantities we can therefore compute a’ = arg min[2(s2 u asA
+ s3 v + s5/2)’/a + la/2],
(6.7)
an equation of optimality derived directly from (3.10) and (6.2), which yields an improved estimate for the design. The quantity I appearing in (6.7) can be computed easily by making use, at each iteration of the process, of Eq. (3.1 1) which, on account of (6.2), reads
1 = (s5(o)/a0(o))2 at z
= 0.
This process is repeated until convergence is achieved.
(6.8)
253
7. NUMERICAL EXAMPLE
EXERCISE How would you proceed to satisfy, at each iteration of the process, the condition
Pu(L)
+ MU@)+
p u dx
(a)
= C,
JoL
where P,M, and c are prescribed quantities? Hint: Instead of (6.8), compute h to satisfy (a).
7. Numerical Example To show the feasibility and accuracy of the method developed in Section 6, we present a numerical example involving a cantilever beam on elastic foundation subject to a load P and a moment M such that the rotation at the free end x = L is zero. This last condition is incorporated for the sole purpose of simplifying the analytical solution. In fact, the exact solution for the optimum design of the present beam configuration and loading condition can be found easily by application of the results in Section 3. From Eq. (3.12) we have K =
+A”’,
(7.1)
i.e., the curvature in absolute value is a constant along the entire beam. Therefore, the elastica consists of two arcs of a parabola with the same curvature intersecting at an inflection point xo = L/2 and satisfying the geometric boundary conditions
w(0) = w’(0) = w’(L) = 0, w’(x0-) = w‘(x, +). By requiring that the slope of the two parabolas be equal at xo, the shear force V at the inflection point can be found easily be integration of all the forces acting perpendicularly to the beam, obtaining in this fashion V = P - 5kL3A“’/48.
(7.3)
Using V furnished by (7.3) we can now easily compute the moments acting along the entire beam, namely,
m(5) = (PL/2)(1 - 25)[1 - (23 - 25 - 45’ - 8t3)>lB],
5 5 3,
(7.4)
3I 5I 1,
(7.5)
05
and
m(5) = (PL/2)(25 - 1)[1
- (21
+ 105 - 285’ + 8t3)/P],
where
p = 192P/kL3A1I2.
(7.6)
254
10 OPTIMAL BEAM DESIGN
In particular, the moment M necessary to ensure the horizontality of the elastica at 5 = 1 is given by Eq. (7.5), namely, M
= m(1) = (PL/2)( 1
- 21/P).
(7.7)
The optimum design follows from Eq. (3.1 I), i.e., a = m(5)/A'I2.
Therefore &(<) = (1
0 < 5 < 3,
(7.8)
+ 105 - 28t2 + 8y3)/P], 3 I 5 I1 ,
(7.9)
- 25)[1 - (23 - 2< - 4 t 2 - 853)/p],
and
Cr(5)
= (25
- 1)[1 - (21
where Cr is the dimensionless quantity given by Cr = ( 2 F / P L ) u *
(7.10)
Clearly, P = 23 is the smallest value with physical meaning that this parameter cantake. The longitudinal stresses in the flanges, in absolute value, are constant over the entire beam and their value is given by
I o I = A1/'Eh/2,
(7.1 1)
where E is the modulus of elasticity and h is the height of the beam, i.e., the distance between the flanges. Hence, for a prescribed value of the stress u, A l l 2 can be computed from (7.1 1) and its value substituted into (7.6) to compute the dimensionless quantity P. The optimum dimensionless design ti, for several values of the parameter P, is presented in Fig. 10-1. X
h
I .o
0.8
dko,6 ,I
IU
2 04 ' cn W
0.2 0 0
0.1
0.2
0.3
0.4
0.5
[=
xL
0.6
0.7
0.8
0.9 I 3
Fig. 10-1. Optimal beam designs, where a = El, p = 192P/kCL3, C = 2u/Eh, and h is the height of the beam. (After Distkfano [1972].)
TABLE 10-1 Comparison of Exact and Computed Designs &ompvtcd
5
(constraint: Emin= 0.005)
L a c *
Initial estimate
1st iteration
2nd iteration
3rd iteration
5th iteration
7th iteration
9th iteration
11th iteration
0.77718 0.47106 0.16139 0.16090 0.50843
0.77199 0.46722 0.15887 0.16220 0.50880
0.89000
0.89000
0.77055 0.46615 0.15817 0.16265 0.50890 0.89000
0.77015 0.46586 0.15798 0.16267 0.50893 0.89000
12th iteration
~
0.00 0.20 0.40 0.60 0.80 1.0
0.77000 0.46574 0.15790 0.16270 0.50894 0.89000
1.0 1.0 1.0 1.0 1.0 1.0
0.89812 0.55810 0.21538 0.13570 0.50212 0.89000
0.82566 0.50691 0.18487 0.14876 0.50495 0.89000
0.79773 0.48631 0.17144 0.15566 0.50692 0.89000
0.77008 0.46580 0.15794 0.16268 0.50894 0.89000
256
10 OPTIMAL BEAM DESIGN
The solution of the same problem, using the method of successive approximations developed in Section 6, can be obtained as follows. At a generic iteration, using the available design a', we integrate Eqs. (6.3a)-(6.3e) in the backward direction, obtaining the values of si,i = 1, . . . , 5, which substituted into (6.5)-(6.6) yield functions u and v by forward integration. Using these quantities we compute an improved design a' by making use of Eq. (6.7). In the numerical example presented in Table 10-1 and corresponding to fi = 100, we started the process by considering a uniform design a' = I . The integration was performed using an Adams-Moulton scheme with step size H = L/200. It is noticed that the singular behavior of the derivatives in Eqs. (6.3), in the neighborhood of the inflection point, prevents in general the routine use of standard methods of numerical integration. A number of asymptotic solutions near the singularity can be obtained and the corresponding results can be matched to the numerical solution furnished by the Adams-Moulton integration scheme, in the neighborhood of the singularity. In the present case it was found that the simple device of imposing a constraint on the design of the form c12 0.005 worked very well and avoided the need of complicating the routine of integration. The results in Table 10-1 show the accuracy of the method.
11. INVARIANT IMBEDDING 8. Alternative Formulation
In order to incorporate displacement constraints and additional local restrictions on the design and the state variables, the remainder of this chapter is devoted to an alternative formulation of the beam-optimization problem. In view of the different imbedding used here, we shall modify the notation slightly. To illustrate the procedures we consider a beam of finite length L, laying on elastic foundation with coefficient k(x) and subject to external forces q(x) that might eventually contain a singularity due to a concentrated load. If u(x) denotes the deflection, v(x) the slope, m(x) the bending moment, and t(x) the shear, the constitutive equations of a Euler-Bernoulli beam are given by duldx = v, dvldx = - (1/a)m, (8.1) and the equilibrium equations by
dmldx
=
-t,
dtldx
= q - ku,
(8.2)
where
u
= EZ
(8.3)
257
8. ALTERNATIVE FORMULATION
is the stiffness, I is the moment of inertia, and E is Young’s modulus. We choose the stiffness a as the design variable, and assume that the crosssectional area A of the beam is given by the formula
where g is a function that depends on the particular geometry of the cross section. For example, for rectangular beams with constant width b and variable height h, g = (12bZa/E)’/3.The volume of the beam is given by .L
Considering u, v , m, and t subject to an appropriate set of boundary conditions, we can now formulate the following minimum volume problem for a prescribed displacement u1 at x = x l , and additional inequality constraints: Minimize the quantity V given by ( 8 . 5 ) subject to the conditions
and possibly to a number of additional constraints cpi(m,t,a,x)
i=1,2
,....
(8.8)
We assume that this optimization problem is well posed; i.e., a solution exists, is unique, and is continuous with respect to boundary conditions and constraints. The study of classes of constraints under which a given optimization problem is well posed is one of great interest and importance, but it is beyond the scope of this chapter whose main purpose is the discussion of numerical computational procedures. The derivation of the necessary conditions for this problem may be done using dynamic programming as outlined in Chapter 9. We leave this derivation as an exercise to the reader. Here we proceed directly, using the formalism of the minimum principle. In order to incorporate the constraint on the deflection u, instead of (8.6) we consider the equivalent integral expression
where 6 is the Dirac delta. Now we form the Hamiltonian
258
10 OPTIMAL BEAM DESIGN
where 1and I l to 1, are Lagrange multipliers used to incorporate the constraints (8.1), (8.2), and (8.9). It is well known that the multipliers satisfy the adjoint differential equations
(8.1 1)
From the first equation of (8.11) we see that 1is a constant, to be chosen to enforce condition (8.6). Comparison of (8.1) and (8.2) with Eqs. (8.11) shows that the Lagrange multipliers I , to I , are the forces and displacements of the beam subject to a virtual concentrated load of intensity I at x = xi. More precisely,
I,
= AU,
2,
= 16,
12
=
-26,
I1 =
-If,
(8.12)
where U , 6, E , and t are the displacements and forces due to a unit virtual force at-xl. Minimization of (810), taking into account (8.12), yields the optimality condition aOpt= arg min[l(mE/a)
+ g ( 4 41,
(8.13)
a
where clop, denotes the optimum design. EXERCISES 1. Let f(u, u, t , rn, x ) be the minimum volume function given by
f(u, u, t , m, x ) = min a
S:
g(a,x ) dx,
(a)
subject to the differential constraints (8.1) and (8.2) and to the deflection constraint (8.6). Show that f satisfies the partial differential equation
-f = min[g(a, x ) + puIS(x - X I ) +f.u m
- (l/a)fum +fh- ku) - A t ] ,
(b)
where p is a constant Lagrange multiplier. 2. Using the method outlined in Chapter 9, derive ordinary differential equations for f., fu, fc, and fm . How are these quantities related to the adjoint variables X I to satisfying (8.1l ) ? 3. How would you include consideration of constraints such as:
Ilongitudinal stress I < u,, ,
I slope I
=L
< uL ?
9. AN ALTERNATIVE DERIVATION OF THE OPTIMALITY CONDITION
259
4. Consider a sandwich beam as indicated in Fig. 8-1. Let E = F(U)
be the (nonlinear) stress-strain law of the flanges. Therefore the curvature K will be given bY K = -(1/2h)[F(m/Ah) - F(-m/Ah)]. (4 If, additionally, we consider a nonlinear elastic foundation characterized by the forcedisplacement relation p=ku-pu', 820, (6) the differential equations of the beam will be given by duldx = U, = - ( l / X ) [ F ( m / A h ) - F(-m/Ah)],
du/&
dm/& = -t, dtldx = q - ku
(0
+ Pu'.
Assuming F to be continuously differentiable with respect to A, derive the pertinent necessary conditions for the following optimization problem: min /'A dx, A20
0
subject to the deflection constraint (8.6) and the differential constraints (f). Use A as the pertinent design variable.
9. An Alternative Derivation of the Optimality Condition Instead of incorporating the differential constraints (8.1) and (8.2) using the Lagrange multipliers I1 to I,, we can proceed in the following way: In place of (8.9) we use the integral representation "L
+
u1 = ~o(mfi/cc kuii) dx,
(9.1)
furnished by the theorem of virtual work, where Z and i7 are the moment and the displacement, respectively, of the beam subject to a virtual load applied at x1 and in the direction of the prescribed displacement. Combining (8.5) and (9.1) we form the Hamiltonian 231 = g(a, x )
+ I(rnE/a + h a ) ,
(9.2)
where 1 2 0 is a Lagrange multiplier that satisfies the equation
dI/ax = - aH1laul = 0.
(9.3) Hence I is a positive constant which is to be chosen to satisfy (8.6). Clearly, minimization of (9.2) yields (8.13), as expected.
260
10 OPTIMAL BEAM DESIGN
EXERCISES 1. Derive the optirnality condition from Eq. (b) of Exercise 1, Section 8.
2. Assuming a not subject to constraints andg(a, x ) = k ~ ( ' /derive ~ , the pertinent condition of optirnality starting from (8.13).
10. Optimum Cantilever under Prescribed Displacement
To illustrate the method and the problems associated with the numerical computations, we consider the case of a cantilever beam laying on elastic foundation and subject to a prescribed displacement at x = xl. The boundary conditions of Eqs. (8.1) and (8.2) are u(0) = 0,
~ ( 0=) 0,
m(L) = M , t(L) = T.
(10.1)
For simplicity we rewrite Eqs. (8.11) in terms of U, V , E,and i given by
dU/dx = i5, dV/dx - (1/a)E, 1
dE/dx = - i , di/dx = 6(x - xl) - ku,
(10.2)
subject to the homogeneous boundary conditions
U(0) = V(0) = 0,
E(L) = t(L)= 0.
(10.3)
So formulated, the solution of the optimum beam reduces to the task of integrating Eqs. (8.1), (8.2), and (10.2) subject to boundary conditions (10.1) and (10.3), respectively. These two systems of equations are coupled together through the optimality condition (8.13). In a number of important cases in the applications we possess an explicit representation for the minimum operation in (8.13). This simplifies in some sense the solution of the system. In any case, this nonlinear boundary-value problem can be integrated using a quasilinearization scheme. (See the exercises below.) We do not pursue this path here, i.e., a direct treatment of the nonlinear boundary-value problem, in favor of the implementation of a simple, first-order, stable iterative method based on ideas of invariant imbedding. This is done in the following section. EXERCISES 1. Assuming u, u, t , m subject to boundary conditions (10.1) show thatfin Eq. (b), Exercise 1, Section 8, satisfies the conditions
fmb, u, t , m,0) = 0, fXu, u, t, m,0) = 0,
A h , 0, t , m,0 = 0, f d u , u, I, m,L) = 0.
(a)
2. Write the boundary conditions for the adjoint variables 8,6, i, and 177,when the beam is (a) clamped-clamped, (b) clamped-hinged, (c) hinged-hinged.
26 1
11. INVARIANT IMBEDDING
3. Derive the conditions on f satisfying the partial differential equation (b) in Exercise 1, Section 8, for the cases indicated above.
4. Implement a quasilinearizationscheme for the solution of Eqs. (S.l), (8.2), (10.1)-(10.3), and (8.13). 5. Implement a quasilinearizationscheme for the solution of Exercise 5, Section 8.
11. Invariant Imbedding
For a given nominal design c1, we consider the uncoupled linear boundaryvalue systems given by Eqs. (8.1), (8.2), (lO.l), and (10.2), (10.3), respectively. First we consider system (8.1)-(8.2). Instead of boundary conditions (10.1) we set m(L) = M , u(X) = w, (11.1) t(L)= T, v(X) = Z, i.e., we consider the families of beams of length L - X subject to arbitrary displacements w and z at the end x = X . This is an “imbedding” procedure that in the fashion of invariant imbedding affords the property of reducing the computation of the original boundary-value problem (8. l), (8.2), (lO.l), to a stable, two-sweep procedure. We seek solutions of the form
t ( X ) = r,(X)w + r12(X)z + S l W , m ( X ) = rzl(X)w r,(X)z s2(X).
+
+
(11.2)
Differentiation of (1 1.2) with respect to X and elimination of the derivatives
w’ = u’(X) and z’ = v’(X) using Eqs. (8.1) and (8.2) evaluated at x = X and further elimination of t(X) and m ( X ) using (1 1.2) yields (rl’
+ k - r12(1/cOr21)w+ (G2+ rl - rl2(l/a)r2>z+ (sl’ - q - rlz(l/4sz) = 0,
and
(rL + rl - r 2 ( 1 / 4 r 2 d w + h’+ r12 + r21 - r,(l/4rz)z
+ (s,’ + s1 - r2( l/u)s,)
= 0,
a system of equations that must be valid for any w and z. Therefore
+
rl’ = - k rl2(1/c1)rZ1, r2’ = -rlz - r21 + r2(1/u)r2, (11.3) r i 2 = -rl + rl2(1/4r2 , s1’ = q + r12U/a)s2, s2’ = -sJ + r2(l/u)s2, ril = -rl r2(1/u)r21, a system of Riccati equations subject to the initial conditions at X = L,
+
r,(L) = rI2(L)= r21(L)= r,(L) = 0,
s1(L) = T,
s2(L)= M ,
(11.4)
262
10 OPTIMAL BEAM DESIGN
obtained upon consideration of Eqs. (1 1.1) and (1 1.2). Consideration of the second and third equation in (1 1.3) and corresponding initial conditions readily yields r12 = r21, (11.5) an expression of Maxwell’s theorem derived from invariant imbedding. Therefore Eqs. (1 1.3) and (1 1.4) reduce to
+ (l/a)r;,, = -r, + (1/.)rlZ~Zr = -2r1, + (1/a)rZ2, 81’ = q + (l/a)r12sz -k
r,’ r;, r,‘
=
s‘,
= -sl
r,(L) = 0, r,,(L) = 0, r,(L) = 0,
(11.6)
s,(L) = T, s,(L) = M .
9
+ (l/a)rzs,,
Substitution of m(X) given by the second equation of (11.2), into Eqs. (8.1) evaluated at x = X and due consideration of Eqs. (10.1) yield u(0) = 0, du/dX = V , (11.7) dv/dX = -(l/a)(r,, u + r2 v + s2), v ( 0 ) = 0, an initial-value problem in the forward direction for the deflection u and slope v of the beam, where the quantities r , , , r , , and s, are given by the integration of (1 1.6) in the backward direction. Substitution of u and v given by (1 1.7) into (1 1.2), taking into account that w = u ( X ) and z = v ( X ) , finally yields the remaining state variables of the beam, i.e., the moment m and the shear force t in the interval 0 I X 5 L . We can treat the virtual system (10.2) in a similar fashion. We consider U, 6, iii, and t satisfying Eqs. (10.2) to be subject to the boundary conditions
U(X)= w,
E(L) = 0, t(L)= 0,
q x ) = 5,
(11.8)
and write for t ( X ) and % ( X ) ,the missing boundary conditions of the imbedded beam of length L - X , equations similar to (1 1.2), i.e.,
f ( X )= T;,(X)W+ f,,(X)Z + S,(X), E ( X ) = T‘,l(X)W r,(X)z S,(X).
+
+
(11.9)
Carrying out the same perturbation analysis done to Eqs. (1 1.2), on Eqs. (1 1.9), we obtain
+ (l/~)?;,, i ; , = - r, + (l/a)r,, r,, 7,’ = -2r12 + (l/a)F22, F,’= -k
+
6,’ = 6 ( X - x1) (l/a)?,, 6,‘ = - 6 , ( l / a ) 7 , s,,
+
f,(L) = 0, = 0, F,(L) = 0, S,(L) = 0, S,(L) = 0.
F12(L)
s, ,
(11.10)
263
12. A TWO-SWEEP ITERATIVE PROCEDURE
Similarly, we can write dU/dX = O, dO/dX = -(l/a)(F12 U + i 2 ij + iz),
U(0) = 0,
(1 1.11)
O(0) = 0,
an initial-value problem for U and ij obtained after substitution of E,given by the second equation of (1 1.9), into the first two equations of (10.2) evaluated at x = X . Finally, t and E can be obtained from (1 1.9) recalling that W = U(X) and Z = tj(X). EXERCISES 1. If, instead of (ll.l), the boundary conditions were u(X) = w, m(X)= M,
u(L)= z, t(L) = T,
how would the initial conditions of (11.6) change? 2. Repeat the previous analysis for the conditions u ( X ) = w, v ( X ) = 2,
u(L) = 0, u(L) = 0.
In this case, some of the r's in (11.3) are not defined at x = L. Derive approximate asymptotic expressions for those quantities at x = L - A, A < 1.
12. A Two-Sweep Iterative Procedure
Using the results of the preceding section we can now implement a simple iterative method for the solution of the optimization problem. To this end let dn)denote the value of the quantity a at the nth iteration. Then at a generic iteration (n + 1) we use the currently available design a(") to integrate (11.6) in the backward direction and subsequently (1 1.7) in the forward direction, computing in turn m("+')given by (1 1.2). Similarly, we compute the values of E'"+ by using the two-sweep process given by (1 1.9)-(11.11). The improved value of the design a("+') follows from the optimality condition 1)
= arg min[l(n+
')m("+1 )/a) + da,41,
(12.1)
a
where the new estimate of 1= A("+') needs to be taken so as to ensure an appropriate convergence of the sequence u(")(xl) to the prescribed value ul. This can be done in a number of ways. In general we shall use formulas of the type
A("+1) = F(1("), u(")(xl),u("-l)(xl),...).
(12.2)
264
10 OPTIMAL BEAM DESIGN
(12.3)
The procedure is repeated until convergence is achieved. Initially we need an a priori estimate of the design a(’). In the absence of special information, a uniform design ~(‘’(x) = constant is usually taken as the initial design. A numerical example in Section 15 illustrates the application of the method. EXERCISES 1. Consider a circular plate of radius a, thickness h, Young’s modulus E, and Poisson’s ratio v. If u denotes the deflection and u and m the radial slope and bending moment, respectively, the equations of equilibrium of the plate undergoing bending may be written u‘
=u
u’ = -(v/r)u - ( l / a ) m m‘ = -((I - v)/r)m- ((1
(a) - v2)/r’)au - Jqr,
where r is the radial coordinate, q(r) is the load, and the stiffness a is given by a = Eh3/(12(l
- v’)).
(b)
Assuming one of the following boundary conditions: Clamped Simply supported
u(a) = 0,
u(0)
u(a) = 0,
u(0) = 0,
= u(a) = 0,
m(a) = 0,
(4
and using the method outlined in this and foregoing sections, solve the following optimization problem: min 1;ra1l3 dr,
subject to the displacement constraint u(0) = 1.
(d)
Clearly, the integral in Eq. (d) is a quantity proportional to the volume of the plate. 2. A circular footing with outer radius a is carrying a column of radius 6, with axial load P. Assuming thin-plate theory and Winkler foundation, the associated boundaryvalue problem for the bending of the footing is given by P
u’ = u,
u(b) = 0,
- ( l / a ) m - (v/r)u, m‘= -((I - v)/r)m - ((1 - v2)/r2)au- ikur,
u(a) = 0,
U’ =
I
2
a
I
and the additional equilibrium condition P = nb’ku(b)
+ 2njbakurdr,
m(a) = 0,
(4
265
14. STABILITY CONSIDERATIONS
where k is the coefficient of the foundation. Derive the optimality conditions and implement a two-sweep method for the solution of the following optimization problem: max P,
(g)
subject to
dr = C, where C is a quantity proportional to the volume of the plate, and to the displacement constraint 0 Iu(b) 5 c. 3. In Exercise 2 above, include a constraint on the absolute values of the shear and radial stresses.
13. Inertia Forces When the weight of the beam is to be taken into account, q in Eqs. (8.2), (8.10), and (11.6) must be replaced by 4=p
+ YA,
(13.1)
where p is the external forces, y is the specific weight of the beam, and A is the cross-sectional area given by Eq. (8.4). Therefore the optimality condition (8.13) must be substituted by uOpt= arg min[A(m%/u)
+ (1 + y/lU)g(u,XI],
(13.2)
a
and Eq. (13.1) must be taken into account when integrating Eqs. (11.6). Similarly, other mass forces can be taken into account. EXERCISE
In Exercise 1 of Section 12, incorporate the weight of the plate in the formulation of the optimization problem.
14. Stability Considerations
In order to prove the stability of the method, we should show that u and u given by (1 1.6) and ( 1 1.7) are stable with respect to small errors introduced in the computational process. To this end it is enough to prove the stability of the differential system (1 1.6) and, in particular, of the quantities rl, r12, and r 2 . We rewrite the first three equations of (1 1.6) in the matrix form R' = - A - BR - RBT + RCR,
where
R(L) = 0,
(14.1)
266
10 OPTIMAL BEAM DESIGN
It is easy to show, by recalling the results of Section 3-12, that the stability of R in the backward direction will depend on the stability of the differential equation dz/dx = ( B - RC)Z
(14.3)
in the forward direction, which in turn follows from consideration of the Liapunov function V ( Z ( X )= ) (z(x), R - ’ ( L - x)z(x))
(14.4)
which clearly exists, is bounded, and is positive definite for x > 0, since R-’ is a flexibility matrix. We leave the details of the demonstrations to the reader. EXERCISES 1 . What are matrices A , B, C, and D in (14.2) when, instead of (8.1), we use the constitutive equations
+
a l l t + a12m, u’=a21t+a22rn,
u’= u
where ax2 = a z l ? 2. When we use Eq. (a) above and assume that the matrix (a,,) is positive definite, do we still enjoy the stability of the Riccati equation (14.1)? 3. Prove the numerical stability of all the quantities involved in Sections 11 and 12.
15. Numerical Examples a. Cantilever on Elastic Foundation
We consider a beam of the sandwich type such as that whose cross section is indicated in Fig. 8-1. In this case if 2h is a fixed quantity denoting the distance between the covering sheets and A/2 is the area of each one of the sheets, we have tl = EZ = EAh’, and g in (8.4) reduces to g(a, X) = UlEh’.
(15.1)
The volume is therefore proportional to a, i.e., L
V = ( l / E h 2 ) I adx.
(15.2)
0
In addition we shall take x1 = L and q = P6(x - L ) ; i.e., we prescribe the displacement u1 in correspondence with a concentrated load applied at the free end of the cantilever. The reason to consider this simplified example is because under the present assumptions we can afford a closed-form solution
267
15. NUMERICAL EXAMPLES
of the problem that can be used to compare the accuracy of the numerical procedures. In fact, in our present case we have m = P E and Eq. (8.13) reduces to
+ E),
aOpt = arg min(p(m2/a) U
(15.3)
where p = (EhZ/P)Ais a constant to be determined such that u(L) = ul. Minimizing by taking derivatives, Eq. (15.3) reduces to =p
(15.4)
2 .
Combining (8.1) and (1 5.4) we obtain (u")2
= lip.
(15.5)
A solution that satisfies the nonlinear differential equation (15.5) and the boundary conditions u(0) = u'(0) = 0 and u(L) = u1 is given by (15.6)
u(x) = U 1 X 2 / L Z .
Equation (15.6) is valid if sign m = sign u", a condition that is clearly fulfilled in our example. The curvature is given by lu"l = (l/p)l'Z
= 2u,/L2.
(15.7)
Combining (15.4) and (15.7), aopt= LZm/2ul,
(15.8)
where the bending moment m can be computed readily by direct integration, namely, m = P ( L - x) -
1°C.- x)ku
dz
0
= P L ( ~- rl) - kU1L2(3- 4q
+ 14)/12,
(15.9)
where we put q = x/L. In a similar fashion we can derive for the shear force t the expression t =~
[-i4 j ( i - 191,
(15.10)
where
j= ku1L/12P.
(15.11)
Combination of (15.8) and (15.9) yields Lop, = 1 - q - p(3 - 41
+ q4),
(15.12)
where c1 is a dimensionless design given by
L =2u,a/P~~.
(1 5.1 3)
268
10 OPTIMAL BEAM DESIGN
0.8
0.6
a OPT. 0.4
0.2
0.0
0.2
0.4 0.6 r] =x/L
0.8
I .o
Fig. 10-2. Optimal beam designs, where EOp,= (2ul/PL3)aOp, and (After Distefano and Todeschini [1972].)
= kLul/12P.
Clearly, the range of p for which clopt is positive is 0 Ip IQ. In Fig. 10-2, Eopt given by (15.12) is presented for several values of p. For purposes of comparison, the same example was solved using the procedure outlined in Section 12. At a generic iteration we integrate Eqs. (11.6) in the backward direction using thecurrently available estimate for the design. There is no need to integrate Eqs. (1 1.10) since in this particular case we have m = PE. Subsequently, using the values of the r's previously computed, we integrate (1 1.7) in the forward direction. In turn we calculate a new, upgraded estimate of the design by means of &+I) = (/p+l) l / z m c n + l , ( 15.14) 1 9
an expression for u("+') that follows from (12.1) by taking derivatives and making m = PE and g = a/EhZ. In Eq. (15.14), p ( " + l ) ,given by p(n+l)
= (Eh2/p)A(n+l),
(15.15)
is a constant at each iteration that must be chosen to ensure the convergence of the sequence u(")(x,) to the prescribed value ul. We have taken in the present case p(n+l) = p(")u(")(xl)/ul.
(1 5.16)
The procedure is continued until the following convergence criterion Ek=max(lEkI, IEk+l[,IEk+21)
(1 5.17)
269
15. NUMERICAL EXAMPLES
where k --
1 - ,$k)/ki'k
+ 1)
(1 5.18)
is fulfilled. All the integrations were performed numerically using an AdamsMoulton scheme with step size 0.005 on a CDC 6400 computer. The resulting values for three different values of and at four equidistant sections of the cantilever are presented in Table 10-2. It is seen that in order to reach the same accuracy, the number of iterations increases for increasing values of the design parameter p. On the other hand, it is of interest to note that the convergence of the process is of an oscillatory type. This can be appreciated in Fig. 10-3 where the relative error &k versus the number of iterations k for two different values of has been plotted. Methods to improve the convergence properties of the process can be devised for the present problem but we do not enter into that discussion here. 0.02
0.01
-0.001
-0.01
-0.02 Fig. 10-3. Convergence of approximations where (After Distkfano and Todeschini [1972].)
E~
= 1 - 6,ck)/GckI ) and +
7 = 0.0.
N
4
0
TABLE 10-2 Comparison of Exact and Computed Designs. Cantilever Beam on Elastic Foundation"
1 12
0.00 0.25 0.50 0.75
0.75000 0.58301 0.41146 0.22363
2.0 2.0 2.0 2.0
0.74922(8) 0.58246(8) 0.41 114(8) 0.22351(8)
0.74983(9) 0.58290(9) 0.41 14Q(9) 0.22362(9)
0.82 x 0.75 x 0.64 x 0.48 x
10-3(8) 10-3(8) 10-3(8) 10-3(8)
0.78 x 0.55 x 0.32 x 0.12 x
10-3(8) 10-3(8) 10-3(8) 10-3(8)
1 6
0.00 0.25
0.5oooO
0.75
0.41602 0.32292 0.19727
2.0 2.0 2.0 2.0
0.5oooS(15) 0.41604(15) 0.32315(14) 0.19732(14)
0.49979(16) 0.41584(16) 0.32288(15) 0.19721(15)
10-3(15) 10-3(15) 10-3(14) 10-3(14)
0.25000 0.24902 0.23438 0.17090
2.0 2.0 2.0 2.0
0.25026(28) 0.24907(27) 0.23432(26) 0.17084(25)
0.25040(29) 0.24927(28) 0.23450(27) 0.17097(26)
10-3(15) 10-3(15) 10-3(14) 10-3(14) 10-3(28) 10-3(27) 10-3(26) 10-3(25)
0.09 x 0.02 x 0.23 x 0.05 x
0.00 0.25 0.50 0.75
0.60 x 0.48 x 0.83 x 0.55 x 0.59 x 0.80 x 0.80 x 0.74 x
0.26 x 0.05 x 0.06 x 0.06 x
10-3(28) 10-3(27) 10-3(26) 10-3(25)
0.50
1
7
1
0
0
4
5
r W
Numbers in parentheses denote the number of iterations.
K? n
27 1
15. NUMERICAL EXAMPLES
A further application of the method in connection with piecewise constant design is presented below. The design is taken in the form
c N
u=
(15.19)
C i H ( X - Xi),
i=l
where H is the usual unit step function. The problem consists in determining the cross-sectional stiffness ci and the lengths x i which minimize the volume of the beam. The values of ci are constrained to be one of any possible combinations of a given set of values a l , a 2 , . . . , a M . This is a version of a piecewise constant optimum design problem that occurs when the flanges of the beam must be constructed using a number of available sections a l , u 2 , . . . , aM. Substituting a given by (15.19) into the optimality condition (15.3), we can solve this problem by minimizing with respect to all possible values of ci. In the present example we have considered a1 = u2 = ... = us = 0.125. Therefore ci = 0.125j(i) where the possible values of j are 1, . . ., 6. The results are shown in Fig. 10-4 where a comparison with the unconstrained solution for B = & is possible. LO\
I
I
I
I
I
\
0.6
-
\
I
I
I
-
- \ \ 0.8 \up 0 \
I
-
\
\
\
212
10 OPTIMAL BEAM DESIGN
b. Clamped Beam under Uniformly Distributed Load An optimal design is characterized by in&
> 0,
(15.20)
a convexity condition that follows from consideration of the optimality condition (8.13). During the computation of the successive approximations following the procedure developed in Section 12, Eq. (1 5.20) might be violated leading to an indeterminacy in Eq. (8.13), unless additional information is furnished. This can be done in a number of ways. For example, we can require a positive lower bound for the design E, i.e., (15.21)
Ci26>0,
where 6 is usually taken to be the order of magnitude of the step of integration of the equations. This problem did not occur in our previous example involving the cantilever on elastic foundation, because in that case, Eq. (15.20) was identically satisfied since m = PE. The purpose of the present example is to show that Eq. (1 5.21) is enough to bypass the difficulties created by a possible violation of (15.20) during the first iterations of the process. To this end we consider a sandwich-clamped beam of length 2L subject to uniformly distributed load q. It is required that the deflection at the middle be u l . The exact optimal solution can be obtained without difficulties. In fact, assuming E to be continuously differentiable, the exact solution of this problem has been given in Section 8-16. The optimal design obtained using the method of successive approximations may be compared with the exact one, given in Chapter 8, in Fig. 10-5 and Table 10-3. In this example, the dimensionless design c( = (10u1/qL4)ccand the dimensionless quantities q = xlL and qo = X[L were introduced. The procedure used to TABLE 10-3
Comparison of Exact and Computed Designs. Clamped-Clamped Beam under Uniformly Distributed Load cy(rlj
C(O’(7lj
7
exact
initial approximation
0.00 0.20 0.40 0.60 0.80 1.00
0.75560 0.42326 0.13168 0.11387 0.30576 0.43171
0.5 0.5 0.5 0.5 0.5 0.5
0.58926 0.74725 0.30957 0.39508 0.05271 0.08591 0.13437 0.17484 0.30277 0.37943 0.41667 0.5 1566
0.72994 0.39632 0.10335 0.14341 0.33713 0.46570
0.74267 0.74948 0.40962 0.41649 0.11719 0.12414 0.12904 0.12198 0.32215 0.31492 0.44985 0.44228
NOTES, COMMENTS, AND BIBLIOGRAPHY
I
*
O
*
213
0' .0
'OPT.
q=x/L
Fig. 10-5. Comparison of successive approximations with optimal design in a clamped beam under uniformly distributed load, where GOpt= ( ~ O U ~ / ~ L and ~ ) LT~ Y= ~ ~0.50224. , (After DistCfano and Todeschini [1972].)
compute the solution using the method of successive approximations is similar to that presented in Section 12, making k = 0, and where the Riccati equations (1 l .6) and (1 l . 10) were subject to appropriate initial conditions in order to account for the difference in boundary conditions of the present beam. NOTES, COMMENTS, AND BIBLIOGRAPHY
2-7. In Part I of this chapter we present a dynamic programming treatment of the min-max problem which results from constraining the work of deformation of the given external loads on their own displacements. Here we follow closely N. DistCfano, Dynamic Programming and a Max-Min Problem in the Theory of Structures, J. Franklin Inst. 294, No. 5 (1972), 339-350.
In addition to the work by Wasiutyriski quoted in Chapter 8, the criterion of the work of deformation has been applied to some problems involving elastic plates. For a theoretical treatment of elastic plates, prestressed or not, according to the design criterion based on the work of deformation, see W. Dzieniszewski, Optimum Design of Plates of Variable Thickness for Minimum Potential Energy, Bull. Academie Polon. Sci., Sir.Sci. Tech. 13, Nc. 6 (1965).
274
10 OPTIMAL BEAM DESIGN
A numerical application of this design criterion to circular plates under uniform pressure is found in N. C . Huang, Optimal Design of Elastic Structures for Maximum Stiffness, Internat. J. Solids Structures 4 (1968), 689-700.
8-15. In Part I1 of this chapter, we follow closely N. Distkfano and R. Todeschini, Invariant Imbedding and Optimum Beam Design with Displacement Constraints, Internat. J . Solids Structures 8 (1972), 1073-1087.
Deflections at specified points of the structure are natural design constraints in most engineering applications. See R. L. Barnett, Minimum-Weight Design of Beams for Deflection, Proc. Amer. SOC.Civil Engrs. Engrg. Mech. Div. 87 (1961), 75-109.
As noted by Barnett, the problem resulting from prescribing the deflection of a beam at a single point might not be well posed. The indeterminacy is generally released, however, by incorporating additional constraints. For an example of this, see G. A. Dupuis, Optimal Design of Statically Determinate Beams Subject to Displacement and Stress Constraints, Div. of Engrg., Brown Univ., Rep. F33615-1826/1 (July 1970).
The first attempt of a systematic treatment of statically determinate beams using the methods of the calculus of variations appears to have been done by E. J. Haug, Jr. and P. G. Kirmser, Minimum Weight Design of Beams with Inequality Constraints on Stress and Deflection, J . Appl. Mech. 34 (Dec. 1967), 999-1004.
The problem of body forces in beams has been treated by invariant imbedding in the paper by Disttfano and Todeschini quoted above. See also R. L. Barnett, Minimum Deflection Design of a Uniformly Accelerating Cantilever Beam, J . Appl. Mech. 30 (19631, 466-467; J. M. Chern, Optimal Structural Design for Given Deflections in Presence of Body Forces, Internat. J . Solid Structures 7 (1971), 373-382.
In this paper the author derives the pertinent necessary condition from the principle of mutual potential energy introduced by Shield and Prager (see reference below) to treat a class of optimal design problems. In the paper by Chern, the author points out the inadequacy of the necessary condition used by Barnett to treat the problem of body forces. Additional references involving beams with deflection constraints are N. C. Huang, Optimal Design of Elastic Beams for Minimum-Maximum Deflection, Trans. ASME (Dec. 1971). 1078-1081 ;
R. T. Shield and W. Prager, Optimal Structural Design for Given Deflection, 2. Angew. Math. P h r ~21 . (1970), 513-523.
NOTES, COMMENTS, AND BIBLIOGRAPHY
275
Miscellaneous References Although not treated in this chapter, the problem of minimum-weight beam design with prescribed eigenvalues is intimately related to the problem discussed here. See, for example, J. B. Keller, The Shape of the Strongest Column, Arch. Rational Mech. Anal. 5 , Number 4 (1960), 275-285; J. B. Keller and F. I. Niordson, The Tallest Column, J. Math. Mech. 16, No. 5 (1966), 433446; C. Y . Sheu, Elastic Minimum-Weight Design for Specified Fundamental Frequency, Internat. J. Solids Structures 4 (1968), 953-958, 1968; N. C. Huang and C. Y . Sheu, Optimal Design of an Elastic Column of Thin-Walled Cross Section, J. Appl. Mech. 35 (1968), 285-288; B. L. Karihaloo and F. I. Niordson, Optimum Design of Vibrating Cantilevers, The Danish Center for Appl. Math. and Mech., Rep. No. 15 (May 1971).
See also J. L. Armand, Applications of Optimal Control Theory to Structural OptimiLation: Analytical and Numerical Approach, preprint of the IUTAM Symp. Optimization Structural Design, Polish Acad. Sci., Warsaw, Poland, Aug. 21-25, 1973.
Optimal Beam Design
1. Introduction
In this and the following chapter we present a unified methodology for the analysis of a class of optimal design problems. The main idea of these two chapters is to treat, within the same conceptual framework, the seemingly related problems of formulating the model, deriving the optimality conditions, and constructing suitable numerical algorithms. In so doing we are naturally enlarging the scope and results of Chapter 8. Here we treat elastic beams and in the next chapter rotating disks. In the first part of this chapter we present a dynamic programming version of the max-min problem for minimum-weight beam design that results from constraining the value of the potential energy. The thorough treatment of this specific problem should be taken more as an example of the use of dynamic programming procedures to a larger class of problems in structural optimization rather than as an isolated case of theoretical importance. Interest in a dynamic programming approach to structural optimization stems from both practical and theoretical considerations. Some of these aspects are already apparent to the reader. 244
2. MAX-MIN OPTIMIZATION PROBLEM
245
The problem is formulated in Section 2, while in Section 3 the basic dynamic programming equations are obtained. In Section 5 the classical EulerLagrange equations for the beam are derived from the fundamental BellmanHamilton-Jacobi equation. It is shown that the conditions of optimality associated with the minimum operation are expressions of the theorem of Castigliano, an expected result repeatedly obtained. An analytical solution for the optimum cantilever laying on elastic foundation is found in Section 4, while a method of successive approximations consisting in a stable, twosweep iterative procedure is developed in Section 6. In order to illustrate the theory, numerical examples are presented in Section 7. In the second part of this chapter, we use the calculus of variations in the fashion of Pontryagin’s minimum principle to derive optimality conditions for a beam under elastic foundation subject to a displacement constraint. Again, this problem is presented as representative of a larger class of problems in structural mechanics. Using ideas of invariant imbedding and the method of successive approximations, the pertinent nonlinear mixed boundary-value problem is reduced to a two-sweep iterative procedure in terms of a system of Riccati differential equations subject to initial values and exhibiting favorable numerical stability properties. Two examples are finally presented to illustrate the application and accuracy of the method. I. DYNAMIC PROGRAMMING 2. Max-Min Optimization Problem
We consider a beam of stiffness a(x) = EZ, where E is Young’s modulus and I is the moment of inertia, and length L, laying on a Winkler foundation k(x), subject to distributed forcesp(x) and a concentrated force P and moment M at the end x = L. The beam is subject to prescribed deformations, deflection w(O), and slope w’(O), at the origin x = 0. We restrict attention to beams of the sandwich type, so that the volume to be minimized is a quantity proportional to the integral jt a dx. The strain energy is given by the functional
where a prime indicates derivative. We define the potential energy V(w, a) = U(w, a) - W(w,a),
where
(2.2)
246
10 OPTIMAL BEAM DESIGN
is the virtual work of the distributed forces p ( x ) and the concentrated forces P and it4 at x = L, over kinematically admissible displacements. By the theorem of minimum potential energy, the function w* that minimizes V in (2.2) and satisfies the boundary conditions is the “true ” displacement function of the beam with a given design a(x). The value of the minimum is min V(w,a) = -+W(w*, a).
(2.4)
W
Clearly, for a given set of forcesp, P,and M , the virtual work W(w*, a) along the true displacement field can be used as a measure of compliance. Therefore, if our aim is to minimize the volume a dx for a fixed value of the compliance W(w*, a), we can formulate the minimization problem:
1
min[JoLa dx - 2p min(U(w, a) - W(w,a)) , a
W
(2.5)
a min-min problem in the deflection w and the design a, where p is a Lagrange multiplier used to incorporate the constraint. Alternatively, if for a given system of forces we wish to determine a design a for which the compliance W(w*,a) is a minimum among all possible designs with the same volume, we formulate the following problem :
or, on account of (2.2) and (2.4),
a noninterchangeable max-min problem in a and w, respectively. We assume to belong to A, the set of all admissible designs. Clearly, the two problems (2.5) and (2.6) are equivalent and L = l/p. In the following we shall use the formulation given by (2.6). 01
EXERCISE Formulate mathematically the following ma& min, problems: (a) maximum buckling load of an axially loaded prismatic bar for a specified volume; (b) largest minimum frequency of a beam for a specified volume. [In both problems assume the volume to be given by V = j!j g(a)dx, whereg is a specified function of the cross-sectionalstiffness a.]
3. Dynamic Programming Instead of a beam of length (0,L), we consider the family of beams of lengths (2,L), where 0 Iz I L, subject to prescribed displacements
w(x)I,=, = 421,
w‘(x)[,=, = 421,
(3.1)
247
3. DYNAMIC PROGRAMMING
at x = z. Recalling Eqs. (2.1), (2.3), and (2.6), we define the optimum value function f(u, u, z ) = max min a.zA
w
[+ (CLW"'+ kw2 L Jz
- 2pw - h)dx
1
- Pw(L) - Mw'(L) , (3.2)
where V ( Z ) = du(z)/dz.
(3.3)
To obtain a differential equation for the optimum value function, we proceed as usual and find the following functional equation
[
f ( u , u, z ) = max min 6 V - 1 5 dE C[z, ) zw(C) +Az]
Jzz+Az + f ( u ( z + Az), u dx
v(z
1
+ Az), z + Az)
,
(3.4)
where
$1
z+Az
6 ~ =
z
(awn2 + kw2 - 2pw) dx
(3.5)
is the potential energy of the slice of beam between z and z + Az. Equation (3.4) together with ( 3 . 9 , which may be easily justified on mechanical grounds, is a local expression of the principle of optimality. Introducing the curvature K
= d2w/dxzl x Z z ,
(3.6)
standard arguments in dynamic programming show that, as Az+O, Eqs. (3.4) and (3.5) yield
fz
= min
max[ - ( u K 2
a~
+ ku2 - 2pu - h ) / 2 -fu
2,
-f,K],
(3.7)
the Bellman-Hamilton-Jacobi equation for the optimum value function, where f,stands for aflaa, a = z, u, u, andf(u, u, z ) is subject to the initial condition f ( u , u, L) = Pu
+ Mu,
(3* 8)
an equation that follows directly from consideration of Eq. (3.2). Minimizing with respect to K in Eq. (3.7) we obtain K =
(3.9)
-fv/u,
that substituted back into Eq. (3.7) yields
fz
= minv;2/2a a
-fuv - ku2/2
+ p u + Au/2].
(3.10)
248
10 OPTIMAL BEAM DESIGN
If the design a is not subject to constraints, we can minimize in (3.10) by taking derivatives, obtaining in this fashion
f,’la’
(3.11)
=I,
an equation that together with (3.9) yields the optimality condition
I
(3.12)
= K2.
Combining Eqs. (3.10) and (3.1 1) we obtain
f, + uf, + ku2/2 - PU
=f:/a,
(3.13)
=A f:,
(3.14)
or, eliminating a using Eq. (3.1 l),
+ ~f.+ ku2/2 - PU)’
(fi
a nonlinear partial differential equation for the optimum value function subject to the initial condition (3.8). EXERCISES 1. It is known that the buckling load P of a prismatic axially loaded bar is given by
P = min w (f fawe2 dx)
/
(4 J:w”
dx) ,
i.e., by the minimum of a Rayleigh quotient. Show that the optimization problem subject to JOLg(a)dx = V,
max P, (I
where V, the volume, is a prescribed quantity and g is a given function of a denoting the cross-sectional area of the bar, may be formulated as max min 4 a
w
:j(awl2- h,w’, -
(4
2h2g(a))dx,
where h, and h, are two constant Lagrange multipliers. 2. In Exercise 1, assume the column clamped at x = 0 and free at x
f(u, u, x)
= max min 0
1
K
where u(x) = w(x), u(x) = u’(x), and differential equation
f
s:
K =
(aK2
-A,
= L.
u2 - 2hg(a)) dx,
Set (4
u”(x), and show that f satisfies the partial
-fx(u, u, x) = max(-(1/2a)fv2 - fhl u2 - A d a ) +f.u), 01
f(u, u, L ) = 0.
(4
249
4. CONSTANT CURVATURE
4. Constant Curvature
If the curvature of the beam does not change the sign, then Eq. (3.14) admits a remarkably simple solution. In fact, assuming K ( Z ) > 0 for 0 Iz < L, Eq. (3.14) reduces to f,
+ Ofu + ku2/2 - p~ = A 1 J 2 f , ,
(4.1)
which admits a solution of the form
f
= rtu2
+ 2r2 uv + r3 v2 + r4 u + r5 v + r 6 ,
(4.2) where r i , i = 1, . . . , 6, are functions of z that satisfy the following initialvalue system :
dr,/dz = -k/2, dr2/dz= - r l ,
r,(L) = 0, r2(L)= 0,
dr4/dz = p + 2A'I2r2, r4(L) = P, dr,/dz = -r4 + 2A'I2r3, r,(L) = M ,
(4.3)
dr,/dz = - 2 r 2 , r3(L)= 0 , dr6/dz = A1l2r5, r 6 ( L ) = O, obtained upon substitution off given by (4.2) into (4.1) and collecting terms in powers of u and v. The initial conditions follow from consideration of Eq. (3.8). Clearly, Eqs. (4.3) admit a solution in quadratures. For example, assume p and k to be constant in 0 I z I L and consider the initial conditions v(0) = 0.
u(0) = 0,
(4.4)
Integration of (4.3) yields
rt = k t / 2 , r2 = kt2/4, r3 = kt3/6, r4 = P - p t - kA'I2t3/6, r5 = M + P t - p t 2 / 2 - kA'I2t4/8, r6 = - M t - P t 2 / 2 p t 3 / 2+ kA'l2t5/40,
(4.5)
+
where t = L - z. On the other hand, integration of (3.12), where by (3.6) subject to the initial conditions (4.4), yields
u(z) = A'/2z2/2, v(z) = A'/2z. The optimal design is given by Eq. (3.1 l), i.e., 01
=fu/A'12
= (2r2u
K
is given (4.6)
+ 2r3 v + r5)/A'J2,
which on account of Eqs. (4.5) and (4.6) yields 01
=
[M
+ PLS - P L ~ ~ ~ / ~+][2( / A1 ' /9)'~ + 85( 1 - 5)/3 - f2][2kL4/8,
(4.7) where [ =
250
10 OPTIMAL BEAM DESIGN
EXERCISES 1 . Consider problem 2 of Section 3. Assumingg(a) = kal/”,where k is a positive constant and n is a positive integer, solve Eq. (e). 2. If n = 1, show that the curvature of the column, in its deformed position, is a constant.
3. Show that when n = 1, the optimum design is proportional to Lz - x z .
5. Euler-Lagrange Equations
We illustrate the procedure outlined in the last chapter, and derive the pertinent necessary conditions of the present optimization problem. Along an optimal design aOpt,Eq. (3.10) holds without the minimum operation, i.e., f z =f,”/2a -fu v - ku2/2+ PU
+ 2~12,
(5.1)
where u satisfies (3.11). On account of (3.11) we can write (5.1) in the form fz
= ACC- f u v - ku2/2
+ PU.
(5.2)
We compute now the total derivatives o f f , and f u with respect to z, i.e., (dldzlfu = f z u +fuuv +fuuv’, (dldzls, = f z u + f u u v + f U U V ’ ,
(5.3)
and also the partial derivatives f , , and fvz from Eq. (5.2), i.e., fuz
= 2% - f u u v - ku
Lz =
-Luv
+ P,
(5.4)
-fu,
which substituted into (5.3) yields (dlWfI4 = P - ku (dldzlf, =
-fu
+
-fufuula,
+ 2%-Lfvul4
(5.5)
where v‘ has been substituted by its equivalent dvldz
=
-fv/U,
(5.6)
obtained recalling (3.3), (3.6), and (3.9). Now, taking partial derivatives with respect to u and v in Eq. (3.11) we have
2% -f v fuula = 0, La” -f u f o v l a = 0, respectively, so that Eqs. (5.5) reduce to (dldz)S, = P - ku, ( 4 W f V
=
-fu
Y
(5.7)
6 . DESIGN CONSTRAINTS. SUCCESSIVE APPROXIMATIONS
25 1
the equations of equilibrium of the beam which together with Eqs. (3.3) and (5.6) are the Euler-Lagrange equations of the original variational problem (2.6). They hold, of course, for any u other than the optimal one given by (5.9) an equation derived directly from (3.11). In Eq. (5.8) werecognizef,, to be the shear force andf, the bending moment of the beam. In fact, the equations (5.10)
f" = -uw" obtained from (3.6) and (3.9), and
f"= -f,' = (olw")',
(5.1 1)
obtained from (5.8) and (5.10), are local expressions of the theorem of Castigliano obtained via dynamic programming. Elimination of fu between (5.8) and (5.1 1) yields (dZ/dx2)(ud2wldxZ)
+ kw = p ,
(5.12)
the classical equation of the beam where u(z) in Eq. (5.8) has been substituted by w ( 4 . EXERCISE Using the method outlined in Chapter 9, discuss the end conditions on the quantities
f. and fu . 6. Design Constraints. Successive Approximations There are several reasons why we wish to solve the problem of optimum beam design using a method of successive approximations. In fact, closedform solutions of the type obtained in Section 4 are the exception rather than the rule. In general we must resort to Eq. (3.14) or to the more complicated ones which are obtained if the hypotheses of beams of sandwich type are relaxed; and although Eq. (3.14) is subject to initial conditions, its numerical solution is not a routine matter in view of the singularities associated with the inflection points of the beam, i.e., where the curvature changes the sign. In addition to this we observe that Eq. (3.14) was derived assuming that there are no constraints on the design, a condition seldom met in practice. It is therefore desirable that the development of successive approximation schemes suitable for the solution of various classes of problems be derived in a unified fashion.
252
10 OPTIMAL BEAM DESIGN
We start by considering Eq. (3.10). For a fixed design oro this equation holds without the minimum operation, i.e.,
f, =fv2/2u0 -f , v
- ku2/2
+ PU + Ia0/2,
(6.1) subject to the initial condition (3.8). Equation (6.1) admits a solution of the form
where si,
f = s1u2 + 2s, uv + s3 v2 + s4 u + s5 v + ss , i = 1, . . . , 6, are functions of z given by ds4/dz= 2s, s5/ao + p (d) ds,/dz= 2s, s5/ao - s4 (e) dS6/dZ= s52/2ao+ h 0 / 2 (f)
dsl/dz= 2szz/u0- k/2 (a) ds2/dz= 2s2s3/a0- s1 (b) ds3/dz= 2s3’/a0 - 2s, (c)
(6.2)
(6.3)
a system of Riccati differential equations subject to the initial conditions S,(L) = S z ( L ) = s3(L) = Sg(L)
s4(L) = P,
= 0,
S,(L) = M ,
(6.4)
obtained from consideration of Eqs. (3.8) and (6.2). Functions u and v can be obtained from Eqs. (3.3) and (5.6), i.e.,
du/dz= V , dvldz = -(2s2 u + 2s3 v
+ s,)/ao,
(6.5)
a coupled system of differential equations subject to the initial conditions
u(0) = uo ,
v(0) = 00,
(6.6)
where uo and vo are prescribed values of the deflection and the slope at z = 0, respectively. Now we are in a position to implement a successive approximation scheme for the computation of the optimum structure. In fact, for a given estimate uo of the design, we integrate Eqs. (6.3a)-(6.3e), and obtain si,i = 1, . . . , 5, which substituted into (6.5)-(6.6) yield functions u and u by forward integration. Using those quantities we can therefore compute a’ = arg min[2(s2 u asA
+ s3 v + s5/2)’/a + la/2],
(6.7)
an equation of optimality derived directly from (3.10) and (6.2), which yields an improved estimate for the design. The quantity I appearing in (6.7) can be computed easily by making use, at each iteration of the process, of Eq. (3.1 1) which, on account of (6.2), reads
1 = (s5(o)/a0(o))2 at z
= 0.
This process is repeated until convergence is achieved.
(6.8)
253
7. NUMERICAL EXAMPLE
EXERCISE How would you proceed to satisfy, at each iteration of the process, the condition
Pu(L)
+ MU@)+
p u dx
(a)
= C,
JoL
where P,M, and c are prescribed quantities? Hint: Instead of (6.8), compute h to satisfy (a).
7. Numerical Example To show the feasibility and accuracy of the method developed in Section 6, we present a numerical example involving a cantilever beam on elastic foundation subject to a load P and a moment M such that the rotation at the free end x = L is zero. This last condition is incorporated for the sole purpose of simplifying the analytical solution. In fact, the exact solution for the optimum design of the present beam configuration and loading condition can be found easily by application of the results in Section 3. From Eq. (3.12) we have K =
+A”’,
(7.1)
i.e., the curvature in absolute value is a constant along the entire beam. Therefore, the elastica consists of two arcs of a parabola with the same curvature intersecting at an inflection point xo = L/2 and satisfying the geometric boundary conditions
w(0) = w’(0) = w’(L) = 0, w’(x0-) = w‘(x, +). By requiring that the slope of the two parabolas be equal at xo, the shear force V at the inflection point can be found easily be integration of all the forces acting perpendicularly to the beam, obtaining in this fashion V = P - 5kL3A“’/48.
(7.3)
Using V furnished by (7.3) we can now easily compute the moments acting along the entire beam, namely,
m(5) = (PL/2)(1 - 25)[1 - (23 - 25 - 45’ - 8t3)>lB],
5 5 3,
(7.4)
3I 5I 1,
(7.5)
05
and
m(5) = (PL/2)(25 - 1)[1
- (21
+ 105 - 285’ + 8t3)/P],
where
p = 192P/kL3A1I2.
(7.6)
254
10 OPTIMAL BEAM DESIGN
In particular, the moment M necessary to ensure the horizontality of the elastica at 5 = 1 is given by Eq. (7.5), namely, M
= m(1) = (PL/2)( 1
- 21/P).
(7.7)
The optimum design follows from Eq. (3.1 I), i.e., a = m(5)/A'I2.
Therefore &(<) = (1
0 < 5 < 3,
(7.8)
+ 105 - 28t2 + 8y3)/P], 3 I 5 I1 ,
(7.9)
- 25)[1 - (23 - 2< - 4 t 2 - 853)/p],
and
Cr(5)
= (25
- 1)[1 - (21
where Cr is the dimensionless quantity given by Cr = ( 2 F / P L ) u *
(7.10)
Clearly, P = 23 is the smallest value with physical meaning that this parameter cantake. The longitudinal stresses in the flanges, in absolute value, are constant over the entire beam and their value is given by
I o I = A1/'Eh/2,
(7.1 1)
where E is the modulus of elasticity and h is the height of the beam, i.e., the distance between the flanges. Hence, for a prescribed value of the stress u, A l l 2 can be computed from (7.1 1) and its value substituted into (7.6) to compute the dimensionless quantity P. The optimum dimensionless design ti, for several values of the parameter P, is presented in Fig. 10-1. X
h
I .o
0.8
dko,6 ,I
IU
2 04 ' cn W
0.2 0 0
0.1
0.2
0.3
0.4
0.5
[=
xL
0.6
0.7
0.8
0.9 I 3
Fig. 10-1. Optimal beam designs, where a = El, p = 192P/kCL3, C = 2u/Eh, and h is the height of the beam. (After Distkfano [1972].)
TABLE 10-1 Comparison of Exact and Computed Designs &ompvtcd
5
(constraint: Emin= 0.005)
L a c *
Initial estimate
1st iteration
2nd iteration
3rd iteration
5th iteration
7th iteration
9th iteration
11th iteration
0.77718 0.47106 0.16139 0.16090 0.50843
0.77199 0.46722 0.15887 0.16220 0.50880
0.89000
0.89000
0.77055 0.46615 0.15817 0.16265 0.50890 0.89000
0.77015 0.46586 0.15798 0.16267 0.50893 0.89000
12th iteration
~
0.00 0.20 0.40 0.60 0.80 1.0
0.77000 0.46574 0.15790 0.16270 0.50894 0.89000
1.0 1.0 1.0 1.0 1.0 1.0
0.89812 0.55810 0.21538 0.13570 0.50212 0.89000
0.82566 0.50691 0.18487 0.14876 0.50495 0.89000
0.79773 0.48631 0.17144 0.15566 0.50692 0.89000
0.77008 0.46580 0.15794 0.16268 0.50894 0.89000
256
10 OPTIMAL BEAM DESIGN
The solution of the same problem, using the method of successive approximations developed in Section 6, can be obtained as follows. At a generic iteration, using the available design a', we integrate Eqs. (6.3a)-(6.3e) in the backward direction, obtaining the values of si,i = 1, . . . , 5, which substituted into (6.5)-(6.6) yield functions u and v by forward integration. Using these quantities we compute an improved design a' by making use of Eq. (6.7). In the numerical example presented in Table 10-1 and corresponding to fi = 100, we started the process by considering a uniform design a' = I . The integration was performed using an Adams-Moulton scheme with step size H = L/200. It is noticed that the singular behavior of the derivatives in Eqs. (6.3), in the neighborhood of the inflection point, prevents in general the routine use of standard methods of numerical integration. A number of asymptotic solutions near the singularity can be obtained and the corresponding results can be matched to the numerical solution furnished by the Adams-Moulton integration scheme, in the neighborhood of the singularity. In the present case it was found that the simple device of imposing a constraint on the design of the form c12 0.005 worked very well and avoided the need of complicating the routine of integration. The results in Table 10-1 show the accuracy of the method.
11. INVARIANT IMBEDDING 8. Alternative Formulation
In order to incorporate displacement constraints and additional local restrictions on the design and the state variables, the remainder of this chapter is devoted to an alternative formulation of the beam-optimization problem. In view of the different imbedding used here, we shall modify the notation slightly. To illustrate the procedures we consider a beam of finite length L, laying on elastic foundation with coefficient k(x) and subject to external forces q(x) that might eventually contain a singularity due to a concentrated load. If u(x) denotes the deflection, v(x) the slope, m(x) the bending moment, and t(x) the shear, the constitutive equations of a Euler-Bernoulli beam are given by duldx = v, dvldx = - (1/a)m, (8.1) and the equilibrium equations by
dmldx
=
-t,
dtldx
= q - ku,
(8.2)
where
u
= EZ
(8.3)
257
8. ALTERNATIVE FORMULATION
is the stiffness, I is the moment of inertia, and E is Young’s modulus. We choose the stiffness a as the design variable, and assume that the crosssectional area A of the beam is given by the formula
where g is a function that depends on the particular geometry of the cross section. For example, for rectangular beams with constant width b and variable height h, g = (12bZa/E)’/3.The volume of the beam is given by .L
Considering u, v , m, and t subject to an appropriate set of boundary conditions, we can now formulate the following minimum volume problem for a prescribed displacement u1 at x = x l , and additional inequality constraints: Minimize the quantity V given by ( 8 . 5 ) subject to the conditions
and possibly to a number of additional constraints cpi(m,t,a,x)
i=1,2
,....
(8.8)
We assume that this optimization problem is well posed; i.e., a solution exists, is unique, and is continuous with respect to boundary conditions and constraints. The study of classes of constraints under which a given optimization problem is well posed is one of great interest and importance, but it is beyond the scope of this chapter whose main purpose is the discussion of numerical computational procedures. The derivation of the necessary conditions for this problem may be done using dynamic programming as outlined in Chapter 9. We leave this derivation as an exercise to the reader. Here we proceed directly, using the formalism of the minimum principle. In order to incorporate the constraint on the deflection u, instead of (8.6) we consider the equivalent integral expression
where 6 is the Dirac delta. Now we form the Hamiltonian
258
10 OPTIMAL BEAM DESIGN
where 1and I l to 1, are Lagrange multipliers used to incorporate the constraints (8.1), (8.2), and (8.9). It is well known that the multipliers satisfy the adjoint differential equations
(8.1 1)
From the first equation of (8.11) we see that 1is a constant, to be chosen to enforce condition (8.6). Comparison of (8.1) and (8.2) with Eqs. (8.11) shows that the Lagrange multipliers I , to I , are the forces and displacements of the beam subject to a virtual concentrated load of intensity I at x = xi. More precisely,
I,
= AU,
2,
= 16,
12
=
-26,
I1 =
-If,
(8.12)
where U , 6, E , and t are the displacements and forces due to a unit virtual force at-xl. Minimization of (810), taking into account (8.12), yields the optimality condition aOpt= arg min[l(mE/a)
+ g ( 4 41,
(8.13)
a
where clop, denotes the optimum design. EXERCISES 1. Let f(u, u, t , rn, x ) be the minimum volume function given by
f(u, u, t , m, x ) = min a
S:
g(a,x ) dx,
(a)
subject to the differential constraints (8.1) and (8.2) and to the deflection constraint (8.6). Show that f satisfies the partial differential equation
-f = min[g(a, x ) + puIS(x - X I ) +f.u m
- (l/a)fum +fh- ku) - A t ] ,
(b)
where p is a constant Lagrange multiplier. 2. Using the method outlined in Chapter 9, derive ordinary differential equations for f., fu, fc, and fm . How are these quantities related to the adjoint variables X I to satisfying (8.1l ) ? 3. How would you include consideration of constraints such as:
Ilongitudinal stress I < u,, ,
I slope I
=L
< uL ?
9. AN ALTERNATIVE DERIVATION OF THE OPTIMALITY CONDITION
259
4. Consider a sandwich beam as indicated in Fig. 8-1. Let E = F(U)
be the (nonlinear) stress-strain law of the flanges. Therefore the curvature K will be given bY K = -(1/2h)[F(m/Ah) - F(-m/Ah)]. (4 If, additionally, we consider a nonlinear elastic foundation characterized by the forcedisplacement relation p=ku-pu', 820, (6) the differential equations of the beam will be given by duldx = U, = - ( l / X ) [ F ( m / A h ) - F(-m/Ah)],
du/&
dm/& = -t, dtldx = q - ku
(0
+ Pu'.
Assuming F to be continuously differentiable with respect to A, derive the pertinent necessary conditions for the following optimization problem: min /'A dx, A20
0
subject to the deflection constraint (8.6) and the differential constraints (f). Use A as the pertinent design variable.
9. An Alternative Derivation of the Optimality Condition Instead of incorporating the differential constraints (8.1) and (8.2) using the Lagrange multipliers I1 to I,, we can proceed in the following way: In place of (8.9) we use the integral representation "L
+
u1 = ~o(mfi/cc kuii) dx,
(9.1)
furnished by the theorem of virtual work, where Z and i7 are the moment and the displacement, respectively, of the beam subject to a virtual load applied at x1 and in the direction of the prescribed displacement. Combining (8.5) and (9.1) we form the Hamiltonian 231 = g(a, x )
+ I(rnE/a + h a ) ,
(9.2)
where 1 2 0 is a Lagrange multiplier that satisfies the equation
dI/ax = - aH1laul = 0.
(9.3) Hence I is a positive constant which is to be chosen to satisfy (8.6). Clearly, minimization of (9.2) yields (8.13), as expected.
260
10 OPTIMAL BEAM DESIGN
EXERCISES 1. Derive the optirnality condition from Eq. (b) of Exercise 1, Section 8.
2. Assuming a not subject to constraints andg(a, x ) = k ~ ( ' /derive ~ , the pertinent condition of optirnality starting from (8.13).
10. Optimum Cantilever under Prescribed Displacement
To illustrate the method and the problems associated with the numerical computations, we consider the case of a cantilever beam laying on elastic foundation and subject to a prescribed displacement at x = xl. The boundary conditions of Eqs. (8.1) and (8.2) are u(0) = 0,
~ ( 0=) 0,
m(L) = M , t(L) = T.
(10.1)
For simplicity we rewrite Eqs. (8.11) in terms of U, V , E,and i given by
dU/dx = i5, dV/dx - (1/a)E, 1
dE/dx = - i , di/dx = 6(x - xl) - ku,
(10.2)
subject to the homogeneous boundary conditions
U(0) = V(0) = 0,
E(L) = t(L)= 0.
(10.3)
So formulated, the solution of the optimum beam reduces to the task of integrating Eqs. (8.1), (8.2), and (10.2) subject to boundary conditions (10.1) and (10.3), respectively. These two systems of equations are coupled together through the optimality condition (8.13). In a number of important cases in the applications we possess an explicit representation for the minimum operation in (8.13). This simplifies in some sense the solution of the system. In any case, this nonlinear boundary-value problem can be integrated using a quasilinearization scheme. (See the exercises below.) We do not pursue this path here, i.e., a direct treatment of the nonlinear boundary-value problem, in favor of the implementation of a simple, first-order, stable iterative method based on ideas of invariant imbedding. This is done in the following section. EXERCISES 1. Assuming u, u, t , m subject to boundary conditions (10.1) show thatfin Eq. (b), Exercise 1, Section 8, satisfies the conditions
fmb, u, t , m,0) = 0, fXu, u, t, m,0) = 0,
A h , 0, t , m,0 = 0, f d u , u, I, m,L) = 0.
(a)
2. Write the boundary conditions for the adjoint variables 8,6, i, and 177,when the beam is (a) clamped-clamped, (b) clamped-hinged, (c) hinged-hinged.
26 1
11. INVARIANT IMBEDDING
3. Derive the conditions on f satisfying the partial differential equation (b) in Exercise 1, Section 8, for the cases indicated above.
4. Implement a quasilinearizationscheme for the solution of Eqs. (S.l), (8.2), (10.1)-(10.3), and (8.13). 5. Implement a quasilinearizationscheme for the solution of Exercise 5, Section 8.
11. Invariant Imbedding
For a given nominal design c1, we consider the uncoupled linear boundaryvalue systems given by Eqs. (8.1), (8.2), (lO.l), and (10.2), (10.3), respectively. First we consider system (8.1)-(8.2). Instead of boundary conditions (10.1) we set m(L) = M , u(X) = w, (11.1) t(L)= T, v(X) = Z, i.e., we consider the families of beams of length L - X subject to arbitrary displacements w and z at the end x = X . This is an “imbedding” procedure that in the fashion of invariant imbedding affords the property of reducing the computation of the original boundary-value problem (8. l), (8.2), (lO.l), to a stable, two-sweep procedure. We seek solutions of the form
t ( X ) = r,(X)w + r12(X)z + S l W , m ( X ) = rzl(X)w r,(X)z s2(X).
+
+
(11.2)
Differentiation of (1 1.2) with respect to X and elimination of the derivatives
w’ = u’(X) and z’ = v’(X) using Eqs. (8.1) and (8.2) evaluated at x = X and further elimination of t(X) and m ( X ) using (1 1.2) yields (rl’
+ k - r12(1/cOr21)w+ (G2+ rl - rl2(l/a)r2>z+ (sl’ - q - rlz(l/4sz) = 0,
and
(rL + rl - r 2 ( 1 / 4 r 2 d w + h’+ r12 + r21 - r,(l/4rz)z
+ (s,’ + s1 - r2( l/u)s,)
= 0,
a system of equations that must be valid for any w and z. Therefore
+
rl’ = - k rl2(1/c1)rZ1, r2’ = -rlz - r21 + r2(1/u)r2, (11.3) r i 2 = -rl + rl2(1/4r2 , s1’ = q + r12U/a)s2, s2’ = -sJ + r2(l/u)s2, ril = -rl r2(1/u)r21, a system of Riccati equations subject to the initial conditions at X = L,
+
r,(L) = rI2(L)= r21(L)= r,(L) = 0,
s1(L) = T,
s2(L)= M ,
(11.4)
262
10 OPTIMAL BEAM DESIGN
obtained upon consideration of Eqs. (1 1.1) and (1 1.2). Consideration of the second and third equation in (1 1.3) and corresponding initial conditions readily yields r12 = r21, (11.5) an expression of Maxwell’s theorem derived from invariant imbedding. Therefore Eqs. (1 1.3) and (1 1.4) reduce to
+ (l/a)r;,, = -r, + (1/.)rlZ~Zr = -2r1, + (1/a)rZ2, 81’ = q + (l/a)r12sz -k
r,’ r;, r,‘
=
s‘,
= -sl
r,(L) = 0, r,,(L) = 0, r,(L) = 0,
(11.6)
s,(L) = T, s,(L) = M .
9
+ (l/a)rzs,,
Substitution of m(X) given by the second equation of (11.2), into Eqs. (8.1) evaluated at x = X and due consideration of Eqs. (10.1) yield u(0) = 0, du/dX = V , (11.7) dv/dX = -(l/a)(r,, u + r2 v + s2), v ( 0 ) = 0, an initial-value problem in the forward direction for the deflection u and slope v of the beam, where the quantities r , , , r , , and s, are given by the integration of (1 1.6) in the backward direction. Substitution of u and v given by (1 1.7) into (1 1.2), taking into account that w = u ( X ) and z = v ( X ) , finally yields the remaining state variables of the beam, i.e., the moment m and the shear force t in the interval 0 I X 5 L . We can treat the virtual system (10.2) in a similar fashion. We consider U, 6, iii, and t satisfying Eqs. (10.2) to be subject to the boundary conditions
U(X)= w,
E(L) = 0, t(L)= 0,
q x ) = 5,
(11.8)
and write for t ( X ) and % ( X ) ,the missing boundary conditions of the imbedded beam of length L - X , equations similar to (1 1.2), i.e.,
f ( X )= T;,(X)W+ f,,(X)Z + S,(X), E ( X ) = T‘,l(X)W r,(X)z S,(X).
+
+
(11.9)
Carrying out the same perturbation analysis done to Eqs. (1 1.2), on Eqs. (1 1.9), we obtain
+ (l/~)?;,, i ; , = - r, + (l/a)r,, r,, 7,’ = -2r12 + (l/a)F22, F,’= -k
+
6,’ = 6 ( X - x1) (l/a)?,, 6,‘ = - 6 , ( l / a ) 7 , s,,
+
f,(L) = 0, = 0, F,(L) = 0, S,(L) = 0, S,(L) = 0.
F12(L)
s, ,
(11.10)
263
12. A TWO-SWEEP ITERATIVE PROCEDURE
Similarly, we can write dU/dX = O, dO/dX = -(l/a)(F12 U + i 2 ij + iz),
U(0) = 0,
(1 1.11)
O(0) = 0,
an initial-value problem for U and ij obtained after substitution of E,given by the second equation of (1 1.9), into the first two equations of (10.2) evaluated at x = X . Finally, t and E can be obtained from (1 1.9) recalling that W = U(X) and Z = tj(X). EXERCISES 1. If, instead of (ll.l), the boundary conditions were u(X) = w, m(X)= M,
u(L)= z, t(L) = T,
how would the initial conditions of (11.6) change? 2. Repeat the previous analysis for the conditions u ( X ) = w, v ( X ) = 2,
u(L) = 0, u(L) = 0.
In this case, some of the r's in (11.3) are not defined at x = L. Derive approximate asymptotic expressions for those quantities at x = L - A, A < 1.
12. A Two-Sweep Iterative Procedure
Using the results of the preceding section we can now implement a simple iterative method for the solution of the optimization problem. To this end let dn)denote the value of the quantity a at the nth iteration. Then at a generic iteration (n + 1) we use the currently available design a(") to integrate (11.6) in the backward direction and subsequently (1 1.7) in the forward direction, computing in turn m("+')given by (1 1.2). Similarly, we compute the values of E'"+ by using the two-sweep process given by (1 1.9)-(11.11). The improved value of the design a("+') follows from the optimality condition 1)
= arg min[l(n+
')m("+1 )/a) + da,41,
(12.1)
a
where the new estimate of 1= A("+') needs to be taken so as to ensure an appropriate convergence of the sequence u(")(xl) to the prescribed value ul. This can be done in a number of ways. In general we shall use formulas of the type
A("+1) = F(1("), u(")(xl),u("-l)(xl),...).
(12.2)
264
10 OPTIMAL BEAM DESIGN
(12.3)
The procedure is repeated until convergence is achieved. Initially we need an a priori estimate of the design a(’). In the absence of special information, a uniform design ~(‘’(x) = constant is usually taken as the initial design. A numerical example in Section 15 illustrates the application of the method. EXERCISES 1. Consider a circular plate of radius a, thickness h, Young’s modulus E, and Poisson’s ratio v. If u denotes the deflection and u and m the radial slope and bending moment, respectively, the equations of equilibrium of the plate undergoing bending may be written u‘
=u
u’ = -(v/r)u - ( l / a ) m m‘ = -((I - v)/r)m- ((1
(a) - v2)/r’)au - Jqr,
where r is the radial coordinate, q(r) is the load, and the stiffness a is given by a = Eh3/(12(l
- v’)).
(b)
Assuming one of the following boundary conditions: Clamped Simply supported
u(a) = 0,
u(0)
u(a) = 0,
u(0) = 0,
= u(a) = 0,
m(a) = 0,
(4
and using the method outlined in this and foregoing sections, solve the following optimization problem: min 1;ra1l3 dr,
subject to the displacement constraint u(0) = 1.
(d)
Clearly, the integral in Eq. (d) is a quantity proportional to the volume of the plate. 2. A circular footing with outer radius a is carrying a column of radius 6, with axial load P. Assuming thin-plate theory and Winkler foundation, the associated boundaryvalue problem for the bending of the footing is given by P
u’ = u,
u(b) = 0,
- ( l / a ) m - (v/r)u, m‘= -((I - v)/r)m - ((1 - v2)/r2)au- ikur,
u(a) = 0,
U’ =
I
2
a
I
and the additional equilibrium condition P = nb’ku(b)
+ 2njbakurdr,
m(a) = 0,
(4
265
14. STABILITY CONSIDERATIONS
where k is the coefficient of the foundation. Derive the optimality conditions and implement a two-sweep method for the solution of the following optimization problem: max P,
(g)
subject to
dr = C, where C is a quantity proportional to the volume of the plate, and to the displacement constraint 0 Iu(b) 5 c. 3. In Exercise 2 above, include a constraint on the absolute values of the shear and radial stresses.
13. Inertia Forces When the weight of the beam is to be taken into account, q in Eqs. (8.2), (8.10), and (11.6) must be replaced by 4=p
+ YA,
(13.1)
where p is the external forces, y is the specific weight of the beam, and A is the cross-sectional area given by Eq. (8.4). Therefore the optimality condition (8.13) must be substituted by uOpt= arg min[A(m%/u)
+ (1 + y/lU)g(u,XI],
(13.2)
a
and Eq. (13.1) must be taken into account when integrating Eqs. (11.6). Similarly, other mass forces can be taken into account. EXERCISE
In Exercise 1 of Section 12, incorporate the weight of the plate in the formulation of the optimization problem.
14. Stability Considerations
In order to prove the stability of the method, we should show that u and u given by (1 1.6) and ( 1 1.7) are stable with respect to small errors introduced in the computational process. To this end it is enough to prove the stability of the differential system (1 1.6) and, in particular, of the quantities rl, r12, and r 2 . We rewrite the first three equations of (1 1.6) in the matrix form R' = - A - BR - RBT + RCR,
where
R(L) = 0,
(14.1)
266
10 OPTIMAL BEAM DESIGN
It is easy to show, by recalling the results of Section 3-12, that the stability of R in the backward direction will depend on the stability of the differential equation dz/dx = ( B - RC)Z
(14.3)
in the forward direction, which in turn follows from consideration of the Liapunov function V ( Z ( X )= ) (z(x), R - ’ ( L - x)z(x))
(14.4)
which clearly exists, is bounded, and is positive definite for x > 0, since R-’ is a flexibility matrix. We leave the details of the demonstrations to the reader. EXERCISES 1 . What are matrices A , B, C, and D in (14.2) when, instead of (8.1), we use the constitutive equations
+
a l l t + a12m, u’=a21t+a22rn,
u’= u
where ax2 = a z l ? 2. When we use Eq. (a) above and assume that the matrix (a,,) is positive definite, do we still enjoy the stability of the Riccati equation (14.1)? 3. Prove the numerical stability of all the quantities involved in Sections 11 and 12.
15. Numerical Examples a. Cantilever on Elastic Foundation
We consider a beam of the sandwich type such as that whose cross section is indicated in Fig. 8-1. In this case if 2h is a fixed quantity denoting the distance between the covering sheets and A/2 is the area of each one of the sheets, we have tl = EZ = EAh’, and g in (8.4) reduces to g(a, X) = UlEh’.
(15.1)
The volume is therefore proportional to a, i.e., L
V = ( l / E h 2 ) I adx.
(15.2)
0
In addition we shall take x1 = L and q = P6(x - L ) ; i.e., we prescribe the displacement u1 in correspondence with a concentrated load applied at the free end of the cantilever. The reason to consider this simplified example is because under the present assumptions we can afford a closed-form solution
267
15. NUMERICAL EXAMPLES
of the problem that can be used to compare the accuracy of the numerical procedures. In fact, in our present case we have m = P E and Eq. (8.13) reduces to
+ E),
aOpt = arg min(p(m2/a) U
(15.3)
where p = (EhZ/P)Ais a constant to be determined such that u(L) = ul. Minimizing by taking derivatives, Eq. (15.3) reduces to =p
(15.4)
2 .
Combining (8.1) and (1 5.4) we obtain (u")2
= lip.
(15.5)
A solution that satisfies the nonlinear differential equation (15.5) and the boundary conditions u(0) = u'(0) = 0 and u(L) = u1 is given by (15.6)
u(x) = U 1 X 2 / L Z .
Equation (15.6) is valid if sign m = sign u", a condition that is clearly fulfilled in our example. The curvature is given by lu"l = (l/p)l'Z
= 2u,/L2.
(15.7)
Combining (15.4) and (15.7), aopt= LZm/2ul,
(15.8)
where the bending moment m can be computed readily by direct integration, namely, m = P ( L - x) -
1°C.- x)ku
dz
0
= P L ( ~- rl) - kU1L2(3- 4q
+ 14)/12,
(15.9)
where we put q = x/L. In a similar fashion we can derive for the shear force t the expression t =~
[-i4 j ( i - 191,
(15.10)
where
j= ku1L/12P.
(15.11)
Combination of (15.8) and (15.9) yields Lop, = 1 - q - p(3 - 41
+ q4),
(15.12)
where c1 is a dimensionless design given by
L =2u,a/P~~.
(1 5.1 3)
268
10 OPTIMAL BEAM DESIGN
0.8
0.6
a OPT. 0.4
0.2
0.0
0.2
0.4 0.6 r] =x/L
0.8
I .o
Fig. 10-2. Optimal beam designs, where EOp,= (2ul/PL3)aOp, and (After Distefano and Todeschini [1972].)
= kLul/12P.
Clearly, the range of p for which clopt is positive is 0 Ip IQ. In Fig. 10-2, Eopt given by (15.12) is presented for several values of p. For purposes of comparison, the same example was solved using the procedure outlined in Section 12. At a generic iteration we integrate Eqs. (11.6) in the backward direction using thecurrently available estimate for the design. There is no need to integrate Eqs. (1 1.10) since in this particular case we have m = PE. Subsequently, using the values of the r's previously computed, we integrate (1 1.7) in the forward direction. In turn we calculate a new, upgraded estimate of the design by means of &+I) = (/p+l) l / z m c n + l , ( 15.14) 1 9
an expression for u("+') that follows from (12.1) by taking derivatives and making m = PE and g = a/EhZ. In Eq. (15.14), p ( " + l ) ,given by p(n+l)
= (Eh2/p)A(n+l),
(15.15)
is a constant at each iteration that must be chosen to ensure the convergence of the sequence u(")(x,) to the prescribed value ul. We have taken in the present case p(n+l) = p(")u(")(xl)/ul.
(1 5.16)
The procedure is continued until the following convergence criterion Ek=max(lEkI, IEk+l[,IEk+21)
(1 5.17)
269
15. NUMERICAL EXAMPLES
where k --
1 - ,$k)/ki'k
+ 1)
(1 5.18)
is fulfilled. All the integrations were performed numerically using an AdamsMoulton scheme with step size 0.005 on a CDC 6400 computer. The resulting values for three different values of and at four equidistant sections of the cantilever are presented in Table 10-2. It is seen that in order to reach the same accuracy, the number of iterations increases for increasing values of the design parameter p. On the other hand, it is of interest to note that the convergence of the process is of an oscillatory type. This can be appreciated in Fig. 10-3 where the relative error &k versus the number of iterations k for two different values of has been plotted. Methods to improve the convergence properties of the process can be devised for the present problem but we do not enter into that discussion here. 0.02
0.01
-0.001
-0.01
-0.02 Fig. 10-3. Convergence of approximations where (After Distkfano and Todeschini [1972].)
E~
= 1 - 6,ck)/GckI ) and +
7 = 0.0.
N
4
0
TABLE 10-2 Comparison of Exact and Computed Designs. Cantilever Beam on Elastic Foundation"
1 12
0.00 0.25 0.50 0.75
0.75000 0.58301 0.41146 0.22363
2.0 2.0 2.0 2.0
0.74922(8) 0.58246(8) 0.41 114(8) 0.22351(8)
0.74983(9) 0.58290(9) 0.41 14Q(9) 0.22362(9)
0.82 x 0.75 x 0.64 x 0.48 x
10-3(8) 10-3(8) 10-3(8) 10-3(8)
0.78 x 0.55 x 0.32 x 0.12 x
10-3(8) 10-3(8) 10-3(8) 10-3(8)
1 6
0.00 0.25
0.5oooO
0.75
0.41602 0.32292 0.19727
2.0 2.0 2.0 2.0
0.5oooS(15) 0.41604(15) 0.32315(14) 0.19732(14)
0.49979(16) 0.41584(16) 0.32288(15) 0.19721(15)
10-3(15) 10-3(15) 10-3(14) 10-3(14)
0.25000 0.24902 0.23438 0.17090
2.0 2.0 2.0 2.0
0.25026(28) 0.24907(27) 0.23432(26) 0.17084(25)
0.25040(29) 0.24927(28) 0.23450(27) 0.17097(26)
10-3(15) 10-3(15) 10-3(14) 10-3(14) 10-3(28) 10-3(27) 10-3(26) 10-3(25)
0.09 x 0.02 x 0.23 x 0.05 x
0.00 0.25 0.50 0.75
0.60 x 0.48 x 0.83 x 0.55 x 0.59 x 0.80 x 0.80 x 0.74 x
0.26 x 0.05 x 0.06 x 0.06 x
10-3(28) 10-3(27) 10-3(26) 10-3(25)
0.50
1
7
1
0
0
4
5
r W
Numbers in parentheses denote the number of iterations.
K? n
27 1
15. NUMERICAL EXAMPLES
A further application of the method in connection with piecewise constant design is presented below. The design is taken in the form
c N
u=
(15.19)
C i H ( X - Xi),
i=l
where H is the usual unit step function. The problem consists in determining the cross-sectional stiffness ci and the lengths x i which minimize the volume of the beam. The values of ci are constrained to be one of any possible combinations of a given set of values a l , a 2 , . . . , a M . This is a version of a piecewise constant optimum design problem that occurs when the flanges of the beam must be constructed using a number of available sections a l , u 2 , . . . , aM. Substituting a given by (15.19) into the optimality condition (15.3), we can solve this problem by minimizing with respect to all possible values of ci. In the present example we have considered a1 = u2 = ... = us = 0.125. Therefore ci = 0.125j(i) where the possible values of j are 1, . . ., 6. The results are shown in Fig. 10-4 where a comparison with the unconstrained solution for B = & is possible. LO\
I
I
I
I
I
\
0.6
-
\
I
I
I
-
- \ \ 0.8 \up 0 \
I
-
\
\
\
212
10 OPTIMAL BEAM DESIGN
b. Clamped Beam under Uniformly Distributed Load An optimal design is characterized by in&
> 0,
(15.20)
a convexity condition that follows from consideration of the optimality condition (8.13). During the computation of the successive approximations following the procedure developed in Section 12, Eq. (1 5.20) might be violated leading to an indeterminacy in Eq. (8.13), unless additional information is furnished. This can be done in a number of ways. For example, we can require a positive lower bound for the design E, i.e., (15.21)
Ci26>0,
where 6 is usually taken to be the order of magnitude of the step of integration of the equations. This problem did not occur in our previous example involving the cantilever on elastic foundation, because in that case, Eq. (15.20) was identically satisfied since m = PE. The purpose of the present example is to show that Eq. (1 5.21) is enough to bypass the difficulties created by a possible violation of (15.20) during the first iterations of the process. To this end we consider a sandwich-clamped beam of length 2L subject to uniformly distributed load q. It is required that the deflection at the middle be u l . The exact optimal solution can be obtained without difficulties. In fact, assuming E to be continuously differentiable, the exact solution of this problem has been given in Section 8-16. The optimal design obtained using the method of successive approximations may be compared with the exact one, given in Chapter 8, in Fig. 10-5 and Table 10-3. In this example, the dimensionless design c( = (10u1/qL4)ccand the dimensionless quantities q = xlL and qo = X[L were introduced. The procedure used to TABLE 10-3
Comparison of Exact and Computed Designs. Clamped-Clamped Beam under Uniformly Distributed Load cy(rlj
C(O’(7lj
7
exact
initial approximation
0.00 0.20 0.40 0.60 0.80 1.00
0.75560 0.42326 0.13168 0.11387 0.30576 0.43171
0.5 0.5 0.5 0.5 0.5 0.5
0.58926 0.74725 0.30957 0.39508 0.05271 0.08591 0.13437 0.17484 0.30277 0.37943 0.41667 0.5 1566
0.72994 0.39632 0.10335 0.14341 0.33713 0.46570
0.74267 0.74948 0.40962 0.41649 0.11719 0.12414 0.12904 0.12198 0.32215 0.31492 0.44985 0.44228
NOTES, COMMENTS, AND BIBLIOGRAPHY
I
*
O
*
213
0' .0
'OPT.
q=x/L
Fig. 10-5. Comparison of successive approximations with optimal design in a clamped beam under uniformly distributed load, where GOpt= ( ~ O U ~ / ~ L and ~ ) LT~ Y= ~ ~0.50224. , (After DistCfano and Todeschini [1972].)
compute the solution using the method of successive approximations is similar to that presented in Section 12, making k = 0, and where the Riccati equations (1 l .6) and (1 l . 10) were subject to appropriate initial conditions in order to account for the difference in boundary conditions of the present beam. NOTES, COMMENTS, AND BIBLIOGRAPHY
2-7. In Part I of this chapter we present a dynamic programming treatment of the min-max problem which results from constraining the work of deformation of the given external loads on their own displacements. Here we follow closely N. DistCfano, Dynamic Programming and a Max-Min Problem in the Theory of Structures, J. Franklin Inst. 294, No. 5 (1972), 339-350.
In addition to the work by Wasiutyriski quoted in Chapter 8, the criterion of the work of deformation has been applied to some problems involving elastic plates. For a theoretical treatment of elastic plates, prestressed or not, according to the design criterion based on the work of deformation, see W. Dzieniszewski, Optimum Design of Plates of Variable Thickness for Minimum Potential Energy, Bull. Academie Polon. Sci., Sir.Sci. Tech. 13, Nc. 6 (1965).
274
10 OPTIMAL BEAM DESIGN
A numerical application of this design criterion to circular plates under uniform pressure is found in N. C . Huang, Optimal Design of Elastic Structures for Maximum Stiffness, Internat. J. Solids Structures 4 (1968), 689-700.
8-15. In Part I1 of this chapter, we follow closely N. Distkfano and R. Todeschini, Invariant Imbedding and Optimum Beam Design with Displacement Constraints, Internat. J . Solids Structures 8 (1972), 1073-1087.
Deflections at specified points of the structure are natural design constraints in most engineering applications. See R. L. Barnett, Minimum-Weight Design of Beams for Deflection, Proc. Amer. SOC.Civil Engrs. Engrg. Mech. Div. 87 (1961), 75-109.
As noted by Barnett, the problem resulting from prescribing the deflection of a beam at a single point might not be well posed. The indeterminacy is generally released, however, by incorporating additional constraints. For an example of this, see G. A. Dupuis, Optimal Design of Statically Determinate Beams Subject to Displacement and Stress Constraints, Div. of Engrg., Brown Univ., Rep. F33615-1826/1 (July 1970).
The first attempt of a systematic treatment of statically determinate beams using the methods of the calculus of variations appears to have been done by E. J. Haug, Jr. and P. G. Kirmser, Minimum Weight Design of Beams with Inequality Constraints on Stress and Deflection, J . Appl. Mech. 34 (Dec. 1967), 999-1004.
The problem of body forces in beams has been treated by invariant imbedding in the paper by Disttfano and Todeschini quoted above. See also R. L. Barnett, Minimum Deflection Design of a Uniformly Accelerating Cantilever Beam, J . Appl. Mech. 30 (19631, 466-467; J. M. Chern, Optimal Structural Design for Given Deflections in Presence of Body Forces, Internat. J . Solid Structures 7 (1971), 373-382.
In this paper the author derives the pertinent necessary condition from the principle of mutual potential energy introduced by Shield and Prager (see reference below) to treat a class of optimal design problems. In the paper by Chern, the author points out the inadequacy of the necessary condition used by Barnett to treat the problem of body forces. Additional references involving beams with deflection constraints are N. C. Huang, Optimal Design of Elastic Beams for Minimum-Maximum Deflection, Trans. ASME (Dec. 1971). 1078-1081 ;
R. T. Shield and W. Prager, Optimal Structural Design for Given Deflection, 2. Angew. Math. P h r ~21 . (1970), 513-523.
NOTES, COMMENTS, AND BIBLIOGRAPHY
275
Miscellaneous References Although not treated in this chapter, the problem of minimum-weight beam design with prescribed eigenvalues is intimately related to the problem discussed here. See, for example, J. B. Keller, The Shape of the Strongest Column, Arch. Rational Mech. Anal. 5 , Number 4 (1960), 275-285; J. B. Keller and F. I. Niordson, The Tallest Column, J. Math. Mech. 16, No. 5 (1966), 433446; C. Y . Sheu, Elastic Minimum-Weight Design for Specified Fundamental Frequency, Internat. J. Solids Structures 4 (1968), 953-958, 1968; N. C. Huang and C. Y . Sheu, Optimal Design of an Elastic Column of Thin-Walled Cross Section, J. Appl. Mech. 35 (1968), 285-288; B. L. Karihaloo and F. I. Niordson, Optimum Design of Vibrating Cantilevers, The Danish Center for Appl. Math. and Mech., Rep. No. 15 (May 1971).
See also J. L. Armand, Applications of Optimal Control Theory to Structural OptimiLation: Analytical and Numerical Approach, preprint of the IUTAM Symp. Optimization Structural Design, Polish Acad. Sci., Warsaw, Poland, Aug. 21-25, 1973.