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Optimal design of beams for moving loads with a deflection constraint
Pergamon
hr.J.
Non-Linear
Mechanics. Vol. 29, No. 2, pp. 205-216, 1994 Copyright Q 1994 Elsevier Science Ltd Printedin Gnat Britain. All rights rewved
0020-7462/94 56.00+ 0.00
OPTIMAL DESIGN OF BEAMS FOR MOVING LOADS WITH A DEFLECTION CONSTRAINT ROBERT H. BRYANT and SARA J. HEINLEIN Department of Civil Engineering, Mechanics and Metallurgy, University of Illinois at Chicago and Harza Engineering Company, U.S.A. (Received 20 January 1993)
Abstract-The problem of minimizing the volume (weight) of an elastic beam subjected to moving loads and to an imposed maximum deflection is the subject of this paper. Specifically, by means of the calculus of variations, the necessary conditions for the minimum volume are established for beams consisting of longitudinal segments of constant cross-sectional area. Two examples are considered, one for a moving concentrated load and the second for a moving distributed load. The results, at optimum, are the load location, the location of the maximum prescribed displacement, the segment lengths, and areas, and consequently the volume of the beam. For the case of the concentrated load, additional results are obtained for predetermined segment lengths. The results are presented for beam cross sections that are of the sandwich type, and for solid cross sections of different widths or depths.
NOMENCLATURE ak Ak Bk c,’
E ;k
L lk m rFl m,
mi
M A7 Mi i, j, n P P r u uo v, vo x, xk x,
:
A
Xk
dimensionless cross-sectional area cross-sectional area of segment k bending stiffness of segment cross section constant modulus of elasticity dimensionless moment of inertia cross-sectional moment of inertia of segment beam length dimensionless segment length dimensionless bending moment due to unit dummy load dimensionless bending moment due to unit couples applied at the hinge dimensionless bending moments bending moment bending moment due to unit dummy load bending moment at Xi integer value uniform load concentrated force integer defining type of cross section actual deflection dimensionless volume volumes dimensionless coordinates coordinates parameter defining extent of uniform load dimensionless displacement displacement constraint parameter defining load coordinate hinge rotation at Xi hinge rotation at Xj due to a unit force at r,-L Lagrange multiplier parameter defining unit dummy load coordinate hinge rotation.
205
R. H. BRYANTand S. J. HEINLEIN
206
INTRODUCTION
The problem of optimizing the weight (volume) of an elastic beam subjected to a displacement constraint has been examined by others. Among them are Shield and Prager [l], Dupuis [2], Masur [3], Rozvany et al. [4], and Haghnaji [S]. The treatment of the problem has been primarily for the case of fixed load location and both discrete and continuously varying cross sections. Rozvany and co-authors considered the general problem of segmentation for both stress and displacement constraints for beams and circular plates. These authors state that their results may be reduced to that of continuously variable cross sections by considering segments of infinitesimal length. Haghnaji considered the problem of moving loads for a beam with a smoothly varying profile. This last study serves as a theoretical reference base for optimal design with moving loads but, from a practical point of view, may be rejected on the basis of ease of fabrication, operational usefulness and/or other considerations. It should be mentioned that the use of finite element methods and the collateral use of sensitivity analysis (see e.g. [6,7]) would provide computational advantages in all but the simplest problems. Other numerical schemes such as the non-linear programming methods discussed by Karihaloo and Kanagasundaram [S] may also be used. Because these methods are approximate, their solutions would constitute upper bounds to the solutions obtained by Haghnaji. The study here is intended to generalize the approach used by Haghnaji to beams with segmentation and moving loads. FORMULATION
OF THE
PROBLEM
We consider the problem of finding a beam of minimum volume subjected to a moving load and a given deflection constraint. That is, we are to determine the coordinates Xk, of the segments of different cross-sectional area Ak, such that the volume I’=
i
Ak(Xk-Xk_l)
(1)
k=l
is minimized. The summation sign indicates that the beam length L is subdivided into n segments within several span lengths or within a span for a single span structure. The displacement constraint is denoted by U(X, vL) I A(& ~9,
(2)
where VL identifies the load location and 5L is the location at which the deflection is measured. Beam without a hinge By use of the unit dummy load method, where M and n;i will represent the moment functions due to the actual and unit dummy load applied at UL and
+ i k=l
&@-I,
-xk-11,
SL)dX
+q2(5L,
vL) - A dud5 1
(5)
207
Optimal design of beams for moving loads
where n(q, 5) is a Lagrange multiplier. The first variation of J, and some algebra, results in 1 1 j=2 nq4drld5+(Ai-Ai+,)~i+((xi-Xi-l)ki ss0 0 xi+*i x,+ti Mn;idX dqdg MAdX-$ MtidX+L i+l s Xi Bi s XI M&idXiq’-A
1
1
dqdc.
(6)
In equation (6), the overdot denotes the variation of the indicated quantity. Stationarity with respect to the slack variable 4 leads to 1
s0
4~9 t%W,
+W(SL
Wddrt d5 = 0
(7)
for all 4. For completeness of this paper, the argument by Haghnaji [S], to determine the meaning of equation (7), is presented: ifq#O
(or U
then1=0,
@a)
ifq=O
(or U=A)
then120.
(gb)
UC&, rlL = voKW1 = A,
PaI
UC54 rlL# ro(t-WI< A.
(W
Let 1 > 0 for q = ~~(0; then
Along the curve q = qo({), which is indicated by the path s in Fig. 1, U has a constant value given by equation (9a). It follows
In the direction n, normal to s, q # qo(r), it follows from equation (9b) that U has a smaller value than along ~~(5). Then U attains a maximum along s and
Fig. 1. q vs &
208
R. H. BRYANTand S. J. HEINLEIN
Since s and n are perpendicular,
it follows for < and rl that
au
au
K=ay = 0,
fl =
1(5).
Then 1 may be written as W/Yt;) = &WC?
-
rlO(Ul? &I2 0.
(12)
Finally assume that Q,(<) shrinks to a single point which corresponds to co, q,,, which in view of experience appears reasonable. Indeed, under some circumstances it may be shown that &, and rlo coincide [9]. It follows for il > 0 that U(rL, vL) I U(
(13)
so for U, a continuous function, it follows from equation (13) that (14) (15) The variation with respect to Ai gives, for 1 > 0, Mii’idX=O
(i= 1,2 ,_..,
n).
(16)
Shield and Prager [l] and Huang [lo] derived equation (16), but in a different form from that presented here, for the case of static loads on beams with continuous profile and special cross sections. The second and third terms in the second integral of equation (6) may be approximated as Xi. The coefficient of pi in equation (6), for all the terms, then gives
(17) for 1 > 0. Finally, the terms multiplying d gives
Equations (8), (14)-(18) constitute the necessary conditions for the minimum weight design of a beam, for a specified maximum displacement. Beam with a hinge For this case, the most general form of the deflection at
5L).
(19)
Here Bj represents the rotation, at a hinge located in the beam at the as not yet determined point, Xi. If slack variables q are again introduced, then equations (l), (2), and (19) combine to form the functional
1
- djh?(Xj, (L) + q2(tL, ?L) - A dn dt.
209
Optimal design of beams for moving loads
The first variation of J, and some algebra, noting M(X,, +5) = 0, results in
When stationarity considerations are investigated, equations (12), (13), (15), and (16), with i = j = 1, are obtained. The equivalent of equation (14) is given by 0
(22)
and equation (18) is replaced by A(<,L, qoL) = i k=l
$ lxk k
MMdX - ejU(Xj, cot).
(23)
Xk-1
Equations (22) and (23) are expressed for the most general form of a statically admissible R. It is easily shown (see e.g. [3]) that for a beam with a hinge at X = Xj, MAdX
= ejfi
(24) X=Xj
and Mk
(25)
dX = 8jS x=x,’
where A=G(x=x,+Xjx
.an;i and
.
M
Note, n;i vanishes at Xj if, for example, the actual structure is used to obtain the dummy load moment. When these results are substituted into the second integral in equation (21) and the variation in Xj is examined, we obtain (Aj-Aj+l)+&
>I
(g8,+%8;
X'Xj
=O.
(26)
It should be noted that ~j is the hinge rotation at Xj due to a unit force at qL. Equation (26) would apply when the two beam segments are connected by a hinge. The terms for the segment cross-sectional areas in equation (26) could not have been found by Masur [3] since, in the case he considered, the cross section vanished at the hinge. Instead, the hinge in this problem is geometrically imposed. Equation (26) was first obtained by Rozvany [ll] where he illustrates a class of segmental interface problems.
APPLICATIONS
The following dimensionless variables are introduced: x = X/L, xk = X,/L, ak = Ah/L2 and ik = Ik/L4 for the geometric properties of the structure. The dimensionless crosssectional moment of inertia may be written as ik = cf(akr
(r = 1, 2, 3),
(27)
where C, equals a constant and the subscript I defines the type of cross section. For example, r = 1 indicates a solid rectangular cross section of constant depth but a stepwise width variation from segment to segment. A sandwich beam of constant depth but stepwise variation in the sheet thickness would also conform to this case. For r = 2, a solid crosssectional variation in width and depth is defined, and for t = 3, a stepwise change in the
R. H. BRYANT and S. J. HEINLEIN
210
depth of a solid cross section is indicated. Only one type of cross section will be considered for any beam in our examples. The dimensionless moments are given by m(x, q) = M(X, r&)/PL and ti(x, 5) = n;i(X, 515)/L, and the hinge rotation is given by mfidx
P-9
and &jPr EL4C2I ’
K2Z-----
where P is the total load on the beam. The dimensionless bending moment due to unit couples applied at the hinge is denoted by 6(x, xj). Equations (14)-(18), and (22), (23) and (26), provided Mlx=x, z 0, may be rewritten as (30) (31)
r
(xi-x+&
mtidx=O
(i= 1,2,. . .,n),
(32)
9
4
(33)
and at an interior hinge, (34a) where +j =
$l (iy s” m(x,V)G(X,xj) dx
(34b)
WC-1
and (34c) Equation (18) or (23) becomes mriidx, where S({, q) = ELC:A(SL, Equations (30)-(33) and n cross-sectional areas and When a hinge is present corresponding location.
(35)
qL)/P. (35) represent (3 + 2n) equations to determine 5, q, rc or A,,; n segment lengths. at x = xj, then equation (34) replaces equation (33) at the
Example 1 The problem to be solved is that of a propped cantilever consisting of two segments joined by a hinge at x = x1 and subjected to a concentrated load P, as shown in Fig. 2(a). The moment rC(x, q) is given by m(x,tj)=
l-: (
m(x, ?) =
x
1-c (
(X’IIf),
> x-(x-q) >
(x2rl).
(36)
Optimal design of beams for moving loads
-1
211
-1
Fig. 2. Example beam and load conditions.
The moment fi(x, 5) is given by the same expressions if q is replaced by 5:in equation (36). Equations (30) and (31) become, when m(x, q) and ni(x, <), given by equation (36), are used, CX1(V2+ 2x: - 6x1t + 352)]a’2 + 2(1 - ~~)~a\ = 0
(37)
x1(x1 - MH2x1 - 4) - 3$]a’z + 2((1 - xJ3a’l = 0
(38)
and
for the condition q I < 5 x1. When equation (37) is multiplied by < and subtracted from equation (38) the result is x1(25 - 3x1)(52 - $) = 0.
(39) Only t = lo = q = u. will satisfy this equation and the physical conditions of the problem. The result is independent of the cross-sectional areas and the segment lengths; then m = ti. Equations (37) and (38) become identical and may be written in the form x1(x1
-
2rlONXl
-
vo)
+
(1
-
d3
2
I=o.
(40)
0
The integration in equation (32) results in the following for a, and a,:
(41’3 When the ratio of the last two equations is taken, the result is
0 al -
a2
‘+l = (Xl
-
?o12
(1 - x1)2 .
(42)
When r is set equal to 1 and equation (42) is substituted into equation (40) by eliminating (a&,), a real solution to the resulting algebraic equation is q. = fi - 1 and xl = d/2. This result yields aI = a, when substituted into equation (42). Again noting m = ti, NLM29:2-H
212
R. H. BRYANTand S. J. HEINLEIN
equation (34a) gives
The results for I = 1 satisfies equation (43) identically. When ai = az, xi = fi/2 and q. = fi - 1, th en equations (40), (41), and (43) are satisfied for all three values of 1. This result is the same as that obtained by Haghnaji [S] for a beam with continuously varying cross section and, not surprising, a beam with a uniform cross section and no hinge. For a, = a2 and m = fi, equation (35) reduces to 6(qo, qo) =
‘&‘dx,
(44)
s0
where u” is the dimensionless curvature. Using equation (36) for rl = ?. < x1, equation (44) gives P
(45)
a’l = 3ELC;A with al = a2, the dimensionless volume is given by UrJ= aixi + a,(1 - Xi).
(46)
The volume for I = 1 is O.O0981P/(ELA)C~, I = 2 is 0.0991P’~2/(ELA)‘~2C2, and I = 3 is 0.214P”3(ELA) 1’3C $I3. If for this structure, factors such as the non-availability of the necessary total length of beam segments need consideration, then al # a, (i.e. xj is specified and %j = 0). The optimum conditions are then given by equations (40), (41), and (35) to determine q. = to, al, a2 and lc2.The results are shown in Figs 3-5 for q,, < xi. Note again q. = to and x1 are independent of I at al/a2 = 1. For q. > xl, the load is beyond the hinge, and the load supporting structure is a cantilever of length (1 - q). The maximum deflection will occur at q. = x1, and the volume of the supporting cantilever just shown is 02 =
a2(1 - Xl).
(47)
For this case, the dimensionless bending moment is m(x, x1) = -(x
0.675____ _..._._. -
0.475-
O&O0.426-'---s
Fig. 3. Dimensionless
load location
vs area ratio.
El r-2 f=3
- x1), I2 = (1 - xi)
213
Optimal design of beams for moving loads
0.75 -
0.70 -
Xl
0.S
0.7
0.6
0.9
0.0
al
_-em
El
_._.s.-.
&
-
N
1.1
1.0
fa,
Fig. 4. Dimensionless hinge location vs area ratio.
_---.-- =, _..._._. p< r-3 0.25 -
_.I_
0.5 ,
0.6
0.7
0:
0.8
CO
111
ajla2
Fig. 5. Dimensionless volume vs area ratio.
and p (1 -x1)? a’2= 3ELC;A
(48)
When the volume given by equation (47) exceeds the volume given by the second term on the right-hand side of equation (46), then equation (47), with u2 given by equation (48), must be used for that second term in computing the total volume. By means of equations (40), (41), (45), and (47), x1, qo, and u1/u2 may be determined when the volumes of the beam, for the two load locations, are equal. This location is shown in Figs 3-5, where the upper curve,
R. H. BRYANTand S. J. HEINLEIN
214
Table 1. Load, hinge location, and area ratio for equal volume I
x1 = ?o
alla2
0.613 0.662 0.655
0.786 0.825 0.852
0.416 0.409 0.405
1 2 3
corresponding to Q, = x1, intersects the lower curve, corresponding the q. < x1. Table 1 lists the values of 11o, x1 and al/a2 corresponding to the intersection points. For values of al/a2 less than the table values, the value of aI corresponding to the intersection of the curves was used in the computation of the total volume along with a2 as given by equation (48). Figure 5 shows that the uniform beam @r/a2 = 1) provides the least volume. This result coincides with those obtained by Bryant [12] for the case of minimum volume, same structure and loading, but with a compliance constraint. Example 2
In this example, we will consider the same propped cantilever beam as in Example 1 but under a distributed uniform moving load of intensity p and length 2aL as shown in Fig. 2(b). The location, qL, acts at the centroid of the loaded region and a is a parameter defining the length of the loaded region. The dimensionless bending moment m(x, q) is given by the following. Case 1: q + u < x1 xsqa: m(x;q)=2a
l-:
x,
(
>
am 2~f ---x. %jx1
m(x; q) = 2a
( > (x- i+ ‘12, T
1 -
x
_
Xl
-=-z dm all
+(x-q++),
(1-C >, am& ( )
m(x;q)=2arj
VW
l-5.
all
Plb)
Xl
Case 2: fj - a < x1 < rj + a xIq-a: m(x.
?)
=
(x1- q + @
3
am -=-
(Xl
>
‘1)
am -=-
all
=
’
rl +
Co
Wb)
X.
Xl
all
mtx.
-
(524
x
2x1
(XI - v + Co”x _ (x - u + 4’ 2
2x1 (x1-q++)
x Xl
-t- (x
-
’ q +
a).
VW
W-9
Optimal design of beams for moving loads Table 2. Load and/or hinge location, 2a
0.25 0.50 0.75
al/a2
rlo
(0
Xl
0.416 0.420 0.437
0.416 0.420 0.426
0.709 0.716 0.727
=
215 1
Table 3. Dimensionless volume for uniform cross section
0.25 0.50 0.75
q+asxs
0.00235 0.00417 0.00515
0.0485 0.0646 0.0717
0.133 0.161 0.173
1:
4x; 4 = Bm -=all
(Xl
rj + a)’
-
x - 2a(x
2x
-
q),
1
h-4+sr)x+2a Xl
Wb)
The expression for ti(x, 5) again represents the dimensionless moment function due to the unit dummy load. Equations (30)-(35) apply, again provided d Ixzx, E 0, and though their application is straightforward, the algebra is rather tedious. The example will restrict itself to the case where al = az. Depending on the value of a, equations (49)-(51) or (52)-(54) were substituted into equations (30), (31), (34), and (35). After integration, the solution for qo, x1, and to was obtained by trial and error, where equations (30), (34) and eventually (31) were satisfied while simultaneously yielding the maximum value of 6 given by equation (35). The results are shown in Table 2 for the indicated values of a. For the beam of constant cross section, the displacement constraint may be expressed as (J=
p
EL2C2a’ r
s 'mfi
:dx.
0
’
(55)
Equation (55) relates the cross-sectional area to the constraint, and the dimensionless volume, uo, is determined, as in Example 1, using equation (46) where al = a2 = a. The dimensionless volume is given in Table 3 for various values of a and r.
CONCLUSION
The necessary conditions for the minimum volume of an elastic beam, consisting of lengthwise subregions of constant cross section, a moving load and subjected to a prescribed displacement constraint, were established. The formulation allowed for the determination of the cross-sectional area, segment length, and the load location. The equations at the change in cross section for a continuous and hinged connection were examined separately. Two examples were considered. The structure was a propped cantilevered beam consisting of two segments separated by a hinge. For the first example, a moving concentrated load, the minimum volume occurred for the case of equal cross sections. For all other cases of specified segment lengths, the corresponding area ratio determined the load location and volume. The second example had a moving uniformly distributed load acting over a finite length. Three different load lengths were considered and the minimum volume was determined for each. For the point load, the location of the load, hinge and the point of maximum deflection were independent of the type of cross section for the optimum design. The results for the point load coincided with the previously obtained results for a compliance constraint.
216
R. H. BRYANTand S. J.
HEINLEIN
REFERENCES 1. R. T. Shield and W. Prager, Optimal structural design for given deflection. 2. Angew. Math. Phys. 21,513-523 (1970). 2. G. Dupuis, Optimal design of statically determinate beam subject to displacement and stress constraints. AIAA J. 9, 981-987 (1971). 3. E. F. Masur, Optimally in the presence of discreteness and discontinuity, in Optimization in Structural Design (Edited by A. Sawczak and Z. Mroz), pp. 441-453. Springer, Berlin. 4. G. I. N. Rozvany, T. G. Ong and B. L. Karihaloo, A general theory of optimal elastic design for structures with segmentation. J. Appl. Mech. 53, 242-248 (1986). 5. P. Haghnaji, Optimal design of elastic structures under moving loads subject to various constraint conditions, Ph.D. Thesis, University of Illinois at Chicago, IL (1987). 6. H. Van Belle, Theory of adjoint structures. AIAA J. 14, 977-979 (1976). 7. N. C. Huang, K. K. Choi and V. Komkov, Design Sensitivity Analysis of StructuralSystems. Academic Press, New York (1986). 8. B. L. Karihaloo and S. Kanagasundaram, Optimum design of statically interminate structures subject to strength and stiffness constraints and multiple loading. Comput. Struct. 30, 563-572 (1988). 9. S. J. Heinlein, Optimal design of beams for moving loads with a deflection constraint. M.S. Thesis, University of Illinois at Chicago, IL (1992). 10. N. C. Huang, On the principle of stationary mutual complementary energy and its application to structural design. Z. Angew. Math. Phys. 22, 608-620 (1971). 11. G. I. N. Rozvany, Structural Design via Optimality Criteria. Kluwer, Dordrecht (1989). 12. R. H. Bryant, Optimal design of elastic beams for moving loads. J. Engng Mech. 117, 154-165 (1991)
hr.J.
Non-Linear
Mechanics. Vol. 29, No. 2, pp. 205-216, 1994 Copyright Q 1994 Elsevier Science Ltd Printedin Gnat Britain. All rights rewved
0020-7462/94 56.00+ 0.00
OPTIMAL DESIGN OF BEAMS FOR MOVING LOADS WITH A DEFLECTION CONSTRAINT ROBERT H. BRYANT and SARA J. HEINLEIN Department of Civil Engineering, Mechanics and Metallurgy, University of Illinois at Chicago and Harza Engineering Company, U.S.A. (Received 20 January 1993)
Abstract-The problem of minimizing the volume (weight) of an elastic beam subjected to moving loads and to an imposed maximum deflection is the subject of this paper. Specifically, by means of the calculus of variations, the necessary conditions for the minimum volume are established for beams consisting of longitudinal segments of constant cross-sectional area. Two examples are considered, one for a moving concentrated load and the second for a moving distributed load. The results, at optimum, are the load location, the location of the maximum prescribed displacement, the segment lengths, and areas, and consequently the volume of the beam. For the case of the concentrated load, additional results are obtained for predetermined segment lengths. The results are presented for beam cross sections that are of the sandwich type, and for solid cross sections of different widths or depths.
NOMENCLATURE ak Ak Bk c,’
E ;k
L lk m rFl m,
mi
M A7 Mi i, j, n P P r u uo v, vo x, xk x,
:
A
Xk
dimensionless cross-sectional area cross-sectional area of segment k bending stiffness of segment cross section constant modulus of elasticity dimensionless moment of inertia cross-sectional moment of inertia of segment beam length dimensionless segment length dimensionless bending moment due to unit dummy load dimensionless bending moment due to unit couples applied at the hinge dimensionless bending moments bending moment bending moment due to unit dummy load bending moment at Xi integer value uniform load concentrated force integer defining type of cross section actual deflection dimensionless volume volumes dimensionless coordinates coordinates parameter defining extent of uniform load dimensionless displacement displacement constraint parameter defining load coordinate hinge rotation at Xi hinge rotation at Xj due to a unit force at r,-L Lagrange multiplier parameter defining unit dummy load coordinate hinge rotation.
205
R. H. BRYANTand S. J. HEINLEIN
206
INTRODUCTION
The problem of optimizing the weight (volume) of an elastic beam subjected to a displacement constraint has been examined by others. Among them are Shield and Prager [l], Dupuis [2], Masur [3], Rozvany et al. [4], and Haghnaji [S]. The treatment of the problem has been primarily for the case of fixed load location and both discrete and continuously varying cross sections. Rozvany and co-authors considered the general problem of segmentation for both stress and displacement constraints for beams and circular plates. These authors state that their results may be reduced to that of continuously variable cross sections by considering segments of infinitesimal length. Haghnaji considered the problem of moving loads for a beam with a smoothly varying profile. This last study serves as a theoretical reference base for optimal design with moving loads but, from a practical point of view, may be rejected on the basis of ease of fabrication, operational usefulness and/or other considerations. It should be mentioned that the use of finite element methods and the collateral use of sensitivity analysis (see e.g. [6,7]) would provide computational advantages in all but the simplest problems. Other numerical schemes such as the non-linear programming methods discussed by Karihaloo and Kanagasundaram [S] may also be used. Because these methods are approximate, their solutions would constitute upper bounds to the solutions obtained by Haghnaji. The study here is intended to generalize the approach used by Haghnaji to beams with segmentation and moving loads. FORMULATION
OF THE
PROBLEM
We consider the problem of finding a beam of minimum volume subjected to a moving load and a given deflection constraint. That is, we are to determine the coordinates Xk, of the segments of different cross-sectional area Ak, such that the volume I’=
i
Ak(Xk-Xk_l)
(1)
k=l
is minimized. The summation sign indicates that the beam length L is subdivided into n segments within several span lengths or within a span for a single span structure. The displacement constraint is denoted by U(X, vL) I A(& ~9,
(2)
where VL identifies the load location and 5L is the location at which the deflection is measured. Beam without a hinge By use of the unit dummy load method, where M and n;i will represent the moment functions due to the actual and unit dummy load applied at UL and
+ i k=l
&@-I,
-xk-11,
SL)dX
+q2(5L,
vL) - A dud5 1
(5)
207
Optimal design of beams for moving loads
where n(q, 5) is a Lagrange multiplier. The first variation of J, and some algebra, results in 1 1 j=2 nq4drld5+(Ai-Ai+,)~i+((xi-Xi-l)ki ss0 0 xi+*i x,+ti Mn;idX dqdg MAdX-$ MtidX+L i+l s Xi Bi s XI M&idXiq’-A
1
1
dqdc.
(6)
In equation (6), the overdot denotes the variation of the indicated quantity. Stationarity with respect to the slack variable 4 leads to 1
s0
4~9 t%W,
+W(SL
Wddrt d5 = 0
(7)
for all 4. For completeness of this paper, the argument by Haghnaji [S], to determine the meaning of equation (7), is presented: ifq#O
(or U
then1=0,
@a)
ifq=O
(or U=A)
then120.
(gb)
UC&, rlL = voKW1 = A,
PaI
UC54 rlL# ro(t-WI< A.
(W
Let 1 > 0 for q = ~~(0; then
Along the curve q = qo({), which is indicated by the path s in Fig. 1, U has a constant value given by equation (9a). It follows
In the direction n, normal to s, q # qo(r), it follows from equation (9b) that U has a smaller value than along ~~(5). Then U attains a maximum along s and
Fig. 1. q vs &
208
R. H. BRYANTand S. J. HEINLEIN
Since s and n are perpendicular,
it follows for < and rl that
au
au
K=ay = 0,
fl =
1(5).
Then 1 may be written as W/Yt;) = &WC?
-
rlO(Ul? &I2 0.
(12)
Finally assume that Q,(<) shrinks to a single point which corresponds to co, q,,, which in view of experience appears reasonable. Indeed, under some circumstances it may be shown that &, and rlo coincide [9]. It follows for il > 0 that U(rL, vL) I U(
(13)
so for U, a continuous function, it follows from equation (13) that (14) (15) The variation with respect to Ai gives, for 1 > 0, Mii’idX=O
(i= 1,2 ,_..,
n).
(16)
Shield and Prager [l] and Huang [lo] derived equation (16), but in a different form from that presented here, for the case of static loads on beams with continuous profile and special cross sections. The second and third terms in the second integral of equation (6) may be approximated as Xi. The coefficient of pi in equation (6), for all the terms, then gives
(17) for 1 > 0. Finally, the terms multiplying d gives
Equations (8), (14)-(18) constitute the necessary conditions for the minimum weight design of a beam, for a specified maximum displacement. Beam with a hinge For this case, the most general form of the deflection at
5L).
(19)
Here Bj represents the rotation, at a hinge located in the beam at the as not yet determined point, Xi. If slack variables q are again introduced, then equations (l), (2), and (19) combine to form the functional
1
- djh?(Xj, (L) + q2(tL, ?L) - A dn dt.
209
Optimal design of beams for moving loads
The first variation of J, and some algebra, noting M(X,, +5) = 0, results in
When stationarity considerations are investigated, equations (12), (13), (15), and (16), with i = j = 1, are obtained. The equivalent of equation (14) is given by 0
(22)
and equation (18) is replaced by A(<,L, qoL) = i k=l
$ lxk k
MMdX - ejU(Xj, cot).
(23)
Xk-1
Equations (22) and (23) are expressed for the most general form of a statically admissible R. It is easily shown (see e.g. [3]) that for a beam with a hinge at X = Xj, MAdX
= ejfi
(24) X=Xj
and Mk
(25)
dX = 8jS x=x,’
where A=G(x=x,+Xjx
.an;i and
.
M
Note, n;i vanishes at Xj if, for example, the actual structure is used to obtain the dummy load moment. When these results are substituted into the second integral in equation (21) and the variation in Xj is examined, we obtain (Aj-Aj+l)+&
>I
(g8,+%8;
X'Xj
=O.
(26)
It should be noted that ~j is the hinge rotation at Xj due to a unit force at qL. Equation (26) would apply when the two beam segments are connected by a hinge. The terms for the segment cross-sectional areas in equation (26) could not have been found by Masur [3] since, in the case he considered, the cross section vanished at the hinge. Instead, the hinge in this problem is geometrically imposed. Equation (26) was first obtained by Rozvany [ll] where he illustrates a class of segmental interface problems.
APPLICATIONS
The following dimensionless variables are introduced: x = X/L, xk = X,/L, ak = Ah/L2 and ik = Ik/L4 for the geometric properties of the structure. The dimensionless crosssectional moment of inertia may be written as ik = cf(akr
(r = 1, 2, 3),
(27)
where C, equals a constant and the subscript I defines the type of cross section. For example, r = 1 indicates a solid rectangular cross section of constant depth but a stepwise width variation from segment to segment. A sandwich beam of constant depth but stepwise variation in the sheet thickness would also conform to this case. For r = 2, a solid crosssectional variation in width and depth is defined, and for t = 3, a stepwise change in the
R. H. BRYANT and S. J. HEINLEIN
210
depth of a solid cross section is indicated. Only one type of cross section will be considered for any beam in our examples. The dimensionless moments are given by m(x, q) = M(X, r&)/PL and ti(x, 5) = n;i(X, 515)/L, and the hinge rotation is given by mfidx
P-9
and &jPr EL4C2I ’
K2Z-----
where P is the total load on the beam. The dimensionless bending moment due to unit couples applied at the hinge is denoted by 6(x, xj). Equations (14)-(18), and (22), (23) and (26), provided Mlx=x, z 0, may be rewritten as (30) (31)
r
(xi-x+&
mtidx=O
(i= 1,2,. . .,n),
(32)
9
4
(33)
and at an interior hinge, (34a) where +j =
$l (iy s” m(x,V)G(X,xj) dx
(34b)
WC-1
and (34c) Equation (18) or (23) becomes mriidx, where S({, q) = ELC:A(SL, Equations (30)-(33) and n cross-sectional areas and When a hinge is present corresponding location.
(35)
qL)/P. (35) represent (3 + 2n) equations to determine 5, q, rc or A,,; n segment lengths. at x = xj, then equation (34) replaces equation (33) at the
Example 1 The problem to be solved is that of a propped cantilever consisting of two segments joined by a hinge at x = x1 and subjected to a concentrated load P, as shown in Fig. 2(a). The moment rC(x, q) is given by m(x,tj)=
l-: (
m(x, ?) =
x
1-c (
(X’IIf),
> x-(x-q) >
(x2rl).
(36)
Optimal design of beams for moving loads
-1
211
-1
Fig. 2. Example beam and load conditions.
The moment fi(x, 5) is given by the same expressions if q is replaced by 5:in equation (36). Equations (30) and (31) become, when m(x, q) and ni(x, <), given by equation (36), are used, CX1(V2+ 2x: - 6x1t + 352)]a’2 + 2(1 - ~~)~a\ = 0
(37)
x1(x1 - MH2x1 - 4) - 3$]a’z + 2((1 - xJ3a’l = 0
(38)
and
for the condition q I < 5 x1. When equation (37) is multiplied by < and subtracted from equation (38) the result is x1(25 - 3x1)(52 - $) = 0.
(39) Only t = lo = q = u. will satisfy this equation and the physical conditions of the problem. The result is independent of the cross-sectional areas and the segment lengths; then m = ti. Equations (37) and (38) become identical and may be written in the form x1(x1
-
2rlONXl
-
vo)
+
(1
-
d3
2
I=o.
(40)
0
The integration in equation (32) results in the following for a, and a,:
(41’3 When the ratio of the last two equations is taken, the result is
0 al -
a2
‘+l = (Xl
-
?o12
(1 - x1)2 .
(42)
When r is set equal to 1 and equation (42) is substituted into equation (40) by eliminating (a&,), a real solution to the resulting algebraic equation is q. = fi - 1 and xl = d/2. This result yields aI = a, when substituted into equation (42). Again noting m = ti, NLM29:2-H
212
R. H. BRYANTand S. J. HEINLEIN
equation (34a) gives
The results for I = 1 satisfies equation (43) identically. When ai = az, xi = fi/2 and q. = fi - 1, th en equations (40), (41), and (43) are satisfied for all three values of 1. This result is the same as that obtained by Haghnaji [S] for a beam with continuously varying cross section and, not surprising, a beam with a uniform cross section and no hinge. For a, = a2 and m = fi, equation (35) reduces to 6(qo, qo) =
‘&‘dx,
(44)
s0
where u” is the dimensionless curvature. Using equation (36) for rl = ?. < x1, equation (44) gives P
(45)
a’l = 3ELC;A with al = a2, the dimensionless volume is given by UrJ= aixi + a,(1 - Xi).
(46)
The volume for I = 1 is O.O0981P/(ELA)C~, I = 2 is 0.0991P’~2/(ELA)‘~2C2, and I = 3 is 0.214P”3(ELA) 1’3C $I3. If for this structure, factors such as the non-availability of the necessary total length of beam segments need consideration, then al # a, (i.e. xj is specified and %j = 0). The optimum conditions are then given by equations (40), (41), and (35) to determine q. = to, al, a2 and lc2.The results are shown in Figs 3-5 for q,, < xi. Note again q. = to and x1 are independent of I at al/a2 = 1. For q. > xl, the load is beyond the hinge, and the load supporting structure is a cantilever of length (1 - q). The maximum deflection will occur at q. = x1, and the volume of the supporting cantilever just shown is 02 =
a2(1 - Xl).
(47)
For this case, the dimensionless bending moment is m(x, x1) = -(x
0.675____ _..._._. -
0.475-
O&O0.426-'---s
Fig. 3. Dimensionless
load location
vs area ratio.
El r-2 f=3
- x1), I2 = (1 - xi)
213
Optimal design of beams for moving loads
0.75 -
0.70 -
Xl
0.S
0.7
0.6
0.9
0.0
al
_-em
El
_._.s.-.
&
-
N
1.1
1.0
fa,
Fig. 4. Dimensionless hinge location vs area ratio.
_---.-- =, _..._._. p< r-3 0.25 -
_.I_
0.5 ,
0.6
0.7
0:
0.8
CO
111
ajla2
Fig. 5. Dimensionless volume vs area ratio.
and p (1 -x1)? a’2= 3ELC;A
(48)
When the volume given by equation (47) exceeds the volume given by the second term on the right-hand side of equation (46), then equation (47), with u2 given by equation (48), must be used for that second term in computing the total volume. By means of equations (40), (41), (45), and (47), x1, qo, and u1/u2 may be determined when the volumes of the beam, for the two load locations, are equal. This location is shown in Figs 3-5, where the upper curve,
R. H. BRYANTand S. J. HEINLEIN
214
Table 1. Load, hinge location, and area ratio for equal volume I
x1 = ?o
alla2
0.613 0.662 0.655
0.786 0.825 0.852
0.416 0.409 0.405
1 2 3
corresponding to Q, = x1, intersects the lower curve, corresponding the q. < x1. Table 1 lists the values of 11o, x1 and al/a2 corresponding to the intersection points. For values of al/a2 less than the table values, the value of aI corresponding to the intersection of the curves was used in the computation of the total volume along with a2 as given by equation (48). Figure 5 shows that the uniform beam @r/a2 = 1) provides the least volume. This result coincides with those obtained by Bryant [12] for the case of minimum volume, same structure and loading, but with a compliance constraint. Example 2
In this example, we will consider the same propped cantilever beam as in Example 1 but under a distributed uniform moving load of intensity p and length 2aL as shown in Fig. 2(b). The location, qL, acts at the centroid of the loaded region and a is a parameter defining the length of the loaded region. The dimensionless bending moment m(x, q) is given by the following. Case 1: q + u < x1 xsqa: m(x;q)=2a
l-:
x,
(
>
am 2~f ---x. %jx1
m(x; q) = 2a
( > (x- i+ ‘12, T
1 -
x
_
Xl
-=-z dm all
+(x-q++),
(1-C >, am& ( )
m(x;q)=2arj
VW
l-5.
all
Plb)
Xl
Case 2: fj - a < x1 < rj + a xIq-a: m(x.
?)
=
(x1- q + @
3
am -=-
(Xl
>
‘1)
am -=-
all
=
’
rl +
Co
Wb)
X.
Xl
all
mtx.
-
(524
x
2x1
(XI - v + Co”x _ (x - u + 4’ 2
2x1 (x1-q++)
x Xl
-t- (x
-
’ q +
a).
VW
W-9
Optimal design of beams for moving loads Table 2. Load and/or hinge location, 2a
0.25 0.50 0.75
al/a2
rlo
(0
Xl
0.416 0.420 0.437
0.416 0.420 0.426
0.709 0.716 0.727
=
215 1
Table 3. Dimensionless volume for uniform cross section
0.25 0.50 0.75
q+asxs
0.00235 0.00417 0.00515
0.0485 0.0646 0.0717
0.133 0.161 0.173
1:
4x; 4 = Bm -=all
(Xl
rj + a)’
-
x - 2a(x
2x
-
q),
1
h-4+sr)x+2a Xl
Wb)
The expression for ti(x, 5) again represents the dimensionless moment function due to the unit dummy load. Equations (30)-(35) apply, again provided d Ixzx, E 0, and though their application is straightforward, the algebra is rather tedious. The example will restrict itself to the case where al = az. Depending on the value of a, equations (49)-(51) or (52)-(54) were substituted into equations (30), (31), (34), and (35). After integration, the solution for qo, x1, and to was obtained by trial and error, where equations (30), (34) and eventually (31) were satisfied while simultaneously yielding the maximum value of 6 given by equation (35). The results are shown in Table 2 for the indicated values of a. For the beam of constant cross section, the displacement constraint may be expressed as (J=
p
EL2C2a’ r
s 'mfi
:dx.
0
’
(55)
Equation (55) relates the cross-sectional area to the constraint, and the dimensionless volume, uo, is determined, as in Example 1, using equation (46) where al = a2 = a. The dimensionless volume is given in Table 3 for various values of a and r.
CONCLUSION
The necessary conditions for the minimum volume of an elastic beam, consisting of lengthwise subregions of constant cross section, a moving load and subjected to a prescribed displacement constraint, were established. The formulation allowed for the determination of the cross-sectional area, segment length, and the load location. The equations at the change in cross section for a continuous and hinged connection were examined separately. Two examples were considered. The structure was a propped cantilevered beam consisting of two segments separated by a hinge. For the first example, a moving concentrated load, the minimum volume occurred for the case of equal cross sections. For all other cases of specified segment lengths, the corresponding area ratio determined the load location and volume. The second example had a moving uniformly distributed load acting over a finite length. Three different load lengths were considered and the minimum volume was determined for each. For the point load, the location of the load, hinge and the point of maximum deflection were independent of the type of cross section for the optimum design. The results for the point load coincided with the previously obtained results for a compliance constraint.
216
R. H. BRYANTand S. J.
HEINLEIN
REFERENCES 1. R. T. Shield and W. Prager, Optimal structural design for given deflection. 2. Angew. Math. Phys. 21,513-523 (1970). 2. G. Dupuis, Optimal design of statically determinate beam subject to displacement and stress constraints. AIAA J. 9, 981-987 (1971). 3. E. F. Masur, Optimally in the presence of discreteness and discontinuity, in Optimization in Structural Design (Edited by A. Sawczak and Z. Mroz), pp. 441-453. Springer, Berlin. 4. G. I. N. Rozvany, T. G. Ong and B. L. Karihaloo, A general theory of optimal elastic design for structures with segmentation. J. Appl. Mech. 53, 242-248 (1986). 5. P. Haghnaji, Optimal design of elastic structures under moving loads subject to various constraint conditions, Ph.D. Thesis, University of Illinois at Chicago, IL (1987). 6. H. Van Belle, Theory of adjoint structures. AIAA J. 14, 977-979 (1976). 7. N. C. Huang, K. K. Choi and V. Komkov, Design Sensitivity Analysis of StructuralSystems. Academic Press, New York (1986). 8. B. L. Karihaloo and S. Kanagasundaram, Optimum design of statically interminate structures subject to strength and stiffness constraints and multiple loading. Comput. Struct. 30, 563-572 (1988). 9. S. J. Heinlein, Optimal design of beams for moving loads with a deflection constraint. M.S. Thesis, University of Illinois at Chicago, IL (1992). 10. N. C. Huang, On the principle of stationary mutual complementary energy and its application to structural design. Z. Angew. Math. Phys. 22, 608-620 (1971). 11. G. I. N. Rozvany, Structural Design via Optimality Criteria. Kluwer, Dordrecht (1989). 12. R. H. Bryant, Optimal design of elastic beams for moving loads. J. Engng Mech. 117, 154-165 (1991)