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Optimal archgrids: Allowance for selfweight
COMPUTER METHODS IN APPLIED @ NORTH-HOLLAND PUBLISHING
OPTIMAL
MECHANICS COMPANY
ARCHGRIDS:
G.I.N. ROZVANY, Faculty of Engineering,
AND ENGINEERING
ALLOWANCE
H. NAKAMURA Monash Received
University,
FOR SELFWEIGHT
and B.T. KUHNELL
Clayton,
26 September
24 (1980) 287-304
Victoria, Australia,
3168
1979
An earlier study by Prager and the first author considered the optimization of archgrids for a given external load. The current paper discusses the optimal design of archgrids for a combination of given external loads and selfweight. Three different methods are described and then illustrated with examples of one- and two-way arch systems. The results furnished by various methods show a good agreement and also converge to earlier obtained analytical solutions.
Introduction
Prager and the first author [l] investigated the optimal design of long span surface structures consisting of a dense system of intersecting arches. Whereas the above study discussed the effect of given external loads only, in the following we consider a combination of external loads and selfweight. Since the weight of the structure itself depends on the design adopted, the formulation of the current problem becomes much more complicated than optimal design for external loads only. Another earlier study [2] obtained closed form analytical optimal solutions for a combination of external loads, selfweight and the weight of roof sheeting of constant thickness but was restricted to one-way systems of parallel arches. Some of these analytical solutions will be used as a basis for comparison in the current study. Archgrids or grid shells (Gitterschalen in German) have been studied extensively and also used in the actual design of long span roofs by Frei Otto [3]. The problem of optimization for selfweight and roof sheeting was considered also, in the context of spherical domes, by Prager and the first author [4]. Some aspects of optimal arch design for combined bending and axial force as well as for alternate load conditions were discussed briefly in a recent note [5]. In this paper, first some existing results are reviewed and then three different methods for the current problem are described. Finally, examples of archgrid optimization are presented.
1. Review of existing results
The optimization of archgrids for external loads [l] was based on a number of simplifying assumptions. It was stipulated that all arches were to resist axial compression only under the single load system considered. Moreover, no allowance was made for elastic compatibility, nor for instability; the resulting solution thus corresponded to an optimal plastic design for a given vertical load system p. In addition, the design was restricted to arches whose horizontal 287
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G.I.N. Rozvany, H. Nakamura,
B.T. Kuhnell, Optimal archgrids: allowance for selfweight
projection was parallel to rectangular co-ordinates (x, y) contained in a horizontal reference plane. With a view to facilitating the mathematical treatment, it was first assumed that at a point (x, y) of the reference plane, the elevation of the x-arch and y-arch is not necessarily the same, but the two arch systems are connected by vertical ties whose weight is disregarded. The total volume of the arches was to be minimized within a permissible stress constraint. It was then found [l] that the “optimal” archgrid satisfies the following three conditions: (a) Each arch centreline has the shape of an appropriately scaled moment diagram which is associated with the “equivalent” simply supported beam of the same span and loading as the considered arch. In other words, each arch must form a funicular curve for the share of load it is resisting and the loads on the x and y arch systems &p,) must add up to the total load [PXkY)+P&,Y)=P(x7Y)l. (b) Along any one arch, the mean value of the square of the slope must be unity. For an x-arch, for example, 2 dx/L, = 1.0,
(1)
where L, is the arch length, z, is the arch elevation, and (x1, x2) are the end co-ordinates of the arch. (c) The elevation of the x and y arches at any one point (x, y) of the system is the same,
&(X9y)= ‘&
Y).
(2)
This means that we have finished up with a single layer of arches in the optimal solution, although initially we permitted two separate layers for the x-arches and y-arches. The above three optimality criteria may be termed (a) zero bending, (b) unit mean square slope and (c) equal elevation conditions. Out of various combinations of loads for one-way arch systems [2], we review here briefly the results for a uniformly distributed vertical load p = const plus selfweight. The relative influence of the selfweight on the solution depends on the nondimensional parameter (r which is defined as a = yucl,
(3)
where y is the specific weight of the arch material, L is the half-span of the arch, and c() is the permissible stress. Introducing then the nondimensional parameters .V= x/L, j.j = y/L. /? = pL/H where H is the horizontal reaction at the arch supports, one finds [2] that the optimal arch shape is given by the relation al = ln{sec[cuZ(l+ p/a)“‘]},
(4)
P = {sin[2a(l+
(5)
where pl(~)“~]}(l + p/2a!)/(l + P/cx)~/~.
GIN.
Rozvany, H. Nakamura, B.T. Kuhnell, Optimal archgrids: allowance for selfweight
289
0 IN--
0.8
a
,
a
1.8
2.0
2.4
Fig. 1. Analytical
optimal
solutions
a =1.50
v
for one-way
arch systems
having various
non-dimensional
spans (Y = yL/ao.
Three optimal arch centrelines for various cu-values are given in fig. 1. The one for (Y= 0 reduces to a parabola which clearly satisfies the unit mean square- slope condition (1) for arches whose selfweight is neglected.
2. Upper bound solutions obtained by an iterative method Reasonably close upper bound solutions for relatively small a-values can be obtained by an iterative method. In each cycle of this procedure, the selfweight is calculated on the basis of the solution in the prior cycle. Then the selfweight is added to the external load and the structure is re-optimized on the basis of conditions (a) through (c) in section 1, as if such combined load was a given external load. For large a-values, this method is clearly inaccurate because the selfweight is a design-dependent variable and hence it cannot be treated as a prescribed load. Even the unit mean square slope condition (1) must give inefficient designs for (Y% 0, since in fig. 1 the mean square slope is clearly much greater than unity for (Y> 0.75. Some upper bound solutions using the foregoing method are given in the examples in section 6.
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3. Method A. Variational formulation The variational formulation gives reasonably good results when the archgrid consists of a large number of arches and hence it can be idealised as an “archgrid-like continuum” [l] in which the arch spacing is infinitesimal. As earlier [l], we consider here an archgrid-like continuum over a domain D of a horizontal plane referred to co-ordinates (x, y). A given single force system p(x, y) defined on D is to be transmitted to rigid supports along the boundary of D by a system of intersecting arches of infinitesimal spacing running in x and y directions. All arches are to be subjected to axial compression only so that all cross-sections are stressed uniformly to a stress value of go under the combined load consisting of the external forces p(x, y) and the selfweight s(x, y). It was established earlier [l, eq. 61 that the selfweight of the archgrid per unit horizontal area is s = c[Hx + H, +
H;‘(Mx.,)2 + H,‘(My,y)2],
(6)
in which c = y/(~~, H, and H, are horizontal reactions per unit width, M, and MY are the bending moments in the “equivalent beams” corresponding to x and y arches [l, fig. 11, and subscripts after commas denote partial derivatives with respect to the variables indicated by such subscripts. Incorporating the equilibrium condition by the use of the Lagrangian multiplier u, we have the variational problem min CD=
{c[Hx + H, + H;‘Mf,x
+ p +
+H;‘MZ,,J+
u[M.,xx +M,w
c(H, + H, + H;‘M:,x + H ;‘MZ,.,)]} dx dy,
(7)
where M& has the meaning (MX,X)2.Noting that HJ,
and
= M,
the Euler-Lagrange
Hyzy = M,,
(8)
equations with respect to variation of M,, My, H,, and H, furnish
- 2u,,t,,, + c-l&X = 0, U)ZX,XX
-2(1+
-2U + u )G.,, - 2u,,z,,, + c-lu,yy = 0,
(9)
and H, =
X2Mf,,(l+u)dX [I XI
H,=
‘=M;,,(l
[f Yl
/I”
XI
+ u) dy /I,:=
(1
+ u) dX]li2,
(1
+ u)dy]“2.
(10)
When s 4 p, the term 1 + u in (10) is replaced by unity; hence (by (8))
“‘Mf, f x,
H:
dx/L, =
x2z:., dx/L, = 1.0, I *I
(11)
G.I.N. Rozuany, H. Nakamura, B.T. Kuhnell, Optimal archgrids: allowance for selfweight
291
which is the unit mean square slope condition in (1). Similarly, for s Q p (9) reduces to 2r,,JX = c-lU,llx,
(12)
2&Y .= c-‘r&Y,
furnishing the equal elevation condition in view of the boundary condition z, = z, = 0 and the transversality condition u = 0 on Bd D [6, p. 221. When the selfweight cannot be neglected, we may put = c-lu,, + K1, 20 + U)ZX,X
20 + r&J
= c-‘u,, + Kzl
(13)
where K1 and Kz are constants. Then differentiation of (13) with respect to x and y, respectively, furnishes (9). Boundary and transversality conditions give K, = KZ = 0. Eq. (13) can also be derived on the basis of the Pruger-Shield opfimulitycriteria [6], [7] and its extension for inclusion of selfweight [8]. The specific cost function for an “equivalent x-beam” [l] is $ = c(H, + H;‘Sz), where S, is the shear force. In the Prager-Shield theory the shear strain u,, is given by the gradient of 4 with respect to SX: U ,x =
2cH;‘Sx = 2cz,,x.
(14)
The effect of selfweight can be taken into consideration by the factor 1+ u. Then (14) furnishes (13). Integration of (13) yields
simply [8] by multiplying such strain
2, = 2, = 2 = c-‘2-l ln(1 + u) + K, where boundary
and transversality
(15)
conditions give K = 0. Hence
u =eZrc-l.
(16)
For small c values the first two terms in the Taylor series for u furnishes u = 2~2, which agrees with earlier results [l]. The relations (10) and (16) reduce to
H,
=
M:,x e2rcd~/~2e2rc
dx]1’*,
and a similar equation gives H,. The equilibrium K{z,,,
+ c[l + (z,x)*]) + H&y
(17) and zero moment condition used earlier is
+ c[I + (z,,)*]> = -P.
(18)
Then an iterative method based on (17) and (18), similar to the one used in the previous study [l], furnishes readily the optimal solution. Each cycle of the iteration consists of two steps. First, finite difference equations based on (18) furnish the z-values at gridpoints on the basis of H, and H, values from the prior cycle. Then the equilibrium and zero bending conditions are temporarily satisfied but the horizontal
292
G.I.N. Rozvany,
H. Nakamura,
B.T. Kuhnell,
Optimal archgrids:
allowance for selfweight
reactions H, and H, violate the optimality condition (17). Second, the new values of H, and H, are calculated by numerical integration from (17). Naturally, (15) implies that the equal elevation criterion (z, = .q,) remains an optimal@ condition for archgrids with selfweight. The main difference between archgrids with purely external loads and those with selfweight is that the unit mean square slope condition changes to [cf. (17)]
zf, e2rc d*>/(l’
e2” dx) = 1.0.
(19)
When c “0, then (19) reduces to (1).
4. Method B. Discrete formulation using differential calculus When the archgrid consists of a small number of arches, then the variational formulation based on a continuum model introduces significant inaccuracies. For such structures, it is advantageous to base the mathematical formulation on a finite number of members and then the number of unknown design parameters will also be finite. We consider a system of arches whose spacing is A in both x and y directions. At a typical nodal point (i, j) (fig. 2a) the vertical external point load is denoted P(i, j) and the selfweight S(i, j). Although the latter is distributed along the members in the real structure, it will be replaced by a point load at nodal points which equals the weight of the four adjacent half
0 i-l,j
Y
i, j-l
i,j
0
L
i,j+l 0
x
---I 0
i+l,j
(a)
9
i-1,j
I i, j-l _--
i,j
+
i,j-1 0
--I
i,j+l 0
9
i+l,j
(b)
Fig. 2. Discrete formulation
(Cl
j
of the archgrid problem.
G.Z.N. Rozvany, H. Nakam~r~, B.T. ~~h~ell, journal
members
archgri~s: alfowunce for seifwe~gh~
293
(thick lines in fig. 2a). The discretized version of the problem in (7) then becomes
(20) The expression behind the summation signs in (20) refers to a typical internal nodal point. Appropriate adjustments must be made for special cases. For example, if the node is adjacent to a boundary in the x-direction (fig. 2b) then the weight of the four adjacent half members would not include the weight of the half member adjacent to the boundary (broken line). The term [(Mfi - M;_l)/A]*, therefore, must be multiplied by 2.0. Naturally, MIT,-1 is zero in this case, since (i, i - 1) is a boundary node. More extensive modifications are necessary when the considered nodal point is on an axis of (symmetry (fig. 2~). Since only the domain on the left side of the axis is considered, the weight of the member (i, j)- (i, j + 1) must be deleted and the weight of the two members along (i - 1, i)- (i + l,i) must be multiplied by l/2. Moreover, the moment value Mjjil is to be deleted from the second difference term, and it is to be replaced by Mi+j-1. Even more complicated modifications are necessary at the intersections of axes of symmetry. One of the advantages of the discrete formulation is that it can handle such corrections of the selfweight along boundary nodes relatively easily. Taking stationarity conditions with respect to H; and M;, we have the following optimality criteria:
I
2 (l+
1’
&.j)
(21)
j=l
(2 + Ui,j)(2Z; +uiJ+*(zf>
- Z&l -Z;+*)
- Z;+l)
+ Uij-*(Ztj
f C-'(Z&+1
- ZrJ-*)
+ Z&j-l
-
2Uij) = 0.
(22)
Similar expressions [denoted by (21a) and (22a)] can be obtained from &@cW{ = 0 and ~~/~M~~ = 0. The conditions (21) and (22) may also be derived directly from (9) and (10) by taking finite differences and cancelling out r&+I - zj) and Uj(zj-1 - zj) in the latter equation. In order to verify the equal elevation condition (z, = z,) independently by method B, the above equations assume different values for z, and z,. Moreover, a separate optimization procedure with an explicit equal elevation constraint was carried out subsequently. This was done by adding
I
(23)
G.I.N. Rozvany, H. Nakamura,
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B.T. Kuhnell, Optimal archgrids: allowance for selfweight
to (20) and this resulted in the extra term (+w,,JWfJcA) term
in the numerator
of (21) and the extra
(+c-‘Hf-‘Awij)
(24)
in (22). The facts that (i) the original formulation (without equal elevation constraint) yielded practically identical values for z, and z, throughout the domain, and (ii) the equal elevation constraint (23) did not cause any significant change in the optimal cost verify the earlier conclusion that zX = z, in the optimal solution (see section 6). Computational procedure. The following major steps were used in the computational procedure: (a) Assume initially zero selfweight [S(i, j) = 0 for all i, j] and an equal load distribution P, = P, = P/2,
(25)
where Px(i, j)A = -MTj+l -Mtj_,+ P,(i,j)A
= -MT+lj -MT-Ii
2MTj, (26)
+ 2ML.
(W Calculate the optimal horizontal forces HI and Hr from (21) and (21a) assuming u(i, j) = 0 for all i and j. (c) Set up matrices to represent
=+A +H++
the equilibrium
equations:
{H+ 2+ ,
[
(z;j-$-9+
(z;+1.j-‘:97},
i =
1,2, . . . , m,
j = 1,2, . . . , n,
(27) and solve these equations for the elevation values 2; and t[j using the Hf and Hr values from step (b). In order to obtain a unique solution of (27) we assume Zfj = P;,jz$r where pi,j
=
ZfjlZt*
elevation values ZTj and Z$ from step (c), calculate the values of Ui,j from (22) and (22a) and denote them by UT,jand U[j, respectively. (d)
Using
the
(e) Check if IUfj
-
U~jl/U~j
s
El
for all (i, j),
(28)
where E~ is a specified tolerance value (in the examples in section 6 &1= 10T6was adopted).
G.Z.N. Rozvany, H. Nakamura,
B.T. Kuhnell, Optimal archgrids: allowance for selfweight
295
(f) If (28) is violated, then adopt UiJ =
(24;
+ U?j)12
(29)
and solve again (22) and (22a) for zt and zt using the Uij values from (29). (g) Repeat steps (b j(f) until (28) is satisfied, except that in step (b) the value of u is in general nonzero in the second and subsequent cycles. (h) When (28) is satisfied, calculate the H; and Hr values again from (21) and (21a) and determine at each node the selfweight which is furnished by the right-hand side of (27). (i) Denoting
the current number of main cycles by N, check if
(@(Iv)- @(Iv - 1)1/@(N) 5 E2.
(30)
In the program used e2 = 10e6 was adopted. (j) If (30) is violated, then the entire procedure involving steps (g j(i) is repeated. When (30) is satisfied, the procedure is terminated. When the equal elevation constraint (23) is enforced, the following modifications are made. In step (b) eqs. (21) and (21a) are modified to allow for the extra term containing Wi,j. In step (c), z l”jand 2:’ are replaced by zij. In step (d), (22) and (22a) with the addition of (24) are solved simultaneously to obtain the Lagrangian multipliers Uij and wij. Steps (e) and (f) are unnecessary. It will be seen that the original problem without the equal elevation constraint consisted of larger iterative cycles each of which contained a series of nested smaller iterative cycles. When the equal elevation constraint (23) is introduced, the inner cycles are automatically eliminated. Extrapolation method. As the value of (Y= CL was increased from zero to 1.0, the convergence became progressively slower. In order to accelerate the convergence, the initial values of H, z and S were determined by quadratic extrapolation from the optimal values for the last three (Yvalues. This has resulted in a considerable saving of computer time.
5. Method C. Weight minimization procedure involving non-linear programming Whereas the computer programs mentioned under method B generated automatically all the matrices representing optimality criteria and equilibrium equations, in the examples involving non-linear programming the derivation of objective functions was carried out manually. Although all symmetry conditions have been taken into consideration in this method, the procedure is only practical for a relatively small number of design parameters. This means that method C is most suitable for comparatively coarse archgrids. Since only uniformly loaded archgrids over a square domain were optimized by this method, the independent variables were (i) the horizontal
arch reactions HI in the x direction; and
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G.I.N. Rozvany, H. Nakamura,
B.T. Kuhnell, Optimal archgrids: allowance for selfweight
(ii) the proportion P$ of the total external load Pij carried by the x-arch at nodal points not contained in an axis of symmetry. Along axes of symmetry PTJ= P[j = Pj,/2 can be adopted, and for i = 1,2, . . . , m, Hf = ff; owing to symmetry. The objective function @ representing the total weight of the archgrid was computed by the following iterative procedure. In the first cycle, the loads on the x and y arches, respectively, are QTd = Pl”j and Of, = P - P$. For these loads and the assumed value of I-If, one can readily determine the selfweight of the arch for any node i,i by calculating the shear forces Sif_1j and SyJ+, for the “equivalent beam” under the above loads [l] WTj = CA[2H; + (H~)-“(SiZ_lj + SX+,)].
(31)
In all subsequent cycles Qtj = P,zi + W& and Q{’ = P -P& + W&, where Wfj and W$ are based on the prior cycle. The iterative procedure was terminated after the Nth cycle if
Iw;j(N)-
w;(N-
l)]/W$(N&
&
for i = 1,2, . . . , m,
i=1,2 ,..., m, (32)
where E = low6 was adopted. Eq. (31) refers to a typical internal nodal point and appropriate modifications were carried out along domain boundaries and axes of symmetry. When (32) was satisfied for a given set of HP and P& values, then the objective fun~ion was given by @=2x
ii
Wzj
(33)
owing to symmetry with respect to x and y. ~o~-~i~eu~ ~~og~~~~i~g u~go~f~~. The algorithm for the minim~ation of the objective function @ in (33) was based on the constrained Rosenbrock method as described in [9], but modified to be used interactively. The program finds the local extremum of any non-linear objective function of TVvariable subject to explicit and/or implicit regional constraints. The objective function (33) and the recurrence relations (31) were programmed as subroutines. The convergence limit for the objective function was 10-‘“. The technique employed is sometimes called the rotating co-ordinate search algorithm and has the following salient steps: (a) A feasible starting point which does not violate the constraints is selected by the user. (b) Initial step sizes are selected by the user. (c) A series of univariable searches are conducted (involving one variable at a time). After a successful step (decrease in @), the length is multiplied by a specified constant a (where a > 1) and after an unsuccessful step (increase in @) the length is multiplied by -b (where 0 < b < 1). (d) Step c is repeated for each variable until a success followed by failure has occurred or any constraint is violated.
(e) At the new point of the design space, the direction of steepest descent is determined making exploratory steps along each axis. (f) The co-ordinate descent.
by
axes are rotated so that one of them lies along the direction of steepest
(g) Steps c-f are repeated until the change in the objective function is smaller than the specified convergence limit or a boundary zone (in the vicinity of a constraint) is entered. In the latter case, the relevant variable is modified so that the considered point lies in between the best value in the boundary zone and the best value outside the boundary zone. The width of the boundary zone was made equal to 10e4 times the feasible range of the considered variable. As in all search methods, it is not certain that the surface is t&modal, nor that the value obtained is a global minimum. The program was, therefore, started from a number of initial points and in each case the objective function converged to the same value. However, such initial points for larger a-values were restricted to a relativety narrow segment of the design space, because for signi~cantly non-optimal H-values the recurrence relations resulted in divergence. The reason for this is that the total weight of the structure is rather sensitive with respect to such H-values when (Y is large [2, fig. 81 and for an inefficient design in terms of fixed horizontal reactions the structure is unable to support its own weight.
6. Numerical examples ~~~~~~~ 1. ~~e-~~y UT& ~~~~~~ wifh ~~~~~ ~er~c~~ totad. Since an anal~ti~l solution is already available for the above problem, this example was computed in order to verify the convergence and accuracy of methods A (variational formulation) and B (discrete formulation using di~erential calculus). The number of internal nodal points with point loads was m = 10, 20 and 30 in three different sets of solutions and the range of cu-values varied from zero to 1.0. The non-dimensional quantities $ = ~~~/~~‘~ and R = ~~~~ were evaluated, where p is the intensity of the uniformly dist~buted vertical load. Results by methods A and B have shown an excellent agreement: e.g. for m = 30, a! = 0.2, both methods gave 6 = 2.820014, whilst for 1y= 1.0 the two resuits were Qi = lL380519 and & = 11.380508. The optimal horizontal force values differed by a slightly greater amount (for a = 1.0, pn = 30, I? = 1.6927 by method A, and I? = 1.6909 by method B, a difference of 0.1%). This is due to the relative insensitivity of 6 with respect to I? around &min. The ratio of the otimal weights by nurn~ri~ methods (@*) and by analytical method (@) is given for various nt and ty value in fig. 3, This dia~am also indicates upper bound solutions obtained by the method described in section 2. The latter are reasonably accurate (error ~1%) for cyZG0.4. The slight negative errors for small ty values are due to discret~tion. Fig. 4 gives the ratio of optimal horizontal reactions by numerical @I*) and analytical (H) methods. We can see that the maximum errors using m = 30 are 0.26% for Q, and about 0.5% for H. The variation of the total weight 6 with the number of main iterative cycles is shown in fig. 5. Since in each main cycle the selfweight is kept constant and the inner subcycles only carry out optimization for a given load at the nodal points, the weight increases monotonically with the number of main cycles. The initial value of 6, computed by quadratic extrapolation,
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1.02!
1.040 i.02a
Approximate upper bound solutions -_---___-__ 1.030
2
-----I
Method
A
-
Method
B
opt
1.020
1.000 0.998 0.998 I
1
I
I
0.2
0.4
0.8
0.8
I ,
1.0 a
Fig. 3. Comparison of errors in optimal weight by various methods for one-way arch systems.
1
1
I
0.2
0.4
0.6
I
0.8
i
Q
1.0
Fig. 4. Comparison of errors in optimal horizontal reactions by various methods for one-way arch systems.
la =l.O]
11.36
Fig. 5. The variation a = 1.0 (method B).
/
5
of total weight with the number
11.3800
I 15
of main iterative cycles for a one-way arch system with
G.I.N. Rozvany, H. Nakamura,
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299
"A
Fig. 6. Plan view of two-way arch system with 9 point loads.
always gives a lower bound on the final value of 5. The number of main cycles for (Y=0.2,0.4, . . . ,l.O was respectively 8, 9, 10, 13 and 20, before the prescribed convergence was achieved. Example 2. Square archgrid with 9 point loads. An archgrid consisting of only three arches in either direction (fig. 6) was optimized by methods B and C. In the latter the recurrence relations (31) reduce to (after adopting single subscript notation for nodal points): a: + QlQZ+ C?:/2 2H.z, where Qi = yp” + PI”
(i =
1’
(34)
1, 2). Similarly
WY-=;[““+-$$I.
(35)
The same expressions can be used for nodal points 3 and 4 if the subscripts 1, 2 and A are replaced by 3,4 and B. The total weight of beam A, for example, then becomes a
A
2H
A
+Q:+X?,Qz+Q% HA
1’
(36)
Both method B (stationarity conditions based on differential calculus) and method C (non-linear programming) were applied to two types of problem. In one the corresponding elevations of x and y arches were permitted to be different, and in the other they were required to be the same (zX = 2,). Method C was used up to an a-value of 0.9 and method B to a = 1.0. The two methods have given identical optimal weight & for six to seven significant digits. Even for (Y= 0.9, the weight values for non-equal elevation were 46 = 64.49896 and 46 = 64.49895 respectively, by methods B and C, and for equal elevations they were 46 = 64.50438 by both methods. The extremely small (0.008%) but consistent difference between the solutions with and without the z, = z, constraint is due to discretization. For the continuum problem, as was shown earlier, the optimal solution automatically satisfies the equal elevation condition whilst the discrete approximation of the same problem produces a minute difference between z, and z,, when no explicit constraint is imposed. The actual shape of the optimal arches for various a-values is given in fig. 7, and the variation of the
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G.I.N. Rozvany, H. Nakamura.
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B.T. Kuhnell, Optimal archgrids: allowance for selfweight
a-=0.0 -// a =0.2
a =0.4
! -
7
0.25L
Arch
i/
A
\
z
F
1.25L ,-a
=0.6
.L--+---L-----
Fig. 7. Optimal arch shapes for archgrids with 9 point loads.
non-dimensional weight 4 = @ao/pL3y is shown in fig. 8a as a continuous line. For a comparison, the weight variation of a one-way arch system with (Yis given in fig. 8b. It can be seen from fig. 7 that the non-convexity of the outer arches disappears at higher (~-values. Example 3. Square archgrids with 49 and 225 point loads. Archgrids with seven arches in both directions were optimized by all three methods for (Y= 0,0.2, . . . ,0.8, and archgrids with fifteen arches in both directions by method A. Methods B and C again yielded almost perfectly identical results. The weight difference between non-equal and equal eleyations was again very small; e.g. for (Y= 0.8, 16@ = 164.8600 for non-equal elevations and 16@ = 164.8637 for equal elevations, a difference of 0.002%. This indicates that the effect of the z, = t, constraint on the weight decreases as the grid becomes denser and disappears when the spacing becomes infinitesimal. Method A yielded slightly higher cost values than the other two methods (1.4% for (Y= 0.6) because of the approximation of the discrete system using a continuum. The weight @ variation as a function of (Y is shown in broken line in fig. 8a. For small a-values, the weight for 49 point loads is higher than for 9 point loads. As explained in [l], this is due to the fact that the distributed load has been replaced by point loads which result in a
GIN.
Rozvany, H. Nakamura,
B.T. KuhneN, Optimal archgrids: allowance for selfweight
One - way arch
Two - way arch system
system
30-
25-
25-
9 Point loads zo-
20-
15-
15-
IO-
/
10-
49 Point loads
5-
01
0
I
0.2
I
I
I
0.4
0.6
0.6
a,
1.0
0
0
I
I
,
0.2
0.4
0.6
I
0.8
Fig. 8. Cost comparison between one-way and two-way arch systems.
0.8
0.6
16 x 16 Grid
0 0
2
4
6
6
Gridline
Fig. 9. Optimal arch shapes for grids with 49 point loads.
aI
1.0
301
302
G.I.N. Rozuany, H. Nakamura, B.T. Kuhnell, Optimal archgrids: allowance for se&weight
H/pL, 1.6-
1.4-
m--’
1.2-
c
1.0-
0.8-
0.8-
8
lpZG?l
i;-i
p= uniform load intensity
0
i
i
Fig. 10. Optimal horizontal
6
6 Gridline
forces for archgrids with 49 point loads.
neglect of a certain part of the loaded area along boundary strips. For finer grids, this effect progressively decreases. However, such finer systems seem to become more and more efficient with an increase in (Y and actually give a lower weight for higher a-values than the coarser grid. In figs. 9-11 we compare the shape of the arches, the distribution of the horizontal reactions H/pL and the sign of P, for (Y= 0.0 and a! = 0.8. The archgrid with 225 point loads was optimized only by method A (variational method). An isometric view of optimal archgrids for fy = 0.0 and (Y= 0.8 is shown in fig. 12.
PX 0
2
4
6
8
Fig. 11. Optimal signs of toads carried by x-arches for various values of (Y.
GIN.
Rozvany, H. Nakamura,
Fig. 12. Comparison
303
B.T. Kuhnell, Optimal archgrids: allowance for selfweight
of optimal archgrids with 225 point loads with and without allowance
for selfweight.
7. Conclusions (a) The shape-optimization of fully stressed archgrids under external load and selfweight was investigated. The centrelines of arches were restricted to two directions (x, y) at right angles. Initially, a difference in elevations (zX# z,) between the two arch systems was permitted. (b) As in the case of archgrids with external load only [l], one optimality condition the elevations of the x and y arches to be equal throughout (z, = 2,).
requires
(c) For external load only, another optimality condition is [l] that the mean square slope of all arches must be unity. For archgrids with selfweight, a similar closed form optimality condition has been derived for the weighted mean square slope of the arches [see (1911. (d) The optimal@ conditions under (b) and (c), together with the equilibrium condition, reduces the optimization problem to an iterative analysis of anisotropic membranes in which the horizontal forces are adjusted after each cycle, using an explicit optimality condition (19). (e) In addition
to the foregoing
variational
solution (termed
“method
A”), two discretized
304
G.I.N. Rorvany, H. Nakamura, B.T. Kuhnell, Optimal archgrids: allowance foq selfweight
methods were used. Method programming.
B employed
differential
calculus, whilst method
C nonlinear
(f) Whereas method C is most suitable for coarse archgrids, method A gives the best results for very dense grids. Method B is most efficient for a medium range. Excellent agreement was obtained by comparing results from different methods. (g) One-way arch systems are less efficient than two-way grids when only external load is considered (fig. 8). However, at high values of the selfweight factor a (i.e. at long spans) the difference becomes much smaller. This is because two-way systems are less efficient in transmitting selfweight than external load. (h) This can also be seen from the fact that the non-convexity decreases as a increases (figs. 7 and 9).
of the arches along the edges
(i) In general, the height/span ratio for optimal archgrid increases with a (figs. 7, 9 and 12). Moreover, the horizontal reactions are much less evenly distributed for higher a-values.
References [l] G.I.N. Rozvany and W. Prager, A new class of optimization problems: optimal archgrids, Comp. Meths. Appl. Mech. Eng. 19 (1979) 127-150. [2] R. Hill, G.I.N. Rozvany, C.M. Wang and K.H. Leong, Optimization spanning capacity and cost sensitivity of fully stressed arches, J. Struct. Mech. 7 (1979) 37wlO. [3] F. Otto (ed.), Grid shells (Inst. of Lightweight Structures, Univ. of Stuttgart, 1974). [4] W. Prager and G.I.N. Rozvany, Optimal Spherical Cupola of Uniform Strength, Ingenieur-Archiv. (scheduled for the Ziegler issue). [5] G.I.N. Rozvany, CM. Wang and M. Dow, Arch optimization using Prager-Shield criteria, J. Eng. Mech. Div. ASCE (submitted). [6] G.I.N. Rozvany, Optimal design of flexural systems (Pergamon Press, Oxford, 1976). [7] W. Prager and R.T. Shield, A general theory of optimal plastic design, J. Appl. Mech. ASME 34 (1967) 184-186. [8] G.I.N. Rozvany, Optimal plastic design: allowance for selfweight, J. Eng. Mech. Div. ASCE 103 (1977) 11651170. [9] J.L. Kuester and J.H. Mize, Optimization techniques with FORTRAN (McGraw-Hill, New York, 1973).
OPTIMAL
MECHANICS COMPANY
ARCHGRIDS:
G.I.N. ROZVANY, Faculty of Engineering,
AND ENGINEERING
ALLOWANCE
H. NAKAMURA Monash Received
University,
FOR SELFWEIGHT
and B.T. KUHNELL
Clayton,
26 September
24 (1980) 287-304
Victoria, Australia,
3168
1979
An earlier study by Prager and the first author considered the optimization of archgrids for a given external load. The current paper discusses the optimal design of archgrids for a combination of given external loads and selfweight. Three different methods are described and then illustrated with examples of one- and two-way arch systems. The results furnished by various methods show a good agreement and also converge to earlier obtained analytical solutions.
Introduction
Prager and the first author [l] investigated the optimal design of long span surface structures consisting of a dense system of intersecting arches. Whereas the above study discussed the effect of given external loads only, in the following we consider a combination of external loads and selfweight. Since the weight of the structure itself depends on the design adopted, the formulation of the current problem becomes much more complicated than optimal design for external loads only. Another earlier study [2] obtained closed form analytical optimal solutions for a combination of external loads, selfweight and the weight of roof sheeting of constant thickness but was restricted to one-way systems of parallel arches. Some of these analytical solutions will be used as a basis for comparison in the current study. Archgrids or grid shells (Gitterschalen in German) have been studied extensively and also used in the actual design of long span roofs by Frei Otto [3]. The problem of optimization for selfweight and roof sheeting was considered also, in the context of spherical domes, by Prager and the first author [4]. Some aspects of optimal arch design for combined bending and axial force as well as for alternate load conditions were discussed briefly in a recent note [5]. In this paper, first some existing results are reviewed and then three different methods for the current problem are described. Finally, examples of archgrid optimization are presented.
1. Review of existing results
The optimization of archgrids for external loads [l] was based on a number of simplifying assumptions. It was stipulated that all arches were to resist axial compression only under the single load system considered. Moreover, no allowance was made for elastic compatibility, nor for instability; the resulting solution thus corresponded to an optimal plastic design for a given vertical load system p. In addition, the design was restricted to arches whose horizontal 287
288
G.I.N. Rozvany, H. Nakamura,
B.T. Kuhnell, Optimal archgrids: allowance for selfweight
projection was parallel to rectangular co-ordinates (x, y) contained in a horizontal reference plane. With a view to facilitating the mathematical treatment, it was first assumed that at a point (x, y) of the reference plane, the elevation of the x-arch and y-arch is not necessarily the same, but the two arch systems are connected by vertical ties whose weight is disregarded. The total volume of the arches was to be minimized within a permissible stress constraint. It was then found [l] that the “optimal” archgrid satisfies the following three conditions: (a) Each arch centreline has the shape of an appropriately scaled moment diagram which is associated with the “equivalent” simply supported beam of the same span and loading as the considered arch. In other words, each arch must form a funicular curve for the share of load it is resisting and the loads on the x and y arch systems &p,) must add up to the total load [PXkY)+P&,Y)=P(x7Y)l. (b) Along any one arch, the mean value of the square of the slope must be unity. For an x-arch, for example, 2 dx/L, = 1.0,
(1)
where L, is the arch length, z, is the arch elevation, and (x1, x2) are the end co-ordinates of the arch. (c) The elevation of the x and y arches at any one point (x, y) of the system is the same,
&(X9y)= ‘&
Y).
(2)
This means that we have finished up with a single layer of arches in the optimal solution, although initially we permitted two separate layers for the x-arches and y-arches. The above three optimality criteria may be termed (a) zero bending, (b) unit mean square slope and (c) equal elevation conditions. Out of various combinations of loads for one-way arch systems [2], we review here briefly the results for a uniformly distributed vertical load p = const plus selfweight. The relative influence of the selfweight on the solution depends on the nondimensional parameter (r which is defined as a = yucl,
(3)
where y is the specific weight of the arch material, L is the half-span of the arch, and c() is the permissible stress. Introducing then the nondimensional parameters .V= x/L, j.j = y/L. /? = pL/H where H is the horizontal reaction at the arch supports, one finds [2] that the optimal arch shape is given by the relation al = ln{sec[cuZ(l+ p/a)“‘]},
(4)
P = {sin[2a(l+
(5)
where pl(~)“~]}(l + p/2a!)/(l + P/cx)~/~.
GIN.
Rozvany, H. Nakamura, B.T. Kuhnell, Optimal archgrids: allowance for selfweight
289
0 IN--
0.8
a
,
a
1.8
2.0
2.4
Fig. 1. Analytical
optimal
solutions
a =1.50
v
for one-way
arch systems
having various
non-dimensional
spans (Y = yL/ao.
Three optimal arch centrelines for various cu-values are given in fig. 1. The one for (Y= 0 reduces to a parabola which clearly satisfies the unit mean square- slope condition (1) for arches whose selfweight is neglected.
2. Upper bound solutions obtained by an iterative method Reasonably close upper bound solutions for relatively small a-values can be obtained by an iterative method. In each cycle of this procedure, the selfweight is calculated on the basis of the solution in the prior cycle. Then the selfweight is added to the external load and the structure is re-optimized on the basis of conditions (a) through (c) in section 1, as if such combined load was a given external load. For large a-values, this method is clearly inaccurate because the selfweight is a design-dependent variable and hence it cannot be treated as a prescribed load. Even the unit mean square slope condition (1) must give inefficient designs for (Y% 0, since in fig. 1 the mean square slope is clearly much greater than unity for (Y> 0.75. Some upper bound solutions using the foregoing method are given in the examples in section 6.
290
G.I.N. Rozvany, H. Nakamura, B.T. Kuhnell, Optimal archgrids: allowance for selfweight
3. Method A. Variational formulation The variational formulation gives reasonably good results when the archgrid consists of a large number of arches and hence it can be idealised as an “archgrid-like continuum” [l] in which the arch spacing is infinitesimal. As earlier [l], we consider here an archgrid-like continuum over a domain D of a horizontal plane referred to co-ordinates (x, y). A given single force system p(x, y) defined on D is to be transmitted to rigid supports along the boundary of D by a system of intersecting arches of infinitesimal spacing running in x and y directions. All arches are to be subjected to axial compression only so that all cross-sections are stressed uniformly to a stress value of go under the combined load consisting of the external forces p(x, y) and the selfweight s(x, y). It was established earlier [l, eq. 61 that the selfweight of the archgrid per unit horizontal area is s = c[Hx + H, +
H;‘(Mx.,)2 + H,‘(My,y)2],
(6)
in which c = y/(~~, H, and H, are horizontal reactions per unit width, M, and MY are the bending moments in the “equivalent beams” corresponding to x and y arches [l, fig. 11, and subscripts after commas denote partial derivatives with respect to the variables indicated by such subscripts. Incorporating the equilibrium condition by the use of the Lagrangian multiplier u, we have the variational problem min CD=
{c[Hx + H, + H;‘Mf,x
+ p +
+H;‘MZ,,J+
u[M.,xx +M,w
c(H, + H, + H;‘M:,x + H ;‘MZ,.,)]} dx dy,
(7)
where M& has the meaning (MX,X)2.Noting that HJ,
and
= M,
the Euler-Lagrange
Hyzy = M,,
(8)
equations with respect to variation of M,, My, H,, and H, furnish
- 2u,,t,,, + c-l&X = 0, U)ZX,XX
-2(1+
-2U + u )G.,, - 2u,,z,,, + c-lu,yy = 0,
(9)
and H, =
X2Mf,,(l+u)dX [I XI
H,=
‘=M;,,(l
[f Yl
/I”
XI
+ u) dy /I,:=
(1
+ u) dX]li2,
(1
+ u)dy]“2.
(10)
When s 4 p, the term 1 + u in (10) is replaced by unity; hence (by (8))
“‘Mf, f x,
H:
dx/L, =
x2z:., dx/L, = 1.0, I *I
(11)
G.I.N. Rozuany, H. Nakamura, B.T. Kuhnell, Optimal archgrids: allowance for selfweight
291
which is the unit mean square slope condition in (1). Similarly, for s Q p (9) reduces to 2r,,JX = c-lU,llx,
(12)
2&Y .= c-‘r&Y,
furnishing the equal elevation condition in view of the boundary condition z, = z, = 0 and the transversality condition u = 0 on Bd D [6, p. 221. When the selfweight cannot be neglected, we may put = c-lu,, + K1, 20 + U)ZX,X
20 + r&J
= c-‘u,, + Kzl
(13)
where K1 and Kz are constants. Then differentiation of (13) with respect to x and y, respectively, furnishes (9). Boundary and transversality conditions give K, = KZ = 0. Eq. (13) can also be derived on the basis of the Pruger-Shield opfimulitycriteria [6], [7] and its extension for inclusion of selfweight [8]. The specific cost function for an “equivalent x-beam” [l] is $ = c(H, + H;‘Sz), where S, is the shear force. In the Prager-Shield theory the shear strain u,, is given by the gradient of 4 with respect to SX: U ,x =
2cH;‘Sx = 2cz,,x.
(14)
The effect of selfweight can be taken into consideration by the factor 1+ u. Then (14) furnishes (13). Integration of (13) yields
simply [8] by multiplying such strain
2, = 2, = 2 = c-‘2-l ln(1 + u) + K, where boundary
and transversality
(15)
conditions give K = 0. Hence
u =eZrc-l.
(16)
For small c values the first two terms in the Taylor series for u furnishes u = 2~2, which agrees with earlier results [l]. The relations (10) and (16) reduce to
H,
=
M:,x e2rcd~/~2e2rc
dx]1’*,
and a similar equation gives H,. The equilibrium K{z,,,
+ c[l + (z,x)*]) + H&y
(17) and zero moment condition used earlier is
+ c[I + (z,,)*]> = -P.
(18)
Then an iterative method based on (17) and (18), similar to the one used in the previous study [l], furnishes readily the optimal solution. Each cycle of the iteration consists of two steps. First, finite difference equations based on (18) furnish the z-values at gridpoints on the basis of H, and H, values from the prior cycle. Then the equilibrium and zero bending conditions are temporarily satisfied but the horizontal
292
G.I.N. Rozvany,
H. Nakamura,
B.T. Kuhnell,
Optimal archgrids:
allowance for selfweight
reactions H, and H, violate the optimality condition (17). Second, the new values of H, and H, are calculated by numerical integration from (17). Naturally, (15) implies that the equal elevation criterion (z, = .q,) remains an optimal@ condition for archgrids with selfweight. The main difference between archgrids with purely external loads and those with selfweight is that the unit mean square slope condition changes to [cf. (17)]
zf, e2rc d*>/(l’
e2” dx) = 1.0.
(19)
When c “0, then (19) reduces to (1).
4. Method B. Discrete formulation using differential calculus When the archgrid consists of a small number of arches, then the variational formulation based on a continuum model introduces significant inaccuracies. For such structures, it is advantageous to base the mathematical formulation on a finite number of members and then the number of unknown design parameters will also be finite. We consider a system of arches whose spacing is A in both x and y directions. At a typical nodal point (i, j) (fig. 2a) the vertical external point load is denoted P(i, j) and the selfweight S(i, j). Although the latter is distributed along the members in the real structure, it will be replaced by a point load at nodal points which equals the weight of the four adjacent half
0 i-l,j
Y
i, j-l
i,j
0
L
i,j+l 0
x
---I 0
i+l,j
(a)
9
i-1,j
I i, j-l _--
i,j
+
i,j-1 0
--I
i,j+l 0
9
i+l,j
(b)
Fig. 2. Discrete formulation
(Cl
j
of the archgrid problem.
G.Z.N. Rozvany, H. Nakam~r~, B.T. ~~h~ell, journal
members
archgri~s: alfowunce for seifwe~gh~
293
(thick lines in fig. 2a). The discretized version of the problem in (7) then becomes
(20) The expression behind the summation signs in (20) refers to a typical internal nodal point. Appropriate adjustments must be made for special cases. For example, if the node is adjacent to a boundary in the x-direction (fig. 2b) then the weight of the four adjacent half members would not include the weight of the half member adjacent to the boundary (broken line). The term [(Mfi - M;_l)/A]*, therefore, must be multiplied by 2.0. Naturally, MIT,-1 is zero in this case, since (i, i - 1) is a boundary node. More extensive modifications are necessary when the considered nodal point is on an axis of (symmetry (fig. 2~). Since only the domain on the left side of the axis is considered, the weight of the member (i, j)- (i, j + 1) must be deleted and the weight of the two members along (i - 1, i)- (i + l,i) must be multiplied by l/2. Moreover, the moment value Mjjil is to be deleted from the second difference term, and it is to be replaced by Mi+j-1. Even more complicated modifications are necessary at the intersections of axes of symmetry. One of the advantages of the discrete formulation is that it can handle such corrections of the selfweight along boundary nodes relatively easily. Taking stationarity conditions with respect to H; and M;, we have the following optimality criteria:
I
2 (l+
1’
&.j)
(21)
j=l
(2 + Ui,j)(2Z; +uiJ+*(zf>
- Z&l -Z;+*)
- Z;+l)
+ Uij-*(Ztj
f C-'(Z&+1
- ZrJ-*)
+ Z&j-l
-
2Uij) = 0.
(22)
Similar expressions [denoted by (21a) and (22a)] can be obtained from &@cW{ = 0 and ~~/~M~~ = 0. The conditions (21) and (22) may also be derived directly from (9) and (10) by taking finite differences and cancelling out r&+I - zj) and Uj(zj-1 - zj) in the latter equation. In order to verify the equal elevation condition (z, = z,) independently by method B, the above equations assume different values for z, and z,. Moreover, a separate optimization procedure with an explicit equal elevation constraint was carried out subsequently. This was done by adding
I
(23)
G.I.N. Rozvany, H. Nakamura,
294
B.T. Kuhnell, Optimal archgrids: allowance for selfweight
to (20) and this resulted in the extra term (+w,,JWfJcA) term
in the numerator
of (21) and the extra
(+c-‘Hf-‘Awij)
(24)
in (22). The facts that (i) the original formulation (without equal elevation constraint) yielded practically identical values for z, and z, throughout the domain, and (ii) the equal elevation constraint (23) did not cause any significant change in the optimal cost verify the earlier conclusion that zX = z, in the optimal solution (see section 6). Computational procedure. The following major steps were used in the computational procedure: (a) Assume initially zero selfweight [S(i, j) = 0 for all i, j] and an equal load distribution P, = P, = P/2,
(25)
where Px(i, j)A = -MTj+l -Mtj_,+ P,(i,j)A
= -MT+lj -MT-Ii
2MTj, (26)
+ 2ML.
(W Calculate the optimal horizontal forces HI and Hr from (21) and (21a) assuming u(i, j) = 0 for all i and j. (c) Set up matrices to represent
=+A +H++
the equilibrium
equations:
{H+ 2+ ,
[
(z;j-$-9+
(z;+1.j-‘:97},
i =
1,2, . . . , m,
j = 1,2, . . . , n,
(27) and solve these equations for the elevation values 2; and t[j using the Hf and Hr values from step (b). In order to obtain a unique solution of (27) we assume Zfj = P;,jz$r where pi,j
=
ZfjlZt*
elevation values ZTj and Z$ from step (c), calculate the values of Ui,j from (22) and (22a) and denote them by UT,jand U[j, respectively. (d)
Using
the
(e) Check if IUfj
-
U~jl/U~j
s
El
for all (i, j),
(28)
where E~ is a specified tolerance value (in the examples in section 6 &1= 10T6was adopted).
G.Z.N. Rozvany, H. Nakamura,
B.T. Kuhnell, Optimal archgrids: allowance for selfweight
295
(f) If (28) is violated, then adopt UiJ =
(24;
+ U?j)12
(29)
and solve again (22) and (22a) for zt and zt using the Uij values from (29). (g) Repeat steps (b j(f) until (28) is satisfied, except that in step (b) the value of u is in general nonzero in the second and subsequent cycles. (h) When (28) is satisfied, calculate the H; and Hr values again from (21) and (21a) and determine at each node the selfweight which is furnished by the right-hand side of (27). (i) Denoting
the current number of main cycles by N, check if
(@(Iv)- @(Iv - 1)1/@(N) 5 E2.
(30)
In the program used e2 = 10e6 was adopted. (j) If (30) is violated, then the entire procedure involving steps (g j(i) is repeated. When (30) is satisfied, the procedure is terminated. When the equal elevation constraint (23) is enforced, the following modifications are made. In step (b) eqs. (21) and (21a) are modified to allow for the extra term containing Wi,j. In step (c), z l”jand 2:’ are replaced by zij. In step (d), (22) and (22a) with the addition of (24) are solved simultaneously to obtain the Lagrangian multipliers Uij and wij. Steps (e) and (f) are unnecessary. It will be seen that the original problem without the equal elevation constraint consisted of larger iterative cycles each of which contained a series of nested smaller iterative cycles. When the equal elevation constraint (23) is introduced, the inner cycles are automatically eliminated. Extrapolation method. As the value of (Y= CL was increased from zero to 1.0, the convergence became progressively slower. In order to accelerate the convergence, the initial values of H, z and S were determined by quadratic extrapolation from the optimal values for the last three (Yvalues. This has resulted in a considerable saving of computer time.
5. Method C. Weight minimization procedure involving non-linear programming Whereas the computer programs mentioned under method B generated automatically all the matrices representing optimality criteria and equilibrium equations, in the examples involving non-linear programming the derivation of objective functions was carried out manually. Although all symmetry conditions have been taken into consideration in this method, the procedure is only practical for a relatively small number of design parameters. This means that method C is most suitable for comparatively coarse archgrids. Since only uniformly loaded archgrids over a square domain were optimized by this method, the independent variables were (i) the horizontal
arch reactions HI in the x direction; and
2%
G.I.N. Rozvany, H. Nakamura,
B.T. Kuhnell, Optimal archgrids: allowance for selfweight
(ii) the proportion P$ of the total external load Pij carried by the x-arch at nodal points not contained in an axis of symmetry. Along axes of symmetry PTJ= P[j = Pj,/2 can be adopted, and for i = 1,2, . . . , m, Hf = ff; owing to symmetry. The objective function @ representing the total weight of the archgrid was computed by the following iterative procedure. In the first cycle, the loads on the x and y arches, respectively, are QTd = Pl”j and Of, = P - P$. For these loads and the assumed value of I-If, one can readily determine the selfweight of the arch for any node i,i by calculating the shear forces Sif_1j and SyJ+, for the “equivalent beam” under the above loads [l] WTj = CA[2H; + (H~)-“(SiZ_lj + SX+,)].
(31)
In all subsequent cycles Qtj = P,zi + W& and Q{’ = P -P& + W&, where Wfj and W$ are based on the prior cycle. The iterative procedure was terminated after the Nth cycle if
Iw;j(N)-
w;(N-
l)]/W$(N&
&
for i = 1,2, . . . , m,
i=1,2 ,..., m, (32)
where E = low6 was adopted. Eq. (31) refers to a typical internal nodal point and appropriate modifications were carried out along domain boundaries and axes of symmetry. When (32) was satisfied for a given set of HP and P& values, then the objective fun~ion was given by @=2x
ii
Wzj
(33)
owing to symmetry with respect to x and y. ~o~-~i~eu~ ~~og~~~~i~g u~go~f~~. The algorithm for the minim~ation of the objective function @ in (33) was based on the constrained Rosenbrock method as described in [9], but modified to be used interactively. The program finds the local extremum of any non-linear objective function of TVvariable subject to explicit and/or implicit regional constraints. The objective function (33) and the recurrence relations (31) were programmed as subroutines. The convergence limit for the objective function was 10-‘“. The technique employed is sometimes called the rotating co-ordinate search algorithm and has the following salient steps: (a) A feasible starting point which does not violate the constraints is selected by the user. (b) Initial step sizes are selected by the user. (c) A series of univariable searches are conducted (involving one variable at a time). After a successful step (decrease in @), the length is multiplied by a specified constant a (where a > 1) and after an unsuccessful step (increase in @) the length is multiplied by -b (where 0 < b < 1). (d) Step c is repeated for each variable until a success followed by failure has occurred or any constraint is violated.
(e) At the new point of the design space, the direction of steepest descent is determined making exploratory steps along each axis. (f) The co-ordinate descent.
by
axes are rotated so that one of them lies along the direction of steepest
(g) Steps c-f are repeated until the change in the objective function is smaller than the specified convergence limit or a boundary zone (in the vicinity of a constraint) is entered. In the latter case, the relevant variable is modified so that the considered point lies in between the best value in the boundary zone and the best value outside the boundary zone. The width of the boundary zone was made equal to 10e4 times the feasible range of the considered variable. As in all search methods, it is not certain that the surface is t&modal, nor that the value obtained is a global minimum. The program was, therefore, started from a number of initial points and in each case the objective function converged to the same value. However, such initial points for larger a-values were restricted to a relativety narrow segment of the design space, because for signi~cantly non-optimal H-values the recurrence relations resulted in divergence. The reason for this is that the total weight of the structure is rather sensitive with respect to such H-values when (Y is large [2, fig. 81 and for an inefficient design in terms of fixed horizontal reactions the structure is unable to support its own weight.
6. Numerical examples ~~~~~~~ 1. ~~e-~~y UT& ~~~~~~ wifh ~~~~~ ~er~c~~ totad. Since an anal~ti~l solution is already available for the above problem, this example was computed in order to verify the convergence and accuracy of methods A (variational formulation) and B (discrete formulation using di~erential calculus). The number of internal nodal points with point loads was m = 10, 20 and 30 in three different sets of solutions and the range of cu-values varied from zero to 1.0. The non-dimensional quantities $ = ~~~/~~‘~ and R = ~~~~ were evaluated, where p is the intensity of the uniformly dist~buted vertical load. Results by methods A and B have shown an excellent agreement: e.g. for m = 30, a! = 0.2, both methods gave 6 = 2.820014, whilst for 1y= 1.0 the two resuits were Qi = lL380519 and & = 11.380508. The optimal horizontal force values differed by a slightly greater amount (for a = 1.0, pn = 30, I? = 1.6927 by method A, and I? = 1.6909 by method B, a difference of 0.1%). This is due to the relative insensitivity of 6 with respect to I? around &min. The ratio of the otimal weights by nurn~ri~ methods (@*) and by analytical method (@) is given for various nt and ty value in fig. 3, This dia~am also indicates upper bound solutions obtained by the method described in section 2. The latter are reasonably accurate (error ~1%) for cyZG0.4. The slight negative errors for small ty values are due to discret~tion. Fig. 4 gives the ratio of optimal horizontal reactions by numerical @I*) and analytical (H) methods. We can see that the maximum errors using m = 30 are 0.26% for Q, and about 0.5% for H. The variation of the total weight 6 with the number of main iterative cycles is shown in fig. 5. Since in each main cycle the selfweight is kept constant and the inner subcycles only carry out optimization for a given load at the nodal points, the weight increases monotonically with the number of main cycles. The initial value of 6, computed by quadratic extrapolation,
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B.T. Kuhnell, Optimal archgrids : allowance for selfweight
1.02!
1.040 i.02a
Approximate upper bound solutions -_---___-__ 1.030
2
-----I
Method
A
-
Method
B
opt
1.020
1.000 0.998 0.998 I
1
I
I
0.2
0.4
0.8
0.8
I ,
1.0 a
Fig. 3. Comparison of errors in optimal weight by various methods for one-way arch systems.
1
1
I
0.2
0.4
0.6
I
0.8
i
Q
1.0
Fig. 4. Comparison of errors in optimal horizontal reactions by various methods for one-way arch systems.
la =l.O]
11.36
Fig. 5. The variation a = 1.0 (method B).
/
5
of total weight with the number
11.3800
I 15
of main iterative cycles for a one-way arch system with
G.I.N. Rozvany, H. Nakamura,
B.T. Kuhnell, Optimal archgrids: allowance for selfweight
299
"A
Fig. 6. Plan view of two-way arch system with 9 point loads.
always gives a lower bound on the final value of 5. The number of main cycles for (Y=0.2,0.4, . . . ,l.O was respectively 8, 9, 10, 13 and 20, before the prescribed convergence was achieved. Example 2. Square archgrid with 9 point loads. An archgrid consisting of only three arches in either direction (fig. 6) was optimized by methods B and C. In the latter the recurrence relations (31) reduce to (after adopting single subscript notation for nodal points): a: + QlQZ+ C?:/2 2H.z, where Qi = yp” + PI”
(i =
1’
(34)
1, 2). Similarly
WY-=;[““+-$$I.
(35)
The same expressions can be used for nodal points 3 and 4 if the subscripts 1, 2 and A are replaced by 3,4 and B. The total weight of beam A, for example, then becomes a
A
2H
A
+Q:+X?,Qz+Q% HA
1’
(36)
Both method B (stationarity conditions based on differential calculus) and method C (non-linear programming) were applied to two types of problem. In one the corresponding elevations of x and y arches were permitted to be different, and in the other they were required to be the same (zX = 2,). Method C was used up to an a-value of 0.9 and method B to a = 1.0. The two methods have given identical optimal weight & for six to seven significant digits. Even for (Y= 0.9, the weight values for non-equal elevation were 46 = 64.49896 and 46 = 64.49895 respectively, by methods B and C, and for equal elevations they were 46 = 64.50438 by both methods. The extremely small (0.008%) but consistent difference between the solutions with and without the z, = z, constraint is due to discretization. For the continuum problem, as was shown earlier, the optimal solution automatically satisfies the equal elevation condition whilst the discrete approximation of the same problem produces a minute difference between z, and z,, when no explicit constraint is imposed. The actual shape of the optimal arches for various a-values is given in fig. 7, and the variation of the
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G.I.N. Rozvany, H. Nakamura.
_._-
B.T. Kuhnell, Optimal archgrids: allowance for selfweight
a-=0.0 -// a =0.2
a =0.4
! -
7
0.25L
Arch
i/
A
\
z
F
1.25L ,-a
=0.6
.L--+---L-----
Fig. 7. Optimal arch shapes for archgrids with 9 point loads.
non-dimensional weight 4 = @ao/pL3y is shown in fig. 8a as a continuous line. For a comparison, the weight variation of a one-way arch system with (Yis given in fig. 8b. It can be seen from fig. 7 that the non-convexity of the outer arches disappears at higher (~-values. Example 3. Square archgrids with 49 and 225 point loads. Archgrids with seven arches in both directions were optimized by all three methods for (Y= 0,0.2, . . . ,0.8, and archgrids with fifteen arches in both directions by method A. Methods B and C again yielded almost perfectly identical results. The weight difference between non-equal and equal eleyations was again very small; e.g. for (Y= 0.8, 16@ = 164.8600 for non-equal elevations and 16@ = 164.8637 for equal elevations, a difference of 0.002%. This indicates that the effect of the z, = t, constraint on the weight decreases as the grid becomes denser and disappears when the spacing becomes infinitesimal. Method A yielded slightly higher cost values than the other two methods (1.4% for (Y= 0.6) because of the approximation of the discrete system using a continuum. The weight @ variation as a function of (Y is shown in broken line in fig. 8a. For small a-values, the weight for 49 point loads is higher than for 9 point loads. As explained in [l], this is due to the fact that the distributed load has been replaced by point loads which result in a
GIN.
Rozvany, H. Nakamura,
B.T. KuhneN, Optimal archgrids: allowance for selfweight
One - way arch
Two - way arch system
system
30-
25-
25-
9 Point loads zo-
20-
15-
15-
IO-
/
10-
49 Point loads
5-
01
0
I
0.2
I
I
I
0.4
0.6
0.6
a,
1.0
0
0
I
I
,
0.2
0.4
0.6
I
0.8
Fig. 8. Cost comparison between one-way and two-way arch systems.
0.8
0.6
16 x 16 Grid
0 0
2
4
6
6
Gridline
Fig. 9. Optimal arch shapes for grids with 49 point loads.
aI
1.0
301
302
G.I.N. Rozuany, H. Nakamura, B.T. Kuhnell, Optimal archgrids: allowance for se&weight
H/pL, 1.6-
1.4-
m--’
1.2-
c
1.0-
0.8-
0.8-
8
lpZG?l
i;-i
p= uniform load intensity
0
i
i
Fig. 10. Optimal horizontal
6
6 Gridline
forces for archgrids with 49 point loads.
neglect of a certain part of the loaded area along boundary strips. For finer grids, this effect progressively decreases. However, such finer systems seem to become more and more efficient with an increase in (Y and actually give a lower weight for higher a-values than the coarser grid. In figs. 9-11 we compare the shape of the arches, the distribution of the horizontal reactions H/pL and the sign of P, for (Y= 0.0 and a! = 0.8. The archgrid with 225 point loads was optimized only by method A (variational method). An isometric view of optimal archgrids for fy = 0.0 and (Y= 0.8 is shown in fig. 12.
PX 0
2
4
6
8
Fig. 11. Optimal signs of toads carried by x-arches for various values of (Y.
GIN.
Rozvany, H. Nakamura,
Fig. 12. Comparison
303
B.T. Kuhnell, Optimal archgrids: allowance for selfweight
of optimal archgrids with 225 point loads with and without allowance
for selfweight.
7. Conclusions (a) The shape-optimization of fully stressed archgrids under external load and selfweight was investigated. The centrelines of arches were restricted to two directions (x, y) at right angles. Initially, a difference in elevations (zX# z,) between the two arch systems was permitted. (b) As in the case of archgrids with external load only [l], one optimality condition the elevations of the x and y arches to be equal throughout (z, = 2,).
requires
(c) For external load only, another optimality condition is [l] that the mean square slope of all arches must be unity. For archgrids with selfweight, a similar closed form optimality condition has been derived for the weighted mean square slope of the arches [see (1911. (d) The optimal@ conditions under (b) and (c), together with the equilibrium condition, reduces the optimization problem to an iterative analysis of anisotropic membranes in which the horizontal forces are adjusted after each cycle, using an explicit optimality condition (19). (e) In addition
to the foregoing
variational
solution (termed
“method
A”), two discretized
304
G.I.N. Rorvany, H. Nakamura, B.T. Kuhnell, Optimal archgrids: allowance foq selfweight
methods were used. Method programming.
B employed
differential
calculus, whilst method
C nonlinear
(f) Whereas method C is most suitable for coarse archgrids, method A gives the best results for very dense grids. Method B is most efficient for a medium range. Excellent agreement was obtained by comparing results from different methods. (g) One-way arch systems are less efficient than two-way grids when only external load is considered (fig. 8). However, at high values of the selfweight factor a (i.e. at long spans) the difference becomes much smaller. This is because two-way systems are less efficient in transmitting selfweight than external load. (h) This can also be seen from the fact that the non-convexity decreases as a increases (figs. 7 and 9).
of the arches along the edges
(i) In general, the height/span ratio for optimal archgrid increases with a (figs. 7, 9 and 12). Moreover, the horizontal reactions are much less evenly distributed for higher a-values.
References [l] G.I.N. Rozvany and W. Prager, A new class of optimization problems: optimal archgrids, Comp. Meths. Appl. Mech. Eng. 19 (1979) 127-150. [2] R. Hill, G.I.N. Rozvany, C.M. Wang and K.H. Leong, Optimization spanning capacity and cost sensitivity of fully stressed arches, J. Struct. Mech. 7 (1979) 37wlO. [3] F. Otto (ed.), Grid shells (Inst. of Lightweight Structures, Univ. of Stuttgart, 1974). [4] W. Prager and G.I.N. Rozvany, Optimal Spherical Cupola of Uniform Strength, Ingenieur-Archiv. (scheduled for the Ziegler issue). [5] G.I.N. Rozvany, CM. Wang and M. Dow, Arch optimization using Prager-Shield criteria, J. Eng. Mech. Div. ASCE (submitted). [6] G.I.N. Rozvany, Optimal design of flexural systems (Pergamon Press, Oxford, 1976). [7] W. Prager and R.T. Shield, A general theory of optimal plastic design, J. Appl. Mech. ASME 34 (1967) 184-186. [8] G.I.N. Rozvany, Optimal plastic design: allowance for selfweight, J. Eng. Mech. Div. ASCE 103 (1977) 11651170. [9] J.L. Kuester and J.H. Mize, Optimization techniques with FORTRAN (McGraw-Hill, New York, 1973).