CoA R e p o r t A e r o 175
THE COLLEGE OF A E R O N A U T I C S
C R A N F I E L D
OPTIMUM STRUCTURAL DESIGN AND LINEAR PROGRAMMING
by
Report A e r o 175
THE COLLEGE OF AERONAUTICS CRANFIELD
Optimum s t r u c t u r a l design and linear programming by
H. S.Y.Chan
CORRIGENDA
Page 4, paragraph 5, line 2 should read 'Ref. 10'. Page 5, paragraph 3, line 1 should read 'Ref. 1'.
Report Aero 175 September 1964
THE COLLEGE OF AERONAUTICS
CRANFIELD
Optimum structural design and linear programming
by
H. S.Y.Chan, B.A.
Summary
The criterion for pin-jointed frame structures to be of minimum weight is formulated. An approximate solution is shown to be obtainable by using the method of linear programming. A three-dimensional example demonstrating such an approach is given, and compared with the exact solution. Finally, optimum s t r u c -t u r e s for mul-tiple loading sys-tems a r e discussed.
CONTENTS
1 Page 1. Introduction 1
2. Optimum design of pin-jointed frames 1
3. Linear programming 2 4. An example of three-dimensional design 3
5. The Michell optimum solution 5 6. Multiple loading systems 6
References 7 Figures
1
-Introduction
The limits of economy of material in structures were first discussed by Michell ( R e f . l ) , who discussed a sufficient condition for frame structures to be of least weight and analysed the geometrical restrictions on such structures. It is not until recently that Cox (Ref. 2), Hopkins and Prager (Ref. 3) were aware of the importance of Michell's contribution. Developments in connection with elastio-plastic solids, especially on the problem of limit design, enabled Drucker and Shield (Refs.4&5) to generalize independently Michell's result to obtain optimum criteria for structures other than frames. Following the original work of Michell, Hemp (Ref. 6) showed that there is a complete analogy between two-dimensional optimum structures (including frames and plates) and the slip-line fields of plane plastic flow.
Use has been made of these results to obtain minimum weight structures in the forms of frames, beams, plates, shells, sandwich structures and filament-a r y s t r u c t u r e s . Most of the work wfilament-as cfilament-arried out filament-anfilament-alyticfilament-ally; the minimum weight being sought r e g a r d l e s s of the problems and costs of manufacture and con-struction. Extensions of the results to multiple loading systems were also attempted by Schmidt (Ref. 8) and Shield (Ref. 9).
An approximate method of designing Michell frames was given by Hemp (Ref. 7), which is completely general in its application. Advances along this direction seem very promising, especially when it is noticed that the correspond-ing computcorrespond-ing problem is almost identical with that of plastic limit analysis(Ref. 10). Dom, Gomory and Greenberg (Ref. 11) used computing techniques to treat the prob-lem of automatic structural design, but without reference to the basic theorems of minimum weight design.
This paper is intended to show the results of computing work based on Reference 7, in which illustrative examples have already been given. However, special emphasis will be placed here on three-dimensional design.
2. Optimum design of pin-jointed frames.
Consider any system of forces defined by a vertor field F in equilibrium, with their points of action contained in a region V of space, within which the s t r u c -ture which balances them is required to lie. Assume that the struc-tures H, which a r e capable of carrying the loading, a r e pin-jointed franaes, and that the yield s t r e s s of the homogeneous material used is the same in tension and_compression
(i f). The problem óf finding the minimum weight structure S in S therefore
becomes the one of determining the minimum volume V .
If T is the end load in any member of any S with cross-section area A, then the design criterion is
| T | < f A ^^j According to the theorem of virtual work,
Fu = XeTL S
2
-for any k i n e m a t i c a l l y a d m i s s i b l e v i r t u a l d i s p l a c e m e n t and s t r a i n field u,e ; w h e r e the s u m m a t i o n i s t a k e n o v e r a l l m e m b e r s L of S.
C o n s i d e r a s t r u c t u r e Sm in S which i s j u s t at the point of c o l l a p s e under the l o a d s F , so that the i n e q u a l i t y (1) i s s a t i s f i e d a s an equation. F r o m (2), one finds for a l l S,
2 e T L = 2 e T L
Sm S (3) If the v i r t u a l d e f o r m a t i o n i s chosen s o that
= [
"^^ along t e n s i o n m e m b e r s in Sm (4) "^ " c o m p r e s s i o n " " " Ul < e in V - Vm w h e r e e (>0) i s a c o n s t a n t , then (3) b e c o m e s e f V m = e f Z A L < 2 | e | • | T | L < efSAL = efVg Sm S S (5) o r Vni< VsT h i s shows that no s t r u c t u r e in S can have a v o l u m e s m a l l e r than
Vm-3 . L i n e a r P r o g r a m m i n g *
F r o m (2), (4) and (5) it follows that the p r o b l e m of finding the o p t i m u m d e s i g n , if it e x i s t s , i s t o d e t e r m i n e a v i r t u a l d i s p l a c e m e n t field u such that no change in length A L of any l i n e a r s e g m e n t of length L in V i s n u m e r i c a l l y g r e a t e r t h a n e L , i . e .
- e L < A L < e L , (6) and such that the work done by the e x t e r n a l f o r c e s i s a m a x i m u m , o r ,
ef Vm = Max. Fu . (7) It i s c l e a r that in g e n e r a l such a p r o b l e m cannot be solved by the c l a s s
-i c a l m e t h o d s of o p t -i m -i z a t -i o n and so a method of obta-in-ing a p p r o x -i m a t e solut-ion i s c o n s i d e r e d .
I m a g i n e the r e g i o n V to be c o v e r e d b y a grid of p o i n t s , which i n c l u d e s the points of a p p l i c a t i o n of F and the points w h e r e s t r u c t u r a l j o i n t s a r e t o be l o c a t e d . G e n e r a l l y , each point i s specified by t h r e e c o o r d i n a t e s and t h e r e a r e t h r e e c o m p o n e n t s of v i r t u a l d i s p l a c e m e n t at each point, u i s t h u s r e d u c e d to a column m a t r i x . P o s s i b l e s t r u c t u r a l m e n i b e r s a r e a s s u m e d to lie along the s e g -m e n t joining any p a i r of t h e s e points for which (6) -m u s t be s a t i s f i e d . F o r e x a -m p l e the m e m b e r AB = L in F i g . 1, with v i r t u a l d i s p l a c e m e n t s (u ,, u^ , uj )^(u^_ us , Ue ) •" T h e t h e o r y and c o m p u t a t i o n a l t e c h n i q u e s of the subject a r e d i s c u s s e d by, for e x a m p l e , G. Zoutendijk in "Methods of F e a s i b l e D i r e c t i o n s " , A m s t e r d a m 1960.
3
-at A,B respectively and direction cosine (a,a.j a,), will be subject to the following restriction:
-eL<ai (u, -U4 ) 4a2(u2- us) 4«j(ui - U6 ) < eL
It follows that, for the present formulation (6) and (7) can be reduced to the following form, writing v = u/e :
To find fVm = Max.Fv (8) subject to: -* < Av<* (9) where A is a coefficient matrix, and * is a column matrix. Once the solution is
obtained, the optimum structure will consist of members whose segments a r e fully strained, i . e . for which (9) is satisfied as an equation. If no structural lay-out is obtained, (9) will give a lower bound to the structural weight.
The problem of (8) and (9) can undoubtedly be solved by the method of linear programming, although the standard methods are not suitable. Consideration has been given to the problem of signed variables and bounded conditions in, for instance, Refs. 10 and 12. Both simplex and dual simplex method can be modified to meet these requirements. A computing programme based on Ref. 12 has been written for the Pegasus digital computer, and one based on Ref. 10 has been written for the Atlas. * These a r e in a sense dual programmes to each other. Computing examples presented in Ref. 7 were completed on Pegasus, and in section 4, a com-puting example will be given using the Atlas Comcom-puting code.
It is of interest to consider also the dual problem of (8)and (9):
M i n i m i z e ^ | t | , (10) subject to: A' t = F , (11) where t is a row matrix.
If either the primal or dual problem has a solution, then both solutions exist and are equal, i . e .
Min.'fcit I = Max.Fv = fV^^
and t can be interpreted as the loads in the members (positive for tension), of the optimum structure which has been designed. Equations (11) are the equilibrium conditions at each joint.
Physically, this means the selection, from all structures obtainable on the grid of points, of the lightest which satisfies the conditions of equilibrium,
4. An example of three-dimensional design.
Consider as an example a force F acting vertically at point P distance 3d above the horizontal plane, and reacted by vertical forces at the corners of a square ABCD, with dimensions 4d x 4d in the horizontal plane, the centre of which
is lying on the line of action of F , as shown in Fig. 2a.
By taking full advantage of the conditions of symmetry, the size of the computing problem can be reduced. In this case, there are four planes of sym-metry for the virtual displacements u, so that only an eighth part of space need be considered.
A grid of 30 points (x' , y*, z') is used, determined by x' = x/d = 0 , 1 , 2 .
y' = y/d = 0 , 1 . 2 . z' = z/d = - 1 , 0 . 1 , 2 , 3 . y'< X'
and a total of 59 variables v = u/e r e s u l t s , with positive directions shown in Fig. 2b. The reacting point B is assumed to be restrained vertically.
The problem (8) , (9) now becomes that of maximizing Fvi in such a way that no member joining two points on the grid has a strain exceeding - 1. For a grid of 30 points, there a r e
3 £ i 3 0 _ ^ ) =435
inequalities of the form (9), but it should be noted that some of them are redund-ant. For example, the constraint for the member (0,0,0) - (2,0,0) will become unnecessary if both constraints for (0,0,0) - ( 1 , 0 , 0 ) and (1,0,0) - ( 2 , 0 , 0 ) are already included.
It results that the linear programming problem consists of 59 variables and 360 constraints of the form (9). For such a problem, the preparation of the data is still a difficult matter. The only feasible way is to write a computing programme, which requires only the locations (x* , y , z') and the directions of v , to generate all the constraints (9) in a form suitable for further calculation.
The computation of the present problem was carried out on the Atlas computer using the computing method of (10). The result is
Max. Fvi „ . Fd f
Vm= ^^^i^^^T^ = 7 . 4 i ^ . (12) The members strained to the limit ±e a r e given by those constraints in (9) which
are satisfied as equations, (in the terminology of linear programming, the c o r r e s -ponding slack variables of those constraints a r e zero). Members of the optimum structure,in equilibrium with the applied loads, must be selected, if possible. from these m e m b e r s .
For the present purpose, consider only those members on the plane DPB which a r e strained to the limit, a s shown in Fig. 2c, in which members in in tension a r e drawn as thick continuous lines, and those in compression as thin continuous lines. The only possible structure in the plane DPB which can be selected from these members is shown in Fig. 2d. By symmetry an identical structure lies in the plane APC and combined together they form the required structure of Fig. 2e, the volume of which is given by (12). Comparison of this result with some more conventional designs in Fig. 2f, shows a reduction in weight of more than 10%.
5
-,5. The Michell optimum solution.
The solution in section 4 is somewhat unexpected in that it is a combin-ation of two plane s t r u c t u r e s . The two-dimensional optimum solution for the force system in Fig. 2d is known to be a fan structure (Fig. 3a), so long as the fan angle 2a is less thanjr/2. The Volume of such a structure is given in Ref. 2 a s
V = 2/2 (2a-l-cota) ^ = 7.2746 ^ . (13) If the solution of section 4 is replaced by a two-fan structure (Fig. 3b),
the volume will decrease by about 2%. It is shown below that the two-fan structure is indeed the theoretical optimum, which has been approximated to within 2% by the calculation of section 4.
The corresponding complete strain field of Fig. 3a was first given in (1), and consists of four quadrantal fields with principal strains lying along the coordin-ate lines of polar coordincoordin-ate system or a cartesian coordincoordin-ate system. Fig. 3c.
The virtual displacements are (i) in the region MPN,
Up = - e r , UQ = 2era (ii) in the region NPS,
Ujj = e (2 - l)z , u^ = -e(l+j)x
Rotating the complete field with respect to the z-axis, gives resulting strain components as follows:
(i) In the region generated by MPN, the strain components in spherical coordinates ( r . e , ^ ) are 3uj. ^TT= 3 ^ = -e , S u e u r « 6 6 = 7 - 3 6 - ^ =e e^^ = ^ c o t e + ~ = e (1 - 2ecote) S i n c e ï * 6< y^ therefore [e^^A < e
(ii) In the region generated by NPS. the strain components in cylindrical coordin-ates (x.^ , z) a r e e =^-i^ = 0 = e ^xx 9x zz ex^= e^z e. ux
e(ï-
1) ^ " ^ ^ X ' 2 X - 1 / 9uz , <iu-x\ exz - 5 < - a - + - a t ) = -eSince -l<-< 1 in NPS, it follows that I e^^ |< e.
The conclusion that the strain nowhere exceeds e holds also for the other regions RPS and RPM. There exists therefore a virtual deformation in accordance with (4), and furthermore, the structural members are lying along principal strain lines. Following the general statements in section 2, the structure of Fig. 3b is thus shown to be the optimum solution.
6
-6. Multiple loading systems.
The fact that a structure might be required to support different sets of loadings at different t i m e s , has led to the consideration of designing multi-purpose structures for minimum weight (Refs. 8,9 & 11) However, owing to the complex-ity of the nature of the problem itself, both theoretical and numerical investigations have not as yet produced a method which is practically acceptable.
It seems reasonable to try a simple-minded approach. This can best be explained by an example which i s also considered in Ref. 8:
A single load P in Fig. 4a is applied at points B and C in turn, and reacted by vertical forces at A and D.
Applying the computing method of section 3 to this two-dimensional case, calculating first the loading system in Fig. 4b. by a grid of points lying within EADF, the resulting structure is found to be as given in Fig. 4c.
Combining this structure with an identical structure for the single load-ing at C, the structural layout of Fig. 4d is obtained after an addition of four mem-b e r s shown as dotted lines, which were found to mem-be fully strained in the computations but which have not been used in the structure of Fig. 4c.
The structure of Fig. 4d can of course carry the multiple loadings at B and C, and it is shown to be four-fold redundant, (taking into account the condition of symmetry). If this structure is now stressed according to the loading at B and C respectively, each structural m e m b e r * j will have two required a r e a s ag^ .^Ci corresponding to loadings at B and C, which a r e linear functions of four
redundances Ri ,R2 ,R3 ,and R4 .
The problem now reduces to that of varying the redundancies so that the required volume is a minimum, i . e .
Min. j_ V=S Max.(ajj| , a^^. ) - * J a g j , a^- > 0 J (14) This can be converted into a standard linear programming form if the
following relationship is applied.
Max. (x.y) = i(x+y+|x-ji) . (15) Calculations based on (14) show that the structure of minimum weight
selected from Fig.4d has
V = 1.796 PL/f ;
the corresponding structural layout is shown in F i g . 4 e .
It cannot of course be claimed that this is the absolute minimum, but it is interesting to compare it with the solution of V = 2.11 PL/f given in Ref. 8.
7 -R E F E -R E N C E S (1) M i c h e l l . A . G . M . (2) Cox. H. L. (3) H o p k i n s , H . G . P r a g e r , W. (4) D r u c k e r , D . C . Shield, R . T . T h e linait of economy of m a t e r i a l in f r a m e s t r u c t u r e s . P h i l . M a g . S . 6 , V . 8 , N o . 4 7 , 1904. T h e t h e o r y of d e s i g n . ARC R e p o r t 19791, 1958. L i m i t s of economy of m a t e r i a l in p l a t e s . J . A p p l . M e c h . . V . 2 2 , 1955. D e s i g n for m i n i m u m weight. 9th I n t e r . Cong. A p p l . M e c h . .Book 5. B r u s s e l s 1957. (5) D r u c k e r . D . C . Shield. R . T . (6) H e m p . W. S. (7) H e m p , W. S. (8) Schmidt, L . C . (9) Shield. R . T . Bounds on m i n i m u m weight d e s i g n . Q u a r t . A p p l . M e c h . , V . 1 5 , 1 9 5 7 . T h e o r y of s t r u c t u r a l d e s i g n . College of A e r o n a u t i c s ' R e p o r t 115, 1958. Studies in the t h e o r y of Michell s t r u c t u r e s . 11th I n t e r . Cong. A p p l . M e c h . ,Munich 1964. M i n i m u m weight layouts of e l a s t i c s s t a t i c -a l l y d e t e r m i n -a t e , t r i -a n g u l -a t e d f r -a m e s under a l t e r n a t i v e load s y s t e m s .
J . M e c h . P h y s . S o l i d s , V . I O , 1962. Optimum design m e t h o d s for m u l t i p l e loading. Z A M P , V . 1 4 , N o . l , 1963. (10) D o m . W . S . G r e e n b e r g . H . J , L i n e a r p r o g r a m m i n g and p l a s t i c l i m i t a n a l y s i s of s t r u c t u r e s . Q u a r t . A p p l . M a t h s . . V . 15. 1957. (11) Dorn. W . S . G o m o r y , R . E . G r e e n b e r g . H . J . (12) W a g n e r . H . M . A u t o m a t i c design of o p t i m a l s t r u c t u r e s . J . d e M é c a n i q u e , V. 3, N o . l , 1964.
T h e dual s i m p l e x a l g o r i t h m for bounded v a r i a b l e s .
Naval R e s e a r c h L o g i s t i c Q u a r t e r l y . V . 5 , N o . 3 , 1958.
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