Optimization and robustness of structural engineering systems

I'ngineering Strm'ture~. Vol.

PIhS0141-0296(96)00072-7

It). No. 4. pp. 289-2~2. 1997 ~' 1"~7 Elsevier Science Lid Printed m Great Britain. All rights reserved I)141 0296/97 $17.00 + ().IX)

ELSEVIER

Optimization and robustness of structural engineering systems John F. Brotchie CSIRO Division of Building, Construction and Engineering, PO Box 56, Highett. Victoria 3190, Australia

This paper outlines several strands in the historical development of structural optimization techniques. The first is indirect via simple optimality conditions such as uniform density of stress or strain for achieving a technical objective like minimum weight, maximum stiffness or minimum strain energy. In the case of prestressed structures, zero net displacement of the concrete in flexure at service loads may be the criterion, and in the case of structural vibration damping, maximum dissipation of energy under dynamic loads. The second is direct via mathematical optimization of a more comprehensive, economic objective such as minimum cost or maximum utility under multiple loading conditions. Further development has led to automated and interactive design and optimization of nonconvex objective functions using improved optimization techniques. A more general, systems theory has been formulated with application to both strongly and weakly fnteracting systems, including structures, taking uncertainty into account. It relates overall utility of the system to its efficiency, diversity and uncertainty or robustness, and enables an overall optimum to be obtained for a given level of uncertainty and diversity. © 1997 Elsevier Science Ltd. All rights reserved.

Keywords: optimality, optimization, structural systems, complexity, interaction, robustness, entropy, utility, energy, uncertainty, diversity, constraint, objective, damping

I.

A more comprehensive systems theory formulation has also been developed with application to both weakly and strongly interacting systems, including structures, taking uncertainty into account. It provides a relationship between the overall utility of the system, its efficiency and its robustness under uncertain futures. It allows a trade-off between these latter two properties - efficiency and robustness - in seeking the overall optimum for a given level of uncertainty and diversity, as later explained. It also applies to selforganizing systems, and dynamic design processes where previous decisions influence future response.

Introduction

The concept of optimization of design and the development of techniques for achieving it has a relatively long history. An early approach was indirect via the satisfaction of optimality conditions such as uniform stress or strain to meet a technical objective, such as minimum weight for a single set of loads. It dates back to the beginning of the century at least. A comparatively more recent and direct approach has been via computer-based search techniques including mathematical programming - which systematically search a design space to maximize a broader economic objective such as minimum cost or maximum utility for multiple loads - subject to safety and serviceability constraints. It began in the second half of the century with the use of computers in engineering. Further developments have enabled the automation of design, and interactive design, and the optimization of nonconvex objective functions with more sophisticated or "intelligent' search techniques.

2.

Optimality conditions

An early study by Michell ~ optimized the form of frameworks under a single loading condition using the optimality criterion of uniform strain to minimize virtual work and frame weight. A variety of frameworks were obtained for different loading and support conditions. The criterion of uniform dissipation of energy in the

289

Optimization and robustness of structural engineering systems: J. F. Brotchie

290

plastic range, implying minimum material volume for plates and shells, was proposed by Drucker and Shield-'. Uniform strain energy density in the elastic range has been suggested by Wasiutynski and Brandt 3 to give minimum total strain energy and hence minimum displacement under load, and maximum stiffness under given loads. The same general approach was later applied to shell structures, including arch dams and submersible shells, and to variable thickness, reinlbrced and prestressed plates 4-~'. It utilized the criterion of uniform stress or strain intensity to satisfy the condition of minimum total strain energy, minimum loss of potential energy or maximum stiffness under a single, sustained system of loads. Lin 7 designed prestressed concrete structures in flexure to minimize displacement under service loads by draping and tensioning the tendons to provide upward vertical tendon forces on the concrete structure to just balance downward service load, resulting in zero net deflection of the structure at this load. (The concrete flexural member was in uniform compression at this load). Balancing (equilibrium) moments in the structure under service loads with tendon forces and their eccentricities was later shown s* to produce the same results and to offer a wider range of solutions including horizontally curved tendons. Maximum dissipation of energy under dynamic loading of a lightweight elastic structure gives the optimality condition '~ for maximum damping of vibration of c= KIw

4.

Automated design and nonconvexity

Cornell et al. ~'- automated the design decision process using utility maximization as the objective. The design system could be on line and interactive, enabling nonquantitative criteria to be taken into consideration. Automated processes were earlier used by Livesley '~ and the French highway authority. For non convex objective functions, stochastic optimization techniques have been developed to enable escape from local optima. Simulated annealing'" and an analogous 'robustness' technique outlined below enable a global optimum to be approached in this way. These techniques have been used to optimize the layout of facilities and urban systems -~° and the three-dimensional location of a road-". They have application to structural engineering design.

5. General systems optimization: a unifying theory A general systems formulation for both weakly, and strongly interacting systems of particles, micro-elements or activities 4.2-" was found to take the form

(I)

where c is the optimal damping coefficient, oJ is the frequency (rad/s) of vibration, and K is the stiffness of the structure at the damping location. Similarly, for forced vibration of a rigid mass, m c = o,n

dition and technical objective have been used to select an initial design for the mathematical programming approach - shortening the iterative process '4.

(4)

U = R + SIA

where U is the total utility or value of the system, its cost effectiveness; and R is the base utility or efficiency where. for weakly interacting systems

R = ~'/,,~:,,

(2)

(5 )

tl

or more generally c=

IKIo.,-

or fl)r strongly interacting systems

(3)

which has broad application to damping of vibration and noise.

3.

Mathematical programming

Computer-based optimization techniques have enabled economic objectives and multiple loading conditions to be handled. Schmidt m has used an iterative gradient search to minimize frame weight in structural design. Moses *~ has used linear programming to optimize the structural design. A generic approach developed at MIT ~2 optimized cost or utility under multiple loading conditions, subject to performance constraints. The objective and constraints are generally nonlinear. Iterative linear programming has been used for solution with a Newton-Raphson linearization of objective and constraints about the current design point in a multidimensional design space. The approach has application to wider structural and other systems problems, but was applied initially to bridge superstructures, trusses and frames,:~ ~7. Second-order approximations have been made to the location of nonlinear constraints to improve convergence". The optimality techniques based on a single loading con-

R = Ea,r~.i~ + Eb,~a,t-,~v,l tl

(6)

IIl~l

where t~lt/

A'it

S

I/A

SIX

mean utility or efficiency of placing unit activity or load i in space j quantity of activity i in space j utility of interaction between unit activities i and k in spaces j and / entropy of the system defined as proportional to the log of the number n, of microstates (e.g. combinatorial patterns of unit activity - zone allocations or load-resistance couplings) the system can occupy for the design [&, 1. S represents dispersal at the macrolevel, uncertainty at the microstate level the diversity (proportional to standard deviation) of the random component ~ of utility b,, with mean a,, 'robustness" of the system: the capacity to internally reorder its microstates within the design Ix,,] to resist future activity or loading variations

The design objective fl~r the system is to select [x,] to

Optimization and robustness of structural engineering systems: J. F. Brotchie maximize U subject to capacity constraints (capacity limits) A, and Zj on total activity i and on total space j, respectively a, e.g.

~'~x,,<-Z~.

~ x , , = A,

i

(7)

i

Equation (4) relates the total utility (U) to the base utility (R), the entropy or uncertainty (S) and diversity (l/A) or the performance (U) to the efficiency (R) and the robustness (S/A) of a system. Equation (4) is also the classic Helmholtz relationship in thermodynamics and statistical mechanics between internal energy (U), free energy (R), entropy (S) and absolute temperature ( T = l/A). The simulated annealing process ~9 for the maximization of nonconvex base utility IR) begins a stochastic search with a high temperature T, and reduces T until a solution is reached (at T = 0). The initially high temperatures and stochastic search enable easy escape from local optima. As the temperature reduces, escape is more difficult. The process is repeated a large number of times until no significant improvement occurs. The robustness approach is mathematically analogous 4 but conceptually different and offers new insights and interpretations. The solution starts with a high diversity !/A for which U is convex, and 1/h is slowly reduced. Moves made randomly about the design space are accepted if R increases and can also be accepted on a stochastic basis - if R decreases but the value of U is thereby increased and S/AU exceeds a random number between 0 and 1. As I/h decreases, U approaches R (the generally nonconvex objective for the case I/A = 0). Retaining a finite value of 1/h allows for a corresponding level of diversity and uncertainty of design conditions and of internal response, and substitutes robustness for efficiency to cater for this uncertainty. (It means distributing the eggs over more than one basket - or introducing redundancy or complementary structural actions in a structure. ) The (indirect) analytical solution of equation (2) for finite l/A, with weakly interacting systems (and single loading) takes the form x,, ~ ~,, e

,,'

1

(8)

+ p

in which ~x means 'is proportional to', p takes the value 1, 0 or -1 depending on the constraint conditions on x~j. For Zj = 1 and x# -< 1, the case of limited zone capacity, or congestion, p = 1; for x o unconstrained, p = 0 the case of unconstrained design elements; and where x 0 increases with previous design decisions x,j,,, i.e. a positive feedback system or increasing returns to scale, p = - I . For the limiting case of 1/h = 0, with p = 1 and Zi = I, the solution x~j takes the value of 0 or 1 (constant). The case p = 1 corresponds to the classical case in economics of decreasing returns to scale, constrained supply (limited capacity) or negative feedback. The case p = 0 is the case where probability x,j is un',fffected by previous decisions xi,,,, or zero feedback. The case p = - 1 is the case of increasing returns to scale, where previous decisions positively influence following ones, i.e. positive feedback applies (e.g. increasing a structural member increases its stiffness and attracts more load to it).

291

Uncertainty requires a design mix, e.g. load sharing between structural subsystems, or redundancy in structures. Consider the following example.

5.1.

Example: floor panel design

In the structural design of a suspended floor with multiple panels (j = I, N), heavy storage loading, i, is planned for one panel (j = 1). The suitability (mean utility or cost effectiveness) a,i of the panels, j, for this loading, i, is a,j = 10,j = I

a,,= 5 j > 1

If Z)= I and total utility h~i is constant over the panel, l / A = 0 and the optimal loading pattern Ix,j] is x, = I, xi,=0, j > l. l n t h i s c a s e S = 0 , S / A = 0 , U = R = 1 0 . However, if utility h,i varies over each panel, i.e. I/A = l/A,, (and p = 0), the optimal load distribution for design for equation (8) is e-X,% z xq

(9) E e A"d /

In this case those parts of each panel are loaded, lor which cost effectiveness h 0 is above a threshold a,, which is chosen to allow the total load to be allocated among the panels. For p = 0, the allocations x,j are independent but sum to I and are given by equation (9). For p = 1, there is a limit to the load that can be applied to one panel (i.e. x~j-< i, e.g. due to congestion of loading elements or of reinforcement in the slab, or deflection constraints). For p = -1, there is a benefit in placing all or nearly all load in one panel (e.g. due to convenience of layout or efficiency of design, perhaps using a thicker slab in that panel). In a more general case, the load i would be shared between design options j to provide a more robust but less efficient design. In an urban system, robustness allows urban restructure (households and firms to relocate or choose a closer trip destination within the existing built environment) to meet future transport energy constraints. Each pattern of trips at the urban system level represents a different microstate, and the more patterns that can be accommodated within the location pattern Ix,j] the more robust the pattern with respect to possible energy futures. Urban systems are essentially self-organizing within the existing built environment in response to external forces. Structural systems also internally "organize" resistance to meet imposed external lbrces within their particular structural form. 6.

Conclusions

The systems theory outlined applies to structural systems as part of a broader set including socio-economic systems. It applies to weakly and strongly interacting systems. One base is utility maximization in economics. Another is energy minimizing - and its relationship to entropy maximizing - in classical statistical mechanics. The three classes of behaviour outlined also correspond to different economic and statistical mechanics cases. The case p = 0 is the norm in behavioural choice where x is independent of previous choices (and of irrelevant alternatives). It corresponds to classical Maxwell-Boltzmann statistics for the thermodynamics of gas molecules. The cases p = I and p = - I corre-

292

Optimization and robustness of structural engineering systems: J. F. Brotchie

spond, respectively, to decreasing returns to scale and increasing returns to scale in economics and to FermiDirac and Bose-Einstein statistics for spin glasses in physics. In the emerging science of complexity -'(, the cases 19 = - I and I / a - - . ~ may be considered to be at "the edge of chaos', in which the solution is sensitive to initial conditions and rapid changes in the solution can occur (in dynamic systems or dynamic or evolutionary design processes). In economics, increasing returns to scale means earlier choices positively influence later ones, leading in the extreme to domination of one element or option x, i, e.g. one technology over another (such as VHS over Beta in video or the internal combustion engine over steam) or one structural design option over another for a particular application (e.g. truss over rigid frame or precast over cast in situ). The new science of complexity is now exploring these unifying relationships and also extending them to other classes of system, particularly biological, including the organization behaviour of various species and their origin, evolution and extinction. Unitication of systems theory can enable developments in one type of system to be more generally evaluated. Structural systems design might benelit from this process.

5 6 7

8 9 I0

II

12

13

t4 15 16 17 IN

References I t) I 2 3

4

Michell. A. C. M. "The limits of economy of material in frame structures'. Phil. Mag. 190 -A. 8(47), Series 6, 589 Drucker, I). C. and Shield, R. T. 'Design for minimum weight'. Proc. 9th Int. Congre.~ of Applied Mechanics, Brusscls. Belgium, 1956 Wasiutynski, Z. and Brandt. A. "The present state of kno,.vlcdgc m the field of optimum design of structures', Appl. Mech. Rev. 1963. 16(5), 341-350 Brotchie, J. F. "A generalised design approach to solution of nonconvex quadratic programming problems', Appl. Math. Mo&'lling 1987, 11. 291-295

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