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Genetic Algorithms in the Optimization of Cable Systems

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30
- Newsletter EnginSoft Year 9 n°1
In recent years, cable-strut assemblies attract a lot of attention from engineers, due to their versatile shapes,their lightweight and architectural impact. Cable-strutstructures have become popular as roofs for arenas,stadiums and sport centers. Yet their working principle isnot so easy to understand since they can carry loadsthanks to prestress, so that their behavior under loadsmust be studied taking into account, at least, thegeometrical non-linearity. For these reasons, designexperience and intuition may not be enough whenengineers work with cable systems.In this article, the matrix theory of a generic three-dimensional pin-jointed structure is first referred. Then,by using modeFRONTIER, a general method able to providea design solution, which is not only technologicallysound, but optimal with respect to the designrequirements, is applied to the design of a Geiger Dome.
Introduction
A structure, with a new particular shape cannot beconsidered as an innovation just because it is verycomplex. An innovation is a new system working with newmechanical principles, for a better use of materials,lightness etc. Cable-strut assemblies are not aninnovation because they are truss systems that are well-known for centuries. However, their last development -the tensegrity systems – can be seen as a real innovation(Motro, 2003). In these systems, the geometrical shapeand the prestress in the elements play a crucial role in thestructure stability. The first civil structure inspired to thetensegrity principle is the cable dome proposed by Geigerand first employed for the roofs of the Olympic GymnasticsHall and the Fencing Hall in Seoul (Geiger et al. 1986). The largest existing cable dome is the Georgia Domedesigned for the Atlanta Olympics in 1996 (Yuan et al.2003).In this article the design of the cable systems isexplained, and the Geiger Dome is used as an example.Particular emphasis is put on a particular optimizationprocedure, based on a genetic algorithm. This allows us tofind a design solution that is not only technologicallysound, but optimal in the design requirements (Biondiniet al. 2011). The genetic algorithm of the modeFRONTIERsoftware will be used.
Matrix Analysis of Pin-joined frameworks
The matrix formulation is based on Pellegrino andCalladine theories (Pellegrino 1986). The hypotheses are:a)members are connected by pin-joints;b)the connectivity between nodes and members isknown;c)self-weight of members is neglected and the additionalloads are applied only in the nodes;d)buckling of the strut is not considered.Hypotheses a) and c) let the members work only with axialforces, either in compression or tension.When we consider a generic three-dimensional pin-jointedstructure, the equilibrium equations can be written in thefollowing form:At =f(1)where A is the equilibrium matrix, t the vector of internalforces and f the vector of nodal forces. In addition, it can be proved through the virtual workprinciple that the compatibility equation is:A
t
d =e(2)where A
t
is the compatibility matrix, d the vector of nodaldisplacements and e the vector of element elongations.
Genetic lgorithms in the Optimization of Cable Systems
Fig. 1 - The Georgia Dome in Atlanta, U.S.A. reproduced by Tibert, 1999
Newsletter EnginSoft Year 9 n°1 -
31
Through the exploration of the balancesubspaces it is possible to classify thepin-jointed framework. In fact, bydefining the number of state of self-stress (s) and the number of internalmechanisms (m), we will see thesituations listed in Table 1.As pointed out in “Pellegrino 1993”, all the informationabout the assembly can be obtained by the singular valuedecomposition (SVD) of the equilibrium matrix, whosedetails are given in “Quarteroni et al. 2008”. The state of self-stress is represented by the solutions At =0, and themechanism by the solution A
t
d =0.
The Geiger Dome
In a Geiger Dome the ridge cables are radially oriented,and the roof is composed of wedge shaped basic units,cyclically distributed around the centre. The Geiger Domehere represented is defined by 84 nodes connected with156 elements, as shown in fig. 2. The structure iscomposed of 36 struts and 120 cables. The 12 externalnodes are fixed. The symmetry of the dome allows tosubdivide the elements into 13 groups, as shown in fig.2(b). Given the connectivity and the fixed nodes, the state of self stress can be computed through the singular valuedecomposition of the equilibrium matrix. The results arereported in table 2. In addition, s=1 and m=61, hence thestructure is statically and kinematically indeterminate.However, the self-stress state can stabilize all the internalmechanisms. Until now, only one balance problem has been solved. Infact, since the vector reported in table 2 is a base, anycoefficient
Ψ
can be chosen, so that
Ψ
t=0. The precisevalue of
Ψ
must consider the performance of the structureunder external loads and the resistance of the material.So, for practical purposes, the introduction of additionaldesign criteria, such as structural performance in terms of rigidity and deformability, is needed. This leads us tointroduce new variables, such as stress intensity and theactual section of the elements that must also match thoseof commercial profiles (Biondini et all 2011). Thealgorithm chosen here is a genetic algorithm implementedin the commercial software modeFRONTIER.
Applied loads and constraints
In addition to the prestress system, two sets of loads areconsidered:a)the structural weight;b)a live vertical load q=0.5kN/m
2
, uniformly distributedover the dome. The constraints of the problem are:1)a constraint on the maximum dome displacements:(3)2)a constraint on cable resistance: the forces mustcomply with the resistance FRd provided by themanufacturer reported in Appendix A, with a safetymargin
γ
s,i
=1,5.3)a constraint on strut instability:
Fig. 2 - Details of the Geiger domeTable 2 - State of self stress of the domeTable 1 - Classification of structural assemblies
(4) The first constraint has to be verified under loads, thesecond and the third have to be verified for both, theprestressing state only and for loads. So, two types of safety margins will be provided: the initial safety marginsand the final safety margins.
Representative design variables of theproblem
The representative variables of theproblem are:• the coefficient
Ψ
(1 variable); • the cable sections divided into groups(10 variables);• the strut sections divided into groups(3 variables). This means that theoptimal solution is searched in a spaceof 14 variables. The area of the cableshas to match those ones of thenormalized product. The list of commercial areas considered by thegenetic algorithm is reported inAppendix A. Therefore, the algorithm considers 64different cable types. For the struts, 50 circularsections with the following defined diameter areconsidered.
Ф
=25 : 5 : 240 [mm](5)
Results of the optimization process
The evaluation of displacements and internal forces in theelements required to assess the fitness of eachindividual element was made possible by Cable3, a finite element program implemented inFortran. The program is able to handle the loadresponse of a general 3D pin-jointed frameworkconsidering both mechanical and geometricalnon-linearities. The commercial softwaremodeFRONTIER as dealt with the structureoptimization problem were p =40, number of individuals in the population; pc =0.85,crossover probability; pm =0.05, mutationprobability; elitism disable. The data flow andthe logic flow of the modeFRONTIER process arereported in Fig. 3. The optimal solution isshown in Table 3.
Table 3: Optimal solution provided by modeFRONTIERTable 4: Initial safety marginsTable 5 - Final safety marginsFig. 3 - Logic and data flow of the optimization process in modeFRONTIERFig. 4 - Probability density function of some variables
32
- Newsletter EnginSoft Year 9 n°1
In Table 4 and 5, the initial and the final safety marginare illustrated. In fact, when dealing with cable systems,not only the final state of the structure (under all loads),but also the initial state (under prestress only) have to beverified. From the tables, we can observe that thedominant condition may be in the initial or in the finalstate. In fact, there are some elements that increase theirforce under loads (Fig. 6), while others decrease it, inaccordance with the working principle of a cable system.For the optimal solution, the maximum deflection underloads is equal to -359.57 mm, practically coincident withthe allowable maximum deflection set equal to -360 mm.Figures 10 and 11 show respectively the deformed shapeand the axial forces for the optimal solution (Straus72004).
Conclusion
In this article, an approach to the problem of optimaldesigns of cable structural systems has been presented. Inthese systems, the solution of the initial balance problemplays a dominant role. In fact, as these structures workonly through axial forces, the geometry and thepretensioning intensity applied to the elements areclosely related. Therefore, the balance configuration must be determinedby specific form-finding techniques that provide both theform and the associated stress state. For practicalpurposes, however, this is not enough and an additionalphase, that takes into account the structural performancein terms of rigidity and deformability, is needed.Experiences and intuition may not be sufficient in thissecond phase because the problem isaffected by the geometrical non-linearitiesand for this reason, a feasible solution mayrequire several trials. The authors have presented here a generalmethod able to provide a design solutionthat is not only technologically sound, butoptimal with respect to the designrequirements. In the suggested formulation, the solutionfor the optimization problem has beenprovided by the genetic algorithm includedin modeFRONTIER. This algorithm has beenapplied to the structural optimization of aGeiger Dome.
A. Albertin, P.G. Malerba,N. Pollini, M. Quagliaroli Department of Structural Engineering, Politecnico di Milano, Milan, Italy
For more information:Francesco Franchini, EnginSoftinfo@enginsoft.it
Fig. 5 - Simulation statistics(a) Deformed configuration(b) Final axial forces.Fig. 6 - Final control of the optimal solution with the commercial FE software Straus7
Newsletter EnginSoft Year 9 n°1 -
33
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