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Faculty of Engineering Optimal Shape Design of Membrane Structures Chin Wei Lim, PhD student1 Professor Vassili Toropov1,2 1School of Civil Engineering 2School of Mechanical Engineering [email protected] [email protected] Introduction 1. To limit deflections and surface wrinkling, a membrane structure can be controlled by means of differential prestressing. 2. Structural wrinkles due to inadequate prestressing can spoil the structural performance and stability by altering the load path and the membrane stiffness. It is also aesthetically unpleasant to have wrinkles. Introduction Example: on 12 December 2010 Minneapolis Metrodome collapsed under the weight of 17 inches of snow General approach To incorporate shape optimization in the design process of membrane roof structures whilst minimizing the wrinkle formation that results in the stress-constrained optimization. To handle a large number of constraints p -norm, p -mean, and Kreisselmeier-Steinhauser (KS) function can be used to aggregate a large number of constraints into a single constraint function. Gradient-based and population-based optimization approaches require many function evaluations that is expensive when FEM is used for analysis. Metamodelling can be used to address this problem. Often the real-life designs problems are multi-objective rather than a single objective. These objectives are usually conflicting hence should be optimized simultaneously. In this study a Multi-Objective Genetic Algorithm (MOGA) is used on the obtained metamodels. Problem Formulation The design is driven by its structural stiffness rather than material strength. Wrinkling occurs due to low stiffness (insufficient prestressing or incorrect differential prestressing ratio). Problem formulation: = + : () ≥ 1 where (x) is the function of the total structural strain energy, to be minimized; is a vector of design variables and = 1 , … , () is a vector of the negative minor principal stresses in elements. Example of a membrane structure and Abaqus FEA Modelling Hyperbolic paraboloid (hypar) Dimensions: L = 3.892 metres; H = 1.216 metres. The membrane is pinned at its corners and supported by flexible (free) pretensioned edging steel cables Finite Elements: i. Membrane: shells (S3R 3node, finite membrane strains). ii. Edging cables: beams (B31 beam element). iii. Mesh:100 x 100 elements L L H Problem Formulation Design variables: 1 (kN/m) 2 3 4 (kN) Nominal 4 3 0 1 Lower bound 4 3 0 1 Upper bound 5.4 5 1 2 1 = principal membrane prestress in the concave direction . 2 = prestressing ratio = ; is the membrane prestress in the convex direction. 3 = edge shape variable . 4 = pretension force in the edging cables T. Shape Design Variable HyperMorph module in Altair HyperMesh was used to parameterize the FE mesh. An edge shape factor was assigned to the morphed shape – used as a design variable for shape optimization performed in Altair HyperStudy. Morphed shape: sag = 15% of L (typical industry designs: 10% - 15%) Nominal shape: sag = 6% of L Abaqus FEA Modelling Material properties: Young’s modulus, E Poisson’s ratio, v Sectional geometry Membrane Cables 1000 kN/m 1.568×108 kN/m2 0.2 0.3 1 mm thickness t 16 mm diameter Ø Conditions for nominal design: i. Membrane: 4 kN/m uniform biaxial prestress. ii. Edging cables: 1 kN pretension force. Loading: 4.8 kPa (static, uniform) surface pressure load. Analysis: Geometrically nonlinear static stress/displacement analysis with adaptive automatic stabilization algorithm. Wrinkling Simulation Nominal design: stress distribution contour plots Tensile stresses in the concave direction. Compressive (wrinkling) stresses in the convex direction. Wrinkling Simulation Nominal design: deformed shape Large wrinkles are formed in the convex direction due to compressive stresses. Metamodel Building Metamodel: Moving Least Squares (Altair HyperStudy) with Gaussian weight decay function Design of Experiments (DoE): optimum Latin hypercube designs • Uniformity-optimized using a Permutation Genetic Algorithm. • Two DoEs are constructed simultaneously: model building DoE (70 points) and validation DoE (30 points). Both DoEs are then merged. Metamodel Building Metamodel quality assessment: Responses Strain energy (kJ) min Metamodel FEA % error 2.323 2.350 1.15 1 1.004 0.40 Stress Constraints Aggregation = 1 , … () are inequality stress constraints; and is a vector of design variables. The following constraint aggregates are defined: 1 −: () = () 1 p−: : = 1 1 () 1 = ln 1 () 1 Stress Constraints Aggregation Influence of the aggregation parameters p for p-norm and p-mean for two variables. KS function is similar to p-norm. Metamodel-based Optimization Variable screening: ANOVA was performed on a polynomial least squares approximation in HyperStudy. Due to the difference in material properties the strain energy induced in the steel cables is several order smaller than in the membrane – pretension force applied to the edging cables can be disregarded as a design variable in the optimization when strain energy minimization is sought. The remaining active design variables in the metamodel-based optimization are: (1) principal membrane prestress in concave direction (1 ), (2) prestressing ratio (2 ), and (3) shape factor (3 ). Metamodel-based Optimization Optimization results for minimum stress function Objective function Constraint function Design variables f(x) g(x) x1 x2 x3 kJ kN/m Nominal Optimum 3.097 0.785 4 3 0 2.323 1 4.663 4.255 1 Metamodel-based Optimization Optimization results for -norm stress function Optimum Nominal -50 Objective f(x) kJ function Constraint g(x) function x1 kN/m Design variables x2 x3 3.097 0.785 4 3 0 -100 -200 -400 -500 2.597 2.461 2.403 2.382 2.378 1 1 1 1 1 4.560 4.428 4.687 4.662 4.663 5 4.774 4.336 4.302 4.291 1 1 1 1 1 Metamodel-based Optimization Optimization results for -mean stress function Optimum Nominal -50 Objective f(x) kJ function Constraint g(x) function x1 kN/m Design variables 3.097 1 4 4 3 x3 0 -200 -400 -500 2.103 2.210 2.280 2.320 2.329 0.785 x2 -100 1 1 1 1 4.210 4.492 4.559 4.572 4.146 4.278 4.182 4.231 4.242 1 1 1 1 1 Metamodel-based Optimization Optimization results for KS stress function KS Optimum Nominal -50 Objective f(x) kJ function Constraint g(x) function x1 kN/m Design x2 variables x3 3.097 0.785 4 -100 -200 -400 -500 2.573 2.458 2.402 2.382 2.378 1 1 1 1 1 4.510 4.391 4.685 4.649 4.656 3 5 0 1 4.809 4.336 4.315 4.297 1 1 1 1 Results and Discussion The -norm and KS-function are conservative – the aggregated minimum stress value is always smaller than min ∈ 1,…, ′ . The value of stress aggregated by -mean is always larger than that in the minimum function – envelopes the feasible solutions. In our case, has to be as smaller as possible (without running into numerical troubles) due to the negative sign when the minimum stress value is approximated. For smaller p the optimization problem can become ill-conditioned – number of iterations increased. Results and Discussion These functions eliminate the discontinuity of deivatives in min ′ ∈ 1,…, The optimization converged after 18 iterations for the constraint on the minimum stress, this was reduced to 13 iterations for the mean; and to 10 and 9 iterations for -norm and KS-function, respectively, when the parameter was taken as -500. Results produced by using the -norm and KS-function are similar. Results and Discussion Tensile stresses in the convex (left) and concave (right) directions. Optimum design obtained with minimum stress constraint . Results and Discussion Deformed shape of the optimum design of the hypar membrane roof – no wrinkles. Multi-objective Optimization A membrane structure with tension everywhere is desired. But … How large the lower bound value of the stress constraint imposed to the minor principal stresses should be in order to eliminate wrinkles and at the same time produce an “optimum” design? Multi-objective Optimization Objective functions: (1) minimize the strain energy, and (2) maximize the minimum minor principal stress. The intersection of the solid red lines shows the location the nominal design. Multi-objective Optimization The obtained Pareto set consists of 500 non-dominated points after 50 iterations, with a total of 17,762 analyses on the metamodel. The trade-offs show that a minimum strain energy design can be achieved and that this maximum stiffness design is not necessarily equivalent to a wrinkle-free membrane. Conclusion A tool set has been established and verified that can be used for practical design of membrane structures