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Optimal Shape Design of Membrane Structures Chin Wei Lim, PhD student 1 Professor Vassili Toropov 1,2 1 School of Civil Engineering 2 School of Mechanical Engineering cncwl@leeds.ac.uk v.v.toropov@leeds.ac.uk Faculty of Engineering

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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.

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Introduction Example: on 12 December 2010 Minneapolis Metrodome collapsed under the weight of 17 inches of snow

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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.

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Problem Formulation

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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 3- node, finite membrane strains). ii.Edging cables: beams (B31 beam element). iii.Mesh:100 x 100 elements H L L

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Problem Formulation Nominal4301 Lower bound4301 Upper bound5.4512

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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

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Abaqus FEA Modelling Material properties: 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. MembraneCables Young’s modulus, E1000 kN/m1.568×10 8 kN/m 2 Poisson’s ratio, v0.20.3 Sectional geometry1 mm thickness t16 mm diameter Ø

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Wrinkling Simulation Nominal design: stress distribution contour plots Compressive (wrinkling) stresses in the convex direction. Tensile stresses in the concave direction.

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Wrinkling Simulation Nominal design: deformed shape Large wrinkles are formed in the convex direction due to compressive stresses.

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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.

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Metamodel Building Metamodel quality assessment: Responses Metamodel FEA% error Strain energy (kJ)2.3232.3501.15 min 11.0040.40

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Stress Constraints Aggregation

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Influence of the aggregation parameters p for p-norm and p-mean for two variables. KS function is similar to p-norm.

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Metamodel-based Optimization

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NominalOptimum Objective functionf(x)kJ3.0972.323 Constraint functiong(x) 0.7851 Design variables x1x1 kN/m44.663 x2x2 34.255 x3x3 01

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Metamodel-based Optimization Nominal Optimum -50-100-200-400 -500 Objective function f(x)kJ3.0972.5972.4612.4032.3822.378 Constraint function g(x) 0.78511111 Design variables x1x1 kN/m44.5604.4284.6874.6624.663 x2x2 354.7744.3364.3024.291 x3x3 01111 1

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Metamodel-based Optimization Nominal Optimum -50-100-200-400-500 Objective function f(x)kJ3.0972.1032.2102.2802.3202.329 Constraint function g(x) 0.78511111 Design variables x1x1 kN/m444.2104.4924.5594.572 x2x2 34.1464.2784.1824.2314.242 x3x3 011111

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Metamodel-based Optimization Nominal Optimum -50-100-200-400-500 Objective function f(x)kJ3.0972.5732.4582.4022.3822.378 Constraint function g(x) 0.78511111 Design variables x1x1 kN/m44.5104.3914.6854.6494.656 x2x2 354.8094.3364.3154.297 x3x3 011111 Optimization results for KS stress function KS

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Results and Discussion

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Deformed shape of the optimum design of the hypar membrane roof – no wrinkles.

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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?

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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.

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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.

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Conclusion A tool set has been established and verified that can be used for practical design of membrane structures

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