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
cncwl@leeds.ac.uk v.v.toropov@leeds.ac.uk
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:
𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑓 𝑥 = 𝑈𝑚𝑒𝑚𝑏𝑟𝑎𝑛𝑒 + 𝑈𝑐𝑎𝑏𝑙𝑒
𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜: 𝑔(𝑥) ≥ 1
where 𝑓(x) is the function of the total structural strain energy, to be
minimized; 𝑥 is a vector of design variables and 𝑔 𝑥 =
𝑔1 𝑥 , … , 𝑔𝑘 (𝑥) 𝑇 is a vector of the negative minor principal stresses in
𝑘 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:
𝑥1 (kN/m)
𝑥2
𝑥3
𝑥4 (kN)
Nominal
4
3
0
1
Lower bound
4
3
0
1
Upper bound
5.4
5
1
2
𝑥1 = principal membrane prestress in the concave direction 𝑃𝑡 .
𝑝
𝑥2 = prestressing ratio 𝜆 = 𝑤 ; 𝑝𝑤 is the membrane prestress in the
𝑃𝑡
convex direction.
𝑥3 = edge shape variable 𝛽.
𝑥4 = 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
𝑔 𝑥 = 𝑔1 𝑥 , … 𝑔𝑘 (𝑥) 𝑇 are 𝑘 inequality stress constraints; and 𝑥 is a
vector of design variables. The following constraint aggregates are
defined:
1
𝜌
𝑘
𝑝−𝑛𝑜𝑟𝑚:
𝑔(𝑥)
𝜌𝑛
=
𝑔(𝑥)
𝜌
1
p−𝑚𝑒𝑎𝑛:
𝐾𝑆 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛:
𝑔 𝑥
𝜌𝑚
𝑔 𝑥
=
𝐾𝑆
1
𝑘
𝑘
1
𝑔(𝑥)
1
= ln
𝜌
𝜌
1
𝜌
𝑘
𝑒 𝜌 𝑔(𝑥)
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 𝑃𝑡 (𝑥1 ), (2) prestressing ratio 𝜆 (𝑥2 ), and (3) shape factor 𝛽
(𝑥3 ).
Metamodel-based Optimization
Optimization results for minimum stress function 𝜎𝑚𝑖𝑛
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 𝜌-norm stress function 𝜎𝜌𝑛
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 𝜌-mean stress function 𝜎𝜌𝑚
Optimum
𝜌
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
𝜌
-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 𝑝-norm and KS-function are conservative – the aggregated
minimum stress value is always smaller than min
𝑒∈ 1,…,𝑛
𝜎𝑒 ′
𝑎
.
 The value of stress aggregated by 𝑝-mean is always larger than that
in the minimum function – envelopes the feasible solutions.
 In our case, 𝑝 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 𝜎𝑒 ′
𝑒∈ 1,…,𝑛
 The optimization converged after 18 iterations for the constraint on
the minimum stress, this was reduced to 13 iterations for the 𝜌mean; and to 10 and 9 iterations for 𝜌-norm and KS-function,
respectively, when the 𝜌 parameter was taken as -500. Results
produced by using the 𝜌-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 𝜎𝑚𝑖𝑛 .
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