Sequential configuration optimization of frame model for anchoring device of membrane structures
Taku Nakajima (Kyoto University)Makoto Ohsaki (Hiroshima University)Jun Fujiwara (Taiyo Kogyo Corporation)Fumiyoshi Takeda (Taiyo Kogyo Corporation)
Outlines
• A method for optimizing cross-sectional shape of anchoring devices (clamping member) for membrane structures.
• Frame model for cross-section of clamping member.• Two-stage extended ground structure approach to
optimization under stress constraints.
• Three types of clamping member: Automatically clamp the membrane as the result of increase of
tensile force. Adjust deformation of membrane with a bolt. Stabilize by buckling and contact utilizing material and
geometrical nonlinearity.
Optimization of mechanical parts
• Practical applications for optimization of mechanical parts. – Optimal shape of beam flange for maximizing
plastic energy dissipation(Ohsaki et al. 2007).
• Optimization approach todesign of compliant bar-joint structures.– Multistable mechanism utilizing
snapthrough behavior(Ohsaki and Nishiwaki, 2005)
Background and Objectives
• Membrane sheets are connected to the boundary frames with anchoring devices.
• Anchoring devices are mass-products.• The total production cost can be reduced by optimizing
shapes and cross-sectional properties of the devices.
Anchoring device of frame-supported membrane
• Increase tensile force → detachment of the membrane from the device
before fracture of membrane.
Tensileforce
Anchoring Device
Boundary framedetachment of the membrane from the device
membranematerial
Clamping member (Type 1)
• Load resistance capacity can be improved by optimizing cross-section of clamping member– Increase tensile force of membrane.
– Increase clamping force.
Tensileforce
Clampingforce
Tensile force
1/2 model ofanchoring device
roller fixed
Frame Model (Type 1)
• Ground structure of frame model (Type 1).• Load P=500(N) at node 1.
R1
P
lower-bound reaction force: 200 Nupper-bound stress : 200.0 N/mm2
Rectangular section with constant width
Frame Model (Type 1)• Minimize total structural volume V.• Constraint :
– absolute value of stress.– clamping force R1 against the membrane.
• Variables : cross-sectional area A of members(height of a section with constant width)
Optimization under stress constraints
• Reversal of the direction of reaction. • Number of members is not drastically reduced.
Initial OptimalR1 -137.7(N) 200.0(N)
(Real scale)
Displacement (stiffness) constraint
• : lower bound (negative) for the x-directional displacement U1 (< 0) of support 1.
1U
Displacement (stiffness) constraint• Bending stiffness is proportional to cubic power of height
• Tight displacement bound
• Increase of height → Small number of members• Bound of displacement is used as an artificial parameter for
controlling the number of members in optimal topology
-0.1 -0.01V 1.678×104 6.759×104
(scaled by 1/5)
Penalization for thin members• Penalize stiffness of thin members (as SIMP method)
– Increase p artificially to 6.
• Increase of p = Reduction of absolute value ofdisplacement bound
12
pbhI
Summary of two-stage approach1. Solve optimization problem with displacement constraint.
2. Solve optimization problem with stress constraint for the optimal topology above.
3. Discretize optimal solution to shorter members.Optimizing again with Y-coordinates of nodes as design variables.
(Real scale)
Optimization result
• A shape that has increasing clamping force with increasing tensile force.
Verification by FE-analysis
Undeformed shape
Deformed shape
Magnified by 10
Clamping member (Type 2)• Adjustment of tensile force is very difficult
– holes are assigned at predetermined locations.
• Temporary supports for obtaining reaction force and tensioning tools.
Temporal supports
Tensioning by tool
holes
Frame Model (Type 2)
• Adjust tensile force by applying vertical force through a bolt.
• Frame model (Type 2)
Tensile force
Force from a bolt
Frame Model (Type 2)1. Apply load P2 (bolt force) at node 2. 2. Fix vertical displacement at node 2.3. Apply horizontal load P1 (membrane tension) at node 1.
• ≦0 : Displacement of node 1 against P1. • ≧0 : Displacement of node 1 against P2.
(1)1U(2)1U
P1: Tensile force
P2: Force from a bolt
Displacement constraint• Optimal solution with sufficiently small number of members.
Optimal (scaled by 1/10)
=-0.01 (Disp. of node 1 against P1)
= 0.1 (Disp. of node 1 against P2)
Stress constraint
• Optimization under stress constraints after subdivision of members.
• Y-coordinates of nodes are also design variables.
(Real scale)
P1
P2
Verification by FE-analysis
Undeformed shape
Deformed shape under P1
Deformed shape under P2
Magnified by 10
Anchoring device (Type 3) Clamp the membrane without external load
utilizing snapthrough and contact. • Maximize downward reaction R4 of node 4 at the
final state. • Constraint : horizontal displacement u2 of node 2.
P
Optimization result
initial optimalReaction Force R4
at node 4 -11.15 -2.63
Horizontal displacement u2at node 2
-2.11(infeasible)
1.00(feasible)
Optimal shape
Deformed state
Optimization result
• Frame is stable by contact to support 3.• Tensile force can be adjusted through
modification of displacement of node 4.
Optimal shape Deformed state
Conclusion
• Optimization of cross-sectional shape of for anchoring device (clamping member) of membrane structures modeled as a frame.
• Reduce number of members by relaxing the stress constraints and assigning displacement constraint.
• Penalization of stiffness of a member with small height.
• Pull the membrane by applying vertical force through a bolt.
• Clamp the membrane without external loadutilizing snapthrough and contact.