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.