Sequential configuration optimization of frame …Sequential configuration optimization of frame...

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.