Design of Structural Braced Frames Using ... - Glaucio Paulino

Design of Structural Braced Frames Using Topology Optimization Lauren L. Stromberg1, Alessandro Beghini2, William F. Baker2, Glaucio H. Paulino1 1Department

of Civil & Environmental Engineering, University of Illinois at Urbana-Champaign

2Skidmore,

Owings & Merrill, LLP

March 30, 2012 Department of Civil & Environmental Engineering University of Illinois at Urbana-Champaign

This presentation focuses on the use of topology optimization with combined elements in the building design industry P/2

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Motivation for new approach

Analytical aspects Combining elements for structural braced frames

Stromberg, Beghini, Baker, Paulino

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Architecture without evident engineering rationale is wasteful of resources and difficult to realize

Railway

Olympic Stadium

Convention Center

http://en.wikipedia.org/wiki /Hadid-Afragola

http://en.wikipedia.org/wiki/Beijing_National_Stadium

http://images.businessweek.com

Stromberg, Beghini, Baker, Paulino

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Buildings can be designed using the lateral bracing for both engineering and architectural criteria

http://en.wikipedia.org/wiki/John_Hancock_Center

http://en.wikipedia.org/wiki/Broadgate_Tower

http://en.wikipedia.org/wiki/Bank_of_China

Stromberg, L.L., Beghini, A., Baker, W.F., Paulino, G.H. “Topology optimization for braced frames: Combining continuum and beam/column elements.” Engineering Structures 2012, 37:106-124. Stromberg, Beghini, Baker, Paulino

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Topology optimization gives practical designs that satisfy principles from both structural engineering and architecture

*Images courtesy of SOM

Design for the “bridges” spanning between towers for the Zendai competition was a collaborative effort between UIUC and SOM

Pallasmaa, J., ed. SOM Journal 7. Ostfildern: Hatje Cantz Verlag, 2012. Stromberg, Beghini, Baker, Paulino

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Resulting designs resemble patterns from nature

Photography.nationalgeographic.com

Design for the “bridges” spanning between towers for the Zendai competition was a collaborative effort between UIUC and SOM

Pallasmaa, J., ed. SOM Journal 7. Ostfildern: Hatje Cantz Verlag, 2012. Stromberg, Beghini, Baker, Paulino

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Current topology optimization techniques demonstrate limitations in design P/2

Quadrilateral (Q4) elements

Beam elements

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Quadrilateral (Q4) elements P

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Continuum approach Stromberg, Beghini, Baker, Paulino

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Proposed Design Process 1.

Size vertical line elements according to gravity load combinations using technique of Baker (1992)1

2.

Run topology optimization on the continuum elements for lateral load combinations (accounting for wind and seismic loads)

3.

Identify optimal bracing layout to create frame model

4.

Optimize final member sizes using virtual work methodology1

5.

Iterate (if necessary)

1Baker,

W.F. “Energy-based design of lateral systems” Struct Eng Int 1992, 2:99-102.

Stromberg, L.L., Beghini, A., Baker, W.F., Paulino, G.H. “Topology optimization for braced frames: Combining continuum and beam/column elements.” Engineering Structures 2012, 37:106-124. Stromberg, Beghini, Baker, Paulino

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Optimal design results in a state of constant stress In terms of displacements 𝒖𝒊 at each point of load application 𝑷𝒊 the compliance can be expressed as 𝑾𝒆𝒙𝒕 =

𝑵𝟐𝒋 𝑳𝒋

𝑷𝒊 𝒖𝒊 = 𝒊

𝒋

𝑬𝑨𝒋

= 𝑾𝒊𝒏𝒕

By introducing the Lagrangian multiplier constraint on the volume, 𝑵𝟐𝒋 𝑳𝒋

𝑾𝒆𝒙𝒕 = 𝒋

𝑬𝑨𝒋

+𝝀

𝑨𝒋 𝑳𝒋 − 𝑽 𝒋

Differentiating with respect to the areas 𝑨𝒊 and solving for the Lagrangian multiplier, 𝑵𝒊 𝝀= 𝑨𝒊

Stromberg, Beghini, Baker, Paulino

𝟐

𝟏 𝝈𝟐 = = 𝒄𝒐𝒏𝒔𝒕 𝑬 𝑬

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Optimal Single Module Bracing The optimal bracing geometry for a single module is considered: 𝐻−𝑧 𝐵2 + 𝑧 2 − 𝐵2 + 𝐻 − 𝑧 𝑓1 = , 𝑓2 = , 𝑓3 = 𝐵 𝐵 𝐵 

1

2

symmetry 3

H  z  load moment arm

2

z



1

H

fi 

f 

L 1  i cos  B

H z B

B

Stromberg, Beghini, Baker, Paulino

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Optimal Single Module Bracing Assuming each member to have a constant stress, Δ= 𝑖

𝑓𝑖 𝐹𝑖 𝐿𝑖 𝜎𝐵 = 𝐸𝐴𝑖 𝐸

𝑖

𝑓𝑖 𝐿𝑖 𝐵

The tip deflection of the frame is minimal when 𝜕Δ 𝜎𝐵 𝜕 = 𝜕𝑧 𝐸 𝜕𝑧

𝑖

𝑓𝑖 𝐿𝑖 =0 𝐵

𝜎𝐵 𝜕 𝐻−𝑧 𝐵2 + 𝑧 2 𝐵2 + 𝐻 − 𝑧 = 𝐻 + + 𝐸 𝜕𝑧 𝐵2 𝐵2 𝐵2

2

=0

Thus, the brace work point height is optimal at 𝑧=

Stromberg, Beghini, Baker, Paulino

3 𝐻 4

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These rules can be generalized for multiple modules Generalizing the previous equations for 𝑵 modules, Δ=

𝜎𝐵 𝐸

𝜎𝐵 = 𝐸 𝜎𝐵 = 𝐸

𝑓𝑖 𝐿𝑖 𝐵

𝑖

𝑖 𝑁

𝐿2𝑖 𝐵2

+ 𝑏𝑟𝑎𝑐𝑒𝑠

𝑧2𝑛 − 𝑧2𝑛−2 𝑛=1

𝑗

𝐻 − 𝑧𝑗 𝐿𝑗 𝐵2

𝑐𝑜𝑙𝑢𝑚𝑛𝑠

𝐻 − 𝑧𝑛−1 𝐵2 + 𝑧2𝑛−1 − 𝑧2𝑛−2 + 𝐵2 𝐵2

2

𝐵2 + 𝑧2𝑛−1 − 𝑧2𝑛−2 + 𝐵2

2

Differentiating, 𝜕 𝜕𝑧2𝑛−1 𝜕 𝜕𝑧2𝑛

𝐸Δ = 0 → −3𝑧2𝑛 + 4𝑧2𝑛−1 − 𝑧2𝑛−2 = 0 𝜎𝐵 𝐸Δ = 0 → −𝑧2𝑛+1 + 4𝑧2𝑛 − 3𝑧2𝑛−1 = 0 𝜎𝐵

Stromberg, L.L., Beghini, A., Baker, W.F., Paulino, G.H. “Topology optimization for braced frames: Combining continuum and beam/column elements.” Engineering Structures 2012, 37:106-124. Stromberg, Beghini, Baker, Paulino

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Optimal Multiple Modules Bracing Rewriting these equations 𝑧2𝑛−1 + 𝑧2𝑛+1 𝑧2𝑛+1 − 𝑧2𝑛−1 𝑧2𝑛 = − 2 4 𝑧2𝑛−2 + 𝑧2𝑛 𝑧2𝑛 − 𝑧2𝑛−2 𝑧2𝑛−1 = + 2 4

This leads to two conclusions: 1. The braced frame central work point is always at 75% of the module height. 2. The module heights are all equal, indicating that patterns are optimal.

Module n  1

z2n 2 z 2 n 1

Module n

z2n z 2 n 1 Module n  1

z2n2 z 2 n 3

Stromberg, L.L., Beghini, A., Baker, W.F., Paulino, G.H. “Topology optimization for braced frames: Combining continuum and beam/column elements.” Engineering Structures 2012, 37:106-124. Stromberg, Beghini, Baker, Paulino

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These results can be verified using Michell solutions

In 1904, Michell derived the mathematics behind structures of least volume, or optimal structures Hemp discusses the example of a cantilever beam subject to a point load with solutions formed by cycloids using the conditions set forth in Michell (1904).

Michell, A.G.M. “The Limits of Economy of Material in Frame-Structures.” Phil Mag 8(47):589-597 Hemp, W.M. Optimum 14 Structures. Oxford: Claredon Press, 1973. Stromberg, Beghini, Baker, Paulino

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These results can be verified using Michell solutions In Hemp (1973), an application of the Michell truss solution is given by the optimal shear bracing with the discretization of cycloids: F

 /3 H z H

 /3

 /6 2B

 /6

Optimal geometry given by angles – how does this correspond to the 75% rule? 15 2B Stromberg, Beghini, Baker, Paulino

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These results can be verified using Michell solutions  /3  /3

H

 6

H

z

h

z

 /6

h  H cos / 6  2B

2

 3 3   z  h cos / 6   H   H  4  2  Same results!

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Topology Optimization Framework Problem statement: Objective function Equilibrium constraint Volume constraint

Using Solid Isotropic Material with Penalization (SIMP) model,

Bendsoe, M.P., Sigmund, O. Topology Optimization: Theory, Methods and Applications. Springer, 2002. Stromberg, Beghini, Baker, Paulino

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The optimal bracing point is highly influenced by the stress levels

Higher areas than constant stress state give lower bracing points

A  A0

A  2A0

A  5A0

A  10A0

Area sized for constant stress Stromberg, L.L., Beghini, A., Baker, W.F., Paulino, G.H. “Topology optimization for braced frames: Combining continuum 18and beam/column elements.” Engineering Structures 2012, 37:106-124. Stromberg, Beghini, Baker, Paulino

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Topology optimization results give practical structures and verify Hemp’s solution P

No pattern constraints H

Each module has a 75% bracing point! Stresses are constant.

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Stromberg, L.L., Beghini, A., Baker, W.F., Paulino, G.H. “Topology optimization for braced frames: Combining continuum 19and beam/column elements.” Engineering Structures 2012, 37:106-124. Stromberg, Beghini, Baker, Paulino

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The combination of different element types is suitable for the design of lateral braced frames Beam elements

75% bracing rule

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Quadrilateral (Q4) Elements

Patterns P

Image courtesy of SOM

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The combination of different element types is suitable for the design of lateral braced frames

Images courtesy of SOM

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In summary, topology optimization with combined element types can and should be used for the practical design of buildings

Results show balance between engineering and architecture

Designs are innovative, aesthetically pleasing and structurally sound

Questions? Stromberg, Beghini, Baker, Paulino

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