Introduction and MotivationToward Group Optimization to Design Building Systems
Toward Group Optimization for the PracticalDesign of Building Systems
Lauren L. Stromberg, Alessandro Beghini, William F. Baker,Glaucio H. Paulino
July 26, 2011
Stromberg, Beghini, Baker, Paulino USNCCM-11 July 26, 2011 1/ 27
Introduction and MotivationToward Group Optimization to Design Building Systems
Architecture without evident engineering rationale
http://en.wikipedia.org/
wiki/File:Hadid-Afragola.jpghttp://en.wikipedia.org/wiki/Beijing_
National_Stadium
http:
//images.businessweek.com/
ss/06/03/italy/source/3.htm
Stromberg, Beghini, Baker, Paulino USNCCM-11 July 26, 2011 2/ 27
Introduction and MotivationToward Group Optimization to Design Building Systems
Combining Engineering and Architecture
*Images courtesy of SOM
Stromberg, Beghini, Baker, Paulino USNCCM-11 July 26, 2011 3/ 27
Introduction and MotivationToward Group Optimization to Design Building Systems
Combining Engineering and Architecture
photography.nationalgeographic.com *Images courtesy of SOM
Stromberg, Beghini, Baker, Paulino USNCCM-11 July 26, 2011 4/ 27
Introduction and MotivationToward Group Optimization to Design Building Systems
Background and motivation
Stromberg, Beghini, Baker, Paulino USNCCM-11 July 26, 2011 5/ 27
Introduction and MotivationToward Group Optimization to Design Building Systems
Combining Engineering and Architecture
www.en.wikipedia.org/wiki/John_
Hancock_Center
“The language of mathematics andrational engineering could not give form toarchitecture of substantive quality on itsown, no more than could ungroundedaesthetic inclination. Rather, by conjoiningcreative energies and different perspectives,better innovative and responsive designsolutions could be developed than eitherarchitect or engineer might conceive inisolation.” Khan [2004].
Stromberg, Beghini, Baker, Paulino USNCCM-11 July 26, 2011 6/ 27
Introduction and MotivationToward Group Optimization to Design Building Systems
Analytical AspectsPractical Examples
Toward group optimization for the practical design ofbuilding systems
Stromberg et al. [2011]
Stromberg, Beghini, Baker, Paulino USNCCM-11 July 26, 2011 7/ 27
Introduction and MotivationToward Group Optimization to Design Building Systems
Analytical AspectsPractical Examples
Overall design process
size vertical line elements (columns) according to gravity loadcombinations (accounting for dead, superimposed dead andlive loads) using technique in Baker [1992]
run topology optimization on the continuum elements forlateral load combinations (accounting for wind and seismicloads)
identify the optimal bracing layout based on results and createframe model
optimize the member sizes using the virtual work methodology
Stromberg, Beghini, Baker, Paulino USNCCM-11 July 26, 2011 8/ 27
Introduction and MotivationToward Group Optimization to Design Building Systems
Analytical AspectsPractical Examples
Numerical example of a bracing system
Stromberg, Beghini, Baker, Paulino USNCCM-11 July 26, 2011 9/ 27
Introduction and MotivationToward Group Optimization to Design Building Systems
Analytical AspectsPractical Examples
Optimal Braced Frames - Constancy of Stress
Constant stress in optimized frame verified in the continuum- theVon-Mises stresses are nearly constant within each optimizedmember
Stromberg, Beghini, Baker, Paulino USNCCM-11 July 26, 2011 10/ 27
Introduction and MotivationToward Group Optimization to Design Building Systems
Analytical AspectsPractical Examples
Optimal Braced Frames - Constancy of Stress
In terms of the displacements ui at each point of load application Pi , thecompliance can be expressed as:
Wext =∑i
Piui =∑j
N2j Lj
EAj= Wint
By introducing the Lagrangian multiplier constraint on the areas of themembers,
Wext =∑j
N2j Lj
EAj+ λ
∑j
AjLj − V
Differentiating with respect to the areas Ai and solving for theLagrangian multiplier λ
λ =
(Ni
Ai
)21
E=σ2
E= const
Stromberg, Beghini, Baker, Paulino USNCCM-11 July 26, 2011 11/ 27
Introduction and MotivationToward Group Optimization to Design Building Systems
Analytical AspectsPractical Examples
Optimal Single Module Bracing
Stromberg, Beghini, Baker, Paulino USNCCM-11 July 26, 2011 12/ 27
Introduction and MotivationToward Group Optimization to Design Building Systems
Analytical AspectsPractical Examples
Optimal Single Module Bracing
The optimal bracing geometry for a single module (even number ofdiagonals) is considered:
From the geometry, the forces in the members are
f1 = H−zB , f2 =
√B2+z2
B , and f3 =−√
B2+(H−z)2B
Stromberg, Beghini, Baker, Paulino USNCCM-11 July 26, 2011 13/ 27
Introduction and MotivationToward Group Optimization to Design Building Systems
Analytical AspectsPractical Examples
Optimal Single Module Bracing
Assuming each member to have a constant stress, σ = FiAi
,
∆ =∑i
fiFiLi
EAi=σB
E
∑i
fiLi
B
The tip deflection of the frame is minimal when
∂∆
∂z=σB
E
∂
∂z
(∑i
fiLi
B
)= 0
=σB
E
∂
∂z
(H
(H − z
B2
)+
B2 + z2
B2+
B2 + (H − z)2
B2
)= 0
Thus, the brace work point height for minimal deflection is
z =3
4H
Stromberg, Beghini, Baker, Paulino USNCCM-11 July 26, 2011 14/ 27
Introduction and MotivationToward Group Optimization to Design Building Systems
Analytical AspectsPractical Examples
Optimal multiple module bracing for a point load
Now, adding one more set of diagonals (odd number of diagonals),
the forces in the diagonals are fi = Li
B while the forces in the columns are
given by fi = (H−zi )B . The displacement at the top of the frame is
∆ =σB
E
∑i
fiLi
B
=σB
E
[∑i
(L2i
B2
)braces
+∑j
((H − zj) Lj
B2
)columns
]
Stromberg, Beghini, Baker, Paulino USNCCM-11 July 26, 2011 15/ 27
Introduction and MotivationToward Group Optimization to Design Building Systems
Analytical AspectsPractical Examples
Optimal multiple module bracing for a point load
The frame with minimal top displacement is defined by thefollowing equations:
∂∆
∂z1= 0⇒ −3z2 + 4z1 = 0
∂∆
∂z2= 0⇒ −H + 4z2 − 3z1 = 0
Therefore,
z1 =3
4z2
z2 =4
7H
Stromberg, Beghini, Baker, Paulino USNCCM-11 July 26, 2011 16/ 27
Introduction and MotivationToward Group Optimization to Design Building Systems
Analytical AspectsPractical Examples
Optimal multiple module bracing for a point load
Generalizing the previous equations,
E∆
σB=
N∑n=1
(z2n − z2n−1)2 + B2
B2+
(z2n−1 − z2n−2)2 + B2
B2
+(H − z2n−1)2
B2(z2n − z2n−2)
Stromberg, Beghini, Baker, Paulino USNCCM-11 July 26, 2011 17/ 27
Introduction and MotivationToward Group Optimization to Design Building Systems
Analytical AspectsPractical Examples
Optimal multiple module bracing for a point load
Here N is the total number of modules and it is assumed thatz2n−2 < z2n−1 < z2n. By differentiating with respect to the nodalelevations z2n (column work point) and z2n−1 (brace work point):
∂
∂z2n−1
(E ∆
σB
)= 0⇒ −3z2n + 4z2n−1 − z2n−2 = 0
∂
∂z2n
(E ∆
σB
)= 0⇒ −z2n+1 + 4z2n − 3z2n−1 = 0
These equations can be rewritten as follow:
z2n =z2n−1 + z2n+1
2− z2n+1 − z2n−1
4
z2n−1 =z2n−2 + z2n
2+
z2n − z2n−24
Stromberg, Beghini, Baker, Paulino USNCCM-11 July 26, 2011 18/ 27
Introduction and MotivationToward Group Optimization to Design Building Systems
Analytical AspectsPractical Examples
Optimal multiple module bracing for a point load
From the previous expressions, two important geometric features ofoptimal braced frames are inferred:
1 The braced frame central work point z2n−1 is always locatedat 75% of the module height.
2 The module heights are all equal indicating that patterns areoptimal
Stromberg, Beghini, Baker, Paulino USNCCM-11 July 26, 2011 19/ 27
Optimal multiple module bracing for a point load
Introduction and MotivationToward Group Optimization to Design Building Systems
Analytical AspectsPractical Examples
Optimal number of diagonals
By minimizing the volume of a frame,
V =∑i
AiLi =∑ Fi
σLi =
P
σ
∑i
fiLi =PB
σ
∑i
fiLi
B
or
V =PE
σ2∆
For the first case (m = 2),
V =PB
σ
∑i
fiLi
B
=PB
σ
(H2
4B2+
B2 + 9H2
16
B2+
H2
16 + B2
B2
)
=PB
σ
(2 +
7
8
(H
B
)2)
Stromberg, Beghini, Baker, Paulino USNCCM-11 July 26, 2011 21/ 27
Introduction and MotivationToward Group Optimization to Design Building Systems
Analytical AspectsPractical Examples
Optimal number of diagonals
Number of Diagonals, m Dimensionless Frame Volume, Vσ/(PB)
1 1 +(
HB
)2
2 2 + 78
(HB
)2
3 3 + 57
(HB
)2
4 4 + 1116
(HB
)2
m (odd) m +[
m+22m+1
] (HB
)2
m (even) m +[12+ 3
4m
] (HB
)2
Stromberg, Beghini, Baker, Paulino USNCCM-11 July 26, 2011 22/ 27
Introduction and MotivationToward Group Optimization to Design Building Systems
Analytical AspectsPractical Examples
Optimal number of diagonals
Dimensionless volume vs. height to width ratio, HB
m odd m even
Stromberg, Beghini, Baker, Paulino USNCCM-11 July 26, 2011 23/ 27
Optimal Braced Frames
Image courtesy of Skidmore, Owings & Merrill, LLP
Optimal Braced Frames
Images courtesy of Skidmore, Owings & Merrill, LLP
Optimal Braced Frames
Images courtesy of Skidmore, Owings & Merrill, LLP
Introduction and MotivationToward Group Optimization to Design Building Systems
Analytical AspectsPractical Examples
Conclusion
Topology optimization, including group optimization, can andshould be used for the practical design of buildings.
Thank you! Questions?
Stromberg, Beghini, Baker, Paulino USNCCM-11 July 26, 2011 27/ 27
Introduction and MotivationToward Group Optimization to Design Building Systems
Analytical AspectsPractical Examples
Conclusion
Topology optimization, including group optimization, can andshould be used for the practical design of buildings.
Thank you! Questions?
Stromberg, Beghini, Baker, Paulino USNCCM-11 July 26, 2011 27/ 27
References
References I
W F Baker. Energy-Based Design of Lateral Systems. Struct Engng Int,2:99–102, 1992.
Yasmin Sabina Khan. Engineering Architecture: The Vision of Fazlur R.Khan. W. W. Norton & Company, New York, 2004. ISBN 0393731073.
L L Stromberg, A Beghini, W F Baker, and G H Paulino. TopologyOptimization for Braced Frames : Combining Continuum and DiscreteElements. Engineering Structures, 2011.
Stromberg, Beghini, Baker, Paulino USNCCM-11 July 26, 2011 28/ 27
