TOPOLOGY OPTIMIZATION OF ROCKING BRACED FRAMES FOR ...€¦ · TOPOLOGY OPTIMIZATION OF ROCKING...
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3. Topology Optimization Results The methodology is applied to a single rocking braced frame to determine the location of the steel bracing members in the elastic rocking spine. Figure 3 shows the (a) design domain, (b) resulting optimized topology, (c) the Von Mises stress distribution along with the convergence history. The decoupled optimization results for each modal load case demonstrate the influence of higher modes, especially the second mode, on the distribution of bracing along the height of the spine. The story shear forces correlate with a high density of bracing and vice-versa (Figure 4). 4. Conclusion • A methodology to extend topology optimization for nonlinear earthquake response is presented using surrogate modal load cases • The method is applied to rocking braced frames and the numerical results demonstrate important conceptual design considerations, especially the contribution of higher modes 1. Introduction Topology optimization is a computational tool that enables the development of efficient and elegant structures. Application of topology optimization for earthquake resistant design is challenging due to the complexity of nonlinear dynamic response under random ground motions. Since inelasticity is concentrated at energy-dissipating elements, modified modal analyses can approximately capture the nonlinear earthquake response using equivalent linear structural properties. Based on these simplified analyses, topology optimization can be extended for nonlinear seismic response of structural systems, such as rocking braced frames, by taking into account the nonlinear response of the rocking base and the frequency characteristic of the design earthquake. 2. Methodology The extended topology optimization problem, using surrogate modes, for nonlinear earthquake response is detailed below. The load cases for the rocking braced frame are shown in Figure 2. TOPOLOGY OPTIMIZATION OF ROCKING BRACED FRAMES FOR NONLINEAR EARTHQUAKE RESPONSE Amory Martin ([email protected]), Prof. Gregory Deierlein (a) (b) (c) F S (T 1 )/R 1 F PT F S (T 2 ) F S (T 3 ) F PT k ED k PT Single rocking frame Steel bracing Post-tensioning cables Energy-dissipating fuses Uplifting base mininimize & ' ( =* +,- . / + 0 + ' 1 + (1) subject to < = >?@ − B @≤0 0≤> E ≤ 1, ∀H ∈ J with 1 + =M EN O- 0 + , for R = 1, … , T 0 + = Γ + VW + XY Z + [ + for R = 1, … , T M EN −\ + ] VW ^ =_, for R = 1, … , T W ^ ` VW a =b +c , for R, d = 1, … , T Figure 1. Rocking braced frame Figure 2. Multiple load cases for topology optimization References 1. Martin A, Deierlein G. Topology Optimization of Elastic Spines in Rocking Braced Frames. Proceedings of the 11th National Conference on Earthquake Engineering, EERI, LA CA (2018) 2. Eatherton M, Ma X, Krawinkler H, Mar D, Billington S, Hajjar J, Deierlein G. Design concepts for controlled rocking of self-centering steel-braced frames. Journal of Structural Engineering (2014) 3. Wiebe L, Christopoulos C. Performance-based seismic design of controlled rocking steel braced frames. II: Design of capacity-protected elements. Journal of Structural Engineering (2013) 4. Sullivan TJ, Priestley MJN, Calvi GM. Estimating the higher-mode response of ductile structures. Journal of Earthquake Engineering (2008) 5. Talischi C, Paulino, G.H., Pereira A. and Menezes I.F. PolyTop: a Matlab implementation of a general topology optimization framework using unstructured polygonal finite element meshes. Structural and Multidisciplinary Optimization, 45(3), pp.329-357. (2012) 6. Stromberg LL, Beghini A, Baker WF, Paulino GH. Topology optimization for braced frames: combining continuum and beam/column elements. Engineering Structures (2012) Figure 3. Topology optimization results Figure 4. Topology optimization modal results and modal story shear forces 0 100 200 300 400 500 600 700 800 900 Iteration 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 Total compliance C T #10 4 Penalization step Convergence ? (a) (b) (c) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Transcript of TOPOLOGY OPTIMIZATION OF ROCKING BRACED FRAMES FOR ...€¦ · TOPOLOGY OPTIMIZATION OF ROCKING...
3. Topology Optimization Results
The methodology is applied to a single rocking braced frame to determinethe location of the steel bracing members in the elastic rocking spine.Figure 3 shows the (a) design domain, (b) resulting optimized topology, (c)the Von Mises stress distribution along with the convergence history.
The decoupled optimization results for each modal load case demonstratethe influence of higher modes, especially the second mode, on thedistribution of bracing along the height of the spine. The story shear forcescorrelate with a high density of bracing and vice-versa (Figure 4).
4. Conclusion• A methodology to extend topology optimization for nonlinear earthquake
response is presented using surrogate modal load cases• The method is applied to rocking braced frames and the numerical
results demonstrate important conceptual design considerations,especially the contribution of higher modes
(a) (b) (c)
1. Introduction
Topology optimization is a computational tool that enables the developmentof efficient and elegant structures. Application of topology optimization forearthquake resistant design is challenging due to the complexity of nonlineardynamic response under random ground motions. Since inelasticity isconcentrated at energy-dissipating elements, modified modal analyses canapproximately capture the nonlinear earthquake response using equivalentlinear structural properties. Based on these simplified analyses, topologyoptimization can be extended for nonlinear seismic response of structuralsystems, such as rocking braced frames, by taking into account thenonlinear response of the rocking base and the frequency characteristic ofthe design earthquake.
2. Methodology
The extended topology optimization problem, using surrogate modes, fornonlinear earthquake response is detailed below. The load cases for therocking braced frame are shown in Figure 2.
TOPOLOGY OPTIMIZATION OF ROCKING BRACED FRAMESFOR NONLINEAR EARTHQUAKE RESPONSE
Amory Martin ([email protected]), Prof. Gregory Deierlein
(a) (b) (c)
FS(T1)/R1
FPT
FS(T2)
FS(T3)
FPT
kED
kPT
Single rocking frame
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Post-tensioning cables
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mininimize &' ( =*
+,-
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/+0+'1+ (1)
subject to <=
>?@ − B@ ≤ 0
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with 1+ = MENO-0+ , for R = 1,… ,T
0+ =Γ+VW+XY Z+
[+for R = 1,… ,T
MEN − \+]V W^ = _ , for R = 1,… ,T
W^`VWa = b+c, for R, d = 1,… ,T
Figure 1. Rocking braced frame
Figure 2. Multiple load cases for topology optimization
References1. Martin A, Deierlein G. Topology Optimization of Elastic Spines in Rocking Braced
Frames. Proceedings of the 11th National Conference on Earthquake Engineering, EERI, LA CA (2018)2. Eatherton M, Ma X, Krawinkler H, Mar D, Billington S, Hajjar J, Deierlein G. Design concepts for
controlled rocking of self-centering steel-braced frames. Journal of Structural Engineering (2014)3. Wiebe L, Christopoulos C. Performance-based seismic design of controlled rocking steel braced
frames. II: Design of capacity-protected elements. Journal of Structural Engineering (2013)4. Sullivan TJ, Priestley MJN, Calvi GM. Estimating the higher-mode response of ductile
structures. Journal of Earthquake Engineering (2008)5. Talischi C, Paulino, G.H., Pereira A. and Menezes I.F. PolyTop: a Matlab implementation of a general
topology optimization framework using unstructured polygonal finite element meshes. Structuraland Multidisciplinary Optimization, 45(3), pp.329-357. (2012)
6. Stromberg LL, Beghini A, Baker WF, Paulino GH. Topology optimization for braced frames: combiningcontinuum and beam/column elements. Engineering Structures (2012)
Figure 3. Topology optimization results
Figure 4. Topology optimization modal results and modal story shear forces
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