Controlling interstory drift ratio profiles via topology optimization strategies

doi:10.1007/s11709-022-0892-3

Front. Struct. Civ. Eng.

2023, Vol. 17

Issue (2) : 165-178 https://doi.org/10.1007/s11709-022-0892-3

RESEARCH ARTICLE

Controlling interstory drift ratio profiles via topology optimization strategies

Wenjun GAO^1,²(

), Xilin LU^1,²

¹. Department of Disaster Mitigation for Structures, College of Civil Engineering, Tongji University, Shanghai 200092, China
². State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University, Shanghai 200092, China

Download: PDF(11973 KB) HTML
Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks

Abstract

An approach to control the profiles of interstory drift ratios along the height of building structures via topology optimization is proposed herein. The theoretical foundation of the proposed approach involves solving a min–max optimization problem to suppress the maximum interstory drift ratio among all stories. Two formulations are suggested: one inherits the bound formulation and the other utilizes a p-norm function to aggregate all individual interstory drift ratios. The proposed methodology can shape the interstory drift ratio profiles into inverted triangular or quadratic patterns because it realizes profile control using a group of shape weight coefficients. The proposed formulations are validated via a series of numerical examples. The disparity between the two formulations is clear. The optimization results show the optimal structural features for controlling the interstory drift ratios under different requirements.

Keywords interstory drift ratio aggregation function bound formulation min–max problem topology optimization

Corresponding Author(s): Wenjun GAO

Just Accepted Date: 25 November 2022 Online First Date: 16 January 2023 Issue Date: 03 April 2023

Cite this article:

Wenjun GAO,Xilin LU. Controlling interstory drift ratio profiles via topology optimization strategies[J]. Front. Struct. Civ. Eng., 2023, 17(2): 165-178.

URL:

https://academic.hep.com.cn/fsce/EN/10.1007/s11709-022-0892-3
https://academic.hep.com.cn/fsce/EN/Y2023/V17/I2/165

Fig.1 Numerical model of the plane frame structure with a design domain for topological design. (a) Elevation view; (b) load distribution.

Tab.1 Shape information of the columns and beams used in the testing frame

Tab.2 Geometry information of the beam and column shapes

Fig.2 Projected physical density fields of optimization results with flexible diaphragms under uniform profile control. (a) p =1; (b) p = 2; (c) p = 4; (d) p = 6; (e) p = 8; (f) p = 10; (g) p = 12; (h) p = 14; (i) p = 16; (j) p = 18; (k) p = 20; (l) p = 22; (m) p = 24; (n) bound formulation.

Fig.3 Projected physical density fields of optimization results with rigid diaphragms under uniform profile control. (a) p =1; (b) p = 2; (c) p = 4; (d) p = 6; (e) p = 8; (f) p = 10; (g) p = 12; (h) p = 14; (i) p = 16; (j) p = 18; (k) p = 20; (l) p = 22; (m) p = 24; (n) bound formulation.

Fig.4 Convergence histories of interstory drift ratio profiles under uniform profile control (f. d.: flexible diaphragm; r. d.: rigid diaphragm; B. F.: bound formulation). (a) f. d., p = 1; (b) f. d., p = 4; (c) f. d., p = 10; (d) f. d., p = 24; (e) f. d., B. F.; (f) r. d., p = 1; (g) r. d., p = 4; (h) r. d., p = 10; (i) r. d., p = 24; (j) r. d., B. F..

Fig.5 Iteration histories of maximum interstory drift ratio (B. F.: bound formulation). (a) Flexible diaphragm model; (b) rigid diaphragm model.

Tab.3 Converged interstory drift ratios generated by bound formulation under uniform profile control (units: × 10^?3)

Fig.6 Comparison of converged interstory drift ratios under uniform profile control (B. F.: bound formulation). (a) Flexible diaphragm model; (b) rigid diaphragm model.

Fig.7 Maximum interstory drift ratios achieved under different aggregation parameters (Agg. F.: aggregation formulation; B. F.: bound formulation). (a) Flexible diaphragm model; (b) rigid diaphragm model.

Tab.4 Shape weight coefficients for inverted triangular and quadratic distributions

Fig.8 Projected physical density fields of optimization results under inverted triangular and quadratic profile controls (f. d.: flexible diaphragm; r. d.: rigid diaphragm; T.: triangular; Q.: quadratic; B. F.: bound formulation). (a) f. d., T., p = 24; (b) f. d., T., B. F.; (c) f. d., Q., p = 24; (d) f. d., Q., B. F. (e) r. d., T., p = 24; (f) r. d., T., B. F.; (g) r. d., Q., p = 24; (h) r. d., Q., B. F..

Fig.9 Convergence histories of interstory drift profiles under inverted triangular and quadratic profile control (f. d.: flexible diaphragm; r. d.: rigid diaphragm; T.: triangular; Q.: quadratic; B. F.: bound formulation). (a) f. d., T., p = 24; (b) f. d., T., B. F.; (c) f. d., Q., p = 24; (d) f. d., Q., B. F.; (e) r. d., T., p = 24; (f) r. d., T., B. F.; (g) r. d., Q., p = 24; (h) r. d., Q., B. F..

Fig.10 Interstory drift profiles under triangular and quadratic profile control (f. d.: flexible diaphragm; r. d.: rigid diaphragm; T.: triangular; Q.: quadratic; B. F.: bound formulation; Agg.: aggregation formulation). (a) f. d., T.; (b) f. d., Q.; (c) r. d., T.; (d) r. d., Q.

Tab.5 Comparison between two diaphragm models

Fig.11 Re-evaluation of optimized results using both flexible and rigid diaphragm models: (a) and (b) are based on the aggregation formulation; (c) and (d) are based on the bound formulation.

1	GB50011-2010. Code for Seismic Design of Buildings. Beijing: Ministry of Construction of China, 2010 (in Chinese)
2	ATC. Guidelines for SEISMIC PERFORMANCE ASSessment of Buildings. Redwood City (CA): Applied Technology Council, 2007
3	SNZ. Concrete Structures Standard. Wellington: Standards New Zealand, 2004
4	7-02 ASCE/SEI. Minimum Design Loads for Buildings and Other Structures. Reston, VA: American Society of Civil Engineers, 2016
5	J P Moehle, S A Mahin. Observations on the behavior of reinforced concrete buildings during earthquakes. American Concrete Institute Special Publication, Earthquake-Resistant Concrete Structures—Inelastic Response and Design, 1991, 127: 67–90
6	R L Mayes. Interstory drift design and damage control issues. Structural Design of Tall Buildings, 1995, 4(1): 15–25 https://doi.org/10.1002/tal.4320040104
7	L G Griffis. Serviceability limit states under wind load. Engineering Journal AISC, 1993, 30(1): 1–16
8	T PaulayM J N Priestley. Seismic Design of reinforced Concrete and Masonry Buildings. New York: John Wiley and Sons, 1992
9	N Kirac, M Dogan, H Ozbasaran. Failure of weak-storey during earthquakes. Engineering Failure Analysis, 2011, 18(2): 572–581 https://doi.org/10.1016/j.engfailanal.2010.09.021
10	J M Jara, E J Hernández, B A Olmos, G Martínez. Building damages during the September 19, 2017 earthquake in Mexico City and seismic retrofitting of existing first soft-story buildings. Engineering Structures, 2020, 209: 109977 https://doi.org/10.1016/j.engstruct.2019.109977
11	H Agha BeigiT J SullivanG M CalviC Christopoulos. Controlled soft storey mechanism as a seismic protection system. In: The 10th International Conference on Urban Earthquake Engineering. Tokyo: Tokyo Institute of Technology, 2013
12	J W Lai, S A Mahin. Strongback system: A way to reduce damage concentration in steel-braced frames. Journal of Structural Engineering, 2015, 141(9): 04014223 https://doi.org/10.1061/(ASCE)ST.1943-541X.0001198
13	B Alavi, H Krawinkler. Strengthening of moment-resisting frame structures against near-fault ground motion effects. Earthquake Engineering & Structural Dynamics, 2004, 33(6): 707–720 https://doi.org/10.1002/eqe.370
14	H Moghaddam, I Hajirasouliha, A Doostan. Optimum seismic design of concentrically braced steel frames: Concepts and design procedures. Journal of Constructional Steel Research, 2005, 61(2): 151–166 https://doi.org/10.1016/j.jcsr.2004.08.002
15	N D Lagaros, M Papadrakakis. Seismic design of RC structures: A critical assessment in the framework of multi-objective optimization. Earthquake Engineering & Structural Dynamics, 2007, 36(12): 1623–1639 https://doi.org/10.1002/eqe.707
16	S Farahmand-Tabar, P Ashtari. Simultaneous size and topology optimization of 3D outrigger-braced tall buildings with inclined belt truss using genetic algorithm. Structural Design of Tall and Special Buildings, 2020, 29(13): e1776 https://doi.org/10.1002/tal.1776
17	C K Kim, H S Kim, J S Hwang, S M Hong. Stiffness-based optimal design of tall steel frameworks subject to lateral loading. Structural Optimization, 1998, 15(3−4): 180–186 https://doi.org/10.1007/BF01203529
18	C M Chan, X K Zou. Elastic and inelastic drift performance optimization for reinforced concrete buildings under earthquake loads. Earthquake Engineering & Structural Dynamics, 2004, 33(8): 929–950 https://doi.org/10.1002/eqe.385
19	X K Zou, C M Chan. An optimal resizing technique for seismic drift design of concrete buildings subjected to response spectrum and time history loadings. Computers & Structures, 2005, 83(19−20): 1689–1704 https://doi.org/10.1016/j.compstruc.2004.10.002
20	V Tomei, M Imbimbo, E Mele. Optimization of structural patterns for tall buildings: the case of diagrid. Engineering Structures, 2018, 171: 280–197 https://doi.org/10.1016/j.engstruct.2018.05.043
21	T Vu-Huu, P Phung-Van, H Nguyen-Xuan, M Abdel Wahab. A polytree-based adaptive polygonal finite element method for topology optimization of fluid-submerged breakwater interaction. Computers & Mathematics with Applications (Oxford, England), 2018, 76(5): 1198–1218 https://doi.org/10.1016/j.camwa.2018.06.008
22	H Ghasemi, H S Park, T Rabczuk. A multi-material level-set based topology optimization of flexoelectric composites. Computer Methods in Applied Mechanics and Engineering, 2018, 332: 47–62 https://doi.org/10.1016/j.cma.2017.12.005
23	H Ghasemi, H S Park, N Alajlan, T Rabczuk. A computational framework for design and optimization of flexoelectric materials. International Journal of Computational Methods, 2020, 17(1): 1850097 https://doi.org/10.1142/S0219876218500974
24	K M Hamdia, H Ghasemi, X Y Zhuang, T Rabczuk. Multilevel Monte Carlo method for topology optimization of flexoelectric composites with uncertain material properties. Engineering Analysis with Boundary Elements, 2022, 134: 412–418 https://doi.org/10.1016/j.enganabound.2021.10.008
25	J Zhang, Q Li. Identification of modal parameters of a 600-m-high skyscraper from field vibration tests. Earthquake Engineering & Structural Dynamics, 2019, 48(15): 1678–1698 https://doi.org/10.1002/eqe.3219
26	L L Beghini, A Beghini, N Katz, W F Baker, G H Paulino. Connecting architecture and engineering through structural topology optimization. Engineering Structures, 2014, 59: 716–726 https://doi.org/10.1016/j.engstruct.2013.10.032
27	J Xu, B F Jr Spencer, X Lu. Performance-based optimization of nonlinear structures subject to stochastic dynamic loading. Engineering Structures, 2017, 134: 334–345 https://doi.org/10.1016/j.engstruct.2016.12.051
28	S Wu, H He, S Cheng, Y Chen. Story stiffness optimization of frame subjected to earthquake under uniform displacement criterion. Structural and Multidisciplinary Optimization, 2021, 63(3): 1533–1546 https://doi.org/10.1007/s00158-020-02761-7
29	F Gomez, B F Jr Spencer, J Carrion. Topology optimization of buildings subjected to stochastic base excitation. Engineering Structures, 2020, 223: 111111 https://doi.org/10.1016/j.engstruct.2020.111111
30	G KreisselmeierR Steinhauser. Systematic control design by optimizing a vector performance index. In: International Federation of Active Controls Symposium on Computer-Aided Design of Control Systems. Zurich: Pergamon Press Ltd., 1979
31	X Lu, Y Cui, J Liu, W Gao. Shaking table test and numerical simulation of a 1/2-scale self-centering reinforced concrete frame. Earthquake Engineering & Structural Dynamics, 2015, 44(12): 1899–1917 https://doi.org/10.1002/eqe.2560
32	W Gao, X Lu. Modelling unbonded prestressing tendons in self-centering connections through improved sliding cable elements. Engineering Structures, 2019, 180: 809–828 https://doi.org/10.1016/j.engstruct.2018.11.078
33	M P Bendsøe, N Kikuchi. Generating optimal topologies in structural design using a homogenization method. Computer Methods in Applied Mechanics and Engineering, 1988, 71(2): 197–224 https://doi.org/10.1016/0045-7825(88)90086-2
34	M P BendsøeO Sigmund. Topology Optimization––Theory, Methods and Applications. Berlin: Springer, 2003
35	F Wang, B S Lazarov, O Sigmund. On projection methods, convergence and robust formulations in topology optimization. Structural and Multidisciplinary Optimization, 2011, 43(6): 767–784 https://doi.org/10.1007/s00158-010-0602-y
36	W Gao, F Wang, O Sigmund. Systematic design of high-Q prestressed micro membrane resonators. Computer Methods in Applied Mechanics and Engineering, 2020, 361: 112692 https://doi.org/10.1016/j.cma.2019.112692
37	A K Chopra. Dynamics of Structures: Theory and Applications to Earthquake Engineering. New Jersey: Prentice-Hall, 1995
38	M P Bendsøe, N Olhoff, J E Taylor. A variational formulation for multicriteria structural optimization. Journal of Structural Mechanics, 1983, 11(4): 523–544 https://doi.org/10.1080/03601218308907456
39	K A James, J S Hansen, J R R A Martins. Structural topology optimization for multiple load cases using a dynamic aggregation technique. Engineering Optimization, 2009, 41(12): 1103–1118 https://doi.org/10.1080/03052150902926827
40	C Le, J Norato, T Bruns, C Ha, D Tortorelli. Stress-based topology optimization for continua. Structural and Multidisciplinary Optimization, 2010, 41(4): 605–620 https://doi.org/10.1007/s00158-009-0440-y
41	W Gao, X Lu, S Wang. Seismic topology optimization based on spectral approaches. Journal of Building Engineering, 2022, 47: 103781 https://doi.org/10.1016/j.jobe.2021.103781
42	N M K Poon, J R R A Martins. An adaptive approach to constraint aggregation using adjoint sensitivity analysis. Structural and Multidisciplinary Optimization, 2007, 34(1): 61–73 https://doi.org/10.1007/s00158-006-0061-7
43	O A Sigmund. 99 line topology optimization code written in Matlab. Structural and Multidisciplinary Optimization, 2001, 21(2): 120–127 https://doi.org/10.1007/s001580050176
44	E Andreassen, A Clausen, M Schevenels, B S Lazarov, O Sigmund. Efficient topology optimization in Matlab using 88 lines of code. Structural and Multidisciplinary Optimization, 2011, 43(1): 1–16 https://doi.org/10.1007/s00158-010-0594-7
45	M Stolpe, K Svanberg. An alternative interpolation scheme for minimum compliance topology optimization. Structural and Multidisciplinary Optimization, 2001, 22(2): 116–124 https://doi.org/10.1007/s001580100129
46	T T Feng, J S Arora, E J Haug. Optimal structural design under dynamic loads. International Journal for Numerical Methods in Engineering, 1977, 11(1): 39–52 https://doi.org/10.1002/nme.1620110106
47	J S Arora, E J Haug. Methods of design sensitivity analysis in structural optimization. AIAA Journal, 1979, 17(9): 970–974 https://doi.org/10.2514/3.61260
48	A R Mijar, C C Swan, J S Arora, I Kosaka. Continuum topology optimization for concept design of frame bracing systems. Journal of Structural Engineering, 1998, 124(5): 541–550 https://doi.org/10.1061/(ASCE)0733-9445(1998)124:5(541
49	L L Stromberg, A Beghini, W F Baker, G H Paulino. Topology optimization for braced frames: combining continuum and beam/column elements. Engineering Structures, 2012, 37: 106–124 https://doi.org/10.1016/j.engstruct.2011.12.034
50	Y Zhou, C Zhang, X Lu. An inter-story drift-based parameter analysis of the optimal location of outriggers in tall buildings. Structural Design of Tall and Special Buildings, 2016, 25(5): 215–231 https://doi.org/10.1002/tal.1236
51	K Svanberg. The method of moving asymptotes—A new method for structural optimization. International Journal for Numerical Methods in Engineering, 1987, 24(2): 359–373 https://doi.org/10.1002/nme.1620240207
52	K Svanberg. A class of globally convergent optimization methods based on conservative convex separable approximations. SIAM Journal on Optimization, 2002, 12(2): 555–573 https://doi.org/10.1137/S1052623499362822
53	S Allahdadian, B Boroomand, A R Barekatein. Towards optimal design of bracing system of multi-story structures under harmonic base excitation through a topology optimization scheme. Finite Elements in Analysis and Design, 2012, 61: 60–74 https://doi.org/10.1016/j.finel.2012.06.010
54	S Allahdadian, B Boroomand. Topology optimization of planar frames under seismic loads induced by actual and artificial earthquake records. Engineering Structures, 2016, 115: 140–154 https://doi.org/10.1016/j.engstruct.2016.02.022

[1]	Aydin HASSANZADEH, Saber MORADI. Topology optimization and seismic collapse assessment of shape memory alloy (SMA)-braced frames: Effectiveness of Fe-based SMAs[J]. Front. Struct. Civ. Eng., 2022, 16(3): 281-301.

Viewed

Full text

Abstract

Cited

Shared

Discussed