Response surface analysis and optimization of controlled rocking steel braced frames


As an alternative to conventional seismic force resisting systems, controlled rocking steel braced frames (CRSBFs) can effectively eliminate permanent structural damage after earthquakes. Together with the rocking action in the braced frame, post-tensioning (PT) elements and fuse members are used to provide self-centering and energy dissipation, respectively. This study firstly aims to assess the influence of design parameters related to the fuse and PT materials on the seismic response of CRSBFs. These factors include the yield strength, initial stiffness, and strain hardening ratio of the fuse, the initial force and modulus of elasticity of the PT strands, and the gravity load on the rocking column. Additionally, different analysis cases are considered to include the effects of frame aspect ratio and earthquake intensity level. Nonlinear response history analyses are performed with factor combinations generated using a design of experiment methodology. The second goal of the study is focused on optimizing the seismic response of CRSBFs with respect to the influential factors identified in the sensitivity analyses. Using a response surface methodology and desirability approach, multiple-response optimization is applied to determine the design variable values needed to simultaneously minimize the maximum transient and residual roof drift ratio and peak floor acceleration. Among other results, it is found that the fuse strain hardening ratio and PT strand modulus of elasticity do not significantly influence the seismic response demands in CRSBFs. The results of the multi-response optimization demonstrate that the initial PT force is most useful for minimizing all three seismic response demand parameters.

This is a preview of subscription content, log in to check access.

Fig. 1

Adapted from Ma et al. (2011) and Eatherton et al. (2014b)

Fig. 2

Adapted from Ma et al. (2011)

Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13


  1. Akbas T, Sause R, Ricles JM et al (2017) Analytical and experimental lateral-load response of self-centering posttensioned CLT walls. J Struct Eng 143:4017019.

    Article  Google Scholar 

  2. American Society of Civil Engineers (ASCE) (2005) Seismic provisions for structural steel buildings. Illinois, Chicago

    Google Scholar 

  3. Azuhata T, Midorikawa M, Wada A (2003) Study on applicability of rocking structural systems to building structures. In: Agnes GS, Wang KW (eds) Smart structures and materials. Damping and Isolation. International Society for Optics and Photonics. Proceedings of SPIE, vol 5052. pp 287–296

  4. Chancellor N, Eatherton M, Roke D, Akbaş T (2014) Self-centering seismic lateral force resisting systems: high performance structures for the city of tomorrow. Buildings 4:520–548.

    Article  Google Scholar 

  5. Derringer G, Suich R (1980) Simultaneous optimization of several response variables. J Qual Technol 12:214–219.

    Article  Google Scholar 

  6. Dowden DM, Clayton PM, Li C-H et al (2015) Full-scale pseudodynamic testing of self-centering steel plate shear walls. J Struct Eng.

    Google Scholar 

  7. DX10 (2016) Design-Expert® Software Version 10

  8. Eatherton MR, Hajjar JF (2010) Large-scale cyclic and hybrid simulation testing and development of a controlled-rocking steel building system with replaceable fuses. Newmark Structural Engineering Laboratory Report, Newmark Structural Engineering Laboratory Report Series, Urbana, IL

  9. Eatherton MR, Hajjar JF (2011) Residual drifts of self-centering systems including effects of ambient building resistance. Earthq Spectra 27:719–744

    Article  Google Scholar 

  10. Eatherton MR, Hajjar JF (2014) Hybrid simulation testing of a self-centering rocking steel braced frame system. Earthq Eng Struct Dyn 43:1725–1742.

    Article  Google Scholar 

  11. Eatherton MR, Fahnestock LA, Miller DJ (2014a) Computational study of self-centering buckling-restrained braced frame seismic performance. Earthq Eng Struct Dyn 43:1897–1914.

    Article  Google Scholar 

  12. Eatherton MR, Ma X, Krawinkler H et al (2014b) Quasi-static cyclic behavior of controlled rocking steel frames. J Struct Eng 140:4014083.

    Article  Google Scholar 

  13. Eatherton MR, Ma X, Krawinkler H et al (2014c) Design concepts for controlled rocking of self-centering steel-braced frames. J Struct Eng 140:4014082.

    Article  Google Scholar 

  14. Erochko J, Christopoulos C, Tremblay R (2014) Design, testing, and detailed component modeling of a high-capacity self-centering energy-dissipative brace. J Struct Eng.

    Google Scholar 

  15. Fang C, Yam M, Ma H, Chung K (2015) Tests on superelastic Ni–Ti SMA bars under cyclic tension and direct-shear: towards practical recentring connections. Mater Struct 48:1013–1030.

    Article  Google Scholar 

  16. FEMA (Federal Emergency Management Agency) (2009) Quantification of building seismic performance factors, FEMA P695. Redwood City, CA

  17. FEMA (Federal Emergency Management Agency) (2012) Seismic performance assessment of buildings. FEMA P-58, Volumes 1 and 2. Redwood City, CA

  18. Guo T, Xu Z, Song L et al (2017) Seismic resilience upgrade of RC frame building using self-centering concrete walls with distributed friction devices. J Struct Eng 143:4017160.

    Article  Google Scholar 

  19. Gupta A, Krawinkler H (1999) Seismic demands for the performance evaluation of steel moment resisting frame structures. Stanford University, Stanford, CA

    Google Scholar 

  20. Hajjar JF, Sesen AH, Jampole E, Wetherbee A (2013) A synopsis of sustainable structural systems with rocking, self-centering, and articulated energy dissipating fuses. Northeastern University.

  21. Hall KS, Eatherton MR, Hajjar JF (2010) Nonlinear behavior of controlled rocking steel-framed building systems with replaceable energy dissipating fuses. Newmark Structural Engineering Laboratory. University of Illinois at Urbana-Champaign

  22. Hassanli R, Youssf O, Mills JE (2017) Seismic performance of precast posttensioned segmental FRP-confined and unconfined crumb rubber concrete columns. J Compos Constr 21:4017006.

    Article  Google Scholar 

  23. Ho TX, Dao TN, Aaleti S et al (2017) Hybrid system of unbonded post-tensioned CLT panels and light-frame wood shear walls. J Struct Eng 143:4016171.

    Article  Google Scholar 

  24. Housner GW (1963) The behavior of inverted pendulum structures during earthquakes. Bull Seismol Soc Am 53:403–417.

  25. Kelly JM, Tsztoo DF (1977) Earthquake simulation testing of a stepping frame with energy-absorbing devices. Earthquake Engineering Research Center, College of Engineering, University of California

  26. Lin Y-C (2015) Steel sliding-controlled coupled beam modules: development and seismic behavior for a moment resisting frame. Eng Struct 99:726–736.

    Article  Google Scholar 

  27. Ma X, Krawinkler H, Deierlein GG (2011) Seismic design and behavior of self-centering braced frame with controlled rocking and energy dissipating fuses. Blume Earthquake Engineering Center

  28. Mazzoni S, McKenna F, Scott MH, Fenves GL et al (2013) OpenSees [Computer Software]: The open system for earthquake engineering simulation. Pacific Earthquake Engineering Research Center, Berkeley, CA

    Google Scholar 

  29. Meek JW (1978) Dynamic response of tipping core buildings. Earthq Eng Struct Dyn 6:437–454.

    Article  Google Scholar 

  30. Midorikawa M, Azuhata T, Ishihara T et al (2002) Earthquake response reduction of buildings by rocking structural systems. In: SPIE’s 9th Annual international symposium on smart structures and materials. International Society for Optics and Photonics, pp 265–272

  31. Miranda E (2000) Inelastic displacement ratios for structures on firm sites. J Struct Eng 126:1150–1159.

    Article  Google Scholar 

  32. Montgomery DC (2013) Design and analysis of experiments. Wiley, Eighth Edi

    Google Scholar 

  33. Moradi S, Alam MS (2014) Feasibility study of utilizing superelastic shape memory alloy plates in steel beam-column connections for improved seismic performance. J Intell Mater Syst Struct 26:463–475.

    Article  Google Scholar 

  34. Moradi S, Alam MS (2017) Lateral load-drift response and limit states of posttensioned steel beam–column connections: parametric study. J Struct Eng 143:4017044.

    Article  Google Scholar 

  35. Moradi S, Alam MS, Asgarian B (2014) Incremental dynamic analysis of steel frames equipped with NiTi shape memory alloy braces. Struct Des Tall Spec Build 23:1406–1425.

    Article  Google Scholar 

  36. Myers RH, Montgomery DC, Anderson-Cook CM (2009) Response surface methodology: process and product optimization using designed experiments, 3rd edn. Wiley, New York

    Google Scholar 

  37. Preciado A, Sperbeck ST, Ramírez-Gaytán A (2016) Seismic vulnerability enhancement of medieval and masonry bell towers externally prestressed with unbonded smart tendons. Eng Struct 122:50–61.

    Article  Google Scholar 

  38. Preciado A, Ramírez-Gaytan A, Gutierrez N et al (2018) Nonlinear earthquake capacity of slender old masonry structures prestressed with steel, FRP and NiTi SMA tendons. Steel Compos Struct 26:213–226

    Google Scholar 

  39. Ruiz-García J, Miranda E (2010) Probabilistic estimation of residual drift demands for seismic assessment of multi-story framed buildings. Eng Struct 32:11–20.

    Article  Google Scholar 

  40. Rutenberg A, Jennings PC, Housner GW (1982) The response of veterans hospital building 41 in the San Fernando earthquake. Earthq Eng Struct Dyn 10:359–379.

    Article  Google Scholar 

  41. Sause R, Ricles JM, Roke DA et al (2010) Seismic performance of a self-centering rocking concentrically-braced frame. In: Proceedings, 9th US National and 10th Canadian conference on earthquake engineering. Toronto, ON, Canada

  42. Sperbeck ST (2008) Seismic risk assessment of masonry walls and risk reduction by means of prestressing. Technical University of Braunschweig, Germany and University of Florence, Italy

  43. Tremblay R, Poirier LP, Bouaanani N et al (2008) Innovative viscously damped rocking braced steel frames. In: Proceedings of the 14th world conference on earthquake engineering, Beijing, China, pp 12–17

  44. Wang W, Kong J, Zhang Y et al (2018) Seismic behavior of self-centering modular panel with slit steel plate shear walls: experimental testing. J Struct Eng 144:4017179.

    Article  Google Scholar 

  45. Wiebe L, Christopoulos C, Tremblay R, Leclerc M (2013) Mechanisms to limit higher mode effects in a controlled rocking steel frame. 1: concept, modelling, and low-amplitude shake table testing. Earthq Eng Struct Dyn 42:1053–1068.

    Article  Google Scholar 

Download references


This research is supported by National Science Foundation Award No. 1554714. The financial support from the Faculty of Engineering and Architectural Science at Ryerson University provided to the first author is also acknowledged.

Author information



Corresponding author

Correspondence to Saber Moradi.



See Figs. 14, 15, 16 and 17.

Fig. 14

Contour plot of desirability at optimal condition for 3d15, optimization 1

Fig. 15

Contour plot of desirability at optimal condition for 3d30, optimization 1

Fig. 16

Contour plot of desirability at optimal condition for 3m15, optimization 1

Fig. 17

Contour plot of desirability at optimal condition for 3m30, optimization 1

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Moradi, S., Burton, H.V. Response surface analysis and optimization of controlled rocking steel braced frames. Bull Earthquake Eng 16, 4861–4892 (2018).

Download citation


  • Steel controlled rocking braced frame
  • Seismic demand
  • Sensitivity
  • Optimization
  • Design of experiment
  • Response surface model