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Evaluation of effective stiffness of RC column sections by support vector regression approach

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Abstract

Effective stiffness of reinforced concrete (RC) members has a very important role in the performance evaluation of RC frame buildings through nonlinear dynamic analyses. The beam effective stiffness can be readily computed using mechanics, but the evaluation of column stiffness is a complicated process and the use of support vector regression helps in this regard. Therefore, in this study, an attempt is made to predict the effective stiffness ratio of reinforced concrete columns using support vector regression (SVR) approach. A data set of 208 samples, which are collected through nonlinear dynamic analysis of reinforced concrete buildings using SAP2000 software, is utilized to develop the SVR model. The input parameters considered are reinforcement percentage, axial load and depth of the column section in both the perpendicular directions, and the output parameter is the effective stiffness ratio of columns. Three different kernel parameters are used, namely exponential radial basis function (ERBF), Gaussian radial basis function and polynomial function for SVR modelling, among which ERBF is found to be the most suitable one. The obtained results indicate that the statistical performance of the SVR-ERBF model is better than the models with other two kernels in predicting the effective stiffness ratio of reinforced concrete columns. Performance of the SVR model is compared with the results of multi-variable regression analysis. In addition to that, a sensitivity analysis is also performed to check the influence of each input parameter on output responses.

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Abbreviations

\(A_{\text{g}}\) :

Gross area of cross section

b :

Bias

C :

Penalty parameter

d :

Degree of polynomial kernel

\({\text{depth}}_{x}\) :

Depth of column in x direction

\({\text{depth}}_{y}\) :

Depth of column in y direction

\({\text{DL}}\) :

Dead load

\(E\) :

Modulus of elasticity of concrete

\(\left( {\frac{{EI_{e} }}{{EI_{g} }}} \right)_{x}\) :

Effective stiffness ratio in x direction

\(\left( {\frac{{EI_{e} }}{{EI_{g} }}} \right)_{y}\) :

Effective stiffness ratio in y direction

\(f_{\text{c}}\) :

Compressive strength of concrete

\(f_{\text{y}}\) :

Yield strength of reinforcing steel

\(f\left( x \right)\) :

Regression function

\(h_{\text{c}}\) :

Column depth

\(I_{\text{eff}}\) :

Effective moment of inertia

\(K(,)\) :

Kernel function

\(L\) :

Lagrange function

\(L_{\varepsilon }\) :

Loss function

\({\text{LL}}\) :

Live load

\(M_{\text{N}}\) :

Nominal moment capacity

\(O_{i}\) :

Calculated effective stiffness ratio

\(\bar{O}\) :

Average values of \(O_{i}\)

\(P\) :

Axial load of column

\(P_{{{\text{DL}} + {\text{LL}}}}\) :

Column axial load due to DL and LL

\(P_{i}\) :

Predicted effective stiffness ratio

\(\bar{P}\) :

Average values of \(P_{i}\)

\(p_{t}\) :

Percentage of reinforcing steel

\(w\) :

Weight vector

\(x\) :

Input vector

\(x_{r} ,\,x_{s}\) :

Support vectors

y :

Output

\(\alpha^{*} ,\alpha\) :

Lagrange multiplier

\(\epsilon_{\text{y}}\) :

Yield strain of steel

\(\phi_{\text{y}}\) :

Yield curvature of the section

\(\sigma\) :

Width of the RBF kernel

\(\left| {\left| w \right|} \right|^{2}\) :

Euclidian norm of weight vector

\(\varepsilon\) :

Allowable error in the loss function

\(\xi ,\xi^{*}\) :

Sack variables

References

  1. Priestley MJN (2003) Myths and fallacies in earthquake engineering. In: Revisited the ninth mallet Milne lecture

  2. Pauley T, Priestley MJN, Paulay T, Priestley N (1992) Seismic design of reinforced concrete and masonry buildings. Wiley, Hoboken

    Book  Google Scholar 

  3. Mehanny SSF, Kuramoto H, Deierlein GG (2001) Stiffness modeling of reinforced concrete beam-columns for frame analysis. ACI Struct J 98:215–225. https://doi.org/10.14359/10190

    Article  Google Scholar 

  4. Panagiotakos TB, Fardis MN (2001) Deformations of reinforced concrete members at yielding and ultimate. ACI Struct J 98:135–148. https://doi.org/10.14359/10181

    Article  Google Scholar 

  5. Khuntia M, Ghosh SK (2004) Flexural stiffness of reinforced concrete columns and beams: analytical approach. ACI Struct J 101:351–363. https://doi.org/10.14359/13095

    Article  Google Scholar 

  6. Pique JR, Burgos M (2008) Effective rigidity of reinforced concrete elements. In: 14th world conference earthquake engineering seismic analysis and design

  7. Elwood KJ, Eberhard MO (2009) Effective stiffness of reinforced concrete columns. ACI Struct J 106:476–484

    Google Scholar 

  8. Kumar R, Singh Y (2010) Stiffness of reinforced concrete frame members for Seismic analysis. ACI Struct J 107:607–615

    Google Scholar 

  9. Bonet JL, Romero ML, Miguel PF (2011) Effective flexural stiffness of slender reinforced concrete columns under axial forces and biaxial bending. Eng Struct 33:881–893. https://doi.org/10.1016/j.engstruct.2010.12.009

    Article  Google Scholar 

  10. American Society of Civil Engineers (ASCE) (2000) FEMA 356 Prestandard and commentary for the seismic rehabilitation of building. Rehabilitation

  11. American Society of Civil Engineers (ASCE) (2014) Seismic evaluation and retrofit of existing buildings (ASCE 41-13)

  12. Chau KW (2007) Reliability and performance-based design by artificial neural network. Adv Eng Softw 38:145–149. https://doi.org/10.1016/j.advengsoft.2006.09.008

    Article  Google Scholar 

  13. Möller O, Foschi RO, Quiroz LM, Rubinstein M (2009) Structural optimization for performance-based design in earthquake engineering: applications of neural networks. Struct Saf 31:490–499. https://doi.org/10.1016/j.strusafe.2009.06.007

    Article  Google Scholar 

  14. de Lautour OR, Omenzetter P (2009) Prediction of seismic-induced structural damage using artificial neural networks. Eng Struct 31:600–606. https://doi.org/10.1016/j.engstruct.2008.11.010

    Article  Google Scholar 

  15. Arslan MH (2010) An evaluation of effective design parameters on earthquake performance of RC buildings using neural networks. Eng Struct 32:1888–1898. https://doi.org/10.1016/j.engstruct.2010.03.010

    Article  Google Scholar 

  16. Yinfeng D, Yingmin L, Ming L, Mingkui X (2008) Nonlinear structural response prediction based on support vector machines. J Sound Vib 311:886–897. https://doi.org/10.1016/j.jsv.2007.09.054

    Article  Google Scholar 

  17. Yan K, Shi C (2010) Prediction of elastic modulus of normal and high strength concrete by support vector machine. Constr Build Mater 24:1479–1485. https://doi.org/10.1016/j.conbuildmat.2010.01.006

    Article  Google Scholar 

  18. Cheng M-Y, Prayogo D, Wu Y-W (2018) Prediction of permanent deformation in asphalt pavements using a novel symbiotic organisms search: least squares support vector regression. Neural Comput Appl 1:1–9. https://doi.org/10.1007/s00521-018-3426-0

    Article  Google Scholar 

  19. Samui P, Kim D (2013) Least square support vector machine and multivariate adaptive regression spline for modeling lateral load capacity of piles. Neural Comput Appl 23:1123–1127. https://doi.org/10.1007/s00521-012-1043-x

    Article  Google Scholar 

  20. Debnath P, Dey AK (2017) Prediction of bearing capacity of geogrid-reinforced stone columns using support vector regression. Int J Geomech 18:1–15. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001067

    Article  Google Scholar 

  21. Boser BE, Guyon IM, Vapnik VN (1992) A training algorithm for optimal margin classifiers. In: Proceedings of the fifth annual workshop on computational learning theory-COLT’92. pp 144–152

  22. Cortes C, Vapnik V (1995) Support-vector networks. Mach Learn 20:273–297. https://doi.org/10.1023/A:1022627411411

    Article  MATH  Google Scholar 

  23. Vapnik V, Golowich SE, Smola A (1997) Support vector method for function approximation, regression estimation, and signal processing. In: Advances in neural information processing systems, vol 9

  24. Smola AJAAJ, Schölkopf B, Sch B, Schölkopf B (2004) A tutorial on support vector regression. Stat Comput 14:199–222. https://doi.org/10.1023/B:STCO.0000035301.49549.88

    Article  MathSciNet  Google Scholar 

  25. Cristianini N, Shawe-Taylor J (2000) An introduction to support vector machines and other kernel based learning methods. Cambridge University Press, Cambridge

    Book  Google Scholar 

  26. Samui P, Kim D, Sitharam TG (2011) Support vector machine for evaluating seismic-liquefaction potential using shear wave velocity. J Appl Geophys 73:8–15. https://doi.org/10.1016/j.jappgeo.2010.10.005

    Article  Google Scholar 

  27. Chen KY (2007) Forecasting systems reliability based on support vector regression with genetic algorithms. Reliab Eng Syst Saf 92:423–432. https://doi.org/10.1016/j.ress.2005.12.014

    Article  Google Scholar 

  28. Neaz Sheikh M, Tsang HH, McCarthy TJ, Lam NTK (2010) Yield curvature for seismic design of circular reinforced concrete columns. Mag Concr Res 62(10):741–748

    Article  Google Scholar 

  29. SAP2000. Structural analysis programme. Computer and Structures Inc.; Berkley

  30. Choudhury S, Singh SM (2013) A unified approach to performance-based design of RC frame buildings. J Inst Eng Ser A 94:73–82. https://doi.org/10.1007/s40030-013-0037-8

    Article  Google Scholar 

  31. Gunn SR (1998) Support vector machines for classification and regression. In: ISIS technical report, University of Southampton, Southampton

  32. MATLAB R2013a. MathWorks, Natick, MA

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  1. Department of Civil Engineering, National Institute of Technology Silchar, Silchar, Assam, 788010, India

    Sourav Das & Satyabrata Choudhury

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  1. Sourav Das
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  2. Satyabrata Choudhury
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Correspondence to Sourav Das.

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Das, S., Choudhury, S. Evaluation of effective stiffness of RC column sections by support vector regression approach. Neural Comput & Applic 32, 6997–7007 (2020). https://doi.org/10.1007/s00521-019-04190-0

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