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

  • Sourav DasEmail author
  • Satyabrata Choudhury
Original Article


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.


Support vector regression Effective stiffness ratio Reinforced concrete columns Nonlinear dynamic analysis 

List of symbols


Gross area of cross section




Penalty parameter


Degree of polynomial kernel


Depth of column in x direction


Depth of column in y direction


Dead load


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


Compressive strength of concrete


Yield strength of reinforcing steel

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

Regression function


Column depth


Effective moment of inertia


Kernel function


Lagrange function

\(L_{\varepsilon }\)

Loss function


Live load


Nominal moment capacity


Calculated effective stiffness ratio


Average values of \(O_{i}\)


Axial load of column

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

Column axial load due to DL and LL


Predicted effective stiffness ratio


Average values of \(P_{i}\)


Percentage of reinforcing steel


Weight vector


Input vector

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

Support vectors



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

Lagrange multiplier


Yield strain of steel


Yield curvature of the section


Width of the RBF kernel

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

Euclidian norm of weight vector


Allowable error in the loss function

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

Sack variables


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflicts of interest.


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Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Civil EngineeringNational Institute of Technology SilcharSilcharIndia

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