Buckling Analysis of Tapered Continuous Columns by Using Modified Buckling Mode Shapes

  • Sina ToosiEmail author
  • Akbar Esfandiari
  • Ahmad Rahbar Ranji
Research Article


Elastic critical buckling load of a column depends on various parameters, such as boundary conditions, material, and cross-section geometry. The main purpose of this work is to present a new method for investigating the buckling load of tapered columns subjected to axial force. The proposed method is based on modified buckling mode shape of tapered structure and perturbation theory. The mode shape of the damaged structure can be expressed as a linear combination of mode shapes of the intact structure. Variations in length in piecewise form can be positive or negative. The method can be used for single-span and continuous columns. Comparison of results with those of finite element and Timoshenko methods shows the high accuracy and efficiency of the proposed method for detecting buckling load.


Buckling analysis Tapered column Continuous columns Finite element method Modified buckling mode shapes 



stiffness matrix


geometric stiffness matrix


eigenvalue of the structure


mode shape of the structure


Kronecker’s delta


variation of eigenvalue


variation of mode shape


variation of stiffness matrix


linear coefficient of jth mode shape

\( {\boldsymbol{\varphi}}_i^{\prime } \)

changes in mode shape of the structure


Young’s modulus of elasticity


column flexural moment of inertia


internal load of elements


beam length


critical buckling load of column


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Copyright information

© Harbin Engineering University and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Sina Toosi
    • 1
    Email author
  • Akbar Esfandiari
    • 1
  • Ahmad Rahbar Ranji
    • 1
  1. 1.Department of Ocean EngineeringAmirKabir University of TechnologyTehranIran

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