Keywords: nonlinear analysis, large deflections, Timoshenko beam, shear deformation coefficients, boundary element method, nonlinear tensionless foundation.

A boundary element method (BEM) is developed for the nonlinear analysis of shear deformable beam-columns of arbitrary doubly symmetric simply or multiply connected constant cross section, partially supported on tensionless three parameter foundations, undergoing moderate large deflections subject to general boundary conditions. The beam-column is subjected to the combined action of arbitrarily distributed or concentrated transverse loading and bending moments in both directions as well as to axial loading. To account for shear deformations, the concept of shear deformation coefficients is used. Five boundary value problems are formulated with respect to the transverse displacements, to the axial displacement and to two stress functions and solved using the analog equation method (AEM) [1], a BEM based method. Application of the boundary element technique yields a system of nonlinear equations from which the transverse and axial displacements are computed by an iterative process. The evaluation of the shear deformation coefficients is accomplished from the aforementioned stress functions using only boundary integration. The essential features and novel aspects of the present formulation compared with previous ones are summarized as follows:

• Shear deformation effect is taken into account on the nonlinear analysis.
• The homogeneous linear half-space is approximated by a tensionless three parameter foundation.
• The beam-column is supported by the most general boundary conditions including elastic support or restrain, while its cross section is an arbitrary doubly symmetric one.
• The proposed model takes into account the coupling effects of bending and shear deformations along the member as well as shear forces along the span induced by the applied axial loading.
• The shear deformation coefficients are evaluated using an energy approach.

The main conclusions that can be drawn are:

• In some cases, the effect of shear deformation is significant, especially for low beam slenderness values.
• The discrepancy of the obtained results performing a linear or a nonlinear analysis is remarkable.
• The significant influence of the unilateral character of the foundation, especially in the case of a stiff soil can be demonstrated.
• The importance of the soil stiffness to the response of the beam-column is verified.
• The developed procedure retains most of the advantages of a BEM solution over a finite element approach, although it requires domain discretization.

1

J.T. Katsikadelis, "The Analog Equation Method. A Boundary-Only Integral Equation Method for Nonlinear Static and Dynamic Problems in General Bodies", Theoretical and Applied Mechanics, 27, 13-38, 2002. doi:10.2298/TAM0227013K

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