Nonlinear analysis of structures on elastic half-space by a FE-BIE approach

Abstract

In the present dissertation, a numerical model able to study the non-linear behaviour of structures on elastic half-space is presented. The work takes into account for first the geometric non-linearity of beams and frames on elastic half-plane, namely in plane strain or plane stress condition. This problem is important in many engineering fields and it has been studied in the past by many researchers for the design of sandwich panels in aerospace industry. Recently, this problem has been studied in relation to the buckling of thin films on elastic supports for electronic design. The problem is solved by means of a mixed variational formulation, which assumes as independent fields the displacements of the structure and the contact pressures between foundation and halfspace. The relation between surface displacement and pressure is given by the Flamant solution, which furnishes the half-plane displacement generated by a concentrated force. The second order effects due to axial loads applied on the structure are added to the total potential energy of the system in order to perform buckling analyses. Then, the model is discretized by subdividing the structure into finite elements (FEs) and simplifying contact pressures by a piecewise constant function. Hence, the stationarity conditions of the total potential energy written in discrete form furnish a system of equations which can be solved easily. A soil-structure interaction (SSI) parameter taking into account both the slenderness of the foundation beam and the stiffness of the soil is introduced. The present model was introduced for the first time by Tullini and Tralli (2010) but it was limited to linear elastic analysis. In this case, the model is extended for studying the stability of beams on elastic half-plane, considering both Euler-Bernoulli and Timoshenko beam. In the first chapter, the present model for an Euler-Bernoulli beam on elastic half-plane is compared with a traditional model characterized by the half-space modelled by two-dimensional (2D) FEs. The present model turns out to be efficient and faster than the traditional model. Then, the stability of beams with finite length and with different end restraints is deeply discussed by determining critical loads and the corresponding mode shapes, varying the SSI parameter. Numerical examples are in good agreement with analytic solutions for the case of the beam with sliding ends. Critical loads converge to the values of a beam with infinite length on elastic half-plane and on a set of equidistant supports. The cases of beam with pinned and free ends furnish new estimates of critical loads, which are less than that for the beam with sliding ends and which are characterized by mode shapes with great deflections close to beam ends. In the second chapter, the stability of Timoshenko beams on elastic half-space is discussed. The present model is compared with a traditional model where both beam and half-plane are modelled by 2D FEs. For stiff or quite stiff beams on soft half-plane, the present model is fast and efficient, whereas for slender beams on stiff half-plane, the present model gives critical loads greater than the ones obtained with the traditional model. Differences are caused by the second order effects of the half-plane, which are taken into account in the traditional model. Then, in the third chapter, structures on half-plane are studied taking into account the material nonlinearity for the structure. A lumped plasticity model is considered and flexural plastic hinges are introduced into the discrete model of slender beams and frames on elastic half-plane. For simplicity, a rigid-perfectly plastic moment-rotation relationship is adopted for describing the behaviour of plastic hinges. Material nonlinearity is introduced into the discrete model following an efficient approach adopted for representing semi-rigid connections of frames. The approach gives the possibility to keep the same number of beam FEs and degrees of freedom of the original model, whereas potential plastic hinges are added to beam FE ends by simply modifying the corresponding stiffness matrices. Hence, incremental analyses of beams and frames are performed by placing potential plastic hinges close to concentrated loads and at beam-column connections. In the fourth chapter of the thesis, the discrete model of a beam on elastic half-plane is extended to the three-dimensional case for performing static and buckling analysis of foundation beams. Beams on 3D half-space are important in civil engineering field and they may adopted for representing shallow foundations on elastic soil. In this case, the relation between surface displacements and contact pressure is given by Boussinesq solution. The flexibility matrix of the soil is first determined for solving the Galerkin boundary element method, in order to study the indentation of the half-space by a rigid square punch and determining the displacements generated by uniform pressure distributions over rectangular areas. In both cases, the half-space surface is discretized in both plane directions adopting power graded meshes characterized by very small surface discretizations close to surface edges. Then, Euler-Bernoulli and Timoshenko beams on 3D halfspace subject to different loads are studied and displacements, surface pressures and bending moments are determined. Finally, the stability of Euler-Bernoulli beams on 3D half-space with finite length and different end restraints is considered. Critical loads and mode shapes are similar to those obtained for the 2D case in the fist chapter, however in this case, results are strictly dependent on the ratio between beam length and width