Enriched beam model for slender prismatic solids in contact with a rigid foundation

Highlights

• Beam model with transverse normal and shear strains in frictionless contact with rigid foundation.

• High order beam theory for receding contact problems.

• Closed-form expression for the size of the non-contact region.

• Analytical expression for the contact pressure distribution.

• Contact pressure distribution predicted by the proposed model compare favorably with refined F.E. analyses.

Abstract

The present paper deals with an enhanced beam model for semi-infinite prismatic solid in contact with a frictionless rigid support. To circumvent the solution of a plane elasticity problem, we propose an extension of the refined beam theory of Baluch et al. (1985) [6] which accounts for the contact interactions with a frictionless rigid foundation. We combine in a certain way two-dimensional elasticity (plane stress) and the classical engineering beam theory with the aim to derive a refined beam model which takes into account the influence of transverse normal strain as well as contact conditions. The results obtained with the proposed analytical model are compared against semi-analytical solutions for receding contact problems and against refined F.E. analyzes. It is shown that the proposed model provides accurate results and is an efficient tool to tackle problems involving semi-infinite beams in frictionless unilateral contact with a rigid foundation.

Introduction

The problem of elastic beams in contact with a rigid support is encountered in several mechanical as well as civil engineering applications. The interaction of an elastic beam in frictionless contact with a rigid support and subjected to monotonic or cyclic loading involves the progressive development of non-contact zones. The size of the contact zone and the contact pressure distribution are the primary unknowns of this problem. A particular class of frictionless contact problems, described as receding contact problems by Dundurs and Stippes [3], [4], [5], [8], [10], [12], [13], [14], [15], [16], are those in which the contact area under load is included within the contact area in the unloaded configuration. These problems, which have attracted much attention over the past years, possess the interesting property that the stress and displacement fields are linearly proportional to the applied load, despite the fact that the problem definition includes unilateral contact inequalities. As shown in [21], [22], the linear proportionality between the applied load and the displacement/stress fields applies also to a restricted but interesting class of frictional problems.

Two main routes have been followed to solve this difficult problem. In the first approach, the elastic beam is considered as a deformable elastic body in smooth contact with a rigid foundation which results in a mixed boundary-value problem.

The contact problem dealing semi-infinite layer in smooth contact with a rigid foundation has been investigated by several authors [3], [4], [5], [8], [10], [12], [13], [14], [15], [16]. Keer and Silva [8] considered a semi-infinite elastic layer initially compressed by a uniform pressure and subjected to an uplift force applied at its end which induces a contact loss over a certain region. Similarly, Civelek and Erdogan considered an infinite elastic layer subjected to gravity load and a concentrated force directed opposite to the gravity field [14], [15]. These mixed boundary-value problem have been solved by means of an appropriate displacement solution to the equations of plane strain linear elasticity [8]. To avoid a singular stress field, the normal displacement to rigid foundation as well as the slope of the bottom surface of the beam must be continuous. The solution strategy leads to a Fredholm integral which can be solved using appropriate numerical techniques. The problem of a steadily moving downward directed concentrated force acting on an infinite elastic strip (plane strain) resting on a smooth rigid foundation was studied by Adams [4], [5]. The resulting mixed boundary value problem is reduced to a pair of coupled Fredholm integral equations. The solution of such a system of singular integral equations can be performed, for instance, with a collocation method [19]. The static solutions have been derived using asymptotic expansion for a null velocity of the load [4]. Other contributions to similar problems can be found in [12], [18]. Due to the complexity of the solution procedure, a direct application of the above solutions to engineering problems seems to be cumbersome and difficult to apply in a day-to-day design office. Furthermore, only semi-infinite beams have been considered.

The second approach employs engineering beam theories (Bernoulli or Timoshenko) instead of continuum mechanics. The solution for a steadily moving upward force on an infinite elastic beam resting in a gravity field on a smooth rigid foundation is given by Adams and Bogy [3]. Both Euler–Bernoulli and Timoshenko beams have been considered but the static problem has not been addressed. This problem has been considered by Timoshenko [7] and Gladwell [9] and later extended by Gao [20] to account for large displacement. Elastic beams on a rigid frictional foundation under both monotonic and cyclic loading have been investigated by Stupkiewicz and Mróz [11]. The evolution of slip zones along the beam is discussed in detail for both monotonic and cyclic loading.

Static problems dealing with beam in contact with planar rigid frictionless foundation have been analyzed by Timoshenko (see [7], pp. 61–64 ). Observing that the beam remain straight within the contact zone, it is straightforward to conclude that there is no bending moment acting in this portion. The standard Bernoulli/Timoshenko beam model seems to be inappropriate when it comes to problems involving contact constraints with a planar rigid foundation. These limitations originate from the constraints placed on the deformation map. Indeed, engineering beam theory assumes that cross-sections that are plane before deformation remain plane after deformation (plane-sections hypothesis). An equally important assumption is that those plane sections do not distort in their own planes, either. Accordingly, the “rigid cross-section” assumption would yield a trivial solution since the transverse displacement is equal to zero in the contact zone, and therefore the curvature is null; and so is the bending moment. Considering the beam equilibrium equation relating the second derivative of the bending moment and the externally applied distributed load (which comprises the contact stresses), it can be seen that a null bending moment within the contact area would give a contact pressure distribution that is pointwise equal in magnitude but with opposite sign to the stresses applied at the beam top face. In other words, the beam behaves like a rigid body in the contact region since no transverse deformation can take place and the contact stresses are computed considering translational equilibrium only.

To ease the determination of the contact stress distribution, the rigid cross-section assumption must be relaxed to permit transverse deformation while preserving the simplicity of an engineering beam theory. One possible way is to replace the rigid foundation by a semi-infinite elastic substrate or an elastic foundation [1], [2], [10] and transfer the beam cross-section deformability to the foundation. There are a large number of papers dealing with beam resting on tensionless elastic foundation. In [17], it has been shown that the behavior of a finite-length beam in contact with a tensionless foundation is significantly different than the response of an infinite beam. In particular, the contact area is a sensitive function of the load pattern and load magnitude as well as the beam length.

In the present paper, an analytical model, based on the refined beam theory of Baluch et al. [6], is proposed to investigate the behavior of prismatic solid in contact with a smooth rigid foundation. The effects of both the transverse and the shear deformations are taken in account. The proposed governing equation bears some similarities with the beam on tensionless elastic foundation. The predictions of the present model are favorably compared against the semi-analytical solutions derived by Keer and Silva [8] for the initially compressed semi-infinite elastic layer subjected to an uplift concentrated force.