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Research Papers

# Variational Approach to Beams Resting on Two-Parameter Tensionless Elastic Foundations

[+] Author and Article Information
A. Nobili

Dipartimento di Ingegneria Meccanica e Civile, Università degli Studi di Modena e Reggio Emilia via Vignolese 905, 41122 Modena, Italyandrea.nobili@unimore.it

J. Appl. Mech 79(2), 021010 (Feb 24, 2012) (10 pages) doi:10.1115/1.4005549 History: Received October 07, 2010; Revised May 09, 2011; Posted January 30, 2012; Published February 13, 2012; Online February 24, 2012

## Abstract

This paper presents a Hamiltonian variational formulation to determine the energy minimizing boundary conditions (BCs) of the tensionless contact problem for an Euler–Bernoulli beam resting on either a Pasternak or a Reissner two-parameters foundation. Mathematically, this originates a free-boundary variational problem. It is shown that the BCs setting the contact loci, which are the boundary points of the contact interval, are always given by second order homogeneous forms in the displacement and its derivatives. This stands for the nonlinear nature of the problem and calls for multiple solutions in the displacement, together with the classical result of multiple solutions in the contact loci position. In particular, it is shown that the Pasternak soil possesses an extra solution other than Kerr’s, although it is proved that such solution must be ruled out owing to interpenetration. The homogeneous character of the BCs explains the well-known load scaling invariance of the contact loci position. It is further shown that the Reissner foundation may be given two mechanical interpretations, which lead to different BCs. Comparison with the established literature is drawn and numerical solutions shown which confirm the energy minimizing nature of the assessed BCs.

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## Figures

Figure 1

Beam resting on a tensionless Pasternak soil in a symmetric framework: [0,X] is (half) the contact interval, X sets the contact locus while (X,L] is the lift-off interval, where the beam is detached from the foundation—the sign convention for the bending moment and shearing force is also presented

Figure 2

Reissner shear-layer model

Figure 3

Beam on a tensionless elastic foundation in a symmetric framework

Figure 4

Energy ΠP(Ξ) for l = 3,α = 2.5 and at three values of the end couple C = 0,±1/50

Figure 5

Plot of the first, sol (a), and second term, sol (b), of Eq. 26 setting the contact locus

Figure 6

Beam in contact, soil and lift-off for sol (b), C = 1/50, Ξ = 1.123826

Figure 7

Beam in contact, soil and lift-off for sol (a), C = -1/50, Ξ = 0.626716

Figure 8

Energy 63 and contact locus Eq. 58 (rescaled) versus Ξ for the half-plain Reissner soil

Figure 9

Beam in contact and lifting-off and soil profile for a Reissner half-plane with α=1.1

Figure 10

Bending moment, shearing force, (negative) contact pressure and its derivative

Figure 11

Beam in contact and lifting-off, soil and shear-layer profile for a Reissner shear-layer soil with α = 1.1 (the shear-layer is arbitrarily displaced)

Figure 12

Bending moment, shearing force, (negative) contact pressure and its derivative for a beam partially supported on a Reissner shear-layer soil

Figure 13

Beam completely supported by a Reissner half-plane (α=1.1,l=0.5)

Figure 14

Bending moment, shearing force, (negative) contact pressure and its derivative for a beam entirely supported on a Reissner half-plane soil

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