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Bending the Foundation Beam on Elastic Base

A new method for analysis of counter beams is presented in the paper. The analysis has taken into account their Foundation Beam stiffness EI, Winkler’s space with modulus of subgrade reaction k and equality deformities of the foundation beam with the ground. The solution is found by using the numerical analysis of the Winkler’s model, with variation of different moduli of the subgrade reaction k2 outside the force zone r, while under the force P exists the modulus of the subgrade reaction k, up to the definition of minimum bending moments. The exponential function k2(r), as the geometric position of the minimum moments is approximately assumed. From the potential energy conditions of the reciprocity of displacement and reaction, the width of the zone r and the modulus of the subgrade reaction k2 are explicitly determined, introducing in the calculation initial and calculation soil displacement wsi successively. At the end of the paper, it presented numerical example in which the influence of k and k2 values on bending moments of the counter beam is analyzed. The essential idea of this paper is to decrease the quantity of the reinforcement in the foundations, beams, i.e. to obtain a cost-efficient foundation construction.

When calculating a beam on a continuous deformable base, it is important to provide a modeling of the foundation beam base as realistic as possible, i.e. its approximation with the actual properties of soil beneath the foundations. Yet, the calculation methods should be kept simplified so that they could be widely implemented in practical applications.

Beam on elastic foundation has been analysed, most usually, based on the Winkler’s model in which the soil is replaced by a bed of elastic springs. The compressive resistance of soil against the beam deflection is quantified in terms of spring constant k [force/length2/length], which is a frequent occurrence in the Euler-Bernoulli beam theory. Shear deformations are neglected and plane cross-section is assumed to remain plane and normal to the longitudinal axis deformation. Many researchers [1] [2] [3] [4] [5] have investigated the modulus of subgrade reaction and found that the geometry, the foundation dimensions and soil layering below the foundation structure are the most important parameters to define the value of this modulus. In the Winkler foundation model, the soil reactive pressure at any arbitrary point x is proportional to the deflection and can be expressed as,

q(x)=kw(x)q(x)=k⋅w(x)(1)

where k is the Winkler’s coefficient of ground reaction at point x. Winkler foundation is a single parameter model, k is used to describe the soil reaction.

The subgrade reaction modulus k is dependent on some parameters like soil type, size and shape of foundations, depth and stress level. The foundation represented by the Winkler model [6] cannot sustain shear stress, and hence a discontinuity of adjacent spring displacement can occur. A different model may result in significant inaccuracies in the evaluated structural response. In order to overcome this problem, many researches have introduced a different mechanical model [7] – [14] . Among them is the class of two parameters foundation. The second parameter introduced the interaction between adjacent springs, in addition to the first parameter from the ordinary Winkler’s model [15] . This procedure is proposed in [13] for homogenous elastic semi space.

In the [13] the elastic base is represented with a layer having thickness H, exposed to pressure, lying on top on an infinitely stiff horizontal base. One dimension of the compressed layer is large, and the load invariable in this direction; the supporting conditions and values of the elastic characteristic are constant, too. An in-plane stress and strain state is considered. The proposed model of soil has two soil characteristics, k (characterising displacement of the elastic base under pressure) and t (and describing behaviour of the subgrade during sliding and base “distribution properties”). Pasternak proposed that both soil characteristics (k and t) are named the subgrade coefficients, i.e. the model of soil of two subgrade coefficients (as cited in [9] ).

In [16] a closed-form analytical solution of the problem of bending of a beam on elastic foundation is proposed. The solution based on the total potential energy functional. In order to eliminated the bearing soil reaction as a variable in the problem solution of beam on elastic foundation, the simplified continuum approach, with a numerical research, is presented in [17] . Study the behaviour of a math foundation on subsoil from the plate theory taking into account the soil-structure interaction, and several model have been described, presented in [18] . Very important work related to subgrade reaction and analysis of beams on elastic foundation is [4] [5] [19] .

The equations available for estimating the soil spring constant k are mostly developed empirically [5] , which is limitation of Winkler’s model. In some instances, plate-load test are used to estimate k, but that estimations are not free from errors because the results depend on size, thickness and stiffness of the plate.

Two-parameter foundation models provide the displacement continuity of the soli medium by adding of a second spring which interacts with the first spring of the Winkler’s model. Displacement continuity is provided for by the introduction of a virtual shear layer which integrates the vertical spring elements and the second foundation parameter k2, is the shear modulus G of the shear layer [16] [17] . The soil reaction q(x) for two parameter foundation model is given in general by:

qs(x)=k1w(x)k2d2w(x)/dx2qs(x)=k1w(x)−k2d2w(x)/dx2(2)

where k1 and k2 are two foundation parameters.

In the paper will be demonstrated that in a case of k and k2 model, bending moments in the counter beam are smaller than in a case of a k model. The change of k values does not significantly affect this difference.