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Beam on Elastic Foundation

General formulation

In this section, we formulate the boundary value problem of beam on an elastic foundation. A beam having some cross section, resting on an elastic support is shown in figure 11.1. We assume that the reaction offered by

**Figure 11.1: Schematic of a long beam on elastic foundation**

the support at any point is directly proportional to the displacement of that point along the y direction and is in a direction opposite to the displacement. Thus, if âˆ† is the vertical displacement of a point in the beam, q_{y} the support reaction per unit width of the beam, then the above assumption that the reaction force is proportional to the displacement mathematically translates into requiring

qy = âˆ’Ksâˆ†. (11.1)

Assuming the beam to be homogeneous, we obtained the equation (8.41)

which we document here again:

(11.2)

where yo is the y coordinate of the centroid of the cross section which can be taken as 0 without loss of generality provided the origin of the coordinate system used is located at the centroid of the cross section, E is the Youngâ€™s modulus, (x, y) is the coordinate of the point along the axis of the beam direction and the y direction, M_{z }is the z component of the bending moment, I_{zz} is the moment of inertia of the section about the z axis. In section 8.1, we integrated the equilibrium equations and obtained equations (8.18) and (8.25) which we record here:

(11.3)

(11.4)

where V_{y }is the shear force along the y direction and q_{y} is the transverse loading along the y direction. Combining the equations (11.3) and (11.4) we obtain,

(11.5)

Substituting equation (11.2) in equation (11.5), we obtain

(11.6)

Assuming the beam to be homogeneous and prismatic, so that EI_{zz }is constant through the length of the beam, and substituting equation (11.1) in equation (11.6), we obtain

(11.7)

Defining,

(11.8)

equation (11.7) can be written as

(11.9)

The differential equation (11.9) has a general solution:

âˆ† = exp(âˆ’Î²x)[C_{1} sin(Î²x) + C_{2 }cos(Î²x)] + exp(Î²x)[C_{3} sin(Î²x) + C_{4} cos(Î²x)], (11.10)

where C_{i} â€™s are constant to be determined from the boundary conditions. Having found the deflection, the stress is estimated from (11.2) as

where we have assumed that the origin is located at the centroid of the cross section and hence have set y_{o} = 0.

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