Beam on Elastic Foundation Civil Engineering (CE) Notes | EduRev

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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

Beam on Elastic Foundation Civil Engineering (CE) Notes | EduRev
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 ∆ is the vertical displacement of a point in the beam, qy 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 = −Ks∆.                 (11.1)

Assuming the beam to be homogeneous, we obtained the equation (8.41)
which we document here again:

Beam on Elastic Foundation Civil Engineering (CE) Notes | EduRev                   (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 Young’s modulus, (x, y) is the coordinate of the point along the axis of the beam direction and the y direction, Mis the z component of the bending moment, Izz 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:

Beam on Elastic Foundation Civil Engineering (CE) Notes | EduRev                      (11.3)

 

Beam on Elastic Foundation Civil Engineering (CE) Notes | EduRev                        (11.4)

 

where Vis the shear force along the y direction and qy is the transverse loading along the y direction. Combining the equations (11.3) and (11.4) we obtain,

Beam on Elastic Foundation Civil Engineering (CE) Notes | EduRev                                              (11.5)

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

Beam on Elastic Foundation Civil Engineering (CE) Notes | EduRev                 (11.6)

 

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

Beam on Elastic Foundation Civil Engineering (CE) Notes | EduRev                    (11.7)

 

Defining,
 

Beam on Elastic Foundation Civil Engineering (CE) Notes | EduRev                            (11.8)

equation (11.7) can be written as

Beam on Elastic Foundation Civil Engineering (CE) Notes | EduRev                    (11.9)

The differential equation (11.9) has a general solution:

∆ = exp(−βx)[C1 sin(βx) + Ccos(βx)] + exp(βx)[C3 sin(βx) + C4 cos(βx)],   (11.10)

where Ci ’s are constant to be determined from the boundary conditions. Having found the deflection, the stress is estimated from (11.2) as

Beam on Elastic Foundation Civil Engineering (CE) Notes | EduRev

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

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