Structural Analysis Methods I: Beam Elements - Mechanics of Structures: Variational and Computational Methods - page 207

p z

P z

M

M ( )

q

x k EI

V (0)

x, u, x =

∆ T

V ( )

y

k w

z, w

FIGURE 4.11
Beam element on elastic foundation. Sign Convention 1.

q

x

p z

y

M ( x )

M + dM
V + dV

z, w

V ( x )

EI

x

k

∆ T

k w w

dx

FIGURE 4.12
Differential element. Sign Convention 1.

EXAMPLE 4.4 Beam Element on Elastic Foundation

A beam on elastic foundation provides an example of the sort of approximation involved
in the use of polynomial trial functions to develop element stiffness matrices. Whereas
the simple Euler-Bernoulli beam of this chapter can be solved “exactly” using a cubic
polynomial as the (assumed) trial function, the same polynomial leads to approximate
element matrices for a beam on an elastic foundation.
Begin with notation for the beam element of Fig. 4.11.

k w is the modulus of the elastic (Winkler) foundation (force/length 2 )
κ T is the prescribed curvature from the temperature difference = α T

h

T , e.g., κ T

where is the coefficient of linear thermal expansion and h is the height of the

beam cross section

α

Fundamental Relations for a Differential Element

The elastic foundation imposes a force of magnitude k w w on the bea m element. This is

introduced as a distributed reaction (force/length), opposite in sign to p z (Fig. 4.12). The
fundamental differential equations for the response of this beam element are derived using
the same procedure employed in Chapter 2 for the simple beam element. The resulting
equations are the same as given by Eq. (1.133) with the addition of the effects of temperature
and the elastic foundation.