A beam of infinite length is attached to an elastic foundation and is subjected to a distr...

A beam of infinite length is attached to an elastic foundation and is subjected to a distributed load w(z) [F/L] over a segment L' of the beam (Figure P10.22a). Consider an element dz of the beam in the loaded region (Figure P10.22b). Derive the equilibrium conditions for the element and show that

Note that w(z) = 0 outside the segment L', and then Eq. (a) reduces to Eq. 10.4. Note also that the solution of Eq. (a) consists of a particular solution Yp and a general solution yh The particular solution is any function yp(z)that satisfies Eq. (a). The general solution is the solution of the homogeneous equation EIx(d4y/dz4) + ky = 0, that is, Eq. 10.6. w(z) is a simple function, say a polynomial, a trigonometric, or an exponential function, the particular solution is a displacement yp of the same form.

FIGURE P10.22a

FIGURE P10.22b

 (10.4)

 (10.6)