When a uniform beam is supported by an elastic foundation,

Chapter , Problem 46RP

(choose chapter or problem)

When a uniform beam is supported by an elastic foundation, the differential equation for its deflectio y(x) is

                                     \(E I \frac{d^{4} y}{d x^{4}}+k y=w(x)\)

where k is the modulus of the foundation and -ky is the restoring force of the foundation that acts in the direction opposite to that of the load w(x). See Figure 7.R.11. For algebraic convenience suppose that the differential equation is written as

                                    \(\frac{d^{4} y}{d x^{4}}+4 a^{4} y=\frac{w(x)}{E I}\),

Where \(a=(k / 4 E I)^{1 / 4}\). Assume \(L=\pi\) and  \(a=1\). Find the deflection y(x) of a beam that is supported on an elastic foundation when

(a) the beam is simply supported at both ends and a constant load \(w_{0}\) is uniformly distributed along its length,

(b) the beam is embedded at both ends and w(x) is a concentrated load \(w_{0}\) applied at \(x=\pi / 2\). [Hint: In both parts of this problem use entries 35 and 36 in the table of Laplace transforms in Appendix III.]

Text Transcription:

E I d^4y/dx^4+k y=w(x)

d^4y/dx^4+4 a^4 y=w(x)/E I

a=k / 4 E I^1 / 4

L=pi

a=1

w_0

x=pi / 2