Question

A cantilever beam of length L is embedded at its right end, and a horizontal tensile force of P pounds is applied to its free left end. When the origin is taken at its free end, as shown in the figure below, the deflection y(x) of the beam can be shown to satisfy the differential equation

EIy″ = Py − w(x)

x

2

.

A cantilever beam is on the x y-coordinate plane.

• The beam of length L runs along the positive x-axis with its embedded end located at some positive value on the x-axis and its free end at the origin.
• A horizontal vector labeled P begins at the free end of the beam and extends left.
• A curve begins at the origin, goes up and right becoming less steep, and ends at a positive y value above where the beam is embedded.
• At some value x on the x-axis between the embedded end and free end of the beam, a vertical vector labeled w₀x begins on the curve and extends down.

Find the general solution of the given differential equation if

w(x) = xw₀.

y =

Find the deflection of the cantilever beam if

w(x) = w₀x, 0 < x < L,

and

y(0) = 0, y'(L) = 0.

y(x) =

  

Answer 1