Consider a cantilever beam of length l loaded by force P at the free end. If the deflection is given by
y(x)=(P/6EI)(x³-3/x²) m
Find the deflection and slope of the deflection at the free end of x=l
To find the deflection at the free end of the beam (x = l), we can substitute the value of x into the equation for deflection y(x):
y(l) = (P/6EI) * (l³ - 3/l²)
Substituting x = l, we have:
y(l) = (P/6EI) * (l³ - 3/l²)
To find the slope of the deflection at the free end of the beam (x = l), we need to calculate the derivative of the deflection equation with respect to x and then evaluate it at x = l:
slope(l) = dy/dx | x=l
First, let's find the derivative of y(x):
dy/dx = (d/dx) [(P/6EI)(x³ - 3/x²)]
= (P/6EI) * (3x² + 6/x³)
Now, let's evaluate the derivative at x = l to find the slope at the free end:
slope(l) = (P/6EI) * (3l² + 6/l³)
So, the deflection at the free end of the beam is given by y(l) = (P/6EI) * (l³ - 3/l²) meters, and the slope of the deflection at the free end is given by slope(l) = (P/6EI) * (3l² + 6/l³).