Chapter 1, Problem 24P

As depicted in Fig. P1.24, the downward deflection $y\left(\text{m}\right)$ of a cantilever beam with a uniform load w (kg/m) can be computed as

$y=\frac{w}{24EI}\left(x^4-4L{x}^3+6L^2{x}^2\right)$

where $x=$ distance $\left(\text{m}\right)$, $E=$ the modulus of elasticity $=2\times {10}^{11}$ Pa, $I=$ moment of inertia $=3.25\times {10}^{-4}{\text{ m}}^4,w=10,000$ N/m, and

$L=$ length $=4$ m. This equation can be differentiated to yield the slope of the downward deflection as a function of x:

$\frac{dy}{dx}=\frac{w}{24EI}\left(4x^3-12L{x}^2+12L^{2}x\right)$

If $y=0$ at $x=0$, use this equation with Euler's method $\left(\Delta x=0.125\text{ m}\right)$ to compute the deflection from $x=0$ to L. Develop a plot of your results along with the analytical solution computed with the first equation.

FIGURE P1.24

A cantilever beam.