PROBLEM $1$: The figure below shows a uniform beam subjected to a distributed load $w$ applied to half

of its length. The beam is simply supported at $x=0$ and at $x=L.$ Every point in the span of the beam

displaces down due to the action of the load and the beam describes a curve called the deflection curve.

The beam deflection $v(x)$ can be computed using the following cquations:

$v(x)=\frac{-wx}{384El}(16x^3-24L{x}^2+9L^3),0x\frac{L}{2}$

$v(x)=\frac{-wL}{384El}(8x^2-24L{x}^2+17L^{2}x-L^2),\frac{L}{2}xL$

Where $L$ is the length of the beam, $E$ is the beam modulus of elasticity $($material$\text{'}$s property$)$ and I is the

beam moment of inertia $($geometric property of the beam's cross sectional area$).$

For $E=200k{10}^5\frac{N}{c}{m}^2,I=4{10}^{4}c{m}^4,L=400cm,$ and $w=5\frac{N}{c}m,$

Create a scatter graph in Excel showing the deflection curve of the beam. $x$ values must be in the

horizontal axis and $v$ values in the vertical one. Correctly label the axis, include units, and add a

title to the graph.

Add to the graph the deflection curve for two other beams with different lengths $L.$ Choose a

longer beam and a shorter one. Increase the range of $x$ values if necessary, maintain the rest of the

data the same.

By inspection, using the graph, estimate the approximate value of $x$ where the maximum

deflection occurs for each of the beams, and determine the magnitude of the maximum deflection.

Write your conclusion about the deflection of a beam as you change its length between supports.

Compare the values estimated in part $($c$)$ with the results using the following equations, are the

estimated values similar?

$v_{\text{m}\text{a}\text{x}}=\frac{-0\mathrm{.}006563w{L}^4}{El}atx=0\mathrm{.}4598L$