Question

Case 1: Uniform beam under distributed load.

In the shown Figure, a uniform beam subject to a linearly increasing distributed load. The deflection \(y(\mathrm{~m})\) can be expressed by \(y=\frac{w_{o}}{120 E I L}\left(-x^{5}+2 L^{2} x^{3}-L^{4} x\right)\)

Where \(E\) is the modulus of elasticity and \(I\) is the moment of inertia \(\left(\mathrm{m}^{4}\right), L\) length of beam.

Use the following parameters \(L=600 \mathrm{~cm}\), \(E=50,000 \mathrm{kN} / \mathrm{cm}^{2}, I=30.000 \mathrm{~cm}^{4}, w_{\mathrm{o}}=2.5\)

\(\mathrm{kN} / \mathrm{cm}\), to find the requirements

1) Develop MATLAB code to determine the point of maximum deflection by using numerical method (bisection, false position method,….). Hint(The value of x where dy/dx=0). 

2) Plot the point of maximum deflection versus iteration number. 

3) Plot the values of the relative approximate error of the point of maximum deflection (𝜖𝑎,𝑥) versus iteration number. 

4) Plot the following quantities versus distance along the beam a) Displacement (y). b) Slope 𝜃(𝑥) = 𝑑𝑦/𝑑𝑥. c) Moment 𝑀(𝑥) = 𝐸𝐼𝑑2𝑦/𝑑𝑥2. d) Shear 𝑉(𝑥) = 𝐸𝐼𝑑3𝑦/𝑑𝑥3. e) Loading 𝑤(𝑥) = −𝐸𝐼𝑑4𝑦/𝑑𝑥4.

Answer 1