For the beam shown below, derive the equation for the elastic deflection curve, $y_{(x)},$ using Bernoulli$-$Euler Beam Theory. What is the equation for the deflection at mid$-$span?

For the beam shown below, derive the equation for the elastic deflection curve, $y_{(x)},$ using Timoshenko Beam Theory. What is the equation for the deflection at mid$-$span?

Plot the ratio of mid$-$span deflection from Problem $1$ to Problem $2$ for $L=25$ to $300$ inches. For this, use $E=29000$ksi,$I=1350i{n}^4,G=11200$ksi,$A_s=9\mathrm{.}322i{n}^2.$ What can be concluded about the effect of shear deformations on the behavior of a beam?

$3.$ For the beam$-$column shown below, derive the equation for the elastic deflection curve using Bernoulli$-$Euler Beam Theory. What is the equation for the maximum deflection at mid$-$span? $[$Just get as far as you can in this problem. There are some trigonometric rules that are required to simplify the final equation using $u=\frac{L}{2}.]$

Comparing the equations for mid$-$span deflection in Problems $1$ and $3,$ what can be concluded about the effect of an axial compression force on the behavior of a beamcolumn? What is your guess if the axial force was tensile?