On all of these problems, do the following:

a$.$ Identiy the four boundary conditions $(2$ at each end, always$).$

b$.$ Write out the governing equation.

c$.$ Identify $q(x).$ Don't be afraid to write $"0"$ if there is no distributed load.

d$.$ Integrate $($a$)$ four times, explicitly identifying $V(x)$ and $M(x)$ after integrating once and twice,

respectively.

e$.$ Impose the four boundary conditions to solve for the four constants of integration. Now you

have $w(x),$ meaning you can get anything.

f$.$ Solve for any other variable $($end shear$/$moment$,$ maximum displacements, etc.$).$

We will start with one of the easiet possible problems of beam deflection and the only one that

deforms the beam into the shape of a circle: pure bending. Consdier the cantilver beam fixed at

the left end and a counter$-$clockwise point moment of $M^{**}$ at the right end $($don$\text{'}$t be confused by

the weird arrow, it is our usual ccw point moment$).$ The beam has constant El and a length of $L.$

Use direct integration of the $4-$th order beam equation to $1)$ find the expression for $w(x)$;$2$ find

the maximum displacement at the right end. Answers: $w(x)=M^{**}\frac{x^2}{2}EI,w_{\text{m}\text{a}\text{x}}=M^{**}\frac{L^2}{2}El.$