Problem $6^{**}$

A uniform beam element defined by Euler$-$Bernoulli beam theory has the shape functions

specified by Eq$.7$:

$N=[(1-3\frac{x^2}{L^2}+2\frac{x^3}{L^3})(x-2\frac{x^2}{L}+\frac{x^3}{L^2})(3\frac{x^2}{L^2}-2\frac{x^3}{L^3})(-\frac{x^2}{L}+\frac{x^3}{L^2})]$

Apply Eq$.8$ below and show that the element matrix $k$ agrees with the matrix determined

by direct method. $($Ref$.$ as given in Textbook Eq$.2\mathrm{.}3-5)$

$k={\displaystyle \underset{0}{\overset{L}{}}}{B}^{T}E{I}_{z}Bdx$

$**$ Problem $6$ The task is time$-$consuming for manual calculations: Please carry out, after your choice a or b below

a$)$ The task can be done in Matlab or other computerprogram and your code$/$input with result being the solution. It

is recommended to use nondimensional variable e$.$g$.$ ksi$=\frac{x}{L}$

b$)$ Do the control for one of the the matrix coefficients, e$.$g$.$ k$44,$ manually or in computerprogram