Consider the triangular axisymmetric element shown below, where z is the axis of symmetry. The element has

material properties: E$=10$GPa,$\backslash$rho $=1000$k$($g$)/($m$^(3)),$ and v$=0\mathrm{.}33.$ The coordinates of nodes i$,$j$,$ and m are shown below

in meters. The element is subject to gravity, where the gravitation constant is g$=10($m$)/($s$^(2)).$ The element is also rotating

about the z$-$axis with and angular velocity of $\backslash$omega $=100$ra$($d$)/($sec$).$ Please answer the following questions:

a$.$ Determine the global stiffness matrix, K$.$

b$.$ Determine the global force matrix, F$.$

c$.$ Determine the displacements of all nodes that are free to move.

d$.$ Identify the four stress states in the element after it has deformed.

e$.$ Determine the reaction forces at each node. Do the net forces $($applied $+$ reaction$)$ in the radial direction

sum to zero? Why or why not?

Consider the function

y$($x$)=$cos$($x$)$

a$\backslash$int$\_(-1)^1$ y$($x$)$dx exactly.

b$\backslash$int$\_(-1)^1$ y$($x$)$dx using the Gaussian Quadrature method with two sampling points.

c$\backslash$int$\_(-1)^1$ y$($x$)$dx using the Gaussian Quadrature method with four sampling points.

d