$($a$)$ State four causes of singular stiffness matrices.

$($b$)$ Briefly explain the benefit of using isoparametric elements, and identify a situation in which a non$-$linear structural FE analysis should be used.

$($c$)$ Two element is shown in Figure $1$ with modulus of elasticity, $\backslash ($ E$=200\backslash$mathrm$\{$CPa$\}\backslash ),$ use direct stiffness method of finite element method to:

$($i$)$ Solve for the two elemental stiffness matrices.

$($ii$)$ Assemble the global stiffness matrix.

$($iii$)$ Compute the global applied force vector.

$($iv$)$ After applying the appropriate restraint condition$($s$),$ solve for the nodal displacements.

$($v$)$ Determine the reaction force$($s$)$ at the restraint$($s$).$

$($vi$)$ Solve for the element strains and stresses.

rigure I