Consider the bar as shown in fig $1.$ For each element $i,A_i$ and $l_i$ are the cross$-$sectional

area and length, respectively. Each element $i$ is subjected to a traction force $T_i$ per unit

length and a body force $f$ per unit volume. The units of $T_i,f,A_i,$ and so on are assumed to

be consistent. The Young's modulus of the material is $E.$ A concentrated load $P_2$ is

applied at node$2.$ Derive the structural stiffness matrix and nodal load vector.