USING MANUAL FINITE ELEMENT ANALYSIS $($stiffness and load matrices, boundary conditions, etc$)$

Q$1.$

The beam shown in the figure below is a wide$-$flange W$16$x$31$ with a cross$-$sectional area of $9\mathrm{.}12$ in $^(2)$ and a depth of $15\mathrm{.}88$ in $.$ The second moment of inertia is $375$in$^(4).$ The beam is subjected to a uniformly distributed load of $1000$ lb ft and a point load of $500$ ib $.$ The modulus of elasticity of the beam is E$=29$ xx$106$lb$//$in$^(2).$ Using manual calculation $($FEA$),$ determine

a$)$ the vertical displacement at node $3$ and

b$)$ the rotations $($slopes$)$ at nodes $2$ and $3.$

c$)$ The vertical displacement at the point where the $500$ lb force is applied.

d$)$ Compute the reaction forces at nodes $1$ and $2$ and reaction moments at node $1.$ Take the length of element $(1)$ as $10$ ft and the length of element $(2)$ as $5$ ft $.$ Note that the $500$lb force is applied at the middle of element $(2).$