Consider the plane frame structure shown in the figure below. The rotational springs attached to the members represent a partially rigid foundation. A concentrated force and a moment are applied as indicated. Taking advantage of structural symmetry, analyze the structure by the direct stiffness method. Clearly indicate the units used. Do NOT neglect the axial deformation of the members.

Young's modulus of the beam elements is $E=30,000$ksi. The moment of inertia of the beam elements is $I_z=200i{n}^4.$ The cross$-$sectional area of the beam elements is $A=20i{n}^2.$ The stiffness coefficient of the rotational spring element is $k_{}={1\mathrm{.}5}^{**}{10}^4$kip$*f\frac{t}{r}ad.$

i$)$ Decompose the external loads to symmetric and anti$-$symmetric components.

ii$)$ Draw $\frac{1}{2}$ structures for symmetric and anti$-$symmetric loading components. $($Clearly indicate the boundary conditions for each case on the plane of symmetry$)$

iii$)$ With the symmetric component, analyze the left $\frac{1}{2}$ of the structure

a$.$ Compute nodal displacements and rotations.

b$.$ Compute member end forces for frame and rotational spring element.

c$.$ Compute reactions.

d$.$ Draw the shear force and bending moment diagrams for the entire structure.

iv$)$ With the anti$-$symmetric component, analyze the left $\frac{1}{2}$ of the structure

a$.$ Compute nodal displacements and rotations.

b$.$ Compute member end forces for frame and rotational spring element.

c$.$ Compute reactions.

d$.$ Draw the shear force and bending moment diagrams for the entire structure.

v$)$ Using the principle of superposition and the results of $($iii$)$ and $($iv$),$ analyze the entire structure with the original loads.

a$.$ Draw the shear force and bending moment diagrams for the entire structure.

b$.$ Sketch the deflection of the entire structure