Problem $12\mathrm{.}3($a$)$ Determine the internal axial force $x_C$ and the bending moment $M_C$ in the section $C$ of the rectangular frame shown in Fig. P$12\mathrm{.}3($a$),$ and the reactions at the fixed ends A and $B.$ The vertical force $F$ acts in the vertical plane of symmetry. The bending stiffness of all parts of the frame is $EI.($b$)$ Determine the deflection at point C$.($c$)$ Evaluate the results from parts $($a$)$ and $($b$)$ in the case $b=$a and $b=2a.[$Hint: Consider only one$-$half of the frame $($Fig$.$ P$12\mathrm{.}3($b$)).$ By symmetry, there is no internal shear force in the cross section $C.$ The axial force and the bending moment can be determined from the conditions that the horizontal displacement and the slope at $C$ must be equal to zero $($by symmetry$).$ Thus, $u_C=$del$\frac{U}{d}el{x}_C=0$ and ${}_C=$del$\frac{U}{d}el{M}_C=0.$ In the expression for the strain energy $U,$ ignore the strain energy contribution from the axial force and the shear force, and use only the bending strain energy.$]$

Figure P$12\mathrm{.}3$