Problem$1(35$ points$)$

The frame shown in Figure $1$ has a pin support at $\backslash ($ A $\backslash ),$ a roller support at $\backslash ($ H $\backslash ),$ and a hinge at $\backslash ($ D $\backslash ).$ In addition, it has a horizontal member $\backslash ($ B G $\backslash )$ that is hinged at its ends, that is at points $\backslash ($ B $\backslash )$ and $\backslash ($ G $\backslash ),$ to the continuous columns $\backslash ($ A B C $\backslash )$ and $\backslash ($ E G H $\backslash ).$ As can be seen from Figure $1$ the loading consists of the following: a$)$ a uniformly distributed horizontal load acting along $\backslash ($ A B C $\backslash )$ of intensity $\backslash (15\backslash$mathrm$\{$kN$\}/\backslash$mathrm$\{$m$\}\backslash )(\backslash (\backslash$mathrm$\{$KN$\}/\backslash )$ vertical meter$)$; b$)$ a vertical concentrated load $12$ kN acting at $\backslash ($ F $\backslash )$

$1.$ Calculate the reactions at $\backslash ($ A $\backslash$& H $\backslash ).($draw any free body diagrams you consider and write the corresponding equations of equilibrium$)$

$2.$ Use an appropriate cut to calculate the axial force $\backslash ($ N$\_\{$C$\}\backslash ),$ shear force $\backslash ($ V$\_\{$C$\}\backslash ),$ and bending moment $\backslash ($ M c $\backslash )$ just to the right of point $\backslash ($ C $\backslash )($along member CDEF$)$

$3.$ Use an appropriate cut to calculate the axial force $\backslash ($ N$\_\{$E$\}\backslash ),$ shear force $\backslash ($ V$\_\{$E$\}\backslash ),$ and bending moment $\backslash ($ M$\_\{$E$\}\backslash )$ just to the right of point $\backslash ($ E $\backslash )($along member $\backslash ($ E F $\backslash ))$

$4.$ Draw the axial force diagram $[\backslash ($ N$]\backslash ).($Give the expression for $\backslash ($ N$($x$)\backslash )$ along each member$)$

$5.$ Draw the shear force diagram $[$V$].($Give the expression for $\backslash ($ V$($x$)\backslash )$ along each member$)$

$(6$points$)$

$6.$ Draw the bending moment diagram $[$M$].($Give the expression for $\backslash ($ M$($x$)\backslash )$ along each member. Clearly show the location$($s$)$ of the point$($s$)$ at which $\backslash ($ M$($x$)\backslash )$ reaches a local or global min. or max. and the corresponding values of $\backslash ($ M$($x$)\backslash )$ at these points$)$

$(9$ points$)$

$7.$ Draw the free body diagram involving only the immediate neighborhood around point E$.$ Clearly show all axial forces, shear forces and acting moments. Confirm that these forces and moments are balancing each other.

$8.$ Draw again the frame showing its members with some thickness and indicate the side $(2$ points$)$ on which you plan to place reinforcement.

Note: The expressions for $\backslash ($ N$($x$),$ V$($x$)\backslash )$ and $\backslash ($ M$($x$)\backslash )$ along the different members should be given in terms $1$