x$=5,$ y$=6,$ z$=0$

Problem $1$:

The ABC beam is supported by roller supports at A and $B$ and a pinned support at $C.$ The loads are uniformly distributed dead load $G_k=20k\frac{N}{m},$ and uniformly distributed imposed load $Q_k=4\mathrm{.}5+(\frac{x}{10})k\frac{N}{m}.$ Consider the length $L_2=6\mathrm{.}1+(\frac{y}{10})m$ and the length $L_1=0\mathrm{.}75L_{2}m.$ The shear forces $[$V$]$ and bending moments $[$M$]$ are as in Figure $1-1.$ Do the following:

a$)$ Define all necessary load combinations for the ultimate limit state design $($note that the structure is not symmetric$).$

b$)$ For each critical load combination, draw the corresponding bending moment and shear force diagrams via hand calculations $($do not use computer solver$),$ and determine the exact location of the maximum moment for both spans $AB$ and $BC.$

c$)$ Draw the shear force and bending moment envelopes derived from the diagrams in part $($b$),$ and explicitly state all the critical design shear and bending moment values.

d$)$

The magnitude of shear and moment:

$V_A=\frac{M_B}{L_1}+\frac{W_1{L}_1}{2}$

$V_{B\text{L}\text{e}\text{f}\text{t}}=\frac{W_1{L}_1}{2}-\frac{M_B}{L_1}$

$V_{B\text{R}\text{i}\text{g}\text{h}\text{t}}=\frac{W_2{L}_2}{2}-\frac{M_B}{L_2}$

$V_C=\frac{M_B}{L_2}+\frac{W_2{L}_2}{2}$