$(60$ points$)$ The simply supported I shaped beam supports the loadings shown.

a$)$ Determine the stress state at point $K.$

b$)$ Determine the principal directions at point $K.$

c$)$ Calculate the principal stresses at point $K.$

d$)$ Show these stresses on drawings of properly oriented elements.

$=+-\frac{M}{\text{I}}y$;$,=\frac{VQ}{It}$;$,I=\frac{b{h}^3}{12}$;$,I_{{}^{\text{'}}}=I_{}+d^{2}A$

$tan2{}_p=\frac{2{}_{xy}}{{}_x-{}_y}$;$,{}_{1,2}=\frac{{}_x+{}_y}{2}+-\sqrt[\phantom{2}]{(\frac{{}_x-{}_y}{2}{)}^2+{}_{xy}^2}$;

${}_{x^{\text{'}}}=\frac{{}_x+{}_y}{2}+\frac{{}_x-{}_y}{2}cos2+{}_{xy}sin2$;${}_{\text{m}\text{a}\text{x}}=\sqrt[\phantom{2}]{(\frac{{}_x-{}_y}{2}{)}^2+{}_{xy}^2}$