$($a$)$ A cantilever beam ABC is loaded with a uniformly distributed load of $4k\frac{N}{m}$ at region BC and an external horizontal load at point C of $100$ kN $.$ Total length of beam is $2$ m $.$ The boundary at point A is a fixed boundary. The dimension of beam and the load distribution are shown in Figure $2($a$).$ The cross$-$section of the beam is shown in Figure $2($b$).$ Neglect self$-$weight of the beam. Neglect buckling in this problem. The allowable yield stress is $250$ MPa after considering safety factor. Assume von$-$Mises criterion is used. The heigh of beam is $h=500mm.$ Width of beam is $b=100mm.$ Top flange and bottom flange are $10$ mm $.$ Web plate thickness is $5$ mm $.$

$($i$)$ Find the maximum tensile normal stress at section A$.$ Find the maximum shear stress at section A $.$

$(5$ marks$)$

$($ii$)$ Check the safety of section A at the connection between top flange plat and web using von$-$Mises criterion. Check safety of material at section A$.$

$(5$ marks$)$

Hint: It is acceptable to assume the shear stress are uniformly distributed in the web plate. During calculation of the moment of inertia of this section as shown in Figure $2($b$),$ it is possible to use the following formula: $I=\frac{1}{12}b{h}^3-\frac{1}{12}(b-t_w)(h-2t_f{)}^3.$ Where $b$ is the total width, $h$ is the total height, $t_w$ is the web thickness, $t_f$ is the flange plate thickness. When the point is only subject to normal stress $$ and shear stress $,$ von$-$Mises criterion is formulated as: $\sqrt[\phantom{2}]{{}^2+3{}^2}[]$