$($a$)$

Let $R$ represent the reaction force at Support B$.$ By releasing the beam at Support B and imposing a

force $R$ at Point $B,$ the deflection of the beam consists of two parts,i$.$e$.$

Part I$-$ the deflection caused by $M_B$;

Part II$-$ the deflection caused by $R$

Please treat R$,$ w$,$ h$,$ L$,$ E as variables in this step, the mathematical equation for the deflection at

Point B caused by $R($ Part II$)$ can be written as

$($Hint : to input equation $\frac{R^{2}L}{Q^{2}wh},$ you can type $\frac{R^{{?}^{{?}^{?}}}{2}^{***}L}{Q^{{?}^{{?}^{?}}}{2}^{**}{w}^{***}h})$A steel cantilever beam, of length $2L,$ has an I$-$section, see Figure Q$3.$ The cantilever beam is under a uniformly distributed load $q=250k\frac{N}{m}$ applied in its vertical plane of symmetry at OA part of the span. The yield strength of the steel is $[]=340$MPa; the Young's modulus $E=249$GPa.,$L=3m$ and the coordinates of Point $H$ are $x=0mm,y=-85mm,z=0mm.$

Figure Q$3$

a$)$

The factors of safety predicted at point $H$ by the maximum shear stress theory of failure $($Tresca criterion$)$