Problem $3(25$ points$)$

The steel curved bar shown has rectangular cross$-$section with a radial height $h=6mm$ and thickness $b=4mm.$ The

radius of the centroidal axis is $R=80mm.$ A force $P=10N$ is applied as shown. Assume the steel modulus of

elasticity and shear modulus $E=207,000$MPa and $G=79\mathrm{.}3({10}^3)MPa,$ repectively.

Find the vertical deflection at point $B.$ Use Castigliano's method for a curved flexural member and since $\frac{R}{h}>10,$

neglect the effect of shear and axial load, thereby assuming that deflection is due to merely the bending moment.

Note the inner and outer radii of the curves bar are:

$r_0=80+\frac{1}{2}(6)=83mm,r_i=80-\frac{1}{2}(6)=77mm$

${\displaystyle \underset{0}{\overset{\frac{}{2}}{}}}si{n}^{2}d={\displaystyle \underset{0}{\overset{\frac{}{2}}{}}}co{s}^{2}d=\frac{}{4}$