Problem $2$:

The double fixed$-$end beam in Figure $2\mathrm{.}1$ must sustain a design ultimate distributed torsional load of $t=(4\mathrm{.}5+0\mathrm{.}05x)kN\frac{m}{m}$ along the span $($due to the eccentric loading$),$ in addition to the superimposed dead load $(30+0\mathrm{.}1y)k\frac{N}{m}($self$-$weight not included$)$ and live load $(25+0\mathrm{.}1x)k\frac{N}{m}.$ Take the density of $\frac{R}{C}$ as $24\mathrm{.}5k\frac{N}{m^3}$ and assume mild exposure conditions.

Material properties:

$f_{cu}=30$MPa;

$f_y=500$MPa;

$f_{yv}=250$MPa

Figure $3$: The double fixed$-$end beam

The cross section of the beam is rectangular and shown in Figure $2\mathrm{.}2.$ The flexural design results indicate that the area of tension reinforcement needed at the support is $A_s=2500m{m}^2$ and the tension reinforcement needed at the midspan is $A_s=1300m{m}^2.$

a$)$ Determine the shear force for the beam in Figure $2\mathrm{.}1$ and draw the shear force envelop.

b$)$ Figure $2\mathrm{.}3$ shows the torsional moment diagram of the fixed end bean in Figure $2\mathrm{.}1.$ Design the section for shear and torsion.

c$)$ Design the beam for serviceability limit state $($SLS$).$ Determine a suitable arrangement for required reinforcement and sketch the steel layout.

Hint: For the seek of simplicity, you don't need to check the deflection of the beam.

Figure $2\mathrm{.}2.$ Rectangular section and its material properties

Figure $2\mathrm{.}3$: Torsional moment diagram of the beam