Problem $2(10$ points$)$

A simple supported beam $12ft$ long carries service dead load $=300l\frac{b}{f}t$ and live loads $=200l\frac{b}{f}t.$ Design an appropriate beam, and check whether the amount of tension steel is within limits of ductile beam behavior. Steel strength is $60$ksi, and concrete strength is $5$ksi.

Note:

$=0\mathrm{.}85$ for $F_c^{\text{'}}4000$ psi

For larger $F_c^{\text{'}}$ : substract $0\mathrm{.}05$ for each additional $1000$ psi up to a minimum of $0\mathrm{.}65.$

Steps:

Calculate the ultimate design load and the moment that the beam needs to carry. Remember the Moment equation for a simple supported beam is:

$M=w\frac{L^2}{8}$

Estimate the gross beam size.

Remember For a simply supported beam, the height of the beam must be at minimum the length of the beam divided by sixteen $(h=\frac{\text{I}}{16})$

Assume gross beam size:

$h=$

$d=$

$b=$

Calculate minimum steel ratio. Assume steel reinforcement ratio and determine $A_s.$

Remember the following equations:

${}_{\text{m}\text{i}\text{n}}=\frac{3\sqrt[\phantom{2}]{?}{F}^{\text{'}}c}{F_t}$

$A_s=bd$

Try steel bar size and determine the Ultimate moment capacity.

Remember the following equations:

$a=A_s\frac{F_t}{0\mathrm{.}85}{F}_c^{\text{'}}b$

$M_u=A_s{F}_t[d-\frac{a}{2}]$

Compare Ultimate moment capacity with the moment that the beam needs to carry.

Check for over reinforcement. Compare $$ with ${}_{\text{m}\text{a}\text{x}}$

Remember $$ max equation:

${}_{\text{m}\text{a}\text{x}}=0\mathrm{.}319(F^{\text{'}}\frac{C}{F}t)$