Problem $1$: $(45$ points$)$

A building owner purchases an apartment building that she is converting into an event space. You are

hired to assess the existing structure for increased load on the floor framing. The existing reinforced

concrete beams are shown below in Figure $1.$

Ignore the compression steel $($top reinforcing steel$)$ when performing calculations.

Stirrups are shear reinforcement and do not contribute to the flexural capacity of the beam.

Assume beam is simply supported.

Reference sections, tables, and equations you use from ACI $318$ to solve this problem.

Dimensional parameters and material properties

Clear span of beam $={23}^{\text{'}}-6^{\text{'}\text{'}}.$

Compressive strength of concrete at $28$ days $=4,000$

Normal weight concrete $($self weight including rebar $=150pcf)$

Yield strength of steel reinforcement $=60$ksi

Service loads on the beam:

Service dead load $($not including self$-$weight $-$ you must calculate this$)=1\mathrm{.}0ki\frac{p}{f}t$

Service live load $=1\mathrm{.}6ki\frac{p}{f}t$

Perform each of the following calculations. Make sure to clearly label each subcalculation and box your

answer after each one.

$($a$)$ Calculate:

a$.$ Dimensions of the beam flange

b$.$ Weight$/$ft of the beam $($including beam flange calculated in $($a$))$

c$.$ Factored load demand on the beam, $w_u$

d$.$ Factored maximum moment demand on the beam, $M_u$

$($b$)$ Draw the factored shear and moment diagram for the factored loading on the beam

$($c$)$ Compute the effective depth $(d)$ for the beam. $($Hint: Another way of phrasing this is $-$ what is the

distance from the compression face of the t$-$beam to the centroid of the tension reinforcement bars?$)$

$($d$)$ Compute the design strength, $M_n$ of the beam ignoring the compression reinforcement. Is the beam

safe?