Problem $1$: A beam has concrete with a compressive strength of $6,000$ psi and ASTM A$615$

Grade $60$ reinforcing steel. The cross section is $18$ in $.$ wide and $24$ in $.$ high. There is a single

layer of four No$.8$ reinforcing bars located $3$ in$.$ above the bottom surface of the beam.

a$.(15$ pts $)$ Calculate the moment at which concrete cracks, considering transformed section

properties. Calculate neutral axis location, maximum concrete and steel stresses using

geometric properties and $=M\frac{y}{\text{I}}.$

b$.(10$ pts $)$ Check the results of part $($a$)$ using section equilibrium.

c$.(15$ pts $)$ Calculate the moment at which concrete reaches its linearity limit in

compression, considering transformed section properties. This limit can be assumed as

$40\%fc.$ Calculate neutral axis location, maximum concrete and steel stresses using

geometric properties and $=M\frac{y}{\text{I}}.$

d$.(10$ pts$)$ Check the results of part $($c$)$ using section equilibrium.

e$.(15$ pts $)$ Calculate the nominal moment capacity, assuming an equivalent rectangular

stress block $($Whitney$\text{'}$s block$)$ for concrete in compression.

f$.(15$ pts $)$ Repeat part $($e$),$ assuming a triangular stress block with a peak value of $f_c.$

g$.(15$ pts $)$ Repeat part $($e$),$ assuming a parabolic stress block with a peak value of $f_c.$

h$.(5$ pts$)$ Compare your results in parts $($e$),($f$)$ and $($g$).$ Assuming the equivalent rectangular

stress block provides the most accurate results, comment on the loss of accuracy if

triangular or parabolic stress distribution was chosen.

You may find centroid locations

chniwn halnw Iispfill.

i$.$ Fill in the table below. Comment on how do $c,$I, fs and curvature change with increasing

load. Curvature is equal to $\frac{}{y}$ in a section.