Problem $1.$ For each of the beam sections shown below find: a$)$ Centroid, b$)$ Iz $($bending about horizontal axis$),$ c$)$ Mcr$+$ d$)$ Mcr$.$ All beams are made with normal weight concrete unless otherwise indicated.

Problem $2.$ For each of the beam sections shown below find: e$)Mn+,Mn+,$ f$)M{n}^{-},Mn-$ ignoring any compression steel and any tension steel that is not located in the extreme layer of the beam. Verify if steel has yielded at Mn$+.$

Problem $3.$ For each of the beam sections shown below find: g$)$ Stresses in concrete and steel when the beam is subjected to the moment $M$ indicated in the figure.

Sip

$f^{\text{'}}c=4,000$

$12$:$n$

fy $=60,000$ psi

$M+=100ft-$ Kip

:ht

$($a$)$