$4\mathrm{.}7$ A T beam $($Fig$.$ P$4\mathrm{.}7)$ supports in addition to its own weight a live load of $60psf.$ The following information is provided: $f_c^{\text{'}}=6000$;$f_{ci}^{\text{'}}=4500\frac{\text{;}}{b}ar({)}_{ti}=-201\frac{\text{;}}{b}ar({)}_{ci}=2700$; $\frac{?}{\text{b}\text{a}\text{r}}({)}_{ts}=-465\frac{\text{;}}{b}ar({)}_{\text{c}\text{s}\text{u}\text{s}}=2700\frac{\text{;}}{b}ar({)}_{cs}=3600$;$=0\mathrm{.}80$;$(d_c{)}_{\text{m}\text{i}\text{n}}=3in$;$(e_0{)}_{mp}=y_b-3in$; ${}_c=150pcf$;$f_{pu}=270$ksi;$f_{pe}=151$ksi; final effective force of $1$ strand $=23\mathrm{.}1$kips. Assume stress$-$relieved bonded strands.

$($a$)$ Assuming you are told there is a wide feasibility domain for $F$ and $e_0,$ determine the value of $F$ necessary at midspan. Round off its value to the nearest integer number of strands. Check that all stresses are within allowable limits$.$

$($b$)$ Determine graphically the feasibility domain for the beam and find graphically the value of $F($Use graph paper$).$ This should lead to the same answer as in $($a$).$

$($c$)$ Assuming the eccentricity is fixed at $e_o=y_b-(d_c{)}_{\text{m}\text{i}\text{n}},$ what is the maximum value of $F$ that the beam can be subjected to$,$ without any of the allowable stresses being exceeded?

Figure P$4.$:

$($d$)$ Let us assume that the live load is not specified. Assuming the eccentricity is fixed at $e_o=y_b(d_c{)}_{\text{m}\text{i}\text{n}},$ what is the maximum value of live load and corresponding $F$ that can be applied to the beam $($from a working stress design approach in flexure only$).$

Going back to question $($a$)$:

$($e$)$ Determine the two limits of the limit kern.

$(1)$ Determine the upper and lower limits of the steel envelopes at every tenth of the span.