$1)$ A typical cross$-$section of a properly reinforced con$-$crete beam is shown in the figure below. The material properties are summarized below. Use an effective depth of $650$ mm $.$

$\backslash [$

$\backslash$begin$\{$aligned$\}$

f$\_\{$y$\}$ & $=400\backslash$mathrm$\{$MPa$\}\backslash \backslash$

$\backslash$phi$\_\{$c$\}$ & $=0\mathrm{.}65\backslash \backslash$

$\backslash$phi$\_\{$s$\}$ & $=0\mathrm{.}85\backslash \backslash$

E$\_\{$s$\}$ & $=200000\backslash$mathrm$\{$MPa$\}$

$\backslash$end$\{$aligned$\}$

$\backslash ]$

a$)$ Is the beam properly reinforced$/$over$-$reinforced$/$ balanced Determine the factored moment resistance? $(15$ marks$)$

b$)$ Consider the beam section $"$a$)$ "above. However, due to embedded items, the reinforcing is raised by $65$ mm $,$ such that the effective depth is reduced to $585$ mm $($Draw the new cross$-$section view$).$ Find the factored moment resistance considering the new reinforcement layout. $(15$ marks$)$

c$)$ Comment on the effect of the change in reinforcement location on the moment resistance $(5$ marks$)$

d$)$ Refer to the beam original effective depth. The same contractor reported that, due to some problems related to batching, the actual $28-$day concrete strength is reduced by $\backslash (10\backslash \%\backslash ).$ What is the new $\backslash ($ F$\_\{$c$\}^\{\backslash$prime$\}\backslash )?$ Find the factored moment resistance for the new strength. $(15$ marks$)$

e$)$ What do you believe has a more substantial influence on the magnitude of the factored moment resistance: a $10\backslash \%$ decrease in the concrete strength $($as solved in part $"$d$)")$ or placement of reinforcement resulting in a $10\backslash \%$ decrease in the effective depth $($as solved in part $"$b$)").$ Why? $(10$ marks$)$