Problem $5(30$ points$)$

A No$.1$ grade of Red Oak $412$ dimension lumber is used for a simply supported beam supporting a point live load as shown in Figure $1.$

a$)$ Determine the maximum allowable load imposed by the hoist based on bending.

b$)$ Determine the maximum allowable load imposed by the hoist based on shear.

c$)$ Determine the maximum load based on bending and shear.

Given:

$-$ Treat the hoist load as live load.

$-$ Moisture content is $\backslash (35\backslash \%\backslash ).$

$-$ Lateral support is provided to prevent buckling.

$-$ Sustained Exposure to $110$ degrees F$.$

Parameters:

$-\backslash (\backslash$mathrm$\{$L$\}=8\backslash$mathrm$\{$ft$\}\backslash )($span$)$

Hint:

$1.$ Use Tabulated Moments and Shears for Beams provided in the lecture notes $3.$

$2.$ Use Table $1$ B for section properties $($lecture notes$).$

$3.$ Design values are provided in table below.

$4.$ For Part $\text{'}\backslash ($ a $\backslash )\text{'},$ the adjusted design bending value shall be equal or greater than the bending stress. From the bending stress equation $($stress$=$ Bending moment$/$S$),$ back$-$calculate $\backslash ($ M $\backslash ),$ and using the moment equation, back$-$calculate P $.$

$5.$ For Part $\text{'}$b$\text{'},$ the adjusted design shear value shall be equal or greater than the shear stress. From the shear stress equation $($stress$=1\mathrm{.}5$xshear$/$A$),$ back$-$calculate $\backslash ($ V $\backslash ),$ and using the shear equation, back$-$calculate $\backslash ($ P $\backslash ).$

$\backslash$begin$\{$tabular$\}\{|$c$|$c$|$c$|$c$|$c$|$c$|$c$|$c$|$c$|$c$|$c$|\}$

$\backslash$hline $\backslash$multirow$\{3\}\{*\}\{$Species and commercial grade$\}$ & $\backslash$multirow$\{3\}\{*\}\{$Size classification$\}$ & $\backslash$multicolumn$\{7\}\{|$c$|\}\{$Design values in pounds per square inch $($psi$)\}$ & $\backslash$multirow$[$b$]\{3\}\{*\}\{\backslash$begin$\{$tabular$\}\{$l$\}$

Specific $\backslash \backslash$

Gravity $\backslash (\{\}^\{4\}\backslash )\backslash \backslash$

$\backslash ($ G $\backslash )$

$\backslash$end$\{$tabular$\}\}$ & $\backslash$multirow$\{3\}\{*\}\{\backslash$begin$\{$tabular$\}\{$l$\}$

Grading $\backslash \backslash$

Rules $\backslash \backslash$

Agency

$\backslash$end$\{$tabular$\}\}\backslash \backslash$

$\backslash$hline & & $\backslash$multirow$[$t$]\{2\}\{*\}\{\backslash [$

$\backslash$begin$\{$array$\}\{$c$\}$

$\backslash$text $\{$ Bending $\}\backslash \backslash$

F$\_\{\backslash$mathrm$\{$b$\}\}\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$array$\}$

$\backslash ]\}$ & $\backslash$multirow$[$t$]\{2\}\{*\}\{$Tension parallel to grain

$\backslash [$

F$\_\{1\}$

$\backslash ]\}$ & $\backslash$multirow$[$t$]\{2\}\{*\}\{$Shear parallel to grain

$\backslash [$

F$\_\{\backslash$mathrm$\{$y$\}\}$

$\backslash ]\}$ & $\backslash$multirow$[$t$]\{2\}\{*\}\{$Compression perpendicular to grain $\backslash ($ F$\_\{$G L$\}\backslash )\}$ & $\backslash$multirow$[$t$]\{2\}\{*\}\{$Compression parallel to grain $\backslash ($ F$\_\{6\}\backslash )\}$ & $\backslash$multicolumn$\{2\}\{|$l$|\}\{$Modulus of Elasticity$\}$ & & $\backslash \backslash$

$\backslash$hline & & & & & & & E & $\backslash (\backslash$mathrm$\{$E$\}\_\{\backslash$min $\}\backslash )$ & & $\backslash \backslash$

$\backslash$hline $\backslash$multicolumn$\{11\}\{|$l$|\}\{$RED OAK$\}\backslash \backslash$

$\backslash$hline Select Structural & $\backslash$multirow$\{4\}\{*\}\{\backslash (2^\{*\}8\backslash )$ wider$\}$ & $1,150$ & $675$ & $170$ & $820$ & $1,000$ & $\backslash (1,400,000\backslash )$ & $510,000$ & $\backslash$multirow$\{8\}\{*\}\{0\mathrm{.}67\}$ & $\backslash$multirow$\{8\}\{*\}\{$NELMA$\}\backslash \backslash$

$\backslash$hline No$.1$ & & $825$ & $500$ & $170$ & $820$ & $825$ & $\backslash (1,300,000\backslash )$ & $470,000$ & & $\backslash \backslash$

$\backslash$hline No$.2$ & & $800$ & $475$ & $170$ & $820$ & $625$ & $\backslash (1,200,000\backslash )$ & $440,000$ & & $\backslash \backslash$

$\backslash$hline No$.3$ & & $475$ & $275$ & $170$ & $820$ & $375$ & $1,100,000$ & $400,000$ & & $\backslash \backslash$

$\backslash$hline Stud & $\backslash (2^\{*\}8\backslash )$ wider & $625$ & $375$ & $170$ & $820$ & $400$ & $\backslash (1,100,000\backslash )$ & $400,000$ & & $\backslash \backslash$

$\backslash$hline Construction & $\backslash$multirow$[$t$]\{3\}\{*\}\{\backslash [$

$2^\{*\}-4^\{*\}\backslash$text $\{$ wide $\}$

$\backslash ]\}$ & $925$ & $550$ & $170$ & $820$ & $850$ & $\backslash (1,200,000\backslash )$ & $440,000$ & & $\backslash \backslash$

$\backslash$hline Standard & & $525$ & $300$ & $170$ & $820$ & $650$ & $\backslash (1,100,000\backslash )$ & $400,000$ & & $\backslash \backslash$

$\backslash$hline Utility & & $250$ & $150$ & $170$ & $820$ & $425$ & $\backslash (1,000,000\backslash )$ & $370,000$ & & $\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\}$