An I$-$beam has a flange width bb $=175$ mmmm $,$ height hh $=175$ mmmm $,$ web thickness twtw $=7\mathrm{.}5$ mmmm $,$ and flange thickness tftf $=11$ mmmm $.$ Use the following steps to calculate the shear stress at a point $60$ mmmm above the neutral axis. Figure An I$-$beam has a flange width $\backslash ($ b$=175\backslash$mathrm$\{$~mm$\}\backslash ),$ height $\backslash ($ h$=175\backslash$mathrm$\{$~mm$\}\backslash ),$ web thickness $\backslash ($ t$\_\{$w$\}=7\mathrm{.}5\backslash$mathrm$\{$~mm$\}\backslash ),$ and flange thickness $\backslash ($ t$\_\{$f$\}=11\backslash$mathrm$\{$~mm$\}\backslash ).$ Use the following steps to calculate the shear stress at a point $60$

Learning Goal:

To calculate the shear stress at a point in the web of an I$-$beam section subjected to a shear force.

When a beam section is subjected to a shear load, a shear stress distribution is developed on the section. The distribution of the shear stress is not linear. Elasticity theory can be used to calculate the shear stress at any point.

However, a simpler method can be used to calculate the average shear stress across the width of the section, a distance $\backslash ($ y $\backslash )$ above or below the neutral axis. The average shear stress is given by $\backslash (\backslash$tau$=\backslash$frac$\{$V Q$\}\{$I t$\}\backslash ).$ Here $\backslash ($ V $\backslash )$ is the shear force on the section, $\backslash ($ I $\backslash )$ is the moment of inertia of the entire section about the neutral axis, and $\backslash ($ t $\backslash )$ is the width of the section at the distance $\backslash ($ y $\backslash )$ where the shear stress is being calculated. $\backslash ($ Q $\backslash )$ is the product of the area of the section above $($or below$)\backslash ($ y $\backslash )$ and the distance from the neutral axis to the centroid of that area $($Figure $1).$ In short, $\backslash ($ Q $\backslash )$ is the moment of the area about the neutral axis.

mm above the neutral axis.

Part A $-$ Moment of inertia

The shear formula includes the moment of inertia of the whole cross section, $\backslash ($ I $\backslash ),$ about the neutral axis. Calculate the moment of inertia.

Express your answer with appropriate units to three significant figures.

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$\backslash [$

I$=$

$\backslash ]$

Units

Part B $-\backslash ($ Q $\backslash )$ for the given point

The shear stress $60$ mm above the neutral axis depends on the value of $\backslash ($ Q $\backslash )$ for the area above that point. Calculate the value of $\backslash ($ Q $\backslash ).$

Express your answer with appropriate units to three significant figures.

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$\backslash [$

Q$=[$

$\backslash ]$ Part B $-\backslash ($ Q $\backslash )$ for the given point

The shear stress $60$ mm above the neutral axis depends on the value of $\backslash ($ Q $\backslash )$ for the area above that point. Calculate the value of $\backslash ($ Q $\backslash ).$

Express your answer with appropriate units to three significant figures.

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$\backslash [$

Q$=\backslash$quad $\backslash |$

$\backslash ]$

Part C $-$ Shear stress

Use the results from Parts A and B to calculate the shear stress at a point $60$ mm above the neutral axis if the shear force on the section is $\backslash ($ V$=8\backslash$mathrm$\{$kN$\}\backslash ).$

Express your answer with appropriate units to three significant figures.

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$\backslash [$

$\backslash$tau$=\backslash$square

$\backslash ]$