Cantilever Beam System: For the $3-$span system shown in the figure below, the middle $-$ span is defined as $L_2=25ft.$ The Glulam combination is $20$F$-$E$3$ of Western Species $($DF$/$DF$).$ Since the top$-$fiber allowable tensile bending strength is about one$-$half of that for the bottom fiber, the length $L_1$ and Hinge Locations should be selected such that the maximum negative bending moment $)$ at the interior support is one$-$half the maximum positive moment $).$

$($a$)$ Show that the end$-$span $L_1$ is equal to $22\mathrm{.}8ft,$ rounded to ${22\mathrm{.}83}^{\text{'}}={274}^{\text{'}\text{'}}.$ Hint: $($i$)$ For the left span $(M+\frac{1}{4}M)W\frac{L_1^2}{8}$; $($ii$)$ for the center span $(M+\frac{1}{2}M)=w\frac{L_2^2}{8}.$ Solve for $L_1$ as function of $L_2=25ft.$ See the lecture notes.

$($b$)$ Show that the length $"$b$"$ of the suspended span is equal to $20\mathrm{.}4ft,$ rounded to ${20\mathrm{.}33}^{\text{'}}={244}^{\text{'}\text{'}}.$ Thus the cantilever length $a=2\mathrm{.}3ft,$ rounded to $a={2\mathrm{.}33}^{\text{'}}=28$ inches.

$($c$)$ Use the dimensions of $L_1=22\mathrm{.}83ft,L_2=25ft,a=2\mathrm{.}33ft.$ Thus the I.P$.$ is at $2\mathrm{.}5ft$ from the support. Sketch the structure and show the Maximum Moments $-$ The Negative $M_{\text{m}\text{a}\text{x}}$ will not be exactly half of the Positive $M_{\text{m}\text{a}\text{x}}$ due to rounding approximations with lengths.

$($d$)$ The total load $($DL including self$-$weight and LL$)$ is equal to $1\mathrm{.}6\frac{k}{f}t,$ normal duration $)=(1\mathrm{.}0,$ and applied to the entire $3-$span system. Consider only this total load combination over the whole structure. Select a beam size for bending, shear, and deflections with the limit of $\frac{L}{240}.$ Specify camber for the left span and the suspended span between hinges. Investigate the displacement of the hinge and make a judgement about design concerns, if any. Check bearing requirement for maximum reaction force. Explore and propose if a shallower section can be used for the suspended section $($relatively typical in construction$).$