extend radially out from a central ring. The resulting idealized live load on each

beam is a linear distributed load.

Prob. $2\mathrm{.}5$

We do not yet know how many beams will be in the final design; therefore, we do

not yet know the distributed load values $w_1$ or $w_2.$

a$.$ To help in the design process, the team wants us to develop an expression

for the vertical displacement along the entire length of the beam. Before

doing so$,$ we want to identify some bounds to help verify our expression.

The location of the peak displacement will change with the ratio $\frac{w_1}{w_2}.$

We can consider two extremes for that ratio: $\frac{w_1}{w_2}=1($uniform

distributed load$)$ and $\frac{w_1}{w_2}=($triangular distributed load$).$ Using the

formulas inside the front cover of this text, find the location of the peak

displacement for each of these extremes measured from end $A.$

b$.$ Determine the expression for the vertical displacement in terms of $x$

measured from end $A.$ Although the displacement can be derived from the

loading $($we get to that in Chapter $5$: Deformations$),$ it is probably easier

in this case to use the formulas from the inside front cover of this text.

c$.$ Using a spreadsheet or another computer tool, plot the displaced shapes

for the following ratios of $\frac{w_1}{w_2}$:$1\mathrm{.}0,2\mathrm{.}5,5\mathrm{.}0.$ Show all three plots on one

graph. To make the plot, a team member recommends setting $E,$I,$l,$ and

$w_2$ all equal to $1.$ Use your results from part $($a$)$ to verify these results.

d$.$ From your results in parts $($a$)$ and $($c$),$ what do you conclude about the

sensitivity of the peak displacement location for this project?