HW$01$

MEMT $212$

Last Revised $9/4/24$

$($adapted from Hibbeler ${10}^{th}$ ed$.$ probs. $4-31$ & $4-32)$

Eight A$992$ steel, $1$ inch diameter, reinforcing rods are embedded into high strength concrete to create the column shown. The column is subjected to an axial load of $200$kip.

a$.$ Find the average stresses carried by the concrete and by each steel rod.

b$.$ Suppose you wanted $60\%$ of the applied force to be carried in the concrete and $40\%$ to be carried in the steel. What diameter should be specified for the steel rods instead of $1$ inch? $($Assume nothing else changes to answer this question.$)$

Answers: $($a$.){}_{\text{c}\text{o}\text{n}\text{c}\text{r}\text{e}\text{t}\text{e}}=2\mathrm{.}29$ksi,${}_{\text{s}\text{t}\text{e}\text{e}\text{l}}=15\mathrm{.}8$ksi; $($b$.)0\mathrm{.}839$ in

$2.($adapted from Hibbeler ${10}^{th}$ ed$.$ probs. $4-51$ & $4-52)$

Assume bar ACEB is perfectly rigid, and is supported by two aluminum rods $($EF and CD$)$ that are connected with pinned ends. The elastic modulus of the rods is $10$ Mpsi $,$ and they each have diameters of $1$inch. Before the $2$ kip load is applied, bar ACEB is perfectly vertical, and no force is carried in the aluminum rods. After the $2$ kip load is applied, determine:

a$.$ the displacement of the tip of the bar $($point B$)$ due to the applied load.

b$.$ the force carried in each rod due to the applied load.

Answers: $($a$.)0\mathrm{.}00257$ in ; $($b$.)F_{CD}=0\mathrm{.}211$kip,$F_{EF}=1\mathrm{.}26$kip

$3.($adapted from Hibbeler ${10}^{th}$ ed$.$ Prob $4-54)$

Rod AC is composed of $2014-$T$6$ aluminum. It is reinforced with an A$992$ steel sleeve that is bonded firmly to it all the way along segment $BC.$ With no load applied, none of the materials are carrying any stress, and a gap of $0\mathrm{.}5$ mm is present relative to the lower rigid support $($at point C $).$ When a $400$ kN load is then applied as shown, determine:

a$.$ both support reactions $($ A and C $).$

b$.$ the displacement of point $B$ relative to its unloaded position.

Answers: $($a$.)F_D=218\mathrm{.}8kN,F_C=181\mathrm{.}2kN$; $($b$.)0\mathrm{.}61$ mm

$4.($adapted from Hibbeler ${10}^{th}$ ed$.$ Prob $4-60$ and $4-61)$

Three $0\mathrm{.}5-$inch diameter bolts composed of A$-36$ steel are used to affix the bracket shown to the wall. When an $8001$b force is applied as shown, determine:

a$.$ The force developed in each bolt

b$.$ The gap that opens between the bracket and the wall at bolt D$\text{'}$s location

For this analysis, assume:

both the bracket and the wall are rigid.

the bolts carry no shear; instead the ledge at A applies all of the required vertical reaction.

the bracket is shown with an exaggerated displacement due to the bolts stretching under the applied load.

the material within the grip of the bolts has $2$ inches of length before the load is applied.

there is no pre$-$load applied to the bolts $($i$.$e$.$ they are installed snug but not "torqued down"$)$

Answers: $($a$.)F_B=54\mathrm{.}2lb,F_C=163lb,F_D=380lb$; $($b$.)0\mathrm{.}000133$ in