Problem $3$

Consider the structure below. The thickness of beam B$-$D is $t_{BD}.$ and it is connected to the support

beam $($C$-$F$)$ by a bolt at C $,$ which acts like a pin in single shear. The diameter of the bolt at C is $d_C.$

a$)$ Compute the average shear stress in the bolt at C and the bearing stress at C in beam $B-D.$

b$)$ Determine the minimum cross$-$section of rod A$-$B if the allowable normal stress of

member $A-B$ is ${}_{\text{a}\text{l}\text{l}\text{o}\text{w}\text{,}AB}.$

c$)$ The pin at $A$ is in double shear and has diameter $d_A.$ Determine the average shear stress

of the pin at $A.$

d$)$ Assuming that the stress is uniform over the cross$-$sections of member AB and the pins,

draw a differential element showing:

The stresses at a cross$-$section of member AB $.$ Use the following coordinate axes to draw the

element: the $x$ axis is parallel to the axis of bar $AB$ with the positive sense going from $A$ to $B,$

the positive axis y is $90$ counterclockwise to the positive x axis and the positive z axis is

perpendicular to the page $($as per the right$-$hand rule$).$

The stresses at a cross$-$section of the pin at point $C.$ Note that the coordinate axes are defined

as: the positive x axis is parallel to the resultant force acting at the pin $($with the same sense as

the force$),$ the positive y axis is $90$ counterclockwise with respect to the positive x axis, and

the positive $z$ axis is perpendicular to the page $($as per the right$-$hand rule$).$

Take $t_{BD}=\frac{3}{4}$ in$,d_A=\frac{1}{4}$ in$,d_C=\frac{1}{2}$ in$,$ and $a=1in.$

Take ${}_1=20,{}_2=70,P=400lb,$ and ${}_{\text{a}\text{l}\text{l}\text{o}\text{w}\text{,}AB}=3\mathrm{.}5$ksi