Question 6.16: Consider yet again the bracket of the foregoing examples, an......
Consider yet again the bracket of the foregoing examples, and as shown again here in Figure a. Let the thickness h of the horizontal portion of the bracket be: 3/16 inch. Let the other dimensions and distances be the same as in Example Analysis 12. That is, a=2 in, b=12 in, c=4 in, d=8 in, e=4 in, h=3/16 in, 2r=1/4 in (a)
Let the depth, or width, t of the bracket be: 1 inch.
Let the weight load W be 100 pounds.
Determine the tensile stress σ_{Q} at point Q. Suggest design alternatives to reduce σ_{Q}.
The stress at Q is simply a bending, or flexural, stress arising due to the moment M of the weight W about Q. Recall then from elementary mechanics of materials that σ_{Q} may be expressed as:
σ_{Q}=M\left\lgroup\frac{h}{2} \right\rgroup \div I (a)
where I is the second moment of area of the bracket which may be expressed as:
I= t\left\lgroup\frac{h}{2} \right\rgroup ^{3}\div 12 = \frac{th^{3}}{96} (b)
By inspection of Figure a we see that the moment M of W about Q is simply:
M =Wb (c)
By substituting from Eqs. (b) and (c) into (a) we find σ_{Q} to be:
σ_{Q}=\frac{48Wb}{th^{2}} (d)
Finally, by substituting the given data values into Eq. (d), σ_{Q} becomes:
σ_{Q}=\frac{\left(48\right)\left(100\right) \left(24\right) }{\left(1\right) \left(3/16\right)^{2} }=3.277\times 10^{6} psi (e)