Problem: The frame below must support a 0.20 kN load at point C. The members have rectangular cross sections 1.2 cm by 2.6 cm, and the bolts are all 8 mm diameter. Which bolts are in double shear? What is the maximum shearing stress? Is member BD in tension or in compression? Determine the normal stress in member BD.
Solution: Since the forces on BD
are applied only at two points, this is a two-force member and the forces
must be axial. Member ABC is a three-force member. Clearly, member BD must
be in compression. The bolt at B is in double shear. The free body diagrams
are:
 |
 |
The angle 
.
Since member ABC is in equilibrium, the three forces must form a triangle,
where the angle 
is
the complement of 36.3o:
 |
 |
Note that both possibilities for FA have the correct x-component, which must balance the x-component of FBD. Initially, I drew the force triangle below (on the left) using the solid FA choice. But the forces must go through a single point, and only the "dashed-line" choice can do that (force triangle on the right). The calculations are exactly the same; it may seem a little odd, but the result is the same regardless of which triangle is used. That's because the x-component is the same. The math would yield a y-component (because of a quadratic) that would be either positive or negative, and we would have to choose the negative from the square root.
It is easier to take moments about A than use the force traingle:
Returning to the force triangle,
The cross-sectional area of a bolt is 
.
Now I am ready to compute stresses...
a) normal stress in BD: 
b) shearing stress in the bolt at B: 
.
The stress is the same at D.
c) shearing stress in the bolt at A: 
.