Stress

Assuming:

• the material is continuous (has no voids)
• the material is cohesive (all portions are connected together)

Normal Stress

$o_z=\underset{\mathrm{\Delta }A\to 0}{lim}\frac{\mathrm{\Delta }{F}_z}{\mathrm{\Delta }A}$

Shear Stress

${\tau }_{zx}=\underset{\mathrm{\Delta }A\to 0}{lim}\frac{\mathrm{\Delta }{F}_x}{\mathrm{\Delta }A}$
${\tau }_{zy}=\underset{\mathrm{\Delta }A\to 0}{lim}\frac{\mathrm{\Delta }{F}_y}{\mathrm{\Delta }A}$

Sign Convention

A shear stress acting on a positive face of an element is positive if it acts in the positive direction of one of the coordinate axes and negative if it acts in the negative direction of an axis. A shear stress acting on a negative face of an element is positive if it acts in the negative direction of an axis and negative if it acts in the positive direction.

Shearing Force and Shear Stress

There is a tendency to shear the bolt along cross-sections $mn$ and $pq$.

• Shear Forces ($V$) act over cut surfaces of the bolt
• Bolt is in "Double shear"
• Simple statics in this case, $V=\frac{P}{2}$
• In this case: $A$ = cross sectional area of the bolt
• So, $\tau =\frac{P}{2A}=\frac{2P}{\pi d^2}$

Average Shear Stress