## Max Shear Stress Theory

In the below figure, a ductile beam is in tension due to an applied load, F, which creates the three corresponding principal stresses. Does failure occur if the materialâ€™s yielding strength is 40 MPa?

##
__
__**Hint**

**Hint**

Max shear stress:

$$$\tau_{max}=\frac{\sigma_1-\sigma_3}{2}$$$

where
$$\sigma$$
are the principal stresses.

##
__
__**Hint 2**

**Hint 2**

Yielding occurs when

$$$\tau_{max}\geq \frac{S_y}{2}$$$

where
$$\tau_{max}$$
is the max shear stress, and
$$S_y$$
is the yield strength.

For a ductile material in tension, the max shear stress theory predicts that yielding will occur whenever

$$$\tau_{max}\geq \frac{S_y}{2}$$$

where
$$\tau_{max}$$
is the max shear stress, and
$$S_y$$
is the yield strength. To find the max shear stress:

$$$\tau_{max}=\frac{\sigma_1-\sigma_3}{2}$$$

where
$$\sigma$$
are the principal stresses. Thus,

$$$\tau_{max}=\frac{24MPa-(-35MPa)}{2}=\frac{59MPa}{2}=29.5\:MPa$$$

$$$\frac{S_y}{2}=\frac{40MPa}{2}=20\:MPa$$$

Because
$$29.5\:MPa\geq 20\:MPa$$
, failure does occur.

Failure occurs.