## 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
Max shear stress:
$$\tau_{max}=\frac{\sigma_1-\sigma_3}{2}$$$where $$\sigma$$ are the principal stresses. 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.