56. A torque  is applied to a steel strip, as shown in the figure. The strip has a length of  and a thickness of . If the allowable shear stress of the steel is  and the maximum probability of failure is designed to be , determine the minimum width of the steel using membrane analogy theory.

Solution

According to the membrane analogy theory,

where  is the shear stress,  is the shear modulus,  is the angle of twist per unit length,  is the torque,  is the length of the median line, and  is the wall thickness.

The limit-state function is the shear stress of the strip subtracted from the allowable shear stress. Failure occurs when.

where .

Using FOSM, we have

The probability of failure is then given by

Thus

Solving for  yields

Ans.