The two portions of member A B are glued together along a...

The two portions of member $A B$ are glued together along a plane forming an angle $\theta$ with the horizontal. Knowing that the ultimate stress for the glued joint is $2.5 \mathrm{ksi}$ in tension and $1.3 \mathrm{ksi}$ in shear, determine (a) the value of $\theta$ for which the factor of safety of the member is maximum, ( $b$ ) the corresponding value of the factor of safety. (Hint: Equate the expressions obtained for the factors of safety with respect to the normal and shearing stresses.)

Key Concepts

Optimization in Structural Design

This concept refers to the process of finding the most favorable design parameters that meet all performance criteria while ensuring safety. In problems like this, it often involves equating expressions for different failure modes—such as tension and shear—to determine the conditions (such as the angle of the joint) that maximize the factor of safety. Such optimization ensures a balanced design where no single mode of failure governs the safety of the system.

Factor of Safety

The factor of safety is a measure of the structural performance that indicates how much stronger a system is compared to the expected loads. It is typically defined as the ratio of the ultimate (or allowable) stress to the actual applied stress. In design, maximizing the factor of safety helps ensure that even with uncertainties in loads or material properties, the risk of failure is minimized.

Failure Criteria

Failure criteria define the conditions under which a material or joint fails. In the context of this problem, the material has specified ultimate strengths in tension and in shear. These criteria are used to determine whether a particular state of stress will cause failure, guiding the design to ensure that neither the tensile nor the shear stresses exceed their respective limits.

Stress Transformation

This concept involves calculating the stress components (normal and shear) acting on a plane that is oriented at an angle relative to the principal axes. The transformation equations allow engineers to understand how an applied load is resolved into stress components on any arbitrary plane, which is critical when analyzing joints or connections that are rotated relative to the loading direction.

Please give Ace some feedback