The bar is connected to the support using a pin having a...

The bar is connected to the support using a pin having a diameter of $d=1$ in. If the allowable tensile stress for the bar is $\left(\sigma_{t}\right)_{\text {allow }}=20 \mathrm{ksi},$ and the allowable bearing stress between the pin and the bar is $\left(\sigma_{b}\right)_{\text {allow }}=30 \mathrm{ksi}$ determine the dimensions $w$ and $t$ so that the gross area of the cross section is $w t=2$ in $^{2}$ and the load $P$ is a maximum. What is this maximum load? Assume the hole in the bar has the same diameter as the pin.

Key Concepts

Allowable Tensile Stress

This concept refers to the maximum tensile (pulling) stress that a material can safely withstand without experiencing failure. It is a critical parameter in the design of structural members under tension, as it determines the minimum cross?sectional area required to carry an applied load without exceeding the material’s strength limit.

Allowable Bearing Stress

Allowable bearing stress is the maximum contact pressure that can be applied between two interfacing parts—such as a pin and the connected member—without causing excessive deformation or damage. This ensures that localized stresses at the interface do not lead to premature failure or material yielding.

Gross Cross-Sectional Area

The gross cross-sectional area is the total area of a member’s cross section before considering any holes or notches. It is used in stress calculations to estimate the average stress experienced by the component under load, and it serves as a basis for ensuring that the structural element has sufficient area to support applied forces.

Design Optimization Under Constraints

Design optimization involves determining the most effective dimensions or configuration of a component—to maximize performance measures like load capacity—while satisfying multiple constraints (e.g., stress limits, geometric requirements). This process is essential in engineering to achieve an efficient and safe design that meets all specified criteria.

Pin Connections in Structural Members

Pin connections are a common method of joining components where a pin is inserted through aligned holes in the connected members. The design of these joints must ensure that both the overall material (through tensile stress) and the interface between the pin and the member (through bearing stress) can safely transfer the applied loads without failure.

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