The rigid lever arm is supported by two A-36 steel wires...

The rigid lever arm is supported by two A-36 steel wires having the same diameter of $4 \mathrm{mm}$. Determine the smallest force $\mathbf{P}$ that will cause (a) only one of the wires to yield; (b) both wires to yield. Consider A-36 steel as an elastic perfectly plastic material.

Key Concepts

Yield Strength of Materials

Yield strength is the stress at which a material begins to deform plastically. Once the yield point is reached, permanent deformation occurs. In the context of the problem, determining the force that causes yielding in the wires involves knowing the yield strength of A-36 steel and the corresponding force required to reach that stress over the cross-sectional area of the wire.

Elastic-Perfectly Plastic Behavior

An elastic-perfectly plastic material exhibits linear elastic behavior up to its yield point, after which it deforms plastically at a constant stress level. This idealization simplifies the analysis because the material does not strain harden; once yield is achieved, any additional deformation occurs without an increase in force. This assumption is crucial for calculating the applied force needed for the wires to yield.

Tensile Stress in Structural Elements

Tensile stress is defined as the force per unit area acting on a material, and it plays a key role in determining when a tension member such as a steel wire yields. For wires with circular cross-sections, the area is determined by the diameter, and the tensile load required to reach yield is calculated by multiplying the yield stress by this area. This concept underpins the derivation of the force conditions for yielding in the problem.

Load Distribution in Supporting Structures

When a lever arm is supported by multiple wires, the manner in which the applied load is shared between them affects when each wire will reach its yield point. Understanding load distribution—how the force P is divided among the supports—is essential for determining cases where only one support may yield or both supports yield simultaneously. This involves geometrical and equilibrium considerations to assess the critical loads.

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