The weight is suspended from steel and aluminum wires, each...

The weight is suspended from steel and aluminum wires, each having the same initial length of $3 \mathrm{m}$ and cross-sectional area of $4 \mathrm{mm}^{2}$. If the materials can be assumed to be elastic perfectly plastic, with $\left(\sigma_{Y}\right)_{\mathrm{st}}=120 \mathrm{MPa}$ and $\left(\sigma_{Y}\right)_{\mathrm{al}}=70 \mathrm{MPa},$ determine the force in each wire if the weight is (a) $600 \mathrm{N}$ and (b) $720 \mathrm{N} . E_{\mathrm{al}}=70 \mathrm{GPa}, E_{\mathrm{st}}=200 \mathrm{GPa}$

The weight is suspended from steel and aluminum wires, each having the same initial length of $3 \mathrm{m}$ and cross-sectional area of $4 \mathrm{mm}^{2}$. If the materials can be assumed to be elastic perfectly plastic, with $\left(\sigma_{Y}\right)_{\mathrm{st}}=120 \mathrm{MPa}$ and $\left(\sigma_{Y}\right)_{\mathrm{al}}=70 \mathrm{MPa},$ determine the force in each wire if the weight is
(a) $600 \mathrm{N}$ and
(b) $720 \mathrm{N} . E_{\mathrm{al}}=70 \mathrm{GPa}, E_{\mathrm{st}}=200 \mathrm{GPa}$

Key Concepts

Elastic Deformation

This concept refers to the reversible deformation of a material when it is subjected to stress. Under elastic deformation, materials obey Hooke’s Law, meaning the strain is directly proportional to the applied stress until the material reaches its yield point. Once the load is removed, the material will return to its original dimensions.

Plastic Deformation

Plastic deformation is the irreversible change in shape that occurs when a material is stressed beyond its elastic limit. In the context of a perfectly plastic material, once the yield stress is reached, the material continues to deform at a constant stress. This behavior is crucial for understanding how different materials respond once their elastic limits are exceeded.

Yield Strength

Yield strength is the maximum stress that a material can withstand while still undergoing elastic deformation. Beyond this point, permanent plastic deformation occurs. In problems involving load sharing between materials, the yield strength indicates which material will yield first and how the load is distributed when deformation becomes plastic.

Young's Modulus

Young's modulus is a measure of stiffness of an elastic material, defined as the ratio of stress to strain in the linear (elastic) part of the stress-strain curve. It determines how much a material deforms under a given load, and differences in Young's modulus between materials affect how loads are shared in composite systems.

Load Sharing in Parallel Systems

In systems where multiple elements (such as wires) are connected in parallel and subjected to the same load, the total applied force is distributed among the elements. The division of load depends on the stiffness (Young's modulus) and yield strength of each component. This concept helps to analyze how different materials with varying elastic and plastic properties share the load when subjected to the same overall force.

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