The 10 -mm-diameter steel bolt is surrounded by a bronze...

The 10 -mm-diameter steel bolt is surrounded by a bronze sleeve. The outer diameter of this sleeve is $20 \mathrm{mm},$ and its inner diameter is $10 \mathrm{mm}$. If the yield stress for the steel is $\left(\sigma_{Y}\right)_{\mathrm{st}}=640 \mathrm{MPa},$ and for the bronze $\left(\sigma_{Y}\right)_{\mathrm{br}}=520 \mathrm{MPa}$ determine the magnitude of the largest elastic load $P$ that can be applied to the assembly. $E_{\mathrm{st}}=200 \mathrm{GPa}, E_{\mathrm{br}}=100 \mathrm{GPa}$

The 10 -mm-diameter steel bolt is surrounded by a bronze sleeve. The outer diameter of this sleeve is $20 \mathrm{mm},$ and its inner diameter is $10 \mathrm{mm}$. If the yield stress for the steel is $\left(\sigma_{Y}\right)_{\mathrm{st}}=640 \mathrm{MPa},$ and for the bronze $\left(\sigma_{Y}\right)_{\mathrm{br}}=520 \mathrm{MPa}$
determine the magnitude of the largest elastic load $P$ that can be applied to the assembly. $E_{\mathrm{st}}=200 \mathrm{GPa}, E_{\mathrm{br}}=100 \mathrm{GPa}$

Key Concepts

Yield Stress

Yield stress is the maximum stress a material can withstand without experiencing permanent (plastic) deformation. In the context of the problem, it is crucial because the applied load must be limited so that neither the steel bolt nor the bronze sleeve surpasses its respective yield stress, ensuring the assembly remains in the elastic regime.

Modulus of Elasticity

The modulus of elasticity, or Young’s modulus, quantifies a material’s stiffness and relates stress to strain within the elastic limit. It is essential in composite systems because different materials (steel and bronze in this case) will deform differently under the same load due to their distinct moduli, influencing how the load is shared between them.

Composite Action in Parallel Members

In a composite structure where different materials are rigidly connected in parallel, such as a bolt with a surrounding sleeve, the key assumption is that all components undergo the same axial strain. This strain compatibility means that the load carried by each material is proportional to its stiffness (modulus times cross-sectional area), and it restricts the maximum load to the point where the weaker material reaches its yield stress first.

Load Distribution Based on Axial Stiffness

Because the materials in a composite member share the same deformation, the load carried by each component is determined by its axial stiffness (the product of its modulus of elasticity and cross-sectional area). This principle allows one to calculate the portion of the total load each material bears, ensuring the overall applied load does not cause any part to exceed its elastic limit.

Cross-Sectional Geometry Analysis

Determining the cross-sectional areas of the individual components (the inner bolt and the surrounding sleeve) is fundamental in analyzing how the load is distributed. The areas directly impact the axial stiffness of each component, thereby influencing the overall load capacity of the composite structure. This geometric evaluation is key to ensuring that the calculated load does not induce stresses beyond the yield point in any material.

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