A steel pipe and an aluminum pipe are securely bonded...

A steel pipe and an aluminum pipe are securely bonded together to form the composite beam shown. The modulus of elasticity is 200 GPa for the steel and 70 GPa for the aluminum. Knowing that the composite beam is bent by a couple of moment $500 \mathrm{N} \cdot \mathrm{m}$ determine the maximum stress $(a)$ in the aluminum, $(b)$ in the steel.

Key Concepts

Section Transformation

Section transformation is a method used in the analysis of composite beams to convert the cross-sectional area of one material into an equivalent area in terms of another material’s properties. This technique simplifies the process of determining the neutral axis and moment of inertia for the composite section, enabling the application of standard bending formulas.

Bending Stress Distribution

Bending stress distribution in beams is governed by the flexural formula, which relates the bending moment, the distance from the neutral axis, and the section’s moment of inertia. In composite beams, this distribution is affected by the differing material properties, and the calculation must account for these differences to identify the maximum stresses in each material.

Moment of Inertia

The moment of inertia is a geometric property that quantifies a beam's resistance to bending by integrating the cross-sectional area relative to the neutral axis. For composite beams, the effective moment of inertia is calculated using the transformed areas of the different materials, which is critical for accurately determining bending stresses.

Modulus of Elasticity

The modulus of elasticity, or Young’s modulus, is a measure of a material's stiffness or resistance to elastic deformation under load. In composite beams, the different moduli of the constituent materials determine how stresses are shared during bending, making it crucial to account for these differences when analyzing the beam.

Composite Beam Analysis

Composite beam analysis involves studying beams made of two or more materials that are bonded together so that they act as a single unit under loads. This analysis requires careful consideration of the differing material properties, especially the moduli of elasticity, and often uses methods such as section transformation to ensure compatibility of strains and accurate prediction of stress distribution.

Neutral Axis

The neutral axis in a bending beam is the line along the cross-section where the material experiences zero stress. In composite beams, the location of the neutral axis is not simply the geometric centroid but must be determined by considering the transformed sections weighted by the differing moduli of elasticity to accurately reflect stress distribution.

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