Five metal strips, each 0.5 ×1.5 -in. cross section, are bonded together to form the composite b

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Five metal strips, each 0.5 ×1.5 -in. cross section, are...

Five metal strips, each $0.5 \times 1.5$ -in. cross section, are bonded together to form the composite beam shown. The modulus of elasticity is $30 \times 10^{6}$ psi for the steel, $15 \times 10^{6}$ psi for the brass, and $10 \times 10^{6}$ psi for the aluminum. Knowing that the beam is bent about a horizontal axis by a couple of moment 12 kip-in., determine (a) the maximum stress in each of the three metals, (b) the radius of curvature of the composite beam.

Five metal strips, each $0.5 \times 1.5$ -in. cross section, are bonded together to form the composite beam shown. The modulus of elasticity is $30 \times 10^{6}$ psi for the steel, $15 \times 10^{6}$ psi for the brass, and $10 \times 10^{6}$ psi for the aluminum. Knowing that the beam is bent about a horizontal axis by a couple of moment 12 kip-in., determine
(a) the maximum stress in each of the three metals,
(b) the radius of curvature of the composite beam.

Key Concepts

Radius of Curvature Calculation

The radius of curvature of a bent beam is derived from the flexural formula M = EI/?, where M is the bending moment, E is the modulus of elasticity, I is the moment of inertia, and ? is the radius of curvature. For composite beams, an equivalent or effective flexural rigidity (EI) is computed to determine the overall curvature under a bending moment.

Bending Stress Formula

The bending stress in a beam is calculated using the formula ? = (M*y)/I, where M is the applied bending moment, y is the distance from the neutral axis, and I is the moment of inertia of the section. In composite beams, this formula is applied separately for each material after accounting for the different moduli of elasticity.

Neutral Axis Determination

In a composite beam, the neutral axis is the line along which the bending stress is zero. Determining its location is essential because the bending stresses in each material depend on their distance from this axis. The location is found by ensuring that the sum of the moments of the transformed areas about the neutral axis is zero.

Composite Beam Analysis

Composite beam analysis involves studying beams made up of two or more materials with different mechanical properties that are bonded together to act as a single unit. This requires accounting for the different moduli of elasticity and stresses in each material while ensuring compatibility of deformations and equilibrium under bending loads.

Modular Ratio

The modular ratio is the ratio of the modulus of elasticity of one material to that of another (often a chosen reference material). It allows for the transformation of areas from one material into an equivalent area of the reference material, enabling analysis of composite structures with varying stiffnesses.

Transformed Section Method

The transformed section method is a technique used to analyze composite beams by converting the cross-sectional areas of different materials into an equivalent area in a reference material using a modular ratio. This simplifies the calculation of properties such as the moment of inertia and assists in finding the neutral axis for a composite section.

Moment of Inertia of Composite Sections

Calculating the moment of inertia of a composite section involves summing the contributions of each transformed sub-area about a common axis, often the neutral axis. This is critical in determining the beam’s resistance to bending and is typically done using the parallel axis theorem for each constituent material's area.

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