# Bending of Members Made of Several Materials

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Mechanical Engineering
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JoVE Core Mechanical Engineering
Bending of Members Made of Several Materials

### Video successivo20.8: Stress Concentrations

Consider a member of two different materials that have the same cross-sectional area. The equation of stress for such a member is expressed in terms of elastic moduli for each segment separately.

The normal strain in the two different segments of the member varies linearly with the distance from the neutral axis.

By writing the expression for the force exerted on the area element for each segment of the member and defining the ratio of elasticity of moduli as a constant, the resistance to the bending can be estimated.

Here, the force exerted on one segment of the member can be expressed in terms of the force exerted on the other member by multiplying it with the ratio of the elastic moduli of the two materials.

If the ratio of elastic moduli is greater than 1, then a widening occurs. If the ratio is less than 1, then a narrowing of the cross-sectional area occurs. This effect occurs parallel to the neutral axis, and the new cross-section is known as the transformed section.

## Bending of Members Made of Several Materials

In analyzing a structural member composed of two different materials with identical cross-sectional areas, it is crucial to understand how their distinct elastic properties affect the member's response under load. The analysis involves assessing stress and strain distributions using the transformed section concept, which accounts for variations in material properties.

Hooke's Law determines stress in each material, stating that stress is proportional to strain but varies due to each material's unique modulus of elasticity. The normal strain changes linearly with distance from the neutral axis, leading to different stress distributions in each material segment and influencing the force exerted on each segment. The composite member's calculation of forces and moments is simplified by relating the force in one material to the other by defining the ratio of their elastic moduli.

The elastic moduli ratio transforms one material's section into an equivalent section of the other, adjusting its contribution to the overall structural behavior. The ratio of elastic moduli significantly influences the geometry of the transformed section. When the ratio is greater than one, the material with the higher modulus appears effectively wider in the transformed section, indicating greater stiffness. Conversely, if the ratio is less than one, the material seems narrower, signifying lower stiffness. This transformation is critical for calculating the neutral axis position and moment of inertia, and it is essential for determining bending stresses and deflections in composite beams, thereby ensuring structural integrity under various loading conditions.