Bending stress

Bending Stress and Shear Stress

Flexural Formula	Assumptions in Simple Bending
M = σ * I / y M = Maximum Bending moment (moment of resistance of a beam) I = Moment of inertia of the beam section about the neutral axis y = Distance of the layer from the neutral axis E = Modulus of elasticity of the beam material R = Radius of curvature of a bent beam	The material of the beam is homogeneous and isotropic. The beam is straight before loading and has a uniform cross section. The beam material is stressed within its elastic limit. The transverse sections remain plane after bending. Shear stress is neglected. Each layer of the beam is free to expand or contract independently. Young's modulus has the same value in tension and compression.

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Shear Stress Distribution for Circular Beam Section

The shear stress distribution diagram for a circular beam section can be represented as follows:

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Formula to calculate average shear stress for a circular section with diameter 'd':

τ_avg = (3/4) * τ_max

Shear Stress Distribution for Rectangular Beam Section

The shear stress distribution diagram for a rectangular beam section can be represented as follows:

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Formula to calculate average shear stress for a rectangular section:

τ_avg = (2/3) * τ_max

Shear Stress for Beam

Shear stress (τ) for a beam can be calculated using the formula:

τ = (Shear Force * Distance to Centroid of Shear Area) / (Shear Area * Shear Width)

Shear Distribution Diagram for T and C Sections

Here is the shear distribution diagram for T and C sections:

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Simple Bending

Term	Explanation
Neutral Axis	The fibers in the lower part of the beam undergo elongation while those in the upper part are shortened. These changes in the lengths of the fibers set up tensile and compressive stresses in the fibers. The fibers in the centroidal layer are neither shortened nor elongated. These centroidal layers which do not undergo any extension or compression is called neutral layer or neutral surface. When the beam is subjected to pure bending, there will always be one layer which will not be subjected to either compression or tension. This layer is called the neutral layer, and the axis passing through this layer is called the neutral axis.
Moment of Resistance	Moment of resistance refers to the resistance offered by a beam to bending. It is the measure of a beam's ability to resist the bending moment applied to it. The moment of resistance depends on the shape, size, and material of the beam. It is directly proportional to the moment arm and the bending stress induced in the beam. The moment of resistance is a crucial parameter in the design and analysis of beams subjected to bending loads.

Bending Types

Bending Type	Characteristics
Pure Bending	The beam deflects into an arc of a circle. The beam is subjected to normal (bending) stresses of tensile or compressive nature.
Ordinary Bending	The beam does not deflect into an arc of a circle. The beam is subjected to both normal and shear stresses.

Bending stress|Mechanical Engineering| Strenght of Material