Bending Stress

When a beam is subjected to external forces, it tends to bend. The bending moment is a measure of this bending effect and represents the tendency of a force to cause rotation about a specific point along the beam. Engineers and physicists use the bending moment to determine how forces cause a beam to bend at different points. This helps them ensure that structures can support loads without excessive deformation or failure. The bending moment in a beam depends on the applied loads, the support conditions, and the geometry of the beam.

To calculate bending stress in a material, engineers and physicists use a fundamental formula that relates the bending moment, the material’s cross-section, and the distance from the neutral axis. The formula for bending stress in a beam is:

\[ \sigma = \frac{M y}{I} \]

Where:

– M: Bending moment

– y: Distance from the neutral axis

– I: Moment of inertia

Moment of inertia is a geometric property that quantifies a beam’s resistance to bending. It depends on the beam’s shape and size. The bending stress is directly proportional to the bending moment and the distance from the neutral axis but inversely proportional to the moment of inertia.

Bending stress can be categorized based on the type of force applied and the resulting stress distribution within the material.

1. Pure Bending Stress: It occurs when a beam or a structural element is subjected to a uniform bending moment without any shear force. 

Example

Imagine holding a plastic ruler horizontally and applying equal forces at both ends, bending it into a smooth curve. The middle portion of the ruler experiences pure bending stress because no additional force is trying to cut or shear through it.

2. Transverse Bending Stress: It occurs when a beam experiences both bending stress and shear stress at the same time.

Example

Consider a cantilever beam, which is rigidly fixed at one end and subjected to a force at the free end. This force causes the beam to bend downward, generating both bending and shear stresses near the fixed end.

Problem 1: A simply supported beam of length 4 meters carries a central load of 500 N. The beam has a rectangular cross-section with a width of 10 cm and a height of 20 cm. Calculate the maximum bending stress in the beam. (For a simply supported beam with a central load, the maximum bending moment is given by \( M = \frac{P L}{4} \))

Solution

Given:

Load: P = 500 N

Length: L = 4 m

Substituting values:

\[ M = \frac{500 \times 4}{4} = 500 N \cdot m \]​

For a rectangular cross-section, the moment of inertia is given by:

\[ I = \frac{b h^3}{12} \]​

Where:

Width: b = 0.1 m

Height: h = 0.2 m

Substituting values:

\[ I = \frac{0.1 \times (0.2)^3}{12} =  6.67 \times 10^{-5}  \, m^4 \]​

The bending stress formula is:

\[ \sigma = \frac{M y}{I} \]

Where:

Distance: y = \( \frac{h}{2} = \frac{0.2}{2} = 0.1 \text{ m} \)

By substituting values, we get:

\[ \sigma = \frac{500 \times 0.1}{6.67 \times 10^{-5} } = 750 \text{ kPa} = 0.75 \text{ MPa}  \]

The maximum bending stress in the beam is 0.75 MPa.