What is maximum bending stress formula?

Short Answer:

The maximum bending stress formula is used to find the highest stress developed in a beam when it bends under a load. It helps to determine whether a beam can safely withstand the applied loads without failing. The formula is derived from the bending equation and is expressed as:

Here,  is the maximum bending stress,  is the maximum bending moment,  is the distance from the neutral axis to the extreme fiber, and  is the moment of inertia of the beam’s cross-section.

Detailed Explanation :

Maximum Bending Stress Formula

When a beam is subjected to loads, it experiences bending moments that create internal stresses within the material. These stresses vary across the cross-section, being maximum at the outermost fibers and zero at the neutral axis. The maximum bending stress formula helps engineers to calculate this extreme stress so that the beam can be designed safely and efficiently.

The general bending equation is:

Where,

  •  = Bending moment at a section,
  •  = Moment of inertia of the beam section about the neutral axis,
  •  = Bending stress at a distance  from the neutral axis,
  •  = Young’s modulus of the material,
  •  = Radius of curvature of the beam.

From this relation, the bending stress at any distance  from the neutral axis can be expressed as:

The maximum bending stress occurs at the outermost fiber of the beam, where . Therefore,

This is the maximum bending stress formula, and it gives the value of the largest stress experienced by the beam material under bending.

Explanation of Each Term

  1. :
    This is the maximum bending stress developed in the beam. It indicates how much internal resistance the beam material offers against bending. If this stress exceeds the material’s yield strength, the beam may fail.
  2. :
    This is the maximum bending moment acting on the beam. The bending moment depends on the type of loading and support conditions. For example, a simply supported beam with a central load will have maximum bending moment at the center.
  3. :
    It is the distance from the neutral axis to the outermost fiber (either top or bottom) of the beam cross-section. The farther the outer fiber, the greater the bending stress.
  4. :
    It is the moment of inertia of the beam cross-section about the neutral axis. A higher moment of inertia means the beam can resist more bending stress.

Derivation Concept

To derive the formula, consider a beam that bends due to a moment . During bending, the beam develops a curvature and the top fibers are compressed while the bottom fibers are stretched. The neutral axis remains unchanged in length. The strain in any fiber is proportional to its distance  from the neutral axis.

The stress in the fiber, according to Hooke’s law, is proportional to the strain. Therefore,

where  is the radius of curvature.

Now, substituting this value of stress into the bending moment equation,

and solving gives,

Here, . Therefore,

At the maximum distance , the maximum stress occurs, and thus,

Practical Importance

The maximum bending stress formula is extremely important in engineering design because it allows engineers to determine:

  • Whether a beam can withstand applied loads safely.
  • The required beam dimensions to prevent bending failure.
  • The selection of proper material that can handle the expected stress.

For example, in bridge design or machine frames, the designer calculates  using load conditions, then uses this formula to ensure that  does not exceed the allowable stress for the material.

If , then the beam must be redesigned — either by increasing its cross-sectional area (which increases ) or by selecting a stronger material.

Units and Dimensions

The units of each term are:

  • : N/m² or MPa (stress),
  • : N·m (moment),
  • : m (distance),
  • : m⁴ (moment of inertia).

When substituted correctly, the equation gives consistent units for stress.

Conclusion :

The maximum bending stress formula  is a key relation in the study of beams. It helps to identify the highest stress developed due to bending moments and is essential for safe and efficient structural design. By ensuring that this stress does not exceed the permissible limit of the material, engineers can prevent bending failure and ensure long-lasting performance of mechanical and structural components.