Bending Stress Formula: Several Use Cases And Examples

This article discusses about bending stress formula for different types of beam configurations. We all know that when an object curves due to application of load then it is said to be subjected under bending.

It is very important to know the amount of bending stress being experienced by the work piece. The work piece will break if the applied bending stress exceeds more than the maximum allowable bending stress. The bending strength of the material is the maximum amount of bending strength that can be applied on the work piece before the work piece starts to fracture.

What is bending stress?

Let us start our discussion with the definition of bending stress. It is simply the stress which is responsible for bending of the work piece.

In further sections we shall see the mathematical forms of bending stress for various beam configurations and cross sectional shapes.

What is a beam?

A beam is a structural element that is mainly used for supporting the primary structure. The beam is not necessarily a support, it can itself be a structure for example bridges and balconies.

Most commonly used beams in industry are cantilever beams, simply supported beams and continuous beams.

Bending stress formula for beam

The bending stress depends on the bending moment moment of inertia of cross section and the distance from the neutral axis where the load is applied.

Mathematically, it can be represented as-

σ = My/I

y it he distance from the neutral axis

I is the moment of inertia of cross section

In terms of section modulus-

σ = M/Z

where,

Z is the section modulus of the beam

M is the bending moment

Bending stress formula units

The formula of bending stress can be given as-

σ = My/I

The formula in terms of units of each quantity can be given as-

Units = N – mm x mm/mm⁴

From above, we can derive that the units of bending stress is-

Units = N/mm²

Allowable bending stress formula

The allowable stress is the value of stress beyond which stress should not be applied for safety reasons. The allowable bending stress depends on the flexural rigidity of the material.

The allowable bending stress formula can be given as-

σ_allowable = σ_max/F_s

where,

Fs is the factor of safety

Bending stress formula derivation

Let us consider a beam section as shown in the diagram below-

Let us assume a moment, M is applied on the beam. The beam curves by an angle theta and makes a radius of curvature R as shown in figure below-

The strain in neutral axis is zero. Whereas the strain acting on the line where force is applied experiences strain. Balancing all strain values we get total strain,

(R + y)θ – Rθ/Rθ = y/R

Strain is also given by-

Strain = σ/E

from above equations we can conclude that,

σ/y = E/R

Now,

M = Σ E/R x y²

and,

δA = E/R Σ y² δA

M = E/R x I

From above equations we conclude that,

σ/y = E/R = M/I

Hence derived.

Bending stress formula for rectangular beam

Depending upon the cross section of the beam, the moment of inertia changes and hence the bending stress formula.

The moment of inertia of rectangle is given as-

I = bd³/12

From above, bending stress formula for a rectangular beam can be written as-

σ = 6M/ bd²

Bending stress formula for hollow rectangular beam

Hollow beams are used to reduce the weight of the beam. These beams can be used in light weight applications.

Let us consider a beam with hollow rectangular cross section with outside length as D and inner length as d, outside breadth as B and inner breadth as b.

The section modulus of this cross section will be-

Z = 1/6D x (BD³ – bd³)

Hence the bending stress formula for a hollow beam can be given by-

σ = 3M/(BD³ – bd³)

Bending stress formula for circular cross section

Let us consider a beam having a circular cross section of diameter D.

The moment of inertia of circular section can be given by-

I = πD⁴/64

From above, we can write the bending stress formula for circular beam as-

σ = 32M/ bd³

Bending stress formula for hollow shaft

Let us consider a hollow circular shaft having inner diameter d and outside diameter D.

The moment of inertia of hollow circular section can be given as-

I = π (D⁴-d⁴)/64

From above, the bending stress can be written as-

σ = 32MD/π(D⁴-d⁴)

Bending stress formula for pipe

A pipe is simply a hollow circular shaft. So the bending stress formula is same as that of hollow circular shaft.

That is,

σ = 32MD/π(D⁴-d⁴)

Maximum bending stress for simply supported beam

The general formula for bending stress remains the same that is-

σ = My/I

However, the formula is modified as per the type of loading. The loading can be in the form of point load, uniformly distributed load or uniformly variable load. In further sections we shall see the different formulae for simply supported beams in different forms of loading.

What is bending moment?

The reaction induced in a structural element or the bending effect produced when an external load is applied on the beam (structural element).

Bending moment formula for different beam configurations under different types of loading is discussed in below sections.

Bending moment formula for fixed beam

A fixed beam is a type of beam which is fixed at both the ends. At both the ends the reaction forces are present. The bending moment formula for fixed beam under different types of loading is given below-

• Bending moment under UDL or Uniformly distributed load

The formula for bending moment of fixed beam under UDL is given as-

M = ωL²/12

• Bending moment under point load

The formula for bending moment of fixed beam under point load is given as-

M = ωL/8

• Bending moment under trapezoidal load or UVL or uniformly variable load

The formula for bending moment of fixed beam under trapezoidal load is given as-

M₁ = ωL²/30

For other side,

M₂ = ωL²/20

Bending moment formula for continuous beam

The bending moment of continuous under different types of loading is shown below-

• Bending moment under UDL

To find the bending moment of continuous beam under uniformly distributed load, we need to find the reaction forces at the end points. After that we have to apply equilibrium conditions that is sum of all horizontal and vertical forces is zero as well as moments is zero. To solve UDL, we multiply the length with the magnitude of UDL. For example, if 2N/m of UDL is applied till 4m length of work piece then the net load acting will be 2×4= 8N at center that is at 2m.

• Bending moment under point load

The procedure is same as for UDL. The only difference is that here we know the magnitude of force and the distance at which it is acting so we need not convert it into point load as we did for UDL.

• Bending moment under UVL or uniformly varied load

To solve UVL, we need to find the area of the triangle formed by UVL. The area is the magnitude of point load that will be acting due to UVL. The distance from vertex will be L/3 at which the point load will act. Rest of the procedure is discussed above.

Bending moment formula for rectangular beam

Bending moment of the beam does not depend on the shape of the beam. The bending moment will change as per the loading conditions and the type of beam (whether continuous, cantilever simply supported etc).

Only the moment of inertia changes with the shape of the cross section of the beam. This way the bending stress formula changes. The bending stress formula for rectangular cross section is discussed in above section.

Bending moment formula for UDL

UDL or uniformly distributed load is the type of load which is applied to a certain length of the work piece and is equal in magnitude wherever applied.

The bending moment formula for UDL of different beam configurations are given below-

• For simply supported beam-

The formula for bending moment of simply supported beam under UDL is given as-

M = ωL²/8

• For cantilever beam-

The formula for bending moment of cantilever beam under UDL is given as-

M = ωL²/2

Bending moment formula for point load

Point load is the type of load which acts only at a particular point on the surface of the work piece.

The bending moment formulae for point loads for different beam configurations are given below-

• For simply supported beam: The formula for bending moment of simply supported beam under point load is given as- M = ωL/4

• For cantilever beam: The formula for bending moment of cantilever beam under point load is given as- M = ωL

For other beam configurations, the formula for bending moment is discussed in above sections.

Bending moment formula for trapezoidal load

Trapezoidal load is a type of load which is applied to a certain length of the work piece and varies linearly with length. Trapezoidal load is combination of both UDL and UVL. Lets assume magnitude of UDL as zero to ease our calculations.

The bending moment for different beam configurations under trapezoidal load are given below-

• For simply supported beam– The bending moment of simply supported beam under trapezoidal load is given as- M = ωL²/12

• For cantilever beam– The bending moment of cantilever beam under trapezoidal load is given as- M = ωL²/6

For other beam configurations, the formula is discussed in above section

Summary of bending moment formula

Table below shows a brief summary of formula for different beam configurations under different types of loading

Summary of bending stress formula

Table below shows a brief summary of formula for bending stresses of different beam cross sections