Beam Deflection Formula: Definition, Types of Beams & Solved examples

Beam Deflection Formula, in structural engineering terms, is the degree to which a part of a structural element like a beam is displaced by a considerable amount of load. It is also referred to as displacement, which can occur from externally applied loads or from the weight of the body structure itself. It can occur in beams, trusses, frames and basically any other body structure. 

Also Read: Mechanical Properties of Solids

Key Terms: Beam Deflection, Cantilever Beam, Simply Supported Beams, Elasticity, Mechanical Properties of Solids, Young’s Modulus

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Beam Deflection

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A beam is a long piece of a body that is capable of holding the load by resisting the bending. The deflection of the beam in a particular direction when force is applied to it is known as Beam Deflection. There are mainly four variables, which can determine the magnitude of the beam deflections:

• Loading on the structure
• The length of unsupported members
• The material (specifically the Young’s Modulus)
• The cross-section size (specifically the Moment of Inertia(I)

Beam Deflection Schematic

The deflection distance of a member under a load can be calculated by integrating the function that mathematically describes the slope of the member under that load. Generally, we calculate it by taking the double integral of the Bending Moment Equation, which means M(x) divided by the product of E (Young’s Modulus) and I (Moment of Inertia).

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Beam Deflection Formula

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Beams can vary greatly in their geometry and composition. Some simple examples are shown below. The formulas expressed are approximations developed for long, slender, homogeneous, prismatic beams with small deflections and linear elastic properties.

Cantilever Beams

Cantilever beams are the special types of beams that are constrained by only one given support. These types of objects would naturally deflect more due to having support at one end only, so that the slope and deflection at that end must be zero. Cantilever beams are further classified as:

• End-Loaded 
• Uniformly Loaded

Cantilever Beam Schematic

End-Loaded Cantilever Beams

In this case of cantilever beams, the load is applied at a single point on the beam. Following is a schematic to understand this a little better. 

End-Loaded Cantilever Schematic

To calculate the deflection of the cantilever beam with force at free end we can use the below equation:

\(\eth_ B = \frac{FL^{3}}{3EI}\)

Where, 

δ_B is Beam deflection

F is Force at one end

L is Length of beam

E is Young’s Modulus

I is Moment of inertia

If the span doubles, the deflection increases eightfold. The deflection at any point x, along the span of an end loaded cantilever beam can be calculated using:

\(\eth_ x = \frac{Fx^{3}}{6EI} \)(3L-x)

When x = L (the end of the beam), δx is identical to δB in the equation above.

Also Read: Tensile Stress

Uniformly-Loaded Cantilever Beams

Uniformly-loaded cantilever beams have the force acting uniformly through out the length of the beam. The schematic is as follows:

Uniformly-Loaded Cantilever Schematic

The deflection, at the free end B, of a cantilever beam under a uniform load is given by:

\(\eth_ B = \frac{qL^{4}}{8EI}\)

Where, 

q is Force at one end

L is Length of beam

E is Young’s Modulus

I is Moment of inertia

The deflection at any point x, along the span of a uniformly cantilevered beam can be calculated using:

\(\eth_ x= \frac{qx^{2}}{24EI}\)(6L² -4Lx + x²)

Simply-Supported Beams

Simply-supported beams have support under their end which allows rotation, but not deflection. The simply supported beams can be further classified as:

Simply-Supported Beams

• Centre-Loaded
• Off-Centre
• Uniformly Loaded

Centre-Loaded Simple Beams

In centre-loaded beams, the force acts at the centre of the beam. The schematic is as follows:

Centre-Loaded Schematic

The special case of elastic deflection at the midpoint C of a beam, loaded at its centre, supported by two simple supports is given by:

\(\eth_ c= \frac{qL^{3}}{48EI}\)

The deflection at any point x, along the span of a centre loaded simply supported beam can be calculated using:

\(\eth_ x = \frac{Fx^{}}{48EI}\)I\((3L^2-4x^2)\)

Off-Centre Loaded Simple Beams

In this type of beam, the force acts at a point slightly away from the centre. 

The maximum elastic deflection on a beam supported by two simple supports, loaded at a distance a from the closest support is given by:

\( \)

Where, a is distance from the load to the closest support

This maximum deflection occurs at a distance x₁ from the closest support and it is given by:

x₁ = \(\sqrt{\frac{L^2-a^2}{3}}\)

Also Read: Thermal Stress

Uniformly-Loaded Simple Beams

In this type of beams, the force acts uniformly along the length of the beam.

The elastic deflection at midpoint C on a beam supported by two simple supports, under a uniform load as shown in the picture is given by:

\(\eth_ c= \frac{5qL^{}}{384EI}\) 

Where, q is uniform load on the beam (force per unit length)

Also Read: Strain Energy

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Beam Deflection Formula Units

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The formulas stated above require the use of a consistent set of units. Most calculations are made in SI units.

• Force: Newtons (N)
• Length: Metres (m)
• Modulus of elasticity: N/m²
• Moment of Inertia: m⁴

Other units may be used as well, as long as they are self-consistent. For example, sometimes the kilogram-force (kgf) unit is used to measure loads. In such cases, the modulus of elasticity must be converted to kgf/m².

Also Read: Stress Vs Pressure

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Solved Example

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Ques: A cantilever beam of length 2 metres is supported at one end only. The Young’s Modulus of a metal is 200 x 10⁹ Nm^-2. The moment of inertia is 50 kg m². If a force of 300 N is applied at one end, calculate the deflection in the metal beam.

Solution:  From the question,

E= 200 x 10⁹ Nm^-2

I= 50 kgm²

L= 2m

F= 300N

Using the formula for calculating deflection of cantilever beam , \(\eth_ B = \frac{FL^{3}}{3EI}\)

= \(\frac{300*2*2*2}{3*200*10^9*50}\)

=0.08m

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Things to Remember

• Deflection Consideration: The maximum deflection of a loaded beam should be within a certain limit so that the strength and efficiency of the beam should not be affected.
• Deflection in beams can be broadly classified as: Cantilevers and Simply Supported Beams.
• Based on the location of application of force, cantilever and simply supported bemas can be further classified. 
• The units of beam deflection formula are expressed in terms of force, length, moment of inertia or elasticity modulus. 
• Young’s modulus of elasticity of the beam is equal to the ratio of stress and strain within the limit of proportion.
• Young’s Modulus is the property of the material which does not depend on the external factors like force, moment etc.
• Three elastic modules which are young’s modulus, shear modulus and bulk modulus are used to describe the elastic behaviour of objects as they respond to deforming forces that act on them.
• The young’s modulus and shear modulus are relevant only for solids.
• The bulk modulus is relevant for solids, liquids and gases.

Also read:

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Previous Year Questions on Beam Deflection Formula

1. Choose the correct relation between the parameters of elesticity. [JEE Main 2021]
2. Ratio of weights of steel and brass wires upon suspension. [NEET 2015]
3. Which graph is a straight line. [NEET 2014]
4. Ratio of increase in length of steel and copper wires. [NEET 2013]
5. Calculate the force required to stretch the wire. [NEET 2018]
6. Calculate the elastic potential energy in the elongated wire. [NEET 2019]
7. Calculate the stress relation between two extended wires. [KCET 2018]
8. What is Young’s Modulus. [KCET 2017]
9. Which substance has highest elasticity? [KCET 2010]
10. Ratio of hydraulic stress to strain is? [KEAM]
11. Determine the ratio of e=increase in length of two wires of same material. [KEAM]
12. Determine the length of the cord when Young’s Modulus is given. [KEAM]
13. Calculate the energy stored in the wire. [KEAM]
14. Calculate the displacement when force is applied on a cube. [JEE Main 2018]
15. Calculate the force needed to push the cork into the bottle. [JEE Main 2016]
16. Determine the Young’s Modulus. [JEE Main 2019]
17. Choose the wrong statement. [KEAM]
18. Determine the stress change factor. [JEE Main 2017]
19. Calculate the tensile stress in the wire. [JEE Main 2019]
20. Calculate the fractional decrease in wire radius. [JEE Main 2013]

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