Deflection, in structural engineering terms, means the movement of a beam or node from its original position. It happens due to the forces and loads being applied to the body. Deflection 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. In this article, we will discuss the beam deflection formula with examples. Let us learn it!

**Beam Deflection Formula**

**What is Beam Deflection?**

Deflection is the degree to which a particular structural element can be displaced with the help of a considerable amount of load. It can also be referred to as the angle or distance. The distance of deflection of a member under a load is directly related to the slope of the deflected shape of the body under that load. It can be computed by integrating the function which is used to describe the slope of the member under that load.

The Beam is a long piece of a body that is capable to hold the load by resisting the bending. The deflection of the beam towards in a particular direction when force is applied to it is known as Beam deflection.

The beam can be bent or moved away from its original position. This distance at each point along the member is the representation of the deflection. There are mainly four variables, which can determine the magnitude of the beam deflections. These include:

- How much loading is on the structure?
- The length of the unsupported member
- The material, specifically the Youngâ€™s Modulus
- The Cross-Section Size, specifically the Moment of Inertia (I)

There is a variety of range of beam deflection equations that can be used to calculate a basic value for deflection in different types of beams. Generally, we calculate deflection by taking the double integral of the Bending Moment Equation means M(x) divided by the product of E and I (i.e. Youngâ€™s Modulus and Moment of Inertia).

The unit of deflection, or displacement, will be a length unit and normally we measure it in a millimetre. This number defines the distance in which the beam can be deflected from its original position.

**The formula for Beam Deflection:**

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. To calculate the deflection of the cantilever beam we can use the below equation:

D= \( \frac{WL^3}{3EI} \)

Where,

D | Beam deflection |

W | Force at one end |

L | Length of beam |

E | Youngâ€™s Modulus |

I | Moment of Inertia |

## Solved Examples

Q: Calculate the deflection of a cantilever beam of length 2 meter which has support at one end only. Youngâ€™s modulus of the metal is \( 200\times 10^9\) and the moment of inertia is 50 Kg mÂ². At the end force applied is 300 N.

Solution:

Given values are,

E= \( 200\times 10^9 Nm^{-2} \)

I = 50 kgmÂ²

L = 2 m

W = 300 N

Now applying the formula,

D= \(\frac{WL^3}{3EI}\)

Substituting the values,

D = \( \frac{300 \times 2^3}{3\times 200 \times 50} \)

D = \( \frac{2400}{30000} \)

So,

D = 0.08 m

Therefore value of beam deflection will be 0.08 m i.e. 8 cm.

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