Deflection at the Centre of a Simply Supported Beam with Uniformly Distributed Load

Introduction:
In civil engineering, the deflection of a beam refers to the vertical displacement that occurs when the beam is subjected to external loads. The deflection at the centre of a simply supported beam with a uniformly distributed load can be calculated using the principles of structural analysis and beam theory.

Given:
- Span of the beam (L)
- Uniformly distributed load (2w/m)
- Moment of inertia (I)

Assumptions:
- The beam is made of a homogeneous material with a constant flexural rigidity.
- The beam is initially straight and has a constant cross-section throughout its length.
- The beam is subjected to a static load.
- The deflection is small compared to the span of the beam.

Procedure:
To calculate the deflection at the centre of the beam, we can follow these steps:

1. Determine the reaction forces:
Since the beam is simply supported, it is supported at both ends. The reaction forces at each support can be calculated using the equilibrium equations. For a uniformly distributed load, the reactions at the supports are equal and half of the total load.

2. Determine the equation of the deflection curve:
The equation of the deflection curve can be obtained by integrating the differential equation of the bending moment. For a simply supported beam with a uniformly distributed load, the equation of the deflection curve is a quadratic function.

3. Apply boundary conditions:
To determine the constants of integration in the equation of the deflection curve, we need to apply the appropriate boundary conditions. In this case, the deflection at the supports is zero since the beam is simply supported.

4. Calculate the deflection at the centre:
Once the equation of the deflection curve is determined, we can calculate the deflection at the centre of the beam by substituting the midpoint of the span into the equation.

Conclusion:
The deflection at the centre of a simply supported beam with a uniformly distributed load can be determined by following the steps mentioned above. The calculation involves determining the reaction forces, obtaining the equation of the deflection curve, applying appropriate boundary conditions, and finally calculating the deflection at the centre. It is important to note that the moment of inertia and the span of the beam influence the magnitude of the deflection.