Deflection Analysis of Simply Supported and Cantilever Beams under Point Load

In this article, we will explore the deflection of different beam types under point loads, focusing on a simply supported beam and a cantilever beam. We will derive the deflection at the free ends of these beams and compare the results.

Beam Deflection Basics

Beam deflection is an important consideration in structural design, particularly for beams subjected to external loads. The deflection can be calculated using specific formulas that depend on various factors such as the type of beam, the load applied, and the material properties of the beam.

Simply Supported Beam

A simply supported beam is supported at both ends such that no moment is transmitted to the supports. When a point load is applied at the mid-span, the beam experiences deflection. This is given by the formula:

[ delta frac{P L^3}{48 E I} ]

Where:

( delta ) deflection at mid-span (m) ( P ) point load (N) ( L ) length of the beam (m) ( E ) modulus of elasticity of the material (Pa) ( I ) moment of inertia of the beam's cross-section (m4)

Given Values for Simply Supported Beam

Consider a simply supported beam of length 3 meters with a point load of 200 kN (200,000 N) at mid-span. The deflection at mid-span is measured to be 4.3 mm.

First, we need to calculate the product ( EI ) (the product of modulus of elasticity and moment of inertia) using the given values:

Substitute the known values into the formula:

[ EI frac{P L^3}{48 delta} ]

Convert all units to consistent SI units:

( P 200,000 ) N ( L 3 ) m ( delta 0.0043 ) m (4.3 mm)

Calculate ( EI ):

[ EI frac{200,000 times 27}{48 times 0.0043} approx 26,150,000 text{ Nm}^2 ]

Cantilever Beam

A cantilever beam is supported at one end and is free at the other. The deflection at the free end for a cantilever beam under a point load is given by:

[ delta frac{P L^3}{3 E I} ]

Where:

( delta ) deflection at the free end (m) ( P ) point load (N) ( L ) length of the beam (m) ( E ) modulus of elasticity of the material (Pa) ( I ) moment of inertia of the beam's cross-section (m4)

Given Values for Cantilever Beam

Now, let's use the ( EI ) value to find the deflection at the free end of a cantilever beam of the same material and length, subjected to a 100 kN (100,000 N) point load at the free end.

Substitute the known values into the formula for a cantilever beam:

[ delta frac{100,000 times 3^3}{3 times 26,150,000} ]

Calculate ( delta ):

[ delta frac{100,000 times 27}{3 times 26,150,000} approx 0.0344 text{ m} 34.4 text{ mm} ]

Alternative Method for Cantilever Beam

An alternative approach to determine the deflection of a cantilever beam can be derived from the deflection of a simply supported beam. If we consider the simply supported beam as two cantilevers with the fixed end in the middle, the deflection of each cantilever at the free end would be same as 4.3 mm. For a 3-meter span, the deflection would be:

[ delta 4.3 times 8 34.4 text{ mm} ]

Both methods yield the same result, confirming the consistency of the calculations.

Conclusion

The deflection at the free end of the cantilever beam carrying a load of 100 kN is approximately 34.4 mm. This analysis demonstrates the importance of understanding beam deflection principles and can be applied to various structural engineering designs.