Strength of a Simply Supported Beam of Circle Cross-Section with Concentrated Load

Introduction:
A simply supported beam is a structural element that is supported at two points, and carries a load over the span between the supports. The cross-section of the beam can have different shapes, such as rectangular, circular, or I-shaped. The strength of a beam is the maximum load it can carry without failing, and it depends on various factors, such as the material, the shape, and the dimensions of the beam.

Beam with Circle Cross-Section:
A beam with a circle cross-section has a diameter d, which is the distance across the widest point of the circle. The area of the cross-section is A=πd²/4, which means that the larger the diameter, the greater the area. The moment of inertia of the cross-section is I=πd⁴/64, which is a measure of the resistance of the beam to bending. The larger the moment of inertia, the stiffer the beam.

Concentrated Load:
A concentrated load is a force that is applied at a single point on the beam, as opposed to a distributed load that is spread over the span of the beam. The magnitude of the concentrated load is W, which can be expressed in units of force, such as Newtons or pounds. The location of the load is at the centre of the beam, which means that it is equidistant from the two supports.

Strength Proportional to d²:
The strength of the beam is proportional to d² because the area and the moment of inertia of the cross-section are both proportional to d². The larger the area, the greater the load that can be supported without exceeding the stress limit of the material. The larger the moment of inertia, the greater the resistance to bending, which means that the deflection of the beam under the load will be smaller.

Conclusion:
In conclusion, a simply supported beam of circle cross-section with a concentrated load at the centre has a strength that is proportional to the square of the diameter. This means that increasing the diameter of the beam will increase its strength, but also its weight and cost. The designer needs to balance these factors to achieve an optimal design that meets the requirements of the application.

L/D³