Stress and Deformation in a simple cantilever beam

Knowing how to calculate stress and deformation of a simple cantilever beam is a very essential part of mechanical engineering. This is because most of the structures on a part can be approximated to cantilever beams and their initial analysis can be done by hand calculation before you go into detailed FEA (which consumes a little more time).

This concept is also called beam theory.

A cantilever beam is a suspended beam with one point of contact. In representations, it looks something like this.

Over here, we are going to discuss calculation of 2 parameters, both of which are the results of a force F applied at a distance L from the base of the cantilever. The representation now looks like this.

See, that dotted line, it is called Neutral axis. This is the line in which there is no elongation of material. Which means, the stress along this line is zero. The reason being that when you apply any force on a cantilever beam, a section of the beam will be in tension and another part would be in compression. The neutral axis separates these two regions.



With force applied, the beam would look something like this.

The stress distribution in the section starting from one extreme points through the neutral axis to the other extreme point.

We need to understand 2 concepts before proceeding. They are the Area Moment of Inertia and Section Modulus.

Area Moment of Inertia (also known as 2nd moment of inertia, moment of inertia of a plane area)

Area moment of inertia is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis. It is usually denoted by the symbol I. The higher this value is, the further the points are from the Neutral Plane of the body.

Let’s see an example.

Your intuition knows which of these parts will be harder to bend. This is because the points in those parts are further away from the neutral plane. It is harder to bend because it will have a higher area moment of Inertia.

(Click here to find the formulae to calculate area moment of Inertia for different surfaces)

Section Modulus

Section modulus is the property that helps us calculate the highest stress in the cross section. The highest stress in the cross section usually occurs at the point that is the furthest away from the neutral axis. This is because that is the point that gets pulled/compressed the most. Refer to the previous images to check it out.



Section Modulus can be calculated with the following formula.

Z = I/Length from Neutral axis to the furthest point in the section

Stress

Stress is the pressure or tension exerted on the cross-section of the beam by force F. Stress is directly proportional to the moment (Force X Distance) and inversely proportional to the section modulus.

Stress = (F x L)/Z

Do note that stress does not depend on the material properties of the beam. It is dependent only on the geometry. The material property only decides whether the part will fail at a given stress or not.

Deformation

Deformation at any section on the cantilever beam can be calculated using the following equation.

Deformation = (F x L3)/(3 x E x I)

E is the Young’s Modulus of the beam. It a material property of the part.

Remember that when learning about concepts like these, remembering the formulae is secondary. Understanding the concept is the most important part.

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