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Cantilever Beams

Formulae for the shear and deflection of Cantilever Beams under a selection of differing loadings.
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Overview

This section covers Beams used as Cantilever. The examples include Beams which are "Built-in" at one end and either supported or guided at the other.

Fixed At One End With A Uniform Load.

23287/Cantilever-Beams-101.png
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The stress is given by: S = \displaystyle\frac{W}{2\;Z\;l}\;\;(l - x)^2

The Stress at the Support: S_s = \displaystyle\frac{W\;l}{2\;Z}

If the cross section is Constant then this is the maximum stress.

The Deflection is given by: y = \displaystyle\frac{W\;x^2}{24\;E\;I\;l}\;\;[2l^2 + (2l - x)^2]

The Maximum deflection is at the end and is: \hat{y} = \displaystyle\frac{W\;l^3}{8\;E\;I}

A beam is a horizontal structural element that is capable of withstanding load primarily by resisting bending. The bending force induced into the material of the beam as a result of the external loads, own weight, span and external reactions to these loads is called a bending moment.

Fixed At One End. Load At The Other

23287/Cantilever-Beams-111.png
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The Stress is given by: S = \displaystyle\frac{W}{Z}\;\;(l - x)

The Stress at the Support: S = \displaystyle\frac{W\;l}{Z}

If the Cross-Section is Constant, then this is the Maximum Stress.

The Deflection at any point is given by: y = \displaystyle\frac{W\;x^2}{6\;E\;I}\;\;(3\,l - x)

The Maximum Deflection is at the end and is: \hat{y} = \displaystyle\frac{W\;l^3}{3\;E\;I}

Deflection is a term that is used to describe the degree to which a structural element is displaced under a load.

Fixed At One End. Load Intermediate.

23287/Cantilever-Beams-112.png
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Between the Fixed End and the Load: S = \displaystyle\frac{W}{Z}\;\;(l - x)

Beyond the Load the stress is zero.

At the Fixed End: S = \displaystyle\frac{W\;l}{Z}

If the Cross-section is Constant, then this is the Maximum Stress.

The Deflection of any point between the Fixed end and the Load is:y = \displaystyle\frac{W\;l^2}{6\;E\;I}\;\;(3\;l - x)

Beyond the load the Deflection is: y = \displaystyle\frac{W\;x^2}{6\;E\;I}\;\;(3\;v - l)

The Maximum Deflection at the "Free" end is: y = \displaystyle\frac{W\;l^2}{6\;E\;I}\;\;(2\;l + 3\;b)

Deflection at the Load: y = \displaystyle\frac{W\;l^3}{3\;E\;I}

Fixed At One End. Supported At The Other. Uniform Load.

23287/Cantilever-Beams-113.png
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The Stress at any point is : S = \displaystyle\frac{W\;(l - x)}{2\;Z\;l}\;\;\left(\displaystyle\frac{1}{4}\;l - x \right)

The Maximum Stress at the Fixed End: y = \displaystyle\frac{W\;l}{8\;Z}

The Stesss is zero at \displaystyle x = \frac{1}{4}\;l. The greatest negative Stres is at \displaystyle x = \frac{5}{8}\;l and is:
S\;= - \frac{9}{128}\;\;\frac{W\'l}{Z}

The deflection is given by: y = \displaystyle\frac{W\;x^2(l - x)}{48\;E\;I\;l}\;\;(3\;l - 2\;x)

The maximum Deflection is at x = 0.5785 l and is: \hat{y} = \displaystyle\frac{W\;l^3}{185\;E\;I}

The Deflection at the centre is: y_c = \displaystyle\frac{W\;l^3}{192\;E\;I}

The Deflection at the point of greatest negative Stress, i.e. at x = \displaystyle\frac{5}{8}\;l, is:
y = \frac{W\;l^3}{187\;E\;I}

Fixed At One End, Supported At The Other With A Central Point Load.

23287/Cantilever-Beams-114.png
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The Stress between the Fixed Point and the Load: S = \displaystyle\frac{W}{16\;Z}\left(3\;l - 11\;x \right)

Between the Support and the Load: S\;= - \displaystyle\frac{5}{16}\;\displaystyle\frac{W\;v}{Z}

Stress at the Fixed end. This is the maximum value S = \displaystyle\frac{3}{16}\;\displaystyle\frac{W\;l}{Z}

The Stress is Zero at \displaystyle x = \frac{3}{11}\;l

The Greatest negative Stress is at the centre and is \displaystyle - \frac{5}{32}\;\;\frac{W\;l}{z}

The Deflection of any Point between the Fixed End and the Load:

y = \frac{W\;x^2}{96\;E\;I}\;\;(9\;l - 11\;x)

The Deflection of any Point between the Support and the Load
y = \frac{W\;v}{96\;E\;I}\;\;(3\;l^2 - 5\;v^2)

The Maximum Deflection is at \displaystyle v = 0.4472\;l
\hat{y} = \frac{W\;l^3}{107.33\;E\;I}

The Deflection of the Load is: \displaystyle\frac{7}{768}\;\;\displaystyle\frac{W\;l^3}{E\;I}

Fixed At One End And Free But Guided At The Other. Uniform Load.

23287/Cantilever-Beams-116.png
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The Stress at any Point S = \displaystyle\frac{W\;l}{Z}\;\;\left\{\displaystyle\frac{l}{3} - \displaystyle\frac{x}{l}\;+\displaystyle\frac{1}{2}\;\left(\displaystyle\frac{x}{l} \right)^2 \right\}

The Maximum Stress is at the support and is \displaystyle \frac{W\;l}{3\;Z}

The Stress is zero when \displaystyle x = 0.4227\;l

The Greatest negative stress is at the free end and is \displaystyle - \frac{W\;l}{3\;Z}

The Deflection at any Point is given by: y = \displaystyle\frac{W\;x^2}{24\;E\;I\;l}\;\;(2l - x)^2

The Maximum Deflection is at the free end and is: \hat{y} = \displaystyle\frac{W\;l^3}{12\;E\;I}

Fixed At One End. Free But Guided At The Other. Point Load.

23287/Cantilever-Beams-018.png
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Stress at any Point: S = \displaystyle\frac{W}{Z}\;\;\;\left(\displaystyle\frac{1}{2}l - x \right)

The Stress at the Support S_s = \displaystyle\frac{W\;l}{2\;Z}

The Stress at the Free End: S_f\;= - \displaystyle\frac{W\;l}{2\;Z}

These are the Maximum Stresses and are equal and opposite. The Stress is zero at the Centre.

The Deflection at any Point is: y = \displaystyle\frac{W\;x^2}{12\;E\;I}\;\;(3l - 2x)

The Maximum Deflection is at the Free End and is: \hat{y} = \displaystyle\frac{W\;l^3}{12\;E\;I}