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# Cantilever Beams

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

## 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.

The stress is given by: $\inline&space;S&space;=&space;\displaystyle\frac{W}{2\;Z\;l}\;\;(l&space;-&space;x)^2$

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

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

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

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

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

The Stress is given by: $\inline&space;S&space;=&space;\displaystyle\frac{W}{Z}\;\;(l&space;-&space;x)$

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

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

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

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

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.

Between the Fixed End and the Load: $\inline&space;\displaystyle&space;S&space;=&space;\frac{W}{Z}\;\;(l&space;-&space;x)$

Beyond the Load the stress is zero.

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

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: $\inline&space;\displaystyle&space;y&space;=&space;\frac{W\;x^2}{6\;E\;I}\;\;(3\;l&space;-&space;x)$

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

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

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

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

The Stress at any point is : $\inline&space;S&space;=&space;\displaystyle\frac{W\;(l&space;-&space;x)}{2\;Z\;l}\;\;\left(\displaystyle\frac{1}{4}\;l&space;-&space;x&space;\right)$

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

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

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

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

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

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

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

The Stress between the Fixed Point and the Load: $\inline&space;S&space;=&space;\displaystyle\frac{W}{16\;Z}\left(3\;l&space;-&space;11\;x&space;\right)$

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

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

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

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

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

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

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

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

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

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

The Stress at any Point $\inline&space;S&space;=&space;\displaystyle\frac{W\;l}{Z}\;\;\left\{\displaystyle\frac{l}{3}&space;-&space;\displaystyle\frac{x}{l}\;+\displaystyle\frac{1}{2}\;\left(\displaystyle\frac{x}{l}&space;\right)^2&space;\right\}$

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

The Stress is zero when $\inline&space;\displaystyle&space;x&space;=&space;0.4227\;l$

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

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

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

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

Stress at any Point: $\inline&space;S&space;=&space;\displaystyle\frac{W}{Z}\;\;\;\left(\displaystyle\frac{1}{2}l&space;-&space;x&space;\right)$

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

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

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

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

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