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Simple Supported Beams

Stress and deflection formulae for simple supported beams

Introduction

The following symbols have been used throughout:
• $\inline&space;S$ is the Stress at any point
• $\inline&space;Z$ is the Section Modulus of beam cross section.
• $\inline&space;y$ is the deflection at any point.
• $\inline&space;W$ is the load on the Beam. Note for uniform loads $\inline&space;W&space;=&space;wl$ where $\inline&space;w$ is the load per unit length
• $\inline&space;E$ is the Modulus of Elasticity ( Young's Modulus)
• $\inline&space;I$ is the Moment of Inertia of the cross-section about the neutral axis

Simply Supported At Both Ends With A Uniform Load.

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.

An elastic modulus, or modulus of elasticity, is the mathematical description of an object or substance's tendency to be deformed elastically (i.e., non-permanently) when a force is applied to it. The elastic modulus of an object is defined as the slope of its stress-strain curve in the elastic deformation region

Stress at any point: $\inline&space;S=-\displaystyle\frac{W}{2zl}\times&space;x(l&space;-&space;x)$

Deflection at any point: $\inline&space;y=\displaystyle\frac{Wx(l&space;-&space;x)}{24\,E\,I\,l}\:\;[l^2&space;+&space;x(l&space;-&space;x)]$

Stress at critical points: $\inline&space;-\displaystyle\frac{Wl}{8Z}$

This is the maximum stress when the cross section is uniform.

Deflection at critical points:

Maximum ( At centre) $\inline&space;\displaystyle\frac{5}{384}\times\displaystyle\frac{W\;l^3}{E\,I}$

Simply Supported At Both Ends . Load At Centre.

Stress between each support and the load: $\inline&space;S=\displaystyle\frac{W\,x}{2\,Z}$

Stress at the Centre: $\inline&space;S=-&space;\displaystyle\frac{W\,l}{4\,Z}$

This is the maximum stress when the cross section is constant.

The deflection between each support and the load: $\inline&space;y=\displaystyle\frac{W\,x}{48\,E\,I}\;(3l^2&space;-&space;4x^2)$

The maximum deflection occurs at the load: $\inline&space;\hat{y}&space;=&space;\displaystyle\frac{W\,l^3}{48\,E\,I}$

Simply Supported At Both Ends. Load At Any Point.

Stress for a portion of length $\inline&space;a$: $\inline&space;S_a\;=&space;-&space;\displaystyle\frac{W\,b\,x}{Z\,l}$

Stress for a portion of length $\inline&space;b$: $\inline&space;S_b&space;=&space;\displaystyle\frac{W\,a\,v}{Z\,l}$

Stress at the point of load. This is the maximum stress if the cross section is constant. $\inline&space;\hat{S}\;=&space;-&space;\displaystyle\frac{W\,a\,b}{Z\,l}$

Deflection for the portion of length $\inline&space;a$ $\inline&space;y_a=\displaystyle\frac{W\;b\;x\;}{6\,E\,I\,l}\times(l^2&space;-&space;x^2&space;-&space;b^2)$

Deflection for the portion of length $\inline&space;b$: $\inline&space;y_b=\displaystyle\frac{W\;a\;v\;}{6\,E\,I\,l}\times(l^2&space;-&space;v^2&space;-&space;a^2)$

Deflection at the load: $\inline&space;y_w=\displaystyle\frac{W\;a^2\;b^2\;}{6\,E\,I\,l}$

When $\inline&space;a$ is the length of the shorter portion and $\inline&space;b$ the longer one, the maximum deflection is in the longer one at:

$\inline&space;v&space;=&space;b\;\sqrt{\displaystyle\frac{1}{3}&space;+&space;\displaystyle\frac{2a}{3b}}\;&space;=&space;v_1$

And the deflection is: $\inline&space;y_{v_1}&space;=&space;\displaystyle\frac{W\;a\;v_i^2}{3\;E\,I\,l}$

Simply Supported At Both Ends. Two Equal Loads Symmetrically Placed

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

The stress in between each support and the adjacent load: $\inline&space;S\;=&space;-&space;\displaystyle\frac{W\;x}{Z}$

Stress at the load points and any point between: $\inline&space;S=-\displaystyle\frac{W\;a}{Z}$

Deflection between each support and the adjacent load: $\inline&space;y=\displaystyle\frac{W\;x}{6\;E\;I}\times&space;[3a(l&space;-&space;a)&space;-&space;x^2]$

Deflection between loads: $\inline&space;y=\displaystyle\frac{W\;a}{6\;E\;I}\times&space;[3v(l&space;-&space;v)&space;-&space;a^2]$

The maximum deflection is at the centre and is: $\inline&space;\hat{y}&space;=&space;\displaystyle\frac{W\;a}{24\;E\;I}\times&space;(3l^2&space;-&space;4a^2)$

Deflection at the loads is: $\inline&space;y=\displaystyle\frac{W\;a^2}{6\;E\;I}\;(3l&space;-&space;4a)$

Simply Supported But With Both Ends Overhanging Symmetrically. Uniform Load.

Stress between each support and the end adjacent: $\inline&space;S=\displaystyle\frac{W}{2\,Z\;L}\;(c&space;-&space;u)^2$

Between the supports: $\inline&space;S=\displaystyle\frac{W}{2\;Z\;L}\;(c^2&space;-&space;x[l&space;-&space;x])$

Stress at each support: $\inline&space;S_s=\displaystyle\frac{W\;c^2}{2\;Z\;L}$

Stress at the centre: $\inline&space;S=\displaystyle\frac{W\;(c^2&space;-&space;l^2/4)}{2\;Z\;L}$

The greater of these is maximum stress when the section is constant.

Should $\inline&space;l&space;>&space;2c$ the stress is zero at points $\inline&space;\displaystyle&space;\sqrt{\frac{l^2}{4}&space;-&space;c^2}$ on each side of the centre. Should the section be constant and $\inline&space;l&space;=&space;2.828\;c$ ,the stresses at the centre and at the supports are equal and opposite. They are:
$S=\pm&space;\frac{W\;L}{46.62\;Z}$
The Deflection between each support and the adjacent end:
$y=\frac{W\;u}{24\;E\;I\;L}\;\times[6c^2(l&space;+&space;u)&space;-&space;u^2(4c&space;-&space;u)&space;-&space;l^3]$
Deflection between the supports
$y=\frac{W\;x(l&space;-&space;x)}{24\;E\;I\;L}\;\times[x(l&space;-&space;x)&space;+&space;l^2&space;-&space;6c^2]$
Deflection at the ends: $\inline&space;y_e=\displaystyle\frac{W\;x}{24\;E\;I\;L}\;\times[3\,c^2(c&space;+&space;2l)&space;-&space;l^3]$

Deflection at the centre: $\inline&space;y_c=\displaystyle\frac{W\;l^2}{384\;E\;I\;L}\;\times[5\;l^2&space;-&space;24\,c^2]$

When $\inline&space;l$ is between $\inline&space;2c$ and $\inline&space;2.449c$ the maximum upward deflections occur at points $\inline&space;\displaystyle&space;\sqrt{3\,\left(\frac{1}{4}l^2&space;-&space;c^2&space;\right)}$ on each side of the centre. The value of these are :

$\hat{y}\;=&space;-&space;\frac{W}{96\;E\;I\;L}\times&space;(6\,c^2&space;-&space;l^2)^2$

Simply Supported With Both Ends Overhanging, Supports Unsymmetrical, Uniform Load.

Stress for the overhanging length $\inline&space;c$: $\inline&space;S_c&space;=&space;\displaystyle\frac{W}{2\;Z\;L}\times&space;(c&space;-&space;u)^2$

Stress between the supports:
$S_s=\frac{W}{2\;Z\;L}\times&space;\left\{&space;c^2\left(\frac{l&space;-&space;x}{l}&space;\right)&space;+&space;\frac{d^2\;x}{l}\;-x(l&space;-&space;x)\right\}$
For the overhang ends of length $\inline&space;d$: $\inline&space;S_d=\displaystyle\frac{W}{2\;Z\;L}\times&space;(d&space;-&space;w)^2$

The stress at the support next to end of length $\inline&space;c$: $\inline&space;S=\displaystyle\frac{W\;c^2}{2\;Z\;L}$

The critical stress between the supports is at $\inline&space;x$: $\inline&space;x_1=\displaystyle\frac{l^2&space;+&space;c^2&space;-&space;d^2}{2\;l}$

The value is: $\inline&space;\displaystyle\frac{W}{2\,Z\,l}(c^2&space;-&space;x_1^2)$ The stress at the support next to end of length $\inline&space;d$ is: $\inline&space;\displaystyle\frac{W\;d^2}{2\;Z\;L}$ If the cross section is constant, the greatest of these three is the maximum stress.

If $\inline&space;\displaystyle&space;x_1\;>\;c$ the stress is zero at points $\inline&space;\displaystyle&space;\sqrt{x_1^2&space;-&space;c^2}$ on both sides of $\inline&space;\displaystyle&space;x&space;=&space;x_1$ The deflection for the overhanging length $\inline&space;c$:
$y&space;=&space;\frac{Wu}{24\;E\,I\,L}\;\;[2l(d^2&space;+&space;2c^2)&space;+&space;6c^2u&space;-&space;u^2(4c&space;-&space;u)&space;-&space;l^3]$

The deflection between the supports:
$y&space;=&space;\frac{Wx(l&space;-&space;x)}{24\;E\,I\,L}\;\left\{x(l&space;-&space;x)&space;+&space;l^2&space;-&space;2(d^2&space;+&space;c^2)&space;-&space;\frac{2}{l}\;[d^2x&space;+&space;c^2(l&space;-&space;x)]&space;\right\}$
The deflection for the overhanging length $\inline&space;d$:
$y&space;=&space;\frac{Ww}{24\;E\,I\,L}\;\;[2l(c^2&space;+&space;2d^2)&space;+&space;6d^2w&space;-&space;w^2(4d&space;-&space;w)&space;-&space;l^3]$
The deflection at end $\inline&space;c$: $\inline&space;y=\displaystyle\frac{Wc}{24\;E\,I\,L}\;[2l(d^2&space;+&space;2c^2)&space;+&space;3c^3&space;-&space;l^3]$

The deflection at end $\inline&space;d$: $\inline&space;y=\displaystyle\frac{Wd}{24\;E\,I\,L}\;[2l(c^2&space;+&space;2d^2)&space;+&space;3d^3&space;-&space;l^3]$

This case is so complicated that conventional general expressions for the critical deflections between the supports can not be obtained.

Simply Unsymmetrically Supported With Both Ends Overhanging And A Load At Any Point.

Between the supports for the segment of length $\inline&space;a$ $\inline&space;S=-&space;\displaystyle\frac{W\;b\;x}{Z\;l}$

For the segment of length $\inline&space;b$: $\inline&space;S=-&space;\displaystyle\frac{W\;a\;v}{Z\;l}$

Beyond the support $\inline&space;S&space;=&space;0$

Stress at the load: $\inline&space;S\;=&space;-&space;\displaystyle\frac{W\;a\;b}{Z\;l}$

If the cross section is constant, this is the maximum stress.

Deflection for overhanging length $\inline&space;c$: $\inline&space;y=-&space;\displaystyle\frac{W\;a\;b\;u}{6\;E\;I\;l}(l&space;+&space;b)$

Deflection for overhanging length $\inline&space;d$: $\inline&space;y=-&space;\displaystyle\frac{W\;a\;b\;w}{6\;E\;I\;l}(l&space;+&space;a)$

For the deflection between the supports see paragraph 3 above

Deflection for the end $\inline&space;c$: $\inline&space;y=-&space;\displaystyle\frac{W\;a\;b\;c}{6\;E\;I\;l}(l&space;+&space;b)$

Deflection for the end $\inline&space;d$: $\inline&space;y=-&space;\displaystyle\frac{W\;a\;b\;d}{6\;E\;I\;l}(l&space;+&space;a)$

Simply Supported With Ends Overhanging. Single Overhanging Load.

Between load and adjacent support: $\inline&space;S=\displaystyle\frac{W}{Z}\;(c&space;-&space;u)$

Between supports: $\inline&space;S=\displaystyle\frac{W\;c}{Z\;l}\;(l&space;-&space;x)$

Between the unloaded end and the adjacent $\inline&space;S&space;=&space;0$

Stress at the support adjacent to the load: $\inline&space;S=\displaystyle\frac{W\;c}{Z}$

If the cross section is Constant, this is the maximum stress. The stress is zero at the other support.

$y=\frac{W\;u}{6\;E\;I}(3cu&space;-&space;u^2&space;+&space;2cl)$
Deflection between the supports: $\inline&space;y=-&space;\displaystyle\frac{W\;c\;x}{6\:E\;I}(l&space;-&space;x)(2l&space;-&space;x)$
Deflection between the loaded and adjacent support: $\inline&space;y=\displaystyle\frac{W\;c\;l\;w}{6\:E\;I}$
Deflection at the load: $\inline&space;y=\displaystyle\frac{W\;c^2}{6\:E\;I}\;\;(c&space;+&space;l)$
The Maximum upwards deflection is at $\inline&space;0.42265\;l$ and is: $\inline&space;\hat{y}&space;=&space;\displaystyle\frac{W\;c\;l^2}{15.55\:E\;I}$
Deflection at the unloaded end: $\inline&space;\displaystyle\frac{W\;c\;l\;d\;}{6\;E\;I}$