Simple Supported Beams

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

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

Stress and deflection formulae for simple supported beams

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Introduction

The stress and deflection for simply supported beams under a number of loading scenarios is illustrated within this page.

The following symbols have been used throughout:

$S$ is the Stress at any point
$Z$ is the Section Modulus of beam cross section.
$y$ is the deflection at any point.
$W$ is the load on the Beam. Note for uniform loads $W = wl$ where $w$ is the load per unit length
$E$ is the Modulus of Elasticity ( Young's Modulus)
$I$ is the Moment of Inertia of the cross-section about the neutral axis

Simply Supported at both ends with a uniform load.

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

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

Stress at critical points: $-\displaystyle\frac{Wl}{8Z}$

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

Deflection at critical points:

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

Simply supported at both ends . Load at centre.

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

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

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

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

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

Simply supported at both ends. Load at any point.

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

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

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

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

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

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

When $a$ is the length of the shorter portion and $b$ the longer one, the maximum deflection is in the longer one at:

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

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

Simply supported at both ends. Two equal loads symmetrically placed

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

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

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

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

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

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

Simply supported but with both ends overhanging symmetrically. Uniform Load.

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

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

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

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

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

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

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

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

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

Simply supported with both ends overhanging, Supports unsymmetrical, uniform load.

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

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

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

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

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

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

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

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

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 $a$ $S=- \displaystyle\frac{W\;b\;x}{Z\;l}$

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

Beyond the support $S = 0$

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

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

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

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

For the deflection between the supports see paragraph 3 above

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

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

Simply Supported with ends overhanging. Single overhanging load.

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

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

Between the unloaded end and the adjacent $S = 0$

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

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

The deflection between the load and the adjacent support $y=\frac{W\;u}{6\;E\;I}(3cu - u^2 + 2cl)$ Deflection between the supports: $y=- \displaystyle\frac{W\;c\;x}{6\:E\;I}(l - x)(2l - x)$

Deflection between the loaded and adjacent support: $y=\displaystyle\frac{W\;c\;l\;w}{6\:E\;I}$

Deflection at the load: $y=\displaystyle\frac{W\;c^2}{6\:E\;I}\;\;(c + l)$

The Maximum upwards deflection is at $0.42265\;l$ and is: $\hat{y} = \displaystyle\frac{W\;c\;l^2}{15.55\:E\;I}$

Deflection at the unloaded end: $\displaystyle\frac{W\;c\;l\;d\;}{6\;E\;I}$