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Simple Supported Beams

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.

23287/Simple-Supported-Beams-101.png
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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: 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.

23287/Simple-Supported-Beams-102.png
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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.

23287/Simple-Supported-Beams-103.png
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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

23287/Simple-Supported-Beams-104.png
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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: 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.

23287/Simple-Supported-Beams-105.png
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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.

23287/Simple-Supported-Beams-106.png
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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.

23287/Simple-Supported-Beams-107.png
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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.

23287/Simple-Supported-Beams-108.png
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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}