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Simple Supported Beams
Stress and deflection formulae for simple supported beams
Contents
- Introduction
- Simply Supported At Both Ends With A Uniform Load.
- Simply Supported At Both Ends . Load At Centre.
- Simply Supported At Both Ends. Load At Any Point.
- Simply Supported At Both Ends. Two Equal Loads Symmetrically Placed
- Simply Supported But With Both Ends Overhanging Symmetrically. Uniform Load.
- Simply Supported With Both Ends Overhanging, Supports Unsymmetrical, Uniform Load.
- Simply Unsymmetrically Supported With Both Ends Overhanging And A Load At Any Point.
- Simply Supported With Ends Overhanging. Single Overhanging Load.
- Page Comments
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
![y=\frac{Wx(l - x)}{24\,E\,I\,l}\:\;[l^2 + x(l - x)]](/images/eqns/4553c7e33a4bcf1165ccede901bdafee.gif){kind=small}
Simply Supported At Both Ends. Two Equal Loads Symmetrically Placed
Click here to expand this hidden section 
![y=\frac{W\;x}{6\;E\;I}\times [3a(l - a) - x^2]](/images/eqns/55ad3cef12037f284ebc68c70ca2b482.gif){kind=small}
![y=\frac{W\;a}{6\;E\;I}\times [3v(l - v) - a^2]](/images/eqns/76cddf4cd1226f1ae1d9d781d5a94ca3.gif){kind=small}
Simply Supported But With Both Ends Overhanging Symmetrically. Uniform Load.
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![S=\frac{W}{2\;Z\;L}\;(c^2 - x[l - x])](/images/eqns/d4cba04bc381f5b792b8418fd5093988.gif){kind=small}
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:-
![y=\frac{W\;u}{24\;E\;I\;L}\;\times[6c^2(l + u)\;'- u^2(4c - u) - l^3]](/images/eqns/b844e36aa64de8ec3b0fc4a8880028f3.gif){kind=small}
![y=\frac{W\;x(l - x)}{24\;E\;I\;L}\;\times[x(l - x) + l^2 - 6c^2]](/images/eqns/f32b7148104dd5a445835d58946927f6.gif){kind=small}
![y_e=\frac{W\;x}{24\;E\;I\;L}\;\times[3\,c^2(c + 2l) - l^3]](/images/eqns/1854adf8b965aacb91c743621c470e95.gif){kind=small}
![y_c=\frac{W\;l^2}{384\;E\;I\;L}\;\times[5\;l^2 - 24\,c^2]](/images/eqns/c08763de41b3937af8765d686bd4ee8a.gif){kind=small}
on each side of the centre.the value of these are :-
the stress is zero at points 
on both sides of 
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]](/images/eqns/209823c7923f9c5a51581a2351ec3713.gif){kind=small}
![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\}](/images/eqns/7addd1e75bbc1d32afe19b71547ddd1d.gif){kind=small}
![y = \frac{Ww}{24\;E\,I\,L}\;\;[2l(c^2 + 2d^2) + 6d^2w - w^2(4d - w) - l^3]](/images/eqns/75df52be6d0fd9ab1ea4f5461134175d.gif){kind=small}
![y=\frac{Wc}{24\;E\,I\,L}\;[2l(d^2 + 2c^2) + 3c^3 - l^3]](/images/eqns/b77fcd7a59d78306b2bd3767ebce96b9.gif){kind=small}
![y=\frac{Wd}{24\;E\,I\,L}\;[2l(c^2 + 2d^2) + 3d^3 - l^3]](/images/eqns/d3942796147407b341bbdd7c22d63ed3.gif){kind=small}
Simply Unsymmetrically Supported With Both Ends Overhanging And A Load At Any Point.
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Simply Supported With Ends Overhanging. Single Overhanging Load.
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