Beam Analysis Tool

This calculator has two sub-pages that evaluate single-span and multi-span beams under user-defined loading conditions. It outputs the maximum moments and deflections along with shear, moment, slope and deflection diagrams. 
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Single-Span Beam Analysis
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Multi-Span Beam Analysis Tool
Calculator assumptions
ExplanationSupportsBeams can have three types of idealised support conditions: pin, roller or fixed. Each support condition restrains the beam in different ways, thereby affecting the force development in the beam.
At a point of support, there are a maximum of 6 degrees of freedom, which describe the number of possible movements: 3 translations in x, y, z and 3 rotations about x, y, z.
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Six degrees of freedom at a beam support
If a certain degree of freedom is restrained at a support, then there is a corresponding reaction. For example, pinned supports allow rotation, so there will be no moment reaction. But since pinned supports restrict translation in both the x and y direction, there will be a reaction in both of those directions. 
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Idealised roller support (left) with roller reaction (right)
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Idealised pin support (left) with pin reactions (right)
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Idealised fixed support (left) with fixed reactions (right)
Internal ForcesWhen a beam is loaded, internal forces develop within it to maintain equilibrium. The internal forces in a beam have two components: shear forces in the vertical direction and normal forces in the axis of the beam. 
We can represent the internal forces at any point in a beam with two resultant forces: a shear force, which is the resultant force of all internal vertical forces at the point of interest and a bending moment which is the resultant of the normal internal forces at the point of interest. Shear force and bending moment vary in magnitude along the length of the beam. 
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Internal forces within a beam
If you apply a load on the topside of a beam, the beam will be curving downwards, also known as sagging action. When a beam is sagging, the top of the beam will shorten and the forces at the top of the beam will be in compression, while the bottom side of the beam will stretch and the fibres at the bottom of the beam will be in tension. The inverse is known as hogging action, when the load is applied from the underside of the beam or when a beam runs continuously over supports.
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Sagging action where top fibres are in compression and bottom fibres are in tension
Beam AnalysisTo design a beam, you need to know what shear forces and bending moments will exist in the beam under your desired loading. Beam analysis is the process to determine these forces. A common method is to plot the Shear Force Diagram (SFD) and Bending Moment Diagram (BMD), which show you the shear force and bending moment at any point along the beam. 
There are several ways to determine the points needed to plot a SFD and BMD. You can use engineering principles, for example that fact that a uniformly distributed load creates a parabolic bending moment profile. You can also analyse the internal forces at any point in a beam directly, by making an imaginary ‘cut’ at a point of interest in the beam and then solve the three equations of equilibrium: ΣFx = 0, ΣFy = 0 and ΣM=0 with the applied loads and reactions/internal forces.
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Imaginary 'cuts' at points of interest along a beam
Sometimes you cannot solve the three equations of equilibrium because there is more than three unknown variables, this means you have a statically indeterminate beam. You must therefore use Finite Element Analysis (FEA) to determine your internal forces by solving for your degrees of freedom in the stiffness equation F = k x. The maths quickly becomes too complex to do by hand, which is where computer FEA packages come into play!
Sign ConventionA common sign convention is as follows:
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Applied forces pointing downwards are negative (-)
Reaction forces pointing upwards are positive (+)
Reaction moments causing the beam to concave upwards and create a sagging action are positive (+). Remember, sagging = smiley face = positive.
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🙏 Credits
Based on the spreadsheet "BEAMANAL.xls" by Alex Tomanovich.
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