Timoshenko Beam Calculator and Plot

Last updated September 22, 2023
By Aidan Phelan

https://www.desmos.com/calculator/i8cxyq9ptf

This Desmos calculator uses the equations laid out in the “Diaphragm Deflection” post following the “Timoshenko Beam Formulas” post derivation. These equations are a derivation of the governing equation for a Timoshenko beam but using a similar method to the NDS in order to find a useful solution with numbers available in a standard diaphragm situation. That being said, unlike the NDS equation, all units are consistent and conversations will need to be made to pounds and inches. 

This calculator provides deflections at key points and provides full plots for both distributed loads and point loads of simply supported Timoshenko beams. The plots also give results with and without nail slip, the lower pair of curves being without nail slip (click the colored circle next to the equation to hide them). This offers some useful insight if the specifics of the chord splices are unknown, allowing you to see exactly how much each element of the chord splices will increase the deflections by. It also shows that when the beam is shear controlled not bending controlled the nail slip approaches zero and the lines converge since bending is needed to induce tension in the chords. 

The inputs are w = distributed load (lb/in), P = point load (lb), L = span length (in), W = depth (in), a = point load location (in), G_a = SDPWS table 4.2 (lb/in^2), E = elastic modulus of chords (lb/in^2), A = cross sectional area of chords (in^2), D = diameter of nails in splice (in), N = number of nails per splice, S = length of each chord board (length between splices) (in). Note, w=2*v*W/L where v is the unit shear. All the variables in Desmos are defined with equations, so rolling unit conversions into the data entry is very simple. The outputs are D = displacement (in) and k = stiffness (lb/in).

An interesting element of the point load graph is that it changes significantly whether or not the beam is bending controlled or shear controlled. The distributed load graph is always a parabola and acts as expected, but the point load graph shifts between linear and parabolic. When the beam is bending controlled, the point load deflection graph is parabolic, as expected for a Euler-Bernoulli bending beam. When the beam is shear controlled, the point load deflection graph is linear up to the point. This may be mathematically explained by how shear is the derivative of moment and this reflects itself in the deflections, and intuitively it makes sense that shear is distributed constantly along the length of the beam between point loads so, without bending, it will deflect the same amount per unit length due to the shear.

Disclaimer

Structural calculators are provided for your education and development. These calculators are provided as-is with no warranty or guarantee whatsoever. From time to time these calculators may contain mistakes or undisclosed assumptions. We strongly recommend you review the calculations, make adjustments to fit your needs, and check the results by hand until you are satisfied with the results. We reserve the right to update these calculators at any time and without notice. Use of these calculators is not intended to circumvent the need for a design professional – the end user is responsible to ensure the design is safe, meets applicable codes, and meets project needs.

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