Kinematics of Euler Bernoulli and Timoshenko Beam Elementsby bugatti79 Tags: beam, bernoulli, elements, euler, kinematics, timoshenko 

#1
Nov2712, 01:50 PM

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Folks,
Trying to get some appreciation for what is going on in the attached schematic of 1)Euler bernoulli and 2) Timoshenko beam elements. For the first one, ie the top picture, how was ##u z \frac{dw}{dx}## arrived at? thanks 



#2
Nov2712, 02:36 PM

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dw/dx is the slope of the beam, which is assumed to be small. So dw/dx is also the angle the beam has rotated, in radians.
The top picture (Euler beam theory) assumes that cross sections of the beam stay perpendicular to the neutral axis. So the angle between a cross section and the vertical is the same as the slope of the beam. The picture is (stupidly, IMHO) drawn with a "left handed" coordinate system (z and w positive downwards not upwards) which is where the minus signs come from. In the bottom picture (Timoshenko beam theory) plane sections of the beam do not stay perpendicular to the neutral axis, so there is an extra shear strain (measured by angle gamma) involved. 



#3
Nov2812, 07:47 AM

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#4
Nov2812, 02:36 PM

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Kinematics of Euler Bernoulli and Timoshenko Beam Elements 



#5
Nov2812, 02:41 PM

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What practical examples are there where one shouldn't use EulerBernouilli to track beam deflection etc. Would it for applications of plastic loading?
Thanks 



#6
Nov2812, 11:28 PM

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For a rectangular section beam, Euler is OK when length/depth > 10 (some people say > 20). For a more complicated criss sections, and/or composite beams made from several materials, you have to consider each case on its own merits. With computer software like finite element analysis, you might as wel always use the Timoshenko formulation. Even if the correction is neglibile, it doesn't cause any numerical problems to include it. 


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