Register to reply 
Kinematics of Euler Bernoulli and Timoshenko Beam Elements 
Share this thread: 
#1
Nov2712, 01:50 PM

P: 660

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

Engineering
Sci Advisor
HW Helper
Thanks
P: 7,177

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

P: 660




#4
Nov2812, 02:36 PM

Engineering
Sci Advisor
HW Helper
Thanks
P: 7,177

Kinematics of Euler Bernoulli and Timoshenko Beam Elements



#5
Nov2812, 02:41 PM

P: 660

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

Engineering
Sci Advisor
HW Helper
Thanks
P: 7,177

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. 


Register to reply 
Related Discussions  
Euler Bernoulli Beam 4th order ODE Balance of Units  General Math  3  
EulerBernoulli Beam Theory and Nonlinear Differential Equations  Classical Physics  9  
Euler Bernoulli BeamFinite Element Method  Engineering, Comp Sci, & Technology Homework  0  
Derivation of Deflection from EulerBernoulli Beam Equation  Engineering, Comp Sci, & Technology Homework  8  
[Vibration analysis] Timoskenko beam Vs.Eulerbernoulli beam ?  Mechanical Engineering  2 