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How to calculate the shear stiffness of a beam

Lately I began to wonder why I hadn't learned how to calculate the beam shear stiffness in college. I vaguely recalled that the teacher told us the shear stiffness could be omitted in the beam calculation compared to the bending stiffness.

Can anybody show me how to calculate the shear stiffness if it is large enough and cannot be ignored?  Like the example shown in the picture below

 

 

 

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Stachiv's picture

Dear James,

1st derive the equation of motion;

2nd decompose the problem (equation) into two parts: 1) with linear force w1 and 2) with linear force w2

3rd the problems has to be combined together so you can apply so called matching conditions.

It is tedious work but can be performed analytically.

Good luck,

Ivo

p.s. you can find many scientific papers solving similar topic.

Dear Stachiv, 

Thank you very much for your in time reply!

I have been searching for the relative papers for a while but sadly, got nothing valuable. I am just wondering whether I have picked up the right searching keywords, "cantilever beam, shear stiffness".

Best,

James

Shear stiffness is a material+geometric property and does not depend on the applied load.  Seep. 92 of

"Mechanics and analysis of beams, columns and cables: a modern introduction" By S. Krenk

 http://books.google.co.nz/books?id=jxOJdcMbw_cC&pg=PA91&dq=shear+stiffness+beam&hl=en&ei=lBX4TdjQBtHWiAK977X9DA&sa=X&oi=book_result&ct=result&resnum=6&ved=0CEYQ6AEwBQ#v=onepage&q=shear%20stiffness%20beam&f=false

Stachiv's picture

In general you can derive the beam equation with due account of rotary inertia and shear deformation. The Euler-Bernoulli beam Eq. is s special case of general beam equation (take for instance Theory of vibration with applications from W. T. Thomson and as to the approach to solve the problems by splitting you can take for instance Worked problems in applied mathematics by lebedev et al or you can read download paper from here: http://www.sciencedirect.com/science/article/pii/S0888327009003811

Hope it can helps.

Regards, 

Ivo

 P.S. I real application I suggest you to rewrite your eq. of motion in dimensionless form and compare the values of each terms. It allows one to neglect some terms in equation.

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