Beam with spring support

[Greg_bob]

Dear all,

Please find attached the system I'm intresting.
Let's consider an infinite beam (b=infinite) supported on springs from x to the end, with uniform load.
From timoshenko beam theory I'm looking for the tensor stress acting at the base on a quasi-static test.

I have to questions:
1. How to compute the length (Lambda) where the beam deflection is acting, I think Lambda should be function of elastic modulus of the beam (E_1) spring elastic modulus (E_2) and beam height (D_1). Assumption: the beam tip never touch the base.

2. How to compute the stress tensor acting at the base (sigma(x)).
for sigma(x<a)=0
for sigma(x>a+Lambda)=rho g D_1

Could you please help me and provide me some lecture/tips to reach the solution?

Best regards,
Greg B

 

[GregLocock]

Ludicrous file size


Cheers

Greg Locock


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[Denial]

I don't know about any tensor approach, but the situation you are exploring requires "beam on elastic foundation" theory.[&nbsp;] The classic text on this is by Hetenyi, and is called something like "Theory of Beams on an Elastic Foundation".[&nbsp;] Roark's formula book contains enough simple results for you to get what you are seeking with a few algebraic manipulations.

Your attached diagram of results is not conceptually correct.[&nbsp;] The deflected shape (where the beam is elastically supported) will deflect in the shape of damped trig curve.