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Beam with spring support

Beam with spring support

Beam with spring support

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

RE: Beam with spring support

I don't know about any tensor approach, but the situation you are exploring requires "beam on elastic foundation" theory.  The classic text on this is by Hetenyi, and is called something like "Theory of Beams on an Elastic Foundation".  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.  The deflected shape (where the beam is elastically supported) will deflect in the shape of damped trig curve.

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