Elastic curve of a spring supported beam

[heitor]

Hello.

I'd like to validate a finite element code using a spring supported beam. It's a beam with a spring at each side.

I'd like to find the elastic curve of this beam, but my memory betrayed me. I forgot how to get this equation.

Maybe by superposition? At first I would find the elastic curve equation of a simply supported beam, after I would add the deflection of the springs?

Thank you.

 

[JStephen]

A simply supported beam, with spring supports at each end?
Calculate the deflection for a simply supported beam. How you get this equation is look it up in AISC manual or "Formulas for Stress and Strain", or hunt up the Strength of Materials textbook from college.
When you have that, add the spring deflection to it. Take the reactions from the beam case, calculate spring deflection, and add linear deflection to the beam deflection.

 

[ccw]

Hi Heitor,

I am not sure what you mean by the first part. I get a picture of a beam simply supported at the ends by resting on top of two equal coil springs, one on each end in compression, or maybe two equal coil springs in tension. Is that correct?

Anyway the second part elastic curve equation of a simply supported level beam, constant cross section, evenly loaded, constant distributed load, say by its own weight is as follows:

Let w = lbs / in. dist. load
Let L = total length of beam in inches.
Let x be the distance along the beam in inches from end.
Let y be the deflection at any point x in inches.
Let I be the section momemt of inertia in inches**4
Let E be the modulus of elasticity in tension lbs/in**2

then y = [wx/24EI][L**3-2Lx**2+x**3] inches.

ymax (obviously at x=L/2) = [5wL**4]/[384EI] inches.

If my interpretation is correct in the first part, then superposition will work. Just add a constant offset to the y = equation above of Ys= wL/2k inches, where k = spring rate of each spring.

 

[israelkk]

heitor

I do not intend to offend you but I would like to say that if you intend to master FEA you must have a very strong knowledge and understanding of mechanics of materials, theoretical elasticity, plasticity, strength of material etc.

FEA is just a tool and a tool in the wrong hands can do a great damage.

I recall an ad from a company who is selling a product for stress and strain testing feature where they use a slogan similar to: beware that your "finite element" want become a "finite elephant".

FEA analysis can produce nonsense and fault results if boundary conditions and loads are not set correctly. Setting those parameters demand deep understanding of the theory of of all the field mentioned earlier.

Interpreting the finite element results is even more sensitive issue and without the theoretical knowledge false results may be considered as real and accurate results.

If you can not produce the deflection of a beam equations analytically from the basic differential equation of a beam y''=M/(E*I) then you must invest more and hard in the theoretical issue if you intend to go along with FEA.

 

[heitor]

Thank you all.

Israelkk, you didn't ofend me. I posted this question because I don't want to blindly trust the finite element results, I think you got the point. Anyway, thanks for such a kind warning. Thanks to you, the elephant is going away

 

[rb1957]

isn't this problem statically determinate ?

so it's easy to calculate the load at each end, and then deflection from the spring constant ??