Solution for beam on elastic beam

You can get some ideas from

Bending of sandwich beams with transversely flexible core

Y Frostig, M Baruch - AIAA
journal, 1990 - pdf.aiaa.org

http://pdf.aiaa.org/getfile.cfm?urlX=8%3CWIG7D%2FQKU%3E6B5%3AKF2Z%5CD%3A%2B82*H%25%5E_SI%0A

Dear parisa,

Assume that the two beams show two different flexural deflections, equal to wup(x) and wdown(x) respectively. Consider first the lower beam. It is simply supported and it has a distributed load along its length, which is equal to the distributed load imposed by the springs. This is equal to {wup(x)-wdown(x)}*k(x), where k(x) is the spring constant which may be independent of x, or may be different for each position on the horizontal axis. Then, solve the simply supported lower beam, and calculate its deflection wdown(x) as a function of the product {wup(x)-wdown(x)}*k(x), then solve the last equation for wup(x), namely express wup(x) as a function of wdown(x). The boundary conditions at the simple supports will be zero moment and zero vertical displacement, as it is well known...

In addition, you must impose the static equilibrium equations, to the whole beam system, namely force equilibrium along the vertical axis and moment equilibrium along on simple support of the lower beam.

Finally, you have to solve the upper beam, by considering it as a beam which carries the following loads: 1-The imposed vertical force P, and 2-The distributed load from the springs, which is equal to {wup(x)-wdown(x)}*k(x) in terms of its absolute value. Then solve the upper beam to find its vertical deflection wup(x). wup(x) will be expressed in terms of wdown(x). So, from the above procedure, two equations result, containing wup(x) and wdown(x). Solving them simultaneously will lead to the calculation of the vertical deflections of the two beams. Remember, boundary conditions for the upper beam at its ends: zero moment and zero shear force.

The above procedure will conclude to the analytical solution that you seek. I hope I helped you. For any other information do not hesitate to contact me at gpapazafeiropoulos@yahoo.gr.

Best regards,

George Papazafeiropoulos

Second Lieutenant, Hellenic Air Force

Civil Engineer, PhD candidate

Dear Manooir,

Provided that the bending moment diagram of a beam is known, you can calculate the deformation of the beam by the equation:

M(x)=-EIw''(x)

The difference between your problem and parisa's problem is that, parisa will take this equation for the whole length of the beam, whereas in your problem, you are going to take tis equation for each part of the beam, which has a constant modulus. As you have stated, your problem involves two beams, each of which comprises of three parts, each of which has a different value of modulus of elasticity E. So you will take (for each of the two beams) that:

[M1(x)]=-[E1][I1][w1''(x)], for 0<x<l1

[M2(x)]=-[E2][I2][w2''(x)], for l1<x<l2

[M3(x)]=-[E3][I3][w3''(x)], for l2<x<l3

where l1 and l2 are the positions on the beam where the modulus of elasticity changes its value, and l3 is the total length of the beam. The remaining procedure is the same.

The load is not supposed in general to be uniformly distributed along the lower beam. The parts of the beam which have increased modulus of elasticity will take over higher bending moments than the others. But there are exceptions, according to the specific problem studied each time..

I hope I helped you. If you have any questions, please contact me at gpapazafeiropoulos@yahoo.gr for more details...

Best regards,

George Papazafeiropoulos

Second Lieutenant, Hellenic Air Force

Civil Engineer, PhD Candidate