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An Elastic Plate Problem

It would be very helpful for me if somebody clarifies the following problem: Suppose a beam with simply-supported boundary condition is loaded by uniformly distributed load. One can find out the moment and shear force at any point on the beam by finding forces at the boundaries and then considering equilibrium of some section of the beam.

Is it possible to apply similar procedure in plate problem? Suppose there is a simply-supported plate loaded by uniformly distributed load. Is it possible to find out shear forces (Vx and Vy) and moments (Mxx, Myy and Mxy) CONSIDERING EQUILIBRIUM OF SOME PLATE SECTION ? It is required to find out the shear forces and moments WITHOUT USING STRESS STRAIN RELATION, i.e. NOT BY FIRST FINDING OUT THE DISPLACEMENTS USING GOVERNING EUATION AND THEN THE MOMENTS FROM MOMENT-DISPLACEMENT RELATION.


You can use exactly the same approach as used in the elementary theory  of beams.  But things get quite a bit more complicated.  If you just want to deal with bending moments you can try to solve the second order differential equation for the moments and then solve for the shear forces.  The problem is the specification of boundary conditions

For a detailed exposition you should read Timoshenko and Woinowsky-Krieger's "Theory pf Plates and Shells" - the material around equation 100.  Another good starting point is "Advanced Mechanics of Materials" by Boresi et al. (chapter on flat plates).


Thank you very much for clearing this doubt and the provided references.

I think the moment and shear force distribution depends only on the loading and boundary conditions. So a change in constitutive relation (stress-strain relation), will not bring change in moment and shear force distribution. One example is nonlocal elasticity theory (differential constitutive relations of nonlocal elasticity theory by A. C. Eringen) where constitutive relation is a bit different from traditional elasticity theory.

So, moment and shear force distribution will be same for traditional and nonlocal elasticity theory. Am I correct? 

This is an interesting question which has been partly addressed in "Non-local elastic plate theories" by Lu, Zhang, Lee, Wang and Reddy, Proc, Royal Society Lond. A, vol. 463, pp. 3225-3240, 2007. 

The easiest way to check your idea is to compare the shear forces and bending monents from that paper with the results for a local plate.  I haven't got access to the paper right now, but if you get hold of it please send a copy over to me.

If the nonlocality is only due to gradients in the strain field, then I don't expect the balance of forces to change due to the local/nonlocal nature of the material.  For an example of equations for shear stresses and bending moments in a nonlocal beam see "Applications of nonlocal continuum mechanics to static analysis of micro- and nanostructures" by Wang and Liew, Physics Letters A, vol. 363, no. 3, pp. 236-242, 2006.

-- Biswajit 



Thank you for your comments.

 I am also trying to get a copy of  "Non-local elastic plate theories" by Pin Lu et. al. for many days. That paper may have addressed these interesting issues. If you get a copy kindly send me one.

In almost all the papers on nonlocal elastic structures (Eringen's nonlocal elasticity model) equilibrium equations are taken same as local one. So if moments and shear forces can be calculated from equilibrium equations and boundary conditions alone, those should be same for local and nonlocal cases.

You can consult following recent paper where the authors showed that moment for clamped-clamped beam do depends on nonlocal paramenter (so not same as local moment):

 "Nonlocal continuum theories of beams for the analysis of carbon nanotubes", J. N. Reddy and S. D. Pang, JOURNAL OF APPLIED PHYSICS 103, 023511, 2008

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