1. The Tri-harmonic Plate Bending Equation
CMEM th International Conference on Computational Methods and Experimental Measurements Opatija, Croatia June
The Tri-harmonic Plate Bending Equation
Ihor D. Kotchergenko
KOTCHERGENKO ENGENHARIA LTDA.
Belo Horizonte – MG - Brasil

2. The areolar strain concept

3. Kolosov’s derivatives

4. Polar representation

5. Strain as a potential function
Is a total derivative Wo W1 C1 C2

6. Compatibility equations

7. Equivalence to Helmholtz theorem
Canonic form of elastic energy (2-dimension)

8. Irrotational and equivoluminal waves

9. Reflection of an irrotational wave

10. Reflection of an irrotational wave

11. The traction vector y X

12. Green’s formula in complex form

13. Gain of area during straining

14. Taylorian expansion of the displacement field

15. The boundary conditions at the surfaces

16. Stresses in a beam

17. Simply supported beam with uniform load

18. Deflection of simple suported beam

19. Plate bending

20. Stresses at a plate section

21. First approach to shear strains

22. Plate bending and torsion moments

23. The refined shear forces

24. The tri-harmonic plate bending equation

25. Bending and shear rigidities

26. Kirshhoff’s plate bending test

27. Elimination of Kirchhoff’s anomally

28. The plate bending boundary conditions

29. Boundary conditions

30. References
[7] Reissner, E., The effect of transverse shear deformation on the bending of elastic plates, ASME Journal of Applied Mechanics, Vol. 12, pp. A68-77, 1945.
[8] Wang, C. M., Lim, G. T., Reddy, J. N, Lee, K. H, Relationships between bending solutions of Reissner and Mindlin plate theories, Engineering Structures, vol. 23, pp ,
[9] Kotchergenko, I.D., The Areolar Strain Concept Applied to Elasticity, WIT Transactions on Modelling and Simulation, Vol. 46, 2007, WIT Press, (Open Access).
[10] Kotchergenko, I.D., The Areolar Strain Concept, Mechanics of Solids,
Structures and Fluids, Volume 12, ASME, IMECE-2008.
[11] Kotchergenko, I.D., The Areolar Strain Approach for grazing waves, , WIT Transactions on Modelling and Simulation, Vol. 55, 2013, WIT Press, (Open Access).
[12] Kotchergenko, I.D., Kolosov-Mushkhelishvili Formulas Revisited, 11thInternational Conference on Fracture, Turin, March 2005.
[13] Mitrinovic, D.S., Keckic, J.D., From the History of Nonanalitic Functions, Série:
Mathématiques et Physique, No , Publications de La Faculté D’Electrotechnique de L’Université à Belgrade, 1969, (Open Access).