The Tri-harmonic Plate Bending Equation CMEM 2015 17th International Conference on Computational Methods and Experimental Measurements Opatija, Croatia June 05-05 2015 The Tri-harmonic Plate Bending Equation Ihor D. Kotchergenko koteng@terra.com.br KOTCHERGENKO ENGENHARIA LTDA. Belo Horizonte – MG - Brasil

The areolar strain concept

Kolosov’s derivatives

Polar representation

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

Compatibility equations

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

Irrotational and equivoluminal waves

Reflection of an irrotational wave

Reflection of an irrotational wave

The traction vector y X

Green’s formula in complex form

Gain of area during straining

Taylorian expansion of the displacement field

The boundary conditions at the surfaces

Stresses in a beam

Simply supported beam with uniform load

Deflection of simple suported beam

Plate bending

Stresses at a plate section

First approach to shear strains

Plate bending and torsion moments

The refined shear forces

The tri-harmonic plate bending equation

Bending and shear rigidities

Kirshhoff’s plate bending test

Elimination of Kirchhoff’s anomally

The plate bending boundary conditions

Boundary conditions

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. 838-849, 2001. [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. 274-371, Publications de La Faculté D’Electrotechnique de L’Université à Belgrade, 1969, (Open Access).