WIT Press

The Tri-harmonic Plate Bending Equation


Free (open access)





Page Range

367 - 378




492 kb

Paper DOI



WIT Press


I. D. Kotchergenko


This is the third paper published at WIT Transactions on Modeling and Simulation focusing on the application of the areolar strain concept to revisit subjects of the theory of elasticity, allowing a deeper insight on some old topics. In Vol. 46, 2007, WIT Press, a new concept of strain is introduced which does not distinguish finite from infinitesimal strain. Besides the classical “forward” strain, it incorporates the “sidelong” strain into its imaginary part. Instead of comparing the change in distance between two contiguous points, this concept describes the complete state of strain in an areola that surrounds a given point. This strain is given by a total derivative, in agreement with its physical meaning. As a consequence, the relative displacement between two arbitrary points of the strained plane is obtained by a line integral of the strain, along any path joining these points. In Vol. 55, 2013, WIT Press, the nature of the propagation of plane shear waves is cleared up, showing that although a point of the plane performs lateral vibration only, the areola centered at this point pulses in the direction of the wave path, through a combination of rotation and shear. In the present paper, a similar behavior is explored to show that the rotation of the neutral planes of beams and plates does not follow the rotation of a point (areola) situated on these planes. A compatible field of strains is defined at the neighborhood of the neutral plane of the plate and then forced to comply with the boundary conditions. A sixth-order differential equation is obtained, which gives exactly the shearing forces, eliminating thus Kirchhoff’s anomaly. This tri-harmonic equation allows the prescription of complete boundary conditions for plate bending analysis.


areolar strain, beam shear, plate shear, tri-harmonic