Computational Mechanics Australia Pty Ltd
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Finite Element Analysis of thin plate: deflections, bending moments, shear forces
Over several years our company devoted its efforts to developement of a very accurate and time-efficent software for modeling of bending of plates using quadrilateral elements based on the Mindlin theory. The result of our efforts has been a release of QUAD-PLATE - a very advanced program for Finite Element Analysis of stresses, strains and deflections arising in complex structures comprised of plates and beams.

QUAD-PLATE is offered with complete source codes written in standard C/C++. It is available without any royalties or additional fees.

This page is devoted to an example for which an analytical solution is known.

Consider analysis of a thin square plate under the action of uniform load. The edges of the plate are simply supported. The analytical soution of the problem is presented in the book "Theory of Plates and Shells" by S. Timoshenko and S. Woinowsky-Krieger.

Calculations were performed for a plate made of material with the following properties: the Young modulus of 200 GPa, the Poisson ratio of 0.27.
The geometry of the plate is square, 1 meter by 1 meter; plate thickness is 4 cm. The applied load (surface pressure) equals 100 MPa.

We will use a mesh deliberately created with a significant degree of irregularity, so that the mesh is not symmetric with respect to any axis. That would allow us to demonstrate excellent precision of the developed finite elements, which deliver quality numerical results regardless of the mesh distortions (the latter are common for domains of irregular shapes).
Finite Element Mesh of Square Plate
We present the following results for:

  • bending moments

  • twisting moment

  • transversal shear forces

  • Each of the images below contains the numerical solution on the left and the corresponding analytical solution on the right.

    It can be readily observed that the numerical solution obtained using the Mindlin element on the one hand and the analytical solution on the other hand are practically identical.
    Bending moment Mx: numerical and analytical solutions
    Bending moment Mx in a square plate under uniform load
    Bending moment My: numerical and analytical solutions
    Bending moment My in a square plate under uniform load
    Twisting moment Mxy: numerical and analytical solutions
    Twisting moment Mxy in a square plate under uniform load
    Transversal shear force Qx: numerical and analytical solutions
    Transversal shear force Qx in a square plate under uniform load
    Transversal shear force Qy: numerical and analytical solutions
    Transversal shear force Qy in a square plate under uniform load
    QUAD-PLATE is available with complete source codes and the full commercial lisence allowing easy incorporation into your own applications. All our codes are written in standard C/C++ and will compile and run on all UNIX and Windows platforms without changes. They are also fully commented, allowing easy modification at user's discretion.

    QUAD-PLATE is available on no-royalties and no-annual-renewals basis: there is a one-off price for an unlimited commercial lisence.

    QUAD-PLATE standard distribution kit includes:

    - fully-commented source codes of the program in C and C++.

    - the product's User Manual containing detailed description of input and output data structures;

    - a graphical application that allows the user to visualize input and output data of QUAD-PLATE for each particular example created by the user.

    QUAD-PLATE has been completely developed by our company and contains no third party code.

    To learn more about the full range of our products please follow this link.

    We also invite you to contact us with any questions, preferably by e-mail on or