Computational Mechanics Australia Pty Ltd
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Finite Element Modeling of bending of plate-and-beam structures on elastic foundation.
QUAD-PLATE is mathematical software 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++ without any maintenance fees or royalties.

To illustrate the accuracy of QUAD-PLATE, consider analysis of a thin square plate resting on elastic supports.

Laterally loaded plate resting on elastic foundation is a common mechanical model; typical examples include a concrete road or an airport runway. We assume that the reaction of the subgrade is proportional to the deflection of the plate. This reaction is given by folliwing product: kw, where k is called a subgrade modulus.

The subgrade modulus is expressed in units of force per volume.

In Finite Element Analysis the subgrade modulus is used to obtain a vertical elastic spring constant (expressed in units of force per length) multiplying the subgrade modulus by the area of the spring support element.

As an example, we consider a square plate resting on elastic subgrade. The plate is under the action of a uniformly distibuted load; all four edges of the plate are simply supported. An analytical soution of the problem is presented in the book "Theory of Plates and Shells" by S. Timoshenko and S. Woinowsky-Krieger.

The calculations were performed for the plate made of a material with the following properties: the Young modulus of 20 GPa, the Poisson ratio of 0.15.
The geometry of the plate is square, 1 meter by 1 meter; plate thickness is 4 cm. The intensity of the applied load is 100 MPa.

Mesh of square plate with elastic supports
Mesh of square plate
The distribution of the bending moments due to loading is shown below. The image contains the numerical solution on the left and the corresponding analytical solution on the right.

It can easily be seen that the numerical solution obtained using the quadrilateral plate element based on the Mindlin theory and the analytical solution are practically equal.
Bending moment Mx in plate on elastic foundation: numerical and analytical solutions
Bending moment Mx in a square plate on elastic foundation
The maximum deflection happens in the middle of the plate and is equal to 3.53 mm. The corresponding maximum analytical deflection is 2.99 mm.

The introduction of the subgrade modulus tends to reduce the deflection and the bending moments of the plate. If we perform computations for the exactly the same plate but without elastic supports, we obtain the the maximum deflection value of 3.75 mm.

Below we present the diagram of the numerical bending moments in the plate with elastic supports (on the left) and without elastic supports (on the right).
It can be seen that the values of the bending moments are also decresed for the plate with elastic supports.
Bending moment Mx in plate on elastic foundation: numerical and analytical solutions
Bending moment Mx in a square plate with or without elastic supports
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 ANSI C and will compile and run on all UNIX and Windows platforms without changes. They are also fully commented, allowing easy modification at user discretion.

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

QUAD-PLATE standard distribution kit includes:

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

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

- a graphical application that allows the user to visualize the input and output data of QUAD-PLATE.

QUAD-PLATE is 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.

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