## Modulus of subgrade reaction method

The program solves the differential equation:

Where:

w  = deflection of the slab [m]

x, y  = coordinates [m]

ks  = modulus of subgrade reaction [kN/m³]

ν  = Poisson's ratio of slab [-]

E  = Young's modulus [kN/m²]

t  = thickness of slab [m]

Analytical solutions for this differential equation only exist for a few special cases. For problems in daily design practice (with variously distributed loads, freely pivoted, restrained or supported boundaries, etc.) numerical methods must be relied on.

The differential equation is solved by the program using finite-element-methods. Triangle elements are used. Simple assumptions are made for these triangle elements with regard to displacements. In the present case, a displacement assumption is used that is described in Zienkiewicz  (Carl-Hanser-Verlag, 1984, Page 236). This displacement approach produces very good results compared to other approaches. The approach adopted leads to equation systems with a number of variables corresponding to three times the number of system nodes. The overall solution is assembled like a mosaic from the many partial solutions via the triangle elements. It is clear that the quality of the solution is increased with increasing finite-element mesh refinement.

The moment distribution is acquired from the two-fold numerical differentiation of the deflection area. The numerical differentiation always produces an undesirable roughening effect. In order to compensate for this the program offers two different methods for determination of moments:

• Method 1:
The moments are determined in the element centre in a post-processing calculation and then proportionally distributed across the neighbouring nodes. Generally delivers the best values. The moments may deviate, but only in the boundary regions.

• Method 2:
The moments are determined in the triangle nodes of each element in a post-processing calculation. The actual value at each node is given by forming the mean. Only produces better values than Method 1 (boundary region) for simple systems.

Shear force calculation represents a further fundamental problem when using the finite-element method because the derivation of the moments cause additional roughening of the function.

The quality of the calculated displacements is, generally, excellent. If you are only interested in the displacements you need not worry about the following explanations.

Remember that all finite-element methods are approximation methods. The quality of the approximation increases with increasing mesh density. In the current version, systems with a maximum of 45,000 triangle elements and nodes can be processed.

• Note on boundary conditions:
The case of a boundary with a free-earth support is automatically considered when the finite-element-method is employed. All system boundaries or partial system boundaries possessing no action or displacement boundary conditions automatically have a free-earth support. In finite-element theory this type of boundary condition is also known as a natural boundary condition.