MatrixFrame® allows elastic foundation calculations according to the Winkler model or the Winkler–Pasternak model.
Winkler model
The analysis of beam bending on an elastic foundation is based on the assumption that the reaction forces of the foundation at each point are proportional to the deflection of the beam at that point. The vertical deformation behaviour of the foundation is represented by identical, independent, closely spaced, discrete and linearly elastic springs. The constant of proportionality of these springs is referred to as the spring (soil) stiffness coefficient Cz, Cy [kN/m³∙m]. This mechanical representation of an elastic foundation was introduced by Winkler:
where:
SoilpressureZ - soil pressure [kN/m²]
Uz - displacement [m]
Cz = SoilpressureZ/Uz - spring (soil) stiffness coefficient [kN/m³∙m]
For 2D analysis, the spring (soil) stiffness coefficient Cz acting along the Z-axis is used. For 3D analysis, the coefficients Cz and Cy acting along the Z-axis and Y-axis, respectively, are used.
Pasternak model
Another foundation model, proposed by Pasternak, considers the shear interaction between adjacent springs. This mathematical model introduces additional parameters Cfz and Cfy [kN/m], which characterise the interaction between springs due to transverse shear, associated with frictional effects.
For the subsoil, Pasternak proposed a two-parameter model characterised by the modulus of subgrade reaction k, representing resistance to vertical loading, and the shear modulus G [N/mm], representing resistance to shear forces.
In this model, the foundation is represented by two functional layers: an upper layer responsible for horizontal forces and a lower layer responsible for vertical forces:
The subsoil reaction p acting on the beam is defined by the following equation:
In this expression, ps= kw.
The differential equation is expressed as:
If:
and 
then:
where:
g - shear ratio
For an axially symmetric circular plate, the solution is:
For a rectangular plate:
For 2D analysis, the spring (soil) stiffness coefficient Cfz acting along the Z-axis is used. For 3D analysis, the coefficients Cfz and Cfy acting along the Z-axis and Y-axis, respectively, are used.
Elastic foundation name
The elastic foundation is assigned a default identifier consisting of the prefix 'E' followed by a numerical index. The naming sequence begins with 'E1' and increments sequentially for each new foundation. The default prefix may be modified in the Options menu.
Elastic foundation calculation models
MatrixFrame® provides two distinct models for elastic foundation calculations:
- Linear calculation: performed without tension elimination using a local coordinate system
- Non-linear calculation: incorporates tension elimination through a global coordinate system
Elastic foundation constant
The elastic foundation constant can be defined using one of three methods:
- Cz (spring value per member): calculation is based on a default unit width of 1 m (NONE).
- Elastic foundation constant + section width: calculation is based on a manually entered cross-section width (ABSOLUTE).
- Elastic foundation constant (width from section): calculation is based on the width defined in the cross section properties (PROJECTION).
Tapered elastic foundation
For tapered elastic foundation calculations, the start and end values of the spring (soil) stiffness coefficients must be specified and are required to be different.
Area-based elastic foundation
Area-based elastic foundations (elastic areas), also referred to as region-type foundations, are available in MatrixFrame® only for 2D-Plate and 2D-Wall projects.