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001893

2024-09-02

Stiffness of Nodal Support via Fictitious Column

This technical article presents the equations utilized by the program to determine support springs from the column parameters.

Modeling Column as Nodal Support

In certain instances, two-dimensional models may prove advantageous over their three-dimensional counterparts. For the in-plane modeling of detached slabs, however, the support conditions that result from the columns and are not represented in the 2D model must be taken into account. A rigid support would lead to stiffness ratios in the area of the nodal supports that usually do not reflect the reality. Furthermore, the use of a rigid nodal support would lead to an increase in singularity effects in the finite element (FE) calculation. Given that the yielding of the column influences the stiffness and internal forces, it is essential to incorporate this into the 2D model.

Tip

In the technical article "Preventing Singularities on Nodal and Line Supports of Plate Structures", the effects of 2D modeling are described using an example for RFEM 5.

Preventing Singularities on Nodal and Line Supports of Plate Structures

Determining Stiffness of Column

It is possible to specify the constants of the springs for displacement and torsion manually when defining the nodal support. However, the program also offers the option of determining the stiffness automatically. For this, select the "Stiffness via Fictitious Column" checkbox in the nodal support dialog box.

Determining Stiffness via Fictitious Column Using Program

In the "Stiffness via Fictitious Column" tab, you can then define the boundary conditions from which the program determines the support stiffness.

Determining Support Springs

First of all, you can select between three support models.

Selecting Model of Support

In the following, however, we will only discuss the determination of the spring constants in the support model using the "Elastic Surface Support" and the "Elastic Nodal Support", since the model of the "Nodal support with adapted FE mesh" is calculated numerically using iterations and stiffness matrices.

The dimensions of the column head determine the boundary conditions of the FE model and the design. This also defines the loaded area.

The column cross-section is governing for the column stiffness required for the calculation.

Elastic Surface Foundations

This model allows for a detailed analysis of the load distribution and the deformations over a surface. This type of analysis is more complex than that of the elastic nodal support, as it models a continuous distribution of reaction forces and the bending behavior in multiple directions.

Elastic surface foundation	Considering Shear Stiffness
Hinged at Column Base	$C_{u X} = \frac{12 \cdot E I_{z}}{h^{3} \cdot A_{head} \cdot (4 + k_{z})}$ $C_{u Y} = \frac{12 \cdot E I_{y}}{h^{3} \cdot A_{head} \cdot (4 + k_{y})}$ $C_{u Z} = \frac{E A}{h \cdot A_{head}}$ $C_{φ Y} = \frac{E I_{z}}{A_{head} \cdot h} \cdot \frac{2}{1 + k_{z}}$ $C_{φ X} = \frac{E I_{y}}{A_{head} \cdot h} \cdot \frac{2}{1 + k_{y}}$ Spring Constants – Hinged Column Head, Elastic Surface Foundation, Considering Shear Stiffness
Semi-Rigid at Column Base	$C_{u X} = \frac{12 \cdot E I_{z}}{h^{3} \cdot A_{head} \cdot (4 + k_{z})} + \frac{X}{100} \cdot \frac{36 \cdot E I_{z}}{A_{head} \cdot h^{3} \cdot (1 + k_{z}) \cdot (4 + k_{z})}$ $C_{u Y} = \frac{12 \cdot E I_{y}}{h^{3} \cdot A_{head} \cdot (4 + k_{y})} + \frac{X}{100} \cdot \frac{36 \cdot E I_{y}}{A_{head} \cdot h^{3} \cdot (1 + k_{y}) \cdot (4 + k_{y})}$ $C_{u Z} = \frac{E A}{h \cdot A_{head}}$ $C_{φ Y} = 2 \cdot \frac{E I_{z}}{\cdot A_{head} \cdot h \cdot (1 + k_{z})} + \frac{X}{100} \cdot \frac{E I_{z}}{A_{head} \cdot h \cdot (1 + k_{z})}$ $C_{φ X} = \frac{2 \cdot E I_{y}}{A_{head} \cdot h \cdot (1 + k_{y})} + \frac{X}{100} \cdot \frac{E I_{y}}{A_{head} \cdot h \cdot (1 + k_{y})}$ Spring Constants – Semi-Rigid Column Head, Elastic Surface Foundation, Considering Shear Stiffness
Restrained at Column Base	$C_{u X} = \frac{12 \cdot E I_{z}}{h^{3} \cdot A_{head} \cdot (1 + k_{z})}$ $C_{u Y} = \frac{12 \cdot E I_{y}}{h^{3} \cdot A_{head} \cdot (1 + k_{y})}$ $C_{u Z} = \frac{E A}{h \cdot A_{head}}$ $C_{φ Y} = \frac{E I_{z}}{A_{head} \cdot h} \cdot \frac{3}{1 + k_{z}}$ $C_{φ X} = \frac{E I_{y}}{A_{head} \cdot h} \cdot \frac{3}{1 + k_{y}}$ Spring Constants – Restrained Column Head, Elastic Surface Foundation, Considering Shear Stiffness

Elastic surface foundation	Without Considering Shear Stiffness
Hinged at Column Base	$C_{u X} = \frac{3 \cdot E I_{z}}{h^{3} \cdot A_{head}}$ $C_{u Y} = \frac{3 \cdot E I_{y}}{h^{3} \cdot A_{head}}$ $C_{u Z} = \frac{E A}{h \cdot A_{head}}$ $C_{φ Y} = 2 \cdot \frac{E I_{z}}{A_{head} \cdot h}$ $C_{φ X} = 2 \cdot \frac{E I_{y}}{A_{head} \cdot h}$ Spring Constants – Hinged Column Head, Elastic Surface Foundation, Neglecting Shear Stiffness
Semi-Rigid at Column Base	$C_{u X} = \frac{3 \cdot E I_{z}}{h^{3} \cdot A_{head}} + \frac{X}{100} \cdot \frac{9 \cdot E I_{z}}{A_{head} \cdot h^{3}}$ $C_{u Y} = \frac{3 \cdot E I_{y}}{h^{3} \cdot A_{head}} + \frac{X}{100} \cdot \frac{9 \cdot E I_{y}}{A_{head} \cdot h^{3}}$ $C_{u Z} = \frac{E A}{h \cdot A_{head}}$ $C_{φ Y} = 2 \cdot \frac{E I_{z}}{A_{head} \cdot h} + \frac{X}{100} \cdot \frac{E I_{z}}{A_{head} \cdot h}$ $C_{φ X} = 2 \cdot \frac{E I_{y}}{A_{head} \cdot h} + \frac{X}{100} \cdot \frac{E I_{y}}{A_{head} \cdot h}$ Spring Constants – Semi-Rigid Column Head, Elastic Surface Foundation, Neglecting Shear Stiffness
Restrained at Column Base	$C_{u X} = \frac{12 \cdot E I_{z}}{h^{3} \cdot A_{head}}$ $C_{u Y} = \frac{12 \cdot E I_{y}}{h^{3} \cdot A_{head}}$ $C_{u Z} = \frac{E A}{h \cdot A_{head}}$ $C_{φ Y} = 3 \cdot \frac{E I_{z}}{A_{head} \cdot h}$ $C_{φ X} = 3 \cdot \frac{E I_{y}}{A_{head} \cdot h}$ Spring Constants – Restrained Column Head, Elastic Surface Foundation, Neglecting Shear Stiffness

Elastic Nodal Support

This model focuses on the deformations and forces at specific nodal points. Thus, this makes it easier to calculate than the previous model.

Elastic Nodal Support	Hinged Column Head with Consideration of Shear Stiffness
Hinged at Column Base	$C_{u X} = 0$ $C_{u Y} = 0$ $C_{u Z} = \frac{E A}{h}$ $C_{φ Y} = 0$ $C_{φ X} = 0$ Spring Constants – Hinged Column Head, Hinged Column Base, Elastic Nodal Support, Considering Shear Stiffness
Semi-Rigid at Column Base	$C_{u X} = \frac{X}{100} \cdot \frac{12 \cdot E I_{z}}{h^{3} \cdot (4 + k_{z})}$ $C_{u Y} = \frac{X}{100} \cdot \frac{12 \cdot E I_{y}}{h^{3} \cdot (4 + k_{y})}$ $C_{u Z} = \frac{E A}{h}$ $C_{φ Y} = 0$ $C_{φ X} = 0$ Spring Constants – Hinged Column Head, Semi-Rigid Column Base, Elastic Nodal Support, Considering Shear Stiffness
Restrained at Column Base	$C_{u X} = \frac{12 \cdot E I_{z}}{h^{3} \cdot (4 + k_{z})}$ $C_{u Y} = \frac{12 \cdot E I_{y}}{h^{3} \cdot (4 + k_{y})}$ $C_{u Z} = \frac{E A}{h}$ $C_{φ Y} = 0$ $C_{φ X} = 0$ Spring Constants – Hinged Column Head, Restrained Column Base, Elastic Nodal Support, Considering Shear Stiffness

Elastic Nodal Support	Hinged Column Head Without Consideration of Shear Stiffness
Hinged at Column Base	$C_{u X} = 0$ $C_{u Y} = 0$ $C_{u Z} = \frac{E A}{h}$ $C_{φ Y} = 0$ $C_{φ X} = 0$ Spring Constants – Hinged Column Head, Hinged Column Base, Elastic Nodal Support, Neglecting Shear Stiffness
Semi-Rigid at Column Base	$C_{u X} = \frac{X}{100} \cdot \frac{3 \cdot E I_{z}}{h^{3}}$ $C_{u Y} = \frac{X}{100} \cdot \frac{3 \cdot E I_{y}}{h^{3}}$ $C_{u Z} = \frac{E A}{h}$ $C_{φ Y} = 0$ $C_{φ X} = 0$ Spring Constants – Hinged Column Head, Semi-Rigid Column Base, Elastic Nodal Support, Neglecting Shear Stiffness
Restrained at Column Base	$C_{u X} = \frac{3 \cdot E I_{z}}{H^{3}}$ $C_{u Y} = \frac{3 \cdot E I_{y}}{H^{3}}$ $C_{u Z} = \frac{E A}{h}$ $C_{φ Y} = 0$ $C_{φ X} = 0$ Spring Constants – Hinged Column Head, Restrained Column Base, Elastic Nodal Support, Neglecting Shear Stiffness

Elastic Nodal Support	Semi-Rigid Column Head with Consideration of Shear Stiffness
Hinged at Column Base	$C_{u, X} = \frac{12 \cdot E I_{z}}{h^{3} \cdot (4 + k_{z})}$ $C_{u, Y} = \frac{12 \cdot E I_{y}}{h^{3} (4 + k_{y})}$ $C_{u, Z} = \frac{E A}{h}$ $C_{φ, X} = \frac{3 + k_{y}}{1 + k_{y}} \cdot \frac{E I_{y}}{h}$ $C_{φ, Y} = \frac{3 + k_{z}}{1 + k_{z}} \cdot \frac{3 \cdot E I_{z}}{h}$ Spring Constants – Semi-Rigid Column Head, Hinged Column Base, Elastic Nodal Support, Considering Shear Stiffness
Semi-Rigid at Column Base	$C_{u, X} = \frac{12 \cdot E I_{z}}{h^{3} \cdot (4 + k_{z})} + \frac{X}{100} \cdot \frac{36 \cdot E I_{z}}{h^{3} \cdot (1 + k_{z}) (4 + k_{z})}$ $C_{u, Y} = \frac{12 \cdot E I_{y}}{h^{3} \cdot (4 + k_{y})} + \frac{X}{100} \cdot \frac{36 \cdot E I_{y}}{h^{3} \cdot (1 + k_{y}) (4 + k_{y})}$ $C_{u, Z} = \frac{E A}{h}$ $C_{φ, X} = \frac{3 + k_{y}}{1 + k_{y}} \cdot \frac{E I_{y}}{h} + \frac{X}{100} \cdot \frac{E I_{y}}{h \cdot (1 + k_{y})}$ $C_{φ, Y} = \frac{3 + k_{z}}{1 + k_{z}} \cdot \frac{E I_{z}}{h} + \frac{X}{100} \cdot \frac{E I_{z}}{h \cdot (1 + k_{z})}$ Spring Constants – Semi-Rigid Column Head, Semi-Rigid Column Base, Elastic Nodal Support, Considering Shear Stiffness
Restrained at Column Base	$C_{u, X} = \frac{12 \cdot E I_{z}}{h^{3} \cdot (1 + k_{z})}$ $C_{u, Y} = \frac{12 \cdot E I_{y}}{h^{3} \cdot (1 + k_{y})}$ $C_{u, Z} = \frac{E A}{h}$ $C_{φ, X} = \frac{4 + k_{y}}{1 + k_{y}} \cdot \frac{E I_{y}}{h}$ $C_{φ, Y} = \frac{4 + k_{z}}{1 + k_{z}} \cdot \frac{E I_{z}}{h}$ Spring Constants – Semi-Rigid Column Head, Restrained Column Base, Elastic Nodal Support, Considering Shear Stiffness

Elastic Nodal Support	Semi-Rigid Column Head Without Considering Shear Stiffness
Hinged at Column Base	$C_{u, X} = \frac{3 \cdot E I_{z}}{h^{3}}$ $C_{u, Y} = \frac{3 \cdot E I_{y}}{h^{3}}$ $C_{u, Z} = \frac{E A}{h}$ $C_{φ, X} = \frac{3 \cdot E I_{y}}{h}$ $C_{φ, Y} = \frac{3 \cdot E I_{z}}{h}$ Spring Constants – Semi-Rigid Column Head, Hinged Column Base, Elastic Nodal Support, Neglecting Shear Stiffness
Semi-Rigid at Column Base	$C_{u, X} = \frac{3 \cdot E I_{z}}{h^{3}} + \frac{X}{100} \cdot \frac{9 \cdot E I_{z}}{h^{3}}$ $C_{u, Y} = \frac{3 \cdot E I_{y}}{h^{3}} + \frac{X}{100} \cdot \frac{9 \cdot E I_{y}}{h^{3}}$ $C_{u, Z} = \frac{E A}{h}$ $C_{φ, X} = \frac{3 \cdot E I_{y}}{h} + \frac{X}{100} \cdot \frac{E I_{y}}{h}$ $C_{φ, Y} = \frac{3 \cdot E I_{z}}{h} + \frac{X}{100} \cdot \frac{E I_{z}}{h}$ Spring Constants – Semi-Rigid Column Head, Semi-Rigid Column Base, Elastic Nodal Support, Neglecting Shear Stiffness
Restrained at Column Base	$C_{u, X} = \frac{12 \cdot E I_{z}}{h^{3}}$ $C_{u, Y} = \frac{12 \cdot E I_{y}}{h^{3}}$ $C_{u, Z} = \frac{E A}{h}$ $C_{φ, X} = \frac{4 \cdot E I_{y}}{h}$ $C_{φ, Y} = \frac{4 \cdot E I_{z}}{h}$ Spring Constants – Semi-Rigid Column Head, Restrained Column Base, Elastic Nodal Support, Neglecting Shear Stiffness

Info

If the shear stiffness is not considered, the actual deformation of the column may be underestimated. This can result in a less accurate analysis, particularly for short, wide columns or in instances where the column is required to carry significant horizontal loads.

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