2024-08-26

Stiffness via Fictitious Column

How are the spring constants of "Stiffness" determined using a fictitious column?

Answer:

FAQ 005600 | Stiffness via Fictitious Column

In the Nodal Support dialog box, you can create this as a fictitious column; in this case, the support conditions, geometric, and material properties of the column are used to determine the stiffnesses, which are used to determine the spring constants of the column. Darüber hinaus wird eine Flächenergebnisanpassung zur Vermeidung von Singularitäten automatisiert entsprechend der Querschnittsfläche der Stütze erstellt.

Um diese Federkonstanten gemäß der fiktiven Stütze abzubilden kann man im Register "Steifigkeit mittels fiktiver Stütze" entsprechende Konfigurationen vornehmen.
Bei der Steifigkeit mittels fiktiver Stütze gilt es zunächst diverse Fallunterscheidungen zu betrachten.

Im ersten werden nur Konfigurationen betrachtet bei welcher die Lagerung anhand einer Flächenbettung modelliert ist.
Ist die Stütze unten eingespannt berechnen sich die Lagerfedern der Stütze gemäß folgenden Formeln
Bei Berücksichtigung der Schubsteifigkeit:

C_{u X} = \frac{12 \cdot E I_{Z}}{H^{3} \cdot A_{h e a d} \cdot (1 + k_{z})}; C_{u Y} = \frac{12 \cdot E I_{Y}}{H^{3} \cdot A_{h e a d} \cdot (1 + k_{y})}; C_{u Z} = \frac{E \cdot A}{h \cdot A_{h e a d}}

C_{φ y} = \frac{E \cdot I_{z}}{A_{h e a d} \cdot h} \cdot \frac{3}{1 + k_{z}}; C_{φ X} = \frac{E \cdot I y}{A_{h e a d} \cdot h} \cdot \frac{3}{1 + k_{y}}

Springs due to Stiffness Restrained by Fictitious Column, Assuming Elastic Surface Foundation and Considering Shear Stiffness

Bei Vernachlässigung der Schubsteifigkeit:

C_{u X} = \frac{12 \cdot E I_{Z}}{H^{3} \cdot A_{h e a d}}; C_{u Y} = \frac{12 \cdot E I_{Y}}{H^{3} \cdot A_{h e a d}}; C_{u Z} = \frac{E \cdot A}{h \cdot A_{h e a d}}

C_{φ y} = 3 \cdot \frac{E \cdot I_{z}}{A_{h e a d} \cdot h}; C_{φ X} = 3 \cdot \frac{E \cdot I y}{A_{h e a d} \cdot h}

Spring Constants of Stiffness Using Fictitious Supports, Assuming Elastic Surface Foundation for Restrained Column and Neglecting Shear Stiffness

Ist die Stütze unten gelenkig gelagert berechnen sich die Lagerfedern wie folgt:
Bei Berücksichtigung der Schubsteifigkeit:

C_{u X} = \frac{12 \cdot E I_{Z}}{H^{3} \cdot A_{h e a d} \cdot (4 + k_{z})}; C_{u Y} = \frac{12 \cdot E I_{Y}}{H^{3} \cdot A_{h e a d} \cdot (4 + k_{y})}; C_{u Z} = \frac{E \cdot A}{h \cdot A_{h e a d}}

C_{φ y} = \frac{E \cdot I_{z}}{A_{h e a d} \cdot h} \cdot \frac{2}{1 + k_{z}}; C_{φ X} = \frac{E \cdot I_{y}}{A_{h e a d} \cdot h} \cdot \frac{2}{1 + k_{y}}

Springs due to Stiffness Simply Supported by Fictitious Column, Assuming Elastic Surface Foundation and Considering Shear Stiffness

Bei Vernachlässigen der Schubsteifigkeit:

C_{u X} = \frac{3 \cdot E I_{Z}}{H^{3} \cdot A_{h e a d}}; C_{u Y} = \frac{3 \cdot E I_{Y}}{H^{3} \cdot A_{h e a d}}; C_{u Z} = \frac{E \cdot A}{h \cdot A_{h e a d}}

C_{φ y} = 2 \cdot \frac{E \cdot I_{z}}{A_{h e a d} \cdot h}; C_{φ X} = 2 \cdot \frac{E \cdot I_{y}}{A_{h e a d} \cdot h}

Springs due to Stiffness Simply Supported by Fictitious Column, Assuming Elastic Surface Foundation and Neglecting Shear Stiffness

Ist die Stütze unten nachgiebig gelagert berechnen sich die Lagerfedern wie folgt:
Bei Berücksichtigung der Schubsteifigkeit:

C_{u X} = \frac{12 \cdot E I_{Z}}{H^{3} \cdot A_{h e a d} \cdot (4 + k_{z})} + \frac{X}{100} \cdot \frac{36 \cdot E I_{Z}}{A_{h e a d} \cdot H^{3} \cdot (1 + k_{z}) \cdot (4 + k_{z})};

C_{u Y} = \frac{12 \cdot E I_{Y}}{H^{3} \cdot A_{h e a d} \cdot (4 + k_{y})} + \frac{X}{100} \cdot \frac{36 \cdot E I_{Y}}{A_{h e a d} \cdot H^{3} \cdot (1 + k y) \cdot (4 + k_{y})};

C_{u Z} = \frac{E \cdot A}{h \cdot A_{h e a d}}

C_{φ Y} = 2 \cdot \frac{E \cdot I_{Z}}{\cdot A_{h e a d} \cdot h \cdot (1 + k_{z})} + \frac{X}{100} \cdot \frac{E I_{Z}}{A_{h e a d} \cdot H (1 + k_{z})}; C_{φ X} = \frac{2 \cdot E \cdot I_{Y}}{A_{h e a d} \cdot h \cdot (1 + k_{y})} + \frac{X}{100} \cdot \frac{E I_{Y}}{A_{h e a d} \cdot H \cdot (1 + k_{y})}

Spring Constants for Stiffness Using Fictitious Column for "Semi-Rigidly" Supported Columns with Elastic Surface Foundation Considering Shear Stiffness

Bei Vernachlässigen der Schubsteifigkeit:

C_{u X} = \frac{3 \cdot E I_{Z}}{H^{3} \cdot A_{h e a d}} + \frac{X}{100} \cdot \frac{9 \cdot E I_{Z}}{A_{h e a d} \cdot H^{3}}; C_{u Y} = \frac{3 \cdot E I_{Y}}{H^{3} \cdot A_{h e a d}} + \frac{X}{100} \cdot \frac{9 \cdot E I_{Y}}{A_{h e a d} \cdot H^{3}}; C_{u Z} = \frac{E \cdot A}{h \cdot A_{h e a d}}

C_{φ Y} = 2 \cdot \frac{E \cdot I_{Z}}{A_{h e a d} \cdot h} + \frac{X}{100} \cdot \frac{E I_{Z}}{A_{h e a d} \cdot H}; C_{φ X} = 2 \cdot \frac{E \cdot I_{Y}}{A_{h e a d} \cdot h} + \frac{X}{100} \cdot \frac{E I_{Y}}{A_{h e a d} \cdot H}

Spring Constants for Stiffness Using Fictitious Column for "Semi-Rigidly" Supported Columns with Elastic Surface Foundation Without Considering Shear Stiffness

Im folgenden sind die Lagerfedern bei einer Modellierung anhand einer elastischen Knotenlagerung gelistet:
Ist die Stütze unten eingespannt berechnen sich die Lagerfedern wie folgt:
Bei Berücksichtigen der der Schubsteifigkeit:

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 \cdot A}{h}

C_{φ y} = 0; C_{φ X} = 0

Springs due to Stiffness Supported by Fictitious Column, Assuming Elastic Nodal Support and Considering Shear Stiffness

Bei Vernachlässigen der Schubsteifigkeit:

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 \cdot A}{h}

C_{φ y} = 0; C_{φ X} = 0

Springs due to Stiffness Supported by Fictitious Column, Assuming Elastic Nodal Support While Neglecting Shear Stiffness

Ist die Stütze unten gelenkig gelagert berechnen sich die Lagerfedern wie folgt:

C_{u X} = 0; C_{u Y} = 0; C_{u Z} = \frac{E \cdot A}{h}

C_{φ y} = 0; C_{φ X} = 0

Springs due to Stiffness Simply Supported by Fictitious Column, Assuming Elastic Nodal Support and Considering Shear Stiffness

Bei Vernachlässigen der Schubsteifigkeit:

C_{u X} = 0; C_{u Y} = 0; C_{u Z} = \frac{E \cdot A}{h}

C_{φ y} = 0; C_{φ X} = 0

Springs due to Stiffness Simply Supported by Fictitious Column, Assuming Elastic Nodal Support and Neglecting Shear Stiffness

Ist die Stütze nachgiebig gelagert berechnen sich die Lagerfedern wie folgt:
Bei Berücksichtigen der der Schubsteifigkeit:

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 \cdot A}{h}

C_{φ y} = 0; C_{φ X} = 0

Springs due to Stiffness Semi-Rigidly Supported by Fictitious Column, Assuming Elastic Nodal Support and Considering Shear Stiffness

Bei Vernachlässigen der Schubsteifigkeit:

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 \cdot A}{h}

C_{φ y} = 0; C_{φ X} = 0

Springs due to Stiffness Semi-Rigidly Supported by Fictitious Column, Assuming Elastic Nodal Support While Neglecting Shear Stiffness

Author

Mr. Böttcher provides technical support for customers of Dlubal Software and takes care of their requests.

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