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005600

2024-08-26

Stiffness via Fictitious Column

How are the spring constants of "Stiffness via Fictitious Column" determined?

Answer:

In the Nodal Support dialog box, you can design it as a fictitious column. In this case, the support conditions due to the geometric and material properties of the column determine the stiffnesses and apply them as the spring constants. To avoid singularities, a surface results adjustment is automatically created according to the cross-sectional area of the column.

FAQ 005600 | How are the spring constants of "Stiffness" determined using a fictitious column?

FAQ 005600 | Stiffness via Fictitious Column

In order to model these spring constants according to the fictitious column, it is possible to make various configurations in the "Stiffness via Fictitious Column" tab. In this case, it is important to consider various case distinctions.

Elastic Surface Foundations

If you want to model the support using a surface elastic foundation, you can use various cases.

If the column is restrained at the bottom, the support springs of the column are calculated according to the following formulas.

When considering the shear stiffness:

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

When neglecting the shear stiffness:

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

If the column has hinged supports at the bottom, the support springs are calculated as follows:

When considering the shear stiffness:

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

When neglecting the shear stiffness:

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

If the column is supported flexibly at the bottom, the support springs are calculated as follows:

When considering the shear stiffness:

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

When neglecting the shear stiffness:

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

Elastic Nodal Support

When modeling as an elastic nodal support, various cases are also possible.

If the column is restrained at the bottom, the support springs are calculated as follows:

When considering the shear stiffness:

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

When neglecting the shear stiffness:

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

If the column has hinged supports at the bottom, the support springs are calculated as follows:

When considering the shear stiffness:

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

When neglecting the shear stiffness:

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

If the column is supported flexibly, the support springs are calculated as follows:

When considering the shear stiffness:

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

When neglecting the shear stiffness:

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

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