Determination of Effective Lengths in RF-/CONCRETE Columns

Technical Article

With RF-/CONCRETE Columns, it is possible to determine effective lengths for columns automatically. This article describes which entries are necessary and how the calculation of the effective lengths is carried out.

Basics

When designing columns, it has to be decided for the buckling safety design if the effects according to the second-order analysis have to be taken into account. The slenderness of the component is compared to the limiting slenderness while considering the adjacent components according to the corresponding standards. The slenderness of the compression element is calculated from

\begin{array}{l}\frac12\;\cdot\;\mathrm\lambda\;=\;\frac{{\mathrm l}_0}{\mathrm i}\\\mathrm{mit}\\{\mathrm l}_0\;=\;\mathrm{effective}\;\mathrm{length}\\\mathrm i\;=\;\sqrt{\frac{\mathrm I}{\mathrm A}}\;=\;\mathrm{radius}\;\mathrm{of}\;\mathrm{gyration}\;\mathrm{of}\;\mathrm{the}\;\mathrm{concrete}\;\mathrm{cross}-\mathrm{section}\end{array}

For normal frames, it is allowed to determine the effective lengths l0 of the columns with the following equations:

  • stiffened components
    ${\mathrm l}_0\;=\;0.5\;\cdot\;\mathrm l\;\cdot\;\sqrt{\left(1\;+\;\frac{{\mathrm k}_1}{0.45\;+\;{\mathrm k}_1}\right)\;\cdot\;\left(1\;+\;\frac{{\mathrm k}_2}{0.45\;+\;{\mathrm k}_2}\right)}$
  • unstiffened components
    ${\mathrm l}_0\;=\;\mathrm l\;\cdot\;\max\;\left\{\begin{array}{l}\sqrt{1\;+\;10\cdot\;\frac{{\mathrm k}_1\;\cdot\;{\mathrm k}_2}{{\mathrm k}_1\;+\;{\mathrm k}_2}}\\\left(1\;+\;\frac{{\mathrm k}_1}{1\;+\;{\mathrm k}_1}\right)\;\cdot\;\left(1\;+\;\frac{{\mathrm k}_2}{1\;+\;{\mathrm k}_2}\right)\end{array}\right.$

k1 and k2 are here the degrees of restraint of both column ends. They are determined according to Figure 01.

Figure 01 - Determination of the Degrees of Restraint of the Column Ends Taking Into Acount the Stiffness of the Connecting Beam

The rotational section modulus of the beam MR is calculated from ${\mathrm M}_\mathrm R\;=\;\mathrm\alpha\;\cdot\;\frac{\mathrm E\;\cdot\;{\mathrm I}_\mathrm R}{{\mathrm l}_\mathrm R}$.

The factor α results from the release conditions of the distant beam end and the moment distribution in the beam. These correlations are also shown in Figure 01.

Basically, the degrees of restraint k should be between 0.1 and ∞. 0.1 represents the rigid restraint and ∞ the hinged support. The theoretical limit for a rigid restraint amounts to 0. Since it is not possible in practice to implement a rigid restraint, it is recommended according to [2] Sec. 5.8.3.2 (3) to use a minimum value of 0.1.

Another option is to determine the effective length with the equation ${\mathrm l}_0\;=\;\mathrm\beta\;\cdot\;{\mathrm l}_\mathrm{col}$. It is based on the nomogram from [1] where the coefficient β can be read off. The coefficient β describes the ratio of the effective length l0 to the real column length lcol.

Example

Figure 02 shows the structure to determine the effective length. The calculation is carried out in this example on member M1 for buckling around the y-axis.

Figure 02 - Structure Example

$\begin{array}{l}{\mathrm k}_\mathrm A\;=\;0.1\;(\mathrm{fixed}\;\mathrm{column}\;\mathrm{base})\\\;{\mathrm k}_\mathrm B\;=\;\frac{\sum{\displaystyle\left(\frac{\mathrm E\;\cdot\;{\mathrm I}_\mathrm{col}}{{\mathrm l}_\mathrm{col}}\right)}}{\sum{\displaystyle\left(\frac{\mathrm\alpha\;\cdot\;\mathrm E\;\cdot\;{\mathrm I}_\mathrm R}{{\mathrm l}_\mathrm R}\right)}}\\\;{\mathrm k}_\mathrm B\;=\;\frac{\displaystyle\frac{3,300\;\mathrm{kN}/\mathrm{cm}^2\;\cdot\;160,000\;\mathrm{cm}^4}{3\;\mathrm m}\;+\;\frac{3,300\;\mathrm{kN}/\mathrm{cm}^2\;\cdot\;160,000\;\mathrm{cm}^4}{1.5\;\mathrm m}}{\displaystyle\frac{3\;\cdot\;3,300\;\mathrm{kN}/\mathrm{cm}^2\;\cdot\;160,000\;\mathrm{cm}^4}{3\;\mathrm m}}\\\;{\mathrm k}_\mathrm B\;=\;1.0\\\;{\mathrm l}_0\;=\;3\;\mathrm m\;\cdot\;\max\;\left\{\begin{array}{l}\sqrt{1\;+\;10\;\cdot\;\frac{0.1\;\cdot\;1.0}{0.1\;+\;1.0}}\;=\;1.38\\\left(1\;+\;\frac{0.1}{1\;+\;0.1}\right)\;\cdot\;\left(1\;+\;\frac{1.0}{1\;+\;1.0}\right)\;=\;1.64\end{array}\right.\\\;{\mathrm l}_0\;=\;4.91\;\mathrm m\end{array}$

Figure 03 shows the values determined with RF-/CONCRETE Columns.

Figure 03 - Results of the Effective Length Calculation in RF-CONCRETE Columns

Keywords

buckling stability effective length concrete column

Reference

[1]   Ehrigsen, O.; Quast, U.: Knicklängen, Ersatzlängen und Modellstützen, Beton- und Stahlbetonbau 5, Seiten 249 - 257. Berlin: Ernst & Sohn, 2003
[2]   Eurocode 2: Design of concrete structures - Part 1-1: General rules and rules for buildings; EN 1992-1-1:2011-01

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