Application of Eccentricities in RF-CONCRETE Columns

Technical Article

When calculating the internal forces for the buckling analysis with the method based on nominal curvature in RF-CONCRETE Columns, the required eccentricities have to be determined.

These result from a planned eccentric load introduction, imperfect structures (geometric and material imperfections), and an additional eccentricity from the calculation according to the second-order analysis.

Planned Eccentricities

The eccentricities from the planned eccentric load introduction can be easily determined with the moments due to an eccentric load application. For a constant moment distribution, the following relation applies:
$\frac{{\mathrm M}_{0\mathrm{Ed}}}{{\mathrm N}_{\mathrm{Ed}}}\;=\;{\mathrm e}_0\;\geq\;{\mathrm e}_\min\;\mathrm{mit}\;{\mathrm e}_{01}\;=\;{\mathrm e}_{02}$

Thus the minimum eccentricity according to Clause 6.1.(4) ${\mathrm e}_\min\;=\;\frac{\mathrm h}{30}\;\geq\;20\;\mathrm{mm}$ where h = cross-section depth.

For a linearly variable moment distribution, an equivalent eccentricity may be determined and the following formula applied:
${\mathrm e}_{\mathrm e}\;=\;\max\;\left\{\begin{array}{l}0.6\;\cdot\;{\mathrm e}_{02}\;+\;0.4\;\cdot\;{\mathrm e}_{01}\\0.4\;\cdot\;{\mathrm e}_{02}\end{array}\right\}\;\mathrm{}\;\mathrm{whereby}:\;\left|{\mathrm e}_{02}\right|\;\geq\;\left|{\mathrm e}_{01}\right|$

Figure 01 - Variable Moment Distribution

The eccentricities have to be applied with algebraic signs. They have the same sign if the related moments create tension on the same side.

If there is any moment distribution, the maximum eccentricity is always used for the calculation so that no columns have to be excluded from the analysis.

Unintentional Eccentricities due to Imperfections

For the calculation of the moments according to the linear static analysis, the geometric and material imperfections have to be taken into account as well. This may occur by applying equivalent geometric imperfections, which are considered as the inclination θi.

The eccentricity is determined in EN 1992-1-1 according to formula (5.2):
${\mathrm e}_{\mathrm i}\;=\;{\mathrm\Theta}_{\mathrm i}\;\cdot\;\frac{{\mathrm l}_0}2$

The inclination θi is calculated according to formula (5.1):
${\mathrm\Theta}_{\mathrm i}\;=\;{\mathrm\Theta}_0\;\cdot\;{\mathrm\alpha}_{\mathrm h}\;\cdot\;{\mathrm\alpha}_{\mathrm m}$

Where:

Fundamental value of the inclination ${\mathrm\Theta}_0\;=\;\frac1{200}$
Reduction factor for the height $\frac23\;\leq\;{\mathrm\alpha}_{\mathrm h}\;=\;\frac2{\sqrt{\mathrm l}}\;\leq\;1.0$
Reduction factor for the number of structural components (m = number of columns) ${\mathrm\alpha}_{\mathrm m}\;=\;\sqrt{0.5\;\cdot\;\frac{1\;+\;1}{\mathrm m}}$

This results in the bending moment for the equivalent imperfection ei:
${\mathrm M}_{\mathrm{Edi}}\;=\;{\mathrm N}_{\mathrm{Ed}}\;\cdot\;{\mathrm e}_{\mathrm i}$

Figure 02 - Intentional and Unintentional Eccentricities (Linear Static Analysis)

Additional Eccentricity from Calculation According to Second-Order Theory

Under the axial force load, a curvature of the column occurs, with the column head being deflected by the path e2. This results in the moment distribution according to the second-order theory.

Figure 03 - Total Moment with Components from Planned Eccentricity, Imperfection and Second-Order Theory

For the method based on nominal curvature, a parabolic moment distribution is assumed. The determination of the eccentricity according to the second-order theory is done according to EN 1992-1-1 5.8.8.2(3).

It can be found in detail in the standard or the manual for (RF-)CONCRETE Columns.

Specifics When Determining the Eccentricities for a Biaxial Load Eccentricity

For columns loaded with biaxial load eccentricity, a separate design in both principal axis orientations must first be performed. For this design, the imperfections have to be applied exclusively in the direction where they lead to the most unfavorable effects. The designs can then be performed in both directions with the entire applied reinforcement. So it is a matter of ruling out that the imperfections necessitate a biaxial analysis.

In order to be allowed to perform the designs in both directions without having to consider further biaxial bending, the conditions as per equation (5.38) have to be fulfilled.

To even be able to perform this seperate design, the corresponding option in (RF-)CONCRETE Columns has to be set active first.

Figure 04 - Activating Separate Design According to EN 1992-1-1 5.8.9(2)

On the one hand, the slenderness ratio in equation (5.38a) has to be taken into account:
$\frac{{\mathrm\lambda}_{\mathrm y}}{{\mathrm\lambda}_{\mathrm z}}\;\leq\;2\;\mathrm{and}\;\frac{{\mathrm\lambda}_{\mathrm z}}{{\mathrm\lambda}_{\mathrm y}}\;\leq\;2$, where λy and λz are the slendernesses in relation to the y- and z-axis.

On the other hand, one of the conditions regarding the related load eccentricities according to equation (5.38a) has to be fulfilled:
$\frac{\left({\displaystyle\frac{{\mathrm e}_{\mathrm y}}{{\mathrm h}_{\mathrm{eq}}}}\right)}{\left({\displaystyle\frac{{\mathrm e}_{\mathrm z}}{{\mathrm b}_{\mathrm{eq}}}}\right)}\;\leq\;0.2\;\mathrm{or}\;\frac{\displaystyle\left(\frac{\displaystyle{\mathrm e}_{\mathrm z}}{{\mathrm b}_{\mathrm{eq}}}\right)}{\displaystyle\left(\frac{\displaystyle{\mathrm e}_{\mathrm y}}{{\mathrm h}_{\mathrm{eq}}}\right)}\;\leq\;0.2$

Where:
${\mathrm b}_{\mathrm{eq}}\;=\;{\mathrm i}_{\mathrm y}\;\cdot\;\sqrt{12}\;\mathrm{and}\;{\mathrm h}_{\mathrm{eq}}\;=\;{\mathrm i}_{\mathrm z}\;\cdot\;\sqrt{12}$ for an equivalent rectangular cross-section
iy and iz are the radii of gyration in relation to the y- and z-axis.
${\mathrm e}_{\mathrm z}\;=\;\frac{{\mathrm M}_{\mathrm{edy}}}{{\mathrm N}_{\mathrm{ed}}}\;\mathrm{and}\;{\mathrm e}_{\mathrm y}\;=\;\frac{\displaystyle{\mathrm M}_{\mathrm{edz}}}{\displaystyle{\mathrm N}_{\mathrm{ed}}}$ are the load eccentricities in the direction of the axes.

When calculating the design moments MEdy and MEdz, the imperfection is not taken into account for the subordinate direction.

The subordinate direction is determined via the ratio of the total eccentricity to the eccentricity according to the linear static analysis.

$\frac{{\mathrm e}_{2\mathrm{tot},\mathrm z}}{{\mathrm e}_{1,\mathrm z}}\;\leq\;\frac{\displaystyle{\mathrm e}_{2\mathrm{tot},\mathrm y}}{\displaystyle{\mathrm e}_{1,\mathrm y}}\;\rightarrow\;{\mathrm e}_{\mathrm i,\mathrm y}\;=\;0$ for the calculation of (5.38b).

Otherwise, ei,z is set to 0.

If this fulfills one of the conditions from equation (5.38b), the design can be separately performed while neglecting the imperfections in the subordinate direction. If (5.38b) is not fulfilled, a biaxial design in consideration of all imperfections must be performed.

Example Bracket

Figure 05 - Bracket System

${\mathrm e}_{0,\mathrm y}\;=\;\frac{\displaystyle2.4\;\mathrm{kNm}}{120\;\mathrm{kNm}}\;=\;20\;\mathrm{mm}$

${\mathrm e}_{0,\mathrm z}\;=\;\frac{\displaystyle12\;\mathrm{kNm}}{120\;\mathrm{kNm}}\;=\;100\;\mathrm{mm}$

${\mathrm e}_{\mathrm i,\mathrm y}\;=\;{\mathrm e}_{\mathrm i,\mathrm z}\;=\;\frac1{200}\;\cdot\;\frac2{\sqrt{4.2\;\mathrm m}}\;\cdot\;1\;\cdot\;\frac{9.156\;\mathrm m}2\;=\;22.3\;\mathrm{mm}$

The eccentricities due to the second-order theory are imported from the program:
e2,y = 238 mm
e2,z = 119.5 mm

With these values, it can now be determined which direction is subordinate.

$\frac{119.5\;+\;22.3\;+\;20}{22.3\;+\;20}\;=\;3.83\;>\;\frac{\displaystyle238\;+\;22.3\;+\;100}{\displaystyle22.3\;+\;100}\;=\;2.95\;\rightarrow\;{\mathrm e}_{\mathrm i,\mathrm z}\;=\;0$

Thus, ei,z can be set to 0 and the values in accordance with Figure 06 result.

Figure 06 - Equation (5.38) Results

Keywords

buckling equivalent member eccentricity nominal curvature

Reference

[1]   Fingerloos, F.; Hegger, J.; Zilch, K.: Eurocode 2 für Deutschland - Kommentierte Fassung, 2., überarbeitete Auflage. Berlin: Beuth, 2016
[2]   Handbuch BETON Stützen. Tiefenbach: Dlubal Software, Januar 2013.
[3]   Handbuch RF-BETON Stützen. Tiefenbach: Dlubal Software, Januar 2013.

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