DC EC 06: 2nd order effect in a column

Description

Geometry	Cross-section:	$b=250mm$ $h=400mm$
	Column height:	$4m$
	Material:	C25/30
	Concrete cover:	$c=40mm$
	Creep factor $\varphi _{t.\infty}$ :	2
	Reinforcement distribution:	Identical Only upper and lower
Load	Design forces:	$N_{Ed.ULS}=300kN$ $M_{Ed.ULS}=120kNm$ $M_{Ed.SLS QP}=88.9kNm$
Standard		EN 1992-1-1 [- -]

How to model a column in Diamonds that can be compared with handcalculations

Diamonds always looks for the most economical reinforcement ratio. This is not always neat according to the local axes of the rod, but often according to the main axes of inertia. While in hand calculations the load is placed along a local axis and the calculation is also performed in that direction. To impose that behavior in Diamonds, you must choose only top and bottom reinforcement.
By choosing an ‘identical’ reinforcement distribution, you ask him to choose the top and bottom reinforcement equally. This is a reinforcement distribution for which graphs/tables are suitable for manual calculations.
In this example we want to focus on the 2nd order effects. Therefor we set the additional eccentricity to zero.
By default, Diamonds will apply the 2nd order eccentricity in different directions, only keeping the worst direction. Since the cross section is not a square, and the loads are applied around the strong axis, we can already predict that the worst direction for the 2nd order effect, will be around the weak axis. When that happens, we have bi-directional bending. Something we don’t want in this example.
So to force Diamonds into applying the 2nd order effect into the same direction as the already present bending moment, we deselect the buckling verification around the weak axis.
According to Eurocode 2, reinforcement is required for the forces in ULS, possibly increased with additional reinforcement to limit the stresses in SLS. It is not known in advance whether additional reinforcement will be required for SLS. Since we are only doing a ULS calculation in this example, we turn off the stress limits in SLS in Diamonds (by making a copy of C25/30 and editing the copy).

Independent reference results

Open handcalculations

Eurocode 2 offers simplified methods (EN 1992-1-1 §5.8.7 and §5.8.8) to include 2nd order effects and imperfections into the results. The method based on nominal curvature (EN 1992-1-1 §5.8.8) is implemented in Diamonds. This method results in an increased bending moment.

Is a 2nd order calculation required? (EN 1992-1-1 §5.8.3.1)

This column has the same ULS loads has validation example DC EC 05. So we just reuse the reinforcement amounts:

$A_s=2 \cdot 514mm^2=1028mm^2$

$A_c=A - As = 250mm \cdot 400 mm - 1028mm^2= 98972mm^2$

$I_{0.c}=\frac{b \cdot h^3}{12}=\frac{250mm \cdot (400m)^3}{12}=133333cm^4$

(5.14) $\begin{align*} \lambda = \frac{l_0}{\sqrt{\frac{I_{0.c}}{A}}}= \frac{4m}{\sqrt{\frac{133333cm^4}{250mm \cdot 400 mm}}}=34.64 \end{align*}$

$n=\frac{N_{Ed.ULS}}{A_c \cdot f_{cd}} = \frac{300\text{kN}}{98972mm^2 \cdot 16.67\text{MPa}}=0.182$

$A=1.7; B=1.1; C=0.7$

(5.13N) $\begin{align*} \lambda_{lim} = \frac{ 20 \cdot A \cdot B \cdot C}{\sqrt{n}}= \frac{ 20 \cdot 0.7 \cdot 1.1 \cdot 0.7}{\sqrt{0.182}}=25.28 \end{align*}$

Because $\lambda$ is bigger than $\lambda_{lim}$ , second order effects need to be taken into account.

Note: these intermediar results cannot be consulted in Diamonds at the moment. However, we’re working on a solution (expected in Diamonds 2025).

Magnitude of 2nd order eccentricity (EN 1992-1-1 §5.8.3.2)

$n_u=1+\frac{A_s \cdot f_{yd}}{A_c \cdot f_{cd}}=1+\frac{1028mm^2 \cdot 434.7\text{MPa}}{98972mm^2 \cdot 16.6\text{MPa}}=1.27$

$n_{bal}=0.4$

Correction factor $K_r$ depending on the axial force (this factor is sometimes called $K_2$ ):

(5.36) $\begin{align*} K_r=min\left ( \frac{n_u-n}{n_u-n_{bal}},1 \right )=1 \end{align*}$

$\beta =0.35+\frac{f_{ck}}{200\text{MPa}}-\frac{\lambda }{150} = 0.35+\frac{25\text{MPa}}{200\text{MPa}}-\frac{34.64}{150} = 0.244$

(5.19) $\begin{align*} \varphi _{ef}=\frac{M_{Ed.SLS QP}}{M_{Ed.ULS}} \cdot \varphi _{t.\infty} = \frac{88.9\text{kNm}}{200\text{kNm}} \cdot 2 = 1.48 \end{align*}$

Correction factor $K_{\varphi}$ depending on the axial force (this factor is sometimes called $K_1$ ):

(5.37) $\begin{align*} K_{\varphi}=max\left (1+ \beta \cdot \varphi _{ef},1 \right )=1.362 \end{align*}$

$c=8 \qquad \qquad \text{because constant bending moment, see EN 1992-1-1 §5.8.8.2 (4)}$

$\begin{align*} e_2 &=\frac{l_{0}^{2}}{c} \cdot K_r \cdot K_{\varphi} \cdot \frac{2 \cdot f_{yd} }{0.9\cdot d \cdot E_s}\nonumber\\ &=\frac{(4m)^{2}}{8} \cdot 1 \cdot 1.362 \cdot \frac{2 \cdot 434.7\text{MPa}}{0.9\cdot 360mm \cdot 200000\text{MPa}}\\ &=36.54mm \end{align*}$

Magnitude of design bending moment (2nd order effect included)

The additional moment for 2nd order effects equals:

(5.33) $\begin{align*} M_{Ed.2nd.order}=N_{Ed.ULS} \cdot e_2 = 300\text{kN} \cdot 36.54mm = 11.0 \text{kNm} \end{align*}$

The total design moment becomes:

$M_{Ed}= M_{Ed.ULS} + M_{Ed.2nd.order} = 120kN \cdot 11 \text{kNm} = 131.0 \text{kNm}$

Reinforcement calculation

The principle for entirly the same as for has validation example DC EC 05. Only $\mu_d$ differs:

$\mu_d=\frac{M_{Ed}}{b\cdot d^2 \cdot f_{cd}}=\frac{131\text{kNm}}{250mm\cdot (360mm)^2\cdot 16,6\text{MPa}}=0.243$

Graph 10.2a from Gewapend beton: numeri (click here to open the graph) shows $\omega _1=\omega _2$ as a function of $\mu_d, v_d$ and $\delta_1$ . Using this graph and the results above, we can deduct:

$\omega _1=\omega _2=0.172$

$A_{s1}=A_{s2}==\omega_1 \cdot b \cdot d\cdot \frac{f_{cd}}{f_{yd}}= 0,172 \cdot 250mm \cdot 360mm \cdot \frac{16.6\text{MPa}}{434,8\text{MPa}} = 593mm^2$

Diamonds results and comparison

Longitudonal reinforcement calculated by Diamonds (EN 1992-1-1 [- -])
The reinforcement due to ULS is shown using a thin line, the reinforcement due to ULS + SLS + buckling with a thick line. Because the SLS verification were turned off, the thick line represents the reinforcement for ULS + buckling.

Results	Independent reference	Diamonds	Difference
Longitudonal reinforcement	593mm²	593mm²	0%

References

EN 1992-1-1: 2005 + AC: 2010
Van Hooymissen, L., Spegelaere, M., Van Gysel, A., & De Vylder, W. (2002). Gewapend beton. Academia Press
Keep in mind that the calculations in this book are made done using the NBN B15-002/ ENV 1992-1. Both old standards. However, this book still remains a good reference if you want to understand how reinforcement calculations work.
Gruyaert, E., & Minne, P. (2019). Gewapend beton: numeri.
This reference is a summary of Gewapend beton (2002) but with updated formula and principles according to EN 1992-1-1. This document contains multiple graphs and tables helping the design of reinforced concrete.

Tested in Diamonds 2024r01.

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