DC EC 13: longitudinal reinforcement in column under pure compression (ULS FI design)

Description

Geometry	Cross-section:	$b=h=250mm$
Geometry	Material:	C35/45 Identical reinforcement pattern Any SLS checks in this material have been turned off
Internal forces		$N_{Ed.ULS FI}=1700\text{kN}$ ULS FC is neglected
Fire conditions		90min of ISO 834 fire
Standard		EN 1992-1-2 [- -]

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

In this example we don’t want 2nd order effects. Therefor we deselect everything.
According to Eurocode 2, reinforcement is required for the forces in ULS, possibly increased with additional reinforcement to limit the stresses in SLS. It’s unknown 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 C35/45 and editing the copy).

Independent reference results

Open handcalculations

Thermal results

The procedure to perform a fire design can be found here. This article supposes you know the workflow.
From the thermal analysis , we can reduct the following results:

Since this column is subjected to pure compression, symmetry in the reduced cross-section (see later) is important for the calculation of the deformation states. Therefor the mesh in the fire analysis has been set to a smaller value than the default.
The area in the concrete cross-section where the temperature exceeds 500°C, doesn’t contribute to the resistance. It is completely neglected, as if it wasn’t there. Resulting in a reduced cross-section. It’s with this reduced cross-section the reinforcement amounts in case of fire are calculated.
The area of the reduced cross-section can be consulted after running a thermal analysis in Diamonds: $A_{c,T}= 32473 mm^2$
The temperature of the tensile reinforcement equals:
$\theta_{s,T}= 579.3^{\circ}C$

$\theta_{s,T}$ will be used to reduce the yielding strength of steel.
The mean temperature of all reinforcement steel equals:

$\theta_{s,mean}= 579.3^{\circ}C$

$\theta_{s,mean}$ will be used to reduce young’s modulus of steel

Using these temperatures we can calculate the reduction on the yielding streng and young’s modulus of steel. The concrete strength is not reduced, because it’s assumed that concrete below 500°C, keeps its full strength.

The reduction factor for the yielding strength (EN 1992-1-2 Table 3.2a) becomes $k_s = 0.535$
The reduction factor for the young’s modulus (EN 1992-1-2 Table 3.2a) becomes $k_E = 0.371$

Reinforcement calculation

The design concrete strength in case of fire:

$f_{cd.T}= \frac{f_{ck}}{\gamma_{c.T}}= \frac{35 \text{MPa}}{1}=35 \text{MPa}$

The reduced young’s modulus equals:

$E_{s.T}=k_E \cdot E_s = 0.371 \cdot 200000 \text{MPa}=74180\text{MPa}$

The reduced yielding strength equals:

$\begin{align*} f_{yd.T} &=min \left(k_s \cdot \frac{f_{yk}}{\gamma_{s.T}}, \varepsilon \cdot E_{s.T} \right) \\ &=min \left(0.535 \cdot \frac{500\text{MPa}}{1}, 0.002 \cdot 74180\text{MPa} \right)\\ &=min \left(267.6\text{MPa}, 148.4\text{MPa} \right)\\ &=148.4\text{MPa}\\ \end{align*}$

The longitudinal reinforcement due to compression is calculated using the formula (see DC EC 03):

$\begin{align*} A_s &=\frac{N_{Ed,ULS} -A_{c.T} \cdot f_{cd.T}}{f_{yd.T}}\\ &=\frac{1700 \text{kN} -32473 mm^2 \cdot 35 \text{MPa} }{148.4\text{MPa}}\\ &=3798mm^2\\ \end{align*}$

Divide reinforcement over 4 sides:

$A_{s, on each side}=\frac{3798mm^2}{4}=949mm^2$

Diamonds results and comparison

Longitudinal reinforcement calculated by Diamonds (EN 1992-1-2 [- -])
Diamonds never shows reinforcement amounts twice. If he shows 4 times 949mm², you should add 949mm² on each side of the column. Not 949mm² divided by 4!

Results	Independent reference	Diamonds	Difference
Longitudinal reinforcement	949mm²	949mm²	0,00%

References

EN 1992-1-1: 2005 + AC: 2010
EN 1992-1-2: 2004/A1:2019
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.

Tested in Diamonds 2025.

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