DC EC 12: longitudinal reinforcement in beam under pure bending (ULS FI design)

Description

Geometry	Cross-section:	$b=150\text{mm}$ $h=350\text{mm}$
	Material:	C35/40 any SLS checks in this material have been turned off
	Concrete cover:	$c=40\text{mm}$
Internal forces		$M_{Ed.ULS FI}=60\text{kNm}$ ULS FC is neglected
Fire conditions		75min of ISO 834 fire
Standard		EN 1992-1-2 [- -]

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:

The area where the temperature exceeds 500°C, doesn’t contribute to the resistance. It is completely neglected, as if it wasn’t there.
The area of the reduced cross-section: $A_{c,T}= 25438.6mm^2$ .
The temperature of the tensile reinforcement equals
$\theta_{s,T}= 540.8^{\circ}C$

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

$\theta_{s,mean}= 0.5 \cdot (540.8^{\circ}C+545.4^{\circ}C)=497.6^{\circ}C$

$\theta_{s,mean}$ 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 $k_s = 0.654$
The reduction factor for the young’s modulus (EN 1992-1-2 Table 3.2a) becomes $k_E = 0.602$

Using Section Utility, you can consult the reduced the cross section. The reduced section is no longer a nice rectangle (see image below), but a “rounded rectangle”. Diamonds will calculate the deformation states of this new shape, but that calculation not suitable for hand calculations. As an alternative, we’ll approximate the rounded rectangle with a rectangle. The hand calculation results will deviate from the Diamonds results, but they’ll be of the same magnitude.

By double clicking a point at the very top of bottom, you can find half of maximum height of the reduced cross-section: 141.36mm. Multiply that by two and you have the height of the recuded cross section (= 282.64mm).

The tensile reinforcement was at the distance of $350mm/2-40mm = 135mm$ from the center line of the cross-section. So the concrete coverage in the reducted cross-section equals $141.32mm-135mm = 6.32mm$ . Resulting in a useful height of $282.64mm-6.32mm=276.3mm$ .

From which you can deduct the ‘mean reduced width’: $b'=\frac{A_{c.T}}{h'}=\frac{25438.6mm}{276.3mm}=90mm$ .

Reinforcement calculation

$\mu _d=\frac{M_{Ed.ULS FI}}{b\cdot d^2\cdot f_{cd}}=\frac{60\text{kNm}}{90.0\text{mm}\cdot (276.3\text{mm})^2\cdot 35\text{MPa}}=0.249$

This is a well chosen example: an example for which the deformation state $\xi$ (NL: vervormingstoestand) will be between the boundaries 0 and 0.617. Within these boundaries the reinforcement efficiancy $s_1$ equals 1 [L. Vanhooymissen, Gewapend beton]. Meaning that we can use the Table 3 from Gewapend beton: numeri to determine the reinforcement amounts. The table (click here to open the Table) shows $\omega _1$ as a function of $\mu _d$ , and we find the values below.
Note that this won’t work for every beam loaded with fire. For deformation states $\xi$ outside those boundaries, the reinforcement efficiancy $s_1$ is a function of steels young’s modulus and yielding strength. But both depend upon the temperature of the reinforcement steel… While Table 3 assumes that the reinforcement steel is at room temperature (cold state)!

$\xi=0.363$

$\omega _1=0.294$

$A_{s1}=\omega _1 \cdot b\cdot d\cdot \frac{f_{cd}}{k_s \cdot f_{yd}}=0.294 \cdot 90.0\text{mm} \cdot 276.4\text{mm} \cdot \frac{16.7\text{MPa}}{0.654 \cdot 500\text{MPa}}=783mm^2$

Open calculation using Section Utility

Because this model falls withing the limits of $\xi$ being between 0 and 0.617 (see handcalculations for explanation), we could also use the Section Utility profile to verify Diamonds. To do so:

Open the reduced cross-section in Section Utility.
Go to File > Save As and save the cross-section somewhere on your computer.
Close the fire results.
Go to Edit > Material library.
- Make a copy of the material C35/45 (only ULS design). This is the material that is currently assigned to the beam. Name the copied material C35/45 (only ULS FI design).
- Set the partial safety factor for concrete equal to 1.
- Set the partial safety factor for the reinforcement steel equal to 1
  Also change the yielding strength of the reinforcement steel to $k_s \cdot f_{yd}=0.654 \cdot 500\text{MPa}}=326.7\text{MPa}}$ .
- Note that this dialog doesn’t allow us to adjust young’s modulus for the reinforcement steel. That’s why we have to make sure $\xi$ remains within the imposed boundaries in order to be able to use this workflow.
- Save the changes and close this dialog.
Go to the geometry configuration . Make a copy of the current beam.
Select the copied beam and open this dialog .
Select a Section Utility cross-section and open Section Utility. Load in the reduced cross section.

Double click the cross-section and change the material to C35/45 (only ULS FI design).
Close Section Utility. Send the cross-section to Diamonds.
Go to the loads configuration and remove the fire load from the copied beam (because the section is already reduced due to fire).
Run the thermal and elastic analysis . Calculate the reinforcement .

Diamonds results and comparison

Longitudinal reinforcement calculated by Diamonds (EN 1992-1-2 [- -])

Results	Independent reference (hand calculations)	Independent reference (Section utility calculation)	Diamonds	Difference to hand calculation	Difference to Section Utility calculation
Maximum longitudinal reinforcement	783 mm²	817 mm²	818 mm²	4,4%	0,25%

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.
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 2025.

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