DC EC 14: longitudinal reinforcement in a plate under pure bending (ULS FC design)

Description

Geometry	Plate section for points A, B:	Preslabs $h=180mm$ $e=50mm$
	Plate section for point C:	Two way slab $h=150mm$
	Material:	C25/30
	Concrete cover:	$c=35mm$
Internal forces	Point A:	$M_{xx}=-0.09\text{kNm}$ $M_{zz}=-0.86\text{kNm}$ $M_{xz}=16.61\text{kNm}$
	Point B:	$M_{xx}=67.81\text{kNm}$ $M_{zz}=24.41\text{kNm}$ $M_{xz}=-0.05\text{kNm}$
	Point C:	$M_{xx}=-14.17\text{kNm}$ $M_{zz}=-0.01\text{kNm}$ $M_{xz}=-0.26\text{kNm}$
Standard		EN 1992-1-1 [- -]

Independent reference results

Open handcalculations

Mesh points and interpolation

Diamonds calculates the reinforcement at each vertex and in the centre of each triangle side of a mesh triangle. The ‘attractive’ colour gradient of the results is created by interpolation.
To avoid having to worry about interpolation here as well, we specifically take internal forces that occur in mesh nodes.

You can easily retrieve the internal forces in the mesh nodes via the results table . Or if you add a point in the geometry, like we did for point C, that becomes a meshnode by default.

Diamonds sign convention

Diamonds calculates the bending moments M_xx, M_zz and the torsional moment M_xz in the plate. When calculating the reinforcement, the contribution of the torsional moment M_xz cannot be ignored. Therefor M_xx and M_zz are not the reinforcement moments ( = the moments based on with the reinforcement is calculated). Instead Diamonds uses Wood-Armer moments to calculate the reinforcement. They are listed in the table below.

*Wood-Armer reinforcement moments written as they are often found in literature*
Bottom		Top
$M_{x}=M_{x}+\left\| M_{xy} \right\|$ $M_{y}=M_{y}+\left\| M_{xy} \right\|$		$M_{x}=M_{x}-\left\| M_{xy} \right\|$ $M_{y}=M_{y}-\left\| M_{xy} \right\|$
If $M_{x*} < 0$ :	$M_{x}=0$ $M_{y}=M_{y}+\left\| \frac{M_{xy}^2}{M_{x}} \right\|$	If $M_{x*} > 0$ :	$M_{x}=0$ $M_{y}=M_{y}-\left\| \frac{M_{xy}^2}{M_{x}} \right\|$
If $M_{y*} < 0$ :	$M_{y}=0$ $M_{x}=M_{x}+\left\| \frac{M_{xy}^2}{M_{y}} \right\|$	If $M_{y*} > 0$ :	$M_{y}=0$ $M_{x}=M_{x}-\left\| \frac{M_{xy}^2}{M_{y}} \right\|$

The sign convention for Wood-Armer moments is:

Positive moment causes sagging ⌣ (tension in the bottom)
Negative moment causes hogging ⌢ (tension in the top)

That is the opposite of the sign convention in Diamonds. As a result, the headers “Bottom” and “Top” should be switched when we want to recalculate Diamonds.

*The first step in understanding how the Wood-Armer reinforcement moments are implemented in Diamonds, is converting the table to the Diamonds sign convention*
Top		Bottom
$M_{x}=M_{x}+\left\| M_{xy} \right\|$ $M_{y}=M_{y}+\left\| M_{xy} \right\|$		$M_{x}=M_{x}-\left\| M_{xy} \right\|$ $M_{y}=M_{y}-\left\| M_{xy} \right\|$
If $M_{x*} < 0$ :	$M_{x}=0$ $M_{y}=M_{y}+\left\| \frac{M_{xy}^2}{M_{x}} \right\|$	If $M_{x*} > 0$ :	$M_{x}=0$ $M_{y}=M_{y}-\left\| \frac{M_{xy}^2}{M_{x}} \right\|$
If $M_{y*} < 0$ :	$M_{y}=0$ $M_{x}=M_{x}+\left\| \frac{M_{xy}^2}{M_{y}} \right\|$	If $M_{y*} > 0$ :	$M_{y}=0$ $M_{x}=M_{x}-\left\| \frac{M_{xy}^2}{M_{y}} \right\|$

Diamonds nomenclature

In Diamonds, the x’-direction is the principal direction of the plate and the z’-direction the secundairy (= local coordinate system of a surface). Also in Diamonds, top is referred to as “Superior” and bottom as “Inferior”. Let’s convert the table again to take these modifications into account.

*The second step in understanding how the Wood-Armer reinforcement moments are implemented in Diamonds, is adjusting the table to the Diamonds nomenclature*
Top (Superior)		Bottom (Inferior)
$M_{zs}=M_{zz}+\left\| M_{xz} \right\|$ $M_{xs}=M_{xx}+\left\| M_{xz} \right\|$		$M_{zi}=M_{zz}-\left\| M_{xz} \right\|$ $M_{xi}=M_{xx}-\left\| M_{xz} \right\|$
If $M_{zs} < 0$ :	$M_{zs}=0$ $M_{xs}=M_{xx}+\left\| \frac{M_{xz}^2}{M_{zz}} \right\|$	If $M_{zi} > 0$ :	$M_{zi}=0$ $M_{xi}=M_{xx}-\left\| \frac{M_{xz}^2}{M_{zz}} \right\|$
If $M_{xs} < 0$ :	$M_{xs}=0$ $M_{zs}=M_{zz}+\left\| \frac{M_{xz}^2}{M_{xx}} \right\|$	If $M_{xi} > 0$ :	$M_{xi}=0$ $M_{zi}=M_{zz}-\left\| \frac{M_{xz}^2}{M_{xx}} \right\|$

$M_{zs}$ is thus the reinforcement moment to calculate the top reinforcement parallel to the local z’-axis of the plate.
$M_{zi}$ is the reinforcement moment to calculate the bottom reinforcement parallel to the local z’-axis.

Determine the reinforcement moments

Point	$M_{xs}$	$M_{zs}$	$M_{xi}$	$M_{zi}$
A	-0.09+\|16.61\|=16.52	-0.86+\|16.61\|=15.75	-0.09-\|16.61\|=-16.70	-0.86-\|16.61\|=-17.47
B	67.81+\|-0.05\|=67.86	24.41+\|-0.05\|=24.46	67.81-\|-0.05\|=67.76	24.41-\|-0.05\|=24.36
B	67.81+\|-0.05\|=67.86	24.41+\|-0.05\|=24.46	Because both $M_{xi}$ and $M_{zi}$ > 0, their value is changed to 0.
C	-14.17+\|-0.26\|=-13.91	-0.01+\|-0.26\|=0.25	-14.17-\|-0.26\|=-14.43	-0.01-\|-0.26\|=-0.27
C	Because $M_{xs}$ < 0, $M_{xs}=0$ and $M_{zs}=M_{zz}+\left\| \frac{M_{xz}^2}{M_{xx}} \right\|=$ $-0.01+\left\| \frac{-0.26^2}{-14.17} \right\|=-0.01$		-14.17-\|-0.26\|=-14.43	-0.01-\|-0.26\|=-0.27

Reinforcement calculation

Based on the names M_xs, M_zs, M_xi, M_zi we know where the reinforcement is going to be placed. Therefor we’re neglecting the signs of the reinforcement moments in the table below.
To calculate the reinforcement in a plate in bending, we use the same formulae as for a beam (see DC EC 02), but with $b$ equal to 1m. We repeat the process from DC EC 02 for $M_{xs}$ of point A. The principle is the same for the other reinforcement moments.
$d=h-c=180mm-35mm=145mm$

$\mu _d=\frac{M_{xs}}{b\cdot d^2\cdot f_{cd}}=\frac{16.52\text{kNm}}{1000mm\cdot (145mm)^2\cdot 16.7\text{MPa}}=0.005$

Table 3 from Gewapend beton: numeri (click here to open the Table) shows $\omega _1$ as a function of $\mu _d$ , and we find:

$\omega _1=0.005$

$A_{s1}=\omega _1 \cdot b\cdot d\cdot \frac{f_{cd}}{f_{yd}}=0.005 \cdot 145mm \cdot 1000mm \cdot \frac{16.7\text{MPa}}{434.8\text{MPa}}=272mm^2$

However, pay attention when calculating $A_{zs}$ in points A & B. The plate is a preslab. The useful height for the secundairy top reinforcement is measured starting from the top of the preslab d= h-(4) (more info).

Results	Reinforcement moments [kNm]	Usefull height $d$ [mm]	Reinforcement amount [mm²]
Point A	$M_{xs}=16.52$	145	272
	$M_{zs}=15.75$	= 180 – 50 – 35= 9	407
	$M_{xi}=16.70$	145	275
	$M_{zi}=17.47$	145	288
Point B	$M_{xs}=67.86$	145	1212
	$M_{zs}=24.46$	= 180 – 50 – 35= 95	653
	$M_{xi}=0$	145	0
	$M_{zi}=0$	145	0
Point C	$M_{xs}=0.01$	115	0
	$M_{zs}=0$	115	0
	$M_{xi}=14.43$	115	302
	$M_{zi}=0.27$	115	5

Diamonds results and comparison

Longitudinal upper reinforcement calculated by Diamonds in Point B (EN 1992-1-1 [- -])
Diamonds calculates the reinforcement required for the ULS. When the SLS requires additional reinforcement, it will be added to the ULS amount and named ‘TOT’. So when comparing results for this validation examole, look at the ULS.

Results		Independent reference	Diamonds	Difference
Point A	$A_{xs}$	272 mm²	273 mm²	≈0%
	$A_{zs}$	407 mm²	407 mm²	0%
	$A_{xi}$	275 mm²	276 mm²	≈0%
	$A_{zi}$	288 mm²	289 mm²	≈0%
Point B	$A_{xs}$	1212 mm²	1214 mm²	≈0%
	$A_{zs}$	653 mm²	653 mm²	0%
	$A_{xi}$	0 mm²	0 mm²	0%
	$A_{zi}$	0 mm²	0 mm²	0%
Point C	$A_{xs}$	0 mm²	0 mm²	0%
	$A_{zs}$	0 mm²	0 mm²	0%
	$A_{xi}$	302 mm²	303 mm²	≈0%
	$A_{zi}$	5 mm²	6 mm²	≈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.
R. H. WOOD, The reinforcement of slabs in accordance with a pre-determined fiel of moments (original paper), Concrete magazine February 1968

Tested in Diamonds 2025.

Article Attachments

.bsf

DC EC 14 126 KB

Was this article helpful?

Yes No

Need Support?

Can't find the answer you're looking for? Don't worry we're here to help!

CONTACT SUPPORT

Point	$M_{xs}$	$M_{zs}$	$M_{xi}$	$M_{zi}$
A	-0.09+\|16.61\|=16.52	-0.86+\|16.61\|=15.75	-0.09-\|16.61\|=-16.70	-0.86-\|16.61\|=-17.47
B	67.81+\|-0.05\|=67.86	24.41+\|-0.05\|=24.46	67.81-\|-0.05\|=67.76	24.41-\|-0.05\|=24.36
B	67.81+\|-0.05\|=67.86	24.41+\|-0.05\|=24.46	Because both $M_{xi}$ and $M_{zi}$ > 0, their value is changed to 0.
C	-14.17+\|-0.26\|=-13.91	-0.01+\|-0.26\|=0.25	-14.17-\|-0.26\|=-14.43	-0.01-\|-0.26\|=-0.27
C	Because $M_{xs}$ < 0, $M_{xs}=0$ and $M_{zs}=M_{zz}+\left\| \frac{M_{xz}^2}{M_{xx}} \right\|=$ $-0.01+\left\| \frac{-0.26^2}{-14.17} \right\|=-0.01$		-14.17-\|-0.26\|=-14.43	-0.01-\|-0.26\|=-0.27