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Hedi Boukraa, B.Sc.

2023-05-24

VE 1022 | Design of Flat Slab According to DIN EN 1992-1-1 with NA: Slab Internal Forces and Bending Design Under Full Load

Description

The model is based on the example 4 of [1]: Point-supported slab.

The flat slab of an office building with crack-sensitive lightweight walls is to be designed. Inner, border and corner panels are to be investigated. The columns and the flat slab are monolithically joined. The edge and corner columns are placed flush with the edge of the slab. The axes of the columns form a square grid. It is a rigid system (building braced with wall plates).

The office building has 5 floors with a floor height of 3.0 m. The environmental conditions to be assumed are defined as "closed interior spaces". There are predominantly static actions.

Reinforced Concrete Slab Supported by Concrete Columns

The focus of this example is to determine the slab moments and the required reinforcement above the columns under full load.

Material	Concrete C35/45	Modulus of Elasticity	E	34000	N/mm²
	Concrete C35/45	Design value of concrete compressive strength	f_cd	19.8	N/mm²
	Reinforcing Steel B500S(B)	Characteristic yield strength	f_yk	500.000	N/mm²
	Reinforcing Steel B500S(B)	Design yield strength	f_yd	435.000	N/mm²
Geometry	Flat slab	Slab thickness	h	240	mm
		Span of a panel in y-direction	l_eff,y	6.750	m
		Span of a panel in z-direction	l_eff,z	6.750	m
	Column cross-section	Width	b	450	mm
	Column cross-section	Height	h	450	mm
Load	Permanant loads	Reinforced concrete slab		6.000	kN/m²
		Covering and suspended ceiling		1.250	kN/m²
		Total characteristic value	g_k	7.250	kN/m²
		Total design value	g_d	9.790	kN/m²
	Variable loads	Live load		2.000	kN/m²
		Partitions aggregate		2.000	kN/m²
		Total characteristic value	q_k,1	3.25	kN/m²
		Total design value	q_d	4.880	kN/m²

Verification Example 1022 | Slab Designed According to DIN EN 1992-1-1 with NA

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Verification Example 1022 | Slab Designed According to DIN EN 1992-1-1 with NA

The approximation method used in [1] is not suitable as a reference solution for the FE calculation. As an alternative, the reference solution is established as a range of solutions derived from various FE calculations, as documented in the EvaDAT website. The acceptable range of results is limited to a +/- 10% deviation from the median of the available results.

RFEM Settings

The columns are modeled as nodal supports with a vertical spring constant C_u,Z of 2295000 kN/m and a torsion spring constant C_Φ of 232368 kNm/rad in x- und y-direction.
The target length of the finite elements l_FE is 0.450 m.
In addition, a FE mesh refinement is applied to each nodal support. The mesh refinement type is circular with a radius R=0.700 m, an inner target FE length L_FE,i=0.100 m and an outter target FE length L_FE,o=0.300 m.
The evaluation of the moments takes place on the column periphery
The first reinforcement layer is assumed to have a concrete cover of d₁ = 3.0 cm and a bar diameter of 20 mm. This results in an effective depth d_x = 200 mm and d_y = 180 mm.

Results

In the following tables, the results of RFEM 6 are compared with the reference solution.

Slab Moment in x-direction

moment [kNm/m] in x-direction
Axis	Result location	RFEM Result	Reference solution
Axis	Result location	RFEM Result	Median	+/- 10% range
Axis 1	Column C	-73.0	-74.4	-67.0	-81.8
	Column B	-93.4	-95.7	-86.1	-105.2
	Column A	-76.9	-78.1	-70.2	-85.9
	Span C-B	37.3	37.5	33.8	41.3
	Span B-A	49.7	50.5	45.4	55.5
Axis 2	Column C	-115.5	-115.8	-104.2	-127.4
	Column B	-143.6	-144.7	-130.2	-159.2
	Column A	-98.5	-99.2	-89.2	-109.1
	Span C-B	34.1	34.6	31.1	38.1
	Span B-A	51.2	52.1	46.8	57.3
Axis 3	Column C	-101.0	-100.7	-90.6	-110.7
	Column B	-130.2	-130.8	-117.7	-143.9
	Column A	-91.2	-91.8	-82.6	-100.9
	Span C-B	31.3	31.6	28.4	34.7
	Span B-A	48.9	49.7	44.7	54.6
Axis 1-2	Column C	-6.7	-7.0	-6.3	-7.7
	Column B	-11.8	-11.9	-10.7	-13.1
	Span C-B	22.0	22.0	19.8	24.2
	Span B-A	37.6	37.7	33.9	41.4
Axis 2-3	Column C	-10.7	-11.1	-9.9	-12.2
	Column B	-17.6	-17.9	-16.1	-19.7
	Span C-B	19.4	19.5	17.5	21.4
	Span B-A	37.9	37.9	34.1	41.7

Verification Example 1022 | Support and Panel Moment in x-Direction in Relevant Axes

Trosional Moment Trosional moment m_xy

Trosional moment m_xy [kNm/m]
Axis	Result location	RFEM Result	Reference solution
Axis	Result location	RFEM Result	Average result	+/- 10% range
Axis 1	Column C	17.62	16.7	15.0	18.3
	Column B	20.4	19.1	17.1	21.0
	Column A	32.8	31.1	28.0	34.2
Axis 2	Column C	2.45	2.4	2.1	2.6
	Column B	1.74	1.7	1.5	1.9
	Column A	2.64	2.7	2.4	3.0

Verification Example 1022 | Torsional Moment in Relevant Axes

Support Forces

Support Forces [kN]
Column	RFEM result	Reference solution
Column	RFEM result	Median	+/- 5% range
C1=A3	300.8	300.9	285.9	316.0
B1=A2	328.8	328.8	312.3	345.2
A1	153.2	153.4	145.7	161.0
C2=B3	707.1	707.0	671.7	742.4
B2	787.3	787.0	747.7	826.4
C3	624.4	624.5	593.3	655.7

Verification Example 1022 | Support Forces

Required Reinforcement

Required Reinforcement a_sx [cm₂/m] in x-direction, d_x = 200 mm
Axis	Result location	RFEM results	Reference solution
Axis	Result location	RFEM results	Median	+/- 10% range
Axis 1	Column C	10.82	10.8	9.7	11.9
	Column B	13.95	13.8	12.4	15.2
	Column A	13.37	13.0	11.7	14.3
	Span C-B	4.51	4.5	4.1	5.0
	Span B-A	6.08	6.1	5.5	6.7
Axis 2	Column C	14.5	16.1	14.4	17.7
	Column B	18.43	20.3	18.3	22.3
	Column A	12.2	12.5	11.3	13.8
	Span C-B	3.89	3.9	3.5	4.3
	Span B-A	5.91	5.9	5.3	6.5
Axis 3	Column C	12.21	13.5	12.1	14.8
	Column B	16.29	17.8	16.0	19.5
	Column A	10.9	11.1	10.0	12.2
	Span C-B	3.56	3.6	3.2	3.9
	Span B-A	5.62	5.7	5.1	6.2
Axis 1-2	Column C	0.8	0.8	0.7	0.9
	Column B	1.33	1.3	1.2	1.5
	Span C-B	2.53	2.5	2.2	2.7
	Span B-A	4.34	4.3	3.9	4.8
Axis 2-3	Column C	1.19	1.2	1.1	1.3
	Column B	1.99	2.0	1.8	2.2
	Span C-B	2.2	2.2	1.9	2.4
	Span B-A	4.34	4.3	3.9	4.8

Verification Example 1022 | Required Reinforcement in x-Direction at Top of Slab

Verification Example 1022 | Required Reinforcement in x Direction at Bottom of Slab

Required Reinforcement a_sy [cm₂/m] in y-direction, d_x = 180 mm
Axis	Result location	RFEM results	Reference solution
Axis	Result location	RFEM results	Median	+/- 10% range
Axis A	Column 3	12.34	12.3	11.0	13.5
	Column 2	15.98	15.7	14.2	17.3
	Column 1	15.31	14.8	13.4	16.3
	Span 2-3	5.04	5.0	4.5	5.5
	Span 1-2	6.81	6.8	6.2	7.5
Axis B	Column 3	16.64	17.9	16.1	19.6
	Column 2	21.31	22.7	20.4	24.9
	Column 1	13.93	13.9	12.5	15.2
	Span 2-3	4.34	4.4	3.9	4.8
	Span 1-2	6.6	6.7	6.0	7.3
Axis C	Column 3	13.93	15.0	13.5	16.4
	Column 2	18.67	19.7	17.7	21.7
	Column 1	12.42	12.3	11.1	13.5
	Span 2-3	3.97	4.0	3.6	4.4
	Span 1-2	6.29	6.4	5.7	7.0
Axis A-B	Column 3	0.85	0.9	0.8	1.0
	Column 2	1.47	1.5	1.3	1.6
	Span 2-3	2.82	2.8	2.5	3.0
	Span 1-2	4.85	4.8	4.4	5.3
Axis B-C	Column 3	1.34	1.4	1.2	1.5
	Column 2	2.22	2.2	2.0	2.4
	Span 2-3	2.46	2.4	2.2	2.7
	Span 1-2	4.83	4.8	4.4	5.3

The internal forces and the required reinforcement calculated by RFEM 6 are consistently within the defined result range of +/- 10% of the median value.

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Verification Example 1022 | Slab Designed According to DIN EN 1992-1-1 with NA

Links

EvaDAT Example 0036: Design of Flat Slab According to DIN EN 1992-1-1 with NA Part 1 Plate Internal Forces and Bending Design under Full Loading

References

German Concrete and Structural Engineering Association E. V, examples for design according to Eurocode 2. Volume 1: Building construction, Berlin: Ernst & Sohn 2012, 1st prize corrected reprint of 1. Edition, 978-3-433-01877-4

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