SLX 10: Global imperfections

Description

Material		Steel	S235
Geometry	Cross-section	Columns	HEA 220
	Cross-section	Beams	IPE 400 (top level) IPE 450 (bottom level)
	Boundary conditions	In points A, B and C	Fixed supports
	Boundary conditions	The entire frame	X rotatie and Z-displacement prevented to impose 2D behaviour
Loads		On top level	q_{ULS FC.1} $= 82.1 \text{kN/m}$
Loads		On bottom level	q_{ULS FC.2} $=107.3 \text{kN/m}$
Calculation options		Global imperfection	$\varphi =3.09 \cdot 10^{-3}=\frac{1}{324}$

Note: for the calculation of the global imperfections $\varphi$ , we refer to the document “SX008a-EN-EU” . Its reference is at the bottom of this article.

Independent reference results

Open handcalculations

Diamonds uses equivalent horizontal forces to take global imperfections into account. But it’s not possible to see how large those equivalent horizontal forces are. However, we can calculate them by hand:

$H_1= \varphi \cdot q_1 \cdot L = 3.09 \cdot 10^{-3} \cdot 82.1 \text{kN/m} \cdot 14m=3.55 \text{kN}$

$H_2= \varphi \cdot q_2 \cdot L = 3.09 \cdot 10^{-3} \cdot 107.3 \text{kN/m} \cdot 14m=4.64 \text{kN}$

In order to test whether the Diamonds results are correct, the Diamonds contians 3 load groups containing ULS loads:

loadgroup “ver”: contains only q_{ULS FC.1}and q_{ULS FC.2}.
loadgroup “ver + hor”: contains q_{ULS FC.1}and q_{ULS FC.2} but also the equivalent horizontal forces $H_1$ and $H_2$ to simulate global imperfections. Because we don’t want the equivalent horizontal forces to affect the reaction forces of the structure, the opposite loads are added to the foundation.
loadgroup “hor”: contains only the equivalent horizontal forces H_1 and H_2. Because we don’t want the equivalent horizontal forces to affect the reaction forces of the structure, the opposite loads are added to the foundation.

Diamonds results and comparison

1st order WITHOUT global imperfections

First we collect the results, which we will compare in the next paragraph.

TABLE A: Horizontal displacement $\delta_x$ in the load group “hor”

Point	Which result	Diamonds
D	$\delta_x$ in load group “hor”	1.231mm
E	$\delta_x$ in load group “hor”	1.924mm

Horizontal deformation $\delta_y$ due to the load group “hor” after a first order calculation

TABLE B: Horizontal displacement $\delta_x$ in the load group “ver”
Due to the vertical loads, the beams deform. Since the connection to the columns is not perfectly rigid, the columns rotate a little around the global X-axis. Causing the upper beam to shorten -0,238mm and the beam on the lower floor to stretch 0.110mm.

Point	Which result	Diamonds
D	$\delta_x$ in load group “ver”	0.110mm
E	$\delta_x$ in load group “ver”	-0.238mm

Horizontal deformation $\delta_y$ due to the load group “ver” after a first order calculation

TABLE C: Horizontal displacement $\delta_x$ in the load group “ver+hor”

Point	Which result	Diamonds
D	$\delta_x$ in load group “ver+hor”	1.231mm+0.110mm = 1.341mm
E	$\delta_x$ in load group “ver+hor”	1.924mm-0.238mm = 1.687mm

Horizontal deformation $\delta_y$ due to the load group “ver + hor” after a first order calculation

1st order WITH global imperfections

TABLE D: Horizontal displacement $\delta_x$ in the load group “ver”

Point	Which result	Diamonds
D	$\delta_x$ in load group “ver”	1.346mm
E	$\delta_x$ in load group “ver”	1.691mm

Horizontal deformation $\delta_y$ due to the load group “ver” after a first order calculation with global imperfections

Comparison

Compare the horizontal deformation of manual applied equivalent horizontal loads (TABLE A) with the horizontal deformation in the reference document (SX008a-EN-EU).

Point	Which result	Independent reference (SX008a-EN-EU)	Diamonds (TABLE A)	Difference
D	$\delta_x$	1.23mm	1.231mm	0,08%
E	$\delta_x$	1.23mm+0.69mm=1.92mm	1.924mm	0,21%

Compare the horizontal deformation of manual applied equivalent horizontal loads (TABLE C) with the global imperfections calculated by Diamonds (TABLE D).

Point	Which result	Independent reference (TABLE C)	Diamonds (TABLE D)	Difference
D	$\delta_x$	1.341mm	1.346mm	0,37%
E	$\delta_x$	1.687mm	1.691mm	0,24%

References

Gozzi, J. (2005). Example: Sway stability. Calculation Sheet Access Steel, SX008a-EN-EU

Tested in Diamonds 2024r01.

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