# RF-/STEEL EC3 Version 5/8

Online manuals, introductory examples, tutorials, and other documentation.

# 8.1 Stability

We will perform some stability analyses for flexural buckling and lateral-torsional buckling for a column with double-bending, considering the interaction conditions.

Design values
 Figure 8.1 Design values of static loadsNd = 300 kNqz,d = 5 kN/mFy,d = 7.5 kN
Internal forces according to linear static analysis
Design locations (governing x-location)

The design is performed for all x-locations (see Chapter 4.5) of the equivalent member. The governing location is x = 2.00 m. RFEM or RSTAB determines the following internal forces:

Table 8.1 Internal forces
N My Mz Vy Vz

-300.00 kN

10.00 kNm

7.50 kNm

3.75 kN

0.00 kN

Cross-section properties HE-B 160, S 235
Table 8.2 Cross-section properties HE-B 160, S 235
Property Symbol Value Unit

Cross-sectional area

A

54.30

cm²

Moment of inertia

Iy

2490.00

cm4

Moment of inertia

Iz

889.00

cm4

iy

6.78

cm

iz

4.05

cm

Polar radius of gyration

ip

7.90

cm

Polar radius of gyration

ip,M

41.90

cm

Section weight

G

42.63

kg/m

Torsional constant

IT

31.40

cm4

Warping constant

Iω

47940.00

cm6

Elastic section modulus

Wy

311.00

cm3

Elastic section modulus

Wz

111.00

cm3

Plastic section modulus

Wpl,y

354.00

cm3

Plastic section modulus

Wpl,z

169.96

cm3

Buckling curve

BCy

b

Buckling curve

BCz

c

Flexural buckling about minor axis (⊥ to z-z axis)

→ Design for flexural buckling must be performed.

cross-sectional geometry: h/b = 1.00 ≤ 1.2; structural steel S 235; t ≤ 100 mm

• [1] Table 6.2, row 3, column 4: buckling curve c
• ⇒ αz = 0.49   ([1] Table 6.1)

Results of RF-/STEEL EC3 calculation
 Iz 889.00 cm4 Effective member length Lcr,z 4.000 m Elastic flexural buckling force Ncr,z 1151.60 kN Slenderness λz 1.053 > 0.2 6.3.1.2(4) Buckling curve BCz c Tab. 6.2 Imperfection factor αz 0.490 Tab. 6.1 Auxiliary factor Φz 1.263 6.3.1.2(1) Reduction factor χz 0.510 Eq. (6.49)
Flexural buckling about major axis (⊥ to y-y axis)

→ Design for flexural buckling must be performed.

cross-sectional geometry: h/b = 1.00 ≤ 1.2; structural steel S 235; t ≤ 100 mm

• [1] Table 6.2, row 3, column 4: buckling curve b
• ⇒ αy = 0.34   ([1] Tabelle 6.1)

Results of RF-/STEEL EC3 calculation
 Second moment of area Iy 2490.00 cm4 Effective member length Lcr,y 4.000 m Elastic flexural buckling force Ncr,y 3225.51 kN Cross-sectional area A 54.30 cm2 Yield strength fy 23.50 kN/cm2 3.2.1 Slenderness λy 0.629 > 0.2 6.3.1.2(4) Buckling curve BCy b Tab. 6.2 Imperfection factor αy 0.340 Tab. 6.1 Auxiliary factor Φy 0.771 6.3.1.2(1) Reduction factor χy 0.822 Eq. (6.49)
Lateral-torsional buckling
Ideal elastic critical moment

In this example, the elastic critical moment for lateral-torsional buckling is determined according to the Austrian National Annex with assumption of hinged supports free to warp.

The load application point is assumed to be in the shear center (you can adjust the application point for transverse loads in the Details dialog box, cf. Chapter 3.1.2).

The program also shows Mcr,0 which is determined on the basis of a constant moment distribution.

For the results by x-location, the program also shows the Mcr,x values. Those are the elastic critical moments at the x-locations relative to the elastic critical moment at the location of the maximum moment. Using Mcr,x, the program then calculates the relative slenderness ƛLT.

Slenderness for lateral-torsional buckling

Calculation according to [1] clause 6.3.2.2 for location with the maximum moment at x = 2.00 m:

HEB-160, cross-section class 1: Wy = Wpl,y = 354.00 cm3

Reduction factor χLT

Calculation according to [1] clause 6.3.2.3

HEB-160: h/b = 1.0 < 2.0 ⇒ buckling curve b according to [1] Table 6.5

• Auxiliary factor:

Limiting slenderness:

Parameter (minimum value):

Imperfection factor according to [1] Table 6.3:

According to [1] clause 6.3.2.3, the reduction factor may be modified as follows:

Correction factor kc according to [1] Table 6.6 for a parabolic moment diagram:

Interaction factors kyy and kyz

Determination according to [4] Annex B, Table B.2, for structural components susceptible to torsional deformations

The equivalent moment factor CmLT is obtained according to Table B.3 for ψ = 0 as:

Interaction factors kzy and kzz

Determination according to [1] Annex B, Table B.2, for structural components susceptible to torsional deformations

The equivalent moment factor CmLT is obtained according to Table B.3 for ψ = 0 as:

Interaction design for buckling about major axis and lateral-torsional buckling

According to [1] Eq. (6.61) the following requirement must be fulfilled:

where

Interaction design for buckling about minor axis and lateral-torsional buckling

According to EN1993-1-1 Eq. (6.62) the following requirement must be fulfilled:

$\frac{300}{0.510·1276.05/1.0}+0.934·\frac{10.0}{0.908·83.19/1.0}+1.481·\frac{7.50}{39.94/1.0}=0.863\le 1$

Results of RF-/STEEL EC3 calculation
 Section depth h 160.0 mm Section width b 160.0 mm Criterion h/b 1.00 ≤ 2 Tab. 6.5 Buckling curve BCLT b Tab. 6.5 Imperfection factor αLT 0.340 Tab. 6.3 Shear modulus G 8100.00 kN/cm3 Length factor kz 1.000 Length factor kw 1.000 Length L 4.000 m Warping constant Iw 47940.00 cm6 Torsional constant It 31.40 cm4 Elastic critical moment for LTB for determination of related slenderness Mcr,0 190.90 kNm Moment distribution Diagr My 6) parabola Maximum sagging moment My,max 10.00 kNm Boundary moment My,A 0.00 kNm Moment ratio ψ 0.000 Moment factor C1 1.130 [2] Ideal elastic critical moment Mcr 215.71 kNm Elastic section modulus Wy 354.00 cm3 Slenderness λLT 0.621 6.3.2.2(1) Parameter λLT,0 0.400 6.3.2.3(1) Parameter β 0.750 6.3.2.3(1) Auxiliary factor φLT 0.682 6.3.2.3(1) Reduction factor χLT 0.908 Eq. (6.57) Correction factor kc 0.940 6.3.2.3(2) Modification factor f 0.972 6.3.2.3(2) Reduction factor χLT,mod 0.934 Eq. (6.58) Moment distribution Diagr My 3) max in span Tab. B.3 Moment factor ψy 1.000 Tab. B.3 Moment Mh,y 0.00 kNm Tab. B.3 Moment Ms,y 10.00 kNm Tab. B.3 Ratio­­ Mh,y / Ms,y αh,y 0.000 Tab. B.3 Load type Load z uniform load Tab. B.3 Moment factor Cmy 0.950 Tab. B.3 Moment distribution Diagr Mz 3) max in span Tab. B.3 Moment factor ψz 1.000 Tab. B.3 Moment Mh,z 0.00 kNm Tab. B.3 Moment Ms,z 7.50 kNm Tab. B.3 Ratio­ Mh,z / Ms,z αh,z 0.000 Tab. B.3 Load type Load y concentrated load Tab. B.3 Moment factor Cmz 0.900 Tab. B.3 Moment distribution Diagr My,LT 3) max in span Tab. B.3 Moment factor ψy,LT 1.000 Tab. B.3 Moment Mh,y,LT 0.00 kNm Tab. B.3 Moment Ms,y,LT 10.00 kNm Tab. B.3 Ratio Mh,y,LT / Ms,y,LT αh,y,LT 0.000 Tab. B.3 Load type Load z uniform load Tab. B.3 Moment factor CmLT 0.950 Tab. B.3 Component type Com-ponent torsionally weak Interaction factor kyy 1.067 Tab. B.2 Interaction factor kyz 0.888 Tab. A.1 Interaction factor kzy 0.934 Tab. A.1 Interaction factor kzz 1.481 Tab. A.1 Axial force (compression) NEd 300.00 kN Governing cross-sectional area Ai 54.30 cm2 Tab. 6.7 Compression resistance NRk 1276.05 kN Tab. 6.7 Partial safety factor γM1 1.000 6.1 Design component for N γNy 0.29 ≤ 1 Eq. (6.61) Design component for N hNz 0.46 ≤ 1 Eq. (6.62) Moment My,Ed 10.00 kNm Moment resistance My,Rk 83.19 kNm Tab. 6.7 Moment component ηMy 0.13 Eq. (6.61) Moment Mz,Ed 7.50 kNm Elastic section modulus WZ 169.96 cm3 Moment resistance Mz,Rk 39.94 kNm Tab. 6.7 Moment component ηMz 0.19 Eq. (6.61) Design 1 η1 0.59 ≤ 1 Eq. (6.61) Design 2 η2 0.86 ≤ 1 Eq. (6.62)
Literatur
 [1] Eurocode 3: Design of steel structures - Part 1-1: General rules and rules for buildings; EN 1993-1-1:2010-12 [4] Johannes Naumes, Isabell Strohmann, Dieter Ungermann and Gerhard Sedlacek. Die neuen Stabilitätsnachweise im Stahlbau nach Eurocode 3. Stahlbau, 77, 2008.