RF-/STEEL EC3 Version 5/8

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RF-/STEEL EC3 Version 5/8

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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
System and loads
Table 8.0 System and design loads (γ times)
Figure 8.1

Design values of static loads

Nd = 300 kN
qz,d = 5 kN/m
Fy,d = 7.5 kN

Internal forces according to linear static analysis
Figure 8.2 Internal forces
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

Radius of gyration

iy

6.78

cm

Radius of gyration

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)

Ncr,z=21000·889.00·π2400.002=1151.60 kN 

λ¯z=A·fyNcr,z=54.30·23.51151.60=1.053>0.2 

→ 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)

Φ=0.5·[1+0.49·(1.053-0.2)+1.0532]=1.263 

χz=11.263+1.2632-1.0532=0.510 

NEdχz·A·fy/γM1=3000.510·54.30·23.5/1.0=0.461 

Results of RF-/STEEL EC3 calculation
Table 8.3 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)

Ncr,y=21000·2490.00·π2400.002=3225.51 kN 

λ¯y=A·fyNcr,y=54.30·23.53225.51=0.629>0.2 

→ 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)

Φ=0.5·[1+0.34·(0.629-0.2)+0.6292]=0.771 

χy=10.771+0.7712-0.6292=0.822 

NEdχy·A·fy/γM1=3000.822·54.30·23.5/1.0=0.286 

Results of RF-/STEEL EC3 calculation
Table 8.4 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).

Mcr=C1·π2·E·Izl2·IωIz+l2·G·Itπ2·E·Iz 

Mcr=1.13·π2·21000·8894002·47940889+4002·8100·31.40π2·21000·889=215.71 kNm 

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

λ¯LT=Wy·fyMcr=354·23.5215.71=0.621 

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:

ΦLT=0.5·1+αLT·λ¯LT-λ¯LT,0+β·λ¯LT2 

ΦLT=0.5·1+0.34·0.621-0.40+0.75·0.6212=0.682 

Limiting slenderness:

λLT,0=0.40 

Parameter (minimum value):

β=0.75 

Imperfection factor according to [1] Table 6.3:

αLT=0.34 

χLT=1ΦLT+ΦLT2-β·λ¯LT2=10.682+0.6822-0.75·0.6212=0.908 

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

χLT,mod=χLTf   mit  f=1-0.5·1-kc·1-2.0·λ¯LT-0.82 

χLT,mod=0.9080.972=0.934 

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

kc=0.94 

f=1-0.5·(1-0.94)·[1-2.0·(0.621-0.8)2]=0.972 

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:

Cmy=CmLT=0.95+0.05·αh=0.95  mit  αh=MhMs=010=0

kyy=Cmy·(1+(λ¯y-0.2)·NEdχy·NRk/γM1)Cmy·(1+0.8·NEdχy·NRk/γM1) 

kyy=0.95·(1+(0.629-0.2)·0.286)0.95·(1+0.8·0.286)=1.0671.167 

kyz=0.60·kzz=0.60·1.481=0.888 

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:

Cmz=0.90+0.01·αh=0.90  mit  αh=MhMs=010=0 

kzy=(1-0.1·λ¯zCmLT-0.25·NEdχz·NRk/γM1)(1-0.1CmLT-0.25·NEdχz·NRk/γM1) 

kzy=(1-0.1·1.0530.95-0.25·0.461)(1-0.10.95-0.25·0.461)=0.8920.934 

kzy=0.934 

kzz=Cmz·(1+(2·λ¯z-0.6)·NEdχz·NRk/γM1)Cmz·(1+1.4·NEdχz·NRk/γM1) 

kzz=0.90·(1+(2·1.053-0.6)·0.461)0.90·(1+1.4·0.461)=1.5251.481 

kzz=1.481 

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

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

NEdχy·NRk/γM1+kyy·My,EdχLT·My,Rk/γM1+kyz·Mz,EdMz,Rk/γM11 

where

My,Rk=Wpl,y·fy=354·23.5=8319 kNcm=83.19 kNm 

Mz,Rk=Wpl,z·fy=169.96·23.5=3994.1 kNcm=39.94 kNm 

3000.822·1276.05/1.0+1.067·10.00.908·83.19/1.0+0.888·7.5039.94/1.0=0.5941 

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:

NEdχz·NRk/γM1+kzy·My,EdχLT·My,Rk/γM1+kzz·Mz,EdMz,Rk/γM11 

3000.510·1276.05/1.0+0.934·10.00.908·83.19/1.0+1.481·7.5039.94/1.0=0.8631

Results of RF-/STEEL EC3 calculation
Table 8.5 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.