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.

Table 8.0 System and design loads (γ times)

Design values of static loads Nd = 300 kN qz,d = 5 kN/m Fy,d = 7.5 kN

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

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

$N_{cr,z}=\frac{21000·889.00·{\pi }^2}{400.{00}^2}=1151.60 \mathrm{kN}$

${\overline{\lambda }}_z=\sqrt{\frac{A·f_y}{N_{cr,z}}}=\sqrt{\frac{54.30·23.5}{1151.60}}=1.053>0.2$

→ 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
• ⇒ α_z = 0.49   ([1] Table 6.1)

$\Phi =0.5·[1+0.49·(1.053-0.2)+1.{053}^2]=1.263$

${\chi }_z=\frac{1}{1.263+\sqrt{1.{263}^2-1.{053}^2}}=0.510$

$\frac{N_{Ed}}{{\chi }_z·A·f_y/{\gamma }_{M1}}=\frac{300}{0.510·54.30·23.5/1.0}=0.461$

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)

$N_{cr,y}=\frac{21000·2490.00·{\pi }^2}{400.{00}^2}=3225.51 \mathrm{kN}$

${\overline{\lambda }}_y=\sqrt{\frac{A·f_y}{N_{cr,y}}}=\sqrt{\frac{54.30·23.5}{3225.51}}=0.629>0.2$

→ 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
• ⇒ α_y = 0.34   ([1] Table 6.1)

$\Phi =0.5·[1+0.34·(0.629-0.2)+0.{629}^2]=0.771$

${\chi }_y=\frac{1}{0.771+\sqrt{0.{771}^2-0.{629}^2}}=0.822$

$\frac{N_{Ed}}{{\chi }_y·A·f_y/{\gamma }_{M1}}=\frac{300}{0.822·54.30·23.5/1.0}=0.286$

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)

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).

$M_{\mathrm{cr}}=C_1·\frac{{\pi }^2·E·I_z}{{\mathcal{l}}^2}·\sqrt{\frac{I_{\omega }}{I_z}+\frac{{\mathcal{l}}^2·G·I_t}{{\pi }^2·E·I_z}}$

$M_{\mathrm{cr}}=1.13·\frac{{\pi }^2·21000·889}{{400}^2}·\sqrt{\frac{47940}{889}+\frac{{400}^2·8100·31.40}{{\pi }^2·21000·889}}=215.71 \mathrm{kNm}$

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

For the results by x-location, the program also shows the M_cr,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 M_cr,x, the program then calculates the relative slenderness ƛ_LT.

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: W_y = W_pl,y = 354.00 cm³

${\overline{\lambda }}_{\mathrm{LT}}=\sqrt{\frac{W_y·f_y}{M_{\mathrm{cr}}}}=\sqrt{\frac{354·23.5}{215.71}}=0.621$

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:

${\Phi }_{\mathrm{LT}}=0.5·\left[1+{\alpha }_{\mathrm{LT}}·\left({\overline{\lambda }}_{\mathrm{LT}}-{\overline{\lambda }}_{\mathrm{LT},0}\right)+\beta ·{\overline{\lambda }}_{\mathrm{LT}}^2\right]$

${\Phi }_{\mathrm{LT}}=0.5·\left[1+0.34·\left(0.621-0.40\right)+0.75·0.{621}^2\right]=0.682$

Limiting slenderness:

${\overline{\lambda }}_{\mathrm{LT},0}=0.40$

Parameter (minimum value):

$\beta =0.75$

Imperfection factor according to [1] Table 6.3:

${\alpha }_{\mathrm{LT}}=0.34$

${\chi }_{\mathrm{LT}}=\frac{1}{{\Phi }_{\mathrm{LT}}+\sqrt{{\Phi }_{\mathrm{LT}}^2-\beta ·{\overline{\lambda }}_{\mathrm{LT}}^2}}=\frac{1}{0.682+\sqrt{0.{682}^2-0.75·0.{621}^2}}=0.908$

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

${\chi }_{\mathrm{LT}\text{,mod}}=\frac{{\chi }_{LT}}{f} \text{mit } f=1-0.5·\left(1-k_c\right)·\left[1-2.0·\left({\overline{\lambda }}_{\mathrm{LT}}-0.8^2\right)\right]$

${\chi }_{\mathrm{LT}\text{,mod}}=\frac{0.908}{0.972}=0.934$

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

$k_c=0.94$

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

Determination according to [6] Annex B, Table B.2, for structural components susceptible to torsional deformations

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

$C_{my}=C_{mLT}=0.95+0.05·{\alpha }_h=0.95 \text{mit } {\alpha }_h=\frac{M_h}{M_s}=\frac{0}{10}=0$

$k_{yy}=C_{my}·(1+({\overline{\lambda }}_y-0.2)·\frac{N_{Ed}}{{\chi }_y·N_{Rk}/{\gamma }_{M1}})\le C_{my}·(1+0.8·\frac{N_{Ed}}{{\chi }_y·N_{Rk}/{\gamma }_{M1}})$

$k_{yy}=0.95·(1+(0.629-0.2)·0.286)\le 0.95·(1+0.8·0.286)=\underline{1.067}\le 1.167$

$k_{yz}=0.60·k_{zz}=0.60·1.481=\underline{0.888}$

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

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

$C_{mz}=0.90+0.01·{\alpha }_h=0.90 \text{mit} {\alpha }_h=\frac{M_h}{M_s}=\frac{0}{10}=0$

$k_{zy}=(1-\frac{0.1·{\overline{\lambda }}_z}{C_{mLT}-0.25}·\frac{N_{Ed}}{{\chi }_z·N_{Rk}/{\gamma }_{M1}})\ge (1-\frac{0.1}{C_{mLT}-0.25}·\frac{N_{Ed}}{{\chi }_z·N_{Rk}/{\gamma }_{M1}})$

$k_{zy}=(1-\frac{0.1·1.053}{0.95-0.25}·0.461)\ge (1-\frac{0.1}{0.95-0.25}·0.461)=\underline{0.892}\le 0.934$

$k_{zy}=\underline{0.934}$

$k_{zz}=C_{mz}·(1+(2·{\overline{\lambda }}_z-0.6)·\frac{N_{Ed}}{{\chi }_z·N_{Rk}/{\gamma }_{M1}})\le C_{mz}·(1+1.4·\frac{N_{Ed}}{{\chi }_z·N_{Rk}/{\gamma }_{M1}})$

$k_{zz}=0.90·(1+(2·1.053-0.6)·0.461)\le 0.90·(1+1.4·0.461)=\underline{1.525}\ge 1.481$

$k_{zz}=\underline{1.481 }$

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

$\frac{N_{Ed}}{{\chi }_y·N_{Rk}/{\gamma }_{M1}}+k_{yy}·\frac{M_{y,Ed}}{{\chi }_{LT}·M_{y,Rk}/{\gamma }_{M1}}+k_{yz}·\frac{M_{z,Ed}}{M_{z,Rk}/{\gamma }_{M1}}\le 1$

where

$M_{y,Rk}=W_{pl,y}·f_y=354·23.5=8319 \mathrm{kNcm}=83.19 \mathrm{kNm}$

$M_{z,Rk}=W_{pl,z}·f_y=169.96·23.5=3994.1 \mathrm{kNcm}=39.94 \mathrm{kNm}$

$\frac{300}{0.822·1276.05/1.0}+1.067·\frac{10.0}{0.908·83.19/1.0}+0.888·\frac{7.50}{39.94/1.0}=0.594\le 1$

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

$\frac{N_{Ed}}{{\chi }_z·N_{Rk}/{\gamma }_{M1}}+k_{zy}·\frac{M_{y,Ed}}{{\chi }_{LT}·M_{y,Rk}/{\gamma }_{M1}}+k_{zz}·\frac{M_{z,Ed}}{M_{z,Rk}/{\gamma }_{M1}}\le 1$

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

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)