10465x

2020-02-26

Stability Analysis of Column Under Axial Force and Bending

In this technical article, a hinged column with a centrally acting axial force and a line load acting on the strong axis will be designed by means of the RF-/STEEL EC3 add-on module according to EN 1993-1-1.

The system assumptions, loadings, internal forces, and cross-section design were explained in an earlier article and are, therefore, not discussed again.

System

Design Under Axial Force and Bending Moment According to EN 1993-1-1, 6.3.3 [1]

Components subjected to bending and compression usually have to fulfill the following requirements.

Flexural buckling design:

\frac{N_{Ed}}{\frac{χ_{z} \cdot N_{Rk}}{γ_{M 1}}} k_{zy} \cdot \frac{M_{y, Ed}}{χ_{LT} \cdot \frac{M_{y, Rk}}{γ_{M 1}}} \leq 1

Formula 1

Lateral-torsional buckling design:

\frac{N_{Ed}}{\frac{χ_{y} \cdot N_{Rk}}{γ_{M 1}}} k_{yy} \cdot \frac{M_{y, Ed}}{χ_{LT} \cdot \frac{M_{y, Rk}}{γ_{M 1}}} \leq 1

Formula 2

Flexural Buckling Design Around the Minor Axis

\frac{N_{Ed}}{\frac{χ_{z} \cdot N_{Rk}}{γ_{M 1}}} k_{zy} \cdot \frac{M_{y, Ed}}{χ_{LT} \cdot \frac{M_{y, Rk}}{γ_{M 1}}} \leq 1

Formula 1

The effective length of the hinged column is L_cr = 6.50 m.

According to EN 1993-1-1, 6.3.1.2:

χ = \frac{1}{ϕ \sqrt{ϕ^{2} - {\bar{λ}}^{2}}} \leq 1

ϕ = 0.5 \cdot [1 α \cdot (\bar{λ} - 0.2) {\bar{λ}}^{2}]

{\bar{λ}}_{z} = \sqrt{\frac{A \cdot f_{y}}{N_{cr, z}}}

N_{cr, z} = \frac{π^{2} \cdot E \cdot I_{z}}{l^{2}} = \frac{π^{2} \cdot 21, 000 kN / {cm}^{2} \cdot 10.140 {cm}^{4}}{{(650 cm)}^{2}} = 4, 974.28 kN

{\bar{λ}}_{z} = \sqrt{\frac{180.6 {cm}^{2} \cdot 23.5 kN / {cm}^{2}}{4, 974.28 kN}} = 0.924

Formula 3

Selection of a buckling curve according to Table 6.2:

\frac{h}{b} = \frac{360 mm}{300 mm} = 1.2 \leq 1.2

t_{f} = 22.5 mm \leq 100 mm

Formula 4

Instability perpendicular to the z-axis: Buckling stress curve BSC_z: c

Table 6.1 shows the imperfection factor α = 0.49.

ϕ = 0.5 \cdot [1 0.49 \cdot (0.924 - 0.2) 0 . 924^{2}] = 1.104

χ_{z} = \frac{1}{1.104 \sqrt{1 . 104^{2} - 0 . 924^{2}}} = 0.585 \leq 1.0

Formula 5

For I, H, and rectangular hollow cross-sections that are only subjected to compression and bending, the coefficient k_zy = 0 may be assumed.

This results in the design as follows:

\frac{N_{Ed}}{\frac{χ_{z} \cdot N_{Rk}}{γ_{M 1}}} \leq 1

N_{Rk} = A \cdot f_{y} = 180.60 {cm}^{2} \cdot 23.5 \frac{kN}{{cm}^{2}} = 4, 244.1 kN

\frac{2, 000 kN}{\frac{0.585 \cdot 4, 244.1 kN}{1}} = 0.81 \leq 1

Formula 6

→ Design is fulfilled.

Lateral-Torsional Buckling Design

The effective length of the hinged column is L_cr = 6.50 m.

According to EN 1993-1-1, 6.3.1.2:

χ = \frac{1}{ϕ \sqrt{ϕ^{2} - {\bar{λ}}^{2}}} \leq 1

ϕ = 0.5 \cdot [1 α \cdot (\bar{λ} - 0.2) {\bar{λ}}^{2}]

{\bar{λ}}_{z} = \sqrt{\frac{A \cdot f_{y}}{N_{cr, y}}}

N_{cr, y} = \frac{π^{2} \cdot E \cdot I_{y}}{l^{2}} = \frac{π^{2} \cdot 21, 000 kN / {cm}^{2} \cdot 43, 190 {cm}^{4}}{{(650 cm)}^{2}} = 21, 187.3 kN

{\bar{λ}}_{z} = \sqrt{\frac{180.6 {cm}^{2} \cdot 23.5 kN / {cm}^{2}}{21, 187.3 kN}} = 0.924

Formula 7

Effective length according to Table 6.2:

\frac{h}{b} = \frac{360 mm}{300 mm} = 1.2 \leq 1.2

t_{f} = 22.5 mm \leq 100 mm

Formula 4

Instability perpendicular to the y-axis: Buckling stress curve BSC_z: b
Table 6.1 shows the imperfection factor α = 0.34.

ϕ = 0.5 \cdot [1 0.34 \cdot (0.448 - 0.2) 0 . 448^{2}] = 0.642

χ_{y} = \frac{1}{0.642 \sqrt{0 . 642^{2} - 0 . 448^{2}}} = 0.907 \leq 1.0

Formula 8

Interaction factor according to Annex B, Table B1:

k_{yy} = C_{my} \cdot (1 ({\bar{λ}}_{y} - 0.2) \cdot \frac{N_{Ed}}{χ_{y} \cdot \frac{N_{Rk}}{γ_{M 1}}}) \leq C_{my} \cdot (1 0.8 \cdot \frac{N_{Ed}}{χ_{y} \cdot \frac{N_{Rk}}{γ_{M 1}}})

Formula 9

Equivalent moment factor C_my according to Table B.3:

α_{h} = \frac{M_{h}}{M_{s}} = \frac{0, 00 kNm}{79, 22 kNm} = 0

C_{my} = 0, 95 + 0, 05 \cdot α_{h} = 0, 95

{\bar{λ}}_{y} = 0, 448

N_{Rk} = A \cdot f_{y} = 180, 60 {cm}^{2} \cdot 23, 5 \frac{kN}{{cm}^{2}} = 4.244, 1 kN

k_{yy} = 0, 95 \cdot (1 (0, 448 - 0, 2) \cdot \frac{2.000 kN}{0, 907 \cdot \frac{4.244, 10 kN}{1, 0}}) = 1, 07

k_{yy, \max} = 0, 95 \cdot (1 0, 8 \cdot \frac{2.000 kN}{0, 907 \cdot \frac{4.244, 10 kN}{1, 0}}) = 1, 34

1, 07 < 1, 34

Formula 10

According to EN 1993-1-1, 6.3.2.3:

χ_{LT} = \frac{1}{ϕ_{LT} \sqrt{ϕ_{LT}^{2} - β \cdot {\bar{λ}}_{LT}^{2}}}

ϕ_{LT} = 0.5 \cdot [1 α_{LT} \cdot ({\bar{λ}}_{LT} - {\bar{λ}}_{LT 0}) β \cdot {\bar{λ}}_{LT}^{2}]

Formula 11

According to EN 1993-1-1, Tab. 6.5:

\frac{h}{b} = \frac{360 mm}{300 mm} = 1.20 < 2

Formula 12

→ Lateral-torsional buckling curve BC_LT: b

According to EN 1993-1-1, Tab. 6.3:

α_{LT} = 0.34

β = 0.75

λ_{LT 0} = 0.40

M_{cr} = C_{1} \cdot \frac{π^{2} \cdot E \cdot I_{z}}{{(k \cdot L)}^{2}} \cdot (\sqrt{(\frac{k}{k_{W}}) \cdot \frac{I_{W}}{I_{z}} \frac{{(k \cdot L)}^{2} \cdot G \cdot I_{t}}{π^{2} \cdot E \cdot I_{z}} {(C_{2} \cdot z_{g})}^{2}} - C_{2} \cdot z_{g})

k = 1.0

k_{w} = 1.0

Formula 13

C₁ and C₂ from Table 3.2 NCCI: Elastic critical torsional buckling moment [5] (compatible additional documents to Eurocode 3):
C₁ = 1.127
C₂ = 0.454

Distance from load application point to shear center z_g = 18 cm.

M_{cr} = 1.127 \cdot \frac{π^{2} \cdot 21, 000 \frac{kN}{{cm}^{2}} \cdot 10, 140 {cm}^{4}}{{(1 \cdot 650 cm)}^{2}} \cdot (\sqrt{(\frac{1}{1}) \cdot \frac{2, 883, 000 {cm}^{6}}{10, 140 {cm}^{4}} \frac{{(1.0 \cdot 650 cm)}^{2} \cdot 8, 076.92 \frac{kN}{{cm}^{2}} \cdot 292.5 {cm}^{4}}{π^{2} \cdot 21, 000 \frac{kN}{{cm}^{2}} \cdot 10, 140 {cm}^{4}} {(0.454 \cdot 18 cm)}^{2}} - 0.454 \cdot 18 cm)

M_{cr} = 115, 310 kNcm = 1, 153.10 kNm

{\bar{λ}}_{LT} = \sqrt{\frac{W_{pl, y} \cdot f_{y}}{M_{cr}}} = \sqrt{\frac{2, 683 {cm}^{3} \cdot 23.5 \frac{kN}{{cm}^{2}}}{115, 310 kNcm}} = 0.739

ϕ_{LT} = 0.5 \cdot [1 0.34 \cdot (0.739 - 0.4) 0.75 \cdot 0 . 739^{2}] = 0.762

χ_{LT} = \frac{1}{0.762 \sqrt{0 . 762^{2} - 0.75 \cdot 0 . 739^{2}}} = 0.85 < 1

Formula 14

According to EN 1993-1-1, Tab. 6.7:

M_{y, Rk} = f_{y} \cdot W_{pl, y} = 23.5 \frac{kN}{{cm}^{2}} \cdot 2, 683 {cm}^{3} = 63, 050.5 kNcm = 630.51 kNm

Formula 15

Buckling design around the major axis:

\frac{N_{Ed}}{\frac{χ_{y} \cdot N_{Rk}}{γ_{M 1}}} k_{yy} \cdot \frac{M_{y, Ed}}{χ_{LT} \cdot \frac{M_{y, Rk}}{γ_{M 1}}} \leq 1

\frac{2, 000 kN}{\frac{0.907 \cdot 4, 244, 10 kN}{1.0}} 1.072 \cdot \frac{79.22 kNm}{0.85 \cdot \frac{630.51 kNm}{1.0}} = 0.67 \leq 1

Formula 16

Buckling design around the minor axis:

\frac{N_{Ed}}{\frac{χ_{z} \cdot N_{Rk}}{γ_{M 1}}} k_{zy} \cdot \frac{M_{y, Ed}}{χ_{LT} \cdot \frac{M_{y, Rk}}{γ_{M 1}}} \leq 1

\frac{2, 000 kN}{\frac{0.585 \cdot 4, 244, 10 kN}{1.0}} 0.894 \cdot \frac{79.22 kNm}{0.85 \cdot \frac{630.51 kNm}{1.0}} = 0.93 \leq 1

Formula 17

→ Checks fulfilled.

Knowledge Base

KB 001622 | Stability Analysis of Column Under Axial Force and Bending

Author

Ms. Matula provides technical support for our customers.

Links

Structural Analysis Software for Steel Structures

References

European Committee for Standardization (CEN). (2010). Eurocode 3: Design of Steel Structures – Part 1‑1: General Rules and Rules for Buildings, EN 1993‑1‑1:2010‑12. Berlin: Beuth Verlag GmbH.
Handbuch RF-/STAHL EC3. Tiefenbach: Dlubal Software, Juni 2020.
Albert, A. (2018). Schneider – Bautabellen für Ingenieure mit Berechnungshinweisen und Beispielen (23rd ed.). Cologne: Bundesanzeiger.
Kuhlmann, U.; Feldmann, M.; Lindner, J.; Müller, C.; & Stroetmann, R. (2014). Eurocode 3 Bemessung und Konstruktion von Stahlbauten – Band 1: General Rules and Building Construction - DIN EN 1993-1-1 with National Annex, Commentary and Examples. Berlin: Beuth.
Bureau, A.: NCCI: Elastisches kritisches Biegedrillknickmoment. Aachen: RWTH, 2010

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I am performing a stability analysis of a beam for lateral-torsional buckling. Why is the modified reduction factor χ_LT,mod used in the design according to DIN EN 1993‑1‑1, 6.3.3 Method 2? Is it possible to deactivate this?

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