Stability Analysis of a Column Under Axial Force and Bending
Technical Article
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 the cross-section design have already been explained in an earlier article and are therefore not discussed again.
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:
$\frac{{\mathrm N}_{\mathrm{Ed}}}{\displaystyle\frac{{\mathrm\chi}_{\mathrm z}\;\cdot\;{\mathrm N}_{\mathrm{Rk}}}{{\mathrm\gamma}_{\mathrm M1}}}\;+\;{\mathrm k}_{\mathrm{zy}\;}\cdot\frac{{\mathrm M}_{\mathrm y,\mathrm{Ed}}}{{\mathrm\chi}_{\mathrm{LT}}\;\cdot\;{\displaystyle\frac{{\mathrm M}_{\mathrm y,\mathrm{Rk}}}{{\mathrm\gamma}_{\mathrm M1}}}}\;\leq\;1$
Lateral-torsional buckling design:
$\frac{{\mathrm N}_{\mathrm{Ed}}}{\displaystyle\frac{{\mathrm\chi}_{\mathrm y}\;\cdot\;{\mathrm N}_{\mathrm{Rk}}}{{\mathrm\gamma}_{\mathrm M1}}}\;+\;{\mathrm k}_{\mathrm{yy}\;}\cdot\frac{{\mathrm M}_{\mathrm y,\mathrm{Ed}}}{{\mathrm\chi}_{\mathrm{LT}}\;\cdot\;{\displaystyle\frac{{\mathrm M}_{\mathrm y,\mathrm{Rk}}}{{\mathrm\gamma}_{\mathrm M1}}}}\;\leq\;1$
Flexural Buckling Design About the Minor Axis
$\frac{{\mathrm N}_{\mathrm{Ed}}}{\displaystyle\frac{{\mathrm\chi}_{\mathrm z}\;\cdot\;{\mathrm N}_{\mathrm{Rk}}}{{\mathrm\gamma}_{\mathrm M1}}}\;+\;{\mathrm k}_{\mathrm{zy}\;}\cdot\frac{{\mathrm M}_{\mathrm y,\mathrm{Ed}}}{{\mathrm\chi}_{\mathrm{LT}}\;\cdot\;{\displaystyle\frac{{\mathrm M}_{\mathrm y,\mathrm{Rk}}}{{\mathrm\gamma}_{\mathrm M1}}}}\;\leq\;1$
The effective length of the hinged column is Lcr = 6.50 m.
According to EN 1993-1-1, 6.3.1.2:
$\mathrm\chi\;=\;\frac1{\mathrm\phi\;+\;\sqrt{\mathrm\phi^{2\;}-\;\overline{\mathrm\lambda}^2}}\;\leq\;1\\\mathrm\phi\;=\;0.5\;\cdot\;\left[1\;+\;\mathrm\alpha\;\cdot\;\left(\overline{\mathrm\lambda}\;-\;0.2\right)\;+\;\overline{\mathrm\lambda}^2\;\right]\\{\overline{\mathrm\lambda}}_{\mathrm z}\;=\;\sqrt{\frac{\mathrm A\;\cdot\;{\mathrm f}_{\mathrm y}}{{\mathrm N}_{\mathrm{cr},\mathrm z}}}\\{\mathrm N}_{\mathrm{cr},\mathrm z}\;=\;\frac{\mathrm\pi^2\;\cdot\;\mathrm E\;\cdot\;{\mathrm I}_{\mathrm z}}{\mathrm l^2}\;=\;\frac{\mathrm\pi^2\;\cdot\;21,000\;\mathrm{kN}/\mathrm{cm}^2\;\cdot\;10.140\;\mathrm{cm}^4}{\left(650\;\mathrm{cm}\right)^2}\;=\;4,974.28\;\mathrm{kN}\\{\overline{\mathrm\lambda}}_{\mathrm z}\;=\;\sqrt{\frac{180.6\;\mathrm{cm}^2\;\cdot23.5\;\mathrm{kN}/\mathrm{cm}^2}{4,974.28\;\mathrm{kN}}}\;=\;0.924$
Selection of buckling curve according to Table 6.2:
$\frac{\mathrm h}{\mathrm b}\;=\;\frac{360\;\mathrm{mm}}{300\;\mathrm{mm}}\;=\;1.2\;\leq\;1.2\\{\mathrm t}_{\mathrm f}\;=\;22.5\;\mathrm{mm}\;\leq\;100\;\mathrm{mm}$
Instability perpendicular to the z-axis: Buckling stress curve BSCz: c
Table 6.1 shows the imperfection factor α = 0.49.
$\mathrm\phi\;=\;0.5\;\cdot\;\left[1\;+\;0.49\;\cdot\;\left(0.924\;-\;0.2\right)\;+\;0.924^2\right]\;=\;1.104\\{\mathrm\chi}_{\mathrm z}\;=\;\frac1{1.104\;+\;\sqrt{1.104^2\;-\;0.924^2}}\;=\;0.585\;\leq\;1.0$
For I, H, and rectangular hollow cross-sections that are only subjected to compression and bending, the coefficient kzy = 0 may be assumed.
This results in the design as follows:
$\frac{{\mathrm N}_{\mathrm{Ed}}}{\displaystyle\frac{{\mathrm\chi}_{\mathrm z}\;\cdot\;{\mathrm N}_{\mathrm{Rk}}}{{\mathrm\gamma}_{\mathrm M1}}}\;\leq\;1\\{\mathrm N}_{\mathrm{Rk}\;}=\;\mathrm A\;\cdot\;{\mathrm f}_{\mathrm y}\;=\;180.60\;\mathrm{cm}^2\;\cdot\;23.5\;\frac{\mathrm{kN}}{\mathrm{cm}^2}\;=\;\;4,244.1\;\mathrm{kN}\\\frac{2,000\;\mathrm{kN}}{\displaystyle\frac{0.585\;\cdot\;4,244.1\;\mathrm{kN}}1}\;=\;0.81\;\leq\;1$
→ Design is fulfilled.
Lateral-Torsional Buckling Design
The effective length of the hinged column is Lcr = 6.50 m.
According to EN 1993-1-1, 6.3.1.2:
$\mathrm\chi\;=\;\frac1{\mathrm\phi\;+\;\sqrt{\mathrm\phi^{2\;}-\;\overline{\mathrm\lambda}^2}}\;\leq\;1\\\mathrm\phi\;=\;0.5\;\cdot\;\left[1\;+\;\mathrm\alpha\;\cdot\;\left(\overline{\mathrm\lambda}\;-\;0.2\right)\;+\;\overline{\mathrm\lambda}^2\;\right]\\{\overline{\mathrm\lambda}}_{\mathrm z}\;=\;\sqrt{\frac{\mathrm A\;\cdot\;{\mathrm f}_{\mathrm y}}{{\mathrm N}_{\mathrm{cr},\mathrm y}}}\\{\mathrm N}_{\mathrm{cr},\mathrm y}\;=\;\frac{\mathrm\pi^2\;\cdot\;\mathrm E\;\cdot\;{\mathrm I}_{\mathrm y}}{\mathrm l^2}\;=\;\frac{\mathrm\pi^2\;\cdot\;21,000\;\mathrm{kN}/\mathrm{cm}^2\;\cdot\;43,190\;\mathrm{cm}^4}{\left(650\;\mathrm{cm}\right)^2}\;=\;21,187.3\;\mathrm{kN}\\{\overline{\mathrm\lambda}}_{\mathrm z}\;=\;\sqrt{\frac{180.6\;\mathrm{cm}^2\;\cdot23.5\;\mathrm{kN}/\mathrm{cm}^2}{21,187.3\;\mathrm{kN}}}\;=\;0.924$
Effective length according to Table 6.2:
$\frac{\mathrm h}{\mathrm b}\;=\;\frac{360\;\mathrm{mm}}{300\;\mathrm{mm}}\;=\;1.2\;\leq\;1.2\\{\mathrm t}_{\mathrm f}\;=\;22.5\;\mathrm{mm}\;\leq\;100\;\mathrm{mm}$
Instability perpendicular to the y-axis: Buckling stress curve BSCz: b
Table 6.1 shows the imperfection factor α = 0.34.
$\mathrm\phi\;=\;0.5\;\cdot\;\left[1\;+\;0.34\;\cdot\;\left(0.448\;-\;0.2\right)\;+\;0.448^2\right]\;=\;0.642\\{\mathrm\chi}_{\mathrm y}\;=\;\frac1{0.642\;+\;\sqrt{0.642^2\;-\;0.448^2}}\;=\;0.907\;\leq\;1.0$
Interaction factor according to Annex B, Table B1:
${\mathrm k}_{\mathrm{yy}}\;=\;{\mathrm C}_{\mathrm{my}}\;\cdot\;\left(1\;+\;\left({\overline{\mathrm\lambda}}_{\mathrm y}\;-\;0.2\right)\;\cdot\;\frac{{\mathrm N}_{\mathrm{Ed}}}{{\mathrm\chi}_{\mathrm y}\;\cdot\;{\displaystyle\frac{{\mathrm N}_{\mathrm{Rk}}}{{\mathrm\gamma}_{\mathrm M1}}}}\right)\;\leq\;{\mathrm C}_{\mathrm{my}}\;\cdot\;\left(1\;+\;0.8\;\cdot\;\frac{{\mathrm N}_{\mathrm{Ed}}}{{\mathrm\chi}_{\mathrm y}\;\cdot\;{\displaystyle\frac{{\mathrm N}_{\mathrm{Rk}}}{{\mathrm\gamma}_{\mathrm M1}}}}\right)$
Equivalent moment factor Cmy according to Table B.3:
${\mathrm\alpha}_{\mathrm h}\;=\;\frac{{\mathrm M}_{\mathrm h}}{{\mathrm M}_{\mathrm s}}\;=\;\frac{0.00\;\mathrm{kNm}}{79.22\;\mathrm{kNm}}\;=\;0\\{\mathrm C}_{\mathrm{my}}\;=\;0.95\;+\;0.05\;\cdot\;{\mathrm\alpha}_{\mathrm h}\;=\;0.95\\{\overline{\mathrm\lambda}}_{\mathrm y}\;=\;0.448\\{\mathrm N}_{\mathrm{Rk}}\;=\;\mathrm A\;\cdot\;{\mathrm f}_{\mathrm y}\;=\;180.60\;\mathrm{cm}^2\;\cdot\;23.5\;\frac{\mathrm{kN}}{\mathrm{cm}^2\;\;}\;=\;4,244.1\;\mathrm{kN}\\{\mathrm k}_{\mathrm{yy}}\;=\;0.95\;\cdot\;\left(1\;+\;\left(0.448\;-\;0.2\right)\;\cdot\;\frac{2,000\;\mathrm{kN}}{0.907\;\cdot\;{\displaystyle\frac{4,244.10\;\mathrm{kN}}{1.0}}}\right)\;=\;1.07\\{\mathrm k}_{\mathrm{yy},\max}\;=\;0.95\;\cdot\;\left(1\;+\;0.8\;\cdot\;\frac{2,000\;\mathrm{kN}}{0.907\;\cdot\;{\displaystyle\frac{4,244.10\;\mathrm{kN}}{1.0}}}\right)\;=\;1.34\\1.07\;<\;1.34$
According to EN 1993-1-1, 6.3.2.3:
${\mathrm\chi}_{\mathrm{LT}}\;=\;\frac1{{\mathrm\phi}_{\mathrm{LT}}\;+\;\sqrt{{\mathrm\phi}_{\mathrm{LT}}^2\;-\;\mathrm\beta\;\cdot\;{\overline{\mathrm\lambda}}_{\mathrm{LT}}^2}}\\{\mathrm\phi}_{\mathrm{LT}}\;=\;0.5\;\cdot\;\left[1\;+\;{\mathrm\alpha}_{\mathrm{LT}}\;\cdot\;\left({\overline{\mathrm\lambda}}_{\mathrm{LT}}\;-\;{\overline{\mathrm\lambda}}_{\mathrm{LT}0}\right)\;+\;\mathrm\beta\;\cdot\;{\overline{\mathrm\lambda}}_{\mathrm{LT}}^2\right]$
According to EN 1993-1-1, Tab. 6.5:
$\frac{\mathrm h}{\mathrm b}\;=\;\frac{360\;\mathrm{mm}}{300\;\mathrm{mm}}\;=\;1.20\;<\;2$ → Buckling stress curve BSCLT: b
According to EN 1993-1-1, Tab. 6.3:
${\mathrm\alpha}_{\mathrm{LT}}\;=\;0.34\\\mathrm\beta\;=\;0.75\\{\mathrm\lambda}_{\mathrm{LT}0}\;=\;0.40\\{\mathrm M}_{\mathrm{cr}}\;=\;{\mathrm C}_1\;\cdot\;\frac{\mathrm\pi^2\;\cdot\;\mathrm E\;\cdot\;{\mathrm I}_{\mathrm z}}{\left(\mathrm k\;\cdot\;\mathrm L\right)^2}\;\cdot\;\left(\sqrt{\left(\frac{\mathrm k}{{\mathrm k}_{\mathrm W}}\right)\;\cdot\;\frac{{\mathrm I}_{\mathrm W}}{{\mathrm I}_{\mathrm z}}\;+\;\frac{\left(\mathrm k\;\cdot\;\mathrm L\right)^2\;\cdot\;\mathrm G\;\cdot\;{\mathrm I}_{\mathrm t}}{\mathrm\pi^2\;\cdot\;\mathrm E\;\cdot\;{\mathrm I}_{\mathrm z}}\;+\left({\mathrm C}_2\;\cdot\;{\mathrm z}_{\mathrm g}\right)^2\;}\;-\;{\mathrm C}_2\;\cdot\;{\mathrm z}_{\mathrm g}\;\right)\\\mathrm k\;=\;1.0\\{\mathrm k}_{\mathrm w}\;=\;1.0$
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 zg = 18 cm.
${\mathrm M}_{\mathrm{cr}}\;=\;1.127\;\cdot\;\frac{\mathrm\pi^2\;\cdot\;21,000\;{\displaystyle\frac{\mathrm{kN}}{\mathrm{cm}^2}}\;\cdot\;10,140\;\mathrm{cm}^4}{\left(1\;\cdot\;650\;\mathrm{cm}\right)^2}\;\cdot\;\left(\sqrt{\left(\frac11\right)\;\cdot\;\frac{2,883,000\;\mathrm{cm}^6}{10,140\;\mathrm{cm}^4}\;+\;\frac{\left(1.0\;\cdot\;650\;\mathrm{cm}\right)^2\;\cdot\;8,076.92\;{\displaystyle\frac{\mathrm{kN}}{\mathrm{cm}^2}}\;\cdot\;292.5\;\mathrm{cm}^4}{\mathrm\pi^2\;\cdot\;21,000\;{\displaystyle\frac{\mathrm{kN}}{\mathrm{cm}^2}}\;\cdot\;10,140\;\mathrm{cm}^4}\;+\;\left(0.454\;\cdot\;18\;\mathrm{cm}\right)^2\;}\;-\;0.454\;\cdot\;18\;\mathrm{cm}\;\right)\\{\mathrm M}_{\mathrm{cr}}\;=\;115,310\;\mathrm{kNcm}\;=\;1,153.10\;\mathrm{kNm}\\{\overline{\mathrm\lambda}}_{\mathrm{LT}}\;=\;\sqrt{\frac{{\mathrm W}_{\mathrm{pl},\mathrm y}\;\cdot\;{\mathrm f}_{\mathrm y}}{{\mathrm M}_{\mathrm{cr}}}}\;=\;\sqrt{\frac{2,683\;\mathrm{cm}^3\;\cdot\;23.5\;{\displaystyle\frac{\mathrm{kN}}{\mathrm{cm}^2}}}{115,310\;\mathrm{kNcm}}}\;=\;0.739\\{\mathrm\phi}_{\mathrm{LT}}\;=\;0.5\;\cdot\;\left[1\;+\;0.34\;\cdot\;\left(0.739\;-\;0.4\right)\;+\;0.75\;\cdot\;0.739^2\right]\;=\;0.762\\{\mathrm\chi}_{\mathrm{LT}}\;=\;\frac1{0.762\;+\;\sqrt{0.762^2\;-\;0.75\;\cdot\;0.739^2}}\;=\;0.85\;<\;1$
According to EN 1993-1-1, Tab. 6.7:
${\mathrm M}_{\mathrm y,\mathrm{Rk}}\;=\;{\mathrm f}_{\mathrm y}\;\cdot\;{\mathrm W}_{\mathrm{pl},\mathrm y}\;=\;23.5\;\frac{\mathrm{kN}}{\mathrm{cm}^{2\;}}\;\cdot\;2,683\;\mathrm{cm}^3\;=\;63,050.5\;\mathrm{kNcm}\;=\;630.51\;\mathrm{kNm}$
Buckling design about the major axis:
$\frac{{\mathrm N}_{\mathrm{Ed}}}{\displaystyle\frac{{\mathrm\chi}_{\mathrm y}\;\cdot\;{\mathrm N}_{\mathrm{Rk}}}{{\mathrm\gamma}_{\mathrm M1}}}\;+\;{\mathrm k}_{\mathrm{yy}\;}\cdot\;\frac{{\mathrm M}_{\mathrm y,\mathrm{Ed}}}{{\mathrm\chi}_{\mathrm{LT}}\;\cdot\;{\displaystyle\frac{{\mathrm M}_{\mathrm y,\mathrm{Rk}}}{{\mathrm\gamma}_{\mathrm M1}}}}\;\leq\;1\\\frac{2,000\;\mathrm{kN}}{\displaystyle\frac{0.907\;\cdot\;4,244,10\;\mathrm{kN}}{1.0}}\;+\;1.072\;\cdot\;\frac{79.22\;\mathrm{kNm}}{0.85\;\cdot\;{\displaystyle\frac{630.51\;\mathrm{kNm}}{1.0}}}\;=\;0.67\;\leq\;1$
Buckling design about the minor axis:
$\frac{{\mathrm N}_{\mathrm{Ed}}}{\displaystyle\frac{{\mathrm\chi}_{\mathrm z}\;\cdot\;{\mathrm N}_{\mathrm{Rk}}}{{\mathrm\gamma}_{\mathrm M1}}}\;+\;{\mathrm k}_{\mathrm{zy}\;}\cdot\;\frac{{\mathrm M}_{\mathrm y,\mathrm{Ed}}}{{\mathrm\chi}_{\mathrm{LT}}\;\cdot\;{\displaystyle\frac{{\mathrm M}_{\mathrm y,\mathrm{Rk}}}{{\mathrm\gamma}_{\mathrm M1}}}}\;\leq\;1\\\frac{2,000\;\mathrm{kN}}{\displaystyle\frac{0.585\;\cdot\;4,244,10\;\mathrm{kN}}{1.0}}\;+\;0.894\;\cdot\;\frac{79.22\;\mathrm{kNm}}{0.85\;\cdot\;{\displaystyle\frac{630.51\;\mathrm{kNm}}{1.0}}}\;=\;0.93\;\leq\;1$
→ Checks fulfilled.
Author
Dipl.-Ing. (BA) Sandy Matula
Customer Support
Ms. Matula provides technical support for our customers.
Keywords
Design Stability Stability analysis of a column Hinged column Axial force Bending Flexural buckling
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