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Keywords: lateral torsional buckling, tapered beam, cantilever beam, buckling curve, design rule, Eurocode 3.

Summary

Adapting the height of a beam to the distribution of the internal forces results in material saving and can lead to an economic advantage. The resulting slender constructions must be verified against lateral torsional buckling. No simple design rules are available for tapered beams in the Eurocode 3 documents. The results presented in this paper show that the clause "General method for lateral torsional buckling of frames" may be used as a safe design criterion for tapered cantilevers loaded in their tip section with a point load at the upper flange.

Simulations are made with Abaqus [1]. Tapered beams are welded and residual stresses in accordance to [2] and [3] are included in the model. A simulation step was added to attain the stress free ends of the beam before adding an end plate at the higher end and a stiffener at the tip. A parabolic imperfection with bow is assumed. Material and geometric non linearities are accounted for whenever necessary so that a realistic beam behaviour is simulated.

The width-to-thickness ratios of the web and the flanges correspond to the limits of class 2 and class 3 sections. Two ratios $h_{mean}/b$ were simulated. A taper factor $\omega$ is defined based on the difference in height between the clamped end and the tip. The simulated $\omega$ -values range from 0 to .

The same dense mesh is used for all the simulations. This ensures that the influence of local buckling modes on the elastic critical load is automatically taken into account. Three magnitudes of the load are defined: the plastic load $P_{pl}$ , the elastic critical buckling load $P_{cr}$ and the ultimate load of the beam $P_{u}$ . They are used to redefine the dimensionless slenderness $\overline \lambda_{LT} = \sqrt{P_{pl}/P_{cr}}$ and the reduction factor $\chi_{LT}= P_{u}/P_{pl}$ .

The results are compared to the 2 sets of buckling curves available in the final draft version of Eurocode 3 part 1-1 (date feb. 2002) [4]; the standard set (STD-set) and the alternative set (ALT-set). For clarity both sets are compared to each other.

The values of $P_{pl}$ result from a calculation without taking into account geometric nonlinear behaviour. These values are compared with a theoretical value that is calculated under the assumption that a plastic hinge occurs at the end section. For low taper factors the beams seem to be stronger than theoretically expected. This is explained by the influence of the boundary conditions used to model the end plate. As lateral contraction is not free to occur the biaxial stress state results in effective yielding at longitudinal stresses higher than $f_{y}$ . The results show that for higher taper factors the plastic hinge effectively moves away from the clamped end.

The lowest positive eigenvalue of a linearised bifurcation analysis is used as $P_{cr}$ . The ratio between the global and local buckling load is used as an indicator for possible mode interaction. Beams with slender flanges seem to be most susceptible.

The value of $P_{u}$ is the maximum load attained during the simulation. No additional conditions - such as e.g. a limit on the lateral displacements - is used to accept or to reject this value.

The lower bound of all the simulation results is compared to the available buckling curves. It is shown that within the ALT-set curve c is the best fit and is always on the safe side. From the STD-set curve b is to be used. The results indicate that there is no need to make a distinction between different values of $h_{mean}/b$ . The curves may be used independent of the slenderness of the flanges and the web. This is demonstrated on plots using a relative distance between the results and the proposed buckling curve. The distance is nearly the same for all simulated taper factors.

References

1: Hibbitt, Karlsson & Sorensen, "Abaqus-manuels", http://www.abaqus.com
2: ECCS TC8 TWG 8.2, "Ultimate Limit State Calculations of Sway Frames with Rigid Joints, Publication 33, First Edition" Brussels, 1984.
3: ECCS TC8, "Reports from the ad-hoc-working group for interaction formulae", 2000
4: CEN, "prEN 1993-1-1: 2002 Eurocode 3 : Design of steel structures, Part 1-1 : General structural rules", Brussels, 2002.

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Front Page Browse CCP CSETS CTR IJRT Other Authors Search Purchase Guide FAQ Contact us	Civil-Comp Proceedings ISSN 1759-3433 CCP: 77 PROCEEDINGS OF THE NINTH INTERNATIONAL CONFERENCE ON CIVIL AND STRUCTURAL ENGINEERING COMPUTING Edited by: B.H.V. Topping Paper 46 Finite Element Simulations of Lateral Torsional Buckling of Tapered Cantilever Beams P. Buffel, G. Lagae, R. Van Impe, W. Vanlaere and M. De Beule Laboratory for Research on Structural Models, Ghent University, Belgium doi:10.4203/ccp.77.46 purchase the full-text of this paper Full Bibliographic Reference for this paper P. Buffel, G. Lagae, R. Van Impe, W. Vanlaere, M. De Beule, "Finite Element Simulations of Lateral Torsional Buckling of Tapered Cantilever Beams", in B.H.V. Topping, (Editor), "Proceedings of the Ninth International Conference on Civil and Structural Engineering Computing", Civil-Comp Press, Stirlingshire, UK, Paper 46, 2003. doi:10.4203/ccp.77.46 Keywords: lateral torsional buckling, tapered beam, cantilever beam, buckling curve, design rule, Eurocode 3. Summary Adapting the height of a beam to the distribution of the internal forces results in material saving and can lead to an economic advantage. The resulting slender constructions must be verified against lateral torsional buckling. No simple design rules are available for tapered beams in the Eurocode 3 documents. The results presented in this paper show that the clause "General method for lateral torsional buckling of frames" may be used as a safe design criterion for tapered cantilevers loaded in their tip section with a point load at the upper flange. Simulations are made with Abaqus [1]. Tapered beams are welded and residual stresses in accordance to [2] and [3] are included in the model. A simulation step was added to attain the stress free ends of the beam before adding an end plate at the higher end and a stiffener at the tip. A parabolic imperfection with bow is assumed. Material and geometric non linearities are accounted for whenever necessary so that a realistic beam behaviour is simulated. The width-to-thickness ratios of the web and the flanges correspond to the limits of class 2 and class 3 sections. Two ratios $h_{mean}/b$ were simulated. A taper factor $\omega$ is defined based on the difference in height between the clamped end and the tip. The simulated $\omega$ -values range from 0 to . The same dense mesh is used for all the simulations. This ensures that the influence of local buckling modes on the elastic critical load is automatically taken into account. Three magnitudes of the load are defined: the plastic load $P_{pl}$ , the elastic critical buckling load $P_{cr}$ and the ultimate load of the beam $P_{u}$ . They are used to redefine the dimensionless slenderness $\overline \lambda_{LT} = \sqrt{P_{pl}/P_{cr}}$ and the reduction factor $\chi_{LT}= P_{u}/P_{pl}$ . The results are compared to the 2 sets of buckling curves available in the final draft version of Eurocode 3 part 1-1 (date feb. 2002) [4]; the standard set (STD-set) and the alternative set (ALT-set). For clarity both sets are compared to each other. The values of $P_{pl}$ result from a calculation without taking into account geometric nonlinear behaviour. These values are compared with a theoretical value that is calculated under the assumption that a plastic hinge occurs at the end section. For low taper factors the beams seem to be stronger than theoretically expected. This is explained by the influence of the boundary conditions used to model the end plate. As lateral contraction is not free to occur the biaxial stress state results in effective yielding at longitudinal stresses higher than $f_{y}$ . The results show that for higher taper factors the plastic hinge effectively moves away from the clamped end. The lowest positive eigenvalue of a linearised bifurcation analysis is used as $P_{cr}$ . The ratio between the global and local buckling load is used as an indicator for possible mode interaction. Beams with slender flanges seem to be most susceptible. The value of $P_{u}$ is the maximum load attained during the simulation. No additional conditions - such as e.g. a limit on the lateral displacements - is used to accept or to reject this value. The lower bound of all the simulation results is compared to the available buckling curves. It is shown that within the ALT-set curve c is the best fit and is always on the safe side. From the STD-set curve b is to be used. The results indicate that there is no need to make a distinction between different values of $h_{mean}/b$ . The curves may be used independent of the slenderness of the flanges and the web. This is demonstrated on plots using a relative distance between the results and the proposed buckling curve. The distance is nearly the same for all simulated taper factors. References 1 Hibbitt, Karlsson & Sorensen, "Abaqus-manuels", http://www.abaqus.com 2 ECCS TC8 TWG 8.2, "Ultimate Limit State Calculations of Sway Frames with Rigid Joints, Publication 33, First Edition" Brussels, 1984. 3 ECCS TC8, "Reports from the ad-hoc-working group for interaction formulae", 2000 4 CEN, "prEN 1993-1-1: 2002 Eurocode 3 : Design of steel structures, Part 1-1 : General structural rules", Brussels, 2002. purchase the full-text of this paper (price £20) go to the previous paper go to the next paper return to the table of contents return to the book description purchase this book (price £123 +P&P)
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