Local Web Buckling in Tapered Composite Beams: A Parametric Study

Local Web Buckling in Tapered Composite Beams - A Parametric Study

P. C. G. da S. Vellasco

State University of Rio de Janeiro, UERJ

Structural Engineering Department

Rua São Francisco Xavier, 524

20550-013 Rio de Janeiro, RJ. Brazil

[e-mail]

R. E. Hobbs

Imperial College of Science Technology and Medicine

Civil Engineering Department

Imperial College Road, London, SW7 2BU. UK

[e-mail]

Composite flooring systems supported by tapered (varying web depth) beams are very attractive from an economic point of view. However, the tapered beam sections are fabricated from plate by welding, and are susceptible to imperfection effects. These may interact with the localised compressive stress field that is generated in the web at a slope change in the lower flange to cause local web buckling. A substantial parametric study using a non-linear elasto-plastic finite element program and covering practical ranges of the important parameters including the area of the tension flange, taper slope and web thickness is reported. Moment-rotation relations, peak moments and failure mechanisms have been predicted. The validity of the work is supported by the good correlation obtained between the results of the parametric study and experimental data.

Keywords: Composite tapered beams, steel structures, non-linear finite element analysis, vertical web buckling

Introduction

Although the potential of composite tapered beams, Fig. 1, has been demonstrated by their incorporation in several major projects, the full economic exploitation of this structural solution has been inhibited by a limited understanding of certain aspects of its behaviour. The local buckling of the relatively deep slender web at changes of slope in the lower flange, in particular at the slope change that occurs at the centre of a beam whose depth increases linearly from each support, is one of these problem areas.

Very few references to the problem of vertical web buckling at the central point of tapered composite beams were found in the literature: Vellasco et al (1993); Vellasco (1992); Vellasco and Hobbs (1996) and only one reference was found on the specific question of local buckling: Hargreaves and Hobbs (1986). In view of this fact, a broader survey was made including other aspects directly related to the study like the patch load phenomenon.

The patch load problem has been studied since 1932 when the first experimental investigations were developed by Ketchum and Draffin (1932), followed by studies of Lyse and Godfrey (1935). Since 1960, several tests have been executed by Bergfelt and Hovick (1968); Bergfelt (1971); Bergfelt (1979); Bergfelt (1983); Skaloud and Novak (1975); Dubas and Gheri (1975); Bagchi and Rockey (1975); Drdacky and Novotny (1977); Roberts and Rockey (1979); Roberts (1981); Roberts (1983); Roberts and Newark (1997); Kennedy (1997). A complete set of references related to this subject can be found in Vellasco (1992); Fonseca (1999); Souza (1995) and Fonseca (2001).

A substantial programme of experiments was then carried out at approximately half scale in order to investigate the influence of the primary parameters, Vellasco and Hobbs (1996). A second aim of the experimental programme was to validate finite element modelling of the problem, with a view to the subsequent parametric study that is reported in the present paper.

The first three models were tested with different web thickness. Two tapered beams with 600mm deep webs, respectively 7 and 5mm thick reached their ultimate loads with no sign of web buckling, and failed by the development of a plastic hinge at the edge of the tapered panel. A third beam, with a 3mm web, did fail by vertical web buckling, Fig. 2. A second group of three tests, relevant to the possibility of unpropped construction demonstrated that the plastic neutral axis position had no significant influence on the occurrence of vertical buckling. Further details of this test programme may be found in Vellasco (1992), Vellasco and Hobbs (1996).

Finite element simulations of the tests using an elasto-plastic large deflection code achieved a good agreement with the experiments, Vellasco and Hobbs (1996). The numerical method predicted the final deformed shape and the web stress patterns reached by the girders at the ultimate loading stage. In the first series, the calculations confirmed that web stiffeners were only needed at the flange slope change for webs that were less than 4mm thick. No stiffeners were required in the second series of beams where the neutral axis position was lower, Vellasco and Hobbs (1996).

This paper describes the subsequent finite element parametric study of local web buckling at the point where a change in flange slope occurs. The investigation focussed on three main parameters. A study of the effects of taper angle and web thickness is first reported. This is followed by an examination of the influence of the tension flange area on the load carrying capacity of the girder. Further details of the validation of the finite element model and a full description of the FINAS program used in the parametric study is presented in Vellasco and Hobbs (1996).

Nomenclature

A_fl = cross-sectional area of lower flange, mm²

b_f = flange width, mm

h = height of web, mm

M_pl = full plastic moment for the beam cross-section, kNm

t_f = thickness of the flange, mm

t_w = thickness of the web, mm

z = direction out of the plane of the web

Greek Symbols

a = slope of lower flange

Description of the Calculation Model

The finite element used in the modelling is an eight-node doubly curved isoparametric shell element with six degrees of freedom per node. The large displacement relationship is governed by an updated Lagrangian technique. The material non-linearity is treated by a Von Mises analysis, with an associated flow rule, on seven layers through the thickness of the shell.

The calculation model exploited symmetry and was composed of 112 shell elements distributed over one half of the beam as shown in Fig. 3. The beam was subjected to uniform bending. Again using symmetry, load was simply applied by three concentrated forces spread across the top of the upper flange and was resisted by three vertical supports across the bottom flange, Fig. 3. Web stiffeners were used at the load points and the supports to prevent spurious local failures of the web. As in the experimental work, the composite (steel/concrete) top flange of the section was modelled for convenience by an appropriately increased steel thickness. The only drawback to this approach was a need to prevent lateral buckling of the top flange, which was achieved by restraining the movement in the out-of-plane, z, direction of all the nodes on the line that joined the web and the top flange. To standardise the analysis, an imperfection shape in the form of a half-sine wave in the longitudinal and vertical directions was adopted. The maximum amplitude was equal to 2.4mm approximately, or h/250. Finite element runs including the experimental measured imperfections and true yield stresses validated this approach, Vellasco and Hobbs (1996).

The literature review on patch loading mentioned earlier suggested that one of the most important parameters was the thickness of the web. The results obtained from the experimental investigation, Vellasco and Hobbs (1996), confirmed that fact. Consideration of the vertical web-buckling phenomenon identified the slope change and the area of the bottom flange as being other pertinent variables.

Influence of the Angle of Taper and Web Thickness

The finite element model used in this group of simulations was based on the leading dimensions of an earlier full-scale composite tapered beam test, Hargreaves and Hobbs (1986). All the beam models had a total span of 4.8m and a moment lever arm of 800mm. The widths of the top and bottom flanges were 250mm and their thicknesses were 25 and 12mm respectively, with the extra thickness of the top flange modelling the effective section of the composite beam as noted earlier. The maximum depth at the centre of the beam was 800mm. The yield stresses of the web, top and bottom flanges were taken as 420MPa, 391MPa and 398MPa, respectively. These relatively high values (based on Hargreaves and Hobbs, 1986) were chosen to induce the most critical condition for local web buckling. The four web thicknesses used were 3, 5, 7 and 10mm. The angles of taper varied from 0° to 14.84° (a = 0 to 1/4). Cases with a zero angle, i.e., a beam with parallel flanges, were also considered because they represent an upper bound to the problem.

All of the beams with 10mm webs failed by the development of a plastic hinge. The exact position of this plastic hinge changed from beam to beam, moving towards the side vertical stiffener as the taper angle increased. The web at the centre of the girder remained undeformed, showing no need for a vertical stiffener even with the steepest slope. For the parallel beam a central plastic hinge formed. For the beams with angles of 1.15°, 1.43°, 1.91° and 2.86° (a = 1/50, 1/40, 1/30 and 1/20), the behaviour was very similar to the previous case. There was significant movement in the position of the plastic hinge, towards the edge of the panel, for the girders with angles of 8.53° and 11.31°, (a = 1/6.67, 1/5). Finally, for the angle of 14.84° (a = 1/4) the plastic hinge moved close to the end stiffener. Figure 4 shows a comparison in terms of vertical displacement.

The beam with the 7mm web and parallel flanges failed by the plastic hinge collapse described in the previous section. At the centre of the beam the web did deform laterally: these deflections were associated with a rotation of the top flange. The same type of failure occurred for the beams with slopes of 1.15° and 1.43°, (a = 1/50, 1/40) and indeed the angles of 1.91°, 2.86°, 8.53° and 11.31° (a = 1/30, 1/20, 1/6.67, 1/5) were also associated with the same collapse pattern. The contribution of the vertical compressive force component that arises from the bottom flange slope change had increased substantially but not by the amount necessary to make the web buckle. For the beam with a taper angle of 14.84° (a = 1/4) the collapse happened by a combination of web buckling and the development of a plastic hinge near the vertical stiffener.

A vertical displacement comparison is portrayed in Fig. 5. Two different regions can be seen in this figure. The first corresponds to the beams with small angles of taper (up to 2.86°, a = 1/20). The other represents the higher slopes in which the final failure is associated with a substantial loss of height in the region near the vertical stiffener. The main conclusion for the 7mm web is that only the steepest angle, 14.84° (a = 1/4), shows a slight web-buckling tendency at the slope change.

In the 5mm beam with parallel flanges the collapse occurred due to the formation of a plastic hinge near the end stiffener. At the centre of the beam, the web deformed laterally associated with a rotation of the top flange. The girder with an angle of 1.15° (a = 1/50) failed by plastic vertical buckling of the web. Its deformed configuration at the point of maximum load is portrayed in Fig. 6(a). Very little web deformation is noticeable. However, at a further stage on the unloading path, the web has become very distorted, as can be seen in Fig. 6(b). The same type of failure occurred for the beams with angles of 1.43° and 2.86° (a = 1/40, 1/30). Both cases had very large associated web deformations although these only appeared at points on the unloading path far beyond the maximum load, confirming the inelastic characteristic of the collapse. Three larger angles (8.53°, 11.31° and 14.84°(a = 1/6.67, 1/5, 1/4)) produced elasto-plastic failures, and by 14.84° (a = 1/4) the web stresses and the buckle have become more concentrated near the bottom flange.

A vertical displacement comparison for all the beams investigated in this thickness group is presented in Fig. 7. The significant drop in ultimate load, as the angle of slope increased, is plain. The two regions of plastic and elasto-plastic failure are also clearly identifiable. With these results in hand, it is possible to state that for the 5mm web all the slopes required a vertical stiffener to avoid local buckling.

The beam with parallel flanges and a 3mm web buckled in the longitudinal direction, because of the high longitudinal compressive bending stresses in the web. The collapse at the ultimate loading stage is described in Fig. 8. No significant equivalent (Von Mises) stresses were found near the bottom flange. The girder with an angle of 1.43° (a = 1/40) failed by a combined longitudinal and vertical buckling of the web. A vertical buckling of the web in the inelastic region was the main cause of failure for the beam with an angle of 2.86° (a = 1/20). The angles of 8.53°, 11.31° and 14.84° (a = 1/6.67, 1/5, 1/4) induced an elastic vertical web buckling collapse similar to the one described in Fig. 9. Figure 10 presents a vertical displacement comparison of all the beams investigated in this section. Again, the different forms of collapse are well characterised by these diagrams. From these results it is evident that if a 3mm web is to be used, vertical stiffeners are required at the centre of the web for any angle of taper.

The ultimate loads achieved by the beams are given in Table 1 in terms of percentage of the central plastic moment capacity. As the angle increases, there is a drop in the maximum load for all thicknesses. For the 10mm beams, this change was due to the type of plastic collapse developed by the beams. With an increase in the taper angle, the cross section depth and the plastic section modulus decrease at the critical section, explaining the loss in moment capacity. The same behaviour happened for the 7mm beams. The 5mm parallel beam failed by the formation of a plastic hinge. All the other 5mm web girders had a vertical web buckling failure in the elasto-plastic and elastic ranges. For the beams with 3mm webs, there is a significant loss of resistance and change in the failure mode as the angle increases. With parallel flanges, longitudinal web buckling caused the failure. As the angle increased, the vertical stresses began to influence the failure in a more significant way, changing completely the form of collapse.

It is interesting to examine the influence of web thickness for a given angle of taper. The thicknesses correspond to central h/tw values of 260.5, 156.3, 111.7 and 78.2. The results obtained for the beams with parallel flanges demonstrated that the girders with 10, 7 and 5mm webs failed by the development of a plastic hinge mechanism near the vertical loading stiffener. The web remained undeformed, showing no necessity for a vertical stiffener although the 3mm beam had a longitudinal web buckling failure. The taper angle of 1.43° (a =1/40) generated the following results: as in the previous case, the 7 and 10mm beams failed by plastic hinge collapse, as already mentioned. The 5 and 3mm beams collapsed under an inelastic vertical web buckling failure. They would both require vertical stiffeners at the centre part of the beam to avoid this failure mode.

The results for the angle of 8.53° (a=1/6.67) are compared in Fig. 11. The 10 and 7mm beam still failed due to the formation of a plastic hinge mechanism near the loaded vertical stiffener. On the other hand, the 5 and 3mm beams developed the elastic vertical buckling failure explained previously. These two last beams certainly require a vertical stiffener to improve their performance. The beams with a taper of 11.31° (a=1/5) reached very similar kinds of failures. Again, the plastic hinge failure was the cause of collapse for the 10 and 7mm beams while vertical stiffeners were again required to minimise the effects of the elastic vertical web buckling failure developed by the other two beams.

Finally, Fig. 12 presents lateral displacement comparisons for the girders with an angle of slope of 14.84° (a=1/4). The 10mm beam collapsed by the plastic hinge mechanism, as in the previous cases. The 7mm beam buckled inelastically in the vertical direction. The last two beams buckled elastically in the vertical direction. All the girders that developed the vertical buckling would have their performance significantly enhanced if a vertical web stiffener were used at the critical central section of the beam.

Influence of the Tension Flange Area

The finite element simulations described in this section are broadly similar to the previous examples. They possess the same loading and support configurations, initial imperfections and material properties. The thicknesses of the web used were again 3, 5, 7 and 10mm, but the angle of slope was now fixed at 5.71° (a = 1/10). The only change from the beams of the previous section was the thickness of the top and bottom flanges, increased from 25 and 12mm to 50 and 25mm, respectively. The bottom flange thickness was changed to increase the area of the bottom flange from 3000 to 6250 mm². The top flange thickness had to be increased to prevent the occurrence of local flange buckling.

The 10 mm web beam collapsed due to the development of a plastic hinge mechanism near the edge vertical stiffener. Despite the presence of very high compressive stresses near the bottom flange at the centre of the beam, only very small lateral deflections developed in the web. With that in mind, it is possible to conclude that stiffeners were not required in this case.

The girder with a 7mm web failed by a plastic vertical buckling of the web. Its deformed configuration at the point of maximum load is portrayed in Fig. 13.a. In this diagram, only small lateral web deflections are noticeable. However, at a point further down the unloading path, the web became very distorted, as shown in Fig. 13.b. The use of a vertical stiffener at this particular point would substantially improve the beam's performance.

The last two beams investigated failed by plastic and elastic vertical web buckling, respectively. Both cases require the presence of a vertical stiffener to improve their performance. Table 2 describes the maximum loads reached by the four simulations performed in this section together with some results taken from (Vellasco and Hobbs, 1996). The percentage moment resistance capacity was substantially reduced when the area of the bottom flange was increased. This feature is explained by the growth in the compressive force applied to the web by the larger tension flange area.

In both cases, the 3mm beam failed by an elastic vertical buckling of the web. The collapse of the 5mm beams is associated with a plastic vertical buckling of the web. The 10mm beams fail by the development of the plastic hinge at a point near the panel edge. The main difference of the two different flange sizes occurs in the beams with a 7mm web. The beam with a 3000mm² tension flange failed by the plastic mechanism described above, while the other case failed by plastic vertical buckling of the web. From these results it is possible to conclude that vertical stiffeners at the centre of the web are required for the web thicknesses less than or equal to 5mm in the 3000mm² cases, and 7mm in the 6250mm² specimens.

Discussion

The influence of the angle of taper and the thickness of the web can be better visualised with the aid of Fig. 14. From that diagram, it is evident that the bending moment resistance for beams with different web thicknesses, with the same slope, increase as the slope becomes steeper. These results confirmed that the angle of slope is a very significant variable in the beam's load carrying capacity.

Vertical web buckling only occurred in one case for the standard flanges for a h/t_w ratio less than 155. This happened when the 7mm beam was combined with the very large angle of 14.84° (a=1/4). Situations with these high h/t_w ratios are rarely found in design practice. A small increase in the thickness of the web would be the most effective and least costly solution to avoid this kind of collapse.

The effect of the tension flange area on the moment capacity of the girders is illustrated by Fig. 15. When the area of the tension flange was doubled, the decrease in moment resistance as a proportion of the fully plastic moment at the centre was up to 45%. These results also demonstrated the strong influence of this parameter on the buckling phenomena. Vertical web buckling only occurred in one case for the standard flange for a h/t_w ratio less than 155.

Conclusions

Vertical stiffeners would only be needed in the web in extreme situations. Most of the cases found in practice will not need stiffeners. Given the high cost of fitted stiffeners, the approach of slightly increasing the thickness of the web to avoid vertical buckling seems likely to be the most satisfactory design expedient in limiting cases.

If the thickness of the web cannot be increased, then a vertical stiffener should be used on both sides of the web over approximately one third of the web's depth. These stiffeners should be designed as normal load bearing stiffeners to an appropriate steel design standard.

The parametric study could also be extended to cover other factors that may also influence the local web buckling such as: the rotation restraint provided by the flange to the web, residual stresses, hybrid (different yield stress for web and flanges) tapered girders and others.

Although the simultaneous occurrence of high shear or normal stresses in high bending moment regions is not common in these beams, it might be worthwhile to extend the present investigation to cover the interaction of slope-change buckling with shear and normal stresses. Certainly the full behaviour is not completely understood and further testing and parametric studies may be justified.

Acknowledgements

The authors wish to thank CNPq. (the Brazilian Science Development Agency) and the EPSRC (U.K. Science and Engineering Research Council) for their financial support to the project. Thanks are also due to all the staff of the Structures Laboratory, Civil Engineering Department, Imperial College London, where the investigation was conducted.

Fonseca, E.T., Vellasco, M.M.B.R., Vellasco, P.C.G. da S., Andrade, S.A.L. de, Pacheco, M.A.C., 2001, "A Neural Network System For Patch Load Prediction", Journal of Intelligent and Robotic Systems, (in print).

Manuscript received: October, 2000. Technical Editor: Átila P.S.Freire.

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Publication Dates

• Publication in this collection
  24 Sept 2002
• Date of issue
  2001

History

• Received
  Oct 2000