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Shear behaviour of trapezoidal column panels. I: Experiments and finite element modelling
JCSR-04099; No of Pages 10 Journal of Constructional Steel Research xxx (2014) xxx–xxx
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Journal of Constructional Steel Research
Shear behaviour of trapezoidal column panels. I: Experiments and finite element modelling Eduardo Bayo a,⁎, Alfonso Loureiro b, Manuel Lopez b a b
University of Navarra, Spain University of A Coruña, Spain
a r t i c l e
i n f o
Article history: Received 12 April 2014 Accepted 18 October 2014 Available online xxxx Keywords: Column shear panel Beams of unequal depths Experimental validation Finite element modelling
a b s t r a c t The behaviour of joints exerts an important influence in the overall performance of steel and composite structures, both in deformations and in ultimate resistance. As a consequence, a considerable effort has been carried out in recent years to better characterize the behaviour of joints in these kinds of structures. Modern codes, including Eurocodes 3 (EC3) and 4 (EC4), have included these research advances so that they may be used in common practice. The method adopted in EC3 and EC4 to characterize the connections is the so-called component method. One of the most important components is the column web panel in shear, and its behaviour has been investigated thoroughly for the case of rectangular shear panels arising when the beams have equal depths. However, the case of trapezoidal column panels, formed by beams of different depths at each side of the column, has not been researched as much. Formulae or design recommendations for the shear behaviour of steel beam-tocolumn joints with beams of unequal size are not currently included in design codes. The aim of this paper is the characterization of the shear behaviour of trapezoidal column panels. In order to isolate the shear behaviour from other components, stiffened beam to column connections with commercial sections are experimentally tested. Also, finite element modelling and analysis are carried out to compare results, and a mechanical model is proposed. Current modelling procedures are tested and the results compared with those coming from the experiments and numerical simulations. © 2014 Elsevier Ltd. All rights reserved.
1. Introduction The characterization of steel joint behaviour and properties has been a matter of research for a number of years, and all the accumulated knowledge has been compiled to a large extent in currently available structural steel design codes. One important aspect is the behaviour of the column panel subjected to the shear forces arising from the moments of the adjacent beams as well as the shear forces acting on the columns. A correct definition of the column panel zone deformations under static conditions is of great importance due to its influence on the overall sway behaviour of the frame. An increase in frame drift due to panel zone shear deformation may render the frame unserviceable. This may even happen for commonly considered rigid joints. Modelling of the column panel is also important for the avoidance of local failure under ultimate limit state conditions. Krawinkler et al. [1] reported the importance that panel shear deformations have on the frame behaviour under lateral loads, and proposed a formulation for the stiffness and resistance of shear panels of beam to column connections with beams of equal depths. An alternative ⁎ Corresponding author. Tel.: +34 948425600; fax: +34 948425629. E-mail address: [email protected] (E. Bayo).
formulation has been proposed in the Eurocode 3, part 1.8 [2] following the component method [3–5]. These and other proposed methods allow introducing the flexibility of joints in both the elastic and inelastic range in order to assess the frame response [6,7] and also including high strength steel [8]. The lines connecting the flanges of beams of unequal depths define a trapezoid within the panel column zone. Hoogenbroom and Blaauwendraad [9], and Curtis and Greiner [10] have proposed analytical and computational methods to characterize the shear behaviour of isolated quadrilateral panels. More recently, Hashemi and Jazany [11] have investigated the connection detailing of joints of unequal beam depths under seismic loads. One of their conclusions is that inclined stiffeners connecting the lower flanges of the beams perform better than the horizontal ones. Jordao et al. [12] have studied the performance of this type of joints for high strength steel without the use of web stiffeners. As a consequence, the stress field at the panel zone becomes rather complex due to the fact that the compression, tension and shear components are all coupled together within the panel zone. The common failure mode of the experiments carried out in [12] was web buckling due to compression. Within the context of the component method, they propose a joint model based on two subpanels delimited by 3 levels of load introduction coming from the beams. A suitable
http://dx.doi.org/10.1016/j.jcsr.2014.10.026 0143-974X/© 2014 Elsevier Ltd. All rights reserved.
Please cite this article as: Bayo E, et al, Shear behaviour of trapezoidal column panels. I: Experiments and finite element modelling, J Constr Steel Res (2014), http://dx.doi.org/10.1016/j.jcsr.2014.10.026
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1930
860
15
2
B
A
1000 15
1000
15
120
DEEP BEAM
SHALLOW BEAM
120 15
15 COLUMN
Fig. 1. Experimental setup.
modelling of the joint for global analysis can be achieved by considering the cruciform element proposed by Bayo et al. [13,14]. Work remains to be done to better characterize the complex behaviour of steel joints with beams of unequal depths and trapezoidal panels. The study involves a large number of variables and intervening factors. In this paper, experimental work and numerical (finite element) results are presented that provide important information to characterize and model this type of joints. The moment rotation diagrams depend on a large number of material, geometrical and loading variables. The research presented concentrates on the shear stiffness and resistance of the trapezoidal panel zone and the additional contribution provided by the elements surrounding the panel such as the column flanges adjacent to the panel. These elements play a major role in the joint postelastic reserve shear strength and stiffness. Since the aim is to study the shear behaviour and component of the panel zone, stiffeners are included to avoid the effects of other components such as the column web in tension and compression, and to prevent their interaction with the shear component as much as possible. Furthermore, an inclined stiffener in the lower part of the panel assures trapezoidal behaviour. Consequently, only stiffened fully welded connections were considered in this investigation. The moments acting at both sides of the joint are the most important contribution to the panel shear force, and they are usually considered as the load variables characterizing the strength and stiffness of the joint.
However, the beneficial effect of the column shear should also be considered (particularly in the case of short columns) in the joint stiffness and strength as proposed by Krawinkler et al. [1]. The deformation is defined in terms of the average shear distortion of the panel zone (PZ), and it is measured in this investigation by means of two inclinometers located on the middle section of the panel. The deformations on both sides of the panel (the shallow and deep sides) will be considered as well. The aim is to understand the mechanisms of deformation and characterize the moment rotation curves for the shear component of the trapezoidal panel. 2. Experimental work and finite element models The experimental work has been carried out in two different prototypes with beams of unequal depth. The overall scheme is illustrated in Fig. 1. The column is pinned at both ends and actuation is applied at points A and B on the attached beams. The distance from these points and the column flanges is exactly 1000 mm. Table 1 shows the beams and column sizes in both tests, and Fig. 2 depicts a picture of the experimental setup. The top horizontal and the inclined stiffeners were welded as depicted in Figs. 1 and 2 to avoid, as mentioned above, any type of failure other than that produced by shear. The stiffeners were 15 mm thick and in all cases were rigid enough to provide the necessary resistance to
Table 1 Characteristics of the tests: beam and column configurations. Test
Column
Deep beam
Shallow beam
Depth ratio
Loading point
Type of loading
E1A E1B E2A E2B
HEA 240 HEA 240 HEA 240 HEA 240
HEB 300 HEB 300 HEB 300 HEB 300
HEB 160 HEB 160 HEB 180 HEB 180
1.88 1.88 1.67 1.67
A B A B
Elastic Failure Elastic Failure
Please cite this article as: Bayo E, et al, Shear behaviour of trapezoidal column panels. I: Experiments and finite element modelling, J Constr Steel Res (2014), http://dx.doi.org/10.1016/j.jcsr.2014.10.026
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3
Fig. 4. Detail of the finite element model. Fig. 2. Experiment E1 during the loading process.
is subtracted from those at the beams to obtain the rotation due to the shear deformations. The finite element analysis was performed using Abacus®. Solid elements with reduced integration (to avoid shear locking) and hourglass control (C3D8R) were adopted. Fig. 4 illustrates the part of the model that corresponds to the column panel. The material behaviour was introduced by means of the true stress–strain data obtained from the coupons. Static nonlinear material and geometric analyses with force control were performed. The Von Misses yield criterion was selected to define the inelastic response. For all the tests, a previous simulation had been done in order to detect the load at which plastic behaviour begins. These values have been used in tests E1A and E2A as the upper limit of the applied forces, with the objective of avoiding damage in the web panels. The experiments were simulated with the exact dimensions of the elements. In all cases, the analysis was extended until the complete plastic behaviour of the panel was reached. The finite element models were refined near the most stressed zones (the panel and adjacent zones) with the aim of assessing correctly the strain and stress fields. The convergence of the model was evaluated to reach good results with a suitable level of meshing, and to allow parametric studies with a reasonable computational cost.
Table 2 Steel properties obtained from coupons. Part
σy (MPa)
σu (MPa)
E (MPa)
HEA 240 HEA 240 HEA 240 HEA 240 Stiffeners Stiffeners
330 328 326 329 300 309
493 494 496 490 446 449
207,000 211,000 209,000 210,000 213,000 211,000
prevent either tension or compression failure of the column webs, as well as out of plane bending of the column flanges. The column web was 8 mm thick. The type of steel used for all the parts was S275. Coupons were extracted from both the columns and stiffeners to obtain the true properties of the material. Table 2 shows those properties that were subsequently used in the finite element analyses for comparison purposes. The panel was instrumented with 4 strain gauges placed at the corners of the panel as shown in Fig. 3. They served to monitor the yielding sequence and the levels of shear strain. Five inclinometers were used as shown in Fig. 3 to obtain the rotations. Two were placed vertically at the beams adjacent to the joint. Two more were placed vertically at the top and bottom of the web panel to capture the possible different rotations at those levels. A fifth one was placed in the middle of the panel to capture the rotation of the column due to bending. This rotation
2
4
3. Description of the results 3.1. Test E1A The first test E1A consisted in loading the prototype E1 at the tip of the shallow beam (point A). The aim was to obtain the elastic stiffness
1
Inclinometer 2
5 1
4
3
Strain gauge
3
Fig. 3. Placement of strain gauges and inclinometers.
Please cite this article as: Bayo E, et al, Shear behaviour of trapezoidal column panels. I: Experiments and finite element modelling, J Constr Steel Res (2014), http://dx.doi.org/10.1016/j.jcsr.2014.10.026
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Fig. 5. Elastic shear stress contours at the end of the elastic range in experiment E1A.
of the trapezoidal panel under the shear coming from loading the shallow beam. The maximum load applied was 30 kN, and afterwards it was unloaded. A previous finite element analysis had predicted pure elastic behaviour up to that loading level. The shear stress distribution within the panel obtained from the finite element analysis is illustrated in Fig. 5, which corresponds to the level of strains just at the end of the elastic range. Fig. 6 shows the strain gauge data for this test. It may be seen that the stress distribution is uniform over an upper rectangular part of the panel. The readings of the inclinometers 3 and 4 were similar, and the inclinometers 1 and 2 showed that the rotation at the shallow beam was approximately 2.4 times the rotation at the deep beam. 3.2. Test E1B The following test, E1B, consisted in loading the prototype E1 at the tip of the deep beam (point B) until failure, which occurred at a load of 230 kN. The shear stress distributions within the panel coming from the finite element analysis during the elastic and inelastic parts of the response are illustrated in Figs. 7 and 8, respectively. It may be seen how even in the elastic range the shear stress field now extends deeper towards the bottom of the panel becoming nonrectangular in shape.
Fig. 6. Results from strain gauges in experiment E1A.
Fig. 7. Elastic shear stress levels in test E1B.
During the elastic part of the response, the readings of the inclinometers 1 and 2 also showed a rotation at the shallow side 1.2 times larger than the rotation at the deep side. However, within the inelastic response the angles at the shallow beam became much larger than those of the deep beam reaching 40% more right before failure. The final failure was due to the cracking of the panel at the upper right corner (right next to strain gauge 1) 3.3. Test E2A The test E2A consisted in loading the prototype E2 at the tip of the shallow beam. The qualitative behaviour of this test was similar to that of E1A. The maximum load applied was 37 kN, and afterwards it was unloaded. Again, a previous finite element analysis had predicted pure elastic behaviour up to that loading level. The shear stress distribution within the panel obtained from the finite element analysis is illustrated in Fig. 9, which corresponds to the level of strains just at the end of the elastic range. Fig. 10 shows the strain gauge data for this test. The readings of inclinometers 3 and 4 were similar, and the rotation at the shallow beam was approximately double the size of the rotation of the deep beam.
Fig. 8. Inelastic shear stress levels in test E1B.
Please cite this article as: Bayo E, et al, Shear behaviour of trapezoidal column panels. I: Experiments and finite element modelling, J Constr Steel Res (2014), http://dx.doi.org/10.1016/j.jcsr.2014.10.026
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Fig. 9. Elastic shear stress levels at the end of the elastic range in test E2A.
5
Fig. 11. Elastic shear stress levels in test E2B.
much more inclined than the deep beam due to the deformation of the trapezoidal panel as an articulated quadrilateral mechanism.
3.4. Test E2B The final test E2B consisted in loading the experiment E2 at the tip of the deep beam (point B) until failure, which occurred at a load of 230 kN. The shear stress distributions within the panel coming from the finite element analysis during the elastic and inelastic parts of the response are illustrated in Figs. 11 and 12 respectively. The rotation of the lower part of the panel was similar to that of the higher part right before failure. The angles at the shallow beam were 1.15 times those of the deep beam during the elastic part of the response. However, within the inelastic response the angles at the shallow beam became much larger than those of the deep beam reaching a difference of 40% right before failure. Figs. 13 and 14 show the readings of the strain gauges corresponding to tests E1B and E2B, respectively. The pictures of the deformed panel and deformed shape of the experiment before failure are shown in Figs. 15 and 16, respectively. The plastic shear stress levels of the column panel as well as the final deformed shape of the specimen obtained by finite element simulation can be seen in Fig. 17. It may be observed how the finite element model replicates the actual deformed shape of the experiment. It is also worth noting how the shallow beam gets
Fig. 10. Results from strain gauges in experiment E2A.
4. Kinematics of the deformation and mechanical model As mentioned above and as observed from tests and finite element models, three different degrees of freedom can be identified in the shear panel during the elastic range loading. Fig. 18 shows an enlarged picture of the finite element panel deformation when the shallow beam is loaded (Test E1A), and it evidences the fact that the rotation of the shallow side is larger than those of the centre and deeper parts of the panel, respectively. Fig. 19 shows the panel deformation when the deep beam is loaded (Test E1B); again the rotation of the shallow side is larger than those of the centre and deeper parts of the panel. Finally, Fig. 20 shows the deformed shape due to shear forces, coming from the column, acting in the panel; and again the same pattern is observed. As explained below, these deformations do not correspond to those of a single degree of freedom articulated quadrilateral, and therefore they are independent. The departure from the kinematics of an articulated quadrilateral in the elastic range is due to the bending deformation produced in the column flanges in both sides of the panel;
Fig. 12. Inelastic shear stress levels in test E2B.
Please cite this article as: Bayo E, et al, Shear behaviour of trapezoidal column panels. I: Experiments and finite element modelling, J Constr Steel Res (2014), http://dx.doi.org/10.1016/j.jcsr.2014.10.026
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Fig. 15. Panel inelastic deformation. Fig. 13. Results from strain gauges in test E1B.
this can be seen by observing the deformation of the shallow side in Fig. 19. In regard to the initial stiffness and deformations, Figs. 21 and 22 show the moment–rotation curves corresponding to the deep beam, the centre of the panel and the shallow beam for tests E1A and E2A, respectively. Figs. 23 and 24 show those corresponding to the tests E2A and E2B, respectively. In all cases, by observing the slopes of the moment–rotation curves one can identify different stiffness and see how the rotation of the shallow beam side is larger than the rotation of the centre of the panel, and this one is in turn larger than the rotation at the deep beam side. As mentioned above the joint shows three independent degrees of freedom in the elastic range, which do not follow the deformations of an articulated quadrilateral. However, once the panel becomes plastic, the incremental rotations follow the kinematics of a one-degree of freedom articulated quadrilateral. The data appearing in Table 3 explains this effect. It can be seen in this table how the ratios between the heights of the deep and shallow beams (the depth ratio) are very different from those of the corresponding incremental rotations in the elastic range. However, when the panel deforms within the plastic range the ratio of the incremental rotations and the depth ratio become quite similar,
thus corroborating the kinematics of a one-degree of freedom articulated quadrilateral mechanism. From the previous discussion it was possible to observe that there are three independent rotations and therefore three degrees of freedom (DOF) during the elastic range: left connection, right connection and centre of panel; and only one DOF in the inelastic range corresponding to an articulated quadrilateral mechanism. In view of this, a mechanical model is presented in Fig. 25. This model is made up of rigid bars connected by springs. The model illustrates the three possible rotations: ϕ1 on the right part of the panel, ϕ2 on the left part of the panel, and ϕp on the panel itself. When the panel becomes plastic the value of Kp becomes significantly lower than K1 and K2, and the model resembles a one-degree of freedom system. The corresponding dual internal forces are M1, M2 and V. M1 is the right side moment, M2 is the left side one, and V is the shear coming from the column. The spring Kp models the stiffness of the panel under shear; the springs K1 and K2 take into account the relative rotation between the centre of the panel and the right and left connections, respectively. The parameters h1, h2 and hp represent the equivalent height of the shallow, centre and deep parts of the joint, respectively. The values of the springs and parameters are defined after a numerical parametric analysis in the companion (Part II) paper [15].
Fig. 14. Results from strain gauges in test E2B.
Fig. 16. Test deformed shape before failure.
Please cite this article as: Bayo E, et al, Shear behaviour of trapezoidal column panels. I: Experiments and finite element modelling, J Constr Steel Res (2014), http://dx.doi.org/10.1016/j.jcsr.2014.10.026
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Fig. 17. Final stage of stress and deformation in test E2B.
5. Moment rotation curves: stiffness and resistance This section presents a comparison of the moment rotation curves obtained from the experiments, the finite element analysis, the Eurocode 3 [2] and the method proposed by Krawinkler et al. [1]. Since the last two methods do not include trapezoidal panels they are applied to each connection (left and right) as if the panel was
rectangular for the corresponding beam depth. It is important to note that the EC3 and Krawinkler's formulae were developed for rectangular panels that correspond to beams of equal depths, and as such do not fit in this study. However, they are included in this comparison to observe their results and assess the possible errors that may derive from their use to trapezoidal panels. The intention is to inform and warn those designers who may use these formulae for cases other than rectangular panels. The comparison is established in terms of the moment at the connection versus the average shear rotation of the panel, as well as the other possible rotations at each side of the joint (additional DOFs). The three experimental rotations of the panel: right, centre and left have been measured using the following relations: Rotation at the right of the panel = inclin(2) − inclin(5) Rotation at the centre of the panel = (inclin(3) + inclin(4)) /2 − inclin(5) Rotation at the left of the panel = inclin(1) − inclin(5) Inclinometers 1 and 2 measure the rotations at the left (deep) and right (shallow) parts of the panel, respectively. Inclinometers 3 and 4 measure the total rotation at the top and bottom parts on the panel along the centre line (see Fig. 3). The reading of inclinometer 5 is subtracted from all the measures to take into account the flexural rotation of the column at the joint level. Figs. 26 and 27 show the moment–rotation curves corresponding to the tests E1A and E2A, respectively. The results from the experiments
Fig. 18. Enlarged panel deformed shape in test E1A (load in shallow beam).
Fig. 19. Enlarged panel deformed shape in test E1B (load in deep beam).
Fig. 20. Enlarged panel deformed shape (shear load in panel).
Please cite this article as: Bayo E, et al, Shear behaviour of trapezoidal column panels. I: Experiments and finite element modelling, J Constr Steel Res (2014), http://dx.doi.org/10.1016/j.jcsr.2014.10.026
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Fig. 21. Shallow, centre and deep side rotations in the FEM for test E1A. Fig. 24. Shallow, centre and deep side rotations in the FEM for test E2B.
Fig. 22. Shallow, centre and deep side rotations in the FEM for test E2A.
only provide the initial stiffness since, as mentioned above, they are only loaded in the linear elastic range (the maximum applied load was 30 kN for test E1A and 37 kN for test E2A), and their plots are hidden underneath the finite element ones. Figs. 28 and 29 show the moment rotation curves corresponding to the tests E1B and E2B, respectively. It may be seen that the predictions of the finite element model in terms of stiffness and resistance are good when considering the rotation at the centre of the panel. The differences at the knee level are due to uncertainties in the modelling of the welding material properties. EC3 provides a good prediction of the stiffness but a poor prediction of the resistance, even when including the additional resistance provided by the column flanges. Krawinkler's model underestimates both the stiffness and resistance, although the latter is better approximated than EC3. Table 4 compares the values of the stiffness obtained by the different methods, as well as the relative errors, when considering the rotation of the central section of the panel as the reference for the rotations (third degree of freedom). It may be observed that the finite element model provides quite good results. The EC3 formulation also gives good results (around 5% error) and Krawinkler's model gives larger errors of up to 20%. The results are similar when considering the deep section of the panel as the reference for the rotations (second degree of freedom). Table 5 shows similar differences for the EC3 case although it provides stiffness values that are non-conservative (note that the cases E1A and E2A do not appear since there is no moment applied at that side). A much worse situation arises when considering the shallow section (first degree of freedom) of the panel as the reference for the rotations. Table 6 shows that the errors arising from the finite element model are within 10%, however the errors introduced by both the EC3 and Krawinkler's models are considerable, 97% and 57% respectively, and also non-conservative. Therefore the application of the formula provided by EC3 and Krawinkler's model is not recommended for the shallow beam side. Table 3 Incremental shallow/deep rotations in the elastic and plastic ranges.
Fig. 23. Shallow, centre and deep side rotations in the FEM for test E1B.
FEM
Depth ratio
E1A E1B E2A E2B
1.87 1.87 1.66 1.66
Incremental shallow/deep rotation Elastic range
Plastic range
2.37 1.19 2.05 1.15
1.78 1.62 1.61 1.48
Please cite this article as: Bayo E, et al, Shear behaviour of trapezoidal column panels. I: Experiments and finite element modelling, J Constr Steel Res (2014), http://dx.doi.org/10.1016/j.jcsr.2014.10.026
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Fig. 25. Proposed mechanical model of the shear panel.
6. Conclusions In this paper we have investigated the shear behaviour (component) of stiffened trapezoidal shear panels appearing in joints with unequal beam depths. Tests and FE simulations have been performed. A
Fig. 26. Moment–rotation at the centre of panel for test E1A (the test plot is underneath the FEM curve).
Fig. 27. Moment–rotation at the centre of panel for test E2A (the test plot is underneath the FEM curve).
mechanical model of rigid bars and springs has also been proposed to capture the kinematics and internal forces acting on the joint. The main conclusions can be summarized as follows: 1. The shear deformation zone corresponds to a part of the upper rectangle when loading the shallow beam, and the whole trapezoid when loading the deep beam. Consequently the mechanics of the deformation and the initial stiffness values of the left and right connections are different. 2. Three independent degrees of freedom are identified in the elastic range: the rotations corresponding to the left, right and central sections of the panel. The inelastic range is well characterized by one degree of freedom, which corresponds to an articulated quadrilateral mechanism. These degrees of freedom should be considered at the time of characterizing the joint stiffness for frame analyses. 3. The finite element analysis predicts the stiffness and resistance with good accuracy. EC3 approximates the initial stiffness reasonably well when using the dimensions of the beam attached to the corresponding connection, and the middle section and deep beam side of the panel as the references for the rotations (degrees of freedom 2 and 3). However, the initial stiffness estimated by EC3 becomes very poor when considering the rotation of the shallow beam (degree of freedom 1). Krawinkler's model provides slightly worse results than the EC3 when using the middle section and deep beam side of the panel as the reference for the rotations. However, the stiffness predicted by Krawinkler's model becomes very poor when considering the rotation at the shallow beam side.
Fig. 28. Moment–rotation at the centre of panel for test E1B.
Please cite this article as: Bayo E, et al, Shear behaviour of trapezoidal column panels. I: Experiments and finite element modelling, J Constr Steel Res (2014), http://dx.doi.org/10.1016/j.jcsr.2014.10.026
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E. Bayo et al. / Journal of Constructional Steel Research xxx (2014) xxx–xxx Table 6 Comparison of rotational stiffness (kNm/mrad) considering the shallow side of the panel.
E1A E2A
Test
FEM
Error (%)
EC3
Error (%)
Krawinkler
Error (%)
16.7 19.7
18.3 21.6
9.6 9.7
31.9 36.0
96.9 88.5
25.4 28.9
56.8 51.3
Acknowledgements The financial support provided by the Spanish Ministerio de Ciencia e Innovación under contract BIA2010-20839-02-C01-C02 is gratefully acknowledged. References
Fig. 29. Moment–rotation at the centre of panel for test E2B.
Table 4 Comparison of rotational stiffness (kNm/mrad) considering the centre of the panel.
E1A E1B E2A E2B
Test
FEM
Error (%)
EC3
Error (%)
Krawinkler
Error (%)
29.1 59.0 31.8 56.7
28.7 57.5 31.4 55.5
−1.4 −2.5 −1.3 −2.2
31.9 60.9 36.0 60.9
−2.0 −1.3 1.8 1.2
25.4 51.7 28.9 51.7
−22.0 −16.2 −18.3 −14.1
Table 5 Comparison of rotational stiffness (kNm/mrad) considering the deep side of the panel.
E1B E2B
Test
FEM
Error (%)
EC3
Error (%)
Krawinkler
Error (%)
59.8 57.5
61.3 59.2
2.5 3.0
60.9 60.9
−1.9 8.9
51.7 51.7
−16.7 −7.5
4. A mechanical model of rigid bars and springs has been proposed. This model includes the three degrees of freedom of the trapezoidal shear panel. The variables that characterize this model will be obtained after a detailed parametric analysis in the companion (Part II) paper [15]. In the context of advanced global analysis allowed by modern codes [16,17] and optimized solutions [18] the use of the proposed joint model could be particularly useful since it allows the joint to be accurately modelled within the structure.
[1] Krawinkler H, Bertero VV, Popov EP. Shear behaviour of steel frame joints. J Struct Div ASCE 1975;101(11):2317–36. [2] CEN. Eurocode 3, EN 1993-1-8:2005: Design of steel structures. Part 1.8: Design of joints. Brussels, Belgium: European Committee for Standardization; 2005. [3] Zoetemeijer P. Summary of the research on bolted beam-to-column connections (period 1978–1983). Technical Report 6-85-M. Delft: Steven Laboratory; 1983. [4] Yee YL, Melchers RE. Moment–rotation curves for bolted connections. J Struct Eng ASCE 1986;112(3):615–35. [5] Faella C, Piluso V, Rizzano G. Structural steel semi-rigid connections. Boca Raton: CRC Press; 2000. [6] Castro JM, Elghazouli AY, Izzuddin BA. Modeling of the panel zone in steel and composite frames. Eng Struct 2005;27:129–44. [7] Charney FA, Downs WM. Modeling procedures for panel zone deformations in moment resisting frames. Connections in Steel Structures V. Delft, the Netherlands: Bouwen met Staal; 2005 121–30. [8] Girão-Coelho A, Bijlaard F, Kolstein H. Numerical modelling of high strength steel column web shear panel behaviour. Eurosteel 2008. Brussels: ECCS European Convention for Constructional Steelwork; 2008 1125–30. [9] Hoogenbroom P, Blaauwendraad J. Quadrilateral shear panel. Eng Struct 2000;22: 1690–8. [10] Curtis H, Greiner G. A stress-based quadrilateral shear panel. Finite Elem Anal Des 1996;21:159–78. [11] Hashemi BH, Jazany RA. Study of connection detailing on SMRF seismic behaviour for unequal beam depths. J Constr Steel Res 2012;68:150–64. [12] Jordao S, Simoes da Silva L, Simoes R. Behaviour of welded beam-to-column joints with beams of unequal depth. J Constr Steel Res 2013;91:42–59. [13] Bayo E, Cabrero JM, Gil B. An effective component based method to model semi-rigid connections for the global analysis of steel and composite structures. Eng Struct 2006;28:97–108. [14] Bayo E, Gracia J, Gil B, Goñi R. An efficient cruciform element to model semi-rigid composite connections for frame analysis. J Constr Steel Res 2012;72:97–104. [15] Lopez M, Loureiro A, Bayo E. Shear behaviour of trapezoidal column panels. II: parametric study and cruciform element. J Constr Steel Res 2014. http://dx.doi. org/10.1016/j.jcsr.2014.10.029. [16] CEN. Eurocode3, EN1993-1-1-2005. Design of steel structures. Part 1.1: General rules and rules for buildings; 2005. [17] AISC. Specification for structural steel buildings. Chicago, IL: American Institute of Steel Construction; 2005. [18] Carbonell-Marquez JF, Gil-Martin LM, Hernandez-Montes E. Strength design optimization of structural steel members according to Eurocode 3. J Constr Steel Res 2013;80:213–23.
Please cite this article as: Bayo E, et al, Shear behaviour of trapezoidal column panels. I: Experiments and finite element modelling, J Constr Steel Res (2014), http://dx.doi.org/10.1016/j.jcsr.2014.10.026
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Journal of Constructional Steel Research
Shear behaviour of trapezoidal column panels. I: Experiments and finite element modelling Eduardo Bayo a,⁎, Alfonso Loureiro b, Manuel Lopez b a b
University of Navarra, Spain University of A Coruña, Spain
a r t i c l e
i n f o
Article history: Received 12 April 2014 Accepted 18 October 2014 Available online xxxx Keywords: Column shear panel Beams of unequal depths Experimental validation Finite element modelling
a b s t r a c t The behaviour of joints exerts an important influence in the overall performance of steel and composite structures, both in deformations and in ultimate resistance. As a consequence, a considerable effort has been carried out in recent years to better characterize the behaviour of joints in these kinds of structures. Modern codes, including Eurocodes 3 (EC3) and 4 (EC4), have included these research advances so that they may be used in common practice. The method adopted in EC3 and EC4 to characterize the connections is the so-called component method. One of the most important components is the column web panel in shear, and its behaviour has been investigated thoroughly for the case of rectangular shear panels arising when the beams have equal depths. However, the case of trapezoidal column panels, formed by beams of different depths at each side of the column, has not been researched as much. Formulae or design recommendations for the shear behaviour of steel beam-tocolumn joints with beams of unequal size are not currently included in design codes. The aim of this paper is the characterization of the shear behaviour of trapezoidal column panels. In order to isolate the shear behaviour from other components, stiffened beam to column connections with commercial sections are experimentally tested. Also, finite element modelling and analysis are carried out to compare results, and a mechanical model is proposed. Current modelling procedures are tested and the results compared with those coming from the experiments and numerical simulations. © 2014 Elsevier Ltd. All rights reserved.
1. Introduction The characterization of steel joint behaviour and properties has been a matter of research for a number of years, and all the accumulated knowledge has been compiled to a large extent in currently available structural steel design codes. One important aspect is the behaviour of the column panel subjected to the shear forces arising from the moments of the adjacent beams as well as the shear forces acting on the columns. A correct definition of the column panel zone deformations under static conditions is of great importance due to its influence on the overall sway behaviour of the frame. An increase in frame drift due to panel zone shear deformation may render the frame unserviceable. This may even happen for commonly considered rigid joints. Modelling of the column panel is also important for the avoidance of local failure under ultimate limit state conditions. Krawinkler et al. [1] reported the importance that panel shear deformations have on the frame behaviour under lateral loads, and proposed a formulation for the stiffness and resistance of shear panels of beam to column connections with beams of equal depths. An alternative ⁎ Corresponding author. Tel.: +34 948425600; fax: +34 948425629. E-mail address: [email protected] (E. Bayo).
formulation has been proposed in the Eurocode 3, part 1.8 [2] following the component method [3–5]. These and other proposed methods allow introducing the flexibility of joints in both the elastic and inelastic range in order to assess the frame response [6,7] and also including high strength steel [8]. The lines connecting the flanges of beams of unequal depths define a trapezoid within the panel column zone. Hoogenbroom and Blaauwendraad [9], and Curtis and Greiner [10] have proposed analytical and computational methods to characterize the shear behaviour of isolated quadrilateral panels. More recently, Hashemi and Jazany [11] have investigated the connection detailing of joints of unequal beam depths under seismic loads. One of their conclusions is that inclined stiffeners connecting the lower flanges of the beams perform better than the horizontal ones. Jordao et al. [12] have studied the performance of this type of joints for high strength steel without the use of web stiffeners. As a consequence, the stress field at the panel zone becomes rather complex due to the fact that the compression, tension and shear components are all coupled together within the panel zone. The common failure mode of the experiments carried out in [12] was web buckling due to compression. Within the context of the component method, they propose a joint model based on two subpanels delimited by 3 levels of load introduction coming from the beams. A suitable
http://dx.doi.org/10.1016/j.jcsr.2014.10.026 0143-974X/© 2014 Elsevier Ltd. All rights reserved.
Please cite this article as: Bayo E, et al, Shear behaviour of trapezoidal column panels. I: Experiments and finite element modelling, J Constr Steel Res (2014), http://dx.doi.org/10.1016/j.jcsr.2014.10.026
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1930
860
15
2
B
A
1000 15
1000
15
120
DEEP BEAM
SHALLOW BEAM
120 15
15 COLUMN
Fig. 1. Experimental setup.
modelling of the joint for global analysis can be achieved by considering the cruciform element proposed by Bayo et al. [13,14]. Work remains to be done to better characterize the complex behaviour of steel joints with beams of unequal depths and trapezoidal panels. The study involves a large number of variables and intervening factors. In this paper, experimental work and numerical (finite element) results are presented that provide important information to characterize and model this type of joints. The moment rotation diagrams depend on a large number of material, geometrical and loading variables. The research presented concentrates on the shear stiffness and resistance of the trapezoidal panel zone and the additional contribution provided by the elements surrounding the panel such as the column flanges adjacent to the panel. These elements play a major role in the joint postelastic reserve shear strength and stiffness. Since the aim is to study the shear behaviour and component of the panel zone, stiffeners are included to avoid the effects of other components such as the column web in tension and compression, and to prevent their interaction with the shear component as much as possible. Furthermore, an inclined stiffener in the lower part of the panel assures trapezoidal behaviour. Consequently, only stiffened fully welded connections were considered in this investigation. The moments acting at both sides of the joint are the most important contribution to the panel shear force, and they are usually considered as the load variables characterizing the strength and stiffness of the joint.
However, the beneficial effect of the column shear should also be considered (particularly in the case of short columns) in the joint stiffness and strength as proposed by Krawinkler et al. [1]. The deformation is defined in terms of the average shear distortion of the panel zone (PZ), and it is measured in this investigation by means of two inclinometers located on the middle section of the panel. The deformations on both sides of the panel (the shallow and deep sides) will be considered as well. The aim is to understand the mechanisms of deformation and characterize the moment rotation curves for the shear component of the trapezoidal panel. 2. Experimental work and finite element models The experimental work has been carried out in two different prototypes with beams of unequal depth. The overall scheme is illustrated in Fig. 1. The column is pinned at both ends and actuation is applied at points A and B on the attached beams. The distance from these points and the column flanges is exactly 1000 mm. Table 1 shows the beams and column sizes in both tests, and Fig. 2 depicts a picture of the experimental setup. The top horizontal and the inclined stiffeners were welded as depicted in Figs. 1 and 2 to avoid, as mentioned above, any type of failure other than that produced by shear. The stiffeners were 15 mm thick and in all cases were rigid enough to provide the necessary resistance to
Table 1 Characteristics of the tests: beam and column configurations. Test
Column
Deep beam
Shallow beam
Depth ratio
Loading point
Type of loading
E1A E1B E2A E2B
HEA 240 HEA 240 HEA 240 HEA 240
HEB 300 HEB 300 HEB 300 HEB 300
HEB 160 HEB 160 HEB 180 HEB 180
1.88 1.88 1.67 1.67
A B A B
Elastic Failure Elastic Failure
Please cite this article as: Bayo E, et al, Shear behaviour of trapezoidal column panels. I: Experiments and finite element modelling, J Constr Steel Res (2014), http://dx.doi.org/10.1016/j.jcsr.2014.10.026
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3
Fig. 4. Detail of the finite element model. Fig. 2. Experiment E1 during the loading process.
is subtracted from those at the beams to obtain the rotation due to the shear deformations. The finite element analysis was performed using Abacus®. Solid elements with reduced integration (to avoid shear locking) and hourglass control (C3D8R) were adopted. Fig. 4 illustrates the part of the model that corresponds to the column panel. The material behaviour was introduced by means of the true stress–strain data obtained from the coupons. Static nonlinear material and geometric analyses with force control were performed. The Von Misses yield criterion was selected to define the inelastic response. For all the tests, a previous simulation had been done in order to detect the load at which plastic behaviour begins. These values have been used in tests E1A and E2A as the upper limit of the applied forces, with the objective of avoiding damage in the web panels. The experiments were simulated with the exact dimensions of the elements. In all cases, the analysis was extended until the complete plastic behaviour of the panel was reached. The finite element models were refined near the most stressed zones (the panel and adjacent zones) with the aim of assessing correctly the strain and stress fields. The convergence of the model was evaluated to reach good results with a suitable level of meshing, and to allow parametric studies with a reasonable computational cost.
Table 2 Steel properties obtained from coupons. Part
σy (MPa)
σu (MPa)
E (MPa)
HEA 240 HEA 240 HEA 240 HEA 240 Stiffeners Stiffeners
330 328 326 329 300 309
493 494 496 490 446 449
207,000 211,000 209,000 210,000 213,000 211,000
prevent either tension or compression failure of the column webs, as well as out of plane bending of the column flanges. The column web was 8 mm thick. The type of steel used for all the parts was S275. Coupons were extracted from both the columns and stiffeners to obtain the true properties of the material. Table 2 shows those properties that were subsequently used in the finite element analyses for comparison purposes. The panel was instrumented with 4 strain gauges placed at the corners of the panel as shown in Fig. 3. They served to monitor the yielding sequence and the levels of shear strain. Five inclinometers were used as shown in Fig. 3 to obtain the rotations. Two were placed vertically at the beams adjacent to the joint. Two more were placed vertically at the top and bottom of the web panel to capture the possible different rotations at those levels. A fifth one was placed in the middle of the panel to capture the rotation of the column due to bending. This rotation
2
4
3. Description of the results 3.1. Test E1A The first test E1A consisted in loading the prototype E1 at the tip of the shallow beam (point A). The aim was to obtain the elastic stiffness
1
Inclinometer 2
5 1
4
3
Strain gauge
3
Fig. 3. Placement of strain gauges and inclinometers.
Please cite this article as: Bayo E, et al, Shear behaviour of trapezoidal column panels. I: Experiments and finite element modelling, J Constr Steel Res (2014), http://dx.doi.org/10.1016/j.jcsr.2014.10.026
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E. Bayo et al. / Journal of Constructional Steel Research xxx (2014) xxx–xxx
Fig. 5. Elastic shear stress contours at the end of the elastic range in experiment E1A.
of the trapezoidal panel under the shear coming from loading the shallow beam. The maximum load applied was 30 kN, and afterwards it was unloaded. A previous finite element analysis had predicted pure elastic behaviour up to that loading level. The shear stress distribution within the panel obtained from the finite element analysis is illustrated in Fig. 5, which corresponds to the level of strains just at the end of the elastic range. Fig. 6 shows the strain gauge data for this test. It may be seen that the stress distribution is uniform over an upper rectangular part of the panel. The readings of the inclinometers 3 and 4 were similar, and the inclinometers 1 and 2 showed that the rotation at the shallow beam was approximately 2.4 times the rotation at the deep beam. 3.2. Test E1B The following test, E1B, consisted in loading the prototype E1 at the tip of the deep beam (point B) until failure, which occurred at a load of 230 kN. The shear stress distributions within the panel coming from the finite element analysis during the elastic and inelastic parts of the response are illustrated in Figs. 7 and 8, respectively. It may be seen how even in the elastic range the shear stress field now extends deeper towards the bottom of the panel becoming nonrectangular in shape.
Fig. 6. Results from strain gauges in experiment E1A.
Fig. 7. Elastic shear stress levels in test E1B.
During the elastic part of the response, the readings of the inclinometers 1 and 2 also showed a rotation at the shallow side 1.2 times larger than the rotation at the deep side. However, within the inelastic response the angles at the shallow beam became much larger than those of the deep beam reaching 40% more right before failure. The final failure was due to the cracking of the panel at the upper right corner (right next to strain gauge 1) 3.3. Test E2A The test E2A consisted in loading the prototype E2 at the tip of the shallow beam. The qualitative behaviour of this test was similar to that of E1A. The maximum load applied was 37 kN, and afterwards it was unloaded. Again, a previous finite element analysis had predicted pure elastic behaviour up to that loading level. The shear stress distribution within the panel obtained from the finite element analysis is illustrated in Fig. 9, which corresponds to the level of strains just at the end of the elastic range. Fig. 10 shows the strain gauge data for this test. The readings of inclinometers 3 and 4 were similar, and the rotation at the shallow beam was approximately double the size of the rotation of the deep beam.
Fig. 8. Inelastic shear stress levels in test E1B.
Please cite this article as: Bayo E, et al, Shear behaviour of trapezoidal column panels. I: Experiments and finite element modelling, J Constr Steel Res (2014), http://dx.doi.org/10.1016/j.jcsr.2014.10.026
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Fig. 9. Elastic shear stress levels at the end of the elastic range in test E2A.
5
Fig. 11. Elastic shear stress levels in test E2B.
much more inclined than the deep beam due to the deformation of the trapezoidal panel as an articulated quadrilateral mechanism.
3.4. Test E2B The final test E2B consisted in loading the experiment E2 at the tip of the deep beam (point B) until failure, which occurred at a load of 230 kN. The shear stress distributions within the panel coming from the finite element analysis during the elastic and inelastic parts of the response are illustrated in Figs. 11 and 12 respectively. The rotation of the lower part of the panel was similar to that of the higher part right before failure. The angles at the shallow beam were 1.15 times those of the deep beam during the elastic part of the response. However, within the inelastic response the angles at the shallow beam became much larger than those of the deep beam reaching a difference of 40% right before failure. Figs. 13 and 14 show the readings of the strain gauges corresponding to tests E1B and E2B, respectively. The pictures of the deformed panel and deformed shape of the experiment before failure are shown in Figs. 15 and 16, respectively. The plastic shear stress levels of the column panel as well as the final deformed shape of the specimen obtained by finite element simulation can be seen in Fig. 17. It may be observed how the finite element model replicates the actual deformed shape of the experiment. It is also worth noting how the shallow beam gets
Fig. 10. Results from strain gauges in experiment E2A.
4. Kinematics of the deformation and mechanical model As mentioned above and as observed from tests and finite element models, three different degrees of freedom can be identified in the shear panel during the elastic range loading. Fig. 18 shows an enlarged picture of the finite element panel deformation when the shallow beam is loaded (Test E1A), and it evidences the fact that the rotation of the shallow side is larger than those of the centre and deeper parts of the panel, respectively. Fig. 19 shows the panel deformation when the deep beam is loaded (Test E1B); again the rotation of the shallow side is larger than those of the centre and deeper parts of the panel. Finally, Fig. 20 shows the deformed shape due to shear forces, coming from the column, acting in the panel; and again the same pattern is observed. As explained below, these deformations do not correspond to those of a single degree of freedom articulated quadrilateral, and therefore they are independent. The departure from the kinematics of an articulated quadrilateral in the elastic range is due to the bending deformation produced in the column flanges in both sides of the panel;
Fig. 12. Inelastic shear stress levels in test E2B.
Please cite this article as: Bayo E, et al, Shear behaviour of trapezoidal column panels. I: Experiments and finite element modelling, J Constr Steel Res (2014), http://dx.doi.org/10.1016/j.jcsr.2014.10.026
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Fig. 15. Panel inelastic deformation. Fig. 13. Results from strain gauges in test E1B.
this can be seen by observing the deformation of the shallow side in Fig. 19. In regard to the initial stiffness and deformations, Figs. 21 and 22 show the moment–rotation curves corresponding to the deep beam, the centre of the panel and the shallow beam for tests E1A and E2A, respectively. Figs. 23 and 24 show those corresponding to the tests E2A and E2B, respectively. In all cases, by observing the slopes of the moment–rotation curves one can identify different stiffness and see how the rotation of the shallow beam side is larger than the rotation of the centre of the panel, and this one is in turn larger than the rotation at the deep beam side. As mentioned above the joint shows three independent degrees of freedom in the elastic range, which do not follow the deformations of an articulated quadrilateral. However, once the panel becomes plastic, the incremental rotations follow the kinematics of a one-degree of freedom articulated quadrilateral. The data appearing in Table 3 explains this effect. It can be seen in this table how the ratios between the heights of the deep and shallow beams (the depth ratio) are very different from those of the corresponding incremental rotations in the elastic range. However, when the panel deforms within the plastic range the ratio of the incremental rotations and the depth ratio become quite similar,
thus corroborating the kinematics of a one-degree of freedom articulated quadrilateral mechanism. From the previous discussion it was possible to observe that there are three independent rotations and therefore three degrees of freedom (DOF) during the elastic range: left connection, right connection and centre of panel; and only one DOF in the inelastic range corresponding to an articulated quadrilateral mechanism. In view of this, a mechanical model is presented in Fig. 25. This model is made up of rigid bars connected by springs. The model illustrates the three possible rotations: ϕ1 on the right part of the panel, ϕ2 on the left part of the panel, and ϕp on the panel itself. When the panel becomes plastic the value of Kp becomes significantly lower than K1 and K2, and the model resembles a one-degree of freedom system. The corresponding dual internal forces are M1, M2 and V. M1 is the right side moment, M2 is the left side one, and V is the shear coming from the column. The spring Kp models the stiffness of the panel under shear; the springs K1 and K2 take into account the relative rotation between the centre of the panel and the right and left connections, respectively. The parameters h1, h2 and hp represent the equivalent height of the shallow, centre and deep parts of the joint, respectively. The values of the springs and parameters are defined after a numerical parametric analysis in the companion (Part II) paper [15].
Fig. 14. Results from strain gauges in test E2B.
Fig. 16. Test deformed shape before failure.
Please cite this article as: Bayo E, et al, Shear behaviour of trapezoidal column panels. I: Experiments and finite element modelling, J Constr Steel Res (2014), http://dx.doi.org/10.1016/j.jcsr.2014.10.026
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Fig. 17. Final stage of stress and deformation in test E2B.
5. Moment rotation curves: stiffness and resistance This section presents a comparison of the moment rotation curves obtained from the experiments, the finite element analysis, the Eurocode 3 [2] and the method proposed by Krawinkler et al. [1]. Since the last two methods do not include trapezoidal panels they are applied to each connection (left and right) as if the panel was
rectangular for the corresponding beam depth. It is important to note that the EC3 and Krawinkler's formulae were developed for rectangular panels that correspond to beams of equal depths, and as such do not fit in this study. However, they are included in this comparison to observe their results and assess the possible errors that may derive from their use to trapezoidal panels. The intention is to inform and warn those designers who may use these formulae for cases other than rectangular panels. The comparison is established in terms of the moment at the connection versus the average shear rotation of the panel, as well as the other possible rotations at each side of the joint (additional DOFs). The three experimental rotations of the panel: right, centre and left have been measured using the following relations: Rotation at the right of the panel = inclin(2) − inclin(5) Rotation at the centre of the panel = (inclin(3) + inclin(4)) /2 − inclin(5) Rotation at the left of the panel = inclin(1) − inclin(5) Inclinometers 1 and 2 measure the rotations at the left (deep) and right (shallow) parts of the panel, respectively. Inclinometers 3 and 4 measure the total rotation at the top and bottom parts on the panel along the centre line (see Fig. 3). The reading of inclinometer 5 is subtracted from all the measures to take into account the flexural rotation of the column at the joint level. Figs. 26 and 27 show the moment–rotation curves corresponding to the tests E1A and E2A, respectively. The results from the experiments
Fig. 18. Enlarged panel deformed shape in test E1A (load in shallow beam).
Fig. 19. Enlarged panel deformed shape in test E1B (load in deep beam).
Fig. 20. Enlarged panel deformed shape (shear load in panel).
Please cite this article as: Bayo E, et al, Shear behaviour of trapezoidal column panels. I: Experiments and finite element modelling, J Constr Steel Res (2014), http://dx.doi.org/10.1016/j.jcsr.2014.10.026
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Fig. 21. Shallow, centre and deep side rotations in the FEM for test E1A. Fig. 24. Shallow, centre and deep side rotations in the FEM for test E2B.
Fig. 22. Shallow, centre and deep side rotations in the FEM for test E2A.
only provide the initial stiffness since, as mentioned above, they are only loaded in the linear elastic range (the maximum applied load was 30 kN for test E1A and 37 kN for test E2A), and their plots are hidden underneath the finite element ones. Figs. 28 and 29 show the moment rotation curves corresponding to the tests E1B and E2B, respectively. It may be seen that the predictions of the finite element model in terms of stiffness and resistance are good when considering the rotation at the centre of the panel. The differences at the knee level are due to uncertainties in the modelling of the welding material properties. EC3 provides a good prediction of the stiffness but a poor prediction of the resistance, even when including the additional resistance provided by the column flanges. Krawinkler's model underestimates both the stiffness and resistance, although the latter is better approximated than EC3. Table 4 compares the values of the stiffness obtained by the different methods, as well as the relative errors, when considering the rotation of the central section of the panel as the reference for the rotations (third degree of freedom). It may be observed that the finite element model provides quite good results. The EC3 formulation also gives good results (around 5% error) and Krawinkler's model gives larger errors of up to 20%. The results are similar when considering the deep section of the panel as the reference for the rotations (second degree of freedom). Table 5 shows similar differences for the EC3 case although it provides stiffness values that are non-conservative (note that the cases E1A and E2A do not appear since there is no moment applied at that side). A much worse situation arises when considering the shallow section (first degree of freedom) of the panel as the reference for the rotations. Table 6 shows that the errors arising from the finite element model are within 10%, however the errors introduced by both the EC3 and Krawinkler's models are considerable, 97% and 57% respectively, and also non-conservative. Therefore the application of the formula provided by EC3 and Krawinkler's model is not recommended for the shallow beam side. Table 3 Incremental shallow/deep rotations in the elastic and plastic ranges.
Fig. 23. Shallow, centre and deep side rotations in the FEM for test E1B.
FEM
Depth ratio
E1A E1B E2A E2B
1.87 1.87 1.66 1.66
Incremental shallow/deep rotation Elastic range
Plastic range
2.37 1.19 2.05 1.15
1.78 1.62 1.61 1.48
Please cite this article as: Bayo E, et al, Shear behaviour of trapezoidal column panels. I: Experiments and finite element modelling, J Constr Steel Res (2014), http://dx.doi.org/10.1016/j.jcsr.2014.10.026
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Fig. 25. Proposed mechanical model of the shear panel.
6. Conclusions In this paper we have investigated the shear behaviour (component) of stiffened trapezoidal shear panels appearing in joints with unequal beam depths. Tests and FE simulations have been performed. A
Fig. 26. Moment–rotation at the centre of panel for test E1A (the test plot is underneath the FEM curve).
Fig. 27. Moment–rotation at the centre of panel for test E2A (the test plot is underneath the FEM curve).
mechanical model of rigid bars and springs has also been proposed to capture the kinematics and internal forces acting on the joint. The main conclusions can be summarized as follows: 1. The shear deformation zone corresponds to a part of the upper rectangle when loading the shallow beam, and the whole trapezoid when loading the deep beam. Consequently the mechanics of the deformation and the initial stiffness values of the left and right connections are different. 2. Three independent degrees of freedom are identified in the elastic range: the rotations corresponding to the left, right and central sections of the panel. The inelastic range is well characterized by one degree of freedom, which corresponds to an articulated quadrilateral mechanism. These degrees of freedom should be considered at the time of characterizing the joint stiffness for frame analyses. 3. The finite element analysis predicts the stiffness and resistance with good accuracy. EC3 approximates the initial stiffness reasonably well when using the dimensions of the beam attached to the corresponding connection, and the middle section and deep beam side of the panel as the references for the rotations (degrees of freedom 2 and 3). However, the initial stiffness estimated by EC3 becomes very poor when considering the rotation of the shallow beam (degree of freedom 1). Krawinkler's model provides slightly worse results than the EC3 when using the middle section and deep beam side of the panel as the reference for the rotations. However, the stiffness predicted by Krawinkler's model becomes very poor when considering the rotation at the shallow beam side.
Fig. 28. Moment–rotation at the centre of panel for test E1B.
Please cite this article as: Bayo E, et al, Shear behaviour of trapezoidal column panels. I: Experiments and finite element modelling, J Constr Steel Res (2014), http://dx.doi.org/10.1016/j.jcsr.2014.10.026
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E. Bayo et al. / Journal of Constructional Steel Research xxx (2014) xxx–xxx Table 6 Comparison of rotational stiffness (kNm/mrad) considering the shallow side of the panel.
E1A E2A
Test
FEM
Error (%)
EC3
Error (%)
Krawinkler
Error (%)
16.7 19.7
18.3 21.6
9.6 9.7
31.9 36.0
96.9 88.5
25.4 28.9
56.8 51.3
Acknowledgements The financial support provided by the Spanish Ministerio de Ciencia e Innovación under contract BIA2010-20839-02-C01-C02 is gratefully acknowledged. References
Fig. 29. Moment–rotation at the centre of panel for test E2B.
Table 4 Comparison of rotational stiffness (kNm/mrad) considering the centre of the panel.
E1A E1B E2A E2B
Test
FEM
Error (%)
EC3
Error (%)
Krawinkler
Error (%)
29.1 59.0 31.8 56.7
28.7 57.5 31.4 55.5
−1.4 −2.5 −1.3 −2.2
31.9 60.9 36.0 60.9
−2.0 −1.3 1.8 1.2
25.4 51.7 28.9 51.7
−22.0 −16.2 −18.3 −14.1
Table 5 Comparison of rotational stiffness (kNm/mrad) considering the deep side of the panel.
E1B E2B
Test
FEM
Error (%)
EC3
Error (%)
Krawinkler
Error (%)
59.8 57.5
61.3 59.2
2.5 3.0
60.9 60.9
−1.9 8.9
51.7 51.7
−16.7 −7.5
4. A mechanical model of rigid bars and springs has been proposed. This model includes the three degrees of freedom of the trapezoidal shear panel. The variables that characterize this model will be obtained after a detailed parametric analysis in the companion (Part II) paper [15]. In the context of advanced global analysis allowed by modern codes [16,17] and optimized solutions [18] the use of the proposed joint model could be particularly useful since it allows the joint to be accurately modelled within the structure.
[1] Krawinkler H, Bertero VV, Popov EP. Shear behaviour of steel frame joints. J Struct Div ASCE 1975;101(11):2317–36. [2] CEN. Eurocode 3, EN 1993-1-8:2005: Design of steel structures. Part 1.8: Design of joints. Brussels, Belgium: European Committee for Standardization; 2005. [3] Zoetemeijer P. Summary of the research on bolted beam-to-column connections (period 1978–1983). Technical Report 6-85-M. Delft: Steven Laboratory; 1983. [4] Yee YL, Melchers RE. Moment–rotation curves for bolted connections. J Struct Eng ASCE 1986;112(3):615–35. [5] Faella C, Piluso V, Rizzano G. Structural steel semi-rigid connections. Boca Raton: CRC Press; 2000. [6] Castro JM, Elghazouli AY, Izzuddin BA. Modeling of the panel zone in steel and composite frames. Eng Struct 2005;27:129–44. [7] Charney FA, Downs WM. Modeling procedures for panel zone deformations in moment resisting frames. Connections in Steel Structures V. Delft, the Netherlands: Bouwen met Staal; 2005 121–30. [8] Girão-Coelho A, Bijlaard F, Kolstein H. Numerical modelling of high strength steel column web shear panel behaviour. Eurosteel 2008. Brussels: ECCS European Convention for Constructional Steelwork; 2008 1125–30. [9] Hoogenbroom P, Blaauwendraad J. Quadrilateral shear panel. Eng Struct 2000;22: 1690–8. [10] Curtis H, Greiner G. A stress-based quadrilateral shear panel. Finite Elem Anal Des 1996;21:159–78. [11] Hashemi BH, Jazany RA. Study of connection detailing on SMRF seismic behaviour for unequal beam depths. J Constr Steel Res 2012;68:150–64. [12] Jordao S, Simoes da Silva L, Simoes R. Behaviour of welded beam-to-column joints with beams of unequal depth. J Constr Steel Res 2013;91:42–59. [13] Bayo E, Cabrero JM, Gil B. An effective component based method to model semi-rigid connections for the global analysis of steel and composite structures. Eng Struct 2006;28:97–108. [14] Bayo E, Gracia J, Gil B, Goñi R. An efficient cruciform element to model semi-rigid composite connections for frame analysis. J Constr Steel Res 2012;72:97–104. [15] Lopez M, Loureiro A, Bayo E. Shear behaviour of trapezoidal column panels. II: parametric study and cruciform element. J Constr Steel Res 2014. http://dx.doi. org/10.1016/j.jcsr.2014.10.029. [16] CEN. Eurocode3, EN1993-1-1-2005. Design of steel structures. Part 1.1: General rules and rules for buildings; 2005. [17] AISC. Specification for structural steel buildings. Chicago, IL: American Institute of Steel Construction; 2005. [18] Carbonell-Marquez JF, Gil-Martin LM, Hernandez-Montes E. Strength design optimization of structural steel members according to Eurocode 3. J Constr Steel Res 2013;80:213–23.
Please cite this article as: Bayo E, et al, Shear behaviour of trapezoidal column panels. I: Experiments and finite element modelling, J Constr Steel Res (2014), http://dx.doi.org/10.1016/j.jcsr.2014.10.026