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Parametric analysis of steel bolted end plate connections using finite element modeling
Journal of Constructional Steel Research 61 (2005) 689–708 www.elsevier.com/locate/jcsr
Parametric analysis of steel bolted end plate connections using finite element modeling Y.I. Maggia, R.M. Gonçalvesa, R.T. Leonb,∗, L.F.L. Ribeiroc a São Carlos School of Engineering, USP, São Carlos, Brazil b School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0355, USA c Federal University of Ouro Preto, Ouro Preto, Brazil
Received 16 March 2004; accepted 19 November 2004
Abstract This paper presents and discusses results of parametric analyses on the behavior of bolted extended end plate connections using Finite Element (FE) modeling tools. The analyses were calibrated to experimental results that are also briefly reviewed in this paper. The analytical models took into account material nonlinearities, geometrical discontinuities and large displacements. Comparisons between numerical and experimental data for moment-rotation curves, displacements of the end plate, and forces on bolts showed satisfactory agreement. Phenomenological T-stub failure models were also used for calculations of the flexural strength for the end plate. These models clearly support the numerical results and show how the interaction between the end plate and the bolts changes the connections’ behavior. The results presented herein show that failures associated with either formation of yield lines in the plate (Mode 1) or bolt tension failure (Mode 3) are well-defined, while failures due to combinations of these mechanisms (Mode 2) represent levels of interaction between the end plate and bolts that are difficult to predict accurately. These results also indicate that the T-stub analogy has limitations in representing the yield lines at the end plate, leading to limitations both in accounting for prying action and in predicting values for strength and stiffness of the connection. © 2005 Elsevier Ltd. All rights reserved. Keywords: End plate connections; Extended end plates; Bolted connections; Finite element modeling; Prying action; T-stub models; Yield lines; Moment–rotation curves; Bolt forces
∗ Corresponding author.
E-mail address: [email protected] (R.T. Leon). 0143-974X/$ - see front matter © 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2004.12.001
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1. Introduction In the past two decades the incorporation of semi-rigid behavior into steel connection design has attracted attention since modeling the real behavior of the connections leads to more reliable designs and economies in construction. Much of the knowledge needed to apply semi-rigid behavior for design has been derived from detailed Finite Element (FE) studies of bolted connections. These FE models have been used frequently to develop moment–rotation curves, to verify design methodologies based on yield lines and other plastic design concepts, and to assess local behavior in the connection components, such as bolts and end plates. Bolted steel connections, such as T-stubs and end plate connections, can be visualized as assemblages of components (plates, bolts, and welds). Because of (a) the large variety of connection configurations possible, (b) the many geometrical discontinuities and associated stress concentrations present in bolted connections, (c) the presence of frictional forces that lead to nonlinear phenomena such as slip, and (d) the need to model uplift and contact forces that lead to prying action, these connections exhibit an overall nonlinear structural behavior commonly classified as “semi-rigid” [19]. The connections of interest in this research are extended end plate connections, which exhibit close to rigid behavior when the plates are thick and stiffened and large bolts are used, but which may show semirigid characteristics as the plates become thinner, stiffeners are eliminated, and smaller bolts are used. Thus their behavior ranges from full-strength, full-restraint (rigid) to partialstrength, partial-restraint (semi-rigid). Existing design procedures for steel frames have been modified during the last three decades to incorporate the semi-rigid behavior of the connections into the frame design process. However, if one considers the large number of variables related to connection geometry, connection components, and constitutive relationships for their materials, the task of deriving simplified guidelines for the incorporation of semi-rigid behavior into design is a formidable analytical assignment. This task is further complicated by the need to treat the problem in three dimensions, to consider nonlinear geometric and material effects, and to include the effects of initial imperfections and residual stresses. Nevertheless, in spite of all these difficulties and complexities, a large number of advanced FE studies have been conducted on end plate connections to provide stiffness, strength, and ductility estimates for a large variety of connection geometries (early investigations include those by [16,8–11]; more recently, [20,4,22,23,14,15], to name a few). In general, these detailed studies have attempted to develop global connection behavior responses, such as moment–rotation curves, that can be readily incorporated into modern structural analysis programs. This paper first presents a short description of the experimental program that was used as the calibration basis for the numerical studies that constitute its main contribution. It then discusses the use of FE models in parametric analyses of bolted end plate connections, with emphasis on the behavior of the extended end plate. Results of parametric studies on extended end plates and their bolts are then presented in order to demonstrate how variations of geometric characteristics in those components can change the connection behavior. Finally, some remarks are made on the “component method” described in Eurocode 3 [6] regarding the flexural strength of end plate connections.
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Fig. 1. Specimen sub-assemblage and instrumentation.
2. Experimental program An experimental program was carried out at the São Carlos School of Engineering, Brazil, utilizing full-scale specimens of bolted extended end plate connections. The specimens were built with beams and columns formed by welded plates, according to the standard Brazilian specifications ([18], Ribeiro et al. 1998). There were no standard rolled steel beams available in Brazil at the time of these tests. A typical specimen, along with its instrumentation, is shown in Fig. 1.
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Table 1 Specimens tested Beam: VS 250×37 (M p = 117 kN m)a Column: CVS 350×105 (M p = 469 kN m) Column flange thickness: 19.0 mm Model ID t p (mm) db (mm)
Beam: VS 350×58 (M p = 284 kN m) Column: CVS 350×128 (M p = 578 kN m) Column flange thickness: 25.0 mm Model ID t p (mm) db (mm)
CT1A-1/CT1B-1 CT1A-2/CT1A-3 CT1B-2/CT1B-3 CT1A-4/CT1A-5 CT1B-4/CT1B-5 CT1A-6/CT1B-6
CT2A-1/CT2B-1 CT2A-2/CT2B-2 CT2A-3/CT2B-3 CT2A-4/CT2B-4 CT2A-5/CT2B-5 CT2A-6/CT2B-6
31.5 25.0 25.0 22.4 22.4 19.0
16.0 16.0 19.0 16.0 19.0 19.0
37.5 31.5 31.5 25.0 25.0 22.4
22.0 22.0 25.0 22.0 25.0 25.0
a VS (beam depth)×(kg per linear meter)/CVS (column depth)×(kg per linear meter).
For each series of connections, the end plate thickness (t p ) and bolt diameter (db ) were chosen as main variables. These parameters were varied with the explicit intent of both calibrating analytical models and observing changes on the end plate behavior rather than with the intent of testing practical configurations. The end plate thickness was varied from 19.0 to 37.5 mm, using combinations with 16.0, 19.0 and 22.0 mm diameter bolts as shown in Table 1. Each specimen had two identical connections, labeled A and B in the table. All specimens were tested in a cruciform configuration (Fig. 1) consisting of 1.50 m long beams attached to a short column by means of extended end plates. The plates were welded with full penetration welds and bolted to the column using ASTM A325 bolts preloaded according to the applicable standards. The continuity of the connections through the column was ensured by 12.5 mm thick stiffeners fillet welded to the column web and flanges. Minimum distances of 3db between the centerline of the bolts and 2db for the end distance to either the end plate edge or the beam flange face were used. Fig. 1 also shows the extensive instrumentation used to track the connection behavior. A typical geometry adopted for these end plates is shown in Fig. 2 for the cases of 16.0 and 19.0 mm bolts and a VS 250 × 37 beam. The main experimental observations included extensive prying action of the bolts (Fig. 3(a)) and double-curvature deformations in the end plate (Fig. 3(b)). Interestingly, the center of rotation for the plate was not located at the level of the compression flange, as usually assumed in design, but at the level of the first bolt in compression. The results also showed that while the variation on the end plate thickness had a great influence on the plastic behavior of the connections using 16.0 mm bolts, this influence decreased markedly as the diameter of the bolts increased. The initial stiffness was also affected by the variation of the plate thickness but to a lesser degree. These results highlighted the importance of using robust FE models to explain these observations and to provide data on which to base simplified design models [18,4]. Detailed results of the experimental program are available in [18]. Some of the moment–rotation relationships from these specimens are shown in Fig. 4. The rotations were calculated by dividing the readings from LVDT No. 1 (see Fig. 1) by the distance between the middle planes of beam flanges. Thus this rotation corresponds to the one due to the deformations of the end plates and bolts only. The experimental program shown
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Fig. 2. Typical end plate and beam geometry for 16.0 and 19.0 mm bolts and VS 250 × 37 beam.
(a) Specimen CT2A-5.
(b) Specimen CT1B-6. Fig. 3. Prying action and plate deformations.
in Table 1 is being expanded by two other ongoing research projects [23,7]. Data from some of those recent tests, with similar specimen geometry to CT1A-4, are also used in this paper to assess the accuracy of the numerical models.
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Fig. 4. Moment–rotation relationships for selected specimens.
3. Numerical modeling The complex behavior observed in the tests described above led to some observations on the merits of using two- vs. three-dimensional models for end plate connections. Krishnamurthy conducted pioneering 2D finite element analyses that were used to develop the original AISC [2] design recommendations for end plate connections [3,10,12]. Although 2D models are easier to mesh, more computationally efficient, and their results easier to interpret [5], there are phenomena that can be better visualized and understood with the use of 3D models. While 3D models are often difficult to mesh, advances in hardware performance allow today for very sophisticated models to be run without a substantial penalty in computation time. Thus the models utilized in this research were carried out with 3D solid and contact elements that can accurately simulate bolted connection behavior [4,14,15]. The numerical models used for the analyses (Table 2) were chosen from some of the monotonically loaded specimens from the experimental program discussed above. ANSYS [1] first-order solid hexahedral elements (SOLID45, 8 nodes, 3 translation DOF’s per node) with incompatible displacement modes were used to model the connection geometry. The geometrical discontinuities were simulated with surface-tosurface contact elements (TARGE170 and CONTA173). The numerical models also incorporate nonlinear constitutive laws for materials using multilinear “true stress” vs. “natural strain” curves [5,4]. To shorten the computation time, beam elements were used to substitute the solid elements far from the connection, and the rotations were transferred to the solid elements by means of constraint equations. Displacements of the column flange in specific regions were restrained to represent the continuity plates and the column web in order to adequately simulate the cruciform configuration used during the experimental tests. Symmetry through the middle plane of the beam web was also used to reduce the model size. Fig. 5 shows
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Table 2 Numerical models from the experimental program Model ID
End plate thickness—t p (mm)
Bolt diameter—db (mm)
CT1A-1 CT1A-2 CT1A-4 CT1B-2 CT1B-4 CT1B-6
31.5 25.0 22.4 25.0 22.4 19.0
16.0 16.0 16.0 19.0 19.0 19.0
Fig. 5. General view of the FE models.
a general view of the meshes used for these analyses, including some of the restraint conditions. A set of four elements through the thickness of the end plate was used to accurately simulate the flexural behavior of this component [4]. The thicknesses of beam flanges and column flange were simulated with a set of only two elements, as flexural deformations on those components were negligible and do not affect the connection behavior [15]. Loading was applied in two steps. First, temperature gradients were used in the bolts to impose pretension forces. The temperature variation was chosen so as to produce the standardized preloads of 85 and 125 kN for 16.0 and 19.0 mm bolts, respectively [3,17]. A displacement at the beam tip was then imposed to generate a bending moment at the connection. Large displacement and large strain features were also used in the analysis to adequately simulate the connection behavior in ultimate states and the membrane effects that were expected for the thin connecting elements [4,15].
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Fig. 6. Moment–rotation curves for variations of model CT1A-4.
4. Assessments of the numerical models To assess the accuracy of any connection model, results obtained with FE models should be compared with actual data at least insofar as initial stiffness, which depends on the connection’s geometry and elastic modulus; ultimate states and, thus, failure modes; and lastly, the stress redistribution when the connection begins plastic work, i.e., the onset and development of nonlinear behavior. In general, it is possible to compare the values for strength and stiffness since they depend on well-defined variables. However, for nonlinear effects, only qualitative comparisons can be made owing to the difficulty in correctly implementing the nonlinear constitutive laws for materials, especially for situations where only uniaxial values of the stress–strain curves are available. All results presented along this paper are focused on verifying that the numerical modeling is a potential tool to satisfactorily replicate the experimental connection behavior and its variations. Fig. 6 shows a comparison of the experimental and analytical results for the moment–rotation curve of model CT1A-4. This figure is a typical example of the quality of the numerical results for the connection stiffness. The curves presented in Fig. 6 clearly indicate the overprediction of the initial stiffness of the numerical model when compared with the experimental specimen. In general, this behavior was observed for all simulations conducted. The differences between numerical and experimental curves can have a variety of sources, often being a direct consequence of the simplifications introduced in the numerical modeling. Among these factors are (1) imperfections originated from the assembly of the tested models; (2) the effects of residual stresses; and (3) representation of stress–strain behavior for materials in the FE models. With respect to initial stiffness, the level of pretension in the bolts can have a significant influence. First, the differences observed can be attributed to the pretension control (the turn-of-nut method) used in the experimental program presented earlier. The effect of variations in pretension on initial stiffness is illustrated by a comparison between models CT1A-4 and CT1A-4T, also presented in Fig. 6. Model CT1A-4T included no pretension
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on the bolts. Observations made during the experimental program revealed that some bolts lost pretension during the assemblage sequence. Thus the pretension load applied to the FE models was probably not achieved in the tests. To emphasize the effect of pretension, the moment–rotation curve for a specimen recently tested (CT1A-4N) is also shown in Fig. 6. For this model, the pretension was applied with torque control and shows better agreement for initial stiffness. Another aspect to take into account is the nonlinear behavior observed in the FE models. As the yielding and hardening of the components depends on constitutive laws based on simple stress–strain relationships, difficulties arise when trying to accurately simulate material behavior from uniaxial experimental tests. Even when data from characterization of materials are available, the theoretical models used to simulate the nonlinear behavior are not allowed to work with all phenomena embodied by materials, such as stress concentrations, some behavioral changes in tension and compression, and softening at the fracture limit. The numerical models presented herein only used the yield stress information from the characterization of materials, taking advantage of standardized information for maximum tensile strengths and strains for the steel of plates and bolts. In a parametric analysis using the FE models, the use of idealized stress–strain relationships is acceptable, considering the variability observed from test to test. The same is true concerning the fracture or ultimate limit state. In general, the failure to converge numerically in the nonlinear range is used as a definition for failure of the connection, as it identifies some model instability. This nonlinear convergence depends on several variables, especially when contact elements are being used, and it does not always indicate the actual failure point. Therefore, one can use simple inspection procedures to identify the ultimate states, mainly looking for the stress and strain states in the bolts. 5. Numerical results The figures in Table 3 illustrate the deformed shapes of the end plates for the six numerical models at a bending moment of about 150 kN m at the end plate. This value corresponds to a moment about 30% higher than the theoretical plastic moment of the beam section (M p = 117 kN m). This large overstrength was achieved due to the large hardening range found between the yield stress (Fy = 288 MPa) and the maximum tensile stress (Fu = 460 MPa) of the steel plates used in the tests. The displacements are magnified for better visualization. Also shown in Table 3 are the cross-sections where the displacements were calculated from the numerical models. In Table 3 one can clearly identify the behavioral changes that may occur depending on the changes in bolt diameter and end plate thickness. Comparisons are made below for three pairs of models. Focusing first on the end plate behavior, the relative displacements between the end plate and the column flange for the model CT1A-1 (t p = 31.5 mm; db = 16.0 mm) are presented in Fig. 7 for section AA (see Table 3 for location), for a sequence of increasing bending moments at the connection. The displacements of the end plate are plotted against the end plate depth showing the deformed shape at the chosen section. One can observe the development of a complete gap between the end plate and the column flange in the
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Table 3 Deformed shapes and calculation sections for end plates
tension zone as the load is increased, due primarily to bolt elongation. The thick end plate leads to this specific behavior as it has a high stiffness in out-of-plane bending. Similar behavior can be found at the plane of symmetry (Section CC ), where displacements are slightly higher. Fig. 8 presents the end plate displacements along the end plate width for the top edge of the end plate (Section DD ). Fig. 8, in conjunction with Fig. 7, emphasizes the bi-directional bending that occurred systematically for all models. Fig. 9 shows a comparison of deformations between CT1A-4 (t p = 22.4 mm; db = 16.0 mm), with a thinner end plate, and CT1A-1 (t p = 31.5 mm; db = 16.0 mm) at section AA for moments equal to M p (117 kN m) and 1.3M p (150 kN m). For the same bolt diameter, as the end plate thickness decreases changes in the deformed shape are clearly observed, reflecting changes in the overall connection behavior. The end plate top edge in model CT1A-4 maintains contact with the column flange even for high bending moments, leading to significant prying effects on the bolts. Moreover, the end plate displacements at the tension beam flange increase for the thinner plate, indicating higher levels of bending in the end plate. As compared to the results observed for model CT1A-1, CT1A-4 shows more interaction between the end plate and bolts. At this point, some discussion about the distribution of forces in the bolts becomes important inasmuch as the prying action increases as the end plate thickness decreases.
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Fig. 7. End plate relative displacements—section AA in model CT1A-1.
Fig. 8. End plate relative displacements—section DD in model CT1A-1.
Fig. 10 shows the variation of the axial forces in the bolts in the tension zone for model CT1A-1. Curves for the two tensioned bolt lines are plotted, one for the upper bolt, located at the end plate extension, and other for the lower bolt, just below the beam flange. The theoretical force F ∗ was obtained by distributing equally the total tension force among all bolts in the tension zone. As one can observe from Fig. 10, the loss of contact between the end plate and the column flange avoids the prying effect, allowing a uniform distribution of forces for all bolts in the tension zone. Some small variations can be observed, but in general the curves in Fig. 10 evidence the high stiffness in bending for the end plate of
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Fig. 9. End plate displacements—section AA in models CT1A-1 and CT1A-4.
Fig. 10. Axial forces in bolts—model CT1A-1.
model CT1A-1. The initial loading, with moments up to 80 kN m, occurs with an almost constant axial force in the bolts, as expected. Fig. 11, which shows the bolt forces for model CT1A-4, clearly indicates that higher forces are transferred to the lower bolts. Although this irregular distribution depends on the relative stiffness between the end plate and bolts, it is clear that the decrease of the end plate thickness leads to a higher stress in the lower bolts. The changes in the force distribution in the bolts are related to the changes in the end plate behavior. For model CT1A-4, additional forces are imposed to the lower bolt line due to the decrease of the end plate stiffness. The mechanism is illustrated in Fig. 12, where
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Fig. 11. Axial forces in bolts—model CT1A-4.
Fig. 12. Development of prying action in model CT1A-4.
Q s and Q i are the reaction forces from the column flange and Fle and Fli are the forces in the upper and lower bolts, respectively. From equilibrium of forces, the longer lever arm between Q i and Fli has to impose more tension forces in the lower bolts, confirming the behavior described above. In the case of model CT1A-4, the forces in the lower bolts are 30% higher than forces in the upper bolts at the end of loading. In general, all trends mentioned above are also valid for an increase in the bolt diameter. The prying effects for CT1A-4 depends on the relative stiffness between end plate and
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Fig. 13. Axial forces in bolts—model CT1B-4.
bolts, although one can observe that the distribution of forces in the bolts tends to be more uniform for CT1B-4 (t p = 22.4 mm; db = 19.0 mm), as shown in Fig. 13, at least for a larger range of loading. The higher tension in the lower bolt at the end of loading for CT1B-4 can be attributed to concentrated yielding at the end plate extension, which is not stiffened by the beam web. The results for CT1B-4 showed that the end plate top edge remained in contact with the column flange even for the maximum applied load. The initial uniformity in the bolt forces occurs because of the concentration of displacements at the tensioned beam flange due to the high stiffness of bolts relative to the end plate. Putting all these results together and looking at Table 3 again, it is possible to identify the three main deformation modes. In model CT1A-1, the thick end plate allows the bolts to be the source of most of the displacements. In model CT1A-4 the interaction between the end plate and the bolts increases and for model CT1B-4 the displacements are concentrated in the end plate tension zone. The observations are also in agreement with the collapse modes presented for T-stub connections [24,13], showing the interdependence between the distribution of forces in bolts and the end plate thickness indicated by a higher or lower degree of prying action. As a general conclusion, the FE models adequately simulate all the observed behavioral changes. In addition, the results reviewed above indicate the feasibility of utilizing FE modeling for the development of a parametric analysis of bolted end plate connections. 6. End plates and T-stub failure modes 6.1. Basic concepts The results presented above indicated that the distribution of forces in the bolts is generally non-uniform, depending on the prying action developed as a consequence of
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Fig. 14. Failure modes for T-stubs in Eurocode-3.
the end plate behavior. Therefore, the possible use of a plastic distribution of tensile forces in the bolts, as opposed to the conventional triangular distribution, may be considered as an improvement for connection design. Such an approach is possible if one takes advantage of the so-called “component method”, which has as its basis dividing the connection into basic components to evaluate its overall behavior. By identifying separately the sources of displacements, this method allows for a more detailed understanding of the behavior of the connection, providing an approach for developing analytical formulations to characterize strength and stiffness. For end plates loaded primarily by flexure and for the column flange, the “component method” suggests the use of equivalent T-stubs, as presented in Eurocode-3 [6,21]. The formulations are based on the energy method, involving a complex set of yield lines around the bolts of the end plate and leading to the effective length of a T-stub, which represents each line of bolts or a group of them by means of equivalent strength and displacements. Fig. 14 illustrates three possible failure modes that can occur in T-stubs representing the area around the bolt lines. To check flexure, the lowest value obtained for the strength capacity of each bolt line should be adopted, considering the following: Mode 1 Complete yielding of the extended end plate or column flange near the bolts; Mode 2 Bolt failure with yielding of the flange (plate or column); Mode 3 Bolt failure. Some comparisons between design and experimental results [4] reveal that this design procedure still requires refinements, mainly for thinner end plates, for which the analytical model used for T-stubs underestimates the strength. This is due to the complex nonlinear phenomena embodied by these bolted connections. The next section provides some discussion about the effectiveness of using T-stub models. 6.2. End plates in bending The failure modes of models CT1A-1, CT1A-4, CT1B-4 and CT1B-6 are discussed below. The upper line of bolts at the end plate extension is labeled “line 1”, and the line located just below the beam’s flange at the tension zone is labeled “line 2” (see Fig. 15). The three failure modes for T-stubs presented in Fig. 14 are also used throughout these discussions. For purposes of comparison, only the strength values for the extended end plate in bending is used, since the column can be considered sufficiently rigid as to not impact behavior significantly.
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Fig. 15. End plate displacements—section BB’ in models CT1A-1 and CT1A-4.
Table 4 Strength of the extended end plate in bending using Eurocode-3 models CT1A-1 and CT1A-4 Model
t p (mm)
db (mm)
Line of bolts
Failure modes—Pr (kN) 1 2
3
CT1A-1
31.5
16.0
1 2
476.5 1414.0
213.4 482.3
180.0 180.0
CT1A-4
22.4
16.0
1 2
240.9 715.0
145.8 281.8
180.0 180.0
The strength of models CT1A-1 (t p = 31.5 mm) and CT1A-4 (t p = 22.4 mm), both with 16.0 mm bolts, is shown in Table 4 for each line of bolts and for the three possible failure modes. The capacity (Pr ) of each line was calculated using the procedures indicated by Eurocode-3 [6], the yield stress data from characterization of the steel plates (Fy = 288 MPa) as presented in the earlier section, and the nominal yield stress for the ASTM-A325 bolts (Fy = 635 MPa). In most cases, bolt tensile failure controls the strength of the connections, as seen for both lines of bolts in CT1A-1. In line 2, a decrease of the plate thickness (CT1A-4) leads to a significant reduction of the strength for failure modes 1 and 2 and increases bending in the end plate. Fig. 15 illustrates the relative displacements of the end plate for models CT1A-1 and CT1A-4 at section BB’ (through the center of holes), for a bending moment of 117 kN m (M p of the beam) in the connection. The end plate of CT1A-1 shows small displacements due to bending, with tensile failure of the bolts’ controlling strength. Model CT1A-4 shows a different behavior, resulting in prying forces and irregular distribution of forces in the bolts. The displacements in lines 1 and 2 are greater, with a noticeable increase for the end plate bending. The experimental results for both CT1A-1 and CT1A-4 showed that the connections failed as a result of bolt failures in the tension zone, which is consistent
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Table 5 Strength of the extended end plate in bending using Eurocode-3 models CT1A-1 and CT1A-4 Model
t p (mm)
db (mm)
CT1B-4
22.4
19.0
CT1B-6
19.0
19.0
Line of bolts 1 2 1 2
Failure Modes—Pr (kN) 1 2
3
275.0 727.4 197.8 523.4
254.0 254.0 254.0 254.0
195.5 309.1 176.1 257.9
Fig. 16. End plate displacements—models CT1B-4 and CT1B-6.
with the predictions for the models. For CT1A-4, the strength of the upper line is not only limited by the failure of the bolts, but also by the end plate bending. The upper line failure observed experimentally in model CT1A-4 is representative of the cases in which there is a strong interaction between the extended end plate yielding and bolt failure. This behavior is also in agreement with the calculations, although it is difficult to clearly identify the relative contribution of these two components to the connection failure. Table 5 lists the strength of models CT1B-4 (t p = 22.4 mm) and CT1B-6 (t p = 19.0 mm), with 19.0 mm bolts, indicating the decrease in strength and the increasing closeness of the values calculated for failure modes 1 and 2 as the end plate thickness decreases. The relative end plate displacements for CT1B-4 and CT1B-6 are presented in Fig. 16 for sections EE’ and GG’ (see Table 3) for a moment equal to M p (117.0 kN m). One can observe the small displacements in the bolts and the predominance of bending in the end plate. For CT1B-6, the calculations predict failure mode 2 governing the strength for line 1, but the value (Pr,2 = 176.1 kN) is very close to the one obtained for failure mode 1 (Pr,1 = 197.8 kN). As the end plate thickness decreases and the bolt diameter increases, the failure mode for line 1 is governed by the bending of the end plate, as indicated in Table 5. For line 2, the values for failure modes 3 and 2 are very close, indicating that the
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beam web stiffens the lower bolts. Because of this, the collapse for line 2 is not governed by the end plate bending. The experimental tests for the CT1B-6 geometry showed a bolt failure in line 2, in qualitative agreement with the predictions. On the other hand, the failure observed experimentally for CT1B-4 occurred at both lines of bolts, indicating the stiffening effect at the end plate extension. Failure mode 2, calculated for line 1 in CT1B-4, shows higher values when compared with CT1B-6, as expected, but it is clear that a larger increase occurs for failure mode 1 from CT1B-6 to CT1B-4. CT1B-4 should be placed into a group of connections in which the failure mode is hard to identify clearly, and is highly dependent on the relative stiffness between the end plate and the bolts. This is true also for specimen CT1A-4 as presented earlier. All these results emphasize that the predictions for the T-stub failure modes are in good qualitative agreement for mode 3 and mode 1, although calculations for the strength in situations where there is strong interaction between the end plate and the bolts (mode 2) still require refinements. This is mainly due to the fact that the yield lines patterns observed in tests with T-stubs do not necessarily match those from extended end plate tests [15]. Thus, the T-stub theory does not allow prying forces to be considered appropriately when applied to the extended end plate connections. 7. Concluding remarks The results presented herein focused on the behavioral variations of bolted extended end plate connections due to changes in plate thickness and bolt diameter. It also discussed the application of FE models as tools to perform parametric analyses in order to assess the accuracy of commonly used design procedures and to provide data for development of new analytical models. Six numerical models, and associated experimental specimens, were discussed from the standpoint of their overall stiffness, displacements of the end plate and axial forces in the bolts. A set of issues was presented for the modeling, and the numerical results demonstrate the feasibility of the FE models to simulate the connection behavior and to capture behavioral changes as consequence of geometric variations. These numerical results were found to be in good qualitative agreement with the actual connection behavior, although they also evidenced remaining difficulties in quantifying ultimate states when the failure mechanism is not well defined. Calculations on the flexural strength for the end plate were made, showing the predictions for the T-stub failure modes applied to end plate connections and emphasizing the interaction between the end plate and the bolts. The strength of each line of bolts in the tension zone was calculated for four numerical models, following the procedures recommended by Eurocode-3. The analytical results were compared qualitatively with the numerical behavior of the end plate in bending, showing that failure modes 1 and 3 are well-defined if the loads required for these modes are well separated. Failure mode 2, however, leads to levels of interaction between the end plate and bolts that are difficult to model accurately. These results allow putting in discussion the reliability of applying the “T-stub theory” for the extended end plate, mainly by means of yielding lines and “equivalent T-stubs”.
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These statements reaffirm that bolted end plate connections involve complex phenomena, clearly hard to simulate with simplified analytical formulations. They also reaffirm the use of FE models as a potential tool for refinements in the current design approaches in order to improve the characterization of the actual behavior of connections. Acknowledgements The authors are grateful for the financial support provided by FAPESP (Brazilian Research Foundation) for the development of all the research work, whose partial results are presented herein. Some of the results were generated during a visit by the senior author to the Georgia Institute of Technology (USA). The help of Ms. Lillie Brantley and Dr. Rami Haj-Ali during that visit was particularly appreciated. References [1] ANSYS Element Reference. ANSYS 6.0 documentation. Online help. [2] American Institute of Steel Construction. Manual of steel construction. 8th ed, Chicago; 1980. [3] American Institute of Steel Construction (1994). Manual of steel construction: load and resistance factor design. 2nd ed., vol. 2, Chicago. [4] Bursi OS, Jaspart JP. Basic issues in the finite element simulation of extended end plate connections. Computer & Structures 1998;69:361–82. [5] Bahaari MR, Sherbourne AN. Computer modeling of an extended end-plate bolted connection. Computers & Structures 1994;52(5):879–93. [6] Eurocode 3. Design of steel structures: Part 1.1—General rules and rules for buildings—Revised annex J: Joints in building frames; 1993. [7] Figueiredo LMB. Composite steel-concrete connections—theoretical and experimental analysis. Doctorate’s thesis. São Carlos. Ph.D thesis, São Carlos School of Engineering, University of São Paulo; 2004 [in Portuguese]. [8] Krishnamurthy N. Two-dimensional finite element analysis of steel end-plate connections: parametric considerations. Auburn, Alabama, Auburn University, Report n. CE-AISC-MBMA-3; 1974. [9] Krishnamurthy N. Parameter study of steel end-plate connections by two-dimensional finite element analysis. Auburn, Alabama, Auburn University, Report n. CE-AISC-MBMA-5; 1974. [10] Krishnamurthy N. Two-dimensional finite element analysis of extended and flush connections with multiple rows of bolts. Auburn, Alabama, Auburn University, Report n. CE-AISC-MBMA-6; 1975. [11] Krishnamurthy N. Tests on bolted end-plate connections and comparisons with finite element analysis. Auburn, Alabama, Auburn University, Report n. CE-AISC-MBMA-7; 1975. [12] Krishnamurthy N. Design of end-plate connections. Vanderbilt, Tennessee, Vanderbilt University, Report n. CE-AISC-MBMA-10; 1976. [13] Kulak GL, Fisher JW, Struik JH. Guide to design criteria for bolted and riveted joints. 2nd ed. New York: John Wiley & Sons; 1987. [14] Maggi YI. Theoretical Analysis, using F.E.M, of Bolted Beam-to-Column Connections with Extended End Plate. São Carlos. Master’s Dissertation, São Carlos School of Engineering, University of São Paulo; 2000 [in Portuguese]. [15] Maggi YI. Analysis of the structural behavior of bolted beam-column extended end plate connections. São Carlos. Ph.D thesis. São Carlos School of Engineering, University of São Paulo; 2004 [in Portuguese]. [16] Nair RS, Birkemoe PC, Munse WH. High-strength bolts subject to tension and prying. Journal of Structural Division 1974;100(ST2):351–72. [17] NBR-8800. Design and Construction of Steel Structures for Buildings. Associação Brasileira de Normas Técnicas (Brazilian Standards Association), Rio de Janeiro; 1986.
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[18] Ribeiro LFL. Study of the Structural Behavior of Bolted Beam-to-Column Connections with Extended End Plate: Theoretical and Experimental Analysis. São Carlos. Ph.D. thesis, São Carlos School of Engineering, University of São Paulo; 1998 [in Portuguese]. [19] Ribeiro LFL, Gonçalves RM, Castiglioni CA. Beam-to-column end plate connections—An experimental analysis. Journal of Constructional Steel Research Surrey—England 1998;46:264–6. [20] Sherbourne AN, Bahaari MR. Finite element prediction of end plate bolted connection behavior. 2: Analytic formulation. Journal of Structural Engineering 1997;123(2):165–75. [21] Steel Construction Institute/British Constructional Steelwork Association. Joints in steel construction: moment connections. London, SCI/BCSA; 1996. [22] Swanson JA. Characterization of the strength, stiffness, and ductility behavior of T-stub connections. Ph.D. dissertation, Georgia Institute of Technology, 1999. [23] Tristão GA, Figueiredo LB, Gonçalves RM. Theoretical and Experimental Analysis on the behavior of composite steel–concrete connections with extended end plate. IV Congresso da Construção Metálica e Mista, Lisboa; 2003 [in Portuguese]. [24] Zoetemeijer P, Back Jde. High strength bolted beam to column connections. The computation of bolts, Tstub flanges and column flanges. Report 6-72-13, Delft University of Technology, Stevin Laboratory; 1972.
Parametric analysis of steel bolted end plate connections using finite element modeling Y.I. Maggia, R.M. Gonçalvesa, R.T. Leonb,∗, L.F.L. Ribeiroc a São Carlos School of Engineering, USP, São Carlos, Brazil b School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0355, USA c Federal University of Ouro Preto, Ouro Preto, Brazil
Received 16 March 2004; accepted 19 November 2004
Abstract This paper presents and discusses results of parametric analyses on the behavior of bolted extended end plate connections using Finite Element (FE) modeling tools. The analyses were calibrated to experimental results that are also briefly reviewed in this paper. The analytical models took into account material nonlinearities, geometrical discontinuities and large displacements. Comparisons between numerical and experimental data for moment-rotation curves, displacements of the end plate, and forces on bolts showed satisfactory agreement. Phenomenological T-stub failure models were also used for calculations of the flexural strength for the end plate. These models clearly support the numerical results and show how the interaction between the end plate and the bolts changes the connections’ behavior. The results presented herein show that failures associated with either formation of yield lines in the plate (Mode 1) or bolt tension failure (Mode 3) are well-defined, while failures due to combinations of these mechanisms (Mode 2) represent levels of interaction between the end plate and bolts that are difficult to predict accurately. These results also indicate that the T-stub analogy has limitations in representing the yield lines at the end plate, leading to limitations both in accounting for prying action and in predicting values for strength and stiffness of the connection. © 2005 Elsevier Ltd. All rights reserved. Keywords: End plate connections; Extended end plates; Bolted connections; Finite element modeling; Prying action; T-stub models; Yield lines; Moment–rotation curves; Bolt forces
∗ Corresponding author.
E-mail address: [email protected] (R.T. Leon). 0143-974X/$ - see front matter © 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2004.12.001
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1. Introduction In the past two decades the incorporation of semi-rigid behavior into steel connection design has attracted attention since modeling the real behavior of the connections leads to more reliable designs and economies in construction. Much of the knowledge needed to apply semi-rigid behavior for design has been derived from detailed Finite Element (FE) studies of bolted connections. These FE models have been used frequently to develop moment–rotation curves, to verify design methodologies based on yield lines and other plastic design concepts, and to assess local behavior in the connection components, such as bolts and end plates. Bolted steel connections, such as T-stubs and end plate connections, can be visualized as assemblages of components (plates, bolts, and welds). Because of (a) the large variety of connection configurations possible, (b) the many geometrical discontinuities and associated stress concentrations present in bolted connections, (c) the presence of frictional forces that lead to nonlinear phenomena such as slip, and (d) the need to model uplift and contact forces that lead to prying action, these connections exhibit an overall nonlinear structural behavior commonly classified as “semi-rigid” [19]. The connections of interest in this research are extended end plate connections, which exhibit close to rigid behavior when the plates are thick and stiffened and large bolts are used, but which may show semirigid characteristics as the plates become thinner, stiffeners are eliminated, and smaller bolts are used. Thus their behavior ranges from full-strength, full-restraint (rigid) to partialstrength, partial-restraint (semi-rigid). Existing design procedures for steel frames have been modified during the last three decades to incorporate the semi-rigid behavior of the connections into the frame design process. However, if one considers the large number of variables related to connection geometry, connection components, and constitutive relationships for their materials, the task of deriving simplified guidelines for the incorporation of semi-rigid behavior into design is a formidable analytical assignment. This task is further complicated by the need to treat the problem in three dimensions, to consider nonlinear geometric and material effects, and to include the effects of initial imperfections and residual stresses. Nevertheless, in spite of all these difficulties and complexities, a large number of advanced FE studies have been conducted on end plate connections to provide stiffness, strength, and ductility estimates for a large variety of connection geometries (early investigations include those by [16,8–11]; more recently, [20,4,22,23,14,15], to name a few). In general, these detailed studies have attempted to develop global connection behavior responses, such as moment–rotation curves, that can be readily incorporated into modern structural analysis programs. This paper first presents a short description of the experimental program that was used as the calibration basis for the numerical studies that constitute its main contribution. It then discusses the use of FE models in parametric analyses of bolted end plate connections, with emphasis on the behavior of the extended end plate. Results of parametric studies on extended end plates and their bolts are then presented in order to demonstrate how variations of geometric characteristics in those components can change the connection behavior. Finally, some remarks are made on the “component method” described in Eurocode 3 [6] regarding the flexural strength of end plate connections.
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Fig. 1. Specimen sub-assemblage and instrumentation.
2. Experimental program An experimental program was carried out at the São Carlos School of Engineering, Brazil, utilizing full-scale specimens of bolted extended end plate connections. The specimens were built with beams and columns formed by welded plates, according to the standard Brazilian specifications ([18], Ribeiro et al. 1998). There were no standard rolled steel beams available in Brazil at the time of these tests. A typical specimen, along with its instrumentation, is shown in Fig. 1.
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Table 1 Specimens tested Beam: VS 250×37 (M p = 117 kN m)a Column: CVS 350×105 (M p = 469 kN m) Column flange thickness: 19.0 mm Model ID t p (mm) db (mm)
Beam: VS 350×58 (M p = 284 kN m) Column: CVS 350×128 (M p = 578 kN m) Column flange thickness: 25.0 mm Model ID t p (mm) db (mm)
CT1A-1/CT1B-1 CT1A-2/CT1A-3 CT1B-2/CT1B-3 CT1A-4/CT1A-5 CT1B-4/CT1B-5 CT1A-6/CT1B-6
CT2A-1/CT2B-1 CT2A-2/CT2B-2 CT2A-3/CT2B-3 CT2A-4/CT2B-4 CT2A-5/CT2B-5 CT2A-6/CT2B-6
31.5 25.0 25.0 22.4 22.4 19.0
16.0 16.0 19.0 16.0 19.0 19.0
37.5 31.5 31.5 25.0 25.0 22.4
22.0 22.0 25.0 22.0 25.0 25.0
a VS (beam depth)×(kg per linear meter)/CVS (column depth)×(kg per linear meter).
For each series of connections, the end plate thickness (t p ) and bolt diameter (db ) were chosen as main variables. These parameters were varied with the explicit intent of both calibrating analytical models and observing changes on the end plate behavior rather than with the intent of testing practical configurations. The end plate thickness was varied from 19.0 to 37.5 mm, using combinations with 16.0, 19.0 and 22.0 mm diameter bolts as shown in Table 1. Each specimen had two identical connections, labeled A and B in the table. All specimens were tested in a cruciform configuration (Fig. 1) consisting of 1.50 m long beams attached to a short column by means of extended end plates. The plates were welded with full penetration welds and bolted to the column using ASTM A325 bolts preloaded according to the applicable standards. The continuity of the connections through the column was ensured by 12.5 mm thick stiffeners fillet welded to the column web and flanges. Minimum distances of 3db between the centerline of the bolts and 2db for the end distance to either the end plate edge or the beam flange face were used. Fig. 1 also shows the extensive instrumentation used to track the connection behavior. A typical geometry adopted for these end plates is shown in Fig. 2 for the cases of 16.0 and 19.0 mm bolts and a VS 250 × 37 beam. The main experimental observations included extensive prying action of the bolts (Fig. 3(a)) and double-curvature deformations in the end plate (Fig. 3(b)). Interestingly, the center of rotation for the plate was not located at the level of the compression flange, as usually assumed in design, but at the level of the first bolt in compression. The results also showed that while the variation on the end plate thickness had a great influence on the plastic behavior of the connections using 16.0 mm bolts, this influence decreased markedly as the diameter of the bolts increased. The initial stiffness was also affected by the variation of the plate thickness but to a lesser degree. These results highlighted the importance of using robust FE models to explain these observations and to provide data on which to base simplified design models [18,4]. Detailed results of the experimental program are available in [18]. Some of the moment–rotation relationships from these specimens are shown in Fig. 4. The rotations were calculated by dividing the readings from LVDT No. 1 (see Fig. 1) by the distance between the middle planes of beam flanges. Thus this rotation corresponds to the one due to the deformations of the end plates and bolts only. The experimental program shown
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Fig. 2. Typical end plate and beam geometry for 16.0 and 19.0 mm bolts and VS 250 × 37 beam.
(a) Specimen CT2A-5.
(b) Specimen CT1B-6. Fig. 3. Prying action and plate deformations.
in Table 1 is being expanded by two other ongoing research projects [23,7]. Data from some of those recent tests, with similar specimen geometry to CT1A-4, are also used in this paper to assess the accuracy of the numerical models.
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Fig. 4. Moment–rotation relationships for selected specimens.
3. Numerical modeling The complex behavior observed in the tests described above led to some observations on the merits of using two- vs. three-dimensional models for end plate connections. Krishnamurthy conducted pioneering 2D finite element analyses that were used to develop the original AISC [2] design recommendations for end plate connections [3,10,12]. Although 2D models are easier to mesh, more computationally efficient, and their results easier to interpret [5], there are phenomena that can be better visualized and understood with the use of 3D models. While 3D models are often difficult to mesh, advances in hardware performance allow today for very sophisticated models to be run without a substantial penalty in computation time. Thus the models utilized in this research were carried out with 3D solid and contact elements that can accurately simulate bolted connection behavior [4,14,15]. The numerical models used for the analyses (Table 2) were chosen from some of the monotonically loaded specimens from the experimental program discussed above. ANSYS [1] first-order solid hexahedral elements (SOLID45, 8 nodes, 3 translation DOF’s per node) with incompatible displacement modes were used to model the connection geometry. The geometrical discontinuities were simulated with surface-tosurface contact elements (TARGE170 and CONTA173). The numerical models also incorporate nonlinear constitutive laws for materials using multilinear “true stress” vs. “natural strain” curves [5,4]. To shorten the computation time, beam elements were used to substitute the solid elements far from the connection, and the rotations were transferred to the solid elements by means of constraint equations. Displacements of the column flange in specific regions were restrained to represent the continuity plates and the column web in order to adequately simulate the cruciform configuration used during the experimental tests. Symmetry through the middle plane of the beam web was also used to reduce the model size. Fig. 5 shows
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Table 2 Numerical models from the experimental program Model ID
End plate thickness—t p (mm)
Bolt diameter—db (mm)
CT1A-1 CT1A-2 CT1A-4 CT1B-2 CT1B-4 CT1B-6
31.5 25.0 22.4 25.0 22.4 19.0
16.0 16.0 16.0 19.0 19.0 19.0
Fig. 5. General view of the FE models.
a general view of the meshes used for these analyses, including some of the restraint conditions. A set of four elements through the thickness of the end plate was used to accurately simulate the flexural behavior of this component [4]. The thicknesses of beam flanges and column flange were simulated with a set of only two elements, as flexural deformations on those components were negligible and do not affect the connection behavior [15]. Loading was applied in two steps. First, temperature gradients were used in the bolts to impose pretension forces. The temperature variation was chosen so as to produce the standardized preloads of 85 and 125 kN for 16.0 and 19.0 mm bolts, respectively [3,17]. A displacement at the beam tip was then imposed to generate a bending moment at the connection. Large displacement and large strain features were also used in the analysis to adequately simulate the connection behavior in ultimate states and the membrane effects that were expected for the thin connecting elements [4,15].
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Fig. 6. Moment–rotation curves for variations of model CT1A-4.
4. Assessments of the numerical models To assess the accuracy of any connection model, results obtained with FE models should be compared with actual data at least insofar as initial stiffness, which depends on the connection’s geometry and elastic modulus; ultimate states and, thus, failure modes; and lastly, the stress redistribution when the connection begins plastic work, i.e., the onset and development of nonlinear behavior. In general, it is possible to compare the values for strength and stiffness since they depend on well-defined variables. However, for nonlinear effects, only qualitative comparisons can be made owing to the difficulty in correctly implementing the nonlinear constitutive laws for materials, especially for situations where only uniaxial values of the stress–strain curves are available. All results presented along this paper are focused on verifying that the numerical modeling is a potential tool to satisfactorily replicate the experimental connection behavior and its variations. Fig. 6 shows a comparison of the experimental and analytical results for the moment–rotation curve of model CT1A-4. This figure is a typical example of the quality of the numerical results for the connection stiffness. The curves presented in Fig. 6 clearly indicate the overprediction of the initial stiffness of the numerical model when compared with the experimental specimen. In general, this behavior was observed for all simulations conducted. The differences between numerical and experimental curves can have a variety of sources, often being a direct consequence of the simplifications introduced in the numerical modeling. Among these factors are (1) imperfections originated from the assembly of the tested models; (2) the effects of residual stresses; and (3) representation of stress–strain behavior for materials in the FE models. With respect to initial stiffness, the level of pretension in the bolts can have a significant influence. First, the differences observed can be attributed to the pretension control (the turn-of-nut method) used in the experimental program presented earlier. The effect of variations in pretension on initial stiffness is illustrated by a comparison between models CT1A-4 and CT1A-4T, also presented in Fig. 6. Model CT1A-4T included no pretension
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on the bolts. Observations made during the experimental program revealed that some bolts lost pretension during the assemblage sequence. Thus the pretension load applied to the FE models was probably not achieved in the tests. To emphasize the effect of pretension, the moment–rotation curve for a specimen recently tested (CT1A-4N) is also shown in Fig. 6. For this model, the pretension was applied with torque control and shows better agreement for initial stiffness. Another aspect to take into account is the nonlinear behavior observed in the FE models. As the yielding and hardening of the components depends on constitutive laws based on simple stress–strain relationships, difficulties arise when trying to accurately simulate material behavior from uniaxial experimental tests. Even when data from characterization of materials are available, the theoretical models used to simulate the nonlinear behavior are not allowed to work with all phenomena embodied by materials, such as stress concentrations, some behavioral changes in tension and compression, and softening at the fracture limit. The numerical models presented herein only used the yield stress information from the characterization of materials, taking advantage of standardized information for maximum tensile strengths and strains for the steel of plates and bolts. In a parametric analysis using the FE models, the use of idealized stress–strain relationships is acceptable, considering the variability observed from test to test. The same is true concerning the fracture or ultimate limit state. In general, the failure to converge numerically in the nonlinear range is used as a definition for failure of the connection, as it identifies some model instability. This nonlinear convergence depends on several variables, especially when contact elements are being used, and it does not always indicate the actual failure point. Therefore, one can use simple inspection procedures to identify the ultimate states, mainly looking for the stress and strain states in the bolts. 5. Numerical results The figures in Table 3 illustrate the deformed shapes of the end plates for the six numerical models at a bending moment of about 150 kN m at the end plate. This value corresponds to a moment about 30% higher than the theoretical plastic moment of the beam section (M p = 117 kN m). This large overstrength was achieved due to the large hardening range found between the yield stress (Fy = 288 MPa) and the maximum tensile stress (Fu = 460 MPa) of the steel plates used in the tests. The displacements are magnified for better visualization. Also shown in Table 3 are the cross-sections where the displacements were calculated from the numerical models. In Table 3 one can clearly identify the behavioral changes that may occur depending on the changes in bolt diameter and end plate thickness. Comparisons are made below for three pairs of models. Focusing first on the end plate behavior, the relative displacements between the end plate and the column flange for the model CT1A-1 (t p = 31.5 mm; db = 16.0 mm) are presented in Fig. 7 for section AA (see Table 3 for location), for a sequence of increasing bending moments at the connection. The displacements of the end plate are plotted against the end plate depth showing the deformed shape at the chosen section. One can observe the development of a complete gap between the end plate and the column flange in the
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Table 3 Deformed shapes and calculation sections for end plates
tension zone as the load is increased, due primarily to bolt elongation. The thick end plate leads to this specific behavior as it has a high stiffness in out-of-plane bending. Similar behavior can be found at the plane of symmetry (Section CC ), where displacements are slightly higher. Fig. 8 presents the end plate displacements along the end plate width for the top edge of the end plate (Section DD ). Fig. 8, in conjunction with Fig. 7, emphasizes the bi-directional bending that occurred systematically for all models. Fig. 9 shows a comparison of deformations between CT1A-4 (t p = 22.4 mm; db = 16.0 mm), with a thinner end plate, and CT1A-1 (t p = 31.5 mm; db = 16.0 mm) at section AA for moments equal to M p (117 kN m) and 1.3M p (150 kN m). For the same bolt diameter, as the end plate thickness decreases changes in the deformed shape are clearly observed, reflecting changes in the overall connection behavior. The end plate top edge in model CT1A-4 maintains contact with the column flange even for high bending moments, leading to significant prying effects on the bolts. Moreover, the end plate displacements at the tension beam flange increase for the thinner plate, indicating higher levels of bending in the end plate. As compared to the results observed for model CT1A-1, CT1A-4 shows more interaction between the end plate and bolts. At this point, some discussion about the distribution of forces in the bolts becomes important inasmuch as the prying action increases as the end plate thickness decreases.
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Fig. 7. End plate relative displacements—section AA in model CT1A-1.
Fig. 8. End plate relative displacements—section DD in model CT1A-1.
Fig. 10 shows the variation of the axial forces in the bolts in the tension zone for model CT1A-1. Curves for the two tensioned bolt lines are plotted, one for the upper bolt, located at the end plate extension, and other for the lower bolt, just below the beam flange. The theoretical force F ∗ was obtained by distributing equally the total tension force among all bolts in the tension zone. As one can observe from Fig. 10, the loss of contact between the end plate and the column flange avoids the prying effect, allowing a uniform distribution of forces for all bolts in the tension zone. Some small variations can be observed, but in general the curves in Fig. 10 evidence the high stiffness in bending for the end plate of
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Fig. 9. End plate displacements—section AA in models CT1A-1 and CT1A-4.
Fig. 10. Axial forces in bolts—model CT1A-1.
model CT1A-1. The initial loading, with moments up to 80 kN m, occurs with an almost constant axial force in the bolts, as expected. Fig. 11, which shows the bolt forces for model CT1A-4, clearly indicates that higher forces are transferred to the lower bolts. Although this irregular distribution depends on the relative stiffness between the end plate and bolts, it is clear that the decrease of the end plate thickness leads to a higher stress in the lower bolts. The changes in the force distribution in the bolts are related to the changes in the end plate behavior. For model CT1A-4, additional forces are imposed to the lower bolt line due to the decrease of the end plate stiffness. The mechanism is illustrated in Fig. 12, where
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Fig. 11. Axial forces in bolts—model CT1A-4.
Fig. 12. Development of prying action in model CT1A-4.
Q s and Q i are the reaction forces from the column flange and Fle and Fli are the forces in the upper and lower bolts, respectively. From equilibrium of forces, the longer lever arm between Q i and Fli has to impose more tension forces in the lower bolts, confirming the behavior described above. In the case of model CT1A-4, the forces in the lower bolts are 30% higher than forces in the upper bolts at the end of loading. In general, all trends mentioned above are also valid for an increase in the bolt diameter. The prying effects for CT1A-4 depends on the relative stiffness between end plate and
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Fig. 13. Axial forces in bolts—model CT1B-4.
bolts, although one can observe that the distribution of forces in the bolts tends to be more uniform for CT1B-4 (t p = 22.4 mm; db = 19.0 mm), as shown in Fig. 13, at least for a larger range of loading. The higher tension in the lower bolt at the end of loading for CT1B-4 can be attributed to concentrated yielding at the end plate extension, which is not stiffened by the beam web. The results for CT1B-4 showed that the end plate top edge remained in contact with the column flange even for the maximum applied load. The initial uniformity in the bolt forces occurs because of the concentration of displacements at the tensioned beam flange due to the high stiffness of bolts relative to the end plate. Putting all these results together and looking at Table 3 again, it is possible to identify the three main deformation modes. In model CT1A-1, the thick end plate allows the bolts to be the source of most of the displacements. In model CT1A-4 the interaction between the end plate and the bolts increases and for model CT1B-4 the displacements are concentrated in the end plate tension zone. The observations are also in agreement with the collapse modes presented for T-stub connections [24,13], showing the interdependence between the distribution of forces in bolts and the end plate thickness indicated by a higher or lower degree of prying action. As a general conclusion, the FE models adequately simulate all the observed behavioral changes. In addition, the results reviewed above indicate the feasibility of utilizing FE modeling for the development of a parametric analysis of bolted end plate connections. 6. End plates and T-stub failure modes 6.1. Basic concepts The results presented above indicated that the distribution of forces in the bolts is generally non-uniform, depending on the prying action developed as a consequence of
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Fig. 14. Failure modes for T-stubs in Eurocode-3.
the end plate behavior. Therefore, the possible use of a plastic distribution of tensile forces in the bolts, as opposed to the conventional triangular distribution, may be considered as an improvement for connection design. Such an approach is possible if one takes advantage of the so-called “component method”, which has as its basis dividing the connection into basic components to evaluate its overall behavior. By identifying separately the sources of displacements, this method allows for a more detailed understanding of the behavior of the connection, providing an approach for developing analytical formulations to characterize strength and stiffness. For end plates loaded primarily by flexure and for the column flange, the “component method” suggests the use of equivalent T-stubs, as presented in Eurocode-3 [6,21]. The formulations are based on the energy method, involving a complex set of yield lines around the bolts of the end plate and leading to the effective length of a T-stub, which represents each line of bolts or a group of them by means of equivalent strength and displacements. Fig. 14 illustrates three possible failure modes that can occur in T-stubs representing the area around the bolt lines. To check flexure, the lowest value obtained for the strength capacity of each bolt line should be adopted, considering the following: Mode 1 Complete yielding of the extended end plate or column flange near the bolts; Mode 2 Bolt failure with yielding of the flange (plate or column); Mode 3 Bolt failure. Some comparisons between design and experimental results [4] reveal that this design procedure still requires refinements, mainly for thinner end plates, for which the analytical model used for T-stubs underestimates the strength. This is due to the complex nonlinear phenomena embodied by these bolted connections. The next section provides some discussion about the effectiveness of using T-stub models. 6.2. End plates in bending The failure modes of models CT1A-1, CT1A-4, CT1B-4 and CT1B-6 are discussed below. The upper line of bolts at the end plate extension is labeled “line 1”, and the line located just below the beam’s flange at the tension zone is labeled “line 2” (see Fig. 15). The three failure modes for T-stubs presented in Fig. 14 are also used throughout these discussions. For purposes of comparison, only the strength values for the extended end plate in bending is used, since the column can be considered sufficiently rigid as to not impact behavior significantly.
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Fig. 15. End plate displacements—section BB’ in models CT1A-1 and CT1A-4.
Table 4 Strength of the extended end plate in bending using Eurocode-3 models CT1A-1 and CT1A-4 Model
t p (mm)
db (mm)
Line of bolts
Failure modes—Pr (kN) 1 2
3
CT1A-1
31.5
16.0
1 2
476.5 1414.0
213.4 482.3
180.0 180.0
CT1A-4
22.4
16.0
1 2
240.9 715.0
145.8 281.8
180.0 180.0
The strength of models CT1A-1 (t p = 31.5 mm) and CT1A-4 (t p = 22.4 mm), both with 16.0 mm bolts, is shown in Table 4 for each line of bolts and for the three possible failure modes. The capacity (Pr ) of each line was calculated using the procedures indicated by Eurocode-3 [6], the yield stress data from characterization of the steel plates (Fy = 288 MPa) as presented in the earlier section, and the nominal yield stress for the ASTM-A325 bolts (Fy = 635 MPa). In most cases, bolt tensile failure controls the strength of the connections, as seen for both lines of bolts in CT1A-1. In line 2, a decrease of the plate thickness (CT1A-4) leads to a significant reduction of the strength for failure modes 1 and 2 and increases bending in the end plate. Fig. 15 illustrates the relative displacements of the end plate for models CT1A-1 and CT1A-4 at section BB’ (through the center of holes), for a bending moment of 117 kN m (M p of the beam) in the connection. The end plate of CT1A-1 shows small displacements due to bending, with tensile failure of the bolts’ controlling strength. Model CT1A-4 shows a different behavior, resulting in prying forces and irregular distribution of forces in the bolts. The displacements in lines 1 and 2 are greater, with a noticeable increase for the end plate bending. The experimental results for both CT1A-1 and CT1A-4 showed that the connections failed as a result of bolt failures in the tension zone, which is consistent
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Table 5 Strength of the extended end plate in bending using Eurocode-3 models CT1A-1 and CT1A-4 Model
t p (mm)
db (mm)
CT1B-4
22.4
19.0
CT1B-6
19.0
19.0
Line of bolts 1 2 1 2
Failure Modes—Pr (kN) 1 2
3
275.0 727.4 197.8 523.4
254.0 254.0 254.0 254.0
195.5 309.1 176.1 257.9
Fig. 16. End plate displacements—models CT1B-4 and CT1B-6.
with the predictions for the models. For CT1A-4, the strength of the upper line is not only limited by the failure of the bolts, but also by the end plate bending. The upper line failure observed experimentally in model CT1A-4 is representative of the cases in which there is a strong interaction between the extended end plate yielding and bolt failure. This behavior is also in agreement with the calculations, although it is difficult to clearly identify the relative contribution of these two components to the connection failure. Table 5 lists the strength of models CT1B-4 (t p = 22.4 mm) and CT1B-6 (t p = 19.0 mm), with 19.0 mm bolts, indicating the decrease in strength and the increasing closeness of the values calculated for failure modes 1 and 2 as the end plate thickness decreases. The relative end plate displacements for CT1B-4 and CT1B-6 are presented in Fig. 16 for sections EE’ and GG’ (see Table 3) for a moment equal to M p (117.0 kN m). One can observe the small displacements in the bolts and the predominance of bending in the end plate. For CT1B-6, the calculations predict failure mode 2 governing the strength for line 1, but the value (Pr,2 = 176.1 kN) is very close to the one obtained for failure mode 1 (Pr,1 = 197.8 kN). As the end plate thickness decreases and the bolt diameter increases, the failure mode for line 1 is governed by the bending of the end plate, as indicated in Table 5. For line 2, the values for failure modes 3 and 2 are very close, indicating that the
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beam web stiffens the lower bolts. Because of this, the collapse for line 2 is not governed by the end plate bending. The experimental tests for the CT1B-6 geometry showed a bolt failure in line 2, in qualitative agreement with the predictions. On the other hand, the failure observed experimentally for CT1B-4 occurred at both lines of bolts, indicating the stiffening effect at the end plate extension. Failure mode 2, calculated for line 1 in CT1B-4, shows higher values when compared with CT1B-6, as expected, but it is clear that a larger increase occurs for failure mode 1 from CT1B-6 to CT1B-4. CT1B-4 should be placed into a group of connections in which the failure mode is hard to identify clearly, and is highly dependent on the relative stiffness between the end plate and the bolts. This is true also for specimen CT1A-4 as presented earlier. All these results emphasize that the predictions for the T-stub failure modes are in good qualitative agreement for mode 3 and mode 1, although calculations for the strength in situations where there is strong interaction between the end plate and the bolts (mode 2) still require refinements. This is mainly due to the fact that the yield lines patterns observed in tests with T-stubs do not necessarily match those from extended end plate tests [15]. Thus, the T-stub theory does not allow prying forces to be considered appropriately when applied to the extended end plate connections. 7. Concluding remarks The results presented herein focused on the behavioral variations of bolted extended end plate connections due to changes in plate thickness and bolt diameter. It also discussed the application of FE models as tools to perform parametric analyses in order to assess the accuracy of commonly used design procedures and to provide data for development of new analytical models. Six numerical models, and associated experimental specimens, were discussed from the standpoint of their overall stiffness, displacements of the end plate and axial forces in the bolts. A set of issues was presented for the modeling, and the numerical results demonstrate the feasibility of the FE models to simulate the connection behavior and to capture behavioral changes as consequence of geometric variations. These numerical results were found to be in good qualitative agreement with the actual connection behavior, although they also evidenced remaining difficulties in quantifying ultimate states when the failure mechanism is not well defined. Calculations on the flexural strength for the end plate were made, showing the predictions for the T-stub failure modes applied to end plate connections and emphasizing the interaction between the end plate and the bolts. The strength of each line of bolts in the tension zone was calculated for four numerical models, following the procedures recommended by Eurocode-3. The analytical results were compared qualitatively with the numerical behavior of the end plate in bending, showing that failure modes 1 and 3 are well-defined if the loads required for these modes are well separated. Failure mode 2, however, leads to levels of interaction between the end plate and bolts that are difficult to model accurately. These results allow putting in discussion the reliability of applying the “T-stub theory” for the extended end plate, mainly by means of yielding lines and “equivalent T-stubs”.
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These statements reaffirm that bolted end plate connections involve complex phenomena, clearly hard to simulate with simplified analytical formulations. They also reaffirm the use of FE models as a potential tool for refinements in the current design approaches in order to improve the characterization of the actual behavior of connections. Acknowledgements The authors are grateful for the financial support provided by FAPESP (Brazilian Research Foundation) for the development of all the research work, whose partial results are presented herein. Some of the results were generated during a visit by the senior author to the Georgia Institute of Technology (USA). The help of Ms. Lillie Brantley and Dr. Rami Haj-Ali during that visit was particularly appreciated. References [1] ANSYS Element Reference. ANSYS 6.0 documentation. Online help. [2] American Institute of Steel Construction. Manual of steel construction. 8th ed, Chicago; 1980. [3] American Institute of Steel Construction (1994). Manual of steel construction: load and resistance factor design. 2nd ed., vol. 2, Chicago. [4] Bursi OS, Jaspart JP. Basic issues in the finite element simulation of extended end plate connections. Computer & Structures 1998;69:361–82. [5] Bahaari MR, Sherbourne AN. Computer modeling of an extended end-plate bolted connection. Computers & Structures 1994;52(5):879–93. [6] Eurocode 3. Design of steel structures: Part 1.1—General rules and rules for buildings—Revised annex J: Joints in building frames; 1993. [7] Figueiredo LMB. Composite steel-concrete connections—theoretical and experimental analysis. Doctorate’s thesis. São Carlos. Ph.D thesis, São Carlos School of Engineering, University of São Paulo; 2004 [in Portuguese]. [8] Krishnamurthy N. Two-dimensional finite element analysis of steel end-plate connections: parametric considerations. Auburn, Alabama, Auburn University, Report n. CE-AISC-MBMA-3; 1974. [9] Krishnamurthy N. Parameter study of steel end-plate connections by two-dimensional finite element analysis. Auburn, Alabama, Auburn University, Report n. CE-AISC-MBMA-5; 1974. [10] Krishnamurthy N. Two-dimensional finite element analysis of extended and flush connections with multiple rows of bolts. Auburn, Alabama, Auburn University, Report n. CE-AISC-MBMA-6; 1975. [11] Krishnamurthy N. Tests on bolted end-plate connections and comparisons with finite element analysis. Auburn, Alabama, Auburn University, Report n. CE-AISC-MBMA-7; 1975. [12] Krishnamurthy N. Design of end-plate connections. Vanderbilt, Tennessee, Vanderbilt University, Report n. CE-AISC-MBMA-10; 1976. [13] Kulak GL, Fisher JW, Struik JH. Guide to design criteria for bolted and riveted joints. 2nd ed. New York: John Wiley & Sons; 1987. [14] Maggi YI. Theoretical Analysis, using F.E.M, of Bolted Beam-to-Column Connections with Extended End Plate. São Carlos. Master’s Dissertation, São Carlos School of Engineering, University of São Paulo; 2000 [in Portuguese]. [15] Maggi YI. Analysis of the structural behavior of bolted beam-column extended end plate connections. São Carlos. Ph.D thesis. São Carlos School of Engineering, University of São Paulo; 2004 [in Portuguese]. [16] Nair RS, Birkemoe PC, Munse WH. High-strength bolts subject to tension and prying. Journal of Structural Division 1974;100(ST2):351–72. [17] NBR-8800. Design and Construction of Steel Structures for Buildings. Associação Brasileira de Normas Técnicas (Brazilian Standards Association), Rio de Janeiro; 1986.
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