EVALUATION OF THE COMPLETE MOMENT-ROTATION CURVE of bolted beam-to-column connections using mechanical models

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1 EUROSTEEL 28, 3-5 September 28, Graz, Austria 579 EVALUATION OF THE COMPLETE MOMENT-ROTATION CURVE of bolted beam-to-column connections using mechanical models Minas Lemonis and Charis Gantes National Technical Univeristy of Athens, School of Civil Engineering, Athens, Greece INTRODUCTION The moment M versus rotation curve represents an essential response characteristic of structural steel joints, since for every load level, it provides the bending moment and the relative rotation between the connected members. The complete M - curve, that describes the full, up to failure, range of the joint response, allows the realisation of nonlinear structural analyses techniques, where the joints are actively simulated, similar to the members, without the traditional idealisations of rigid or pinned joints. Some benefits of this development include the ability to enhance the overall ductility of the structure, since the formation of plastic hinges inside the joints becomes possible, and the more accurate approximation of the actual response, through the global structural analysis. Also, financial savings are available due to the minimised use of rigid joints, as well as by means of a more optimal selection of the connected members. Some existing proposals that allow the estimation of the complete M - curve can be found in references [1-5]. In this study, mechanical models are used for the evaluation of the complete M - curve. These are formed as an assembly of springs and rigid elements, and offer a reasonable compromise between the complex advanced finite element models and the more simple analytical models. Mechanical modeling is based on the characterization of individual components, with a well defined behaviour inside the joint. The advantage of this approach is its flexibility to accommodate different joint typologies under the same basic principles. In this paper joints with bolted connections are examined, and more specifically the commonly used end-plate and cleated connections. 1 MECHANICAL MODELS The stiffness model of Eurocode 3 [6] was adopted as a basis for the mechanical models used in this paper for the calculation of the complete M - curve. In Figure 1, the proposed mechanical models for end-plate connections are presented, with the naming of each component explained in Table 1. The models are composed of two types of springs: springs corresponding to stiffness components, derived directly from the stiffness model of Eurocode 3 [6] and springs corresponding to strength components, namely the ones that are used for strength calculations only. The springs of the first case influence both the deformability of the joint and its strength, and their behaviour is governed by a nonlinear force F - displacement curve. For the second case, the springs do not contribute to the joint deformability, but instead they only limit its available strength. Their behaviour is governed by a rigid plastic F - curve. Fig.1. Proposed aligned (a) and non-aligned (b) mechanical models for end-plate connections

2 Table 1. Naming of the various components used in the mechanical models 58 Component Abbreviation Component Abbreviation Column web panel in shear Column web in compression Beam flange/web in comprssion Column web in tension Column flange in bending Bolts in tension End-plate in bending Beam web in tension cws cwc bfwc cwt cfb bt epb bwt Bolts in shear Angle leg in bearing Beam flange in bearing Seat angle in compression Top angle in bending Top angle in tension Web angle in bending Beam web in bearing bs ab bfb sac tab tat wab bwb The response of bolted beam-to-column joints can be characterized by means of the so called T-stub connection, which is used to model the components of their tension zone [6-9]. For end-plate connections, these components typically are the end-plate and the column flange in bending. It is usually preferable for the design that these components become critical for the response, since they can exhibit high levels of ductility [1] and thus enhance the joint rotational capacity and its effectiveness for plastic design of the structure. This way, the reliable modeling of their response becomes decisive for the accurate estimation of the complete joint M - curve. For this reason a T- stub model was developed. The model is depicted in Figure 2. It has been described in detail in [11] and [12] and due to limited space it will be only briefly discussed here. The complete F - curve is calculated incrementally and, consequently, the model is not intended for hand calculations but for programming in a computer. The flange is modeled as a simple beam and the bolt as an extensional spring attached to the flange. Taking advantage of the symmetry only one half of the connection is modeled. Material nonlinearity is taken into account through bilinear moment-curvature and forcedisplacement relationships for the flange and the bolt, respectively. In the flange certain zones along its length are expected to enter the plastic region. The response of these zones is governed by the plastic material properties (e.g. the hardening modulus E T ). The contact phenomena taking place in the flange are also taken into account. A separation point is designated as the boundary between the deformed part of the flange and the undeformed one, which remains in contact with the underlying base. A key innovation of the model is that the position of the separation point is calculated analytically at every step of the incremental process, allowing it to relocate along the flange length, as additional loading is applied and a growing part of the flange or the bolts enter the plastic region. The model has been evaluated against experimental T-stub tests as well as advanced finite element modeling and showed a quite satisfactory performance in the estimation of the complete F - curve [11, 12]. Fig.2. T-stub connection (a) and its model adopted for the mechanical modeling [11, 12] (b) The difference between the two models of Figure 1 is located in the first spring row, which corresponds to the bolt row at the extended part of the end-plate. For the first model (Figure 1a), all springs of the first row are aligned with the bolt axis. This arrangement is implemented in Eurocode 3 [6] and is referred in this paper as aligned model. Alternatively, in Figure 1b, a modified arrangement is presented, where the spring for the component end-plate in bending is aligned

3 with the beam flange and not with the bolt axis, like the other springs of the bolt row. This arrangement is referred in the paper as non-aligned model. The reasoning behind this proposal is the orientation of the T-stub components for the involved springs of the extended bolt row. It is demonstrated in Figure 3 that at the extended part of the end-plate, the equivalent T-stub for the component end-plate in bending is orientated vertically, in such a way that its web is coincident with the beam flange. Thus, it is considered more consistent to the actual force transfer mechanism that the tensile force F and the resulting displacement for this component is positioned accordingly, aligned with the beam flange. 581 Fig.3. T-stub components for an end-plate connection (a) and a top/seat angle connection (b) For cleated connections the proposed mechanical models for the estimation of the M - curve are presented in Figure 4, with the naming of each component explained in Table 1. The model of Figure 4b has been proposed by Faella et al. [2] for top/seat angle with double web angle connections, and features a non-aligned arrangement in the first spring row, which corresponds to the bolt row of the top angle. The orientation of the T-stub component top angle in bending (Figure 3) suggests for the alignment of the respective spring with the horizontal angle leg, instead of the bolt axis. Furthermore, any other components linked with the horizontal angle leg are aligned to it, as shown in the model of Figure 4b. In the present paper, an aligned arrangement for the first spring row is also examined, as shown in Figure 4a. Fig.4. Proposed aligned (a) and non-aligned (b) mechanical models for cleated connections 2 EVALUATION AGAINST EXPERIMENTAL DATA The performance of the mechanical models has been evaluated against experimental results of beam-to-column joints found in the literature. For end-plate connections, the tests of Bursi and Jaspart [13] and of Coelho et al. [14] were utilized while for top/seat with double web angle connections the monotonic tests carried out by Azizinamini et al. [15] were used. Thus, a sum of 28 individual tests was collected. For the springs of the tension zone, corresponding to T-stub components, the F - curve was computed by means of the T-stub model mentioned in section 1. The equivalent T-stub width was estimated according to Eurocode 3 [6] provisions. For the springs corresponding to non T-stub components, established methodologies, mainly based on Eurocode 3 [6], were adopted for the calculation of the respective F - curves, as described in [12]. It should

4 be noted that for the experimental tests examined here, the T-stub components were always critical for the joint response, with only secondary influence coming from the remaining components. In Figure 5a the M - curves for the two experimental extended end-plate connections, described by Bursi and Japart [13] are shown. The experimental curves are compared against the ones calculated be means of the proposed mechanical models for end-plate connections (aligned and non-aligned), as well as against advanced finite element modeling, performed by the same authors [13]. The performance of the mechanical models allows a satisfactory prediction of the actual curve. The convergence is closer to the advanced finite element results rather than the experimental ones. For the first test, a significant underestimation of the initial stiffness by all computational models, even the advanced finite element one, is observed, probably a result of initial imperfections. Also, for the first test an underestimation by 35% of the rotational capacity is detected, which is attributed to the influence of membrane effects, developing in the weak end-plate. This effect is not predictable by the mechanical models. Nevertheless, the relative error can be considered acceptable for an estimation of the rotational capacity. For the second test, the mechanical models predict quite satisfactorily the M - curve. The non-aligned mechanical model proves more effective and estimates within a narrow error margin the initial stiffness, strength and rotational capacity EP1-1 2 FS1 25 FS EP FS FS (a) (b) Fig.5. M - curves for the tests of Bursi and Japart [13] (a) and Coelho et al. [14] (b) In Figure 5b the M - curves for the four experimental extended end-plate setups described by Coelho et al. [14] are shown. For all end-plates the material constitutive law is given in a quadrilinear form while for the T-stub model, used for the respective components, a bilinear law is required. A fitting of the actual quadrilinear form to a bilinear one was performed, with governing criterion the equivalence of the maximum bending moment in the end-plate cross-section [12]. The performance of the mechanical models leads to a quite satisfactory estimation of the M - curve. For setups FS1 and FS2 the convergence is very good, with only moderate overestimation of initial stiffness. For setup FS3, which features a quite stiff end-plate, a significant overestimation of the initial stiffness is observed, while for strength a close match is achieved, especially by the nonaligned model. Regarding the rotational capacity, the average error for the non-aligned model is about 3% and is considered satisfactory for the estimation of this attribute. Finally, for setup FS4 which features an end-plate of high strength steel (S69), the non-aligned model overestimates initial stiffness by 35%, while rotational capacity is found 35% lower than the experimental values. Strength is predicted within a narrow error margin.

5 In Figure 6, the M - curves by the proposed mechanical models are shown, for a subset of the monotonic tests carried out by Azizinamini et al. [15]. Due to space limitations, the remaining tests are not presented here but are available in [12]. In addition to the experimental results, curves derived by advanced finite element modeling by Kishi et al. [16] are also illustrated in Figure 6. The material properties used in the mechanical models are the ones given in [16] for use in the advanced finite element models. In general, the mechanical models appear reliable in the prediction of the complete M - curve and capable to estimate effectively the initial stiffness, the strength and even the rotational capacity of the joints. The convergence is closer with respect to the advanced finite element models, which was expected since their material properties match exactly and there are no imperfections that usually affect experimental tests. Comparing the two mechanical models, the aligned one demonstrates higher stiffness and strength and less rotational capacity than the nonaligned one. The difference between them is significant, mainly for the initial stiffness and strength. The aligned model proves more effective in general and its performance appears very satisfactory S1 6 8S2 6 8S3 1 14S S2 1 14S Fig.6. M - curves for the monotonic tests of Azizinamini et al. [15] A statistical evaluation of the performance of the proposed mechanical models is presented in Table 2. The mean error and the standard deviation in the estimation of initial stiffness, ultimate strength and rotational capacity are shown, for both cleated connections tests as well as end-plate tests. The results for the cleated connections include all the monotonic tests, carried out by Azizinamini et al. [15] and not only their subset, presented in this paper. It appears from the results that the nonaligned mechanical model achieves overall better agreement with the end-plate connection tests, while the aligned model seems more effective for the cleated connections. The values of mean error can be considered very satisfactory for ultimate strength and rotational capacity and satisfactory for initial stiffness, where the value of standard deviation is higher. In particular, the low value of mean error and standard deviation for the rotational capacity is very promising, since this attribute involves the displacement field of the joint response, both in the elastic and the plastic region, and thus its reliable calculation remains challenging even for advanced computational techniques. Table 2. Mean error and standard deviation (in parenthesis) in the prediction of initial stiffness, ultimate strength and rotational capacity by the proposed mechanical models. End-plate connections (Bursi & Jaspart [13], Coelho et al. [14]) Cleated connections (Azizinamini et al. [15]) Initial Stiffness Ultimate Strength Rotational Capacity Proposed aligned model.32 (.642).2 (.115) -.19 (.256) Proposed non-aligned model.12 (.553) -.4 (.18) -.1 (.35) Proposed aligned model.16 (.444).2 (.121).16 (.297) Proposed non-aligned model -.19 (.317) -.12 (.93).4 (.321)

6 3 CONCLUSIONS The estimation of the complete M - curve of bolted beam-to-column joints using mechanical modeling was examined in this paper. Appropriate mechanical models for end-plate and cleated connections were proposed that include the joint components with their complete F - curve. The key feature of the proposed methodology is the simulation of the T-stub components of the tension zone, using an original incremental T-stub model, specially developed for this purpose. Alternative spring arrangements for the mechanical models were examined, in regard to the alignment of the springs in the first bolt row. The evaluation of the proposed models against experimental joints and advanced finite element models revealed a quite satisfactory and consistent behaviour in the estimation of the complete M - curve. Especially for the rotational capacity, the results appear very satisfactory and prove that the proposed mechanical models offer a promising solution for its reliable estimation. For the end-plate connections the non-aligned models proved more effective overall, while for the cleated connections the aligned models did. Comparison to a larger number of experimental tests or advanced finite element models is always considered useful for a better evaluation of the proposed models. REFERENCES [1] Shi YJ, Chan SL, Wong YL. Modeling for moment-rotation characteristics for end-plate connections. Journal of Structural Engineering ASCE 1996; 122(11): [2] Faella C, Piluso V, Rizzano G. Structural steel semi-rigid connections - Theory, design and software. Boca Raton, Florida: CRC Press; 2. [3] Silva LS, Coelho AMG. A ductility model for steel connections. Journal of Constructional Steel Research 21; 57(1):45-7. [4] Beg D, Zupancic E, Vayas I. On the rotation capacity of moment connections. Journal of Constructional Steel Research 23; 6(3-5): [5] Coelho AMG, Silva LS. Ductility analysis of end plate beam-to-column joints. Eurosteel 25 4th European conference on steel and composite structures; Maastricht; 25. [6] CEN - Comite Europeen de Normalisation. Eurocode 3: Design of steel structures - Part 1.8: Design of joints (pren :23), Stage 49 Draft, Brussels, 23. [7] Zoetemeijer P. A design method for the tension side of statically loaded, bolted beam-to-column connections. Heron 1974; 2(1):1-59 [8] Packer JA, Morris LJ. A limit state design method for the tension region of bolted beam-column connections. The Structural Engineer 1979; 55(1): [9] Mann AP, Morris LJ. Limit design of extended end-plate connections. Journal of the Structural Division ASCE 1979; 15(3): [1] Kuhlmann U, Kuhnemund F. Procedures to verify rotation capacity. In: Ivanyi M, Baniotopoulos CC, editors: Semi-rigid joints in structural steelwork, CISM courses and lectures No.419, Wien-New York, Springer-Verlag, 2. [11] Lemonis ME, Gantes CJ. Incremental modeling of T-stub connections. Journal of Mechanics of Materials and Structures 26; 1(7): [12] Lemonis ME. Beam-to-column joints in steel frames. Ph.D. thesis (in greek): National Technical University of Athens. Athens; 26. [13] Bursi OS, Jaspart JP. Calibration of a finite element model for isolated bolted end-plate steel connections. Journal of Constructional Steel Research 1997; 44(3): [14] Coelho AMG, Bijlaard FSK, Silva LS. Experimental assessment of the ductility of extended end plate connections. Engineering Structures 24; 26(9): [15] Azizinamini A, Radziminski JB. Static and cyclic performance of semirigid steel beam-to-column connections. Journal of Structural Engineering ASCE 1989; 115(12): [16] Kishi N, Ahmed A, Yabuki N, Chen WF. Nonlinear finite element analysis of top- and seat-angle with double web-angle connections. Structural Engineering and Mechanics 21; 12(2):