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Monotonic and cyclic loading models for panel zones in steel moment frames
Journal of Constructional Steel Research 58 (2002) 605–635 www.elsevier.com/locate/jcsr
Monotonic and cyclic loading models for panel zones in steel moment frames Kee Dong Kim a, Michael D. Engelhardt b,* a
Department of Civil Engineering, Kongju National University, Kongju, Chungnam, South Korea Department of Civil Engineering, University of Texas at Austin, Austin, TX 78712-1076, USA
b
Received 14 June 2001; received in revised form 17 August 2001; accepted 5 October 2001
Abstract This paper presents the development of analytical models to predict the elastic and inelastic response of the panel zone portion of columns in steel moment resisting frames. In many practical cases, the panel zone can dominate the inelastic response of a moment frame, and accurate panel zone models are needed to realistically predict overall frame performance. Similar to previous models, the newly proposed models are based on the concept of representing the panel zone as a nonlinear rotational spring. These new models build on previously developed models, and introduce a number of features and refinements that show better correlation with available experimental data. The model for monotonic loading is based on quadrilinear panel zone moment–deformation relations. In this model, both bending and shear deformation modes are considered. The model proposed for cyclic loading is based on Dafalias’ bounding surface theory combined with Cofie’s rules for movement of the bound line. Additional modifications are suggested to the cyclic loading model to account for the influence of a composite floor slab on panel zone response. Extensive comparisons with experimental data are presented. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Seismic; Inelastic; Bounding surface model; Composite; Connections; Joints
* Corresponding author. Tel.: +1-512-471-6837; fax: +1-512-471-1944. E-mail address: [email protected] (M.D. Engelhardt). 0143-974X/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 3 - 9 7 4 X ( 0 1 ) 0 0 0 7 9 - 7
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1. Introduction Mathematical models are developed in this paper for describing the monotonic and cyclic load–deformation response of the panel zone region of beam–column joints of a steel moment resisting frame. The panel zone is the portion of the column contained within the beam–column joint. When a moment frame is subject to lateral loads, high shear forces develop within the panel zone. The resulting deformations of the panel zone can have an important effect on the response of the frame in both the elastic and inelastic ranges of frame behavior [1,2]. Numerous tests have been performed in the past three decades to investigate the load–deformation behavior of the joint panel using connection subassemblies [3–8]. Some significant observations from these tests are: 앫 Joint panel zones often develop a maximum strength that is significantly greater than the strength at first yield. This additional strength has been attributed to strain hardening and to contributions of the column flanges in resisting panel zone shear forces. Large inelastic panel zone deformations are typically required in order to develop the maximum panel zone strength. 앫 Panel zone deformations can add significantly to the overall deformation of a steel moment frame, for both elastic and inelastic ranges of behavior. 앫 Panel zone stiffness and strength can be increased by the attachment of web doubler plates to the column within the joint region. The effectiveness of doubler plates is affected by the method used to connect them to the column. 앫 In the inelastic range, panel zones can exhibit very ductile behavior, both for monotonic and cyclic loading. Experimentally observed hysteresis loops are typically very stable, even at large inelastic deformations. 앫 Large inelastic panel zone deformations can increase the likelihood of fracture occurring in the region of the beam flange to column flange groove welds. This effect has been attributed to the occurrence of large localized deformations or ‘kinks’ in the column flanges at the boundaries of the panel zone. Current US building code provisions [9–11] permit the formation of plastic hinges in the panel zones of steel moment frames under earthquake loading. Thus, rather than forming plastic flexural hinges only in the beams or columns, a primary source of energy dissipation in a steel moment frame can be the formation of plastic shear hinges in the panel zones. Consequently, to accurately predict the response of a steel moment frame under earthquake loading, an accurate analytical model is needed to predict the response of the panel zone. To include panel zone deformation in frame analysis, the traditional center-tocenter line representation of the frame must be modified. Fig. 1 shows a comparison between an experiment on a beam–column subassemblage and an analytical prediction of the subassemblage response. The experiment was specimen A1 reported by Krawinkler et al. [4]. Inelastic response of this specimen was dominated by yielding of the panel zone. The analytical results are obtained from a model of the specimen using center-to-center line dimensions. A beam–column element with plastic hinges
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Fig. 1.
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Comparison of test results [4] and analytical results obtained by using center-to-center line dimension modeling.
was employed in this analysis. From the figure it can be seen that the analysis using the center-to-center line dimension, without explicit modeling of the panel zone, may produce misleading results. Clearly, the panel zone response must be explicitly modeled to obtain realistic predictions of the overall frame behavior. To model the behavior of panel zones in frame analysis, Lui [12] developed a joint model based on the finite element method. The model consists of seven elements for interior beam-to-column joints: one web element, two flange elements (beam elements), and four connection elements. Although capable of representing a variety of deformation modes of panel zones, this model employed a simple hardening rule suitable for monotonic loading and does not realistically model cyclic behavior. Another disadvantage of this model is its high computational cost. Other finite element models using more sophisticated hardening rules could be developed for the analysis of column panel zones. However, in this study, nonlinear rotational springs are used as the basis for modeling the panel zone for nonlinear dynamic analysis of moment resisting frames because of simplicity and computational efficiency. Several researchers, including Fielding and Huang [13], Krawinkler et al. [4] and Wang [14] proposed relationships between panel zone shear force V and panel zone deformation g for monotonic loading. These relationships have been used as the basis of mathematical models for nonlinear rotational springs representing the panel zone. Krawinkler’s V–g relations have been adopted in several building codes [9,10] as a basis for computing the shear strength of panel zones. However, it was pointed out by Krawinkler that a new model might be needed for joints with thick column flanges since his V–g relations were derived from experimental and analytical results for panel zones with relatively thin column flanges. Wang also showed that Krawinkler’s V–g relations may overestimate panel zone shear strength for panel zones with thick column flanges.
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In the remainder of this paper, existing analytical models for panel zones will be examined by making comparisons with experimental data or with other analytical results. A number of refinements and improvements to existing models are then suggested to overcome some of the shortcomings of existing models and to provide better correlation with available experimental data.
2. General characteristics of panel zone element The panel zone element is essentially a rotational spring element, which transfers moment between the columns and beams framing into a joint [15]. The panel element has no dimensions and connects two nodes with the same coordinates. One of these nodes is attached to the elements modeling the columns framing into the joint, as shown in Fig. 2, while the other node is attached to the elements modeling the beams. Therefore, the moment transferred by the panel element is related to the relative rotation between the columns and beams framing into a joint. The vertical and horizontal translations of the two nodes are constrained to be identical. Therefore, one vertical, one horizontal, and two rotational degrees of freedom exist at each joint. The relative rotation between the connected nodes is related to the node rotations as follows: dg ⫽ {1 ⫺1}
再 冎 dqI
(1)
dqJ
or dg ⫽ a·dr
(2)
where dg is the increment of relative rotation, which is the panel element deformation, and dqI, and dqJ, are the increments of rotation of the connected nodes. Then the tangent stiffness relationship for the panel zone element is
Fig. 2.
Idealization of beam-to-column joint.
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dMpa ⫽ Ktdg
609
(3)
pa
where dM is the increment of moment applied on a joint and Kt is the rotational tangent stiffness of the joint. In terms of nodal rotations, the stiffness, KT, is given by KT ⫽ aTKta
(4)
The definable properties of the panel zone element are the rotational stiffnesses and yield moments for monotonic loading, and hysteretic rules for cyclic loading. In the following sections, existing models for monotonic loading are examined and a new model is presented. This is followed by the development of hysteretic rules for a cyclic loading model.
3. Review of existing models for monotonic loading Existing mathematical models for panel zone response under monotonic loading are typically based on a computation of an approximate equivalent shear force acting on the panel zone. The boundary forces on a joint panel, shown in Fig. 3, can be transformed into an approximate equivalent shear force from equilibrium as follows: Veq ⫽
Mbl ⫹ Mbr (Vct ⫹ Vcb) Mbl ⫹ Mbr (1⫺r) pa ⫺ (1⫺r) ⫽ M ⬇ db⫺tbf 2 db⫺tbr db⫺tbf
(5)
where r ⫽ (db⫺tbf) / Hc, Mpa ⫽ Mbl ⫹ Mbr is the panel zone moment, tbf is the thickness of the beam flange, db is the beam depth, and Hc is the column height. A key simplification in this analysis is that the beam moments are replaced by an equivalent couple, with the forces acting at mid-depth of the beam flanges. These forces produce a large shear in the panel zone. The shear in the column segments outside of the panel zone are then subtracted to obtain the net shear force, Veq, acting on the panel zone. In obtaining the shear forces in the column segments outside of the panel zone, it is often assumed that points of inflection in the column occur at a distance Hc/2 above and below the panel zone.
Fig. 3. Boundary forces and equivalent shear forces on panel zone.
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Panel zone shear force V versus panel zone deformation g relations for monotonic loading, which are based on the equivalent shear force, Veq, can be transformed into panel zone moment Mpa versus panel zone deformation g relations by Eq. (5). Fielding and Huang [13] proposed a bilinear relationship consisting of an elastic stiffness Ke followed by a post elastic stiffness K1. Krawinkler et al. [4] and Wang [14] each proposed different tri-linear Mpa–g relations consisting of an elastic stiffness Ke followed by two linear post elastic stiffness values K1 and K2. For all three models, the post elastic stiffness K1 is related to the contribution of the column flanges to the panel stiffness. For the tri-linear models, the second post elastic stiffness K2 is associated with strain hardening. Past researchers computed the elastic stiffness of the panel element by considering pure elastic shear deformation of an effective shear area of the panel zone. Fielding and Krawinkler considered the effective shear area Aeff equal to (dc⫺tcf)tcw, and Wang considered the effective shear area Aeff of (dc⫺2tcf)tcw, where the subscripts ‘c’, ‘f’, and ‘w’ stand for column, flange, and web, respectively. They suggested the yield moment and elastic stiffness of the panel zone be taken as follows: Mpa y ⫽ Ke ⫽
Vydb t¯ yAeffdb ⫽ (1⫺r) (1⫺r)
Mpa GAeffdb y ⫽ gy (1⫺r)
(6a) (6b)
where Vy is the yield shear force of the panel zone, gy ⫽ t¯ y / G, G is the elastic shear modulus, and t¯ y is the Von Mises yield shear stress of the column web, based on shear and axial force interaction. The Von Mises yield shear stress, t¯ y, is taken as: t¯ y ⫽
sy
冑3
冑1⫺(P / P ) y
2
(7)
where P and Py are the axial force and the axial yield force on the column, respectively, and sy is the yield stress of the column web. For the inelastic range, Fielding considered a bilinear model with the following post-elastic stiffness K1: K1 ⫽
5.2Gbcft3cf db(1⫺r)
(8)
where bcf and tcf are width and thickness of column flange, respectively. Krawinkler proposed empirical formulas for the post-elastic stiffness K1 and the second yield moment Mpa sh as follows: K1 ⫽
1.04Gbcft2cf (1⫺r)
pa Mpa sh ⫽ My ⫹
3.12t¯ ybcft2cf (1⫺r)
Wang suggested the post-elastic stiffness K1, as follows:
(9) (10)
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K1 ⫽ 0.7Gbcft2cf
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(11)
Krawinkler and Wang assumed that strain hardening begins at gsh ⫽ 4gy and 3.5gy, respectively. The strain-hardening branch stiffness K2 was suggested as follows: K2 ⫽
GstAeffdb (1⫺r)
(12)
where Gst is the strain hardening shear modulus. The existing models described above are compared to four specimens tested by Krawinkler et al. [4], Fielding and Huang [13], and Slutter [5] in Figs. 4–7. Each of these specimens was a beam–column subassemblage with a weak panel zone. Fig. 4 shows Krawinkler et al.’s specimen A2, which had a column flange thickness of 1 cm. Slutter’s specimen 1, which had a column flange thickness of 1.8 cm, is plotted in Fig. 5. Krawinkler et al.’s specimen B2, with a column flange thickness of 2.37 cm, is shown in Fig. 6. Finally, Fig. 7 shows Fielding and Huang’s test specimen, which had a column flange thickness of 3.5 cm. Additional comparisons are shown in Figs. 8–12. In these figures, FEM analysis predictions are provided for Slutter’s specimen 1 and compared to the simplified nonlinear spring models. This specimen was analyzed a number of times, varying the column flange thickness. These FEM analyses provide an indication of the expected response of panel zones as the column flange thickness varies, but where all other variables remain constant. These finite element analyses were reported by Wang [14]. The specimen tested by Slutter was analyzed by using a two-dimensional FEM, in which the flanges and webs were represented by beam elements and plain stress elements, respectively. The finite element results are compared with the corresponding test data in Fig. 5, for the actual specimen with a 1.8 cm thick column flange.
Fig. 4. Comparison of the monotonic model and test data for Krawinkler et al.’s [4] specimen A2 with tcf ⫽ 1 cm.
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Fig. 5. Comparison of the monotonic model and test data for Slutter’s [5] specimen 1 with tcf ⫽ 1.8 cm.
Fig. 6. Comparison of the monotonic model and test data for Krawinkler et al.’s [4] specimen B2 with tcf ⫽ 2.37 cm.
This comparison shows that Wang’s finite element analysis reasonably predicted the observed panel zone behavior. Figs. 8–12 show finite element predictions for a model based on Slutter’s specimen, but considering column flange thickness values equal to 0.9, 1.35, 2.7, 3.6 and 4.51 cm. In each figure, the corresponding predictions of the simple nonlinear spring elements are also plotted. A number of observations can be made from the comparisons plotted in Figs. 4– 12. Fielding’s bilinear model performs well for small panel zone rotations, but this
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Fig. 7. Comparison of the monotonic model and test data for Fielding and Huang’s [13] specimen with tcf ⫽ 3.5 cm.
Fig. 8. Comparison of the monotonic model and FEM results for Slutter’s specimen 1 with tcf ⫽ 0.9 cm.
model shows rather poor performance at large rotations regardless of column flange thickness because the model neglects strain-hardening effects. The performance of Krawinkler et al.’s model appears reasonable for panel zone joints with column flange thickness less than about 2.5 cm. However, for thicker column flanges, this model somewhat overestimates panel zone strength. Wang’s model generally underestimates panel zone strength regardless of column flange thickness, apparently because in this model the effective shear area of the panel zones is calculated as (dc⫺2tcf)tcw instead of the other models’ effective shear area (dc⫺tcf)tcw.
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Fig. 9. Comparison of the monotonic model and FEM results for Slutter’s specimen 1 with tcf ⫽ 1.35 cm.
Fig. 10. Comparison of the monotonic model and FEM results for Slutter’s specimen 1 with tcf ⫽ 2.7 cm.
4. Proposed model for monotonic loading and comparison with test data As the ratio of column flange thickness to column depth increases, the influence of column flange thickness on panel zone yield moment and elastic stiffness increases [14]. Panel zone models that include shear deformations only cannot account for this increase in yield moment and elastic stiffness according to the increase in the ratio of column flange thickness to column depth. Thus, in this study, both bending and
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Fig. 11. Comparison of the monotonic model and FEM results for Slutter’s specimen 1 with tcf ⫽ 3.6 cm.
Fig. 12. Comparison of the monotonic model and FEM results for Slutter’s specimen 1 with tcf ⫽ 4.51 cm.
shear deformation modes are included in the panel zone model. The resulting monotonic model has quadri-linear Mpa–g relations. It is assumed that the panel zone can be considered as two equivalent beams, which are symmetric with respect to the center of the panel zone and are fixed at the center. The boundary condition at the other end of these beams is considered to be somewhere between free and fixed. Thus, the displacement of the equivalent beam due to the shear force Veq can be described as follows:
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⌬⫽
冉
冊
1 1 ⫹ V S1 S2 eq
(13)
where 1/S1 is the bending flexibility of the equivalent beam and 1/S2 is the shear flexibility. The bending and shear flexibilities can be described as follows: [(db⫺tbf) / 2]3 [(db⫺tbf) / 2] 1 1 ⫽ and ⫽ S1 CrEI S2 G(dctcw ⫹ RfAdp)
(14)
where I is the moment of inertia of the column section, Cr is a constant to be determined according to the degree of end restraint, Adp is the area of a doubler plate, and Rf is the reduction factor to account for the strain incompatibility between a doubler plate and column web. Eq. (13) can be rewritten in terms of the panel zone moment Mpa and the rotation g as follows: S1S2 S1S2 (db⫺tbf) ⌬⫽ g S1 ⫹ S2 S1 ⫹ S2 2
Veq ⫽
S1S2 (db⫺tbf)(db⫺tbf) (db⫺tbf) ⫽ g Mpa ⫽ Veq (1⫺r) S1 ⫹ S2 2 (1⫺r)
(15a) (15b)
Thus, the elastic stiffness is Ke ⫽
S1S2 (db⫺tbf)(db⫺tbf) S1 ⫹ S2 2 (1⫺r)
(16)
The yield moment of panel zone is defined as follows: Mpa y ⫽ KeCygy
(17)
where Cygy is the average shear deformation of panel zone at which shear yielding occurs and Cy is the ratio of the average shear deformation to gy. To describe the behavior of the panel zone in the range from first shear yielding up to the entire shear yielding of the panel zone, it is assumed that for this range the panel zone can be modeled as two separate beams with T-shaped sections, similar to Fielding and Huang’s approach [13]. The web depth of the T-shaped section is a quarter of the column web depth. Then, the post-elastic stiffness K1 can be defined as PS1PS2 (db⫺tbf)(db⫺tbf) K1 ⫽ 2 PS1 ⫹ PS2 2 (1⫺r) PS1 ⫽
CrEIT G[(dc / 2⫺dyw)tcw ⫹ RfAdp / 4] and PS2 ⫽ [(db⫺tbf) / 2]3 [(db⫺tbf) / 2]
(18a) (18b)
where IT is the moment of inertia of the T section and dyw is a quarter of the web depth. The second yield moment Mpa y1 is ¯ y(dctcw ⫹ RfAdp) Mpa y1 ⫽ t
(db⫺tbf) (1⫺r)
(19)
The second post-elastic stiffness after shear yielding of the entire panel zone is
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defined by using an approach similar to Krawinkler et al. [4]. The panel zone, after the shear yielding of the entire web, consists of an elastic–perfectly plastic shear panel surrounded by rigid boundaries with springs at the four corners. It is assumed that these springs simulate the resistance of elements surrounding the panel, in particular the bending resistance of column flanges, and that the spring stiffness can be approximated by Ks ⫽
Ebcft2cf Cs
(20a)
where Cs is a constant to be determined from test results. From the work equation and Eq. (20a), the second post-elastic stiffness K2 is obtained as K2 ⫽
4Ebcft2cf Cs(1⫺r)
(20b)
It is assumed that when strain hardening starts, plastic hinges form in the column flanges at the four corners of the panel [14]. Then, the third yield moment of the panel zone at which strain hardening initiates can be defined as pa ¯ ybcft2cf Mpa y2 ⫽ My1 ⫹ s
(21a)
s¯ y ⫽ s (1⫺(P / Py) )
(21b)
fl y
2
where sfly is the yield stress of the column flange. The strain hardening stiffness K3 is K3 ⫽
Gst(dctcw ⫹ RfAdp)(db⫺tbf) (1⫺r)
(22)
This new proposed monotonic model has been applied to the same four specimens as in the previous section. Figs. 4–12 show comparisons of this new model with test results and FEM results for various values of column flange thickness. Based on calibration to test results (Figs. 4–7), it has been found that Cr ⫽ 5, Cy ⫽ 0.8–0.9, and Cs ⫽ 20 are reasonable. For the comparison of the model and FEM results (Figs. 8–12), the value of Cy is chosen to be Cy ⫽ 1. From the comparison with test results, it can be seen that the performance of the new model shows a smoother transition from elastic to inelastic behavior than the other models because it has quadri-linear relations. From the comparison with FEM results, it can be seen that the new model can reasonably describe the increase of yield strength and elastic stiffness according to an increase in the ratio of column flange thickness to column depth due to the inclusion of the bending deformation mode. The suggested model significantly underestimates the panel zone strength of Krawinkler et al.’s specimen B2 (Fig. 6). This specimen exhibited unusually early strain hardening due to a very short yield plateau (es ⫽ 4.4ey) of the stress–strain relation of the column web material [4]. In fact, all of the models shown in Fig. 6 underestimated the strength of this specimen. However, for the remainder of the comparisons, the correlation between the predictions of this new model and the response obtained by test or FEM analysis is quite good regardless of column flange thickness.
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In this study, the thickest column flange used to evaluate panel zone models was tcf ⫽ 4.5 cm. In actual design practice, even thicker column flanges may be used, perhaps on the order of 8–13 cm. Additional test predictions for such column sections are needed to further verify this monotonic model and to investigate maximum shear strength. No such data were found in the literature. 5. Proposed model for cyclic loading As noted earlier, panel zone response can have a particularly important effect on the behavior of steel moment frames under earthquake loading. Consequently, accurate hysteretic rules are needed to predict panel zone response under cyclic and random loading. A common model used in the past is based on bilinear kinematic hardening, and this model has been widely used for inelastic dynamic analysis of moment resisting frames. Fig. 21 shows a comparison of the bilinear kinematic hardening model and test data for the panel zone of Krawinkler et al.’s specimen A1 [4]. A similar comparison is shown in Fig. 27 for the overall subassemblage response for this test specimen. From both figures, it is clear that the response predictions of the bilinear model do not correlate well with the experimentally observed response. The bilinear model substantially underestimated panel zone strength in the latter cycles of loading. The correlation in the later loading cycles could be improved by increasing the yield strength of the bilinear model, but then the correlation would be particularly poor for the early loading cycles. In general, the ability of the bilinear model to replicate the full loading history of the panel zone is limited. Consequently, a new model was developed to provide improved panel zone response predictions over a wide range of loading histories. In this study, hysteretic rules for the panel zone are developed based on Dafalias’ bounding surface theory [16]. This model also uses Cofie and Krawinkler’s rules for the movement of the bound line [17]. Based on observations from experiments and FEM analyses for panel zones, it has been found that for large plastic rotations, the shear strains in the panel zone are distributed nearly uniformly within the panel, and the value of joint rotation is close to the value of the average shear strain in the panel [4,14]. Therefore, it is assumed that the panel zone moment–rotation relationships can be determined from the material properties of the panel zone using Cofie’s rules. These rules for the movement of the bound line, which were developed for stress–strain relationships, will be adopted for the panel zone moment–rotation relationships. The main feature of Cofie’s model is that the cyclic steady state curve is used to describe the movement of the bounding line. In this study, the same kind of cyclic steady state curve is developed to describe the movement of the bounding line for the cyclic behavior of the panel zone, as follows:
冉 冊
Mpa Mpa g ⫽ pa ⫹ gn Mn xMpa n
c
(23)
where Mpa n and gn are the normalizing panel moment and corresponding elastic
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rotation, respectively. By comparison with available cyclic test data, it has been empirically found that the constant C of the cyclic steady state curve is 7 and x is 1.1–1.2 [2]. The cyclic steady state curve and bound lines are shown together with experimental data [7] in Fig. 13. Experimental and FEM results suggest that column flanges do not significantly influence panel zone stiffness during cyclic loading, but do have a significant effect on panel zone strength. From FEM results for joints with the same dimensions except for the column flange thickness, it has been found that the effect of column flange thickness on the strength of the joint during cyclic loading can be normalized by Mpa n [14] as follows pa Mpa n ⫽ My ⫹ 2Mpcf
(24)
where Mpcf is the plastic moment of the column flange. The elastic rotation corresponding to the normalizing moment Mpa n is gn ⫽
Mpa n Ke
(25)
Panel zone response for the initial half-cycle of loading follows the monotonic loading rules described earlier in this paper. Thereafter, the cyclic behavior of the panel zone is defined by elastic and inelastic curves as shown in Fig. 14. To describe the inelastic curves, the shape factor is employed, which was first used for cyclic stress–strain relationships by Dafalias [16]. The procedure for obtaining the shape factor hˆ is as follows: (i) (ii)
Choose the point A such that 0.1ⱕdA / dinⱕ0.5, as shown in Fig. 15. calculate the shape factor from
Fig. 13.
Cyclic steady curve and bound line.
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Fig. 14.
Fig. 15.
Hysteretic rules for panel zones.
Shape factor for inelastic behavior.
h ⫽ dA / gAp ⫹ (din / gAp)[ln(din / dA)⫺1]. (iii)
Normalize the shape factor by the plastic stiffness of the bound line hˆ ⫽ h / Kbl p.
It has been found that a shape factor of hˆ ⫽ 20 for small rotation amplitude cycles and hˆ ⫽ 40 for large rotation amplitude cycles provide a good correlation with experimental data [2]. Thus, in this study a varying shape factor hˆ is used to describe the inelastic curves of the panel zone response. The varying shape factor hˆ is updated according to the accumulated plastic rotation qp only when unloading occurs and is
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otherwise kept constant along each loading path. The varying shape factor hˆ consists of a Boltzman function with the initial shape factor 20 (at qp ⫽ 0) and the final shape factor 40 (at qp ⫽ ⬁) as follows: hˆ ⫽ 40 ⫹
(20⫺40) [1 ⫹ e(qp⫺0.213)/0.074]
(26)
It has also been found that an elastic limit factor a of 1.4 and a plastic stiffness of the bound line of Kbl p ⫽ 0.008Ke provide good correlation with experimental data [2]. The position of the initial bound line is determined by drawing the line with the slope of the bound line at the point with the corresponding slope on the cyclic steady state curve and by making the resulting line intersect the moment axis, as shown in Fig. 13. The plastic stiffness KAp at the point A as shown in Fig. 15 is calculated by using the shape factor hˆ and the plastic stiffness of the bound line Kbl p , as follows:
冋
ˆ dA KAp ⫽ Kbl p 1 ⫹ h din⫺dA
册
(27)
The corresponding tangent stiffness KAt is determined by using the elastic stiffness Ke and the plastic stiffness KAp as follows: KAt ⫽
KeKAp Ke ⫹ KAp
(28)
The bounding line is updated whenever load reversals occur. The procedure for shifting the bounding line is presented below. (i)
Whenever unloading occurs, the mean values and the amplitude for the last half cycle of the loading history, as shown in Fig. 16, are calculated. pa pa M pa m ⫽ 0.5(M A ⫹ M B )
⫽ 0.5(g
(29b)
pa pa M pa a ⫽ 0.5兩M A ⫺M B 兩
(30a)
g
pa a
⫽ 0.5兩g
pa A
⫹g
pa B
g
pa m
(29a)
⫺g
pa A
)
兩
pa B
(30b)
where the subscripts ‘m’ and ‘a’ stand for a mean value and an amplitude, respectively. and the (ii) Calculate the difference between the moment amplitude Mpa a moment Mpa s on the cyclic steady curve corresponding to the rotation amplitude, gpa a pa ⌬Mpa ⫽ Mpa s ⫺Ma
(ii)
(31)
If ⌬Mpa ⬎ 0, cyclic hardening is predicted to take place in the next excursion. Update the bound line by moving it outward by an amount equal to 2FH(⌬Mpa / Mpa n ), where FH is the hardening factor.
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Fig. 16.
冉 冊 冉 冊 Mpa bl Mpa n
(iv)
⫽
new
Mpa bl Mpa n
old
冉 冊 ⌬Mpa Mpa n
(32)
If ⌬Mpa ⬍ 0, cyclic softening is predicted to take place in the next excursion. Update the bound line by moving it inward by an amount equal to 2FS(⌬Mpa / Mpa n ), where Fs is the softening factor.
冉 冊 冉 冊 Mpa bl Mpa n
(v)
⫺2FH
Movement of bound line.
⫽
new
Mpa bl Mpa n
⫺2FS
old
冉 冊 ⌬Mpa Mpa n
(33)
Further move the bound by an amount equal to FRMpa m , where FR is the mean value relaxation factor.
冉 冊 冉 冊 Mpa bl Mpa n
⫽
new
Mpa bl Mpa n
⫺FRMpa m
(34)
old
The same values used in Cofie’s study, FH ⫽ 0.45, FS ⫽ 0.07, and FR ⫽ 0.05, are adopted in the proposed model.
6. Comparison of cyclic loading model with test data The panel zone model for cyclic loading described above is compared with test results for ten specimens tested by Krawinkler et al. [4], Slutter [5], Popov et al.
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[18], and Engelhardt et al. [7]. For some specimens a doubler plate is used to increase the capacity of the panel zone. Test results by Becker [3] showed that for every load level, except maximum load, the strain in the doubler plate was significantly less than that in the column web. Thus, the doubler plates were not fully effective. To account for the limited participation of a doubler plate in resisting panel shear, a reduction factor was considered in calculating the yield moments and stiffness values of the panel zone with a doubler plate. The effectiveness of doubler plates is affected by the method used to connect them to the column (one side attachment, both sides attachment, welding details, etc.). In this paper, the case with a doubler plate attached to only one side of the panel zone is studied. Figs. 17–22 show comparisons of the analytical response predicted by the proposed model and test results for panel zones with no doubler plate. These test data include specimens with a column flange thickness up to 5.3 cm. In Figs. 20–22, the test results for Popov et al.’s specimen 6, Krawinkler et al.’s specimen A1, and Engelhardt et al.’s specimen DBWP are plotted against the model predictions. The match is good for the cycles in which large deformations are imposed. For the first few cycles in which small deformations are imposed, the predictions are not as good, but still reasonable. For Krawinkler et al.’s specimens A2 and B2 and Slutter’s specimen 1 (Figs. 17–19), a large displacement amplitude was applied for the first half cycle of loading, causing large plastic deformations, far beyond the onset of strain hardening, in the panel zone. The model appears to work better for a large displacement amplitude for which strain hardening effects are fully developed than for a small displacement amplitude. Nonetheless, in spite of the simplicity of the model, reasonable agreement has been achieved between model predictions and test results for these six specimens without doubler plates. In Figs. 23–25, model predictions are compared with test results for specimens with doubler plates. In these specimens, the yield stress of the doubler plates was
Fig. 17.
Comparison of the hysteretic rules and test data for Krawinkler et al.’s [4] specimen A2.
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Fig. 18.
Comparison of the hysteretic rules and test data for Krawinkler et al.’s [4] specimen B2.
Fig. 19.
Comparison of the hysteretic rules and test data for Slutter’s [5] specimen 1.
approximately the same as that of the column web. In the analyses, a reduction factor of Rf ⫽ 0.4 was used to account for strain incompatibility between the column web and the doubler plate. As indicated by these figures, reasonable agreement was achieved between the model and the test data. The fact that a reduction factor of Rf ⫽ 0.4 resulted in a good match with the experimental data suggests that the effectiveness of doubler plates for increasing panel zone strength can be quite limited. Fig. 26 shows a comparison of the analytical and experimental results for Popov et al.’s specimen 8. For this specimen, the reported yield stress of the doubler plate was 338 MPa, and for the column web was 413 MPa. Since the yield stress of the
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Fig. 20.
Fig. 21.
625
Comparison of the hysteretic rules and test data for Popov et al.’s [18] specimen 6.
Comparison of the hysteretic rules and test data for Krawinkler et al.’s [4] specimen A1.
doubler plate was significantly different from that of the column web, two panel elements were employed in parallel to obtain the analytical results. To obtain better correlation between the analytical results and the test data, a reduction factor of Rf ⫽ 0.10 was used. Once again, this suggests very limited effectiveness of the doubler plate. Figs. 17–26 above presented model predictions and test results for local panel zone response for a number of test specimens reported in the literature. Fig. 27 shows a comparison between model predictions and test results for the overall load– displacement response for Krawinkler et al.’s specimen A1. The panel zone model
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Fig. 22.
Comparison of the hysteretic rules and test data for Engelhardt et al.’s [7] specimen DBWP.
Fig. 23.
Comparison of the hysteretic rules and test data for Popov et al.’s [18] specimen 2.
described above was combined with a beam–column element by Kim and Engelhardt [19]. Results are also compared with the simpler bilinear panel zone model. The newly proposed panel zone model clearly provides significantly better correlation with the test data as compared with the bilinear model. 7. Modification of cyclic loading model for composite floor slabs The panel zone model discussed above was developed for the case where no composite floor slab is present. Likewise, the comparisons between the model and experi-
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Fig. 24.
Comparison of the hysteretic rules and test data for Popov et al.’s [18] specimen 3.
Fig. 25.
Comparison of the hysteretic rules and test data for Popov et al.’s [18] specimen 4.
627
mental data were all for bare steel specimens, i.e., specimens in which no composite floor slab was present. However, since most steel moment frames have composite floor slabs, the effects of the slab on panel zone response are of interest. To investigate the effect of composite slabs on the behavior of panel zones, a composite panel zone model was developed for the monotonic and cyclic behavior of beam-to-column joints in steel moment frames with composite floor slabs. The composite panel zone model is the same as the bare steel panel zone model described above, except that the effective depth of the panel zone, and therefore the area of the panel zone, is increased due to the presence of the concrete slab. This, in turn,
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Fig. 26.
Comparison of the hysteretic rules and test data for Popov et al.’s [18] specimen 8.
Fig. 27. Comparison of test and the analysis using the developed panel zone model for overall response for Krawinkler et al.’s [4] specimen A1.
affects the stiffness, yield moments, and cyclic steady curves of the panel zone model. For the bare steel case, a statically equivalent force couple replaces the moment in the beam, with the forces assumed to be acting at mid-depth of the beam flanges. When a slab is present, and positive moment is applied to the beam, the effective location of the force resultant at the beam’s top flange will move up towards the slab, thereby increasing the effective panel zone depth. The bending moment
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applied by the composite beam onto the panel zone will also increase for positive bending. The boundary forces on an interior joint panel, shown in Fig. 28, can be transformed into an approximate equivalent shear force from equilibrium as follows: Veq ⫽ ⫺ ⫽
冉
冊
冉
M+b M⫺ M+b ⫹ M⫺ 1 Vct ⫹ Vcb M+b M⫺ 1 b b b ⫹ ⫺ ⬇ ⫹ ⫺ ⫽ +⫹ ⫺ + ⫺ + ⫺ db db 2 db db Hc d b db
冊 冉
冊
+ ⫺ 1 M+b 1⫺(d+b⫺d⫺ M⫺ b b ) / (2d b )⫺db / Hc pa Mpa⫺ + ⫹ ⫺ ⬇ M ⫺ Hc db db db
(35)
+ 1⫺(d+b⫺d⫺ b ) / (2db )⫺r pa M db⫺tbf
where db⫹ ⫽ dcom⫺ts / 2⫺tbf / 2, dcom is the composite beam depth, ts is the solid slab ⫹ is equal to d⫺ thickness, and d⫺ b ⫽ db⫺tbf. If db b then Eq. (35) reduces to Eq. (5) for the bare steel beam-to-column joint. For a composite beam-to-column exterior joint, Eq. (35) can be reduced to Veq ⫽
1⫺d+b / Hc pa M for positive moment d+b
(36a)
Veq ⫽
1⫺d⫺ b / Hc pa M for negative moment d⫺ b
(36b)
The hysteretic rules for the composite panel zone element are based on those for the bare steel element and are shown in Fig. 29. For the interior composite joint, the cyclic behavior of panel zone follows path O–B–E–H, and for the exterior composite joint it follows the path O–B–E⬘–H because the cyclic behavior of the exterior joint depends on the direction of moment. To account for cracking of the concrete slab under negative moment the unloading behavior follows the inelastic curves of
Fig. 28.
Boundary forces on composite beam-to-column interior joint.
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Fig. 29.
Hysteretic rules for composite beam-to-column joint.
the bare steel beam-to-column joint. The cyclic behavior of the panel zone under positive moment after crack closing in the concrete slab is described by a linear crack closing segment followed by a nonlinear composite cyclic curve (paths C–E and F–H in Fig. 29). The factors f and b shown in Fig. 29, were determined empirically to be f ⫽ 0.5 and b ⫽ 0.1. In Fig. 30 the cyclic steady state curves for a
Fig. 30.
Cyclic steady state curve for composite beam-to-column joint.
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Fig. 31.
631
Comparison of the hysteretic rules and test data for Lee’s [6] specimen EJ-FC.
composite beam-to-column joint tested by Lee [6] are shown. From this figure it can be seen that due to the presence of a composite slab, behavior of the panel zone differs significantly under positive and negative moments. The proposed composite panel zone model is compared with test results for four specimens tested by Lee [6] and Engelhardt et al. [7,8]. Also shown in these figures are the analytical predictions using the bare steel panel zone model. Figs. 31–33 show these comparisons for specimens with no doubler plate. Specimens tested by Lee [6] are shown in Figs. 31 and 32. The beams in these specimens were W18 × 35 sections. For these specimens, the composite panel zone model shows
Fig. 32. Comparison of the hysteretic rules and test data for Lee’s [6] specimen IJ-FC.
632
Fig. 33.
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Comparison of the hysteretic rules and test data for Engelhardt et al.’s [7] specimen DBWPC.
better correlation with the test data than the bare steel panel zone model for both small and large amplitudes of deformations. It is also clear from Figs. 31 and 32 that there is a significant difference in predicted response between the bare steel and composite panel zone elements. From the comparison of the model predictions and the experimental results, it can be seen that the effects of the concrete slab are negligible for Engelhardt et al.’s specimen DBWP-C (Fig. 33). This specimen was constructed using W36 × 150 beams. The similarity between the bare steel and composite model predictions may be attributed to the fact that the increase of panel zone area due to the composite slab was not significant because the steel beam depth was much larger than the slab depth. The increase in panel zone area obtained by using Eq. (35) instead of Eq. (5) is about 36, 16, and 8% for specimens EJ-FC (Fig. 31), IJ-FC (Fig. 32), and DBWPC (Fig. 33), respectively. In Fig. 34, model predictions are compared with test results for Engelhardt and Venti’s [8] specimen UTA-FF. This specimen was constructed with a doubler plate. In the model, a reduction factor of Rf ⫽ 0.60 was used to account for strain incompatibility between the column web and the doubler plate. The performance of the model provides reasonable correlation with experimental data. Further, there is once again little effect of the composite slab due to the rather deep W36 × 150 beams used for this specimen. Only very limited experimental data are available on composite specimens with weak panel zones. To refine mathematical models for composite panel zone response, further experimental data are needed for specimens with composite slabs under large inelastic excursions of monotonic loading as well as cyclic loading.
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Fig. 34.
633
Comparison of the hysteretic rules and test data for Engelhardt and Venti’s [8] specimen UTAFF.
8. Conclusions The objective of the study presented in this paper was to develop models to describe the monotonic and cyclic loading behavior of panel zones in steel moment resisting frames. These new models build on previously developed models, and introduce a number of features and refinements that show better correlation with available experimental data. The model for monotonic loading is based on quadri-linear panel zone moment– deformation relations. In this model, both bending and shear deformation modes are considered. The model proposed for cyclic loading is based on Dafalias’ bounding surface theory combined with Cofie’s rules for movement of the bound line. Finally, additional modifications are suggested for the cyclic loading model to account for the influence of a composite floor slab on panel zone response. For all of the proposed models, extensive comparisons with experimental data were presented to demonstrate the capabilities and limitations of the models. In spite of the simplicity of the proposed models, reasonable agreement between model predictions and test data was achieved over a broad range of experiments. In the process of developing these panel zone models, several issues were identified which appear to merit further research. One of these issues is the effect of very thick column flanges on panel zone strength. Much of the available experimental data are for panel zones in columns with a flange thickness less than about 3–4 cm. In actual practice, columns with flange thickness values in excess of 10 cm are not uncommon. Further experimental data is needed to better quantify the contribution of very thick column flanges to overall panel zone strength. An additional issue of concern is the effectiveness of doubler plates. Based on comparisons between model predictions and experimental data, it appears that
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doubler plates are, in many cases, not fully effective in contributing to panel zone strength and stiffness. In many past experiments, it appears that doubler plates were less than 50% effective. Additional studies are needed to better quantify the contribution of doubler plates to panel zone strength and stiffness for various attachment details. This is an issue of considerable practical importance, since doubler plates are commonly used in practice to augment panel zone strength. Current building code regulations in the US appear to assume that doubler plates will be fully effective. Further studies are also needed to better quantify the effects of a composite floor slab on panel zone behavior. The majority of past tests and past panel zone models have considered the case of a bare steel frame. However, limited data suggests that the presence of a composite concrete floor slab can significantly affect panel zone behavior, particularly for relatively shallow beams. Some simple modifications to the panel zone model were suggested in this paper to account for composite slab effects. However, further experimental data are needed to better understand slab effects and to further refine panel zone response models. Finally, the models presented in this paper are intended to represent the effects of inelasticity on panel zone response, i.e. the models account for the effects of material yielding and strain hardening. However, after yielding and strain hardening, the panel zone will ultimately degrade in strength either due to shear buckling of the panel zone or due to fracture of the column or beam flanges at the corners of the panel zone. The current models do not predict strength degradation due to instability or fracture. Further studies are needed to better understand the ultimate failure modes of the panel zone and to quantify the available deformation capacity.
Acknowledgements The writers gratefully acknowledge support for this work from the National Science Foundation (grant no. CMS-9358186) and from the American Institute of Steel Construction, Inc.
References [1] Tsai KC, Popov EP. Steel beam–column joints in seismic moment resisting frames. Report no. EERC 88/19, University of California, Berkeley, CA; 1988. [2] Kim K, Engelhardt, MD. Development of analytical models for earthquake analysis of steel moment frames. Report no. PMFSEL 95-2, Phil M. Ferguson Structural Engineering Laboratory, The University of Texas at Austin, Austin, TX; 1995. [3] Becker ER. Panel zone effect on the strength and stiffness of rigid steel frames. Research report, Mechanical Lab., University of Southern California; 1971. [4] Krawinkler H, Bertero VV, Popov EP. Inelastic behavior of steel beam-to-column subassemblages. Report No. EERC 71/07, University of California, Berkeley, CA; 1971. [5] Slutter RG. Tests of panel zone behavior in beam–column connections. Report no. 403.1, Fritz Engineering Lab., Lehigh University, Bethlehem, PA; 1981. [6] Lee SJ. Seismic behavior of steel building structures with composite Slabs. PhD thesis, Department of Civil Engineering, Lehigh University, Bethlehem, PA; 1987.
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[7] Engelhardt MD, Fry GT, Jones S, Venti M, Holliday S. Behavior and design of radius-cut reduced beam section connections. Report no. SAC/BD-00/17, SAC Joint Venture, Sacramento, CA; 2000. [8] Engelhardt MD, Venti MJ. Test of a free flange connection with a composite floor slab. Report no. SAC/BD-00/18, SAC Joint Venture, Sacramento, CA; 2000. [9] International Conference of Building Officials. Uniform building code, ICBO, Whittier, CA; 1997. [10] American Institute of Steel Construction. Seismic provisions for structural steel buildings. Chicago, IL; 1997. [11] Federal Emergency Management Agency. Recommended seismic design criteria for new steel moment-frame buildings. Report no. FEMA-350; 2000. [12] Lui EM. Effects of connection flexibility and panel zone deformation on the behavior of panel steel frames. PhD thesis, Department of Civil Engineering, Purdue University, West Lafayette, IN; 1985. [13] Fielding DJ, Huang JS. Shear in steel beam-to-column connections. Welding J. 1971;50(7):313s– 26 (research supplement). [14] Wang SJ. Seismic response of steel building frames with inelastic joint deformation. PhD thesis, Department of Civil Engineering, Lehigh University, Bethlehem, PA; 1988. [15] Kanaan AE, Powell GH. DRAIN-2D—a general purpose computer program for dynamic analysis of inelastic plane structures with user’s guide. Report no. EERC 73/6 and 73/22, University of California, Berkeley, CA; 1973. [16] Dafalias YF. On cyclic and anisotropic plasticity: (I) A general model including material behavior under stress reversals. (II) Anisotropic hardening for initially orthotropic materials. PhD thesis, Department of Civil Engineering, University of California, Berkeley, CA; 1975. [17] Cofie NG, Krawinkler H. Uniaxial cyclic stress–strain behavior of structural steel. J. Engng. Mech., ASCE 1985;111(9):1105–20. [18] Popov EP, Amin NR, Louie JC, Stephen RM. Cyclic behavior of large beam–column assemblies. Earthquake Spectra 1985;1(2):201–37. [19] Kim K, Engelhardt MD. Beam–column element for nonlinear seismic analysis of steel frames. J. Struct. Engng, ASCE 2000;126(8):916–25.
Monotonic and cyclic loading models for panel zones in steel moment frames Kee Dong Kim a, Michael D. Engelhardt b,* a
Department of Civil Engineering, Kongju National University, Kongju, Chungnam, South Korea Department of Civil Engineering, University of Texas at Austin, Austin, TX 78712-1076, USA
b
Received 14 June 2001; received in revised form 17 August 2001; accepted 5 October 2001
Abstract This paper presents the development of analytical models to predict the elastic and inelastic response of the panel zone portion of columns in steel moment resisting frames. In many practical cases, the panel zone can dominate the inelastic response of a moment frame, and accurate panel zone models are needed to realistically predict overall frame performance. Similar to previous models, the newly proposed models are based on the concept of representing the panel zone as a nonlinear rotational spring. These new models build on previously developed models, and introduce a number of features and refinements that show better correlation with available experimental data. The model for monotonic loading is based on quadrilinear panel zone moment–deformation relations. In this model, both bending and shear deformation modes are considered. The model proposed for cyclic loading is based on Dafalias’ bounding surface theory combined with Cofie’s rules for movement of the bound line. Additional modifications are suggested to the cyclic loading model to account for the influence of a composite floor slab on panel zone response. Extensive comparisons with experimental data are presented. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Seismic; Inelastic; Bounding surface model; Composite; Connections; Joints
* Corresponding author. Tel.: +1-512-471-6837; fax: +1-512-471-1944. E-mail address: [email protected] (M.D. Engelhardt). 0143-974X/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 3 - 9 7 4 X ( 0 1 ) 0 0 0 7 9 - 7
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1. Introduction Mathematical models are developed in this paper for describing the monotonic and cyclic load–deformation response of the panel zone region of beam–column joints of a steel moment resisting frame. The panel zone is the portion of the column contained within the beam–column joint. When a moment frame is subject to lateral loads, high shear forces develop within the panel zone. The resulting deformations of the panel zone can have an important effect on the response of the frame in both the elastic and inelastic ranges of frame behavior [1,2]. Numerous tests have been performed in the past three decades to investigate the load–deformation behavior of the joint panel using connection subassemblies [3–8]. Some significant observations from these tests are: 앫 Joint panel zones often develop a maximum strength that is significantly greater than the strength at first yield. This additional strength has been attributed to strain hardening and to contributions of the column flanges in resisting panel zone shear forces. Large inelastic panel zone deformations are typically required in order to develop the maximum panel zone strength. 앫 Panel zone deformations can add significantly to the overall deformation of a steel moment frame, for both elastic and inelastic ranges of behavior. 앫 Panel zone stiffness and strength can be increased by the attachment of web doubler plates to the column within the joint region. The effectiveness of doubler plates is affected by the method used to connect them to the column. 앫 In the inelastic range, panel zones can exhibit very ductile behavior, both for monotonic and cyclic loading. Experimentally observed hysteresis loops are typically very stable, even at large inelastic deformations. 앫 Large inelastic panel zone deformations can increase the likelihood of fracture occurring in the region of the beam flange to column flange groove welds. This effect has been attributed to the occurrence of large localized deformations or ‘kinks’ in the column flanges at the boundaries of the panel zone. Current US building code provisions [9–11] permit the formation of plastic hinges in the panel zones of steel moment frames under earthquake loading. Thus, rather than forming plastic flexural hinges only in the beams or columns, a primary source of energy dissipation in a steel moment frame can be the formation of plastic shear hinges in the panel zones. Consequently, to accurately predict the response of a steel moment frame under earthquake loading, an accurate analytical model is needed to predict the response of the panel zone. To include panel zone deformation in frame analysis, the traditional center-tocenter line representation of the frame must be modified. Fig. 1 shows a comparison between an experiment on a beam–column subassemblage and an analytical prediction of the subassemblage response. The experiment was specimen A1 reported by Krawinkler et al. [4]. Inelastic response of this specimen was dominated by yielding of the panel zone. The analytical results are obtained from a model of the specimen using center-to-center line dimensions. A beam–column element with plastic hinges
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Fig. 1.
607
Comparison of test results [4] and analytical results obtained by using center-to-center line dimension modeling.
was employed in this analysis. From the figure it can be seen that the analysis using the center-to-center line dimension, without explicit modeling of the panel zone, may produce misleading results. Clearly, the panel zone response must be explicitly modeled to obtain realistic predictions of the overall frame behavior. To model the behavior of panel zones in frame analysis, Lui [12] developed a joint model based on the finite element method. The model consists of seven elements for interior beam-to-column joints: one web element, two flange elements (beam elements), and four connection elements. Although capable of representing a variety of deformation modes of panel zones, this model employed a simple hardening rule suitable for monotonic loading and does not realistically model cyclic behavior. Another disadvantage of this model is its high computational cost. Other finite element models using more sophisticated hardening rules could be developed for the analysis of column panel zones. However, in this study, nonlinear rotational springs are used as the basis for modeling the panel zone for nonlinear dynamic analysis of moment resisting frames because of simplicity and computational efficiency. Several researchers, including Fielding and Huang [13], Krawinkler et al. [4] and Wang [14] proposed relationships between panel zone shear force V and panel zone deformation g for monotonic loading. These relationships have been used as the basis of mathematical models for nonlinear rotational springs representing the panel zone. Krawinkler’s V–g relations have been adopted in several building codes [9,10] as a basis for computing the shear strength of panel zones. However, it was pointed out by Krawinkler that a new model might be needed for joints with thick column flanges since his V–g relations were derived from experimental and analytical results for panel zones with relatively thin column flanges. Wang also showed that Krawinkler’s V–g relations may overestimate panel zone shear strength for panel zones with thick column flanges.
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In the remainder of this paper, existing analytical models for panel zones will be examined by making comparisons with experimental data or with other analytical results. A number of refinements and improvements to existing models are then suggested to overcome some of the shortcomings of existing models and to provide better correlation with available experimental data.
2. General characteristics of panel zone element The panel zone element is essentially a rotational spring element, which transfers moment between the columns and beams framing into a joint [15]. The panel element has no dimensions and connects two nodes with the same coordinates. One of these nodes is attached to the elements modeling the columns framing into the joint, as shown in Fig. 2, while the other node is attached to the elements modeling the beams. Therefore, the moment transferred by the panel element is related to the relative rotation between the columns and beams framing into a joint. The vertical and horizontal translations of the two nodes are constrained to be identical. Therefore, one vertical, one horizontal, and two rotational degrees of freedom exist at each joint. The relative rotation between the connected nodes is related to the node rotations as follows: dg ⫽ {1 ⫺1}
再 冎 dqI
(1)
dqJ
or dg ⫽ a·dr
(2)
where dg is the increment of relative rotation, which is the panel element deformation, and dqI, and dqJ, are the increments of rotation of the connected nodes. Then the tangent stiffness relationship for the panel zone element is
Fig. 2.
Idealization of beam-to-column joint.
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dMpa ⫽ Ktdg
609
(3)
pa
where dM is the increment of moment applied on a joint and Kt is the rotational tangent stiffness of the joint. In terms of nodal rotations, the stiffness, KT, is given by KT ⫽ aTKta
(4)
The definable properties of the panel zone element are the rotational stiffnesses and yield moments for monotonic loading, and hysteretic rules for cyclic loading. In the following sections, existing models for monotonic loading are examined and a new model is presented. This is followed by the development of hysteretic rules for a cyclic loading model.
3. Review of existing models for monotonic loading Existing mathematical models for panel zone response under monotonic loading are typically based on a computation of an approximate equivalent shear force acting on the panel zone. The boundary forces on a joint panel, shown in Fig. 3, can be transformed into an approximate equivalent shear force from equilibrium as follows: Veq ⫽
Mbl ⫹ Mbr (Vct ⫹ Vcb) Mbl ⫹ Mbr (1⫺r) pa ⫺ (1⫺r) ⫽ M ⬇ db⫺tbf 2 db⫺tbr db⫺tbf
(5)
where r ⫽ (db⫺tbf) / Hc, Mpa ⫽ Mbl ⫹ Mbr is the panel zone moment, tbf is the thickness of the beam flange, db is the beam depth, and Hc is the column height. A key simplification in this analysis is that the beam moments are replaced by an equivalent couple, with the forces acting at mid-depth of the beam flanges. These forces produce a large shear in the panel zone. The shear in the column segments outside of the panel zone are then subtracted to obtain the net shear force, Veq, acting on the panel zone. In obtaining the shear forces in the column segments outside of the panel zone, it is often assumed that points of inflection in the column occur at a distance Hc/2 above and below the panel zone.
Fig. 3. Boundary forces and equivalent shear forces on panel zone.
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Panel zone shear force V versus panel zone deformation g relations for monotonic loading, which are based on the equivalent shear force, Veq, can be transformed into panel zone moment Mpa versus panel zone deformation g relations by Eq. (5). Fielding and Huang [13] proposed a bilinear relationship consisting of an elastic stiffness Ke followed by a post elastic stiffness K1. Krawinkler et al. [4] and Wang [14] each proposed different tri-linear Mpa–g relations consisting of an elastic stiffness Ke followed by two linear post elastic stiffness values K1 and K2. For all three models, the post elastic stiffness K1 is related to the contribution of the column flanges to the panel stiffness. For the tri-linear models, the second post elastic stiffness K2 is associated with strain hardening. Past researchers computed the elastic stiffness of the panel element by considering pure elastic shear deformation of an effective shear area of the panel zone. Fielding and Krawinkler considered the effective shear area Aeff equal to (dc⫺tcf)tcw, and Wang considered the effective shear area Aeff of (dc⫺2tcf)tcw, where the subscripts ‘c’, ‘f’, and ‘w’ stand for column, flange, and web, respectively. They suggested the yield moment and elastic stiffness of the panel zone be taken as follows: Mpa y ⫽ Ke ⫽
Vydb t¯ yAeffdb ⫽ (1⫺r) (1⫺r)
Mpa GAeffdb y ⫽ gy (1⫺r)
(6a) (6b)
where Vy is the yield shear force of the panel zone, gy ⫽ t¯ y / G, G is the elastic shear modulus, and t¯ y is the Von Mises yield shear stress of the column web, based on shear and axial force interaction. The Von Mises yield shear stress, t¯ y, is taken as: t¯ y ⫽
sy
冑3
冑1⫺(P / P ) y
2
(7)
where P and Py are the axial force and the axial yield force on the column, respectively, and sy is the yield stress of the column web. For the inelastic range, Fielding considered a bilinear model with the following post-elastic stiffness K1: K1 ⫽
5.2Gbcft3cf db(1⫺r)
(8)
where bcf and tcf are width and thickness of column flange, respectively. Krawinkler proposed empirical formulas for the post-elastic stiffness K1 and the second yield moment Mpa sh as follows: K1 ⫽
1.04Gbcft2cf (1⫺r)
pa Mpa sh ⫽ My ⫹
3.12t¯ ybcft2cf (1⫺r)
Wang suggested the post-elastic stiffness K1, as follows:
(9) (10)
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K1 ⫽ 0.7Gbcft2cf
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(11)
Krawinkler and Wang assumed that strain hardening begins at gsh ⫽ 4gy and 3.5gy, respectively. The strain-hardening branch stiffness K2 was suggested as follows: K2 ⫽
GstAeffdb (1⫺r)
(12)
where Gst is the strain hardening shear modulus. The existing models described above are compared to four specimens tested by Krawinkler et al. [4], Fielding and Huang [13], and Slutter [5] in Figs. 4–7. Each of these specimens was a beam–column subassemblage with a weak panel zone. Fig. 4 shows Krawinkler et al.’s specimen A2, which had a column flange thickness of 1 cm. Slutter’s specimen 1, which had a column flange thickness of 1.8 cm, is plotted in Fig. 5. Krawinkler et al.’s specimen B2, with a column flange thickness of 2.37 cm, is shown in Fig. 6. Finally, Fig. 7 shows Fielding and Huang’s test specimen, which had a column flange thickness of 3.5 cm. Additional comparisons are shown in Figs. 8–12. In these figures, FEM analysis predictions are provided for Slutter’s specimen 1 and compared to the simplified nonlinear spring models. This specimen was analyzed a number of times, varying the column flange thickness. These FEM analyses provide an indication of the expected response of panel zones as the column flange thickness varies, but where all other variables remain constant. These finite element analyses were reported by Wang [14]. The specimen tested by Slutter was analyzed by using a two-dimensional FEM, in which the flanges and webs were represented by beam elements and plain stress elements, respectively. The finite element results are compared with the corresponding test data in Fig. 5, for the actual specimen with a 1.8 cm thick column flange.
Fig. 4. Comparison of the monotonic model and test data for Krawinkler et al.’s [4] specimen A2 with tcf ⫽ 1 cm.
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Fig. 5. Comparison of the monotonic model and test data for Slutter’s [5] specimen 1 with tcf ⫽ 1.8 cm.
Fig. 6. Comparison of the monotonic model and test data for Krawinkler et al.’s [4] specimen B2 with tcf ⫽ 2.37 cm.
This comparison shows that Wang’s finite element analysis reasonably predicted the observed panel zone behavior. Figs. 8–12 show finite element predictions for a model based on Slutter’s specimen, but considering column flange thickness values equal to 0.9, 1.35, 2.7, 3.6 and 4.51 cm. In each figure, the corresponding predictions of the simple nonlinear spring elements are also plotted. A number of observations can be made from the comparisons plotted in Figs. 4– 12. Fielding’s bilinear model performs well for small panel zone rotations, but this
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Fig. 7. Comparison of the monotonic model and test data for Fielding and Huang’s [13] specimen with tcf ⫽ 3.5 cm.
Fig. 8. Comparison of the monotonic model and FEM results for Slutter’s specimen 1 with tcf ⫽ 0.9 cm.
model shows rather poor performance at large rotations regardless of column flange thickness because the model neglects strain-hardening effects. The performance of Krawinkler et al.’s model appears reasonable for panel zone joints with column flange thickness less than about 2.5 cm. However, for thicker column flanges, this model somewhat overestimates panel zone strength. Wang’s model generally underestimates panel zone strength regardless of column flange thickness, apparently because in this model the effective shear area of the panel zones is calculated as (dc⫺2tcf)tcw instead of the other models’ effective shear area (dc⫺tcf)tcw.
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Fig. 9. Comparison of the monotonic model and FEM results for Slutter’s specimen 1 with tcf ⫽ 1.35 cm.
Fig. 10. Comparison of the monotonic model and FEM results for Slutter’s specimen 1 with tcf ⫽ 2.7 cm.
4. Proposed model for monotonic loading and comparison with test data As the ratio of column flange thickness to column depth increases, the influence of column flange thickness on panel zone yield moment and elastic stiffness increases [14]. Panel zone models that include shear deformations only cannot account for this increase in yield moment and elastic stiffness according to the increase in the ratio of column flange thickness to column depth. Thus, in this study, both bending and
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Fig. 11. Comparison of the monotonic model and FEM results for Slutter’s specimen 1 with tcf ⫽ 3.6 cm.
Fig. 12. Comparison of the monotonic model and FEM results for Slutter’s specimen 1 with tcf ⫽ 4.51 cm.
shear deformation modes are included in the panel zone model. The resulting monotonic model has quadri-linear Mpa–g relations. It is assumed that the panel zone can be considered as two equivalent beams, which are symmetric with respect to the center of the panel zone and are fixed at the center. The boundary condition at the other end of these beams is considered to be somewhere between free and fixed. Thus, the displacement of the equivalent beam due to the shear force Veq can be described as follows:
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⌬⫽
冉
冊
1 1 ⫹ V S1 S2 eq
(13)
where 1/S1 is the bending flexibility of the equivalent beam and 1/S2 is the shear flexibility. The bending and shear flexibilities can be described as follows: [(db⫺tbf) / 2]3 [(db⫺tbf) / 2] 1 1 ⫽ and ⫽ S1 CrEI S2 G(dctcw ⫹ RfAdp)
(14)
where I is the moment of inertia of the column section, Cr is a constant to be determined according to the degree of end restraint, Adp is the area of a doubler plate, and Rf is the reduction factor to account for the strain incompatibility between a doubler plate and column web. Eq. (13) can be rewritten in terms of the panel zone moment Mpa and the rotation g as follows: S1S2 S1S2 (db⫺tbf) ⌬⫽ g S1 ⫹ S2 S1 ⫹ S2 2
Veq ⫽
S1S2 (db⫺tbf)(db⫺tbf) (db⫺tbf) ⫽ g Mpa ⫽ Veq (1⫺r) S1 ⫹ S2 2 (1⫺r)
(15a) (15b)
Thus, the elastic stiffness is Ke ⫽
S1S2 (db⫺tbf)(db⫺tbf) S1 ⫹ S2 2 (1⫺r)
(16)
The yield moment of panel zone is defined as follows: Mpa y ⫽ KeCygy
(17)
where Cygy is the average shear deformation of panel zone at which shear yielding occurs and Cy is the ratio of the average shear deformation to gy. To describe the behavior of the panel zone in the range from first shear yielding up to the entire shear yielding of the panel zone, it is assumed that for this range the panel zone can be modeled as two separate beams with T-shaped sections, similar to Fielding and Huang’s approach [13]. The web depth of the T-shaped section is a quarter of the column web depth. Then, the post-elastic stiffness K1 can be defined as PS1PS2 (db⫺tbf)(db⫺tbf) K1 ⫽ 2 PS1 ⫹ PS2 2 (1⫺r) PS1 ⫽
CrEIT G[(dc / 2⫺dyw)tcw ⫹ RfAdp / 4] and PS2 ⫽ [(db⫺tbf) / 2]3 [(db⫺tbf) / 2]
(18a) (18b)
where IT is the moment of inertia of the T section and dyw is a quarter of the web depth. The second yield moment Mpa y1 is ¯ y(dctcw ⫹ RfAdp) Mpa y1 ⫽ t
(db⫺tbf) (1⫺r)
(19)
The second post-elastic stiffness after shear yielding of the entire panel zone is
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defined by using an approach similar to Krawinkler et al. [4]. The panel zone, after the shear yielding of the entire web, consists of an elastic–perfectly plastic shear panel surrounded by rigid boundaries with springs at the four corners. It is assumed that these springs simulate the resistance of elements surrounding the panel, in particular the bending resistance of column flanges, and that the spring stiffness can be approximated by Ks ⫽
Ebcft2cf Cs
(20a)
where Cs is a constant to be determined from test results. From the work equation and Eq. (20a), the second post-elastic stiffness K2 is obtained as K2 ⫽
4Ebcft2cf Cs(1⫺r)
(20b)
It is assumed that when strain hardening starts, plastic hinges form in the column flanges at the four corners of the panel [14]. Then, the third yield moment of the panel zone at which strain hardening initiates can be defined as pa ¯ ybcft2cf Mpa y2 ⫽ My1 ⫹ s
(21a)
s¯ y ⫽ s (1⫺(P / Py) )
(21b)
fl y
2
where sfly is the yield stress of the column flange. The strain hardening stiffness K3 is K3 ⫽
Gst(dctcw ⫹ RfAdp)(db⫺tbf) (1⫺r)
(22)
This new proposed monotonic model has been applied to the same four specimens as in the previous section. Figs. 4–12 show comparisons of this new model with test results and FEM results for various values of column flange thickness. Based on calibration to test results (Figs. 4–7), it has been found that Cr ⫽ 5, Cy ⫽ 0.8–0.9, and Cs ⫽ 20 are reasonable. For the comparison of the model and FEM results (Figs. 8–12), the value of Cy is chosen to be Cy ⫽ 1. From the comparison with test results, it can be seen that the performance of the new model shows a smoother transition from elastic to inelastic behavior than the other models because it has quadri-linear relations. From the comparison with FEM results, it can be seen that the new model can reasonably describe the increase of yield strength and elastic stiffness according to an increase in the ratio of column flange thickness to column depth due to the inclusion of the bending deformation mode. The suggested model significantly underestimates the panel zone strength of Krawinkler et al.’s specimen B2 (Fig. 6). This specimen exhibited unusually early strain hardening due to a very short yield plateau (es ⫽ 4.4ey) of the stress–strain relation of the column web material [4]. In fact, all of the models shown in Fig. 6 underestimated the strength of this specimen. However, for the remainder of the comparisons, the correlation between the predictions of this new model and the response obtained by test or FEM analysis is quite good regardless of column flange thickness.
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In this study, the thickest column flange used to evaluate panel zone models was tcf ⫽ 4.5 cm. In actual design practice, even thicker column flanges may be used, perhaps on the order of 8–13 cm. Additional test predictions for such column sections are needed to further verify this monotonic model and to investigate maximum shear strength. No such data were found in the literature. 5. Proposed model for cyclic loading As noted earlier, panel zone response can have a particularly important effect on the behavior of steel moment frames under earthquake loading. Consequently, accurate hysteretic rules are needed to predict panel zone response under cyclic and random loading. A common model used in the past is based on bilinear kinematic hardening, and this model has been widely used for inelastic dynamic analysis of moment resisting frames. Fig. 21 shows a comparison of the bilinear kinematic hardening model and test data for the panel zone of Krawinkler et al.’s specimen A1 [4]. A similar comparison is shown in Fig. 27 for the overall subassemblage response for this test specimen. From both figures, it is clear that the response predictions of the bilinear model do not correlate well with the experimentally observed response. The bilinear model substantially underestimated panel zone strength in the latter cycles of loading. The correlation in the later loading cycles could be improved by increasing the yield strength of the bilinear model, but then the correlation would be particularly poor for the early loading cycles. In general, the ability of the bilinear model to replicate the full loading history of the panel zone is limited. Consequently, a new model was developed to provide improved panel zone response predictions over a wide range of loading histories. In this study, hysteretic rules for the panel zone are developed based on Dafalias’ bounding surface theory [16]. This model also uses Cofie and Krawinkler’s rules for the movement of the bound line [17]. Based on observations from experiments and FEM analyses for panel zones, it has been found that for large plastic rotations, the shear strains in the panel zone are distributed nearly uniformly within the panel, and the value of joint rotation is close to the value of the average shear strain in the panel [4,14]. Therefore, it is assumed that the panel zone moment–rotation relationships can be determined from the material properties of the panel zone using Cofie’s rules. These rules for the movement of the bound line, which were developed for stress–strain relationships, will be adopted for the panel zone moment–rotation relationships. The main feature of Cofie’s model is that the cyclic steady state curve is used to describe the movement of the bounding line. In this study, the same kind of cyclic steady state curve is developed to describe the movement of the bounding line for the cyclic behavior of the panel zone, as follows:
冉 冊
Mpa Mpa g ⫽ pa ⫹ gn Mn xMpa n
c
(23)
where Mpa n and gn are the normalizing panel moment and corresponding elastic
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rotation, respectively. By comparison with available cyclic test data, it has been empirically found that the constant C of the cyclic steady state curve is 7 and x is 1.1–1.2 [2]. The cyclic steady state curve and bound lines are shown together with experimental data [7] in Fig. 13. Experimental and FEM results suggest that column flanges do not significantly influence panel zone stiffness during cyclic loading, but do have a significant effect on panel zone strength. From FEM results for joints with the same dimensions except for the column flange thickness, it has been found that the effect of column flange thickness on the strength of the joint during cyclic loading can be normalized by Mpa n [14] as follows pa Mpa n ⫽ My ⫹ 2Mpcf
(24)
where Mpcf is the plastic moment of the column flange. The elastic rotation corresponding to the normalizing moment Mpa n is gn ⫽
Mpa n Ke
(25)
Panel zone response for the initial half-cycle of loading follows the monotonic loading rules described earlier in this paper. Thereafter, the cyclic behavior of the panel zone is defined by elastic and inelastic curves as shown in Fig. 14. To describe the inelastic curves, the shape factor is employed, which was first used for cyclic stress–strain relationships by Dafalias [16]. The procedure for obtaining the shape factor hˆ is as follows: (i) (ii)
Choose the point A such that 0.1ⱕdA / dinⱕ0.5, as shown in Fig. 15. calculate the shape factor from
Fig. 13.
Cyclic steady curve and bound line.
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Fig. 14.
Fig. 15.
Hysteretic rules for panel zones.
Shape factor for inelastic behavior.
h ⫽ dA / gAp ⫹ (din / gAp)[ln(din / dA)⫺1]. (iii)
Normalize the shape factor by the plastic stiffness of the bound line hˆ ⫽ h / Kbl p.
It has been found that a shape factor of hˆ ⫽ 20 for small rotation amplitude cycles and hˆ ⫽ 40 for large rotation amplitude cycles provide a good correlation with experimental data [2]. Thus, in this study a varying shape factor hˆ is used to describe the inelastic curves of the panel zone response. The varying shape factor hˆ is updated according to the accumulated plastic rotation qp only when unloading occurs and is
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otherwise kept constant along each loading path. The varying shape factor hˆ consists of a Boltzman function with the initial shape factor 20 (at qp ⫽ 0) and the final shape factor 40 (at qp ⫽ ⬁) as follows: hˆ ⫽ 40 ⫹
(20⫺40) [1 ⫹ e(qp⫺0.213)/0.074]
(26)
It has also been found that an elastic limit factor a of 1.4 and a plastic stiffness of the bound line of Kbl p ⫽ 0.008Ke provide good correlation with experimental data [2]. The position of the initial bound line is determined by drawing the line with the slope of the bound line at the point with the corresponding slope on the cyclic steady state curve and by making the resulting line intersect the moment axis, as shown in Fig. 13. The plastic stiffness KAp at the point A as shown in Fig. 15 is calculated by using the shape factor hˆ and the plastic stiffness of the bound line Kbl p , as follows:
冋
ˆ dA KAp ⫽ Kbl p 1 ⫹ h din⫺dA
册
(27)
The corresponding tangent stiffness KAt is determined by using the elastic stiffness Ke and the plastic stiffness KAp as follows: KAt ⫽
KeKAp Ke ⫹ KAp
(28)
The bounding line is updated whenever load reversals occur. The procedure for shifting the bounding line is presented below. (i)
Whenever unloading occurs, the mean values and the amplitude for the last half cycle of the loading history, as shown in Fig. 16, are calculated. pa pa M pa m ⫽ 0.5(M A ⫹ M B )
⫽ 0.5(g
(29b)
pa pa M pa a ⫽ 0.5兩M A ⫺M B 兩
(30a)
g
pa a
⫽ 0.5兩g
pa A
⫹g
pa B
g
pa m
(29a)
⫺g
pa A
)
兩
pa B
(30b)
where the subscripts ‘m’ and ‘a’ stand for a mean value and an amplitude, respectively. and the (ii) Calculate the difference between the moment amplitude Mpa a moment Mpa s on the cyclic steady curve corresponding to the rotation amplitude, gpa a pa ⌬Mpa ⫽ Mpa s ⫺Ma
(ii)
(31)
If ⌬Mpa ⬎ 0, cyclic hardening is predicted to take place in the next excursion. Update the bound line by moving it outward by an amount equal to 2FH(⌬Mpa / Mpa n ), where FH is the hardening factor.
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Fig. 16.
冉 冊 冉 冊 Mpa bl Mpa n
(iv)
⫽
new
Mpa bl Mpa n
old
冉 冊 ⌬Mpa Mpa n
(32)
If ⌬Mpa ⬍ 0, cyclic softening is predicted to take place in the next excursion. Update the bound line by moving it inward by an amount equal to 2FS(⌬Mpa / Mpa n ), where Fs is the softening factor.
冉 冊 冉 冊 Mpa bl Mpa n
(v)
⫺2FH
Movement of bound line.
⫽
new
Mpa bl Mpa n
⫺2FS
old
冉 冊 ⌬Mpa Mpa n
(33)
Further move the bound by an amount equal to FRMpa m , where FR is the mean value relaxation factor.
冉 冊 冉 冊 Mpa bl Mpa n
⫽
new
Mpa bl Mpa n
⫺FRMpa m
(34)
old
The same values used in Cofie’s study, FH ⫽ 0.45, FS ⫽ 0.07, and FR ⫽ 0.05, are adopted in the proposed model.
6. Comparison of cyclic loading model with test data The panel zone model for cyclic loading described above is compared with test results for ten specimens tested by Krawinkler et al. [4], Slutter [5], Popov et al.
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[18], and Engelhardt et al. [7]. For some specimens a doubler plate is used to increase the capacity of the panel zone. Test results by Becker [3] showed that for every load level, except maximum load, the strain in the doubler plate was significantly less than that in the column web. Thus, the doubler plates were not fully effective. To account for the limited participation of a doubler plate in resisting panel shear, a reduction factor was considered in calculating the yield moments and stiffness values of the panel zone with a doubler plate. The effectiveness of doubler plates is affected by the method used to connect them to the column (one side attachment, both sides attachment, welding details, etc.). In this paper, the case with a doubler plate attached to only one side of the panel zone is studied. Figs. 17–22 show comparisons of the analytical response predicted by the proposed model and test results for panel zones with no doubler plate. These test data include specimens with a column flange thickness up to 5.3 cm. In Figs. 20–22, the test results for Popov et al.’s specimen 6, Krawinkler et al.’s specimen A1, and Engelhardt et al.’s specimen DBWP are plotted against the model predictions. The match is good for the cycles in which large deformations are imposed. For the first few cycles in which small deformations are imposed, the predictions are not as good, but still reasonable. For Krawinkler et al.’s specimens A2 and B2 and Slutter’s specimen 1 (Figs. 17–19), a large displacement amplitude was applied for the first half cycle of loading, causing large plastic deformations, far beyond the onset of strain hardening, in the panel zone. The model appears to work better for a large displacement amplitude for which strain hardening effects are fully developed than for a small displacement amplitude. Nonetheless, in spite of the simplicity of the model, reasonable agreement has been achieved between model predictions and test results for these six specimens without doubler plates. In Figs. 23–25, model predictions are compared with test results for specimens with doubler plates. In these specimens, the yield stress of the doubler plates was
Fig. 17.
Comparison of the hysteretic rules and test data for Krawinkler et al.’s [4] specimen A2.
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Fig. 18.
Comparison of the hysteretic rules and test data for Krawinkler et al.’s [4] specimen B2.
Fig. 19.
Comparison of the hysteretic rules and test data for Slutter’s [5] specimen 1.
approximately the same as that of the column web. In the analyses, a reduction factor of Rf ⫽ 0.4 was used to account for strain incompatibility between the column web and the doubler plate. As indicated by these figures, reasonable agreement was achieved between the model and the test data. The fact that a reduction factor of Rf ⫽ 0.4 resulted in a good match with the experimental data suggests that the effectiveness of doubler plates for increasing panel zone strength can be quite limited. Fig. 26 shows a comparison of the analytical and experimental results for Popov et al.’s specimen 8. For this specimen, the reported yield stress of the doubler plate was 338 MPa, and for the column web was 413 MPa. Since the yield stress of the
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Fig. 20.
Fig. 21.
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Comparison of the hysteretic rules and test data for Popov et al.’s [18] specimen 6.
Comparison of the hysteretic rules and test data for Krawinkler et al.’s [4] specimen A1.
doubler plate was significantly different from that of the column web, two panel elements were employed in parallel to obtain the analytical results. To obtain better correlation between the analytical results and the test data, a reduction factor of Rf ⫽ 0.10 was used. Once again, this suggests very limited effectiveness of the doubler plate. Figs. 17–26 above presented model predictions and test results for local panel zone response for a number of test specimens reported in the literature. Fig. 27 shows a comparison between model predictions and test results for the overall load– displacement response for Krawinkler et al.’s specimen A1. The panel zone model
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Fig. 22.
Comparison of the hysteretic rules and test data for Engelhardt et al.’s [7] specimen DBWP.
Fig. 23.
Comparison of the hysteretic rules and test data for Popov et al.’s [18] specimen 2.
described above was combined with a beam–column element by Kim and Engelhardt [19]. Results are also compared with the simpler bilinear panel zone model. The newly proposed panel zone model clearly provides significantly better correlation with the test data as compared with the bilinear model. 7. Modification of cyclic loading model for composite floor slabs The panel zone model discussed above was developed for the case where no composite floor slab is present. Likewise, the comparisons between the model and experi-
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Fig. 24.
Comparison of the hysteretic rules and test data for Popov et al.’s [18] specimen 3.
Fig. 25.
Comparison of the hysteretic rules and test data for Popov et al.’s [18] specimen 4.
627
mental data were all for bare steel specimens, i.e., specimens in which no composite floor slab was present. However, since most steel moment frames have composite floor slabs, the effects of the slab on panel zone response are of interest. To investigate the effect of composite slabs on the behavior of panel zones, a composite panel zone model was developed for the monotonic and cyclic behavior of beam-to-column joints in steel moment frames with composite floor slabs. The composite panel zone model is the same as the bare steel panel zone model described above, except that the effective depth of the panel zone, and therefore the area of the panel zone, is increased due to the presence of the concrete slab. This, in turn,
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Fig. 26.
Comparison of the hysteretic rules and test data for Popov et al.’s [18] specimen 8.
Fig. 27. Comparison of test and the analysis using the developed panel zone model for overall response for Krawinkler et al.’s [4] specimen A1.
affects the stiffness, yield moments, and cyclic steady curves of the panel zone model. For the bare steel case, a statically equivalent force couple replaces the moment in the beam, with the forces assumed to be acting at mid-depth of the beam flanges. When a slab is present, and positive moment is applied to the beam, the effective location of the force resultant at the beam’s top flange will move up towards the slab, thereby increasing the effective panel zone depth. The bending moment
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applied by the composite beam onto the panel zone will also increase for positive bending. The boundary forces on an interior joint panel, shown in Fig. 28, can be transformed into an approximate equivalent shear force from equilibrium as follows: Veq ⫽ ⫺ ⫽
冉
冊
冉
M+b M⫺ M+b ⫹ M⫺ 1 Vct ⫹ Vcb M+b M⫺ 1 b b b ⫹ ⫺ ⬇ ⫹ ⫺ ⫽ +⫹ ⫺ + ⫺ + ⫺ db db 2 db db Hc d b db
冊 冉
冊
+ ⫺ 1 M+b 1⫺(d+b⫺d⫺ M⫺ b b ) / (2d b )⫺db / Hc pa Mpa⫺ + ⫹ ⫺ ⬇ M ⫺ Hc db db db
(35)
+ 1⫺(d+b⫺d⫺ b ) / (2db )⫺r pa M db⫺tbf
where db⫹ ⫽ dcom⫺ts / 2⫺tbf / 2, dcom is the composite beam depth, ts is the solid slab ⫹ is equal to d⫺ thickness, and d⫺ b ⫽ db⫺tbf. If db b then Eq. (35) reduces to Eq. (5) for the bare steel beam-to-column joint. For a composite beam-to-column exterior joint, Eq. (35) can be reduced to Veq ⫽
1⫺d+b / Hc pa M for positive moment d+b
(36a)
Veq ⫽
1⫺d⫺ b / Hc pa M for negative moment d⫺ b
(36b)
The hysteretic rules for the composite panel zone element are based on those for the bare steel element and are shown in Fig. 29. For the interior composite joint, the cyclic behavior of panel zone follows path O–B–E–H, and for the exterior composite joint it follows the path O–B–E⬘–H because the cyclic behavior of the exterior joint depends on the direction of moment. To account for cracking of the concrete slab under negative moment the unloading behavior follows the inelastic curves of
Fig. 28.
Boundary forces on composite beam-to-column interior joint.
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Fig. 29.
Hysteretic rules for composite beam-to-column joint.
the bare steel beam-to-column joint. The cyclic behavior of the panel zone under positive moment after crack closing in the concrete slab is described by a linear crack closing segment followed by a nonlinear composite cyclic curve (paths C–E and F–H in Fig. 29). The factors f and b shown in Fig. 29, were determined empirically to be f ⫽ 0.5 and b ⫽ 0.1. In Fig. 30 the cyclic steady state curves for a
Fig. 30.
Cyclic steady state curve for composite beam-to-column joint.
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Fig. 31.
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Comparison of the hysteretic rules and test data for Lee’s [6] specimen EJ-FC.
composite beam-to-column joint tested by Lee [6] are shown. From this figure it can be seen that due to the presence of a composite slab, behavior of the panel zone differs significantly under positive and negative moments. The proposed composite panel zone model is compared with test results for four specimens tested by Lee [6] and Engelhardt et al. [7,8]. Also shown in these figures are the analytical predictions using the bare steel panel zone model. Figs. 31–33 show these comparisons for specimens with no doubler plate. Specimens tested by Lee [6] are shown in Figs. 31 and 32. The beams in these specimens were W18 × 35 sections. For these specimens, the composite panel zone model shows
Fig. 32. Comparison of the hysteretic rules and test data for Lee’s [6] specimen IJ-FC.
632
Fig. 33.
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Comparison of the hysteretic rules and test data for Engelhardt et al.’s [7] specimen DBWPC.
better correlation with the test data than the bare steel panel zone model for both small and large amplitudes of deformations. It is also clear from Figs. 31 and 32 that there is a significant difference in predicted response between the bare steel and composite panel zone elements. From the comparison of the model predictions and the experimental results, it can be seen that the effects of the concrete slab are negligible for Engelhardt et al.’s specimen DBWP-C (Fig. 33). This specimen was constructed using W36 × 150 beams. The similarity between the bare steel and composite model predictions may be attributed to the fact that the increase of panel zone area due to the composite slab was not significant because the steel beam depth was much larger than the slab depth. The increase in panel zone area obtained by using Eq. (35) instead of Eq. (5) is about 36, 16, and 8% for specimens EJ-FC (Fig. 31), IJ-FC (Fig. 32), and DBWPC (Fig. 33), respectively. In Fig. 34, model predictions are compared with test results for Engelhardt and Venti’s [8] specimen UTA-FF. This specimen was constructed with a doubler plate. In the model, a reduction factor of Rf ⫽ 0.60 was used to account for strain incompatibility between the column web and the doubler plate. The performance of the model provides reasonable correlation with experimental data. Further, there is once again little effect of the composite slab due to the rather deep W36 × 150 beams used for this specimen. Only very limited experimental data are available on composite specimens with weak panel zones. To refine mathematical models for composite panel zone response, further experimental data are needed for specimens with composite slabs under large inelastic excursions of monotonic loading as well as cyclic loading.
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Fig. 34.
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Comparison of the hysteretic rules and test data for Engelhardt and Venti’s [8] specimen UTAFF.
8. Conclusions The objective of the study presented in this paper was to develop models to describe the monotonic and cyclic loading behavior of panel zones in steel moment resisting frames. These new models build on previously developed models, and introduce a number of features and refinements that show better correlation with available experimental data. The model for monotonic loading is based on quadri-linear panel zone moment– deformation relations. In this model, both bending and shear deformation modes are considered. The model proposed for cyclic loading is based on Dafalias’ bounding surface theory combined with Cofie’s rules for movement of the bound line. Finally, additional modifications are suggested for the cyclic loading model to account for the influence of a composite floor slab on panel zone response. For all of the proposed models, extensive comparisons with experimental data were presented to demonstrate the capabilities and limitations of the models. In spite of the simplicity of the proposed models, reasonable agreement between model predictions and test data was achieved over a broad range of experiments. In the process of developing these panel zone models, several issues were identified which appear to merit further research. One of these issues is the effect of very thick column flanges on panel zone strength. Much of the available experimental data are for panel zones in columns with a flange thickness less than about 3–4 cm. In actual practice, columns with flange thickness values in excess of 10 cm are not uncommon. Further experimental data is needed to better quantify the contribution of very thick column flanges to overall panel zone strength. An additional issue of concern is the effectiveness of doubler plates. Based on comparisons between model predictions and experimental data, it appears that
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doubler plates are, in many cases, not fully effective in contributing to panel zone strength and stiffness. In many past experiments, it appears that doubler plates were less than 50% effective. Additional studies are needed to better quantify the contribution of doubler plates to panel zone strength and stiffness for various attachment details. This is an issue of considerable practical importance, since doubler plates are commonly used in practice to augment panel zone strength. Current building code regulations in the US appear to assume that doubler plates will be fully effective. Further studies are also needed to better quantify the effects of a composite floor slab on panel zone behavior. The majority of past tests and past panel zone models have considered the case of a bare steel frame. However, limited data suggests that the presence of a composite concrete floor slab can significantly affect panel zone behavior, particularly for relatively shallow beams. Some simple modifications to the panel zone model were suggested in this paper to account for composite slab effects. However, further experimental data are needed to better understand slab effects and to further refine panel zone response models. Finally, the models presented in this paper are intended to represent the effects of inelasticity on panel zone response, i.e. the models account for the effects of material yielding and strain hardening. However, after yielding and strain hardening, the panel zone will ultimately degrade in strength either due to shear buckling of the panel zone or due to fracture of the column or beam flanges at the corners of the panel zone. The current models do not predict strength degradation due to instability or fracture. Further studies are needed to better understand the ultimate failure modes of the panel zone and to quantify the available deformation capacity.
Acknowledgements The writers gratefully acknowledge support for this work from the National Science Foundation (grant no. CMS-9358186) and from the American Institute of Steel Construction, Inc.
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