Monotonic and cyclic loading models for panel zones in...

Journal of Constructional Steel Research 58 (2002) 605–635www.elsevier.com/locate/jcsr

Monotonic and cyclic loading models for panelzones in steel moment frames

Kee Dong Kima, Michael D. Engelhardtb,*

a Department of Civil Engineering, Kongju National University, Kongju, Chungnam, South Koreab Department of Civil Engineering, University of Texas at Austin, Austin, TX 78712-1076, USA

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 inelasticresponse of the panel zone portion of columns in steel moment resisting frames. In manypractical cases, the panel zone can dominate the inelastic response of a moment frame, andaccurate panel zone models are needed to realistically predict overall frame performance. Simi-lar to previous models, the newly proposed models are based on the concept of representingthe panel zone as a nonlinear rotational spring. These new models build on previouslydeveloped models, and introduce a number of features and refinements that show better corre-lation 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 defor-mation 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 influenceof a composite floor slab on panel zone response. Extensive comparisons with experimentaldata 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: S0143 -974X(01)00079-7

606 K.D. Kim, M.D. Engelhardt / Journal of Constructional Steel Research 58 (2002) 605–635

1. Introduction

Mathematical models are developed in this paper for describing the monotonicand cyclic load–deformation response of the panel zone region of beam–columnjoints of a steel moment resisting frame. The panel zone is the portion of the columncontained within the beam–column joint. When a moment frame is subject to lateralloads, high shear forces develop within the panel zone. The resulting deformationsof the panel zone can have an important effect on the response of the frame in boththe elastic and inelastic ranges of frame behavior [1,2].

Numerous tests have been performed in the past three decades to investigate theload–deformation behavior of the joint panel using connection subassemblies [3–8].Some significant observations from these tests are:

� Joint panel zones often develop a maximum strength that is significantly greaterthan the strength at first yield. This additional strength has been attributed to strainhardening and to contributions of the column flanges in resisting panel zone shearforces. Large inelastic panel zone deformations are typically required in order todevelop the maximum panel zone strength.

� Panel zone deformations can add significantly to the overall deformation of a steelmoment frame, for both elastic and inelastic ranges of behavior.

� Panel zone stiffness and strength can be increased by the attachment of webdoubler plates to the column within the joint region. The effectiveness of doublerplates is affected by the method used to connect them to the column.

� In the inelastic range, panel zones can exhibit very ductile behavior, both formonotonic and cyclic loading. Experimentally observed hysteresis loops are typi-cally very stable, even at large inelastic deformations.

� Large inelastic panel zone deformations can increase the likelihood of fractureoccurring in the region of the beam flange to column flange groove welds. Thiseffect 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 hingesin the panel zones of steel moment frames under earthquake loading. Thus, ratherthan forming plastic flexural hinges only in the beams or columns, a primary sourceof energy dissipation in a steel moment frame can be the formation of plastic shearhinges in the panel zones. Consequently, to accurately predict the response of a steelmoment frame under earthquake loading, an accurate analytical model is needed topredict the response of the panel zone.

To include panel zone deformation in frame analysis, the traditional center-to-center line representation of the frame must be modified. Fig. 1 shows a comparisonbetween an experiment on a beam–column subassemblage and an analytical predic-tion of the subassemblage response. The experiment was specimen A1 reported byKrawinkler et al. [4]. Inelastic response of this specimen was dominated by yieldingof the panel zone. The analytical results are obtained from a model of the specimenusing center-to-center line dimensions. A beam–column element with plastic hinges

607K.D. Kim, M.D. Engelhardt / Journal of Constructional Steel Research 58 (2002) 605–635

Fig. 1. Comparison of test results [4] and analytical results obtained by using center-to-center line dimen-sion modeling.

was employed in this analysis. From the figure it can be seen that the analysis usingthe center-to-center line dimension, without explicit modeling of the panel zone, mayproduce misleading results. Clearly, the panel zone response must be explicitly mod-eled to obtain realistic predictions of the overall frame behavior.

To model the behavior of panel zones in frame analysis, Lui [12] developed ajoint model based on the finite element method. The model consists of seven elementsfor interior beam-to-column joints: one web element, two flange elements (beamelements), and four connection elements. Although capable of representing a varietyof deformation modes of panel zones, this model employed a simple hardening rulesuitable for monotonic loading and does not realistically model cyclic behavior.Another disadvantage of this model is its high computational cost. Other finiteelement models using more sophisticated hardening rules could be developed for theanalysis of column panel zones. However, in this study, nonlinear rotational springsare used as the basis for modeling the panel zone for nonlinear dynamic analysis ofmoment resisting frames because of simplicity and computational efficiency.

Several researchers, including Fielding and Huang [13], Krawinkler et al. [4] andWang [14] proposed relationships between panel zone shear force V and panel zonedeformation g for monotonic loading. These relationships have been used as the basisof 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 abasis for computing the shear strength of panel zones. However, it was pointed outby Krawinkler that a new model might be needed for joints with thick column flangessince his V–g relations were derived from experimental and analytical results forpanel zones with relatively thin column flanges. Wang also showed that Krawinkler’sV–g relations may overestimate panel zone shear strength for panel zones with thickcolumn flanges.


In the remainder of this paper, existing analytical models for panel zones will beexamined by making comparisons with experimental data or with other analyticalresults. A number of refinements and improvements to existing models are thensuggested to overcome some of the shortcomings of existing models and to providebetter correlation with available experimental data.

2. General characteristics of panel zone element

The panel zone element is essentially a rotational spring element, which transfersmoment between the columns and beams framing into a joint [15]. The panel elementhas no dimensions and connects two nodes with the same coordinates. One of thesenodes is attached to the elements modeling the columns framing into the joint, asshown 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 relativerotation between the columns and beams framing into a joint. The vertical and hori-zontal translations of the two nodes are constrained to be identical. Therefore, onevertical, one horizontal, and two rotational degrees of freedom exist at each joint.

The relative rotation between the connected nodes is related to the node rotationsas follows:

dg � {1 �1}�dqI

dqJ� (1)

or

dg � a·dr (2)

where dg is the increment of relative rotation, which is the panel element defor-mation, 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.


dMpa � Ktdg (3)

where dMpa is the increment of moment applied on a joint and Kt is the rotationaltangent 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 stiffnessesand yield moments for monotonic loading, and hysteretic rules for cyclic loading.In the following sections, existing models for monotonic loading are examined anda new model is presented. This is followed by the development of hysteretic rulesfor a cyclic loading model.

3. Review of existing models for monotonic loading

Existing mathematical models for panel zone response under monotonic loadingare typically based on a computation of an approximate equivalent shear force actingon the panel zone. The boundary forces on a joint panel, shown in Fig. 3, can betransformed into an approximate equivalent shear force from equilibrium as follows:

Veq �Mbl � Mbr

db�tbf

�(Vct � Vcb)

2�

Mbl � Mbr

db�tbr

(1�r) �(1�r)db�tbf

Mpa (5)

where r � (db�tbf) /Hc, Mpa � Mbl � Mbr is the panel zone moment, tbf is the thick-ness of the beam flange, db is the beam depth, and Hc is the column height. A keysimplification in this analysis is that the beam moments are replaced by an equivalentcouple, with the forces acting at mid-depth of the beam flanges. These forces producea large shear in the panel zone. The shear in the column segments outside of thepanel zone are then subtracted to obtain the net shear force, Veq, acting on the panelzone. 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/2above and below the panel zone.

Fig. 3. Boundary forces and equivalent shear forces on panel zone.


Panel zone shear force V versus panel zone deformation g relations for monotonicloading, which are based on the equivalent shear force, Veq, can be transformed intopanel zone moment Mpa versus panel zone deformation g relations by Eq. (5). Field-ing and Huang [13] proposed a bilinear relationship consisting of an elastic stiffnessKe 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 tothe panel stiffness. For the tri-linear models, the second post elastic stiffness K2 isassociated with strain hardening.

Past researchers computed the elastic stiffness of the panel element by consideringpure elastic shear deformation of an effective shear area of the panel zone. Fieldingand Krawinkler considered the effective shear area Aeff equal to (dc�tcf)tcw, and Wangconsidered 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 yieldmoment and elastic stiffness of the panel zone be taken as follows:

Mpay �

Vydb

(1�r)�tyAeffdb

(1�r)(6a)

Ke �Mpa

y

gy�

GAeffdb

(1�r)(6b)

where Vy is the yield shear force of the panel zone, gy � ty /G, G is the elastic shearmodulus, and ty is the Von Mises yield shear stress of the column web, based onshear and axial force interaction. The Von Mises yield shear stress, ty, is taken as:

ty �sy

�3�1�(P /Py)2 (7)

where P and Py are the axial force and the axial yield force on the column, respect-ively, and sy is the yield stress of the column web.

For the inelastic range, Fielding considered a bilinear model with the followingpost-elastic stiffness K1:

K1 �5.2Gbcft3

cf

db(1�r)(8)

where bcf and tcf are width and thickness of column flange, respectively. Krawinklerproposed empirical formulas for the post-elastic stiffness K1 and the second yieldmoment Mpa

sh as follows:

K1 �1.04Gbcft2cf

(1�r)(9)

Mpash � Mpa

y �3.12tybcft2

cf

(1�r)(10)

Wang suggested the post-elastic stiffness K1, as follows:


K1 � 0.7Gbcft2cf (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. Eachof 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 of1 cm. Slutter’s specimen 1, which had a column flange thickness of 1.8 cm, is plottedin Fig. 5. Krawinkler et al.’s specimen B2, with a column flange thickness of 2.37cm, 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 analysispredictions are provided for Slutter’s specimen 1 and compared to the simplifiednonlinear spring models. This specimen was analyzed a number of times, varying thecolumn flange thickness. These FEM analyses provide an indication of the expectedresponse of panel zones as the column flange thickness varies, but where all othervariables 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 stresselements, respectively. The finite element results are compared with the correspond-ing 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 withtcf � 1 cm.


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 withtcf � 2.37 cm.

This comparison shows that Wang’s finite element analysis reasonably predicted theobserved panel zone behavior. Figs. 8–12 show finite element predictions for a modelbased on Slutter’s specimen, but considering column flange thickness values equalto 0.9, 1.35, 2.7, 3.6 and 4.51 cm. In each figure, the corresponding predictions ofthe 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


Fig. 7. Comparison of the monotonic model and test data for Fielding and Huang’s [13] specimen withtcf � 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 flangethickness because the model neglects strain-hardening effects. The performance ofKrawinkler et al.’s model appears reasonable for panel zone joints with columnflange thickness less than about 2.5 cm. However, for thicker column flanges, thismodel somewhat overestimates panel zone strength. Wang’s model generally under-estimates panel zone strength regardless of column flange thickness, apparentlybecause 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.




4. Proposed model for monotonic loading and comparison with test data

As the ratio of column flange thickness to column depth increases, the influenceof column flange thickness on panel zone yield moment and elastic stiffness increases[14]. Panel zone models that include shear deformations only cannot account for thisincrease in yield moment and elastic stiffness according to the increase in the ratioof column flange thickness to column depth. Thus, in this study, both bending and




shear deformation modes are included in the panel zone model. The resulting mono-tonic 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 atthe center. The boundary condition at the other end of these beams is considered tobe somewhere between free and fixed. Thus, the displacement of the equivalent beamdue to the shear force Veq can be described as follows:


� � � 1S1

�1S2�Veq (13)

where 1/S1 is the bending flexibility of the equivalent beam and 1/S2 is the shearflexibility. The bending and shear flexibilities can be described as follows:

1S1

�[(db�tbf) / 2]3

CrEIand

1S2

�[(db�tbf) /2]

G(dctcw � RfAdp)(14)

where I is the moment of inertia of the column section, Cr is a constant to be determ-ined according to the degree of end restraint, Adp is the area of a doubler plate, andRf is the reduction factor to account for the strain incompatibility between a doublerplate and column web. Eq. (13) can be rewritten in terms of the panel zone momentMpa and the rotation g as follows:

Veq �S1S2

S1 � S2

� �S1S2

S1 � S2

(db�tbf)2g (15a)

Mpa � Veq

(db�tbf)(1�r)

�S1S2

S1 � S2

(db�tbf)2

(db�tbf)(1�r)

g (15b)

Thus, the elastic stiffness is

Ke �S1S2

S1 � S2

(db�tbf)2

(db�tbf)(1�r)

(16)

The yield moment of panel zone is defined as follows:

Mpay � KeCygy (17)

where Cygy is the average shear deformation of panel zone at which shear yieldingoccurs 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 yieldingup to the entire shear yielding of the panel zone, it is assumed that for this rangethe panel zone can be modeled as two separate beams with T-shaped sections, similarto Fielding and Huang’s approach [13]. The web depth of the T-shaped section is aquarter of the column web depth. Then, the post-elastic stiffness K1 can be defined as

K1 � 2PS1PS2

PS1 � PS2

(db�tbf)2

(db�tbf)(1�r)

(18a)

PS1 �CrEIT

[(db�tbf) /2]3 and PS2 �G[(dc /2�dyw)tcw � RfAdp / 4]

[(db�tbf) / 2](18b)

where IT is the moment of inertia of the T section and dyw is a quarter of the webdepth. The second yield moment Mpa

y1 is

Mpay1 � ty(dctcw � RfAdp)

(db�tbf)(1�r)

(19)

The second post-elastic stiffness after shear yielding of the entire panel zone is


defined by using an approach similar to Krawinkler et al. [4]. The panel zone, afterthe shear yielding of the entire web, consists of an elastic–perfectly plastic shearpanel surrounded by rigid boundaries with springs at the four corners. It is assumedthat these springs simulate the resistance of elements surrounding the panel, in parti-cular the bending resistance of column flanges, and that the spring stiffness can beapproximated by

Ks �Ebcft2cf

Cs(20a)

where Cs is a constant to be determined from test results. From the work equationand 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 columnflanges at the four corners of the panel [14]. Then, the third yield moment of thepanel zone at which strain hardening initiates can be defined as

Mpay2 � Mpa

y1 � sybcft2cf (21a)

sy � sfly(1�(P /Py)2) (21b)

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 specimensas in the previous section. Figs. 4–12 show comparisons of this new model with testresults and FEM results for various values of column flange thickness. Based oncalibration 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 transitionfrom elastic to inelastic behavior than the other models because it has quadri-linearrelations. From the comparison with FEM results, it can be seen that the new modelcan reasonably describe the increase of yield strength and elastic stiffness accordingto an increase in the ratio of column flange thickness to column depth due to theinclusion of the bending deformation mode. The suggested model significantly under-estimates the panel zone strength of Krawinkler et al.’s specimen B2 (Fig. 6). Thisspecimen 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, allof 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 thisnew model and the response obtained by test or FEM analysis is quite good regard-less of column flange thickness.


In this study, the thickest column flange used to evaluate panel zone models wastcf � 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 sectionsare needed to further verify this monotonic model and to investigate maximum shearstrength. 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 onthe behavior of steel moment frames under earthquake loading. Consequently, accur-ate hysteretic rules are needed to predict panel zone response under cyclic and ran-dom loading. A common model used in the past is based on bilinear kinematichardening, and this model has been widely used for inelastic dynamic analysis ofmoment resisting frames. Fig. 21 shows a comparison of the bilinear kinematic hard-ening 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 responsefor this test specimen. From both figures, it is clear that the response predictions ofthe bilinear model do not correlate well with the experimentally observed response.The bilinear model substantially underestimated panel zone strength in the lattercycles of loading. The correlation in the later loading cycles could be improved byincreasing the yield strength of the bilinear model, but then the correlation wouldbe particularly poor for the early loading cycles. In general, the ability of the bilinearmodel to replicate the full loading history of the panel zone is limited. Consequently,a new model was developed to provide improved panel zone response predictionsover 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 forthe movement of the bound line [17]. Based on observations from experiments andFEM analyses for panel zones, it has been found that for large plastic rotations, theshear strains in the panel zone are distributed nearly uniformly within the panel, andthe value of joint rotation is close to the value of the average shear strain in thepanel [4,14]. Therefore, it is assumed that the panel zone moment–rotation relation-ships can be determined from the material properties of the panel zone using Cofie’srules. These rules for the movement of the bound line, which were developed forstress–strain relationships, will be adopted for the panel zone moment–rotationrelationships.

The main feature of Cofie’s model is that the cyclic steady state curve is used todescribe the movement of the bounding line. In this study, the same kind of cyclicsteady state curve is developed to describe the movement of the bounding line forthe cyclic behavior of the panel zone, as follows:

ggn

�Mpa

Mpan

� � Mpa

xMpan�c

(23)

where Mpan and gn are the normalizing panel moment and corresponding elastic


rotation, respectively. By comparison with available cyclic test data, it has beenempirically found that the constant C of the cyclic steady state curve is 7 and x is1.1–1.2 [2]. The cyclic steady state curve and bound lines are shown together withexperimental data [7] in Fig. 13.

Experimental and FEM results suggest that column flanges do not significantlyinfluence panel zone stiffness during cyclic loading, but do have a significant effecton panel zone strength. From FEM results for joints with the same dimensions exceptfor the column flange thickness, it has been found that the effect of column flangethickness on the strength of the joint during cyclic loading can be normalized byMpa

n [14] as follows

Mpan � Mpa

y � 2Mpcf (24)

where Mpcf is the plastic moment of the column flange. The elastic rotation corre-sponding to the normalizing moment Mpa

n is

gn �Mpa

n

Ke(25)

Panel zone response for the initial half-cycle of loading follows the monotonicloading rules described earlier in this paper. Thereafter, the cyclic behavior of thepanel zone is defined by elastic and inelastic curves as shown in Fig. 14. To describethe inelastic curves, the shape factor is employed, which was first used for cyclicstress–strain relationships by Dafalias [16]. The procedure for obtaining the shapefactor h is as follows:

(i) Choose the point A such that 0.1�dA /din�0.5, as shown in Fig. 15.(ii) calculate the shape factor from

Fig. 13. Cyclic steady curve and bound line.


Fig. 14. Hysteretic rules for panel zones.

Fig. 15. Shape factor for inelastic behavior.

h � dA / gAp � (din / gAp )[ln(din /dA)�1].

(iii) Normalize theshapefactorby theplasticstiffnessof theboundline h �h /Kbl

p .

It has been found that a shape factor of h � 20 for small rotation amplitude cyclesand h � 40 for large rotation amplitude cycles provide a good correlation withexperimental data [2]. Thus, in this study a varying shape factor h is used to describethe inelastic curves of the panel zone response. The varying shape factor h is updatedaccording to the accumulated plastic rotation qp only when unloading occurs and is


otherwise kept constant along each loading path. The varying shape factor h consistsof a Boltzman function with the initial shape factor 20 (at qp � 0) and the finalshape factor 40 (at qp � �) as follows:

h � 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 stiffnessof 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 withthe slope of the bound line at the point with the corresponding slope on the cyclicsteady state curve and by making the resulting line intersect the moment axis, asshown in Fig. 13. The plastic stiffness KA

p at the point A as shown in Fig. 15 iscalculated by using the shape factor h and the plastic stiffness of the bound lineKbl

p , as follows:

KAp � Kbl

p �1 � hdA

din�dA� (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 forshifting the bounding line is presented below.

(i) Whenever unloading occurs, the mean values and the amplitude for the lasthalf cycle of the loading history, as shown in Fig. 16, are calculated.

M pam � 0.5(M pa

A � M paB ) (29a)

g pam � 0.5(g pa

A � g paB ) (29b)

M paa � 0.5M pa

A �M paB (30a)

g paa � 0.5g pa

A �g paB (30b)

where the subscripts ‘m’ and ‘a’ stand for a mean value and an amplitude, respect-ively.(ii) Calculate the difference between the moment amplitude Mpa

a and themoment Mpa

s on the cyclic steady curve corresponding to the rotation ampli-tude, gpa

a

�Mpa � Mpas �Mpa

a (31)

(ii) If �Mpa � 0, cyclic hardening is predicted to take place in the next excur-sion. Update the bound line by moving it outward by an amount equal to2FH(�Mpa /Mpa

n ), where FH is the hardening factor.


Fig. 16. Movement of bound line.

�Mpabl

Mpan�

new

� �Mpabl

Mpan�

old

�2FH��Mpa

Mpan� (32)

(iv) 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 to2FS(�Mpa /Mpa

n ), where Fs is the softening factor.

�Mpabl

Mpan�

new

� �Mpabl

Mpan�

old

�2FS��Mpa

Mpan� (33)

(v) Further move the bound by an amount equal to FRMpam , where FR is the mean

value relaxation factor.

�Mpabl

Mpan�

new

� �Mpabl

Mpan�

old

�FRMpam (34)

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 testresults for ten specimens tested by Krawinkler et al. [4], Slutter [5], Popov et al.


[18], and Engelhardt et al. [7]. For some specimens a doubler plate is used to increasethe capacity of the panel zone. Test results by Becker [3] showed that for every loadlevel, except maximum load, the strain in the doubler plate was significantly lessthan that in the column web. Thus, the doubler plates were not fully effective. Toaccount for the limited participation of a doubler plate in resisting panel shear, areduction factor was considered in calculating the yield moments and stiffness valuesof the panel zone with a doubler plate. The effectiveness of doubler plates is affectedby the method used to connect them to the column (one side attachment, both sidesattachment, welding details, etc.). In this paper, the case with a doubler plate attachedto only one side of the panel zone is studied.

Figs. 17–22 show comparisons of the analytical response predicted by the pro-posed model and test results for panel zones with no doubler plate. These test datainclude specimens with a column flange thickness up to 5.3 cm. In Figs. 20–22, thetest results for Popov et al.’s specimen 6, Krawinkler et al.’s specimen A1, andEngelhardt et al.’s specimen DBWP are plotted against the model predictions. Thematch is good for the cycles in which large deformations are imposed. For the firstfew 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 speci-men 1 (Figs. 17–19), a large displacement amplitude was applied for the first halfcycle of loading, causing large plastic deformations, far beyond the onset of strainhardening, in the panel zone. The model appears to work better for a large displace-ment amplitude for which strain hardening effects are fully developed than for asmall displacement amplitude. Nonetheless, in spite of the simplicity of the model,reasonable agreement has been achieved between model predictions and test resultsfor these six specimens without doubler plates.

In Figs. 23–25, model predictions are compared with test results for specimenswith 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.


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 factorof Rf � 0.4 was used to account for strain incompatibility between the column weband the doubler plate. As indicated by these figures, reasonable agreement was achi-eved 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 effective-

ness 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 platewas 338 MPa, and for the column web was 413 MPa. Since the yield stress of the


Fig. 20. Comparison of the hysteretic rules and test data for Popov et al.’s [18] specimen 6.

Fig. 21. 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 panelelements were employed in parallel to obtain the analytical results. To obtain bettercorrelation between the analytical results and the test data, a reduction factor ofRf � 0.10 was used. Once again, this suggests very limited effectiveness of thedoubler plate.

Figs. 17–26 above presented model predictions and test results for local panelzone response for a number of test specimens reported in the literature. Fig. 27shows 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


Fig. 22. Comparison of the hysteretic rules and test data for Engelhardt et al.’s [7] specimen DBWP.


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. Thenewly proposed panel zone model clearly provides significantly better correlationwith 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 com-posite floor slab is present. Likewise, the comparisons between the model and experi-




mental data were all for bare steel specimens, i.e., specimens in which no compositefloor slab was present. However, since most steel moment frames have compositefloor 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, acomposite panel zone model was developed for the monotonic and cyclic behaviorof beam-to-column joints in steel moment frames with composite floor slabs. Thecomposite panel zone model is the same as the bare steel panel zone model describedabove, except that the effective depth of the panel zone, and therefore the area ofthe panel zone, is increased due to the presence of the concrete slab. This, in turn,



Fig. 27. Comparison of test and the analysis using the developed panel zone model for overall responsefor Krawinkler et al.’s [4] specimen A1.

affects the stiffness, yield moments, and cyclic steady curves of the panel zonemodel. For the bare steel case, a statically equivalent force couple replaces themoment in the beam, with the forces assumed to be acting at mid-depth of the beamflanges. When a slab is present, and positive moment is applied to the beam, theeffective location of the force resultant at the beam’s top flange will move up towardsthe slab, thereby increasing the effective panel zone depth. The bending moment


applied by the composite beam onto the panel zone will also increase for positivebending.

The boundary forces on an interior joint panel, shown in Fig. 28, can be transfor-med into an approximate equivalent shear force from equilibrium as follows:

Veq �M+

b

d+b

�M�

b

d�b

��Vct � Vcb

2 ��M+

b

d+b

�M�

b

d�b

�M+

b � M�b

Hc� � 1

d+b

�1

d�b

�1

Hc�Mpa��M�

b

d+b

�M+

b

d�b��

1�(d+b�d�

b ) / (2d+b)�d�

b /Hc

d�b

Mpa (35)

�1�(d+

b�d�b ) / (2d+

b)�rdb�tbf

Mpa

where d �b � dcom�ts /2�tbf /2, dcom is the composite beam depth, ts is the solid slab

thickness, and d�b � db�tbf. If d �

b is equal to d�b then Eq. (35) reduces to Eq. (5)

for the bare steel beam-to-column joint. For a composite beam-to-column exteriorjoint, Eq. (35) can be reduced to

Veq �1�d+

b /Hc

d+b

Mpa for positive moment (36a)

Veq �1�d�

b /Hc

d�b

Mpa for negative moment (36b)

The hysteretic rules for the composite panel zone element are based on those forthe 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 com-posite joint it follows the path O–B–E–H because the cyclic behavior of the exteriorjoint depends on the direction of moment. To account for cracking of the concreteslab under negative moment the unloading behavior follows the inelastic curves of

Fig. 28. Boundary forces on composite beam-to-column interior joint.


Fig. 29. Hysteretic rules for composite beam-to-column joint.

the bare steel beam-to-column joint. The cyclic behavior of the panel zone underpositive moment after crack closing in the concrete slab is described by a linearcrack closing segment followed by a nonlinear composite cyclic curve (paths C–Eand F–H in Fig. 29). The factors f and b shown in Fig. 29, were determined empiri-cally 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.


Fig. 31. 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 canbe seen that due to the presence of a composite slab, behavior of the panel zonediffers significantly under positive and negative moments.

The proposed composite panel zone model is compared with test results for fourspecimens tested by Lee [6] and Engelhardt et al. [7,8]. Also shown in these figuresare the analytical predictions using the bare steel panel zone model. Figs. 31–33show these comparisons for specimens with no doubler plate. Specimens tested byLee [6] are shown in Figs. 31 and 32. The beams in these specimens wereW18 × 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.


Fig. 33. Comparison of the hysteretic rules and test data for Engelhardt et al.’s [7] specimen DBWP-C.

better correlation with the test data than the bare steel panel zone model for bothsmall and large amplitudes of deformations. It is also clear from Figs. 31 and 32that there is a significant difference in predicted response between the bare steel andcomposite panel zone elements.

From the comparison of the model predictions and the experimental results, it canbe seen that the effects of the concrete slab are negligible for Engelhardt et al.’sspecimen DBWP-C (Fig. 33). This specimen was constructed using W36 × 150beams. The similarity between the bare steel and composite model predictions maybe attributed to the fact that the increase of panel zone area due to the compositeslab was not significant because the steel beam depth was much larger than the slabdepth. 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 DBWP-C (Fig. 33), respectively.

In Fig. 34, model predictions are compared with test results for Engelhardt andVenti’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 incom-patibility between the column web and the doubler plate. The performance of themodel provides reasonable correlation with experimental data. Further, there is onceagain little effect of the composite slab due to the rather deep W36 × 150 beamsused for this specimen.

Only very limited experimental data are available on composite specimens withweak panel zones. To refine mathematical models for composite panel zone response,further experimental data are needed for specimens with composite slabs under largeinelastic excursions of monotonic loading as well as cyclic loading.


Fig. 34. Comparison of the hysteretic rules and test data for Engelhardt and Venti’s [8] specimen UTA-FF.

8. Conclusions

The objective of the study presented in this paper was to develop models todescribe the monotonic and cyclic loading behavior of panel zones in steel momentresisting frames. These new models build on previously developed models, and intro-duce a number of features and refinements that show better correlation with availableexperimental 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 areconsidered. The model proposed for cyclic loading is based on Dafalias’ boundingsurface theory combined with Cofie’s rules for movement of the bound line. Finally,additional modifications are suggested for the cyclic loading model to account forthe influence of a composite floor slab on panel zone response. For all of the proposedmodels, extensive comparisons with experimental data were presented to demonstratethe capabilities and limitations of the models. In spite of the simplicity of the pro-posed models, reasonable agreement between model predictions and test data wasachieved over a broad range of experiments.

In the process of developing these panel zone models, several issues were ident-ified which appear to merit further research. One of these issues is the effect of verythick column flanges on panel zone strength. Much of the available experimentaldata 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 notuncommon. Further experimental data is needed to better quantify the contributionof very thick column flanges to overall panel zone strength.

An additional issue of concern is the effectiveness of doubler plates. Based oncomparisons between model predictions and experimental data, it appears that


doubler plates are, in many cases, not fully effective in contributing to panel zonestrength and stiffness. In many past experiments, it appears that doubler plates wereless than 50% effective. Additional studies are needed to better quantify the contri-bution of doubler plates to panel zone strength and stiffness for various attachmentdetails. This is an issue of considerable practical importance, since doubler platesare commonly used in practice to augment panel zone strength. Current building coderegulations 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 floorslab on panel zone behavior. The majority of past tests and past panel zone modelshave considered the case of a bare steel frame. However, limited data suggests thatthe presence of a composite concrete floor slab can significantly affect panel zonebehavior, particularly for relatively shallow beams. Some simple modifications tothe panel zone model were suggested in this paper to account for composite slabeffects. However, further experimental data are needed to better understand slabeffects and to further refine panel zone response models.

Finally, the models presented in this paper are intended to represent the effectsof inelasticity on panel zone response, i.e. the models account for the effects ofmaterial yielding and strain hardening. However, after yielding and strain hardening,the panel zone will ultimately degrade in strength either due to shear buckling ofthe panel zone or due to fracture of the column or beam flanges at the corners ofthe panel zone. The current models do not predict strength degradation due to insta-bility or fracture. Further studies are needed to better understand the ultimate failuremodes of the panel zone and to quantify the available deformation capacity.

Acknowledgements

The writers gratefully acknowledge support for this work from the NationalScience Foundation (grant no. CMS-9358186) and from the American Institute ofSteel Construction, Inc.

References

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[2] Kim K, Engelhardt, MD. Development of analytical models for earthquake analysis of steel momentframes. Report no. PMFSEL 95-2, Phil M. Ferguson Structural Engineering Laboratory, The Univer-sity 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, FritzEngineering Lab., Lehigh University, Bethlehem, PA; 1981.

[6] Lee SJ. Seismic behavior of steel building structures with composite Slabs. PhD thesis, Departmentof Civil Engineering, Lehigh University, Bethlehem, PA; 1987.


[7] Engelhardt MD, Fry GT, Jones S, Venti M, Holliday S. Behavior and design of radius-cut reducedbeam 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 ofCalifornia, Berkeley, CA; 1973.

[16] Dafalias YF. On cyclic and anisotropic plasticity: (I) A general model including material behaviorunder 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.

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