Panel Zones

8/11/2019 Panel Zones

1/10

MODELING PROCEDURES FOR PANEL ZONE DEFORMATIONS INMOMENT RESISTING FRAMES

Finley A. Charney, Virginia Tech, U.S.A.William M. Downs, Simpson Strong Tie, Inc., U.S.A.

ABSTRACT

Elastic and inelastic deformations in the panel zone of the beam-column jointregion of moment resisting frames are responsible for a very significantportion of the lateral flexibility of such systems. This paper provides a brieftheoretical basis for computing panel zone deformations, and comparesresults obtained from two simple mechanical models to each-other and to

those obtained using detailed finite element analysis. It is shown that thesimplest mechanical model, referred to as the Scissors model, producesresults that are comparable to those obtained from the more complexmechanical model, and also correlates well with the results computed fromthe detailed finite element model.

INTRODUCTION

The influence of panel zone deformations on the flexibility of steel moment resisting framesis very significant. This is true for elastic response, and particularly for inelastic responsewhen yielding occurs in the panel zone. Structural analysis should always include such

deformations.

While the state of stress in the panel zone is extremely complex, the sources of deformationcan be divided into three parts: axial, flexural, and shear. For low to medium rise frames,axial deformations are negligible, flexural deformations are minor but significant, and sheardeformations are dominant. This paper concentrates on the shear component of panelzone deformation. See Downs (1) for a detailed discussion on modeling approaches foraxial and flexural deformations within the panel zone.

Mathematical modeling procedures for panel zone deformation are typically based onsimple mechanical analogs which consist of an assemblage of rigid links and rotationalsprings. The principal challenge in the derivation of such models is the development of thetransformations from shear in the panel zone to rotation in the springs of the analog. Twomechanical models were studied in the research reported herein. These are theKrawinkler Model (2) and the Scissors Model. When properly used, the results obtainedfrom these models are essentially identical, even though the kinematics of the Krawinklermodel is significantly different than that of the Scissors model. Unfortunately the Scissorsmodel is often misused in practice because analysts tend to compute spring properties thatwere derived for the Krawinkler model and use them in the Scissors model. A completedescription of the mechanical models is presented in the next section of this paper.

Results obtained from structures implementing the simple mechanical models werecompared with those computed from a detailed finite element model of a full beam-column

subassemblage. The detailed model was created using ABAQUS (3). Both elastic andinelastic analysis was performed. It was found that good correlation was obtained between

Connections in Steel Structures V - Amsterdam - June 3-4, 2004 121


2/10

the mechanical models and the finite element model. The results from the finite elementmodel were also compared to experimental results obtained during the SAC Project (4, 5)and reasonably good correlation was obtained. The detailed analysis is briefly described inthe second main part of this paper.

MECHANICS OF BEAM-COLUMN JOINT

A typical interior beam-column subassemblage of a moment resisting frame is shown infigure 1. The subassemblage is in equilibrium under the loads shown if the moments at themid-span of the girders and mid-height of the columns are zero. It is assumed that size andspan of the girders on either side of the column are same, and that a single column sectionis used over the full height. The girders are welded to the column flanges. A doubler platemay be used to reinforce the panel zone.

Terms and represent the ratios of the effective depth of the column to the span length,and the effective depth of the girder to the column height, respectively. The effective depth

of a section is defined as the distance between the centers of the flanges. Use of theseterms in lieu of the actual physical dimensions greatly simplifies the derivation of theproperties of the models.

L

L

HH

VC

VC

VC/ H

VC/ H

L

L

HH

VC

VC

VC/ H

VC/ H

L

L

HH

VC

VC

VC/ H

VC/ HCV H

L

CV HL

L

L

HH

VC

VC

VC/ H

VC/ H

L

L

HH

VC

VC

VC/ H

VC/ H

L

L

HH

VC

VC

VC/ H

VC/ HCV H

L

CV H

L

CV HL

CV HL

Figure 1. Typical interior beam-column subassemblage.

Total subassemblage drift

The total drift in the subassemblage, , is defined as the lateral displacement of the top ofthe column with respect to the bottom of the column under the load VC. Following theprocedure described by Charney (6), this drift may be divided into three components, onefor the column, one for the girder, and one for the panel zone.

PGC ++= (1)

The column and girder displacement components are due to axial, flexural and sheardeformations occurring in the clear span region of the respective sections. The panelcontribution to displacement may also be divided into axial, flexural, and shear components:

PVPFPAP ++= (2)

122 Connections in Steel Structures V - Amsterdam - June 3-4, 2004


3/10

As stated earlier, this paper concentrates on the development of the panel zone shearcomponent of subassemblage displacement. It should be noted that this componentincludes localized bending in the flanges of the column, but bending through the depth of

the panel is represented by PF.

Panel zone participation in total subassemblage drift

If it is assumed that the moment in the girder at the face of the column is resisted entirely bythe flanges of the girder, it can be shown by simple statics that the horizontal shear force inthe panel zone is

)1( = CP

VV (3)

The corresponding shear stress in the panel is

P

CP

HV

=

)1( (4)

This shear stress is uniform throughout the volume of the panel zone. The term P , whichrepresents the volume of the panel zone, appears repeatedly in the following derivations.

To determine the panel zone contribution to subassemblage drift, equal and opposite unitvirtual forces are applied in lieu of the actual column shears VC. The shear stress in thepanel due to the unit virtual shear force is

P

H

=

)1(1

(5)

The contribution of panel zone shear strain to subassemblage drift is obtained by integratingthe product of the real strains and the virtual stresses over the volume of the panel. Theuniformity of stress and strain over the volume of the panel simplifies the integration.

P

C

V

PPV

G

HVdV

G

== 22

1 )1( (6)

The Krawinkler model and the Scissors model must have the same panel zone shearcontribution to displacement as given by equation 6.

Panel zone shear strength

Research performed by Krawinkler (2) has shown that the strength of the panel zoneconsists of two components; shear in the panel itself, and flexure in the column flanges.The larger of these components is the panel zone shear, which is resisted by the web of thecolumn acting in unison with the doubler plate, if present. If it is assumed that the yield

stress in shear is 3/1 0.6 times the uniaxial yield stress and that the column and doublerplate are made from the same material, the yield strength of the panel in shear is

H

FLtFV

Py

PyYP

==6.0

6.0 (7)



4/10

The second component of strength arises from flexural yielding of the flanges of the column.This phenomenon, which is most significant for W14 and W18 columns with very thickflanges, has been observed from tests (2), and may be computed using the principle ofvirtual displacements. The computed shear strength due to column flange yielding in thejoint region is

HtbFV CfCfy

YF

2

81.= (8)

where the 1.8 multiplier is a calibration factor based on test results.

Force-deformation response

The assumed force-deformation behavior of the beam column joint is illustrated in figure 2.In the figure the deformation is the racking displacement over the height of the panel. Themoment-rotation aspect of figure 2 is used later.

Shear Displacement Spring Rotation

YY

4Y4Y

Shear, VMoment,M

VYPMYP

VYFMYF

Panel

Flange

Total

VYP

MYP

H

H

Y

Y

Figure 2. Force-deformation relationship for beam-column joint.

The total response is equal to the sum of the response of the panel and the column flanges.Following Krawinkler (2) it is assumed that the flange component yields at four times theyield deformation of the panel component. It should be noted that figure 2 shows that theflange component of the resistance is effective immediately upon loading. Krawinklerassumes that this component of resistance does not occur until the panel yields in shear.

We have used the relationship shown in the figure as it simplifies the implementation of themechanical models without compromising accuracy.

The Krawinkler and Scissors models must be proportioned such that yielding is consistentwith figure 2.

The Krawinkler model

The Krawinkler model is shown in figure 3. The model consists of four rigid links connectedat the corners by rotational springs. The springs at the lower left and upper right cornershave no stiffness, and thereby act as true hinges. The spring at the upper left is used torepresent panel zone shear resistance, and the spring at the lower right is used to represent

column flange bending resistance. A total of twelve nodes are required for the model (thereare two nodes at each corner). The number of degrees of freedom in the model depends



5/10

on the use of nodal constraints or slaving. The minimum number of DOF required to modelthe panel is four in a planar structure. The maximum is twenty-eight.

Rigid Link

Real

Hinge

Rotational Spring

for Panel Shear

Rotational Spring

for Column Flange

Bending

Figure 3. The Krawinkler model.

The properties of the springs in the Krawinkler model are easily computed in terms of thephysical properties. Looking at only the panel spring, for example, the moment in the springis equal to the panel shear times the height of the panel. (See the diagram at the right offigure 2.) The rotation in the spring is equal to the shear displacement in the panel dividedby the panel height. Hence,

HVM PKP =, (9)

P

P

P

PKP

G

HV

HLtG

HV

==

1, (10)

Note that the K subscript in the above expressions refers to the Krawinkler model.

The stiffness of the rotational spring representing the panel in the Krawinkler model is themoment divided by the rotation;

P

KP

KP

KP GM

S ==,

,

,

(11)

The yield moment in the spring is simply the panel shear strength times the height of the

panel. Using equation 7,

PZYYPKYP FHVM == 6.0, (12)

As seen in figure 2, the stiffness of the flange bending component of the Krawinkler modelis equal to the yield moment in the flange bending component divided by 4.0 times the yieldrotation of the panel component. The yield rotation of the spring representing the panelcomponent is

G

F

K

MY

KYP

KYP

KYP 6.0,

,

, == (13)

The yield moment is equal to the yield strength times the panel height;



6/10

281 ))((.

, CfCfYYFKYFtbFHVM == (14)

and the resulting stiffness is

GtbM

SCfCf

KYP

KYF

KF

27504

))((.

,

,

, ==

(15)

In summary, Expressions 11 and 12 and 14 and 15 are all that are needed to model thepanel spring and the flange spring, respectively, in the Krawinkler model. If desired, a strainhardening component may be added.

The Scissors model

The Scissors model is shown in figure 4. This model derives its name from the fact that themodel acts as a scissors, with a single hinge in the center. Only two nodes are required tomodel the joint if rigid end zones are used for the column and girder regions inside thepanel zone. The model has four degrees of freedom. As with the Krawinkler model, onerotational spring is used to represent the panel component and the other is used torepresent the flange component of behavior.

The properties of the Scissors model are determined in terms of those derived previouslyfor the Krawinkler model. First, consider the displacement participation factor for panelshear as derived in equation 6. Noting that the denominator of this equation is the same asthe panel spring stiffness for the Krawinkler model, equation 6 may be rewritten as

KP

C

P S

HV

,

22 )1( = (16)

For the Scissors model, the moment in the spring under the column shear VCis simply VCH.If the Scissors spring has a stiffness SP,S, the rotation in the spring is VCH/SP,S. The drift overthe height of the column is the rotation times the height, thus for the Scissors model,

SP

CScissorsP

S

HV

,

2

, = (17)

As this displacement must be identical to that given in equation 16, it is evident that the

relationship between the Krawinkler spring and the Scissors spring is as follows:

2

,

,)1(

= KPSPS

S (18)

Similarly, when the moment in the Krawinkler spring is VPH, the moment in the Scissorsspring is VCH. Using equations 3 and 9

)1(, == HVHVM CPKP (19)

)1(

,

,=

KP

SP

MM (20)



7/10

The relationships given by equations 18 and 20 hold also for the column flange componentsof the models:

2

,

,)1(

= KFSFS

S (21)

)1(

,

,

= KFSFM

M (22)

Rotational Spring

For Column Flange

Bending

Boundary of

Panel Zone

Real

Hinge

Rigid

Link

Rotational Spring

For Panel Shear

Rotational Spring

For Column Flange

Bending

Boundary of

Panel Zone

Real

Hinge

Rigid

Link

Rotational Spring

For Panel Shear

Figure 4. The Scissors model.

As an example, consider the case where andare 0.1 and 0.2, respectively, the Scissorspring must be approximately twice as stiff and 1.43 times stronger than the Krawinklerspring. Many analysts erroneously use the springs derived for the Krawinkler model in theScissors model. This will produce models that are more flexible than the true structure, andthat prematurely yield in the panel zone regions.

Comparisons between the Krawinkler and Scissors models

One should note from equations 18 and 20 that while the properties of the Scissors models

are dependent on the quantities and , those of the Krawinkler model are not. Since itwas explicitly assumed that the columns and girders on both sides of the joint are of equal

height and span, and these terms are reflected in and , the Scissors model may not beused when this condition is violated. There is no such restriction on the use of theKrawinkler model.

The deformed shape of the Krawinkler and Scissors models are shown in figure 5. In thisfigure all of the deformation is assumed to be in the panel, with the girder and column rigid.

The most striking difference in the behavior between the two models is the offset in thecentrelines of the columns and girders in the Krawinkler model, which are not present in theScissors model.

A series of analyses were carried out using DRAIN-2DX (7) to determine the effect of thekinematic differences on the pushover response of a series of assemblages and planarframes which had yielding in the panel zone and at the ends of the girders. A variety ofgirder spans were used, but the column height remained constant. Analysis was performedwith and without gravity load, and with and without P-Delta effects. For simplesubassemblages analyzed using the Krawinkler and the Scissors models, the pushoverresponses were identical. For structures created by assembling subassemblages into arectilinear frame, but with real hinges at the midspan of the girders and midheight of the

columns, the pushover responses were again identical. Minor differences in the pushoverresponses were obtained when the midspan/midheight hinges were removed. It was



8/10

concluded, therefore, that the Scissors model, when properly used, is generally as effectivefor analysis as is the Krawinkler model, given the approximations in the derivations and theuncertainties involved in the analysis.

Offsets

Figure 5. Kinematics of Krawinkler model (left) and Scissors model (right).

COMPARISON WITH DETAILED FINITE ELEMENT ANALYSIS AND SAC RESEARCH

To evaluate the effectiveness of the elastic modeling techniques developed above, a seriesof comparisons was performed using test results provided by Ricles (5) from Phase II of theSAC Steel Project. The properties used for one of the test specimens are provided in Table1. Additionally, the SAC subassemblage was modeled using ABAQUS. The ABAQUSmodel of the subassemblage is shown in figure 6.

Table 1. Lehigh test C1: Geometric and material properties.

Yield Stress (ksi)Member Size Length (in.) GradeMill Certs. Coupon Test

Girder W36x150 354 A572 Grade 50 5756.7 flange

62.9 web

Column W14x398 156 A572 Grade 50 5453.2 flange

52.2 web

Continuity Plate N/A N/A N/A N/A

Doubler Plate (2) @ A572 Grade 50 57 57.1

Figure 6. ABAQUS model of SAC subassemblage.



9/10


10/10

VC average shear force in columns above and below the jointVP horizontal shear force in panel zone

P volume of panel zone =LHtP

REFERENCES

(1) Downs, William M, (2002). Modeling and Behavior of the Beam/Column Joint Regionof Steel Moment Resisting Frames, M.S.Thesis, Department of Civil and EnvironmentalEngineering, Virginia Tech, Blacksburg, Virginia.

(2) Krawinkler, H., (1978), Shear in Beam-Column Joints in Seismic Design of Frames,Engineering Journal, v15, n3, American Institute of Steel Construction, Chicago,Illinois.

(3) Hibbit, Karlson, and Sorensen, (2001). ABAQUS Users Manual, Verson 6.2.

(4) FEMA (2000). Recommended Seismic Design Criteria for New Steel Moment FrameBuildings, FEMA-350.Federal Emergency Management Agency, Washington D.C.

(5) Ricles, J. M., (2002). Inelastic Cyclic Testing of Welded Unreinforced MomentConnections Journal of Structural Engineering, ASCE,v128, n4.

(6) Charney, Finley A., (1993). "Economy of Steel Frame Buildings Through Identificationof Structural Behavior", Proceedings of the Spring 1993 AISC Steel ConstructionConference, Orlando, Florida.

(7) Prakesh, V. and Powell, G. H., (1993). DRAIN 2D-X Users Guide, University ofCalifornia, Berkeley, California.


Documents

Panel Zones