Sources of elastic deformation in steel frame and framed-tube structures: Part 1: Simplified subassemblage models

Journal of Constructional Steel Research 64 (2008) 87–100 www.elsevier.com/locate/jcsr

Sources of elastic deformation in steel frame and framed-tube structures: Part 1: Simplified subassemblage models Finley A. Charney ∗ , Rakesh Pathak Department of Civil and Environmental Engineering, Virginia Tech, Blacksburg, VA 24061, United States Received 4 August 2006; accepted 8 May 2007

Abstract Three simple models for including the effect of beam–column joint deformation in the analysis of steel moment resisting frame and framed tube structures are presented. The first model, called the Fictitious Joint model, is based on two-dimensional frame analysis and is useful for preliminary analysis only. The second model, called the Krawinkler Joint model, and the third model, known as the Scissors Joint model, use an assembly of rigid links and rotational springs to represent the joint, and may be used in preliminary and final analysis of full structural systems. All derivations are provided in the form of “displacement participation factors”, which allow a detailed breakdown of the various components of subassemblage displacement. When applied to isolated beam–column subassemblages, it is shown that all three modeling approaches produce the same general expression for computing deflections arising from shear deformations in the panel zone region. However, the Krawinkler and Scissors models do not include the effects of flexural deformation within the beam–column joint, whereas the Fictitious Joint model does. While not the dominant source of deformation, it is shown in the paper that the effects of flexural deformations in the beam–column joint should not be ignored. It is also shown in this paper that the overall displacements predicted by the simplified models correlate very well with displacements computed from detailed three dimensional finite element analysis of the same subassemblage. However, the finite element analysis approach, taken alone, is not capable of providing a breakdown of the subassemblage displacements into components, such as panel zone shear, or column joint flexure. Part 2 of the paper presents a method for providing this information from the results of detailed finite element analysis. c 2007 Elsevier Ltd. All rights reserved.

Keywords: Steel structures; Moment resisting frames; Panel zone deformations; Structural analysis

1. Introduction and background The effect of beam–column joint deformations on the lateral load response of steel moment resisting frames and framed tubes has been studied in detail [1–16]. In these references, there is general agreement that it is essential that beam–column joint deformations be included in structural analysis, and various approaches are provided for including such effects. These approaches may basically be broken into two types. In the first type, the beam–column joint region is modeled explicitly, using an assemblage of rigid links, rotational springs, and shear panels. The second approach uses standard frame analysis, with joint deformations being considered through the use of a modified force distribution in the beam–column joint region. ∗ Corresponding address: Department of Civil and Environmental Engineering, 200 Patton Hall, Mail Stop 105D, Blacksburg, VA 24061, United States. Tel.:+1 540 231 1444; fax: +1 540 231 7532. E-mail address: [email protected] (F.A. Charney).

c 2007 Elsevier Ltd. All rights reserved. 0143-974X/$ - see front matter doi:10.1016/j.jcsr.2007.05.008

Krawinkler [5] was the first to propose a mechanical beam–column joint model, which is applicable to both linear and nonlinear analysis. A similar, but less complex model, called the Scissors model has been proposed by a variety of researchers. As documented by Charney and Marshall [16], there appears to be confusion as to the properties to use in the mechanical models, particularly in the Scissors model. However, when properly used, the mechanical models are reasonably accurate when compared to experimental results [12]. One of the key advantages of the mechanical models is that they may be used in the structural analysis of full structural systems. In the standard frame analysis approach, the deformations in the joint region are considered through a modification of the forces in the joint regions of the structure. This technique is most often applied to isolated subassemblages, and is not applicable to full frames unless certain compatibility requirements are relaxed. For example, the method may be used

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if a two-dimensional frame is represented as an assemblage of subassemblages, where compatibility is relaxed by assuming that there are points of inflection at the mid-span of girders and mid-height of columns. The standard frame analysis approach may also be used to estimate the effect of beam–column joint deformations on a structure which has been analyzed without explicit modeling of the joints. In this case, a virtual work based post-processor is used, as described in Charney [8]. There is one significant difference between the mechanical models and the frame models, and this has to do with the modeling of flexural deformations in the beam–column joint. In the mechanical models, deflection resulting from flexural flexibility in the joint is ignored, and stiffness associated with bending in the column flanges is included. In the frame model, exactly the opposite is done; the bending in the joint region is considered, and added stiffness associated with column flange bending is ignored. The result is that a given subassemblage modeled with the two different approaches will have different stiffness, with the mechanical model being the stiffer of the two. Because the effect of flexural deformations in the joint region is significant [9], it is desirable to modify the mechanical models to include such deformations. Downs [12] provided preliminary recommendations incorporating such deformations in the Krawinkler model, but final implementation of the approach requires a better understanding of the true pattern and effect of deformations within the beam–column joint. This understanding is obtained through detailed finite element analysis, and is presented in Part 2 of this paper [17]. 2. Objectives and scope The objective of the work presented in this paper is to present and determine the overall accuracy of three simple models for computing the effect of beam–column joint deformations on lateral load induced displacements of steel moment resisting frames and framed tube structures. The accuracy is assessed by comparing the total subassemblage drift computed by the simplified models with results obtain from detailed threedimensional finite element analysis. It is important to note that this paper does not specifically address the influence of axial deformations in the columns. This can be a very significant source of deformation in tall steel frames, and particularly in framed-tubes. An earlier conference paper by Charney [8] does address axial deformations in frames and tube structures as tall as 50 stories. Axial deformations are included in a similar manner in part 3 of the current paper. Part 3 will be published in a different issue of the same journal in which parts 1 and 2 appeared. The first two simplified models are mechanical models, and the third is a frame model. The mechanical models are referred herein as the Krawinkler Joint (KJ) model and the Scissors Joint (SJ) model. A detailed history and description of the mechanical models is provided in Charney and Marshall [16]. The frame model is called the Fictitious Joint (FJ) model. The name for this model comes from the fact that the beam–column joint is not modeled explicitly. Instead, it is represented as a part of the column, and the deformation in the joint region is based

Table 1 Basic subassemblages considered for analysis Designation

Beam size

Beam span (m (ft))

Column size

Column height (m (ft))

A

W770 × 220 (W30 × 148)

4.57 (15) 6.10 (20) 7.62 (25)

W360 × 382 (W14 × 257)

3.81(12.5)

B

W770 × 220 (W30 × 148)

4.57 (15) 6.10 (20) 7.62 (25)

W610 × 372 (W24 × 250)

3.81(12.5)

on (gross) simplifications of the pattern of shear and flexural stresses within the joint region. The three different joint models are presented through the use of a virtual work procedure, called the Displacement Participation Factor (DPF) approach [8,18,19]. After the theoretical background is presented, the DPF procedure is used to analyze two basic subassemblages. Each is run with three different beam spans (4.57 m (15 ft), 6.10 m (20 ft), and 7.62 m (25 ft)), and with and without beam flange continuity plates. The basic list of subassemblages considered is shown in Table 1. Individual subassemblages are designated by a three part symbol consisting of basic designation, beam span (in feet), and presence of continuity plates. For example, B20c represents basic subassemblage B with a 20 ft. (6.10 m) beam span, with continuity plates. The same subassemblage without continuity plates is B20n. When continuity plates are used, they are positioned top and bottom, on each side of the column web, and are of the same thickness as the flange of the beam. The width of the continuity plate is taken as the lesser of the width of the projecting flange width of the beam or column. Web doubler plates are not considered in the analysis performed herein, but they are included in all derivations. Even though the doubler plates are not explicitly included in the examples, the benefits of using doubler plates to reduce drift are briefly discussed. After the derivations are presented, the results of the analyses of the subassemblages indicated in Table 1 are presented and discussed. 3. Overview of displacement participation factor (DPF) approach Consider the structure of Fig. 1(a), which is subjected to the “Real” lateral loads shown on the left side of the structure. A simple two-dimensional frame analysis program has been used to determine the lateral (x) deflection at the upper right of the structure (node 28). This deflection is designated as ∆28x. The analysis was based on centerline dimensions, with axial, flexural, and shear deformations included. It is now desired to determine how much the highlighted column (C10) contributes to the designated deflection. To do so, the real loads are removed, and a virtual force Qˆ is applied at the location of and in the direction of ∆28x. The column’s contribution to the displacement, called the column’s

F.A. Charney, R. Pathak / Journal of Constructional Steel Research 64 (2008) 87–100

89

Fig. 1. Planar frame under real or virtual loading.

Displacement Participation Factor, is given by the familiar virtual work expression "Z Z H H P(x) P(x) ˆ ˆ M(x) M(x) 1 ∆28x dx + ∇C10,T = E AC E IC Qˆ 0 0 # Z H V (x)Vˆ (x) (1) + dx G AC S 0 where P(x), V (x) and M(x) are the “real” axial force, shear, and moment functions along the length of the element, ˆ ˆ P(x), Vˆ (x) and M(x) are the “virtual” force functions in the ˆ H is the length element (arising from the application of Q), of the element, E and G are the elastic and shear modulus, respectively, and AC , IC and AC S , are the cross sectional properties of the column. It is clear from Eq. (1) that the column’s contribution to the deflection consists of axial, flexural, and shear components. If DPFs for all of the beams and columns of the structure were available, it would be seen that ∆28x =

ncols X i=1

∆28x ∇Ci,T +

nbeams X

28x ∇ B∆j,T

(2)

j=1

where the summation ranges over the total number of columns and beams. As seen in Eq. (1), the basic symbol for a DPF is the inverted triangle, with a double subscript and a single superscript, ∆ . The superscript represents the “designated as follows: ∇C,S displacement”, which in the case of the previous example, was ∆28x. The first subscript, C, represents the named component of the structure for which the DPF is computed. There are numerous possibilities here, but B, C, and J serve as examples for beam, column, and joint, respectively. The subscript, S, represents the deformation source, and can be either A, F, S, or T for axial, flexure, shear, or total, respectively. For example, from Eq. (1) it is seen that ∆28x ∆28x ∆28x ∆28x ∇C10,T = ∇C10,A + ∇C10,F + ∇C10,S .

(3)

Fig. 2. Typical beam–column subassemblage.

A similar approach could be used to determine how the highlighted subassemblage (Sub. A) of Fig. 1(a) contributes to a global displacement, such as the x-direction displacement at node 28. In this case the contribution of the beam, column, and beam–column joint region would be computed. A different virtual load pattern, such as that shown in Fig. 1(b), could be used to determine subassemblage A’s contribution to the interstory drift (δ3) at level three of the structure. It is also of interest to analyze an isolated subassemblage that is not part of a larger structure. Such a subassemblage (which might be a laboratory specimen) is shown in Fig. 2. Here, DPFs are used to determine how the column, beam, and joint of the subassemblage contribute to the total drift, δ, as illustrated in Fig. 3. This total drift is the designated displacement for the subassemblage, and hence, the DPF terms for isolated subassemblage analysis will have δ as a superscript. All derivations presented in this paper are based on the assumption that the subassemblage is isolated. In the remainder of this paper, derivations are provided for computing DPFs for the FJ, KJ, and SJ models. When doing so, the subassemblage is divided into column, beam, and joint components. DPFs for the portions of the column and beam

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Fig. 3. Beam–column subassemblage used in analysis.

outside the joint region are identical for each of the models, and these are called the clear span DPF components. The DPF for the joint region represents the portion of the column which passes through the joint, the beam flange continuity plates, and the doubler plates. After the simple model derivations are presented, the results of the analysis of subassemblage A20c and A20n are described in detail. The purpose of this analysis and related discussion is to describe basic features of the behavior of the subassemblage, and to set up the discussion for the detailed finite element analysis. Finally, comparisons for the simplified and detailed analysis of all of the subassemblages are presented and discussed. A general statement on the accuracy of the analysis is made, and recommendations are provided for use by the design profession. 4. Computation of DPF for the fictitious panel zone model The subassemblage to be analyzed is shown in Fig. 2. This subassemblage is extracted from a frame by assuming that inflection points occur at mid span of the beams on either side of the joint, and at mid-height of the columns above and below the joint. It is further assumed that the beams on either side of the joint are of the same section and length, and that the columns above and below the joint are the same section and length (but not necessarily the same as the beam). To simplify the derivations, the dimensionless parameters α and β are defined: dCe (4) L d Be β= (5) H where L is the length of the beam (the bay width) and H is the height of the column (story height), and dCe and d Be are the effective depths of the column and beam, respectively. As shown in Fig. 2, these depths are taken as the distance between the centers of the flanges of the sections. Another possible definition of the effective depth, using out-to-out dimensions, is discussed later. α=

It is desired to determine how much the various components of the subassemblage contribute to the drift along the height of the subassemblage, so equal and opposite virtual forces Qˆ are applied at the top and bottom of the column, as shown in Fig. 3. The shear in the column resulting from this force is designated as VˆC , and all other virtual subassemblage forces are related to this force. A carat (ˆ) over the appropriate symbol identifies quantities as virtual. While the derivation would be simplified by using a unit value for Qˆ this is not done because the presence of Qˆ in the various equations provides force unit consistency. Maintaining Qˆ in the derivations also allows the formulas to be used to determine a subassemblage’s contribution to a global displacement as shown in Fig. 1. In this case, VC is taken as the average of the real shears in the upper and lower half column, and VˆC is the average of the virtual shears in the upper and lower half column. Note, however, that an additional source of deformation, due to beam and column axial deformations, would need to be included. The components analyzed consist of the clear span region of the column, the clear span region of the beam, and the portion of the column that is described as the joint. The joint includes beam flange continuity plates and doubler plates, when present. Only shear and flexural deformations need to be considered because there are no axial forces acting on the isolated subassemblage. The total displacement along the height of the subassemblage is the same as the total DPF for the subassemblage, as follows: δ δ δ δ δ = ∇Tδ = ∇C,F + ∇C,S + ∇ B,F + ∇ B,S + ∇ δJ,F + ∇ δJ,S . (6)

The shears and moments in the clear spans of the beam and column are easily determined because the subassemblage is statically determinate. The forces inside the joint region are indeterminate, however, and a simplifying assumption is required to resolve these forces. The assumption made is that the beam and column moments that enter the joint are completely resolved into flange force couples. This is shown in the free-body diagram of Fig. 4, where the subassemblage has been divided into the component parts. Note that the lower halfcolumn and the left half-beam have been omitted from Fig. 4 for clarity. The flange forces are designated as FBF for the beams and FCF for the columns, and are computed from equilibrium as follows: 0.5L(1 − α)(VC H/L) 0.5VC (1 − α) = βH β 0.5VC H (1 − β) . = αL

FBF =

(7)

FCF

(8)

Force, shear, and moment diagrams for the full height of the column are shown in Fig. 5. Similar diagrams for the beam are presented in Fig. 6. The force diagram for the column shows the left and right beam flange forces entering the column at the top and bottom of the joint region. Using Fig. 5, it is seen that the horizontal shear in the joint region of the column is VCJ = VC − 2FBF , which simplifies to

F.A. Charney, R. Pathak / Journal of Constructional Steel Research 64 (2008) 87–100

VCJ = VC

(1 − α − β) . β

91

(9)

Note that the shear force is taken as a positive quantity. The vertical shear force, VBJ , in the joint region of the column is similarly determined. From Fig. 6, it is seen that VBJ = VB − 2FCF , which after some substitution and manipulation results in VBJ =

VC H (1 − α − β) . L α

(10)

It is important to note that the shear stress in the joint region is the same whether computed from the horizontal or the vertical shear force: τJ =

Fig. 4. Free body diagram of subassemblage components.

VCJ VBJ VC (1 − α − β) = = αLt p β Htp αβ Lt p

(11a)

where t p is the thickness of the panel zone including doubler plates, if present. If the numerator and denominator of Eq. (11a) are multiplied by H , the expression for shear stress may be simplified to τJ =

VC H (1 − α − β) VC H (1 − α − β) = αβ L H t p vp

(11b)

where v p = αβ L H t p is the effective volume of the panel zone. From the moment diagrams of Figs. 5 and 6, it is seen that the moments at the center of the column and beam, MCC and MBC , respectively, are not exactly zero. The column moment is MCC = 0.5H VC − 0.5β H (2FBF ) which simplifies to MCC = 0.5VC H α.

(12)

Similarly, the moment at the center of the beam is VC H − 0.5αL(2FCF ) L which simplifies to MBC = 0.5L

Fig. 5. Column force diagrams.

Fig. 6. Beam force diagrams.

MBC = 0.5VC Hβ.

(13)

Clearly, equilibrium is not exactly satisfied at the center of the joint because MCC and MBC are not equal. This small error is due to the assumption that all of the moment is resisted by the flange of the beam and column. Given the moments and forces along the length of the elements, it is now possible to derive the displacement participation factors. This will be done first for the clear span portions of the beam and column, and then for the joint region. The real shears and moments in the clear span region of the subassemblage are shown in Figs. 5 and 6. The virtual shears and moments are identical, with the exception that (for example) VˆC is used in lieu of VC . Using these forces, the DPFs for the portion of the beams and columns outside the joint are as follows: Z 2 0.5H (1−β) VC VˆC VˆC VC H (1 − β) δ ∇C,S = dx = (14) G A SC G A SC Qˆ 0 Qˆ

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δ ∇ B,S =

2 Qˆ

0.5L(1−α) VC H VˆC H L L

Z

G AS B

0

dx =

VˆC VC H 2 (1 − α) (15) LG A S B Qˆ

2 0.5H (1−β) VC x VˆC x dx E IC Qˆ 0 VˆC VC H 3 (1 − β)3 = 12E IC Qˆ Z

δ ∇C,F =

(16)

Z V H x VˆC H x 2 0.5L(1−α) CL L dx ˆ E I B Q 0 VˆC VC H 2 L(1 − α)3 = . 12E I B Qˆ

δ ∇ B,F =

(17)

The terms A SC and A S B represent the effective shear areas of the column and beam cross sections. For all computations presented in this paper, these areas are computed using Cowper’s detailed formula [20]. A complete summary of the computation of shear areas in wide flange sections is provided in Charney et al. [21]. The DPFs as presented in Eqs. (14) through (17) include columns and beams on both sides of the joint (hence the “2” on the left of the integral sign). DPFs for a centerline analysis of the subassemblage may be obtained from Eqs. (14) through (17) by setting α and β to zero. The DPF for the joint region of the subassemblage consists of three parts: shear in the panel, column flexure in the panel, and beam flexure in the panel. The DPF due to shear is based on the horizontal shear force in the column part of the joint. Using Eq. (9) and its virtual counterpart ∇ δJ,S

1 = VˆC =

ˆ β H VC (1−α−β) VC (1−α−β) β β

Z 0

GαLt p

VˆC VC H (1 − α − β)2 . αβG Lt p Qˆ

Fig. 7. Effective section for beam joint flexure.

dx (18a)

Eq. (18a) may be simplified somewhat by multiplying the numerator and denominator by H , and by recognizing the product αLβ H t p as the volume of the panel, v p resulting in Fig. 8. Effective section for column joint flexure.

∇ δJ,S

VˆC VC H 2 (1 − α − β)2 = . Gv p Qˆ

(18b)

Note that there is no additional contribution from the vertical beam shear as expressed in Eq. (10) because the shear stresses in the beam–column joint region can only be counted once. In fact, Eq. (18) could have been derived using the beam joint shear. This was not done, however, because the joint is physically part of the column, not the beam. Derivation of the DPF due to joint flexure is based on the moments and shears shown in Figs. 5 and 6 for the column and beam, respectively. Before the derivations are presented, it is noted that there is significant uncertainty as to the proper moment of inertia to use in the joint region. For beam joint flexure, the moment of inertia depends on whether or not beam flange continuity plates are present. When such plates are present, the effective bending section is assumed to be as shown

in Fig. 7(a), and the moment of inertia is computed using Eq. (19a). When the continuity plates are not present, the effective section is as shown in Fig. 7(b), and the moment of inertia is computed from Eq. (19b). If doubler plates are used, they should be included in the computation of effective moment of inertia within the joint. This is shown schematically in Fig. 7, and mathematically in Eqs. (19a) and (19b) by the addition of the term IBDP . It is noted that the schematic representation of the doubler plate in Fig. 7 is somewhat different than may be used in practice, where for example, the plate extends above and below the beam. For joint column flexure, the effective cross section is shown in Fig. 8, and the related moment of inertia, in absence of doubler plates, is taken to be equal to IC . As with the beam, doubler plates should be included when present. The plate is shown in Fig. 8, and is represented by the term ICDP in Eq. (19c).

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F.A. Charney, R. Pathak / Journal of Constructional Steel Research 64 (2008) 87–100 Table 2 Ratios of actual to simplified DPFs for column and beam joint flexure

α α α α α

= 0.10 = 0.15 = 0.20 = 0.25 = 0.30

IBJ

IBJ ICJ

Columns β = 0.10

β = 0.15

β = 0.20

β = 0.25

β = 0.30

Beams β = 0.10

β = 0.15

β = 0.20

β = 0.25

β = 0.30

1.12 1.19 1.27 1.35 1.44

1.13 1.21 1.29 1.38 1.48

1.14 1.22 1.31 1.41 1.52

1.15 1.24 1.34 1.44 1.56

1.16 1.26 1.37 1.48 1.61

1.12 1.13 1.14 1.15 1.16

1.19 1.21 1.22 1.24 1.26

1.27 1.29 1.31 1.34 1.37

1.35 1.38 1.41 1.44 1.48

1.44 1.48 1.52 1.56 1.61

" # 3 b tcp d B3 twC cp = +4 + tcp bcp (0.5(d B − tcp ))2 12 12 + IBDP d 3 twC + IBDP = B 12 = IC + ICDP .

insets of Figs. 5 and 6. If this is done, it can be shown [12] that the joint flexure DPFs are as follows: (19a)

1 Qˆ

βH 2

Z 0

(19c)

  VC VˆC H α (1 − α − β) 2 + x dx E ICJ 2 β

which when simplified is ∇ δJ,C F =

  (1 − α − β)2 VˆC VC H 3 β α(1 − β) + . 3 Qˆ 4E ICJ

(20)

The beam joint flexure DPF is based on the moment shown in Fig. 6, where the moment is MBC + VBJ x. Substituting from Eqs. (10) and (13), the DPF for beam joint flexure is  Z αL ˆC  Hβ 2 VC V 1 H (1 − α − β) 2 ∇ δJ,B F = 2 + x dx. E IBJ 2 αL Qˆ 0 Upon solving the integral and simplifying, ∇ δJ,B F =

  VˆC VC H 2 Lα (1 − α − β)2 β(1 − α) + . 3 Qˆ 4E IBJ

VˆC VC H 3 β(1 − β)2 12E ICJ Qˆ

(22)

∇ δJ,B F(simple) =

VˆC VC H 2 Lα(1 − α)2 . 12E IBJ Qˆ

(23)

(19b)

Another factor to consider in determining the effective moments of inertia in the joint region is the fact that the portions of the column and beam outside the joint provide flexural restraint for the joint. This is shown schematically by the “possible zone of influence” in Figs. 7 and 8. To account for this effect Downs [12] recommended that a multiplier of 1.5 be applied to the effective moments of inertia in the joint region. Multipliers were not used in the examples shown in this paper. However, the use of the multipliers is investigated in part 2 of this paper [17]. The column joint flexure DPF is based on the moment shown in Fig. 5, where the moment is MCC + VCJ x. Substituting from Eqs. (9) and (12), the DPF for column joint flexure is ∇ δJ,C F = 2

∇ δJ,C F(simple) =

(21)

Given the uncertainty of the moment distribution and effective section properties within the joint, it is reasonable to adopt a simpler expression for the joint flexure components, wherein the moment in the joint varies linearly from zero at the center of the joint to MCJ or MBJ for the column and beam, respectively. These revised moment diagrams are shown in the

Before proceeding, it is interesting to compare the joint flexural contributions based on the Actual or the Simplified moment diagram within the joint. For columns, the ratio of the Actual to Simplified DPF is RATIOCJF =

3β(1 − α) + (1 − α − β)2 (1 − α)2

(24)

and the ratio for beams is RATIOBJF =

3α(1 − β) + (1 − α − β)2 . (1 − β)2

(25)

Table 2 lists the computed ratios for α and β ranging from 0.10 to 0.30 in increments of 0.05. As may be seen, there is a significant difference in the DPFs for flexure based on the actual and simplified moments in the joint region, particularly for the larger values of α and β. Without a detailed finite element analysis of the subassemblage, there is no way to know which approach produces the more accurate result. 5. Mechanical joint models In both the KJ and SJ models the DPFs for the clear span regions of the column and beam are the same as that given in Eqs. (14)–(17). And, as shown below, the DPF for the panel shear is the same as that given in Eq. (18a) or (18b). The KJ model is shown in Fig. 9. As may be seen, the model consists of four rigid links, connected at the corners by hinges. Mathematical constraints may be used in lieu of the links, but links are used herein because the links provide a better conceptual model of the joint. Rotational springs at the upper left and the lower right provide the necessary stiffness. The upper left spring accounts for panel zone shear stiffness and the lower right spring accounts for column flange stiffness. It is very important to note that the column flange stiffness is not included in the FJ model, and that the beam–column joint bending flexibility is not included in the KJ model. Hence, the KJ model is stiffer than the FJ model. A similar problem exists for the SJ model.

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F.A. Charney, R. Pathak / Journal of Constructional Steel Research 64 (2008) 87–100 Table 3 Summary of DPFs for isolated subassemblage DPF component

Column shear Beam shear Column flexure Beam flexure

Fig. 9. Krawinkler Joint model.

Joint shear Joint beam flexure (actual) Joint column flexure (actual) Joint beam flexure (simplified) Joint column flexure (simplified)

Fig. 10. Scissors Joint model.

The KJ model, as originally developed by Krawinkler [5], was slightly different than that depicted in Fig. 9, and was intended to represent inelastic behavior. In the original model, the shear stiffness was modeled by a shear membrane (instead of a rotational spring), and the column flange spring did not engage (had zero stiffness) until the panel yields in shear. Hence, for linear elastic analysis, the column flange spring had no effect. Because all of the analysis reported in the current paper is for linear elastic systems, the column flange springs are not considered. According to Charney and Marshall [16] the required rotational spring stiffness of the KJ model is K K J = αLβ H t p G = v p G.

(26)

When the column shear is VC , the moment in the spring is M K J = VCJ β H = V H (1 − α − β).

(27)

The virtual moment is the same, with VˆC being used in lieu of VC . The displacement participation for a rotational spring is ∇ Kδ J =

1 M Mˆ . Qˆ K K J

(28)

VˆC VC H 2 (1 − α − β)2 Gv p Qˆ

VC H (1−β) G A SC VC H 2 (1−α) LG A S B VC H 3 (1−β)3 12E IC

Component applicable to model (Y/N) FJ KJ SJ Y

Y

Y

Y

Y

Y

Y

Y

Y

VC H 2 L(1−α)3 12E I B VC H 2 (1−α−β)2 Gv p VC H 2 Lα (1−α−β)2 ] 4E IBJ [β(1−α)+ 3

Y

Y

Y

Y

Y

Y

Y

Y

N

VC H 3 β (1−α−β)2 ] 4E ICJ [α(1 − β) + 3

Y

Y

N

VC H 2 Lα(1−α)2 12E IBJ

Y

N

N

VC H 3 β(1−β)2 12E ICJ

Y

N

N

Given expressions are for isolated subassemblage with VˆC / Qˆ = 1.0.

column flange stiffness. As with the KJ model, the column flange effect will not be included herein. As shown in Charney and Marshall [16], the required rotational spring stiffness for the SJ model is KSJ =

Gv p KK J = . 2 (1 − α − β) (1 − α − β)2

(30)

In this case, the beam moment in the spring is M S J = VC H.

(31)

Using Eq. (28), and recognizing that the virtual moment is the same as above with VˆC substituted for VC , it is seen that ∇ Sδ J =

VˆC VC H 2 (1 − α − β)2 Gv p Qˆ

(32)

which is exactly the same as for the FJ and KJ models. For the reader’s convenience, the DPF expressions for all three of the models are summarized in Table 3. Note that VˆC / Qˆ was taken as 1.0 for all expressions listed in the table. 6. Three dimensional finite element analysis

Substituting from Eqs. (26) and (27) ∇ Kδ J =

Applicable equation

(29)

which is exactly the same as that shown in Eq. (18a) for the FJ model. The SJ model is shown in Fig. 10, where it may be seen that the column and beam are connected through two rotational springs, one representing panel shear, and the other representing

To establish the relative accuracy of the derivations presented above, the subassemblages indicated in Table 1 were analyzed using the expressions summarized in Table 3, as well as through a detailed three dimensional finite element analysis (FEA). In the FEA, the subassemblages were modeled via 8node three dimensional solid elements. The beam and column of each model consisted of an assemblage of rectangular flange and web plates, and as such, fillets were not included

F.A. Charney, R. Pathak / Journal of Constructional Steel Research 64 (2008) 87–100

Fig. 11. One-eighth finite element model.

in the finite element analysis. In order to produce a model with maximum accuracy (a highest possible resolution mesh), only 1/8 of the total subassemblage was modeled. Boundary conditions used in the 1/8 model analysis were verified through the analysis of full subassemblages. The detailed FEA was performed using the program WoodFrameSolver [22] and was verified using SAP2000 [23]. Fig. 11 shows the 1/8 model for the subassemblage with continuity plates. While the simplified models are able to provide a breakdown of the contribution of subassemblage drift to the various components of the subassemblage and to the different sources of deformation, the FEA approach can only provide total subassemblage drift. Hence, the only displacement comparisons that are made are for total drift. Comparisons are also provided for panel zone shear stress as provided by the simplified formulas and the FEA method. It is noted that it is possible to provide more detailed comparisons between the simplified models and the FEA approach by extending the DPF concept to the FEA results [20]. In fact, such a presentation is provided in the companion paper [17] that is published in the same journal in which the current paper appears. 7. Presentation of results The results of the analysis are presented in two parts. First, detailed results are presented and discussed for Subassemblages A20n and A20c. This is followed by a summary of the results of all subassemblage analysis, as well as correlation with the results of the detailed finite element analysis. Subassemblages A20n and A20c have a beam span of 6.10 m (20 ft) and a column height of 3.81 m (12.5 ft). The column and beam sections are W360 × 382 (W14 × 257) and W770 × 220 (W30 × 148), respectively. The modulus of elasticity of the steel is taken as 200.0 GPa (29 000 ksi), and Poisson’s ratio is 0.3. The column shear used in the analysis was 444.82 kN (100 kips). Section properties used in the analysis were based on the overall section dimensions, and did not include fillets. This was done to provide consistency with the FEA which also did not include fillets. The results for subassemblage A20n are given in Table 4, which has five parts. Part (a) shows the actual DPF values, part (b) presents the values as a percentage of the total

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subassemblage drift, part (c) gives the shear stresses in the panel zone computed with Eq. (11), and part (d) provides the results from the finite element analysis. Part (e) of the table presents ratios of the simplified model results to the FEA results. The columns in parts (a), (b), (c) and (e) of the tables represent different assumptions used in the simplified analysis. The two columns with the heading “Using Actual Joint Moments” computes the joint flexure DPFs using Eqs. (20) and (21). The two columns under the heading “Using Simplified Joint Moments” are based on the simplified flexural formula of Eqs. (22) and (23). Analysis was also run with center-tocenter joint dimensions, or out-to-out joint dimensions. Centerto-center analysis uses the distance between the centers of the flanges to determine the parameters α and β (see Eqs. (4) and (5)), while the out-to-out method uses the total depth. As may be seen from Table 4(a), the various assumptions used in the analysis have a marginal effect on the total deflection, which is in the range of 1.752–1.898 cm. From Table 4(b), it may be seen that the combined shear deformations in the clear span of the beams and columns accounts for about 15% of the total drift, and that shear deformation in the panel zone alone is responsible for approximately 28% of the total drift. Hence, all sources of shear deformation produce 43% of the displacement in the subassemblage. Joint beam flexure and joint column flexure, taken together, are responsible for about 10–11% of the total drift. While smaller than the joint shear contribution, this could not be considered as a negligible source of deformation. The deflection due to all sources of joint deformation is as high as 39.9% of the total subassemblage drift. When beam flange continuity plates are included, there is a reduction in the beam joint flexure component only. This is shown in Table 5, where it is seen that the contribution by beam joint flexure has reduced to an average of about 2%. However, the total joint flexural component is still not negligible, at about 8%. The results from the finite element analysis are provided in Tables 4(d), 5(d) for models A20n and A20c, respectively. For model A20n, the FEA drift is 1.768 cm, which is within the range of values (1.752–1.898 cm) computed from the simplified model. The model using the out-to-out joint dimensions produced the best correlation with the FEA. The shear stress in the panel zone of Model A20n is 153.0 MPa for the center-to-center model, and 127.4 MPa for the outto-out model. The FEA shear stress 140.3 MPa, which is within the range predicted by the simplified model. Similar agreement is seen for subassemblage A20c (see Table 5(a) through 5(d)), and for all of the other models analyzed. It is not apparent that either approach produces better results than the other. It is noted, however, that the average shear stress from the two simplified approaches is almost identical to that computed using FEA. While not the main subject of this paper, it is interesting to compute the Sensitivity Indices (SI) for the various components of the subassemblage. These indices are equal to a DPF divided by the volume of material in the component that contributed to the DPF. Various SIs (multiplied by 100 000) are presented in

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Table 4 Results of analysis of subassemblage A20n (a) Actual DPF values Deformation source (cm) Beam flexure Beam shear Column flexure Column shear Joint beam flexure Joint column flexure Panel zone shear Total

Using actual joint moments Center to center 0.495 0.106 0.377 0.164 0.093 0.100 0.563 1.898

Out to out 0.482 0.105 0.366 0.162 0.105 0.103 0.459 1.782

Using simplified joint moments Center to center Out to out 0.495 0.106 0.377 0.164 0.074 0.092 0.563 1.872

0.482 0.105 0.366 0.162 0.083 0.094 0.459 1.752

(b) Percent of total Deformation source (%) Beam flexure Beam shear Column flexure Column shear Joint beam flexure Joint column flexure Panel zone shear Total

Using actual joint moments Center to center

Out to out

Using simplified joint moments Center to center Out to out

26.1 5.6 19.9 8.6 4.9 5.3 29.7 100.0

27.1 5.9 20.5 9.1 5.9 5.8 25.8 100.0

26.4 5.7 20.1 8.8 4.0 4.9 30.1 100.0

Using actual joint moments Center to center 153.0 (22.2)

Out to out 127.4 (18.5)

Using simplified joint moments Center to center Out to out 153.0 (22.2) 127.4 (18.5)

Out to out

Using simplified joint moments Center to center Out to out

27.5 6.0 20.9 9.3 4.7 5.4 26.2 100.0

(c) Shear stress Panel zone shear stress (MPa (ksi))

(d) Results from finite element analysis Total displacement (cm) Average panel zone shear stress (MPa)

1.768 140.3

(e) Simple model to FEA model ratios Deformation source (%) Total displacement Panel shear stress

Using actual joint moments Center to center 1.073 1.095

Table 6 for subassemblage A20c (using actual joint moments and center-to-center dimensions). As may be seen in the table, the SI for the shear panel alone is 18 approximately times that for the column or beam. Based on the idea that an optimum structure has equal SI for all components [19], it is seen that adding one kilogram of steel to the panel zone (by adding a doubler plate) is more than 18 times as effective in reducing drift than would be adding the same kilogram of steel to the column. However, the cost of fabricating and attaching that extra kilogram of steel needs to be considered as well. It might be more economical to increase the size of the entire column (picking a column with a thicker web), than it would be to add a doubler plate. A summary of the results for all “A” subassemblages is presented in Table 7. Similar results are presented in Table 8 for the “B” subassemblages. In each case, the analysis used the simplified expressions for moments in the joint region and the out-to-out joint dimensions. As seen in Table 7(a), the component of drift due to beam column joint deformation is very significant in all cases, and

1.008 0.908

1.0294 1.095

0.991 0.908

joint flexural deformation is not negligible. It is also clear that the addition of the continuity plates had a marginal effect on reducing the subassemblage displacement. This fact is also evident from Table 7(b), where it is seen that joint flexure (Beam + Column) was responsible for about 11% of total drift when continuity plates are not present, and about 8% of total drift when continuity plates are added. Similar results are seen from Table 8(a) and (b). However, in this case, the portion of subassemblage drift due to joint flexure has increased to as much as 15% for Subassemblage B15n. This drops to about 8% when the continuity plates are added. Hence, the continuity plates appear to be more effective in reducing displacements when the column is deeper. In this case, the reduction in drift associated with the addition of the continuity plates is 7%. Tables 7(c) and 8(c) compare the drifts computed using the simplified formulas with those found from detailed finite element analysis. For the A subassemblages, the agreement is excellent, with the maximum difference being only about 2.4%. Differences are slightly greater for the B subassemblages,

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F.A. Charney, R. Pathak / Journal of Constructional Steel Research 64 (2008) 87–100 Table 5 Results of analysis of subassemblage A20c (a) Actual DPF values Deformation source (in.)

Using actual joint moments Center to center

Beam flexure Beam shear Column flexure Column shear Joint beam flexure Joint column flexure Panel zone shear Total

0.495 0.106 0.377 0.164 0.035 0.100 0.563 1.839

Out to out 0.482 0.105 0.366 0.162 0.039 0.103 0.459 1.717

Using simplified joint moments Center to center Out to out 0.495 0.106 0.377 0.164 0.028 0.092 0.563 1.825

0.482 0.105 0.366 0.162 0.031 0.094 0.459 1.700

(b) DPFs as percent of total Deformation source (%) Beam flexure Beam shear Column flexure Column shear Joint beam flexure Joint column flexure Panel zone shear Total

Using actual joint moments Center to center

Out to out

Using simplified joint moments Center to center Out to out

26.9 5.8 20.5 8.9 1.9 5.4 30.6 100.0

28.1 6.1 21.3 9.4 2.3 6.0 26.7 100.0

27.1 5.8 20.7 9.0 1.5 5.1 30.9 100.0

Using actual joint moments Center to center

Out to out

Using simplified joint moments Center to center Out to out

153.0 (22.2)

127.4 (18.5)

153.0 (22.2)

Out to out

Using simplified joint moments Center to center Out to out

28.4 6.2 21.5 9.5 1.8 5.5 27.0 100.0

(c) Shear stress

Panel zone shear stress (MPa (ksi))

127.4 (18.5)

(d) Results from finite element analysis Total displacement (cm) Average shear stress (MPa)

1.738 141.0

(e) Simple model to FEA model ratios Deformation source (%)

Using actual joint moments Center to center

Total displacement Panel shear stress

0.945 1.085

0.988 0.903

Table 6 Sensitivity indices for subassemblage A20c Item

Volume (cm3 )

DPF (cm)

100 000∗ S I

S IPanel /S I

Total Beam Column Joint Panel only

690 452 318 794 295 656 75 970 16 452

1.839 0.601 0.541 0.698 0.563

0.266 0.188 0.183 0.918 3.425

12.86 18.18 18.72 3.73 1

where the maximum difference is 3%. Shear stresses computed from the simple and detailed analysis are shown in Tables 7(d) and 8(d). Again, the agreement is quite good with differences being consistently less than 10%. 8. Use of centerline analysis in lieu of explicit joint modeling This paper has shown that the influence of deformations in the beam–column joint region of moment frame structures

1.050 1.085

0.978 0.903

is very significant. Clearly, analytical models that assume that the entire joint region is “rigid” will underestimate the lateral displacements. When centerline analysis is used, the flexural deformations in the beam–column region will be overestimated, and the shear deformations in the joint region will be underestimated. This may be seen from Fig. 5, where the column force diagrams are shown for the fictitious joint model. If centerline analysis were used, the shear, Vc , would be constant through the joint, and the bending moment would be linear from the inflection point of the column to the centerline of the joint. Considering subassemblage A20n, for example, the centerline displacements are as follows: Beam Flexure: 0.596 cm Beam Shear: 0.112 cm Column Flexure: 0.727 cm Column Shear: 0.204 cm Total: 1.639 cm.

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Table 7 Summary of A-model DPFs from simplified subassemblage analysis using simplified moments and out-to-out dimensions (a) Displacement participation factors Deformation source Beam flexure Beam shear Column flexure Column shear Joint beam flexure Joint column flexure Panel zone shear Total

Model A15n 0.336 0.137 0.366 0.162 0.079 0.094 0.431 1.604

A20n

A25n

0.482 0.105 0.366 0.162 0.083 0.094 0.459 1.752

0.630 0.085 0.366 0.162 0.085 0.094 0.477 1.899

A15c 0.336 0.137 0.366 0.162 0.029 0.094 0.431 1.555

A20c 0.482 0.105 0.366 0.162 0.031 0.094 0.459 1.700

A25c 0.630 0.085 0.366 0.162 0.032 0.094 0.477 1.846

(b) DPFs as percent of total Deformation source Beam flexure Beam shear Column flexure Column shear Joint beam flexure Joint column flexure Panel zone shear Total

Model A15n

A20n

A25n

A15c

A20c

A25c

20.9 8.5 22.8 10.1 4.9 5.9 26.9 100.0

27.5 6.0 20.9 9.3 4.7 5.4 26.2 100.0

33.2 4.5 19.3 8.5 4.5 5.0 25.1 100.0

21.6 8.8 23.5 10.4 1.9 6.1 27.7 100.0

28.4 6.2 21.5 9.5 1.8 5.5 27.0 100.0

34.1 4.6 19.8 8.8 1.7 5.1 25.8 100.0

Model A15n

A20n

A25n

A15c

A20c

A25c

(c) Total displacements Item Simple displacement FEA displacement Simple/FEA

1.604 1.623 0.988

1.752 1.768 0.991

1.899 1.915 0.992

1.555 1.594 0.976

1.700 1.738 0.978

1.846 1.883 0.980

(d) Shear stresses in panel zone Item Simple shear stress FEA shear stress Simple/FEA

Model A15n

A20n

A25n

A15c

A20c

A25c

123.4 136.3 0.905

127.4 140.3 0.908

129.8 142.7 0.910

123.4 137.0 0.901

127.4 141.0 0.904

129.8 143.4 0.905

All DPFs in cm units, all displacement is cm units, and all shear stresses in MPa units.

This total is less than that produced by any of the mechanical models (see Table 4(a)), indicating that even centerline analysis is unconservative (too stiff) with respect to predicting drift. It is important to note, however, that the rigid end zone analysis of the same subassemblage would produce a total drift of 1.142 cm, which is only 63% of the displacement produced by the models that explicitly include beam–column joint deformation. While centerline analysis is somewhat unconservative in the example given, it is clearly an improvement over rigid end zone analysis. Hence, centerline analysis should be used whenever the more accurate modeling approaches are not available. For existing buildings that were analyzed using rigid end zones, it is clear that the lateral drift will be underestimated. This will have an effect on the serviceability of the structure, and may have an adverse effect on the strength as well. Strength would be influenced most significantly for structures which are sensitive to P-delta effects. The seriousness of such problems can only be assessed on a case-by-case basis.

9. Summary, conclusions, and recommendations In this paper, three simplified models for computing displacement participation factors (DPFs) in beam–column joint subassemblages were derived. The models presented were the Fictitious Joint (FJ) model, the Krawinkler Joint (KJ) model, and the Scissors Joint (SJ) model. The FJ model includes the effect of flexural deformations in the beam–column joint, but the KJ and SJ models do not. Based on the results obtained from the analysis presented herein, and recognizing that only twelve individual subassemblages were analyzed, the following conclusions and recommendations are made. 1. When compared to the results of detailed finite element analysis, the use of the formulas provided in Table 3 provide reasonably accurate estimates of the displacements in beam–column subassemblages. The best agreement appears to be obtained when the out-to-out joint dimensions are used. Use of actual versus simplified moments in the joint made little difference with regards to predicted deflection.

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F.A. Charney, R. Pathak / Journal of Constructional Steel Research 64 (2008) 87–100 Table 8 Summary of B-model DPFs from simplified subassemblage analysis using simplified moments and out-to-out dimensions (a) Displacement participation factors Deformation source Beam flexure Beam shear Column flexure Column shear Joint beam flexure Joint column flexure Panel zone shear Total

Model B15n 0.278 0.128 0.147 0.108 0.126 0.038 0.257 1.081

B20n

B25n

0.421 0.100 0.147 0.108 0.137 0.038 0.287 1.237

0.566 0.082 0.147 0.108 0.144 0.038 0.305 1.389

B15c 0.278 0.128 0.147 0.108 0.043 0.038 0.257 0.998

B20c 0.421 0.100 0.147 0.108 0.047 0.038 0.287 1.146

B25c 0.566 0.082 0.147 0.108 0.049 0.038 0.305 1.294

(b) DPFs as percent of total Deformation source Beam flexure Beam shear Column flexure Column shear Joint beam flexure Joint column flexure Panel zone shear Total

Model B15n

B20n

B25n

B15c

B20c

B25c

25.7 11.9 13.6 9.9 11.7 3.5 23.8 100.0

34.0 8.1 11.9 8.7 11.1 3.0 23.2 100.0

40.7 5.9 10.6 7.7 10.4 2.7 22.0 100.0

27.9 12.8 14.7 10.8 4.3 3.8 25.7 100.0

36.7 8.7 12.8 9.4 4.1 3.3 25.0 100.0

43.7 6.4 11.3 8.3 3.8 2.9 23.6 100.0

Model B15n

B20n

B25n

B15c

B20c

B25c

(c) Total displacements Item Simple displacement FEA displacement Simple/FEA

1.081 1.050 1.030

1.237 1.202 1.029

1.389 1.352 1.028

0.998 1.006 0.992

1.146 1.153 0.994

1.294 1.301 0.995

(d) Panel zone shear stresses Item Simple shear stress FEA shear stress Simple/FEA

Model B15n 79.9 85.8 0.931

B20n

B25n

84.4 90.3 0.935

87.1 93.0 0.937

B15c 79.9 87.0 0.918

B20c 84.4 91.4 0.923

B25c 87.1 94.3 0.924

All DPFs in cm units, all displacement is cm units, and all shear stresses in MPa units.

2. The use of center-to-center joint dimensions overestimates the average joint shear stress, and the use of centerline dimensions underestimates the average joint shear stress. However, errors in average shear stress were generally less than 10%. 3. Shear deformations in the beam, the column, and particularly in the panel zone of the beam–column joint are very significant, and should not be ignored. 4. Deflections due to flexural deformations in the beam–column joint are smaller than those due to shear deformations, but are not negligible, and should not be ignored. Such deformations were responsible for as much as 15% of the drift in the subassemblages without beam-flange continuity plates. Addition of such plates reduced the drift by as much as 7% in some subassemblages, and as little as 3% in others. 5. There is considerable uncertainty regarding the appropriate section property to use for computing flexural deformations in the beam–column joint. This is particularly true for beam flexure when beam-flange continuity plates are not provided.

The basic conclusion of this paper is that beam–column joint flexural deformations are significant, and should not be ignored. As mentioned in the beginning of this paper, such deformations are ignored in both the Krawinkler and Scissors models. In fact, these models typically include an added stiffness component (due to column flange bending) that was not included in the current analysis. Because the Krawinkler and Scissors models are most often used in practice, it is desirable to modify these models to account for flexural deformations in the joint. It is suggested in Part 2 of this paper that this may be accomplished by providing some flexural flexibility in the rigid links used in these models. It is also noted that while the detailed FEA produced total displacements that are very consistent with those obtained from the simplified methods, there was no way to determine, on a one-to-one basis, how the individual components of displacement compare. For example, it may be the case that the subassemblage models underestimate panel zone shear deformation while overestimating joint flexural deformation. In

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order to clarify this point, Part 2 of this paper extends the DPF approach to the detailed finite element analysis. This analysis shows that the simplified model consistently overestimates the displacement due to flexure in the joint, although the errors produced are generally small. References [1] Bertero VV, Popov EP, Krawinkler H. Beam–column subassemblies under repeated load. Journal of the Structural Division 1972;98(ST5): 1137–59. [2] Becker R. Panel zone effect on the strength and stiffness of steel rigid frames. Engineering Journal 1975;19–29. [3] Krawinkler H, Bertero VV, Popov EV. Shear behavior of steel frame joints. Journal of the Structural Division 1975;101(11):2317–36. [4] AISC. Tests and analysis of panel zone behavior in beam-to-column connections. American Institute of Steel Construction and the Structural Steel Educational Council; 1980. [5] Krawinkler H. Shear in beam–column joints in seismic design of frames. Engineering Journal 1978;15(3):82–91. [6] Charney FA, Johnson R. The effect of panel zone deformations on the drift of steel framed structures. In: Presented at the ASCE structure congress. 1986. [7] Krawinkler H, Mohassebs. Effects of panel zone deformation on seismic response. Journal of Constructional Steel Research 1987;8:233–50. [8] Charney FA. Sources of elastic deformations in laterally loaded steel frame and tube structures. In: Proceedings of the fourth world congress on tall buildings. 1990. p. 893–915. [9] Horvilleur JF, Charney FA. The effect of beam–column joint deformation on the lateral response of steel frame building structures. In: III Simposio Internacional Y VI Simposio Natcional de Estructueas de Acero. 1993. p. 269–303. [10] Kim K, Englehardt MD. Development of analytical models for earthquake analysis of steel frames. Austin (TX): PMFSEL 95-2, Phil M. Ferguson Structural Engineering Lab, University of Texas; 1995.

[11] Schneider SP, Amidi A. Seismic behavior of steel frames with deformable panel zones. Journal of Structural Engineering 1998;124(1):35–42. [12] Downs WM. Modeling and behavior of beam/column joint regions of steel frames. M.S. thesis. Blacksburg (VA): Department of Civil and Environmental Engineering, Virginia Tech; 2002. [13] Foutch DA, Yun S. Modeling of steel moment resisting frames for seismic loads. Journal of Constructional Steel Research 2002;58:529–64. [14] Kim K, Engelhardt MD. Monotonic and cyclic loading models for panel zones in steel moment frames. Journal of Constructional Steel Research 2002;56:605–35. [15] Gupta A, Krawinkler H. Relating seismic drift demands of SMRFs to element deformation demands. Engineering Journal 2002;39(2):100–8. [16] Charney FA, Marshall JA. Comparison of the Krawinkler and Scissors models for including beam–column joint deformations in the analysis of moment resisting frames. Engineering Journal 2006;43(1):31–48. [17] Charney FA, Pathak R. Sources of elastic deformations in steel frame and framed-tube structures: Part 2: Detailed subassemblage models. Journal of Constructional Steel Research 2008;64(1):101–17. [18] Charney FA. The use of displacement participation factors in the optimization of wind drift controlled buildings. In: Proceedings of the second conference on tall buildings in seismic regions. Council of Tall Buildings and Urban Habitat; 1991. p. 91–8. [19] Charney FA. Economy of steel frames through identification of structural behavior. In: Proceedings of the national steel construction conference. 1993. p. 12-1–13-33. [20] Cowper GR. The shear coefficient in Timoshenko’s beam theory. Journal of Applied Mechanics 1966;33:335–40. [21] Charney FA, Iyer H, Spears PW. Computation of major axis shear deformations in wide flange steel girders and columns. Journal of Constructional Steel Research 2005;61(11):1525–58. [22] Pathak R. Development of finite element analysis modeling mesh generation and analysis software for light wood frame houses. Master of Science thesis. Blacksburg (VA): Department of Civil and Environmental Engineering, Virginia Tech; 2004. [23] SAP2000. Integrated software for structural analysis and design, v7. 44. Berkeley: Computers and Structures, Inc.; 2000.