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Modelling of the panel zone in steel and composite moment frames
Engineering Structures 27 (2005) 129–144 www.elsevier.com/locate/engstruct
Modelling of the panel zone in steel and composite moment frames J.M. Castro, A.Y. Elghazouli∗, B.A. Izzuddin Department of Civil and Environmental Engineering, Imperial College London, UK Received 19 April 2004; received in revised form 29 September 2004; accepted 30 September 2004
Abstract This paper deals with the modelling of the panel zone region within beam-to-column connections in steel and composite moment-resisting frames. Existing analytical models for representing the panel zone response are first reviewed and their scope and limitations are discussed. A new approach, which is particularly suited for modelling steel and composite joints within frame analysis procedures, is then proposed and described. The method rationally accounts for the effect of different boundary conditions, as well as shear and flexural deformation modes, in evaluating the elastic and inelastic response. Validation of the proposed approach is carried out through comparisons against available experimental results in addition to more detailed continuum finite element analyses. The results demonstrate that the approach developed provides a more realistic representation of the behaviour in comparison with available models, especially in the case of composite connections. It is shown that, for composite joints, commonly used simple moment–distortion relationships may not be adequate. This is primarily due to the dependency of the behaviour on the internal force distribution at the joint. The study describes the implementation of the suggested approach within frame analysis procedures, and substantiates the important role played by the panel zone in the response of moment frames under lateral loading conditions. © 2004 Elsevier Ltd. All rights reserved. Keywords: Panel zone; Shear panel; Composite joints; Composite frames; Ductility
1. Introduction Under lateral loading, such as seismic conditions, beamto-column joints in moment-resisting frames are largely subjected to unbalanced moments which cause shear deformations in the panel zones of the columns. The behaviour of the panel zone plays a significant role in determining the overall stiffness and capacity of the frame. Additionally, in terms of seismic design, the panel zone can have a significant influence on the distribution of plasticity and energy dissipation within various structural components. Previous research investigations have indicated that the panel zone has a ductile and stable behaviour [1–6]. The concentration of some inelasticity in this region may also be employed to relieve the demand imposed on the ∗ Corresponding address: Department of Civil and Environmental Engineering, South Kensington Campus, Imperial College London, London SW7 2AZ, UK. Tel.: +44 20 75946021; fax: +44 20 75945934. E-mail address: [email protected] (A.Y. Elghazouli).
0141-0296/$ - see front matter © 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2004.09.008
beams [7]. Excessive inelastic deformations in the panel zone may however impair the global structural behaviour and introduce additional second-order effects. Therefore, the extent of plastic deformations in the panel needs to be adequately assessed and controlled [7]. Due to the significance of panel zone behaviour, its inclusion in analytical models of moment-resisting frames is essential for an adequate assessment of seismic response. A number of techniques have been suggested by several researchers [1,8,9]. More recent work has also addressed improved modelling representation for haunched connections [10]. Available models have been largely derived for steel joints for which the transferred beam-tocolumn moment is converted into a force-couple subjecting the panel zone to pure shear strain conditions. Whilst this approach may be suitable for a steel joint, the extension of its use to composite joints may not be appropriate. In the case of composite floors, the panel zone is usually subjected to more irregular stress conditions which depend on the location of the neutral axes in the connected beams.
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In this paper, existing methods for modelling panel zone behaviour are summarised and their relative merits are discussed. A new proposed approach, based on a realistic assessment of load distribution and boundary conditions within the panel zone, is described. Validation of the suggested approach is carried out by comparison against experimental and numerical results. The suggested approach is applicable to both steel and composite joints in momentresisting frames. 2. Panel zone behaviour When a steel joint is subjected to an unbalanced bending moment, a complex stress state develops in the panel region. This consists of normal stresses, mainly originating from the column, and shear stresses resulting from the moment transmitted from the beams. Experimental studies have shown that the panel behaviour in the elastic range is mainly governed by shear deformations [1,2]. After yielding, the shear resistance reduces significantly and the frame surrounding the panel zone, defined by the column flanges and the web stiffeners, provides additional resistance. On the basis of experimental observation, it is common practice to empirically assume that full yielding takes place at a level of distortion four times that causing first yield [1]. Typical panels are able to undergo considerable inelastic deformations before suffering from web buckling in shear. However, excessive distortions in the joint may cause weld fracture in beam flanges in the case of welded connections [11,12]. Additionally, undesirable column performance may occur due to formation of kinks in the column flanges. Under cyclic loading conditions, panel zones normally exhibit very stable hysteretic behaviour and significant hardening contributions [1,5,13]. Consequently, the panel zone can be relied upon as an effective component when energy dissipation is sought. The inelastic behaviour of the panel zone can generally contribute to relieving the demand on other structural elements. Such stable behaviour must however be considered with care since large panel deformations may impair the overall structural response as it may be coupled with significant second-order effects. Therefore the design should include appropriate measures to limit the extent of severe inelastic deformations in the panel [7]. 3. Existing analytical models As mentioned before, different approaches have been proposed to account for panel zone deformations in the analytical modelling of moment-resisting frames. Two main approaches have been suggested. The first is commonly referred to as the scissors model, as indicated in Fig. 1. In this case, a rotational spring is introduced between the beam and the column, representing the relative rotation between the two elements. Links in the vicinity of the joint are often
Fig. 1. The scissors model.
Fig. 2. The frame model.
employed in order to model the rigid extension of the beam and column. Another representation is offered by the frame model [1], shown in Fig. 2. This approach adopts a set of rigid links, and includes a diagonal translational spring as a modification of an earlier version which incorporated rotational springs at the intersections of the rigid links. The frame model can reproduce the relative rotation between the beams and the column as well as the relative vertical translation between the beams, although the latter effect usually has a relatively insignificant influence on the joint response. In the above-described models, appropriate relationships need to be assigned to the idealised springs. Expressions were derived by a number of researchers [1,2,14–16], which mainly differ in addressing the post-elastic range. All expressions are nevertheless deduced on the basis of a common assumption for the mechanism of beam-tocolumn moment transfer. The panel is typically assumed to have rigid boundaries and to behave under a pure shear stress state. This simplification allows the conversion of the bending moment into horizontal forces, which leads to a set of simple analytical expressions for the idealised springs. In subsequent sections, a brief review of existing representations for moment–distortion (M–γ ) relationships available for both the elastic and post-yield ranges is presented. These relationships can be directly adopted for use in both the scissors and frame models.
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in which the presence of the axial load in the column (P) is accounted for by employing the von Mises yield criterion (Py represents the axial capacity of the column and f y the yield stress of the steel), it is possible to define the unbalanced moment in the joint causing yielding of the panel (M y, pz ) as M y, pz =
A v db τy . 1−ρ
(3)
Irrespective of the minor differences described above, the available proposals for defining the elastic stiffness of the panel generally provide a good representation of steel panel zones. 3.2. Post-elastic range Whereas, for the elastic range, the behaviour has been expressed in largely similar terms by various researchers, some differences are evident when describing the postelastic stage. A number of proposals, with increasing refinement, have been suggested as described below. Fielding and Huang [2] proposed a bi-linear relationship (Fig. 3(a)) for the panel zone behaviour in which the postelastic stiffness (K p-el ) is defined (assuming Poisson’s ratio ν = 0.3) as K p-el =
Fig. 3. Bi-linear and tri-linear relationships.
3.1. Elastic range For the elastic behaviour of the panel zone, the following expression is commonly adopted [1,14]: G A v db M= γ 1−ρ
(1)
where G is the shear modulus of the material, Av is the shear area, and db is the steel beam height. The parameter ρ accounts for the beneficial effect of the shear force in the column (Vcol ), where ρ is defined as Vcol db /M. The main difference between the various proposals for the elastic stiffness is concerned with determining the shear area (Av ). For example, Krawinkler et al. [1,14] assumed Av = [dc − tc f ]tcw , whereas Fielding and Huang [2] proposed Av = dc tcw , in which tc f and tcw are the thicknesses of the column flange and web, respectively. It is worth noting that notable differences would only arise from the two assumptions when relatively deep columns are considered. On the basis of Eq. (1) and limiting the yielding shear stress (τ¯y ) to 2 fy P τy = √ 1 − (2) P 3 y
5.2Gbc tc3f M = γ db [1 − ρ]
(4)
where bc and tc f are the width and thickness of the column flange, respectively. No limits were considered for the postelastic range which is unrealistic since, at a certain stage, inelasticity takes place in the column flanges. On the basis of experimental and analytical results, Krawinkler et al. [1] proposed a tri-linear representation (Fig. 3(b)) in which the post-yielding stiffness (K p-el ) is defined as K p-el =
1.04Gbc tc2f M = . γ 1−ρ
(5)
Beyond an assumed inelastic distortion of four times the yield value, a strain-hardening stiffness corresponding to that of the steel material was suggested. More recently, further refinement to existing models was proposed [16]. A quadri-linear model including both bending and shear deformation modes was suggested. The contribution of column flanges to the panel resistance at the onset of yielding was considered. The model was also extended to account for cyclic conditions. Comparisons with tests showed that the proposed refinement provided more accurate results compared to those obtained from previous models, especially in the case of joints with relatively thick column flanges. 3.3. Composite joints The analytical models described above proved to be sufficiently accurate for steel joints when compared to
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Fig. 4. A modified version of the frame model.
results obtained from experimental tests. However, the applicability of these models to composite joints has not been adequately assessed and validated. Suggestions have been made [5,16] for an arbitrary modification of the panel height in order to account for the presence of the concrete slab in the joint. These empirical approaches have not however been appropriately examined. This is partly attributed to the limited availability of relevant experimental test data, but also largely due to the need for a more fundamental treatment of specific aspects related to composite joints. In subsequent sections, a new approach based on underlying behavioural mechanisms is suggested for modelling the panel zone region in composite joints. This is followed by comparisons against available experimental results as well as detailed numerical simulations.
be explained by the shear stress magnitude and distribution within the panel zone and not through a change of the panel dimensions. In the approach proposed herein, realistic stress distributions in the edges of the panel are considered, accounting for the variable location of the neutral axis in the composite beam. This methodology enables an assessment of the shear stress distribution through the panel depth which, in turn, realistically represents the spread of plasticity in that region. The procedure considers both shear and bending deformations in the elastic and post-elastic stages. The contribution of the column flanges to the extra resistance of the panel zone is also accounted for, with due consideration of the column depth and flange thickness. 4.1. Procedures and details
4. Proposed approach For a steel joint, it seems rational to adopt a moment–distortion relationship for the panel, through a force-couple, since the steel beam is predominantly in bending. This is notably different however from the case of a composite joint where the steel beam is also subjected to a significant axial force due to composite action. Consequently, the establishment of a relationship between the composite beam moment and the distortion of the panel becomes a more intricate task. Through an assumed increase in panel depth, models developed for steel joints can be empirically extended for composite frames. However, this simplified approach may be inadequate since it involves an unrealistic modification in the panel size in the composite case in comparison with a steel counterpart. Experimental observations also clearly indicate that the physical geometries of the panel zone are the same in steel and composite frames. The experimental increase in strength observed in composite joints [5] can therefore only
As noted before, the approach proposed in this paper is aimed primarily at implementation within computationally efficient frame analysis programs. As shown in Fig. 4, the joint is represented by a modified version of the ‘frame model’ described previously. The actual physical dimensions of the panel zone (db × dc ) are considered. In addition, an assemblage of links is included on top of the panel to model the column region in contact with the slab. Relative vertical translations, between diagonally opposite panel nodes, are ignored on the assumption that column rotations remain comparatively small. In deriving the expressions and procedures for the panel zone, the composite beam is assumed to be elastic until the panel yields. This is largely in agreement with observations from detailed analysis. It can also be shown that the stress distribution in the panel is not notably sensitive to this assumption. On this basis, the location of the neutral axis in the composite beam can be determined considering linearity of the normal stress distribution.
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Fig. 5. Analytical and numerical representations for an external joint.
The main aspect of the proposed approach is essentially to determine the spring properties for both the ‘panel zone’ and the ‘top panel’. These properties are derived analytically, in such a way that they are suitable for implementation in frame analysis programs. The procedure establishes an analogy between the analytical model consisting of the actual joint, on one hand, and the corresponding numerical model for frame analysis, on the other hand. Fig. 5(a) and (b) illustrate the ‘analytical’ and ‘numerical’ models, respectively. For simplicity, the description is given for the case of an external joint in which the composite beam, incorporating a profiled slab, is subjected to positive (i.e. sagging) moment. The application to the more general case of an internal joint is described later. This procedure evidently requires an assessment of the effective width of the slab in the vicinity of the joint. This is assumed to be limited to the contact width with the column flange width (bc ) for positive moment. In the case of a negative (i.e. hogging) moment, the slab is not considered, assuming that the reinforcement is not anchored/welded to the column. It is also assumed that the slab does not extend beyond the column face, although the procedure can be easily modified to account for these situations. With reference to Fig. 5, for a given moment (M) carried by the composite beam, a corresponding equivalent shear (Veq,num) is applied to the panel in the numerical model. If the distortion caused by this moment to the panel is known for the analytical model, the corresponding stiffness to use in the numerical model can be readily determined. Moreover, if the shear stress distribution through the panel depth is known, the difference between the load level corresponding to first and full yielding of the panel can be derived. In subsequent sections, the detailed procedure is described for both the elastic and post-elastic ranges. Elastic range As mentioned previously, for the purpose of assessing the panel response, the beam is assumed to be largely elastic up
to yielding of the panel zone. Accordingly, the neutral axis location of the beam can be estimated, together with the ratio (RNM ) between the axial force (Ns ) and bending moment (Ms ) carried by the steel beam, given by Ns As db = RNM = yG − (6) Ms Is 2 where As and Is are the cross-sectional area and second moment of area, respectively, of the steel beam, and yG represents the location of the centroid of the composite beam. It should also be noted that the moment developed in the slab (Mc ) is considered to have an insignificant influence on the behaviour and is hence ignored in this treatment. From the analytical model, the normal stress distribution σ (y) through the composite beam cross-section is given by σ (y) =
M Icomp
y
(7)
where M is the total moment applied to the joint and Icomp is the second moment of area of the composite beam crosssection. On the other hand, in the numerical model shown in Fig. 5(b), the same moment (M) can be represented as db tslab + ds + . (8) M = Ms + Ns 2 2 Moreover, the equivalent shear (Veq,num) applied to the panel zone in the numerical model is defined by Ms Ns − Vcol + (9) db 2 where Vcol = M/ h s is the actual shear in the column and h s is the storey height. On the basis of the above relationships, the equivalent shear (Veq,num) acting in the panel can be represented as a function of the total moment transferred to the joint, such as 1 2 + RNM db 1 Veq,num = M − . db 2 + RNM (db + tslab + 2ds ) hs (10) Veq,num =
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AB = ∆rel,bend
Fig. 6. The virtual system used to find panel deformations.
It is interesting to note that the term within the inner parentheses can be interpreted as a modification factor for the panel zone depth (db ). More appropriately, as discussed in detail later, it should be viewed as a reduction factor to the demand imposed on the panel. From the analytical model, with reference to Fig. 5(a) and knowing the normal stress distribution in the composite beam, it is possible to define both the shear force (Vi ) and bending moment (Mi ) distributions within the panel: For y = [yG ; yG − tb f ] y σ (y) dy (11) V1 (y) = Vcol + bb yG y h s − db − M1 (y) = Vcol V1 (y) dy. (12) 2 yG For y = [yG − tb f ; yG − db + tb f ] y σ (y) dy V2 (y) = V1 (yG − tb f ) + tbw M2 (y) = M1 (yG − tb f ) −
(13)
yG −tb f
y yG −tb f
V2 (y) dy.
For y = [yG − db + tb f ; yG − db ] y V3 (y) = V2 (yG − db + tb f ) + bb M3 (y) = M2 (yG − db + tb f ) −
yG −db +tb f y
yG −db +tb f
(14)
σ (y) dy
(15)
V3 (y) dy. (16)
In the expressions above, bb and tb f are the flange width and thickness of the beam respectively whereas tbw represents the thickness of the beam web. On the basis of the above internal force distribution in the panel, the relative horizontal translation of points A and B in Fig. 5a can be determined. To achieve this, the principle of virtual work (PVW) is applied using a virtual system represented in Fig. 6. By applying a couple of opposite unit forces in the virtual system, the internal virtual forces (V and M in Fig. 6) are readily obtained. The PVW allows the calculation of the relative drift of the panel accounting for shear and bending deformations: db /2−tb f V1 (y) AB V dy ∆rel,shear = G Av −db /2+tdbb/2 −db /2 f V2 (y) V3 (y) + V dy + V dy (17) G A G Av v −db /2+tbf db /2−tbf
db /2−tbf
M1 (y) dy E Ic db /2 −db /2+tb f M2 (y) + M dy E Ic db /2−tb f −db /2 M3 (y) M dy + E Ic −db /2+tb f M
(18)
where Av is the shear area of the column which is considered to be equal to [dc − tc f ]tcw , and Ic is the second moment of area of the column cross-section. On the basis of the relative deformation of the panel, the actual elastic stiffness of the panel zone can be determined for use in the numerical model, as follows: K el,num =
Veq,num AB AB |∆rel,shear | + |∆rel,bend |
.
(19)
Considering that the shear force required for yielding a panel under constant shear stress is given by Vy,panel = τ y Av
(20)
where τ y is the shear stress at yield (defined by Eq. (2) which accounts for the presence of axial force in the column), it is now important to define the relative distortion of the panel at which yielding occurs. This is a relatively complex issue since yielding of the panel does not take place at the same load level. The shear stress distribution in the analytical model is quadratic, whilst in the numerical model it is assumed to be constant throughout the panel zone. A ratio (RV ) can accordingly be defined as RV =
|maxV2 (y)| . Veq,num
(21)
Close examination of the above ratio indicates a value of about 1.15 for typical steel joints. This ratio can be employed in the numerical model to determine the relative drift of the panel at the onset of yielding, such that ∆ y,num =
Vy,panel/RV . K el,num
(22)
It is important to note at this stage that the actual shear stress distribution in the panel zone of a composite joint is usually largely uniform in comparison with a steel case, as discussed further in subsequent parts of this paper. For a steel joint, it is unrealistic to consider full yielding of the panel at the same instant. A value of unity for RV is therefore suggested for use with steel joints, implying an overestimation of the effective yield point, which is also in agreement with experimental test results [1]. After determining the elastic stiffness of the panel zone and its yield point, the next step is to assign these properties to the diagonal spring of the numerical model, as follows: K el,num cos2 α ∆ y,num ∆ y,spring = cos α K el,spring =
(23) (24)
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can be readily derived as ∆p−el,num = ∆ y,num +
f y db2 6EdC G
(26)
where dC G represents the distance from the centroid of the T-section to the external fibre of the column flange and f y is the reserve stress in the same fibre after shear yielding of the panel zone. Recalling the existing models available for steel joints where the yield drift of the panel (∆ y ) is defined as τ y db /G, the ratio between the second and first yield deformations of the panel (assuming Poisson’s ratio ν = 0.3) is given by Fig. 7. The cross-section definition for the post-elastic range.
where α is the angle of the spring, as indicated in Fig. 5(b). With respect to the ‘top panel’, the response is assumed to be predominantly in shear and within the elastic range, and hence its stiffness can be determined on the basis of a proper modification of Eq. (1). In the following section, the procedure for determining the spring properties for the post-elastic range of the panel zone is described. Post-elastic range Beyond yielding of the panel, the shear stiffness provided by the column web effectively drops to that corresponding to strain-hardening of the material. However, an additional source of stiffness is provided by the flanges and web stiffeners delimiting the panel zone. It is assumed that, beyond the yield point, the column web resistance is independent of the column flanges [2]. Consequently, a system of two components in parallel is considered, representing the strain-hardening stiffness of the panel, and two idealised T-sections, respectively. As indicated in Fig. 7, each T-section consists of one column flange and a portion of the web considered as 0.9tc f + 0.05dc . It should be noted that the effective part of the web, which is a function of both the flange thickness and column depth, is included since the shear stress distribution along the column web is not constant throughout the web depth. Assuming that the composite beam remains largely elastic, a strategy similar to that used for the elastic range can be adopted. Therefore, the post-elastic stiffness of the panel, for use in the numerical model, is determined as follows: Veq,num Veq,num 2IT -sec + (25) K p−el,num = µ AB AB Icol ∆rel,shear ∆rel,bend where µ is the strain-hardening parameter and IT -sec is the second moment of area of one T-section. The relative deformation of the panel corresponding to the formation of hinges in the flanges can be derived using a simple approach in which the column flanges of the idealised T-sections are represented by two vertical members fixed at both ends (similar to that adopted in Fig. 6). On the basis of the above, the relative drift of the panel zone in the post-elastic range
f y db ∆ p-el =1+ . ∆y 15.6dC G τ y
(27)
Table 1 The ratio between post-yield and yield deformation in panel zones Beam
Column
Column flange thickness (mm)
UB 305×127×37 UB 610×229×140 UB 914×305×253
UC 203×203×60 UC 305×305×283 UC 356×406×634
14.2 44.1 77.0
∆ p-el ∆y
3.89 2.56 2.07
Close examination of Eq. (27) indicates that this ratio decreases with the increase in member size. In Table 1, this ratio is determined for three different steel joints. It is evident this ratio reduces significantly for thick column flanges. The application of a unique ratio between the post-elastic and the yield drift of the panel can therefore considerably overestimate the response in cases of very thick flanges. However, for the normal range of joints, this drift ratio is between 3.5 to 4.0. As indicated in Table 1, a comparatively small value would need to be considered if relatively large column flanges are utilised. Although Eqs. (26) and (27) above can be readily incorporated within the proposed model, the post-elastic displacement (∆p−el,num) is assumed herein for simplicity to be three and a half times that at yield (∆ y,num). The properties of the diagonal spring in the numerical model in the post-elastic range can therefore be determined as K p−el,num cos2 α 3.5∆ y,num . ∆p−el,spring = cos α K p−el,spring =
(28) (29)
Beyond the second yield point, the only stiffness provided by the panel zone is considered to be that from strainhardening in shear, given as K sh,num = µ
Veq,num AB |∆rel,shear |
.
(30)
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Table 2 Models adopted for validation Model name
Joint type
Location
Node controlled
Validation type
Beam(s) span (m)
Storey height (m)
EXT_SL45 INT_SA2 INT_SB2 EXT_CFC EXT_CL45 EXT_CL25 INT_CL45 INT_CL25
Steel Steel Steel Composite Composite Composite Composite Composite
External Internal Internal External External External Internal Internal
Column Column Column Beam Beam Beam Column Column
Numerical Experimental Experimental Experimental Numerical Numerical Numerical Numerical
4.5 4.064 4.064 2.3 4.5 2.5 4.5 2.5
3.0 2.032 2.032 3.4 3.0 3.0 3.0 3.0
Table 3 Member sizes and details Model EXT_SL45 INT_SA2 INT_SB2 EXT_CFC EXT_CL45 EXT_CL25 INT_CL45 INT_CL25
Beams Section
bb
db
tb f
tbw
Columns Section
bc
dc
tc f
tcw
Slab ds (mm)
tslab (mm)
UB 457×191×82 10 B 15 14 B 22 W 18×35 UB 457×191×82 IPE 300 UB 457×191×82 IPE 300
191.3 101.6 127.9 152.0 191.3 150.0 191.3 150.0
460.0 254.0 348.5 450.0 460.0 300.0 460.0 300.0
16.0 6.8 8.3 10.8 16.0 10.7 16.0 10.7
9.9 5.8 6.1 7.6 9.9 7.1 9.9 7.1
UC 356×368×177 8 WF 24 8 WF 67 W 10×60 UC 356×368×177 HEB 260 UC 356×368×177 HEB 280
372.6 147.8 207.1 256.0 372.6 260.0 372.6 280.0
368.2 201.4 230.9 260.0 368.2 260.0 368.2 280.0
23.8 10.1 23.1 17.3 23.8 17.5 23.8 18.0
14.4 6.2 15.9 10.7 14.4 10.0 14.4 10.5
– – – 76 0 0 0 0
– – – 89 120 100 120 100
Therefore, the strain-hardening stiffness of the diagonal spring is determined from K sh,spring =
K sh,num . cos2 α
(31)
Although the above procedure is described for the case of an external joint subjected to positive moment, the process is similar for a negative moment situation. If no effective width is considered, the procedure reduces to that for an external bare steel joint. In the next section, specific considerations are described for the case of internal composite joints. 4.2. Application to internal joints The same procedure as described above can also be applied to internal joints. However, additional considerations are required, particularly in terms of the moment distribution at the joint. In the case of internal joints, it is necessary to assess the ratio between the positive and negative moments transmitted to the column. To achieve this, the neutral axis locations for both connected beams are evaluated with appropriate consideration of the effective width. The moment distribution between the two beams is then based on the ratio of the second moments of area of the beam cross-sections near the joint. This is a valid assumption for beams with similar dimensions and boundary conditions, and is shown to provide an appropriate assessment of the behaviour, as illustrated in subsequent sections. After determining the moment distribution at the joint, the same equivalence between the analytical and numerical
Fig. 8. Types of substructure used for validation.
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Table 4 Material properties of the models Model
EXT_SL45 INT_SA2 INT_SB2 EXT_CFC EXT_CL45 EXT_CL25 INT_CL45 INT_CL25
Steel E (kN/mm2 )
f y (N/mm2 )
µ (%)
Concrete E (kN/mm2 )
f c (N/mm2 )
210 208 205.5 219a 210 210 210 210
275 282.7a 324a 244a 275 275 275 275
1.0 2.185 2.685 1.0b 1.0 1.0 1.0 1.0
– – – 23 30 30 30 30
– – – 43.8 30.0 30.0 30.0 30.0
a Value based on column properties. b Assumed value.
Fig. 9. The EXT_SL45 model prepared in ANSYS (a) and in ADAPTIC (b).
model can be adopted as was done previously for external joints. From the total moment applied to the panel zone, the stress boundary conditions on the two sides of the joint can be assessed. In addition, the distribution of shear and bending within the panel depth can be evaluated. Accordingly, the deformation of the panel accounting for both shear and flexural modes can be determined. For a total moment imposed on the joint, the equivalent shear force for use in the numerical model is obtained as
Fig. 10. The shear stress distribution obtained in the proposed approach (a) and in ANSYS (b) for a bare steel joint.
Veq,num =
Ms,right Ms,left Ns,left + + db 2 db Ns,right − Vcol + 2
(32)
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Fig. 11. Global and local responses of model EXT_SL45.
where (Ms,left ; Ns,left ) and (Ms,right ; Ns,right ) are the bending moments and axial forces carried by the steel beams on both sides of the joint, respectively. From this, the stiffness of the spring representing the panel zone can then be readily evaluated, as in Eqs. (19) and (23) before. The relative drift corresponding to the yield point can also be determined according to Eqs. (22) and (24). It should be noted however that a value of unity is used for RV , as the stress distribution in the panel zone of an internal composite joint has more resemblance to that obtained for a steel joint than that of an external composite joint. Finally, the spring properties for the post-elastic range are obtained using Eqs. (25) to (31), as before. The full procedure described above, to determine the spring properties for both external and internal joints, can be directly applied within a spreadsheet or a mathematical programming package. In the following section, a number of comparisons are made with available experimental results as well as detailed numerical simulations in order to assess the validity and accuracy of the proposed approach. 5. Validation studies This section presents several validation studies in which the proposed approach, which is developed for the purpose
Fig. 12. Global and local responses of model INT_SA2 (Krawinkler A2 specimen [1]).
of frame analysis representations, is compared to existing experimental results. However, due to the limited availability of test data, validation is also carried out against the results from more detailed three-dimensional numerical models adopting shell and solid elements, using ANSYS [17]. The proposed panel zone models are implemented in conjunction with the ‘frame analysis’ elasto-plastic elements available within the nonlinear analysis program ADAPTIC [18]. To this end, distinction should be made at the onset between the numerical models of ADAPTIC on the one hand, which are 2D frame models incorporating the proposed panel zone models, and those of ANSYS on the other hand, which are detailed 3D simulations employed for validation purposes. The validation is carried out for external as well as internal substructures, as indicated in Fig. 8. These systems represent typical idealised arrangements within moment frames subjected to lateral loading conditions, where hinges are assumed to form at mid-length of the beams and columns. The analysis is carried out by controlling the vertical end displacement of the beam or the top horizontal displacement of the column. Table 2 summarises all the substructures used in the validation, Table 3 provides the
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Fig. 13. Global and local responses of model INT_SB2 (Krawinkler B2 specimen [1]).
Fig. 14. Global and local responses of model EXT_CFC (Lee and Lu EJ–FC specimen [5]).
geometry and sizes of members and Table 4 presents the material properties adopted for the models. Hereafter, the details of the numerical models are first presented, followed by a discussion of the results of validation studies for various steel and composite substructures.
the column flange. Elsewhere, the full width of the slab is accounted for, in accordance with evidence from a detailed assessment of this aspect of behaviour [20]. For the contact region, rigid plastic behaviour for concrete is assumed with due consideration of confinement effects. On the other hand, as noted before, detailed 3D continuum models are utilised for validation purposes using ANSYS. In this case, the beams and columns are modelled using shell elements (SHELL 43) whilst, for composite substructures, the slab is represented using solid elements (SOLID 65). Material nonlinearity is included by the adoption of a bi-linear model with strain-hardening for steel and a tri-axial model with smeared cracking for concrete. Full interaction between the steel beam and the slab is assumed for the numerical models of both ADAPTIC and ANSYS.
5.1. Numerical models As mentioned before, the proposed panel models are utilised within the structural analysis program ADAPTIC [18], which accounts for material and geometric nonlinearities. In the idealised frame substructures, the steel and the column members are modelled using 1D Eulerian cubic elasto-plastic elements which account for the spread of plasticity across section and along the length. For composite frames, the beam and slab are modelled with two lines of cubic elements interconnected by rigid links. This approach can represent accurately the behaviour of a composite beam [19]. Clearly, in the case of 2D frame analysis representations, a number of idealisations are needed with respect to the effective size of the slab as well as its interaction with other frame members. In this study, the effective slab width at the beam ends is considered to be the same as the width of
5.2. Steel frames Although existing analytical models can adequately represent the panel zone in bare steel frames, it is useful to demonstrate the general applicability of the proposed approach prior to dealing with composite cases. In this section, a comparative assessment is carried out for
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Fig. 16. Global and local responses of model EXT_CL45.
Fig. 15. The shear stress distribution obtained in the proposed approach (a) and in ANSYS (b) for an external composite joint.
EXT_SL45, which represents an external substructure, with full details given in Tables 2–4. A view of the continuum mesh prepared in ANSYS is depicted in Fig. 9(a), whilst the frame model used in ADAPTIC is shown in Fig. 9(b). As noted before, the shear stress distribution within the panel zone of a steel joint is highly non-uniform. This is illustrated in Fig. 10(a) and (b) which depict the theoretical elastic stress distribution from the proposed approach as well as the detailed ANSYS representation, respectively. The results clearly demonstrate the discrepancy between the peak and average stresses, and the need to account for this in defining the effective yield point of the panel. On the other hand, Fig. 11(a) and (b) depict the global as well as local panel response, respectively. The results illustrate the accuracy of the proposed frame-modelling approach in comparison with the detailed 3D representation. Fig. 11(b) also indicates the panel response obtained from the expressions suggested by Krawinkler et al. [1,14], described previously. The comparison shows that the
proposed approach and that suggested for steel joints are in close agreement, and provide a good prediction of the panel response in the case of bare steel frames. Additional comparisons are also made with the experimental and analytical results obtained by Krawinkler et al. [1] for two internal substructures (specimens A2 and B2, described in Tables 2–4). These components represent steel joints within top and lower parts of a tall building where different cross-section combinations are incorporated, noting that an axial load representing about 30% of the column cross-section axial capacity was applied. Figs. 12 and 13 illustrate the accuracy, at both local and global levels, provided by the proposed frame models of ADAPTIC (INT_SA2 and INT_SB2) in comparison with the experimental results. 5.3. Composite frames This section presents a number of validation studies focusing on the response of composite frame substructures. The assessment carried out for external joint models is first discussed, followed by frames involving internal joints. As noted before, the details of all models utilised are described in Tables 2–4.
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Fig. 17. Global and local responses of model EXT_CL25.
External joints An ADAPTIC frame model representing an external substructure (EXT_CFC) is first compared with the cyclic test undertaken by Lee and Lu [5] on a composite specimen (EJ–FC). An asymmetric tri-linear curve is used for the spring representing the panel zone and the results are compared with the response envelope from the cyclic test. For the positive moment case, the spring properties are derived as described previously whilst, for negative moment, the bare steel joint is considered due to the lack of reinforcement anchorage. The experimental and numerical results are compared in Fig. 14, in which good agreement is generally observed. However, some differences are noticed at large drift levels which may be attributed to cyclic hardening effects exhibited in the test. In Fig. 14(b), the panel response obtained from available expressions for steel joints [1] is also included. Evidently, these expressions do not provide an adequate prediction of the response of the composite joint, as illustrated by the significant discrepancy on the positive moment side. Two additional substructures (EXT_CL45 and EXT _CL25) are employed to further examine the validity of the proposed approach for external composite joints. For these two substructures, the frame analysis is validated against
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Fig. 18. (a) The change in RNM during panel distortion; (b) the effect of RNM on the panel demand.
detailed ANSYS models. Before presenting the comparative results, it is interesting to examine the theoretical elastic shear stress distribution in the panel, both analytically and from ANSYS. As depicted in Fig. 15, the shear stress is largely uniform as compared to the case of steel joints, as discussed in earlier sections of this paper. The global frame and local panel responses for both substructures are presented in Figs. 16 and 17 including the results from ADAPTIC and ANSYS. The results clearly indicate the good agreement obtained between the frame models and the detailed 3D representation. The global response in these two figures also includes a frame analysis incorporating a fully rigid joint in order to illustrate the key role played by the panel flexibility and capacity. In terms of the local panel response, Figs. 16(b) and 17(b) also indicate that the proposed model provides a better prediction for both the yield point and the ultimate capacity of the panel, in comparison with available expressions for steel joints. In the light of the above discussion, it is important to examine the axial force-to-bending moment ratio in the steel beam, represented by RNM , in relation to the shear distortion imposed on the panel zone. This is illustrated in Fig. 18 for the frame models of the two substructures EXT_CL25 and EXT_CL45. As shown in Fig. 18(a), it is evident that as the
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Fig. 19. Bending moment ratios (M+/M−) for INT_CL45 (a) and INT_CL25 models (b).
panel zone attains its yield, a sudden change in the variation of RNM with panel distortion occurs. From examining Eq. (10), it is clear that the demand on the panel, for a specific overall moment, is reduced for increasing values of RNM . This is also illustrated in Fig. 18(b) which depicts the change in panel demand (normalised by that of a steel beam) for different RNM ratios and for various composite beam sizes. Consequently, since EXT_CL25 is subjected to higher RNM ratios, relatively high bending moments need to be mobilised in the composite beam in order to develop full plasticity in the panel in comparison to the case of EXT_CL45. Consequently, it is to be expected that predictions achieved using conventional steel models [1] for EXT_CL45 appear to be more favourable than that for EXT_CL25. Internal joints The validity of the approach is finally assessed by considering two substructures involving internal joint configurations, namely INT_CL45 and INT_CL25. As in previous cases, ADAPTIC frame models, incorporating the proposed procedures for the panel zone, are compared to detailed 3D analysis carried out in ANSYS.
Fig. 20. Global and local responses of model INT_CL45.
From the results of the two internal substructures, it is firstly useful to assess the positive-to-negative moment distribution between the beams, in comparison to the initial assumption employed in deriving the joint properties. This moment ratio is depicted in Fig. 19 as extracted from the frame (ADAPTIC) and 3D (ANSYS) analyses. As expected, the results indicate that the relative moments are closely related to the ratio of cross-section inertias, estimated as 1.92 for INT_CL45 and 2.14 for INT_CL25. The global frame and local panel responses obtained from ADAPTIC and ANSYS, for the two internal substructures, are presented in Figs. 20 and 21, respectively. The global response obtained from the frame models with a rigid joint assumption is also shown on the figures to illustrate the importance of accounting for the panel zone stiffness and strength. The two figures illustrate very good agreement for the results of ADAPTIC, incorporating the proposed panel model, in comparison with the detailed 3D analysis, at both local and global levels. In contrast, the local response obtained from available expressions for steel joints [1] significantly underestimates the response of the panel, as indicated by the local response shown in Figs. 20(b) and 21(b). In this respect, it should be noted that the use of conventional steel models, in conjunction with a modified panel depth (db ) reflecting the composite beam depth, would
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beams. This approach enables an accurate assessment of the shear stress distribution through the panel depth which allows a realistic representation of the spread of plasticity. The contribution of the column flanges to the extra resistance of the panel zone is also accounted for. The procedure considers both shear and bending deformations, and addresses the elastic and inelastic stages. Validation of the proposed approach is carried out by comparison against available experimental results coupled with detailed numerical simulations. The comparisons illustrate the accuracy and reliability of the approach developed, and its general applicability to both steel and composite frames as well as external and internal joints. It is shown that whilst conventional moment–distortion expressions may be adequate for steel joints, the basis of such relationships is unrealistic for composite frames. The panel zone response within a composite frame is significantly affected by several geometric and loading parameters, including the influence of beam-to-slab interaction on the stress distribution and distortional demand imposed on the panel. The proposed approach, which is suitable for implementation within frame analysis techniques, captures the influence of these key parameters and provides a rational procedure that accounts for the underlying behavioural mechanisms. References
Fig. 21. Global and local responses of model INT_CL25.
also grossly misestimate the behaviour. Moreover, from comparing Figs. 16(b) to 20(b), it is evident that for the same configuration, the actual capacities are markedly different in external and internal joints where both the force distributions and the confinement levels in the vicinity of the joint are different. This provides further illustration of the inadequacy of empirical approaches solely based on geometric and material properties of the panel. The comparisons described in this section demonstrate the suitability of the proposed approach for modelling both steel and composite joints. For steel frames, the method provides equally accurate predictions to those obtained from conventional approaches. On the other hand, for composite frames, the new approach offers a more rational and accurate assessment of the behaviour due to its ability to realistically account for the actual boundary conditions and load distributions at the panel zone. 6. Conclusion A new approach for representing the panel zone component in steel and composite moment-resisting frames is proposed in this paper. Realistic stress distributions in the edges of the panel are considered, which incorporate the variable location of the neutral axis in the case of composite
[1] Krawinkler H, Bertero VV, Popov EP. Inelastic behaviour of steel beam-to-column subassemblages. Report No. EERC 71-07, University of California Berkeley, Berkeley, California, 1971. [2] Fielding DJ, Huang JS. Shear in steel beam-to-column connections. Welding Journal 1971;50(7):S313–26 [research supplement]. [3] Becker R. Panel zone effect on strength and stiffness of steel rigid frames. Engineering Journal, AISC 1975;12(1):19–29. [4] Lee SJ. Seismic behaviour of steel building structures with composite slabs. Lehigh University, Bethlehem, PA, 1987. [5] Lee SJ, Lu LW. Cyclic tests of full-scale composite joint subassemblages. Journal of Structural Engineering, ASCE 1989; 115(8):1977–98. [6] Dubina D, Ciutina A, Stratan A. Cyclic tests of double-sided beam-tocolumn joints. Journal of Structural Engineering, ASCE 2001;127(2): 129–36. [7] FEMA. FEMA 350—Recommended seismic design criteria for new steel moment-resisting frames. Washington (DC): Federal Emergency Management Agency; 2000. [8] Lui EM, Chen WF. Frame analysis with panel zone deformation. International Journal of Solids and Structures 1986;22(12):1599–627. [9] Foutch DA, Yun SY. Modeling of steel moment frames for seismic loads. Journal of Constructional Steel Research 2002;58(5–8):529–64. [10] Lee CH, Uang CM. Analytical modeling of dual panel zone in haunch repaired steel MRF’s. Journal of Structural Engineering, ASCE 1997; 123(1):20–9. [11] Krawinkler H, Popov EP. Seismic behavior of moment connections and joints. Journal of the Structural Division, ASCE 1982;108(2): 373–91. [12] El-Tawil S, Vidarsson E, Mikesell T, Kunnath SK. Inelastic behavior and design of steel panel zones. Journal of Structural Engineering, ASCE 1999;125(2):183–93. [13] Bertero VV, Krawinkler H, Popov EP. Further studies on seismic behaviour of steel beam–column subassemblages. Report No. EERC
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73-27, University of California Berkeley, Berkeley, California, 1973. [14] Krawinkler H, Bertero VV, Popov EP. Shear behavior of steel frame joints. Journal of the Structural Division, ASCE 1975;101(11): 2317–36. [15] Schneider SP, Amidi A. Seismic behavior of steel frames with deformable panel zones. Journal of Structural Engineering, ASCE 1998;124(1):35–42. [16] Kim KD, Engelhardt MD. Monotonic and cyclic loading models for panel zones in steel moment frames. Journal of Constructional Steel Research 2002;58(5–8):605–35.
[17] SAS IP. ANSYS help system release 7.0. Canonsburg (PA): ANSYS, Inc.; 2002. [18] Izzuddin BA. Nonlinear dynamic analysis of framed structures. Ph.D. thesis, Imperial College of Science, Technology and Medicine, University of London, UK, 1991. [19] Elghazouli AY, Izzuddin BA. Analytical assessment of the structural fire response of composite floors. Fire Safety Journal 2001 36:769–93. [20] Castro JM. Seismic behaviour of composite structures. Ph.D. thesis, Imperial College London, University of London, 2005 [in preparation].
Modelling of the panel zone in steel and composite moment frames J.M. Castro, A.Y. Elghazouli∗, B.A. Izzuddin Department of Civil and Environmental Engineering, Imperial College London, UK Received 19 April 2004; received in revised form 29 September 2004; accepted 30 September 2004
Abstract This paper deals with the modelling of the panel zone region within beam-to-column connections in steel and composite moment-resisting frames. Existing analytical models for representing the panel zone response are first reviewed and their scope and limitations are discussed. A new approach, which is particularly suited for modelling steel and composite joints within frame analysis procedures, is then proposed and described. The method rationally accounts for the effect of different boundary conditions, as well as shear and flexural deformation modes, in evaluating the elastic and inelastic response. Validation of the proposed approach is carried out through comparisons against available experimental results in addition to more detailed continuum finite element analyses. The results demonstrate that the approach developed provides a more realistic representation of the behaviour in comparison with available models, especially in the case of composite connections. It is shown that, for composite joints, commonly used simple moment–distortion relationships may not be adequate. This is primarily due to the dependency of the behaviour on the internal force distribution at the joint. The study describes the implementation of the suggested approach within frame analysis procedures, and substantiates the important role played by the panel zone in the response of moment frames under lateral loading conditions. © 2004 Elsevier Ltd. All rights reserved. Keywords: Panel zone; Shear panel; Composite joints; Composite frames; Ductility
1. Introduction Under lateral loading, such as seismic conditions, beamto-column joints in moment-resisting frames are largely subjected to unbalanced moments which cause shear deformations in the panel zones of the columns. The behaviour of the panel zone plays a significant role in determining the overall stiffness and capacity of the frame. Additionally, in terms of seismic design, the panel zone can have a significant influence on the distribution of plasticity and energy dissipation within various structural components. Previous research investigations have indicated that the panel zone has a ductile and stable behaviour [1–6]. The concentration of some inelasticity in this region may also be employed to relieve the demand imposed on the ∗ Corresponding address: Department of Civil and Environmental Engineering, South Kensington Campus, Imperial College London, London SW7 2AZ, UK. Tel.: +44 20 75946021; fax: +44 20 75945934. E-mail address: [email protected] (A.Y. Elghazouli).
0141-0296/$ - see front matter © 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2004.09.008
beams [7]. Excessive inelastic deformations in the panel zone may however impair the global structural behaviour and introduce additional second-order effects. Therefore, the extent of plastic deformations in the panel needs to be adequately assessed and controlled [7]. Due to the significance of panel zone behaviour, its inclusion in analytical models of moment-resisting frames is essential for an adequate assessment of seismic response. A number of techniques have been suggested by several researchers [1,8,9]. More recent work has also addressed improved modelling representation for haunched connections [10]. Available models have been largely derived for steel joints for which the transferred beam-tocolumn moment is converted into a force-couple subjecting the panel zone to pure shear strain conditions. Whilst this approach may be suitable for a steel joint, the extension of its use to composite joints may not be appropriate. In the case of composite floors, the panel zone is usually subjected to more irregular stress conditions which depend on the location of the neutral axes in the connected beams.
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In this paper, existing methods for modelling panel zone behaviour are summarised and their relative merits are discussed. A new proposed approach, based on a realistic assessment of load distribution and boundary conditions within the panel zone, is described. Validation of the suggested approach is carried out by comparison against experimental and numerical results. The suggested approach is applicable to both steel and composite joints in momentresisting frames. 2. Panel zone behaviour When a steel joint is subjected to an unbalanced bending moment, a complex stress state develops in the panel region. This consists of normal stresses, mainly originating from the column, and shear stresses resulting from the moment transmitted from the beams. Experimental studies have shown that the panel behaviour in the elastic range is mainly governed by shear deformations [1,2]. After yielding, the shear resistance reduces significantly and the frame surrounding the panel zone, defined by the column flanges and the web stiffeners, provides additional resistance. On the basis of experimental observation, it is common practice to empirically assume that full yielding takes place at a level of distortion four times that causing first yield [1]. Typical panels are able to undergo considerable inelastic deformations before suffering from web buckling in shear. However, excessive distortions in the joint may cause weld fracture in beam flanges in the case of welded connections [11,12]. Additionally, undesirable column performance may occur due to formation of kinks in the column flanges. Under cyclic loading conditions, panel zones normally exhibit very stable hysteretic behaviour and significant hardening contributions [1,5,13]. Consequently, the panel zone can be relied upon as an effective component when energy dissipation is sought. The inelastic behaviour of the panel zone can generally contribute to relieving the demand on other structural elements. Such stable behaviour must however be considered with care since large panel deformations may impair the overall structural response as it may be coupled with significant second-order effects. Therefore the design should include appropriate measures to limit the extent of severe inelastic deformations in the panel [7]. 3. Existing analytical models As mentioned before, different approaches have been proposed to account for panel zone deformations in the analytical modelling of moment-resisting frames. Two main approaches have been suggested. The first is commonly referred to as the scissors model, as indicated in Fig. 1. In this case, a rotational spring is introduced between the beam and the column, representing the relative rotation between the two elements. Links in the vicinity of the joint are often
Fig. 1. The scissors model.
Fig. 2. The frame model.
employed in order to model the rigid extension of the beam and column. Another representation is offered by the frame model [1], shown in Fig. 2. This approach adopts a set of rigid links, and includes a diagonal translational spring as a modification of an earlier version which incorporated rotational springs at the intersections of the rigid links. The frame model can reproduce the relative rotation between the beams and the column as well as the relative vertical translation between the beams, although the latter effect usually has a relatively insignificant influence on the joint response. In the above-described models, appropriate relationships need to be assigned to the idealised springs. Expressions were derived by a number of researchers [1,2,14–16], which mainly differ in addressing the post-elastic range. All expressions are nevertheless deduced on the basis of a common assumption for the mechanism of beam-tocolumn moment transfer. The panel is typically assumed to have rigid boundaries and to behave under a pure shear stress state. This simplification allows the conversion of the bending moment into horizontal forces, which leads to a set of simple analytical expressions for the idealised springs. In subsequent sections, a brief review of existing representations for moment–distortion (M–γ ) relationships available for both the elastic and post-yield ranges is presented. These relationships can be directly adopted for use in both the scissors and frame models.
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in which the presence of the axial load in the column (P) is accounted for by employing the von Mises yield criterion (Py represents the axial capacity of the column and f y the yield stress of the steel), it is possible to define the unbalanced moment in the joint causing yielding of the panel (M y, pz ) as M y, pz =
A v db τy . 1−ρ
(3)
Irrespective of the minor differences described above, the available proposals for defining the elastic stiffness of the panel generally provide a good representation of steel panel zones. 3.2. Post-elastic range Whereas, for the elastic range, the behaviour has been expressed in largely similar terms by various researchers, some differences are evident when describing the postelastic stage. A number of proposals, with increasing refinement, have been suggested as described below. Fielding and Huang [2] proposed a bi-linear relationship (Fig. 3(a)) for the panel zone behaviour in which the postelastic stiffness (K p-el ) is defined (assuming Poisson’s ratio ν = 0.3) as K p-el =
Fig. 3. Bi-linear and tri-linear relationships.
3.1. Elastic range For the elastic behaviour of the panel zone, the following expression is commonly adopted [1,14]: G A v db M= γ 1−ρ
(1)
where G is the shear modulus of the material, Av is the shear area, and db is the steel beam height. The parameter ρ accounts for the beneficial effect of the shear force in the column (Vcol ), where ρ is defined as Vcol db /M. The main difference between the various proposals for the elastic stiffness is concerned with determining the shear area (Av ). For example, Krawinkler et al. [1,14] assumed Av = [dc − tc f ]tcw , whereas Fielding and Huang [2] proposed Av = dc tcw , in which tc f and tcw are the thicknesses of the column flange and web, respectively. It is worth noting that notable differences would only arise from the two assumptions when relatively deep columns are considered. On the basis of Eq. (1) and limiting the yielding shear stress (τ¯y ) to 2 fy P τy = √ 1 − (2) P 3 y
5.2Gbc tc3f M = γ db [1 − ρ]
(4)
where bc and tc f are the width and thickness of the column flange, respectively. No limits were considered for the postelastic range which is unrealistic since, at a certain stage, inelasticity takes place in the column flanges. On the basis of experimental and analytical results, Krawinkler et al. [1] proposed a tri-linear representation (Fig. 3(b)) in which the post-yielding stiffness (K p-el ) is defined as K p-el =
1.04Gbc tc2f M = . γ 1−ρ
(5)
Beyond an assumed inelastic distortion of four times the yield value, a strain-hardening stiffness corresponding to that of the steel material was suggested. More recently, further refinement to existing models was proposed [16]. A quadri-linear model including both bending and shear deformation modes was suggested. The contribution of column flanges to the panel resistance at the onset of yielding was considered. The model was also extended to account for cyclic conditions. Comparisons with tests showed that the proposed refinement provided more accurate results compared to those obtained from previous models, especially in the case of joints with relatively thick column flanges. 3.3. Composite joints The analytical models described above proved to be sufficiently accurate for steel joints when compared to
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Fig. 4. A modified version of the frame model.
results obtained from experimental tests. However, the applicability of these models to composite joints has not been adequately assessed and validated. Suggestions have been made [5,16] for an arbitrary modification of the panel height in order to account for the presence of the concrete slab in the joint. These empirical approaches have not however been appropriately examined. This is partly attributed to the limited availability of relevant experimental test data, but also largely due to the need for a more fundamental treatment of specific aspects related to composite joints. In subsequent sections, a new approach based on underlying behavioural mechanisms is suggested for modelling the panel zone region in composite joints. This is followed by comparisons against available experimental results as well as detailed numerical simulations.
be explained by the shear stress magnitude and distribution within the panel zone and not through a change of the panel dimensions. In the approach proposed herein, realistic stress distributions in the edges of the panel are considered, accounting for the variable location of the neutral axis in the composite beam. This methodology enables an assessment of the shear stress distribution through the panel depth which, in turn, realistically represents the spread of plasticity in that region. The procedure considers both shear and bending deformations in the elastic and post-elastic stages. The contribution of the column flanges to the extra resistance of the panel zone is also accounted for, with due consideration of the column depth and flange thickness. 4.1. Procedures and details
4. Proposed approach For a steel joint, it seems rational to adopt a moment–distortion relationship for the panel, through a force-couple, since the steel beam is predominantly in bending. This is notably different however from the case of a composite joint where the steel beam is also subjected to a significant axial force due to composite action. Consequently, the establishment of a relationship between the composite beam moment and the distortion of the panel becomes a more intricate task. Through an assumed increase in panel depth, models developed for steel joints can be empirically extended for composite frames. However, this simplified approach may be inadequate since it involves an unrealistic modification in the panel size in the composite case in comparison with a steel counterpart. Experimental observations also clearly indicate that the physical geometries of the panel zone are the same in steel and composite frames. The experimental increase in strength observed in composite joints [5] can therefore only
As noted before, the approach proposed in this paper is aimed primarily at implementation within computationally efficient frame analysis programs. As shown in Fig. 4, the joint is represented by a modified version of the ‘frame model’ described previously. The actual physical dimensions of the panel zone (db × dc ) are considered. In addition, an assemblage of links is included on top of the panel to model the column region in contact with the slab. Relative vertical translations, between diagonally opposite panel nodes, are ignored on the assumption that column rotations remain comparatively small. In deriving the expressions and procedures for the panel zone, the composite beam is assumed to be elastic until the panel yields. This is largely in agreement with observations from detailed analysis. It can also be shown that the stress distribution in the panel is not notably sensitive to this assumption. On this basis, the location of the neutral axis in the composite beam can be determined considering linearity of the normal stress distribution.
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Fig. 5. Analytical and numerical representations for an external joint.
The main aspect of the proposed approach is essentially to determine the spring properties for both the ‘panel zone’ and the ‘top panel’. These properties are derived analytically, in such a way that they are suitable for implementation in frame analysis programs. The procedure establishes an analogy between the analytical model consisting of the actual joint, on one hand, and the corresponding numerical model for frame analysis, on the other hand. Fig. 5(a) and (b) illustrate the ‘analytical’ and ‘numerical’ models, respectively. For simplicity, the description is given for the case of an external joint in which the composite beam, incorporating a profiled slab, is subjected to positive (i.e. sagging) moment. The application to the more general case of an internal joint is described later. This procedure evidently requires an assessment of the effective width of the slab in the vicinity of the joint. This is assumed to be limited to the contact width with the column flange width (bc ) for positive moment. In the case of a negative (i.e. hogging) moment, the slab is not considered, assuming that the reinforcement is not anchored/welded to the column. It is also assumed that the slab does not extend beyond the column face, although the procedure can be easily modified to account for these situations. With reference to Fig. 5, for a given moment (M) carried by the composite beam, a corresponding equivalent shear (Veq,num) is applied to the panel in the numerical model. If the distortion caused by this moment to the panel is known for the analytical model, the corresponding stiffness to use in the numerical model can be readily determined. Moreover, if the shear stress distribution through the panel depth is known, the difference between the load level corresponding to first and full yielding of the panel can be derived. In subsequent sections, the detailed procedure is described for both the elastic and post-elastic ranges. Elastic range As mentioned previously, for the purpose of assessing the panel response, the beam is assumed to be largely elastic up
to yielding of the panel zone. Accordingly, the neutral axis location of the beam can be estimated, together with the ratio (RNM ) between the axial force (Ns ) and bending moment (Ms ) carried by the steel beam, given by Ns As db = RNM = yG − (6) Ms Is 2 where As and Is are the cross-sectional area and second moment of area, respectively, of the steel beam, and yG represents the location of the centroid of the composite beam. It should also be noted that the moment developed in the slab (Mc ) is considered to have an insignificant influence on the behaviour and is hence ignored in this treatment. From the analytical model, the normal stress distribution σ (y) through the composite beam cross-section is given by σ (y) =
M Icomp
y
(7)
where M is the total moment applied to the joint and Icomp is the second moment of area of the composite beam crosssection. On the other hand, in the numerical model shown in Fig. 5(b), the same moment (M) can be represented as db tslab + ds + . (8) M = Ms + Ns 2 2 Moreover, the equivalent shear (Veq,num) applied to the panel zone in the numerical model is defined by Ms Ns − Vcol + (9) db 2 where Vcol = M/ h s is the actual shear in the column and h s is the storey height. On the basis of the above relationships, the equivalent shear (Veq,num) acting in the panel can be represented as a function of the total moment transferred to the joint, such as 1 2 + RNM db 1 Veq,num = M − . db 2 + RNM (db + tslab + 2ds ) hs (10) Veq,num =
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AB = ∆rel,bend
Fig. 6. The virtual system used to find panel deformations.
It is interesting to note that the term within the inner parentheses can be interpreted as a modification factor for the panel zone depth (db ). More appropriately, as discussed in detail later, it should be viewed as a reduction factor to the demand imposed on the panel. From the analytical model, with reference to Fig. 5(a) and knowing the normal stress distribution in the composite beam, it is possible to define both the shear force (Vi ) and bending moment (Mi ) distributions within the panel: For y = [yG ; yG − tb f ] y σ (y) dy (11) V1 (y) = Vcol + bb yG y h s − db − M1 (y) = Vcol V1 (y) dy. (12) 2 yG For y = [yG − tb f ; yG − db + tb f ] y σ (y) dy V2 (y) = V1 (yG − tb f ) + tbw M2 (y) = M1 (yG − tb f ) −
(13)
yG −tb f
y yG −tb f
V2 (y) dy.
For y = [yG − db + tb f ; yG − db ] y V3 (y) = V2 (yG − db + tb f ) + bb M3 (y) = M2 (yG − db + tb f ) −
yG −db +tb f y
yG −db +tb f
(14)
σ (y) dy
(15)
V3 (y) dy. (16)
In the expressions above, bb and tb f are the flange width and thickness of the beam respectively whereas tbw represents the thickness of the beam web. On the basis of the above internal force distribution in the panel, the relative horizontal translation of points A and B in Fig. 5a can be determined. To achieve this, the principle of virtual work (PVW) is applied using a virtual system represented in Fig. 6. By applying a couple of opposite unit forces in the virtual system, the internal virtual forces (V and M in Fig. 6) are readily obtained. The PVW allows the calculation of the relative drift of the panel accounting for shear and bending deformations: db /2−tb f V1 (y) AB V dy ∆rel,shear = G Av −db /2+tdbb/2 −db /2 f V2 (y) V3 (y) + V dy + V dy (17) G A G Av v −db /2+tbf db /2−tbf
db /2−tbf
M1 (y) dy E Ic db /2 −db /2+tb f M2 (y) + M dy E Ic db /2−tb f −db /2 M3 (y) M dy + E Ic −db /2+tb f M
(18)
where Av is the shear area of the column which is considered to be equal to [dc − tc f ]tcw , and Ic is the second moment of area of the column cross-section. On the basis of the relative deformation of the panel, the actual elastic stiffness of the panel zone can be determined for use in the numerical model, as follows: K el,num =
Veq,num AB AB |∆rel,shear | + |∆rel,bend |
.
(19)
Considering that the shear force required for yielding a panel under constant shear stress is given by Vy,panel = τ y Av
(20)
where τ y is the shear stress at yield (defined by Eq. (2) which accounts for the presence of axial force in the column), it is now important to define the relative distortion of the panel at which yielding occurs. This is a relatively complex issue since yielding of the panel does not take place at the same load level. The shear stress distribution in the analytical model is quadratic, whilst in the numerical model it is assumed to be constant throughout the panel zone. A ratio (RV ) can accordingly be defined as RV =
|maxV2 (y)| . Veq,num
(21)
Close examination of the above ratio indicates a value of about 1.15 for typical steel joints. This ratio can be employed in the numerical model to determine the relative drift of the panel at the onset of yielding, such that ∆ y,num =
Vy,panel/RV . K el,num
(22)
It is important to note at this stage that the actual shear stress distribution in the panel zone of a composite joint is usually largely uniform in comparison with a steel case, as discussed further in subsequent parts of this paper. For a steel joint, it is unrealistic to consider full yielding of the panel at the same instant. A value of unity for RV is therefore suggested for use with steel joints, implying an overestimation of the effective yield point, which is also in agreement with experimental test results [1]. After determining the elastic stiffness of the panel zone and its yield point, the next step is to assign these properties to the diagonal spring of the numerical model, as follows: K el,num cos2 α ∆ y,num ∆ y,spring = cos α K el,spring =
(23) (24)
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can be readily derived as ∆p−el,num = ∆ y,num +
f y db2 6EdC G
(26)
where dC G represents the distance from the centroid of the T-section to the external fibre of the column flange and f y is the reserve stress in the same fibre after shear yielding of the panel zone. Recalling the existing models available for steel joints where the yield drift of the panel (∆ y ) is defined as τ y db /G, the ratio between the second and first yield deformations of the panel (assuming Poisson’s ratio ν = 0.3) is given by Fig. 7. The cross-section definition for the post-elastic range.
where α is the angle of the spring, as indicated in Fig. 5(b). With respect to the ‘top panel’, the response is assumed to be predominantly in shear and within the elastic range, and hence its stiffness can be determined on the basis of a proper modification of Eq. (1). In the following section, the procedure for determining the spring properties for the post-elastic range of the panel zone is described. Post-elastic range Beyond yielding of the panel, the shear stiffness provided by the column web effectively drops to that corresponding to strain-hardening of the material. However, an additional source of stiffness is provided by the flanges and web stiffeners delimiting the panel zone. It is assumed that, beyond the yield point, the column web resistance is independent of the column flanges [2]. Consequently, a system of two components in parallel is considered, representing the strain-hardening stiffness of the panel, and two idealised T-sections, respectively. As indicated in Fig. 7, each T-section consists of one column flange and a portion of the web considered as 0.9tc f + 0.05dc . It should be noted that the effective part of the web, which is a function of both the flange thickness and column depth, is included since the shear stress distribution along the column web is not constant throughout the web depth. Assuming that the composite beam remains largely elastic, a strategy similar to that used for the elastic range can be adopted. Therefore, the post-elastic stiffness of the panel, for use in the numerical model, is determined as follows: Veq,num Veq,num 2IT -sec + (25) K p−el,num = µ AB AB Icol ∆rel,shear ∆rel,bend where µ is the strain-hardening parameter and IT -sec is the second moment of area of one T-section. The relative deformation of the panel corresponding to the formation of hinges in the flanges can be derived using a simple approach in which the column flanges of the idealised T-sections are represented by two vertical members fixed at both ends (similar to that adopted in Fig. 6). On the basis of the above, the relative drift of the panel zone in the post-elastic range
f y db ∆ p-el =1+ . ∆y 15.6dC G τ y
(27)
Table 1 The ratio between post-yield and yield deformation in panel zones Beam
Column
Column flange thickness (mm)
UB 305×127×37 UB 610×229×140 UB 914×305×253
UC 203×203×60 UC 305×305×283 UC 356×406×634
14.2 44.1 77.0
∆ p-el ∆y
3.89 2.56 2.07
Close examination of Eq. (27) indicates that this ratio decreases with the increase in member size. In Table 1, this ratio is determined for three different steel joints. It is evident this ratio reduces significantly for thick column flanges. The application of a unique ratio between the post-elastic and the yield drift of the panel can therefore considerably overestimate the response in cases of very thick flanges. However, for the normal range of joints, this drift ratio is between 3.5 to 4.0. As indicated in Table 1, a comparatively small value would need to be considered if relatively large column flanges are utilised. Although Eqs. (26) and (27) above can be readily incorporated within the proposed model, the post-elastic displacement (∆p−el,num) is assumed herein for simplicity to be three and a half times that at yield (∆ y,num). The properties of the diagonal spring in the numerical model in the post-elastic range can therefore be determined as K p−el,num cos2 α 3.5∆ y,num . ∆p−el,spring = cos α K p−el,spring =
(28) (29)
Beyond the second yield point, the only stiffness provided by the panel zone is considered to be that from strainhardening in shear, given as K sh,num = µ
Veq,num AB |∆rel,shear |
.
(30)
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Table 2 Models adopted for validation Model name
Joint type
Location
Node controlled
Validation type
Beam(s) span (m)
Storey height (m)
EXT_SL45 INT_SA2 INT_SB2 EXT_CFC EXT_CL45 EXT_CL25 INT_CL45 INT_CL25
Steel Steel Steel Composite Composite Composite Composite Composite
External Internal Internal External External External Internal Internal
Column Column Column Beam Beam Beam Column Column
Numerical Experimental Experimental Experimental Numerical Numerical Numerical Numerical
4.5 4.064 4.064 2.3 4.5 2.5 4.5 2.5
3.0 2.032 2.032 3.4 3.0 3.0 3.0 3.0
Table 3 Member sizes and details Model EXT_SL45 INT_SA2 INT_SB2 EXT_CFC EXT_CL45 EXT_CL25 INT_CL45 INT_CL25
Beams Section
bb
db
tb f
tbw
Columns Section
bc
dc
tc f
tcw
Slab ds (mm)
tslab (mm)
UB 457×191×82 10 B 15 14 B 22 W 18×35 UB 457×191×82 IPE 300 UB 457×191×82 IPE 300
191.3 101.6 127.9 152.0 191.3 150.0 191.3 150.0
460.0 254.0 348.5 450.0 460.0 300.0 460.0 300.0
16.0 6.8 8.3 10.8 16.0 10.7 16.0 10.7
9.9 5.8 6.1 7.6 9.9 7.1 9.9 7.1
UC 356×368×177 8 WF 24 8 WF 67 W 10×60 UC 356×368×177 HEB 260 UC 356×368×177 HEB 280
372.6 147.8 207.1 256.0 372.6 260.0 372.6 280.0
368.2 201.4 230.9 260.0 368.2 260.0 368.2 280.0
23.8 10.1 23.1 17.3 23.8 17.5 23.8 18.0
14.4 6.2 15.9 10.7 14.4 10.0 14.4 10.5
– – – 76 0 0 0 0
– – – 89 120 100 120 100
Therefore, the strain-hardening stiffness of the diagonal spring is determined from K sh,spring =
K sh,num . cos2 α
(31)
Although the above procedure is described for the case of an external joint subjected to positive moment, the process is similar for a negative moment situation. If no effective width is considered, the procedure reduces to that for an external bare steel joint. In the next section, specific considerations are described for the case of internal composite joints. 4.2. Application to internal joints The same procedure as described above can also be applied to internal joints. However, additional considerations are required, particularly in terms of the moment distribution at the joint. In the case of internal joints, it is necessary to assess the ratio between the positive and negative moments transmitted to the column. To achieve this, the neutral axis locations for both connected beams are evaluated with appropriate consideration of the effective width. The moment distribution between the two beams is then based on the ratio of the second moments of area of the beam cross-sections near the joint. This is a valid assumption for beams with similar dimensions and boundary conditions, and is shown to provide an appropriate assessment of the behaviour, as illustrated in subsequent sections. After determining the moment distribution at the joint, the same equivalence between the analytical and numerical
Fig. 8. Types of substructure used for validation.
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Table 4 Material properties of the models Model
EXT_SL45 INT_SA2 INT_SB2 EXT_CFC EXT_CL45 EXT_CL25 INT_CL45 INT_CL25
Steel E (kN/mm2 )
f y (N/mm2 )
µ (%)
Concrete E (kN/mm2 )
f c (N/mm2 )
210 208 205.5 219a 210 210 210 210
275 282.7a 324a 244a 275 275 275 275
1.0 2.185 2.685 1.0b 1.0 1.0 1.0 1.0
– – – 23 30 30 30 30
– – – 43.8 30.0 30.0 30.0 30.0
a Value based on column properties. b Assumed value.
Fig. 9. The EXT_SL45 model prepared in ANSYS (a) and in ADAPTIC (b).
model can be adopted as was done previously for external joints. From the total moment applied to the panel zone, the stress boundary conditions on the two sides of the joint can be assessed. In addition, the distribution of shear and bending within the panel depth can be evaluated. Accordingly, the deformation of the panel accounting for both shear and flexural modes can be determined. For a total moment imposed on the joint, the equivalent shear force for use in the numerical model is obtained as
Fig. 10. The shear stress distribution obtained in the proposed approach (a) and in ANSYS (b) for a bare steel joint.
Veq,num =
Ms,right Ms,left Ns,left + + db 2 db Ns,right − Vcol + 2
(32)
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Fig. 11. Global and local responses of model EXT_SL45.
where (Ms,left ; Ns,left ) and (Ms,right ; Ns,right ) are the bending moments and axial forces carried by the steel beams on both sides of the joint, respectively. From this, the stiffness of the spring representing the panel zone can then be readily evaluated, as in Eqs. (19) and (23) before. The relative drift corresponding to the yield point can also be determined according to Eqs. (22) and (24). It should be noted however that a value of unity is used for RV , as the stress distribution in the panel zone of an internal composite joint has more resemblance to that obtained for a steel joint than that of an external composite joint. Finally, the spring properties for the post-elastic range are obtained using Eqs. (25) to (31), as before. The full procedure described above, to determine the spring properties for both external and internal joints, can be directly applied within a spreadsheet or a mathematical programming package. In the following section, a number of comparisons are made with available experimental results as well as detailed numerical simulations in order to assess the validity and accuracy of the proposed approach. 5. Validation studies This section presents several validation studies in which the proposed approach, which is developed for the purpose
Fig. 12. Global and local responses of model INT_SA2 (Krawinkler A2 specimen [1]).
of frame analysis representations, is compared to existing experimental results. However, due to the limited availability of test data, validation is also carried out against the results from more detailed three-dimensional numerical models adopting shell and solid elements, using ANSYS [17]. The proposed panel zone models are implemented in conjunction with the ‘frame analysis’ elasto-plastic elements available within the nonlinear analysis program ADAPTIC [18]. To this end, distinction should be made at the onset between the numerical models of ADAPTIC on the one hand, which are 2D frame models incorporating the proposed panel zone models, and those of ANSYS on the other hand, which are detailed 3D simulations employed for validation purposes. The validation is carried out for external as well as internal substructures, as indicated in Fig. 8. These systems represent typical idealised arrangements within moment frames subjected to lateral loading conditions, where hinges are assumed to form at mid-length of the beams and columns. The analysis is carried out by controlling the vertical end displacement of the beam or the top horizontal displacement of the column. Table 2 summarises all the substructures used in the validation, Table 3 provides the
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Fig. 13. Global and local responses of model INT_SB2 (Krawinkler B2 specimen [1]).
Fig. 14. Global and local responses of model EXT_CFC (Lee and Lu EJ–FC specimen [5]).
geometry and sizes of members and Table 4 presents the material properties adopted for the models. Hereafter, the details of the numerical models are first presented, followed by a discussion of the results of validation studies for various steel and composite substructures.
the column flange. Elsewhere, the full width of the slab is accounted for, in accordance with evidence from a detailed assessment of this aspect of behaviour [20]. For the contact region, rigid plastic behaviour for concrete is assumed with due consideration of confinement effects. On the other hand, as noted before, detailed 3D continuum models are utilised for validation purposes using ANSYS. In this case, the beams and columns are modelled using shell elements (SHELL 43) whilst, for composite substructures, the slab is represented using solid elements (SOLID 65). Material nonlinearity is included by the adoption of a bi-linear model with strain-hardening for steel and a tri-axial model with smeared cracking for concrete. Full interaction between the steel beam and the slab is assumed for the numerical models of both ADAPTIC and ANSYS.
5.1. Numerical models As mentioned before, the proposed panel models are utilised within the structural analysis program ADAPTIC [18], which accounts for material and geometric nonlinearities. In the idealised frame substructures, the steel and the column members are modelled using 1D Eulerian cubic elasto-plastic elements which account for the spread of plasticity across section and along the length. For composite frames, the beam and slab are modelled with two lines of cubic elements interconnected by rigid links. This approach can represent accurately the behaviour of a composite beam [19]. Clearly, in the case of 2D frame analysis representations, a number of idealisations are needed with respect to the effective size of the slab as well as its interaction with other frame members. In this study, the effective slab width at the beam ends is considered to be the same as the width of
5.2. Steel frames Although existing analytical models can adequately represent the panel zone in bare steel frames, it is useful to demonstrate the general applicability of the proposed approach prior to dealing with composite cases. In this section, a comparative assessment is carried out for
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Fig. 16. Global and local responses of model EXT_CL45.
Fig. 15. The shear stress distribution obtained in the proposed approach (a) and in ANSYS (b) for an external composite joint.
EXT_SL45, which represents an external substructure, with full details given in Tables 2–4. A view of the continuum mesh prepared in ANSYS is depicted in Fig. 9(a), whilst the frame model used in ADAPTIC is shown in Fig. 9(b). As noted before, the shear stress distribution within the panel zone of a steel joint is highly non-uniform. This is illustrated in Fig. 10(a) and (b) which depict the theoretical elastic stress distribution from the proposed approach as well as the detailed ANSYS representation, respectively. The results clearly demonstrate the discrepancy between the peak and average stresses, and the need to account for this in defining the effective yield point of the panel. On the other hand, Fig. 11(a) and (b) depict the global as well as local panel response, respectively. The results illustrate the accuracy of the proposed frame-modelling approach in comparison with the detailed 3D representation. Fig. 11(b) also indicates the panel response obtained from the expressions suggested by Krawinkler et al. [1,14], described previously. The comparison shows that the
proposed approach and that suggested for steel joints are in close agreement, and provide a good prediction of the panel response in the case of bare steel frames. Additional comparisons are also made with the experimental and analytical results obtained by Krawinkler et al. [1] for two internal substructures (specimens A2 and B2, described in Tables 2–4). These components represent steel joints within top and lower parts of a tall building where different cross-section combinations are incorporated, noting that an axial load representing about 30% of the column cross-section axial capacity was applied. Figs. 12 and 13 illustrate the accuracy, at both local and global levels, provided by the proposed frame models of ADAPTIC (INT_SA2 and INT_SB2) in comparison with the experimental results. 5.3. Composite frames This section presents a number of validation studies focusing on the response of composite frame substructures. The assessment carried out for external joint models is first discussed, followed by frames involving internal joints. As noted before, the details of all models utilised are described in Tables 2–4.
J.M. Castro et al. / Engineering Structures 27 (2005) 129–144
Fig. 17. Global and local responses of model EXT_CL25.
External joints An ADAPTIC frame model representing an external substructure (EXT_CFC) is first compared with the cyclic test undertaken by Lee and Lu [5] on a composite specimen (EJ–FC). An asymmetric tri-linear curve is used for the spring representing the panel zone and the results are compared with the response envelope from the cyclic test. For the positive moment case, the spring properties are derived as described previously whilst, for negative moment, the bare steel joint is considered due to the lack of reinforcement anchorage. The experimental and numerical results are compared in Fig. 14, in which good agreement is generally observed. However, some differences are noticed at large drift levels which may be attributed to cyclic hardening effects exhibited in the test. In Fig. 14(b), the panel response obtained from available expressions for steel joints [1] is also included. Evidently, these expressions do not provide an adequate prediction of the response of the composite joint, as illustrated by the significant discrepancy on the positive moment side. Two additional substructures (EXT_CL45 and EXT _CL25) are employed to further examine the validity of the proposed approach for external composite joints. For these two substructures, the frame analysis is validated against
141
Fig. 18. (a) The change in RNM during panel distortion; (b) the effect of RNM on the panel demand.
detailed ANSYS models. Before presenting the comparative results, it is interesting to examine the theoretical elastic shear stress distribution in the panel, both analytically and from ANSYS. As depicted in Fig. 15, the shear stress is largely uniform as compared to the case of steel joints, as discussed in earlier sections of this paper. The global frame and local panel responses for both substructures are presented in Figs. 16 and 17 including the results from ADAPTIC and ANSYS. The results clearly indicate the good agreement obtained between the frame models and the detailed 3D representation. The global response in these two figures also includes a frame analysis incorporating a fully rigid joint in order to illustrate the key role played by the panel flexibility and capacity. In terms of the local panel response, Figs. 16(b) and 17(b) also indicate that the proposed model provides a better prediction for both the yield point and the ultimate capacity of the panel, in comparison with available expressions for steel joints. In the light of the above discussion, it is important to examine the axial force-to-bending moment ratio in the steel beam, represented by RNM , in relation to the shear distortion imposed on the panel zone. This is illustrated in Fig. 18 for the frame models of the two substructures EXT_CL25 and EXT_CL45. As shown in Fig. 18(a), it is evident that as the
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Fig. 19. Bending moment ratios (M+/M−) for INT_CL45 (a) and INT_CL25 models (b).
panel zone attains its yield, a sudden change in the variation of RNM with panel distortion occurs. From examining Eq. (10), it is clear that the demand on the panel, for a specific overall moment, is reduced for increasing values of RNM . This is also illustrated in Fig. 18(b) which depicts the change in panel demand (normalised by that of a steel beam) for different RNM ratios and for various composite beam sizes. Consequently, since EXT_CL25 is subjected to higher RNM ratios, relatively high bending moments need to be mobilised in the composite beam in order to develop full plasticity in the panel in comparison to the case of EXT_CL45. Consequently, it is to be expected that predictions achieved using conventional steel models [1] for EXT_CL45 appear to be more favourable than that for EXT_CL25. Internal joints The validity of the approach is finally assessed by considering two substructures involving internal joint configurations, namely INT_CL45 and INT_CL25. As in previous cases, ADAPTIC frame models, incorporating the proposed procedures for the panel zone, are compared to detailed 3D analysis carried out in ANSYS.
Fig. 20. Global and local responses of model INT_CL45.
From the results of the two internal substructures, it is firstly useful to assess the positive-to-negative moment distribution between the beams, in comparison to the initial assumption employed in deriving the joint properties. This moment ratio is depicted in Fig. 19 as extracted from the frame (ADAPTIC) and 3D (ANSYS) analyses. As expected, the results indicate that the relative moments are closely related to the ratio of cross-section inertias, estimated as 1.92 for INT_CL45 and 2.14 for INT_CL25. The global frame and local panel responses obtained from ADAPTIC and ANSYS, for the two internal substructures, are presented in Figs. 20 and 21, respectively. The global response obtained from the frame models with a rigid joint assumption is also shown on the figures to illustrate the importance of accounting for the panel zone stiffness and strength. The two figures illustrate very good agreement for the results of ADAPTIC, incorporating the proposed panel model, in comparison with the detailed 3D analysis, at both local and global levels. In contrast, the local response obtained from available expressions for steel joints [1] significantly underestimates the response of the panel, as indicated by the local response shown in Figs. 20(b) and 21(b). In this respect, it should be noted that the use of conventional steel models, in conjunction with a modified panel depth (db ) reflecting the composite beam depth, would
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beams. This approach enables an accurate assessment of the shear stress distribution through the panel depth which allows a realistic representation of the spread of plasticity. The contribution of the column flanges to the extra resistance of the panel zone is also accounted for. The procedure considers both shear and bending deformations, and addresses the elastic and inelastic stages. Validation of the proposed approach is carried out by comparison against available experimental results coupled with detailed numerical simulations. The comparisons illustrate the accuracy and reliability of the approach developed, and its general applicability to both steel and composite frames as well as external and internal joints. It is shown that whilst conventional moment–distortion expressions may be adequate for steel joints, the basis of such relationships is unrealistic for composite frames. The panel zone response within a composite frame is significantly affected by several geometric and loading parameters, including the influence of beam-to-slab interaction on the stress distribution and distortional demand imposed on the panel. The proposed approach, which is suitable for implementation within frame analysis techniques, captures the influence of these key parameters and provides a rational procedure that accounts for the underlying behavioural mechanisms. References
Fig. 21. Global and local responses of model INT_CL25.
also grossly misestimate the behaviour. Moreover, from comparing Figs. 16(b) to 20(b), it is evident that for the same configuration, the actual capacities are markedly different in external and internal joints where both the force distributions and the confinement levels in the vicinity of the joint are different. This provides further illustration of the inadequacy of empirical approaches solely based on geometric and material properties of the panel. The comparisons described in this section demonstrate the suitability of the proposed approach for modelling both steel and composite joints. For steel frames, the method provides equally accurate predictions to those obtained from conventional approaches. On the other hand, for composite frames, the new approach offers a more rational and accurate assessment of the behaviour due to its ability to realistically account for the actual boundary conditions and load distributions at the panel zone. 6. Conclusion A new approach for representing the panel zone component in steel and composite moment-resisting frames is proposed in this paper. Realistic stress distributions in the edges of the panel are considered, which incorporate the variable location of the neutral axis in the case of composite
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