Title no. 110-S87 Reinforced Concrete Coupling · PDF fileReinforced Concrete Coupling Beams—Part II: ... research interests include behavior and modeling of reinforced concrete

ACI Structural Journal/November-December 2013 1067

Title no. 110-S87

ACI STRUCTURAL JOURNAL TECHNICAL PAPER

ACI Structural Journal, V. 110, No. 6, November-December 2013.MS No. S-2012-011 received January 4, 2012, and reviewed under Institute

publication policies. Copyright © 2013, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion including author’s closure, if any, will be published in the September-October 2014 ACI Structural Journal if the discussion is received by May 1, 2014.

Reinforced Concrete Coupling Beams—Part II: Modelingby David Naish, Andy Fry, Ron Klemencic, and John Wallace

Reinforced concrete (RC) shear walls connected by coupling beams form an efficient structural system for tall buildings to resist earthquake loads. Understanding the load-deformation character-istics of coupling beams is essential to modeling the overall system responses of coupled walls to seismic loading. Using new test data presented in a companion paper, modeling studies are performed to evaluate the effectiveness of current modeling approaches with respect to key parameters, including effective elastic stiffness, deformation capacity, and residual strength. Modeling approaches commonly employed in engineering practice are investigated and are shown to reasonably capture measured force-versus-defor-mation behavior. The impact of the test specimen scale factor is reviewed and shown to be potentially significant on the member effective bending stiffness and deformation capacity. The modeling studies are expanded to include beam aspect ratios not consid-ered in the test program to develop modeling parameters for use in seismic rehabilitation and performance-based seismic design.

Keywords: ASCE 41; aspect ratio; coupling beam; diagonal reinforcement; effective stiffness; modeling parameters; nonlinear modeling.

INTRODUCTIONTall building construction is common in metropolitan

areas and it has become increasingly important to provide methods of construction that improve both seismic perfor-mance and constructibility. Core walls, with coupling beams above openings to accommodate doorways, are an effi-cient lateral-force-resisting system for tall buildings. When subjected to strong shaking, the coupling beams act as fuses and typically undergo large inelastic rotations.

Nonlinear modeling of coupling beams has received increased attention as the use of performance-based design for tall core wall buildings has become more common.1 Modeling parameters for diagonally reinforced coupling beams were introduced into Table 6-18 of FEMA 3562; given limited test data, only one row of modeling parameters is provided. No changes were incorporated when the FEMA 356 pre-standard was reformatted into the ASCE 41-06 standard. Although ASCE 41-06 focuses on rehabilitation, the modeling parame-ters associated with well-detailed components are commonly used for design of new buildings. For coupling beams, impor-tant modeling parameters include effective bending stiffness EcIeff, allowable plastic rotation prior to significant lateral strength degradation, and residual strength. The concrete provisions in ASCE 41-06 were updated in Supplement 1.3 The effective bending stiffness for beams in Table 6-4 was reduced from 0.5EcIg to 0.3EcIg, primarily based on a review of concrete column test data at low axial loads, to account for the added flexibility due to reinforcement slip/extension; however, modeling parameters in Table 6-18 for reinforced concrete (RC) coupling beams are unchanged. The value used for coupling beam bending stiffness has a significant impact on the degree of coupling,4 and the appropriateness of the lower bending stiffness value for coupling beams has not been reviewed. Verifying that the relatively simple modeling approaches commonly used in commercially available

computer programs adequately capture coupling beam load-deformation responses, as well as recommending parameters associated with unloading/reloading and pinching behavior, are important issues that have not been adequately investi-gated. The test results presented in the companion paper, as well as test data reported in the literature, provide the infor-mation needed to assess these important topics.

RESEARCH SIGNIFICANCEModeling studies of 1/2-scale test specimens, along with

analytical studies, are used to assess whether common nonlinear modeling approaches used are capable of capturing the load-deformation responses of coupling beams. Specific attention is paid to the selection of modeling parameters to produce the best correlation between test and model results. The research provides vital information to fill critical knowl-edge gaps related to selection of effective elastic stiffness, plastic rotation capacity, residual strength, and the impact of the test scale on these modeling parameters. Findings indicate that the effective bending stiffness at yield is lower than commonly used and modeling approaches used by practicing engineers reasonably capture the measured load-deformation responses.

MODELING PARAMETERSIn the following sections, modeling parameters for

coupling beams are systematically studied using the test results presented in the companion paper.

Effective stiffnessElastic analysis approaches require estimation of the effec-

tive elastic bending and shear stiffness values. In FEMA 356, stiffness values of 0.5EcIg and 0.4EcAcw are recommended for bending and shear, respectively. ASCE 41-06, including Supplement 1, incorporates a lower value for effective stiff-ness of 0.3EcIg, with a mean value obtained from column tests at low axial stress of 0.2EcIg.5 The New Zealand Code (NZS-3101 1995)6 includes an equation to estimate the effective bending stiffness that depends on the expected ductility demand as

2( / )c g

c effn

A E IE I

B C h l×

=+ × (1)

where A, B, and C vary with ductility (A = 1.0 and 0.40; B = 1.7 and 1.7; C = 1.3 and 2.7; for ductility = 1.25 and 6.0).

1068 ACI Structural Journal/November-December 2013

David Naish is an Assistant Professor in the Department of Civil Engineering at California State University, Fullerton, CA. He received his PhD in civil engineering from the University of California, Los Angeles (UCLA), Los Angeles, CA, in 2010. His research interests include behavior and modeling of reinforced concrete and masonry elements and systems.

Andy Fry is Chief Operating Officer at Magnusson Klemencic Associates, Inc., Seattle, WA. He received his BS in civil engineering from the University of Illinois Urbana-Champaign, Urbana, IL, and his MS in structural engineering at the Univer-sity of California, Berkeley, Berkeley, CA. His research interests include performance-based seismic design methodology and related issues, including performance of post-tensioned slab/core wall connections.

Ron Klemencic, FACI, is President of Magnusson Klemencic Associates, Inc. He received his BS in civil engineering from Purdue University, West Lafayette, IN, and his MS in structural engineering from the University of California, Berkeley. He is a member of ACI Committee 318, Structural Concrete Building Code, and 374, Perfor-mance-Based Seismic Design of Concrete Buildings.

John Wallace, FACI, is a Professor of civil engineering at UCLA. He is a member of ACI Committees 318-H, Seismic Provisions (Structural Concrete Building Code); 374, Performance-Based Seismic Design of Concrete Buildings; and Joint ACI-ASCE Committee 352, Joints and Connections in Monolithic Concrete Structures. His research interests include response and design of buildings and bridges to earthquake actions, laboratory and field testing of structural components and systems, and seismic structural health monitoring.

For beams with aspect ratio ln/h = 2.4, Eq. (1) yields a beam with effective elastic stiffness close to 0.5EcIg, whereas for a ductility ratio of 6, the effective (secant) stiffness drops to 0.18EcIg.

Figure 1 plots the secant stiffness normalized with respect to the concrete gross section stiffness versus the chord rotation for test results presented in the companion paper.7-9 Secant stiffness is calculated from the test results assuming fixed end conditions according to

3

12n

c eff

V lE I

×=

× D

The initial stiffness of each residential beam is approxi-mately 0.25EcIg, with an effective stiffness at the yield rota-tion (approximately 1.0% rotation) of 0.12EcIg. The effective stiffness ratio (Ieff/Ig) does not vary significantly for the three different configurations (Fig. 1)—that is, beams without slab (CB24F), beam with RC slab (CB24F-RC), and beams with PT slab (CB24F-PT). The initial stiffness ratio for the beams with slabs is moderately higher (approximately 25%) for rotations up to approximately 2%; however, after significant flexural cracks form at the slab-wall interface, generally at approximately 3% rotation, the stiffness ratio is nearly the same for all three beam test configurations.

The low stiffness ratios (Ieff/Ig) observed for the tests rela-tive to recommended values (Table 1) require additional study, as the effective bending stiffness at yield for the tests are close to the value recommended in NZS 3101:19956 for ductility of 6. Photos of damage,7-9 for the beams without slabs and for the beams with slabs, do not show significant concrete spalling, and diagonal crack widths along the beam are limited to 1/32 in. (0.8 mm) even at 6% total rotation; damage is concentrated at the beam-wall interface in the form of reinforcement slip/extension cracks. The photos also indicate that the quantity of beam transverse reinforcement is sufficient to keep crack widths small for peak shear stresses as large as 10.5 to 13.8√fc′ psi (0.88 to 1.55√fc′ MPa). The larger diagonal crack widths observed for CB24F-1/2-PT, with only one-half the quantity of transverse reinforce-ment used in CB24F, CB24F-RC, and CB24F-PT, indicate

that the quantity of transverse reinforcement required by ACI 318-08 could likely be reduced moderately without compromising deformation capacity. Current modeling of the load-deformation response of coupling beams tends to focus on shear behavior10; however, for the 2.4 and 3.33 aspect ratio beams tested, flexural and reinforcement slip/extension deformations at and adjacent to the beam-wall interface generally accounted for more than 85% of the total rotation.

It is important to note that beam axial deformations were not restrained in the tests, whereas redistribution of shear between walls might lead to axial compression in coupling beams.11 This was not considered in the tests. However, two specimens, CB24F-PT and CB24F-1/2-PT, were constructed with 150 psi (1.03 MPa) post-tensioned strands as a means to provide some restraint on axial growth. The impact of this axial restraint on the effective stiffness is evident in Fig. 1, which shows a moderate increase in stiffness (approximately 0.15EcIg versus 0.12EcIg at yield). Bower12 investigated the effect of axial restraint on stiffness and ductility of diagonally reinforced coupling beams using finite element analysis; results indicate that initial, precracked stiffness can as much as double, but that the impact after cracking on the system response is minimal.

Of the various approaches used to estimate the effective stiffness at yield—that is, FEMA 356 (0.5EcIg), ASCE 41-06 (0.3EcIg), and NZS 3101:1995 for low ductility (0.5EcIg), only ASCE 41-06 (2007) addresses the impact of slip/extension on the effective stiffness at yield. (It is noted that median effective stiffness reported by Elwood et al.5 is actu-ally 0.2EcIg at low axial load. The value of 0.3EcIg is used as a compromise to address issues associated with deformation compatibility checks for gravity columns.)

The contribution of reinforcement slip/extension to the yield rotation is estimated for the beams tested using the

Fig. 1—Effective secant stiffness values derived from test results: ln/h = 2.4.

Table 1—Effective stiffness values

EIeff, % EIg θy, % drift

Test results 14.0 (12.5)* 0.70 (1.00)*

FEMA 356 50.0 0.23

ASCE 41 30.0 0.39

ASCE 41 S1, with slip hinge 16.5 (13.0)* 0.75 (0.95)*

NZS 3101:1995 (m = 1) 50.0 0.23*Modifications for 1/2-scale.


approach recommended by Alsiwat and Saatcioglu,13 where the crack width that develops at the beam-wall interface depends on bar slip and bar extension (integration of strain along the embedment). Using a coupling beam effective stiff-ness derived from a moment-curvature analysis of the beam cross section at the beam-wall interface (approximately 0.5EcIg) and the slip/extension model noted previously, the effective stiffness at yield reduces to 0.12EcIg, which is consistent with the effective stiffness at yield (approximately 1.0% for all beams) derived for the tests (Fig. 1). Additional details of the slip/extension calculations are summarized in the next section.

Table 1 provides a summary of the effective stiffness and yield rotation for each of the different models discussed previously. Based on these results, use of the model detailed in ASCE 41-06 Supplement 1 is recommended—that is, use a moment-curvature analysis to define the secant stiff-ness at the yield point and include a slip/extension spring. Alternatively, as noted in ASCE 41-06 (2007), the effective bending stiffness can be defined to provide an equivalent stiffness that combines both curvature and slip deformations (~0.12EcIg for the test beams).

Reinforcement slip/extension calculationsAs stated in the previous section, slip and extension of the

flexural reinforcement contributes significantly (40 to 50%) to beam deformations. The approach developed by Alsiwat and Saatcioglu13 is used to model this contribution to yield rotation. The calculations are provided herein for the case of determining the slip/extension rotations at yield point of the flexural reinforcement. If adequate embedment of the flex-ural reinforcement is provided, then the slip contribution is negligible. However, this calculation also is included for the sake of completeness. The parameters that are required for this calculation are db, A, ld, fc′, fy, My, and fs. All dimensions are in mm and MPa (1 mm = 0.0397 in., 1 MPa = 0.145 ksi).

Preliminary calculations

[MPa]4y b

ed

f du

l×

=×

(2)

[mm]4s b

ee

f dL

u×

=×

(3)

20 [MPa]4 30

b cu

d fu

′ = - × (4)

130 [mm]s

cfd =

′ (5)

In these calculations, ue represents the elastic bond stress, Le represents the elastic region length, uu represents the peak bond stress, and ds1 represents the local slip at the peak bond stress.

Calculations for fs = fy

2.5

1 [mm]es s

u

uu

d = d ×

(6)

1.25 [mm]2

eexty y

Ld = × e × (7)

dtoty = ds + dexty [mm] (8)

@ [rad]toty

toty

d xd

dθ =

- (9)

@

[mm-MPa]toty

yMK

d

=θ (10)

In these calculations, ds represents the slip of the reinforce-ment, dext represents the extension of the bar due to accu-mulation of strain along its length, dtot represents the total displacement of the bar at the beam-wall interface, θ@dtot represents the angle of the crack that opens at the beam-wall interface due to the slip/extension of the bar, and K repre-sents the stiffness of the corresponding moment-rotation hinge that can be implemented in a structural model.

In addition, the slip/extension model can be used to modify post-yield behavior. In this case, slip is considered negligible. Calculations for es = eu

5.5 0.07 [MPa]27.6

cLf

L

fSu

H ′

= - × × (11)

[mm]4

s bpy

f

f dL

uD ×

=×

(12)

( )[mm]

2y u py

ext exty

Le + e ×d = d + (13)

@ [rad]ext

ext

d xd

dθ =

- (14)

In these calculations, uf represents the frictional bond stress, SL and HL are the spacing and height of the lugs on the reinforcement, respectively, and Lpy is the post-yield length. The end result is the rotation at the beam-wall interface due to slip/extension of the reinforcement.

Effect of scaleAs previously stated, the tests were conducted at 1/2- scale;

therefore, it is important to understand the potential impact of scale on the effective yield stiffness as well as the overall load-deformation behavior. The relative contribu-tion of flexural deformations (curvature) and slip/exten-sion to yield rotation of the test beams at full scale (that is, prototype beams) is assessed using the same approach as noted in the previous section for the 1/2-scale beams. The study is extended to consider coupling beam aspect ratios beyond those tested, by varying the beam length. Results are reported in Fig. 2, where the effective yield rotation due to slip/extension is plotted against beam aspect ratio (ln/h) for various scale factors.


For a given scale factor, variation of the aspect ratio has only a modest impact on the slip/extension rotation, producing roughly a 15 to 20% increase as the aspect ratio increases from 1.0 to 3.0. However, for a given aspect ratio, slip/extension rotation at yield is significantly impacted by scale, with a 35 to 40% reduction for beams at one-half versus full scale. The effective bending stiffness at yield for the 1/2-scale tests of 0.12EcIg increases to 0.14EcIg for the full-scale prototypes due to the reduction in the relative

contribution of slip rotation. Based on these results, use of an effective yield stiffness value of 0.15EcIg for full-scale coupling beams, due to stiffness provided by the slab, is recommended (ln/h = 2.4). Further details of these calcula-tions are provided by Naish et al.7,9

Load-deformation backbone relationsIn this section, backbone relations are derived both from

test results (Fig. 3) and using ASCE 41 (Fig. 4) to provide a simple yet accurate method for estimating the overall load-deformation behavior of coupling beams. Linearized back-bone relations for normalized shear strength versus rotation are plotted in Fig. 4 as dotted lines for the three configura-tions of beams tested—that is, beams with no slab (CB24F, CB24D, CB33F, CB33D), beam with RC slab (CB24F-RC), and beams with PT slab (CB24F-PT and CB24F-1/2-PT). These backbone relations are determined as shown in Fig. 3, which plots the peaks of the load-deformation curves for CB24F and CB24D. The smooth backbone relations are determined using Vave, defined as the average shear force resisted by the beam between the yield point and the onset of significant lateral strength degradation, with an elastic (secant) stiffness defined at 2/3 of Vave/Vn; refer to Naish9 for additional details. The backbone relations that are modi-fied to represent full-scale beams, as discussed in the prior subsection, also are plotted in Fig. 4. For configurations with multiple tests, an average relation is plotted. The results for all seven tests are very consistent, with a yield rotation of approximately 1.0%, initiation of shear strength degradation at 8.0% rotation, and the residual shear strength reached at 12.0% rotation. Backbone relations modified to represent full-scale beams indicate that the total rotations at yield, strength degradation, and residual strength are reduced to 0.70%, 6.0%, and 9.0%, respectively (from 1.0%, 8.0%, and 12.0%). The impact of slab on shear strength also is apparent in Fig. 4, with the ratios of Vave/Vn being approximately 1.1 (no slab), 1.3 (RC slab), and 1.4 (PT slab).

ASCE 41-06 with Supplement 1 modeling parameters also are plotted on Fig. 4 and indicate that the test beams are more flexible at yield and that they attain substantially higher deformation capacity prior to lateral strength degra-dation than the ASCE 41-06 backbone relation. The elastic stiffness of the ASCE 41 (including Supplement 1) rela-tion is based on a bending stiffness of 0.3EcIg, or approxi-mately double that derived for full-scale beams from the test data. The plastic rotation capacity given by ASCE 41-06, Table 6-18 is limited to 3%, whereas the backbone relations for the full-scale beams derived from the test data yield at approximately 0.7% rotation and reach 6.0% rotation prior to significant strength degradation, or a plastic rotation of 5.3%. Therefore, relative to ASCE 41-06, the relations derived for the full-scale beams have a lower effective yield stiffness (0.15EcIg/0.3EcIg = 0.50) and substantially greater deformation capacity (5.3%/3.0% = 1.77). The tests also reveal that a residual strength equal to 0.3Vn can be main-tained to very large rotations (10 to 12%) compared to the ASCE 41-06 residual strength ratio of 0.8 at a plastic rota-tion value of 5.0%. Therefore, it is reasonable to use a plastic rotation value of 5.0% with no strength degradation, with moderate residual strength (0.3Vn) up to a plastic rotation of 7.0%. It is noted that the ASCE 41-06 relation applies to all diagonally reinforced coupling beams, including beams with aspect ratios significantly less than the values of 2.4 and 3.33 investigated in this test program. Results presented in

Fig. 2—Yield rotation due to slip/extension for various aspect ratios and testing scales.

Fig. 3—Determination of linearized backbone relation from test data.

Fig. 4—Backbone load-deformation for full-scale beam models and ASCE 41-06 model (1/2-scale test results from Part 1 are dotted lines).


Fig. 4 apply for the beam aspect ratios tested (2.4 and 3.33), as well as to beams between these ratios. Also, it is reason-able to assume these values can be extrapolated modestly to apply to beams with 2.0 ≤ ln/h ≤ 4.0. Ratios less than 2.0 are addressed later.

APPLICATION TO COMPUTER MODELINGBased on the backbone and effective stiffness rela-

tions discussed previously, nonlinear modeling approaches commonly used by practicing engineers were investigated to assess how well they were able to represent the measured test results. Two models were considered: one using a rotational spring at the ends of the beam to account for both nonlinear flexural and shear deformations (Mn hinge), and one using a nonlinear shear-displacement spring at beam midspan to account for both flexural and shear deformations (Vn hinge) (Fig. 5). In addition, both of the models incorporate elastic slip/extension springs to account for softening due to slip/extension deformations at the beam-wall interface. Both models were subjected to the same loading protocol used in the tests.7-9 In this study, CSI Perform 3D was used (CSI 2006). Naish7 provides detailed information on modeling parameters used to generate results for Fig. 6 to 10.

Diagonally reinforced coupling beams (2.0 < ln/h < 4.0)The Mn hinge model consists of an elastic beam cross

section with EcIeff = 0.5EcIg, elastic-rotation springs (hinges) at each beam end to simulate the effects of reinforcement slip/extension deformations, and rigid plastic rotational springs (hinges) at each beam end to simulate the effects of nonlinear deformations. The stiffness of the slip/extension hinges are defined using the Alsiwat and Saatcioglu13 model discussed previously, whereas the plastic rotations of the nonlinear flexural hinges are modeled using the backbone relations derived from test results (Fig. 4, excluding the elastic deformation), and shear strength is defined using ACI 318-08, Eq. (21-9). The Vn hinge model also consists of an elastic beam cross section and slip/extension hinges; however, instead of using flexural hinges at the beam ends, a shear-force-versus-displacement hinge (spring) is used at beam midspan to simulate the effects of nonlinear deforma-tions. The shear hinge properties are defined using the back-bone relations derived from the test results (Fig. 4).

Figure 6 shows cyclic load-deformation plots for the two models and the test results for CB24F. Both models accu-rately capture the overall load-displacement response of the member; however, the Mn hinge model (Fig. 6(a)) captures the unloading characteristics better than the Vn hinge model (Fig. 6(b)), due to the fact that unloading stiffness modeling parameters, which help to adjust the slope of the unloading curve, are available for the flexural hinges in the commercial computer program used, but not for the shear hinges (refer to Naish et al.6 for a complete description of the modeling parameters and assigned values). As noted in the companion paper (Part I), for the beam test aspect ratios (2.4 and 3.33), flexural and slip/extension deformations account for approximately 80 to 85% of total deformation, whereas

shear deformations generally account for only 15 to 20% of total deformation. Therefore, in both models, the flexural and shear hinges are used to represent overall (combined) deformations—that is, shear deformations are not consid-ered. Therefore, depending on the computer program used, similar modeling studies should be conducted to cali-brate available model parameters with test results prior to modeling complete buildings.

Beams at aspect ratio ln/h ≥ 3.0 exhibit predominately flexural behavior; therefore, only the Mn hinge model is presented for this case. The properties of the slip/extension hinge, plastic moment-rotation hinge, and elastic concrete cross section are determined in the same way as for the 2.4 aspect ratio beam, but for the CB33F cross section. As was observed for residential beams CB33F or CB33D, the Mn hinge provides a good approximation of the overall load-deformation response of the member when compared with test results (Fig. 7).

Based on the analysis and test results, the main impact of the slab on the behavior of the coupling beams is an increase in beam strength of approximately 20%. As noted previously, this increase in beam strength is consistent with the sectional analysis of the beam at the beam-wall inter-face considering the impact of the slab on the positive and

Fig. 5—Modeling components: (a) Mn hinge model; and (b) Vn hinge model.

Fig. 6—Cyclic load-deformation modeling results (ln/h = 2.4): (a) CB24F versus moment hinge model; and (b) CB24F versus shear hinge model.


negative moment strengths of the coupling beam, with shear yield strength calculated as Vy = 2Mn/ln. Model and test results for the coupling beam with the RC slab are compared in Fig. 8, and show that the model accurately captures the measured responses.

Conventionally reinforced coupling beams (3.0 < ln/h < 4.0)

In the testing program, in addition to the diagonally reinforced coupling beams tested, one conventionally reinforced beam also was tested. Results of modeling studies were compared with test results for FB33 as well as with results from previous

tests on similar beams. The model is composed similarly to the Mn-hinge model for Beams CB33F and CB33D, with changes made only to the shear strength (different cross section), plastic rotation capacity (different slip/extension parameters), and the energy dissipation factors (straight bars instead of diagonal bars). The load-deformation response of the model is plotted against test results for FB33 in Fig. 9, revealing good correspondence with test results, specifi-cally in the slope of the loading and unloading curves, and in the pronounced pinching of the cyclic load-deformation plot. The commercial computer program used allowed the shape of the load-deformation loops to be manipulated through specifying energy dissipation parameters to simu-late the pinching of the load-deformation plots of the test beams. A model also is constructed for a test beam of similar geometry, aspect ratio ln/h = 4.0 (Fig. 10), tested by Xiao et al.14 The model accurately captures the elastic behavior of the beam, and follows very closely the unloading character-istics in the nonlinear range. For beams with an aspect ratio greater than 3.0, placement of diagonal reinforcement can become difficult, and the potential increase in deformation and energy dissipation capacity associated with the use of diagonal reinforcement as opposed to longitudinal reinforce-ment may not be justified. Thus, for beams with low shear-stress demands (≤5√fc′ psi [0.42√fc′ MPa]), use of longitu-dinal reinforcement may be appropriate as long as modeling parameters that account for the lower deformation capacity at strength degradation and reduced energy dissipation capacity (more pinching), compared to diagonally reinforced coupling beams, are used.9

Lower aspect ratios (1.0 < ln/h < 2.0)Thus far, the modeling studies have focused on beams with

aspect ratios between approximately 2.0 and 4.0 because the tests were conducted on beams within this range (2.4 and 3.33) and because these beams make up the majority of coupling beam geometries in current tall building construc-tion. Further analyses of these data are beyond the scope of the work presented herein; results of analytical studies for coupling beams with aspect ratios between 1.0 and 4.0 are presented by Naish.9 Data are used in the next section to reassess modeling parameters and acceptance criteria for low-aspect-ratio (1.0 < ln/h < 2.0) coupling beams.

FRAGILITY RELATIONSFragility curves were developed for both diagonally and

conventionally RC coupling beams for beam aspect ratios

Fig. 7—Cyclic load-deformation modeling results (ln/h = 3.33): CB33F versus moment hinge model.

Fig. 8—Cyclic load-deformation modeling results (ln/h = 2.4): CB24F-RC versus moment hinge model.

Fig. 9—Cyclic load-deformation modeling results (ln/h = 3.33): FB33 versus moment hinge model.

Fig. 10—Cyclic load-deformation modeling results (ln/h = 4.0). Xiao HB4-6L-T100 versus moment hinge model.


ln/h between 1.0 and 4.0 using available test results. The objectives of this effort were to provide information needed for performance-based earthquake engineering of tall build-ings15 and to update ASCE 41-063 modeling parameters and acceptance criteria. Given the lack of data, it was not possible to create various bins to address design parameters such as variations in transverse steel reinforcement ratio, maximum shear stress, and aspect ratio. Available test data were collected and organized first simply by aspect ratio; the potential to sort data by maximum shear stress was consid-ered, but not implemented.

Data sources and approachAs noted by Naish,9 a majority of coupling beam tests have

been performed at low aspect ratios (ln/h < 2.0), with detailing that did not conform to modern codes (ACI 318-99 and later versions). Therefore, the first set of fragility relations were developed for beams with relatively higher aspect ratios (2.0 < ln/h < 4.0) on diagonally reinforced coupling beams. Even within this range of aspect ratios, there are relatively few tests of diagonally reinforced beams besides the ones reported in this study. The only relevant study identified was conducted at the University of Cincinnati by Fortney.16

Two sets of fragility relations were developed for conven-tionally reinforced coupling beams: one for higher aspect ratios (3.0 < ln/h < 4.0) and one for lower aspect ratios (1.0 < ln/h < 2.0). Test data were taken from results reported in the literature.7,14,17-20 The final set of fragility curves was devel-oped for diagonally reinforced coupling beams at low aspect ratios (1.0 < ln/h < 2.0). Test data were taken from test results reported in the literature.17-19,21

Four damage states were defined (Table 2): 1) yielding; 2) DS1-minor damage; 3) DS2-major damage I; and 4) DS3-major damage II. These damage states were determined for each test specimen based on investigation of load-defor-mation relations as well as photographs and descriptions of damage provided in the papers/reports by authors.

Fragility curves are defined herein as cumulative density functions plotted against beam chord rotation (θ), defined as the relative displacement (D) normalized by beam span (ln). The data were fit with lognormal distributions, which are commonly used and found to fit the data reasonably well. The parameters θm and s.d. are defined as the mean rotation of the distribution and logarithmic standard deviation for a given damage level, respectively. The results for all studies are provided in Table 3 and plots of the relations against test data are shown in Fig. 11.

Modeling parameters and acceptance criteriaAs stated previously, DS3 represents substan-

tial damage and member strength loss; therefore, DS3 is analogous to the collapse prevention (CP) limit state

in ASCE 41.3 Table 4 provides a comparison of the rotation values for CP and the mean rotation values for DS3. The CP values for conventionally reinforced coupling beams are organized based on shear demand and conforming transverse reinforcement. However, the CP values for diagonally rein-forced beams are independent of aspect ratio, shear stress, or transverse reinforcement. In addition, there is a significant difference between the CP and DS3 values, most notably for the diagonally reinforced beams. This indicates the poten-tial to provide more rows of values for the diagonally rein-forced beams. Values for conventionally reinforced beams are fairly thorough, with distinctions made based on shear demand and conforming reinforcement. Based on the results, possible modifications to the modeling parameters for

Table 2—Details of damage states for fragility relationsDamage state Definition of damage Repair procedures

Yield Substantial change in stiffness of load-deformation plot None

DS1-minor damage Residual cracks greater than 1/16 in. Epoxy injection of cracks (200 to 240 in. in length)

DS2-major damage (I) Residual cracks greater than 1/8 in.; minor spalling of concreteEpoxy injection of cracks in beam (600 to 720 in.) and slab (300 in.); replacement of spalled concrete

DS3-major damage (II)Significant strength degradation (<0.8Vn); buckling/fracture of

reinforcement; crushing of concrete

Chip away damaged concrete; attach mechanical couplers to remaining bars; replace damaged/fractured reinforcement;

replace damaged concrete

Note: 1 in. = 25.4 mm.

Table 3—Summary of fragility function parameters for coupling beams

Beam category Damage state θm, % s.d., %

Diagonally reinforced 1.0 < ln/h < 2.0

Yielding 0.84 0.39

DS1 1.79 0.38

DS2 3.52 0.44

DS3 5.43 0.95

Diagonally reinforced 2.0 < ln/h < 4.0

Yielding 0.97 0.26

DS1 2.03 0.39

DS2 3.94 0.35

DS3 6.02 1.00

Conventionally reinforced 1.0 < ln/h < 2.0

Yielding 0.85 0.25

DS1 1.37 0.21

DS2 2.64 0.33

DS3 4.28 0.74

Conventionally reinforced 2.0 < ln/h < 4.0

Yielding 0.72 0.20

DS1 1.37 0.21

DS2 2.64 0.33

DS3 4.07 0.75

Table 4—Limit/damage state comparisons (plastic hinge rotations)

Coupling beams CP, rad DS3, rad

Conventionally reinforced

1.0 < ln/h < 2.00.020 to 0.025*

0.035

2.0 < ln/h < 4.0 0.034

Diagonally reinforced

1.0 < ln/h < 2.00.030

0.046

2.0 < ln/h < 4.0 0.050*Represents range of values depending on shear-stress demand.


diagonally reinforced coupling beams in ASCE 41-063 are shown in Table 5. These values are based on the median values of the fragility curves, consistent with current ASCE 41-063 values.5 The rotation values are plastic defor-mation values—that is, neglecting elastic deformations.

CONCLUSIONS1. Effective elastic stiffness values for both diagonally

and conventionally reinforced test beams are determined to be approximately 0.15 to 0.20EcIg, or significantly less than FEMA 356 and ASCE 41 prescribed values of 50% and 30%, respectively. Designers should therefore use the Slip/Extension Hinge Model detailed in Supplement 1 of ASCE 41 to better approximate the elastic stiffness of the coupling beam, or use a lower effective stiffness, as given in this paper.

2. While flexural and shear-deformation contributions are equivalent regardless of the specimen scale, deforma-tions due to slip and extension of the flexural reinforcement at the beam-wall interface must be modified to account for

the scale of the given specimen. Thus, the behavior of beam specimens tested at less than full scale must be modified to account for the scale at which the test was conducted.

3. Simple nonlinear models, either moment-hinge or shear-hinge, accurately represent the load-deformation behavior of test beams. The flexural hinge model better matches the test results in the unloading and reloading range, due to the specific modeling parameters available in the computer software used (unloading stiffness modeling parameters), although both models produce acceptable results up to 3% total rotation for beams with ln/h between 2.0 and 4.0. There-fore, depending on the computer program used, the influence of modeling parameters on the load-versus-deformation responses should be compared with test results to ensure that they adequately represent observed behavior.

4. Fragility relations were developed for diagonally reinforced and conventionally reinforced coupling beams using data from a spectrum of tests reported in the literature. The relations provide valuable information for use in perfor-mance-based design and to update modeling parameters and acceptance criteria in ASCE 41-06.

ACKNOWLEDGMENTSThe research has been funded by the Charles Pankow Foundation, with

significant in-kind support provided by Webcor Concrete; this support is gratefully acknowledged. In addition, material contributions from Catalina Pacific Concrete, SureLock, and Hanson Pacific are appreciated. Thanks are extended to laboratory assistants J. Park, N. Lenahan, and C. Sanford, as well as UCLA laboratory technicians S. Keowen, A. Salamanca, S. Kang, and H. Kasper, for help in test preparation and completion.

NOTATIONAcw = cross-sectional area of concrete beam webAsh = area of transverse reinforcement provided within given spacing sAvd = cross-sectional area of each diagonal group of barsbw = width of beam web

Fig. 11—Fragility curves for diagonally RC coupling beams at high aspect ratio: (a) diag-onally reinforced: 2.0 < ln/h < 4.0; (b) conventionally reinforced: 2.0 < ln/h < 4.0; (c) diagonally reinforced: 1.0 < ln/h < 2.0; and (d) conventionally reinforced: 1.0 < ln/h < 2.0.

Table 5—ASCE 41-06 modeling parameters and numerical acceptance criteria for nonlinear procedures: diagonally reinforced coupling beams

ConditionsPlastic hinge rotation, rad

Residual strength ratio

Acceptable plastic hinge rotation, rad

ln/h V/twlw√fc′ a b c IO LS CP

≤2.0 ≤6.0 0.045 0.065 0.30 0.007 0.020 0.045

≤2.0 ≥8.0 0.035 0.055 0.30 0.006 0.018 0.035

≥3.0 ≤6.0 0.050 0.070 0.30 0.009 0.022 0.050

≥3.0 ≥8.0 0.045 0.065 0.30 0.007 0.020 0.045


d = distance between extreme tension steel and extreme compres-sion fiber

db = diameter of reinforcing barEc = modulus of elasticity of concretefc′ = concrete compressive strengthfy = yield strength of reinforcementh = beam depthIeff = effective section moment of inertiaIg = gross section moment of inertialn = clear span of beamMn = moment capacity of beamMy = yield moment of beams = longitudinal spacing of transverse reinforcements.d. = logarithmic standard deviation for given damage levelV = beam shearVave = average beam shear between yield and onset of strength degradationVmax = maximum shear force applied during testVn = nominal shear capacity of beamVy = yield strength of beamx = depth of neutral axisa = angle between diagonal bars and longitudinal axis of beamD = relative displacement of beam endDy = relative displacement at yieldey = yield strainθ = beam chord rotationθm = mean rotation of distribution for given damage levelθy = beam chord rotation at yield

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Buildings,” Structural Design of Tall and Special Buildings, V. 16, No. 5, 2007, pp. 615-632.

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3. ASCE/SEI 41-06, “Seismic Rehabilitation of Existing Buildings,” American Society of Civil Engineers, Reston, VA, 2007, 428 pp.

4. Coull, A., “Stiffening of Coupled Shear Walls against Foundation Movement,” Structural Engineer, V. 52, No. 1, 1974, pp. 23-26.

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Concrete Coupling Beams Subjected to Reversed Cyclic Loading,” UCLA-SGEL Report 2009/06, 2009.

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