Reinforced Concrete Link Beams: Alternative Details for Improved Constructability

David Naish Department of Civil and Environmental Engineering

University of California, Los Angeles

J. Andrew Fry Magnusson Klemencic Associates, Inc.

Ron Klemencic

Magnusson Klemencic Associates, Inc.

John Wallace Department of Civil and Environmental Engineering


Report to Charles Pankow Foundation School of Engineering and Applied Science


August 11, 2009

iii

ABSTRACT

An efficient structural system for tall building construction to resist earthquake loads consists of

reinforced concrete shear walls connected by diagonally reinforced coupling beams.

Construction of coupling beams that satisfy the strength and detailing requirements set forth in

ACI 318-05 for diagonally reinforced coupling beams is cumbersome and costly; therefore, ACI

318-08 provides a new detailing option which aims to improve the constructability while

maintaining adequate strength and ductility. Eight half-scale specimens were tested to compare

the performance of beams constructed utilizing new and old detailing options, to evaluate

common modeling approaches, and to assess the impact of reinforced and post-tensioned slabs.

Test results indicate that the new detailing approach provides equal, if not improved behavior as

compared to the alternative detailing approach, that simple modeling approaches reasonably

capture measured force versus deformation behavior, and that including a slab had only a modest

impact on strength, stiffness, ductility, and observed damage. This report summarizes the results

of these eight tests.

iv

ACKNOWLEDGEMENTS

The research has been funded by the Charles Pankow Foundation, with significant in-kind

support provided by Webcor Concrete; this support is gratefully acknowledged. As well, material

contributions from Catalina Pacific Concrete, SureLock, and Hanson Pacific are appreciated.

Thanks are extended to laboratory assistants Joy Park, Nolan Lenahan, and Cameron Sanford, as

well as UCLA students Anne Lemnitzer, Sarah Taylor-Lange, and Derek Skolnik and UCLA

laboratory technicians Steve Keowen, Alberto Salamanca, Steve Kang, and Harold Kasper, for

help in test preparation and completion.

v

TABLE OF CONTENTS

ABSTRACT.................................................................................................................................. iii ACKNOWLEDGEMENTS ........................................................................................................ iv TABLE OF CONTENTS ............................................................................................................. v LIST OF FIGURES .................................................................................................................... vii LIST OF TABLES ....................................................................................................................... ix LIST OF SYMBOLS ................................................................................................................... xi

1 INTRODUCTION................................................................................................................. 1

2 EXPERIMENTAL PROGRAM.......................................................................................... 5

2.1 Beam Design................................................................................................................... 5 2.2 Material Properties.......................................................................................................... 6 2.3 Test Setup...................................................................................................................... 16 2.4 Loading Protocol........................................................................................................... 16 2.5 Instrumentation ............................................................................................................. 18

3 EXPERIMENTAL RESULTS AND DISCUSSION........................................................ 25

3.1 Detailing........................................................................................................................ 27 3.1.1 Full Section vs. Diagonal Confinement ................................................................ 27 3.1.2 Full vs. Half Confinement .................................................................................... 27

3.2 Slab Impact ................................................................................................................... 31 3.3 Frame Beam.................................................................................................................. 35 3.4 Damage ......................................................................................................................... 36

4 MODELING ........................................................................................................................ 47

4.1 Effective Stiffness......................................................................................................... 47 4.2 Effect of Scale............................................................................................................... 49 4.3 Backbone Relations ...................................................................................................... 53 4.4 Model vs. Test Results.................................................................................................. 56

5 CONCLUSIONS ................................................................................................................. 59

REFERENCES............................................................................................................................ 61 APPENDIX A .............................................................................................................................. 63 APPENDIX B .............................................................................................................................. 92 APPENDIX C .............................................................................................................................. 95 APPENDIX D .............................................................................................................................. 96 APPENDIX E ............................................................................................................................ 100

vii

LIST OF FIGURES

Figure 1.1 Confinement options: (a) Diagonal confinement; and (b) Full section confinement ........4

Figure 2.1 Test beam geometries (ln/h = 2.4) full section confinement: (a) CB24F, CB24F-RC, CB24F-PT, CB24F-1/2-PT elevation; (b) CB24F cross section; and (c) CB24F-RC, CB24F-PT, CB24F-1/2-PT cross section. (Dimensions are inches. 1in = 25.4mm) ...........................................8

Figure 2.2 Photographs of test specimens (ln/h = 2.4) full section confinement (clockwise from top left): (a) CB24F beam construction; (b) CB24F-1/2-PT beam construction; (c) CB24F-PT beam elevation; and (d) CB24F-RC beam and slab construction.......................................................9

Figure 2.3 - Slab geometry and reinforcement for CB24F-RC, CB24F-PT, and CB24F-1/2-PT: (a) Elevation view; and (b) plan view. (Dimensions are inches. 1in = 25.4mm) ............................10

Figure 2.4 - Slab geometry and PT reinforcement for CB24F-PT and CB24F-1/2-PT: (a) Plan view; and (b) photo of post-tensioning load application. (Dimensions are inches. 1in = 25.4mm)11

Figure 2.5 Test beam geometries (ln/h = 2.4) diagonal confinement (clockwise from top left): (a) CB24D elevation; (b) cross section; (c) diagonal bundle section dimensions; and (d) beam construction (Dimensions are inches. 1in = 25.4mm) ....................................................................12

Figure 2.6 Test beam geometries (ln/h = 3.33) full section confinement (clockwise from top left): (a) CB33F elevation; (b) cross-section; and (c) beam construction (Dimensions are inches. 1in = 25.4mm) ............................................................................................................................................13

Figure 2.7 Test beam geometries (ln/h = 3.33) diagonal confinement (clockwise from top left): (a) CB33D elevation; (b) cross-section; (c) diagonal bundle section dimensions; and (d) beam construction (Dimensions are inches. 1in = 25.4mm) ....................................................................14

Figure 2.8 Test beam geometries (ln/h = 3.33) frame beam: (a) FB33 elevation; and (b) cross-section. (Dimensions are inches. 1in = 25.4mm)............................................................................15

Figure 2.9 Laboratory test setup .......................................................................................................16

Figure 2.10 Loading protocol: (a) Load-controlled; and (b) Displacement-controlled. (1k = 4.45kN) .............................................................................................................................................17

Figure 2.11 Sensor layout for: (a) CB24F and CB24D, and (b) CB33F, CB33D, and FB33 ..........19

Figure 2.12 Sensor layout for (a) CB24F-RC, and (b) CB24F-PT and CB24F-1/2-PT ...................20

Figure 2.13 Strain gauge layout for CB24F and CB33F. SG 12 and SG 14 are on horizontal crossties.............................................................................................................................................21

Figure 2.14 Strain gauge layout for CB24D and CB33D. SG 15 and SG 16 are located on horizontal crossties............................................................................................................................22

Figure 2.15 Strain gauge layout for CB24F-RC, CB24F-PT, and CB24F-1/2-PT. SG 12 and SG 16 are located on horizontal crossties. ..............................................................................................23

Figure 2.16 Strain gauge layout for FB33. SG 12 and SG 16 are located on horizontal crossties. ..24

Figure 3.1 Cyclic load-deformation: CB24F vs. CB24D (1in = 25.4mm) .......................................29

Figure 3.2 Cyclic load-deformation: CB33F vs. CB33D (1in = 25.4mm) .......................................29

Figure 3.3 Cyclic load-deformation: CB24F-PT vs. CB24F-1/2-PT (1in = 25.4mm) .....................30

viii

Figure 3.4 Moment curvature analysis summary for beam with and without slab (clockwise from top left): (a) Beam cross section with and without slab; (b) beam elevation with positive and negative moment capacities shown; (c) plot of Mn

- vs. curvature; and (d) plot of Mn+ vs.

curvature ...........................................................................................................................................32

Figure 3.5 Cyclic load-deformation: CB24F vs. CB24F-RC (1in = 25.4mm) .................................33

Figure 3.6 Axial elongation vs. rotation: CB24F vs. CB24F-RC (1in = 25.4mm)...........................33

Figure 3.7 Cyclic load-deformation: CB24F-RC vs. CB24F-PT (1in = 25.4mm) ...........................34

Figure 3.8 Axial elongation vs. rotation: CB24F-PT vs. CB24F-RC (1in = 25.4mm).....................34

Figure 3.9 Cyclic load-deformation: FB33 (1in = 25.4mm).............................................................35

Figure 3.10 CB24F damage photos: (a) 0.075% rotation; (b) 1% rotation; (c) 2% rotation; and (d) 3% rotation........................................................................................................................................38

Figure 3.11 CB24F damage photos: (a) 4% rotation; (b) 6% rotation; (c) 8% rotation; and (d) 10% rotation......................................................................................................................................39

Figure 3.12 CB24D damage photos: (a) 0.075% rotation; (b) 1% rotation; (c) 2% rotation; and (d) 3% rotation ..................................................................................................................................40

Figure 3.13 CB24D damage photos: (a) 4% rotation; (b) 6% rotation; (c) 8% rotation; and (d) 10% rotation......................................................................................................................................41

Figure 3.14 CB24F-PT damage photos: (a) 0.075% rotation; (b) 1% rotation; (c) 2% rotation; and (d) 3% rotation ...........................................................................................................................42

Figure 3.15 CB24F-PT damage photos: (a) 4% rotation; (b) 6% rotation; (c) 8% rotation; and (d) 10% rotation......................................................................................................................................43

Figure 3.16 CB24F-1/2-PT damage photos: (a) 0.075% rotation; (b) 1% rotation; (c) 2% rotation; and (d) 3% rotation ...........................................................................................................................44

Figure 3.17 CB24F-1/2-PT damage photos: (a) 4% rotation; (b) 6% rotation; (c) 8% rotation; and (d) 10% rotation ................................................................................................................................45

Figure 4.1 Effective stiffness plotted as a function of aspect ratio for various levels of displacement ductility (NZS 3101-1995). Included on the plot are test results at the corresponding ductility levels. ..........................................................................................................51

Figure 4.2 Effective stiffness vs. rotation: ln/h = 2.4 .......................................................................51

Figure 4.3 Yield rotation due to slip/extension vs. aspect ratio for various testing scales ...............52

Figure 4.4 Effective elastic stiffness as a function of gross section stiffness calculated for various aspect ratios and testing scales..........................................................................................................52

Figure 4.5 Determination of backbone relation from test data .........................................................55

Figure 4.6 Backbone load-deformation for full-scale beam models and ASCE 41-06 model (1/2-scale test results are dotted lines)......................................................................................................55

Figure 4.7 Cyclic load-deformation: CB24F vs. moment hinge model............................................58

Figure 4.8 Cyclic load-deformation: CB24F vs. shear hinge model ................................................58

ix

LIST OF TABLES

Table 2.1 - Test Matrix and Material Properties............................................................................. 7

Table 3.1 - Moment and Shear Strength Capacities ..................................................................... 26

Table 3.2 - Crack Widths [in.] ...................................................................................................... 37

Table 4.1 - Effective Stiffness Values .......................................................................................... 50

xi

LIST OF SYMBOLS

Acw = cross-sectional area of concrete beam web

Ash = area of transverse reinforcement provided within given spacing, s

Avd = cross-sectional area of each diagonal group of bars

bw = width of beam web

db = diameter of rebar

Ec = modulus of elasticity of concrete

f’c = concrete compressive strength

fy = yield strength of reinforcement

h = beam depth

Ieff = effective section moment of inertia

Ig = gross section moment of inertia

ln = clear span of beam

Mn = moment capacity of beam

My = yield moment of beam

s = longitudinal spacing of transverse reinforcement

V = beam shear

Vave = average beam shear between yield and onset of strength degradation

Vmax = max shear force applied during test

Vn = nominal shear capacity of beam

Vy = yield strength of beam

α = angle between diagonal bars and longitudinal axis of beam

Δ = relative displacement of beam end

Δy = relative displacement at yield

θ = beam chord rotation

θy = beam chord rotation at yield

1

1 Introduction

Tall building construction is common in metropolitan areas and it has become increasingly

important to provide methods of construction that improve both seismic performance and

constructability. Reinforced concrete core walls, with coupling beams above openings to

accommodate doorways, are an efficient lateral-force-resisting system for tall buildings. When

subjected to strong shaking, coupling beams act as fuses and typically undergo large inelastic

rotations.

Various testing programs have been carried out to assess the load – deformation behavior

of coupling beams.1-5 Primary test variables in these studies were the ratio of the beam clear span

to the beam total depth (commonly referred to as the beam aspect ratio) and the arrangement of

the beam reinforcement. In a majority of these studies, the load – deformation behavior of low-

aspect ratio beams (1.0 to 1.5) constructed with beam top and bottom longitudinal reinforcement

were compared with beams constructed with diagonal reinforcement. Concrete compressive

strengths for most tests were around 4 ksi (~25-30 MPa). Although these tests provided valuable

information, they do not address issues for current tall building construction, where beam aspect

ratios are typically between 2.0 and 3.5 and concrete strengths are in the range of 6 to 8 ksi (~40-

55 MPa). In addition, in none of the prior studies was a slab included as part of the test

specimen; whereas the slab might restrain axial elongations and impact stiffness, strength, and

deformation capacity.6-8 Use of post-tensioned slabs is common for current construction.

Use of diagonal reinforcement in coupling beams with clear length to total depth less than

four was introduced into ACI 318-95.9 Two groups of diagonal bars are commonly assumed to

form a truss, with one group serving as the tension member and the other as the compression

member. To enhance the compressive strength and deformation capacity of the diagonal truss

2

members as well as to suppress buckling of the diagonal bars, use of transverse reinforcement

around the diagonal bar groups is required. The quantity of transverse reinforcement required is

the same as that used for columns, which is substantially more than that used in most of the prior

coupling beam test programs. Nominal transverse reinforcement also is required around the

entire beam cross section. Providing transverse reinforcement around the diagonal bar bundles as

detailed in ACI 318-0510 S21.7.7 is difficult where the diagonal groups intersect at the beam

mid-span, particularly for shallow beams, as well as at the beam-wall interface due to

interference with the wall boundary vertical reinforcement (Fig. 1-1(a)). To combat these issues,

ACI 318-0811 S21.9.7 introduced an alternative detailing option, where transverse reinforcement

is placed around the beam cross section to provide confinement and suppress buckling, and no

transverse reinforcement is provided directly around the diagonal bar bundles (Fig. 1-1(b)). Use

of this detailing option avoids the problems noted where the diagonal bars intersect and at the

beam-wall interface, reducing the construction time for a typical floor by a day or two.12

In beams with aspect ratio (ln/h) approaching four, the angle of inclination (α) of the

diagonal reinforcement is often very small (~10°), making placement of the diagonal

reinforcement more difficult, as the diagonal bars are more likely to be obstructed by transverse

reinforcement. Use of straight (longitudinal) flexural reinforcement is common in these

situations, if the shear demand and required ductility are low.

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.13 Modeling

parameters for diagonally-reinforced coupling beams were introduced into Table 6-18 of FEMA

35614; given the limited test data available, only one row of modeling parameters is provided,

and these parameters remain unchanged in ASCE 41-06 (2007).15 Of particular interest is the

selection of the effective secant bending stiffness at yield c effE I and the allowable plastic rotation

prior to significant lateral strength degradation. The value used for coupling beam bending

stiffness has a significant impact on degree of coupling.16

Based on investigation of prior studies, the following parameters were deemed particularly

important for study:

1) Aspect ratio

2) Residual capacity/failure

3

3) Slab inclusion (RC and PT)

4) Detailing/confinement steel

a. Diagonal confinement

b. Full-section confinement

c. ½-section confinement

4

SECTION

Spacing measured perpendicular to the axis of the diagonal bars not to exceed 14 in., typical

*

*

(a)

SECTION

Spacing measured perpendicular to the axis of the diagonal bars not to exceed 14 in., typical

*

*

(a)

Alternate consecutive crosstie 90-deg hooks, both horizontally and vertically, typical

Spacing not to exceed 8 in., typical

SECTION

*Spacing not to exceed 8 in., typical

* *

(b)Alternate consecutive crosstie 90-deg hooks, both horizontally and vertically, typical

Spacing not to exceed 8 in., typical

SECTION

*Spacing not to exceed 8 in., typical

* *

(b)

Figure 1.1 Confinement options: (a) Diagonal confinement; and (b) Full section

confinement

5

2 Experimental Program

2.1 BEAM DESIGN

The test beam prototypes were based on two common tall building configurations for

residential and office construction. Typical wall openings and story heights produce coupling

beams with aspect ratios of approximately 2.4 for residential buildings and 3.33 for office

buildings. A coupling beam with cross-section dimensions of 24" 30"x ( )94.5 118mm x mm

and 24" 36"x ( )94.5 142mm x mm reinforced with two bundles of 8-#11 diagonal bars is

common for residential and office construction, respectively. The nominal shear strengths of the

residential and office beams, determined using ACI 318-08 equation 21-9

( )2 sin 10 'n vd y c cwV A f f Aα= ≤ , are 7.3 'c cvf A and 4.8 'c cvf A , for aspect ratios of 2.4

(α=15.7) and 3.33 (α=12.3), respectively, for Grade 60 reinforcement. Due to geometric and

strength constraints of an existing reaction frame, tests were conducted on one-half scale replicas

of the prototype beams. Thus the test specimens were either 12" 15"x ( )47 59mm x mm or

12" 18"x ( )47 71mm x mm with two bundles of 6-#7 diagonal bars (Figs. 2.1-2.5), for the

residential and office beams, respectively. For aspect ratio 3.33, a 12" 18"x ( )47 71mm x mm

specimen with two groups of 3-#6 straight (longitudinal) flexural reinforcement (referred to as

“frame beam”) was also tested (Fig. 2.8). The maximum shear stress expected for the frame

beam, based on reaching prM at the beam-wall interface at the beam ends, was 3.6 'cf . This

limit was selected based on input from practicing engineers; at higher shear stresses, use of

diagonal reinforcement is common.

6

As stated previously, the configuration of the transverse reinforcement was a primary

variable of the test program. Beams with transverse reinforcement provided around the bundles

of diagonal bars (referred to as “diagonal confinement”) were designed according to ACI 318-05

S21.7.7.4, whereas beams with transverse reinforcement provided around the entire beam cross

section (referred to as “full section confinement”) were designed according to ACI 318-08

S21.9.7.4(d). Volumetric ratios of transverse reinforcement and the ratios bar spacing to bar

diameter ( )/ bs d for the one-half scale test beams were selected to be similar to the prototype

beams. Due to maximum spacing requirements, the volumetric ratios of transverse reinforcement

provided in both the prototype and test beams exceed that calculated using the requirement for

columns (ACI 318-08 21.6.4.4). The test beam geometries and reinforcement configurations are

summarized in Table 2.1 and Figures 2.1-2.8.

Three test specimens with aspect ratio of 2.4 were constructed with 4”-thick slabs. One

specimen (CB24F-RC) contained a slab reinforced with #3 bars @12” spacing, on the top and

bottom in the transverse direction, and on the top only in the longitudinal direction, without post-

tensioning strands (Fig. 2.3). Two specimens (CB24F-PT and CB24F-1/2-PT) both contained a

similar reinforced-concrete slab, but also were reinforced with 3/8” 7-wire strands post-tensioned

to apply 150 psi to the slab in the longitudinal direction (Figs. 2.3-2.4). Specimen notation is

given in Table 2.1.

2.2 MATERIAL PROPERTIES

Material samples were taken and tested in order to determine actual properties for both concrete

compressive strength and steel tensile strengths. Concrete cylinders were tested to determine f’c

for each test specimen on the day of testing. Concrete cylinders were tested both in the UCLA

material testing laboratory as well as at Twining Testing Labs in Long Beach, CA, in order to

provide redundancy, and to help avoid error in the material testing process. Rebar coupons were

tested in order to determine yield and ultimate tensile strengths for steel in the coupling beam

specimens. Rebar in each specimen was taken from the same batch to ensure consistency from

test to test. These material properties are summarized in Table 2.1.

7

Table 2.1 - Test Matrix and Material Properties

Transverse Reinforcement

Beam ln/h type α[°]

Full Section Diagonals

AshactAshreq x

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

AshactAshreq y

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

f’c[psi] fy[psi] fu[psi] Description

CB24F #3 @ 3" N.A. 1.34 (1.25)1

1.24 (1.09)1 6850

Full section confinement ACI 318-08

CB24D #2 @ 2.5" #3 @ 2.5" 1.92 2.44 6850 Diagonal confinement ACI 318-05

CB24F-RC #3 @ 3" N.A. 1.34 (1.25)1

1.24 (1.09)1 7305

Full section conf. w/ RC slab ACI 318-08

CB24F-PT #3 @ 3" N.A. 1.34 (1.25)1

1.24 (1.09)1 7242

Full section conf. w/ PT slab

ACI 318-08

CB24F-1/2-PT

2.4 residential 15.7

#3 @ 6" N.A. 0.67 (0.63)1

0.62 (0.55)1 6990

Full section conf. (reduced) w/ PT slab

ACI 318-08

CB33F #3 @ 3" N.A. 1.34 (1.25)1

1.26 (1.06)1 6850

Full section confinement ACI 318-08

CB33D

12.3

#2 @ 2.5" #3 @ 2.5" 1.92 2.44 6850 Diagonal confinement ACI 318-05

FB33

3.33 office

0 #3 @ 3” N.A. - - 6000

70000 90000

Frame beam with 6-#6 straight bars

()1 Calculations for full-scale prototype beams [1psi = 0.0069MPa]

8

Section A-ASection A-A Section A-ASection A-A Figure 2.1 Test beam geometries (ln/h = 2.4) full section confinement: (a) CB24F, CB24F-RC, CB24F-PT, CB24F-1/2-PT

elevation; (b) CB24F cross section; and (c) CB24F-RC, CB24F-PT, CB24F-1/2-PT cross section. (Dimensions are inches. 1in

= 25.4mm)

9

Figure 2.2 Photographs of test specimens (ln/h = 2.4) full section confinement (clockwise from top left): (a) CB24F beam

construction; (b) CB24F-1/2-PT beam construction; (c) CB24F-PT beam elevation; and (d) CB24F-RC beam and slab

construction

10

Figure 2.3 - Slab geometry and reinforcement for CB24F-RC, CB24F-PT, and CB24F-1/2-PT: (a) Elevation view; and (b) plan

view. (Dimensions are inches. 1in = 25.4mm)

11

Figure 2.4 - Slab geometry and PT reinforcement for CB24F-PT and CB24F-1/2-PT: (a) Plan view; and (b) photo of post-

tensioning load application. (Dimensions are inches. 1in = 25.4mm)

12

Section B-BSection B-B

Figure 2.5 Test beam geometries (ln/h = 2.4) diagonal confinement (clockwise from top left): (a) CB24D elevation; (b) cross

section; (c) diagonal bundle section dimensions; and (d) beam construction (Dimensions are inches. 1in = 25.4mm)

13

Section C-CSection C-C

Figure 2.6 Test beam geometries (ln/h = 3.33) full section confinement (clockwise from top left): (a) CB33F elevation; (b) cross-

section; and (c) beam construction (Dimensions are inches. 1in = 25.4mm)

14

Section D-DSection D-D

Figure 2.7 Test beam geometries (ln/h = 3.33) diagonal confinement (clockwise from top left): (a) CB33D elevation; (b) cross-

section; (c) diagonal bundle section dimensions; and (d) beam construction (Dimensions are inches. 1in = 25.4mm)

15

Section E-ESection E-ESection E-ESection E-E

Figure 2.8 Test beam geometries (ln/h = 3.33) frame beam: (a) FB33 elevation; and (b) cross-section. (Dimensions are inches.

1in = 25.4mm)

16

2.3 TEST SETUP

The setup shown in Figure 2.9, where the test specimen was placed in a vertical position with

end blocks simulating wall boundary zones at each end, was used for all tests. The end blocks

were grouted and post-tensioned to the laboratory strong floor (bottom) and to the steel reaction

frame (top) to minimize slip between the surfaces as well as to provide for fixed end conditions.

Two vertical hydraulic actuators were used to ensure zero rotation at the top of the specimen,

while maintaining constant (zero) axial force in the beam.

The lateral load was applied via a horizontal actuator, with the line of action of the

actuator force passing through the mid-span (mid-height) of the test specimen to achieve zero

moment at the beam mid-span. To prevent out-of-plane rotation or twisting, a sliding truss

system was attached between the steel reaction frame and the reinforced concrete reaction wall.

Figure 2.9 Laboratory test setup

2.4 LOADING PROTOCOL

The testing procedure included load-controlled and displacement-controlled cycles (Fig. 2.10).

Load-control was performed at 0.125, 0.25, 0.50, and 0.75Vy, where 2y y nV M l= to ensure that

the load-displacement behavior prior to yield was captured. For residential beams, Vy was

17

assumed to be 120k using nominal material properties; for office beams, Vy was assumed to be

100k. Beyond 0.75Vy, displacement-control was used in increments of percent chord rotation (θ),

defined as the relative lateral displacement over the clear span of the beam (Δ) divided by the

beam clear span (ln) (excluding any contribution of slip and rotation of the bottom support

block). Three cycles were applied at each load increment for load controlled testing, and three

cycles were applied in displacement-control at each increment of chord rotation up to 3%, which

is approximately the allowable collapse prevention (CP) limit state for ASCE 41-06.15 Two

cycles were applied at each increment of chord rotation exceeding 3%.

-100

-50

0

50

100

Late

ral L

oad

[k]

ln/h = 2.4ln/h = 3.33

-12

-8

-4

0

4

8

12

Rota

tion

[%]

Figure 2.10 Loading protocol: (a) Load-controlled; and (b) Displacement-controlled. (1k =

4.45kN)

18

2.5 INSTRUMENTATION

Each of the test specimens was heavily instrumented. Linear Variable Differential Transformers

(LVDTs) were placed on the specimen to measure key deformation quantities; Figures 2.11 and

2.12 show the sensor layouts for the different test specimens. Vertical sensors (#1-12) measured

flexural response, diagonal sensors (#13-24) measured shear response, vertical sensors (#54-57)

at the beam-wall interface measured slip/extension deformations, horizontal sensors (#50-53) at

the beam-wall interface measured any sliding of the beam with respect to the wall, vertical

sensors (#40-41) spanning the full length of the beam measured axial elongation of the beam,

and all other sensors (#30-33 and AC-1,2) were used to measure the tip displacement of the

beam. As well, strain gauges were placed on diagonal, transverse, and longitudinal rebar (Fig.

2.13-2.16). Data from several different sensors was used to calculate values plotted in all results.

Individual sensor data are available from the authors. Eventually, the data will be uploaded to

the Network for Earthquake Engineering Simulation data repository. Data also will be stored on

a data server at UCLA.

19

Figure 2.11 Sensor layout for: (a) CB24F and CB24D, and (b) CB33F, CB33D, and FB33

20

Figure 2.12 Sensor layout for (a) CB24F-RC, and (b) CB24F-PT and CB24F-1/2-PT

21

Figure 2.13 Strain gauge layout for CB24F and CB33F. SG 12 and SG 14 are on horizontal

crossties

22

Figure 2.14 Strain gauge layout for CB24D and CB33D. SG 15 and SG 16 are located on

horizontal crossties.

23

Figure 2.15 Strain gauge layout for CB24F-RC, CB24F-PT, and CB24F-1/2-PT. SG 12 and

SG 16 are located on horizontal crossties.

24

Figure 2.16 Strain gauge layout for FB33. SG 12 and SG 16 are located on horizontal

crossties.

25

3 Experimental Results and Discussion

Results from the tests are presented and discussed. Overall load-displacement relations are

compared to assess the impact of providing full section confinement as opposed to confinement

around the diagonals for both residential- and office-use beams. The role of transverse

reinforcement is examined by comparing load-displacement relations for the beams, including

one beam with only one-half of the required transverse reinforcement. Other comparisons are

made that examine the effect of the floor slab (both reinforced concrete (RC) and post-tensioned

reinforced concrete (PT)) on the beam load-deformation response, including the effective elastic

bending stiffness at yield as well as the influence of scale on the test results. Table 3.1

summarizes the calculated strengths, as well as the actual strengths and deformations of each test

specimen at major points.

26

Table 3.1 - Moment and Shear Strength Capacities

Beam Mn+ [in-k] Mn

- [in-k] V@Mn [k]@

'n

c cv

V Mf A Vn(ACI)[k]

( )'

n

c cv

V ACIf A Vave [k] '

ave

c cv

Vf A Vy [k] Δy [in] Vmax [k] Δ@Vmax [in]

CB24F 2850 2850 158.3 10.65 136.3 9.15 154.9 10.40 121.3 0.360 171.0 1.08

CB24D 2850 2850 158.3 10.65 136.3 9.15 150.7 10.12 128.8 0.363 159.2 2.16

CB24F-RC

2890 (3550)1

2890 (3350)1

160.6 (191.7)1

10.45 (12.50)1 136.3 8.87 181.0 11.77 147.2 0.362 190.8 2.16

CB24F-PT

3160 (3960)1

3160 (3625)1

175.6 (210.7)1

11.45 (13.75)1 136.3 8.90 198.9 12.98 163.2 0.361 211.8 2.16

CB24F-1/2-PT

3145 (3940)1

3145 (3610)1

174.7 (209.7)1

11.61 (13.90)1 136.3 9.06 182.4 12.12 158.1 0.365 189.6 1.08

CB33F 3615 3615 120.5 6.77 107.8 6.03 118.3 6.62 107.7 0.600 124.0 1.80

CB33D 3615 3615 120.5 6.77 107.8 6.03 114.7 6.42 95.94 0.601 120.6 3.60

FB33 1450 1450 48.3 2.89 - - 56.3 3.37 47.86 0.306 58.1 1.20

1Calculations that consider the impact of the slab [1 in-k = 113 mm-kN, 1 in = 25.4 mm, 1 k = 4.45 kN]

27

3.1 DETAILING

3.1.1 Full Section vs. Diagonal Confinement

Load-deformation responses of CB24F and CB24D are very similar over the full range of

applied rotations (Fig. 3.1). Notably, both beams achieve large rotation (~8%) without significant

degradation in the lateral load carrying capacity, and the beams achieve shear strengths of 1.25

and 1.17 times the ACI nominal strength (Table 3.1). The shear strength of CB24D degraded

rapidly at around 8% rotation, whereas CB24F degraded more gradually, maintaining a residual

shear capacity of ~80% at rotations exceeding 10%.

Figure 3.2 plots load vs. rotation relations for the 3.33 aspect ratio beams with full section

confinement (CB33F) vs. diagonal confinement (CB33D). Similar to the 2.4 aspect ratio beams,

Figure 3.2 reveals that the beams have similar strength (Table 3.1), stiffness, deformation, and

damage (Table 3.2) characteristics.

The test results presented in Figures 3.1-3.2 indicate that the full section confinement

option of ACI 318-08 provides equivalent, if not improved performance, compared to

confinement around the diagonals per ACI 318-05.

3.1.2 Full vs. Half Confinement

The transverse reinforcement used for CB24F-1/2-PT was one-half that used for CB24F-PT to

assess the impact of using less than the code-required transverse reinforcement given that the

requirements of S21.6.4 are based on column requirements. Figure 3.3 plots load-deformation

responses and reveals similar loading and unloading relations up to 3% total rotation, which

approximately corresponds to the Collapse Prevention limit state per ASCE 41-06. At higher

rotations (θ≥4%), modest strength degradation is observed for CB24F-1/2-PT, whereas the

strength of CB24F-PT continues to increase slightly; however, both beams achieve rotations of

~8% before significant lateral strength degradation (<0.8Vave). Vave is defined as the average

shear force resisted by the beam between the yield point and the onset of significant lateral

strength degradation.

28

The results indicate that the one-half scale coupling beams tested with ACI 318-08

detailing are generally capable of achieving total rotations exceeding 8%, whereas ASCE 41

limits plastic rotation to 3% without strength degradation and 5% with 20% strength degradation.

The potential influence of scale on the test results is discussed later (Section 4.2). The test results

indicate that there is little difference in load-deformation response between CB24F-PT and

CB24F-1/2-PT; therefore, the potential to reduce the quantity of required transverse

reinforcement exists, but requires further study since only one beam test was conducted.

29

-4.32 -2.16 0 2.16 4.32Relative Displacement [in]

-200

-100

0

100

200

Late

ral L

oad

[k]

-12 -6 0 6 12Beam Chord Rotation [%]

-890

-445

0

445

890

Late

ral L

oad

[kN

]

CB24FCB24D

Vn (ACI)

Vn (ACI)

Figure 3.1 Cyclic load-deformation: CB24F vs. CB24D (1in = 25.4mm)

-6 -3 0 3 6Relative Displacement [in]

-150

-100

-50

0

50

100

150

Late

ral L

oad

[k]


-670

-335

0

335

670

Late

ral L

oad

[kN

]CB33FCB33D

Vn (ACI)

Vn (ACI)

*

**

* Stroke of controlling sensor exceeded

** Stroke of LVDT exceeded

Figure 3.2 Cyclic load-deformation: CB33F vs. CB33D (1in = 25.4mm)

30

-5 -2.5 0 2.5 5Relative Displacement [in]

-200

-100

0

100

200La

tera

l Loa

d [k

]


-980

-490

0

490

980

Late

ral L

oad

[kN

]

CB24F-PTCB24F-1/2-PT Vn (ACI)

Vn (ACI)

Figure 3.3 Cyclic load-deformation: CB24F-PT vs. CB24F-1/2-PT (1in = 25.4mm)

31

3.2 SLAB IMPACT

Four beams with aspect ratio of 2.4 were tested to systematically assess the impact of a slab on

the load-deformation responses. CB24F did not include a slab, whereas CB24F-RC included an

RC slab, and CB24F-PT and CB24F-1/2-PT included PT slabs (with 150 psi of prestress).

Comparing the load-displacement responses of CB24F vs. CB24F-RC, Figure 3.5 reveals that

the slab increases shear strength by 17% (155 k to 181 k); however, this strength increase can be

taken into account by considering the increase in nominal moment strength due to the presence

of the slab, i.e. slab concrete in compression at the beam-wall interface at one end, and slab

tension reinforcement at the beam-wall interface at the other end (Figure 3.4 and Table 3.1). For

example, consideration of the slab produces increases of approximately 20% in the nominal

moment capacities, which also provide similar increases in beam shear (since yielding of

diagonal reinforcement limits the shear forces on the beams). The results indicate that the higher

test shear strength observed is primarily due to the increase in nominal moment capacity when a

slab is present.

The presence of a slab, and in particular, a post-tensioned slab, might impact the load-

deformation behavior by restraining the axial growth along the member length. Figure 3.6 plots

the axial growth of CB24F vs. CB24F-RC and reveals that the axial growth is very similar for

the two tests. Both beams grow approximately one inch over the course of the test, with

relatively large cracks observed at the beam-wall interface. Strength degradation for CB24F is

noted at 8%, due to the buckling and eventual fracture of the diagonal bars, leading to axial

shortening, whereas the axial extension in CB24F-RC remains stable over the entire test due to

the presence of the slab.

Load-deformation responses for CB24F-RC vs. CB24F-PT are compared in Figure 3.7

and display similar overall behavior, with CB24F-PT experiencing higher shear forces

(13.0 'c cwf A ) than CB24F-RC (11.8 'c cwf A ). This increase in strength is primarily due to the

axial force applied to the specimen by the tensioned strands, which provided approximately 150

psi stress to the slab and increased the nominal moment strength (Table 3.1). Between 8% and

10% rotations, strength degradation is more pronounced for CB24F-PT than CB24F-RC, with

32

30% reduction for CB24F-PT vs. 10% for CB24F-RC, possibly due to the presence of pre-

compression.

A plot of axial elongation of CB24F-RC vs. CB24F-PT, (Fig. 3.8), indicates that the PT

slab with 150 psi prestress grows 30-40% less than the RC slab. As well, the PT slab, like the RC

slab in CB24F-RC, helps to maintain the axial integrity of the beam for rotations exceeding 6%.

0 0.00216 0.00432Curvature [in-1]

0

1000

2000

3000

4000

Mom

ent [

in-k

]

No SlabSlab

Mn+ Mn

-

0 0.0005 0.001 0.0015 0.002 0.0025Curvature [in-1]

No SlabSlab

Mn+ (slab) = 3550 in-k

Mn+(no slab) = 2850 in-k

Mn- (slab) = 3350 in-k

Mn-(no slab) = 2850 in-k

0 0.00216 0.00432Curvature [in-1]

0

1000

2000

3000

4000

Mom

ent [

in-k

]

No SlabSlab

Mn+ Mn

-

0 0.0005 0.001 0.0015 0.002 0.0025Curvature [in-1]

No SlabSlab

Mn+ (slab) = 3550 in-k

Mn+(no slab) = 2850 in-k

Mn- (slab) = 3350 in-k

Mn-(no slab) = 2850 in-k

Figure 3.4 Moment curvature analysis summary for beam with and without slab (clockwise

from top left): (a) Beam cross section with and without slab; (b) beam elevation with

positive and negative moment capacities shown; (c) plot of Mn- vs. curvature; and (d) plot

of Mn+ vs. curvature

33


-200

-100

0

100

200La

tera

l Loa

d [k

]


-980

-490

0

490

980

Late

ral L

oad

[kN

]

CB24FCB24F-RC Vn (ACI)

Vn (ACI)

Figure 3.5 Cyclic load-deformation: CB24F vs. CB24F-RC (1in = 25.4mm)


-0.4

0

0.4

0.8

1.2

Axi

al elo

ngat

ion

[in]


-3

-1.5

0

1.5

3

Axi

al el

onga

tion

[cm

]

CB24FCB24F-RC

Figure 3.6 Axial elongation vs. rotation: CB24F vs. CB24F-RC (1in = 25.4mm)

34


-200

-100

0

100

200La

tera

l Loa

d [k

]-14 -7 0 7 14

Beam Chord Rotation [%]

-980

-490

0

490

980

Late

ral L

oad

[kN

]CB24F-RCCB24F-PT

Vn (ACI)

Vn (ACI)

Figure 3.7 Cyclic load-deformation: CB24F-RC vs. CB24F-PT (1in = 25.4mm)


-0.4

0

0.4

0.8

1.2

Axi

al elo

ngat

ion

[in]


-3

-1.5

0

1.5

3

Axi

al el

onga

tion

[cm

]

CB24F-RCCB24F-PT

Figure 3.8 Axial elongation vs. rotation: CB24F-PT vs. CB24F-RC (1in = 25.4mm)

35

3.3 FRAME BEAM

FB33 was tested to assess the impact of providing straight bars as flexural reinforcement instead

of diagonal bars in beams with relatively low shear stress demand (< 4.0 'cf ). A plot of load

vs. deformation for FB33 (Fig. 3.9) indicates that plastic rotations greater than 4% can be

reached prior to strength degradation. These results correspond well with prior test results5 on

similarly sized beams, which achieved maximum shear stresses of about 4.7 'cf and plastic

chord rotations greater than 3.5%. Compared with CB33F and CB33D (Fig. 3.2), FB33

experiences pinching in the load-deformation plot, indicating that less energy is dissipated. As

well, the beams with diagonal reinforcement exhibited higher ductility, reaching plastic rotations

exceeding 7% prior to strength degradation. However, for beams that are expected to experience

shear forces less than 5.0 'c cwf A , frame beams with straight bars can provide significant

ductility (θp > 4%), and are much easier to construct than diagonally-reinforced beams.

Therefore, adding a shear stress limit of 5.0 'cf for conventionally-reinforced coupling beams

with aspect ratio between 2 and 4 to ACI 318-08 21.9.7 might be prudent. At a minimum, ACI

318 should add commentary to note the significant difference in deformation capacity between

diagonally- and longitudinally-reinforced coupling beams.


-80

-40

0

40

80

Late

ral L

oad

[k]


-356

-178

0

178

356

Late

ral L

oad

[kN

]

Figure 3.9 Cyclic load-deformation: FB33 (1in = 25.4mm)

36

3.4 DAMAGE

Figures 3.10-11 and 3.12-13 are photos of CB24F and CB24D at the peak of every displacement

stage between 0.075% and 10% total rotations, respectively, and reveal that maximum diagonal

crack widths for CB24F were less than 0.02” and flexural crack widths of 0.08 and 0.125” were

measured at 3 and 6% rotations (Table 3.2). In general, diagonal crack widths for CB24D were

larger than for CB24F, possibly due to the reduced transverse reinforcement around the full

section. The results indicate beams detailed with full section confinement might require less

repair than beams detailed with diagonal confinement following an earthquake.

Diagonal crack widths for CB24F-1/2-PT (Figs. 3.16-17) are much larger than those

observed for CB24F-PT (Figs. 3.14-15), especially for rotations exceeding 6%. At 4% rotation,

1/16” diagonal cracks were noted in CB24F-1/2-PT, whereas diagonal cracks were still hairline

in CB24F-PT. Beyond 4% rotation, for CB24F-1/2-PT, spalling of cover concrete was noted,

with 1/4” diagonal cracks noted at 6% rotation; buckling and fracture of reinforcement, and

crushing of the core concrete were noted for rotations between 8 and 10%. In contrast, minimal

damage was observed for CB24F-PT (Figs. 3.14-15), with hairline diagonal cracks and flexural

crack widths of less than 1/4”, with most of the rotation due to rebar slip/pullout at the beam-wall

interface (approximately 1/2” at 6% rotation). Crack widths for all specimens are summarized in

Table 3.2. More photos of damage for all specimens are provided in Appendix A. Particularly, it

is of interest to know the degree of residual damage (i.e. at zero rotation) for repair purposes.

Pictures showing the residual damage of each beam after each rotation level are also shown in

Appendix A.

37

Table 3.2 - Crack Widths [in.]

1% 3% 6% Beam

Slip/ext Flexure Shear Slip/ext Flexure Shear Slip/ext Flexure Shear

CB24F 0.125 0.065 hairline 0.400 0.080 hairline 0.750 0.125 0.015

CB24D 0.125 0.095 hairline 0.375 0.125 0.016 0.500 0.250 0.125

CB24F-RC 0.095 0.045 hairline 0.500 0.125 0.016 0.500 0.375 0.065

CB24F-PT 0.065 0.030 hairline 0.250 0.190 hairline 0.500 0.250 hairline

CB24F-1/2-PT 0.065 0.015 hairline 0.375 0.190 0.031 0.625 0.375 0.250

CB33F 0.125 0.065 hairline 0.315 0.065 hairline 0.500 0.250 0.015

CB33F 0.125 0.065 hairline 0.250 0.125 0.016 0.500 0.190 0.125

FB33 0.060 0.030 hairline 0.250 0.250 0.125 - - -

[1 in = 25.4 mm]

38

Rotation = 0.0075

Rotation = 0.01

Rotation = 0.02

Rotation = 0.03

Figure 3.10 CB24F damage photos: (a) 0.075% rotation; (b) 1% rotation; (c) 2% rotation; and (d) 3% rotation

39

Rotation = 0.04

Rotation = 0.06

Rotation = 0.08

Rotation = 0.10

Figure 3.11 CB24F damage photos: (a) 4% rotation; (b) 6% rotation; (c) 8% rotation; and (d) 10% rotation

40

Rotation = 0.0075

Rotation = 0.01

Rotation = 0.02

Rotation = 0.03

Figure 3.12 CB24D damage photos: (a) 0.075% rotation; (b) 1% rotation; (c) 2% rotation; and (d) 3% rotation

41

Rotation = 0.04

Rotation = 0.06

Rotation = 0.08

Rotation = 0.10

Figure 3.13 CB24D damage photos: (a) 4% rotation; (b) 6% rotation; (c) 8% rotation; and (d) 10% rotation

42

Rotation = 0.0075

Rotation = 0.01

Rotation = 0.02

Rotation = 0.03

Figure 3.14 CB24F-PT damage photos: (a) 0.075% rotation; (b) 1% rotation; (c) 2% rotation; and (d) 3% rotation

43

Rotation = 0.04

Rotation = 0.06

Rotation = 0.08

Rotation = 0.10

Figure 3.15 CB24F-PT damage photos: (a) 4% rotation; (b) 6% rotation; (c) 8% rotation; and (d) 10% rotation

44

Rotation = 0.0075

Rotation = 0.01

Rotation = 0.02

Rotation = 0.03

Figure 3.16 CB24F-1/2-PT damage photos: (a) 0.075% rotation; (b) 1% rotation; (c) 2% rotation; and (d) 3% rotation

45

Rotation = 0.04

Rotation = 0.06

Rotation = 0.08

Rotation = 0.10

Figure 3.17 CB24F-1/2-PT damage photos: (a) 4% rotation; (b) 6% rotation; (c) 8% rotation; and (d) 10% rotation

47

4 Modeling

Typical modeling procedures are discussed and results generated with models are compared to

test results. Specifically, models for effective secant stiffness at yield are presented to provide a

direct comparison between typical parameters used by engineers and values obtained via testing.

As well, the impact of scaling test specimens is investigated to allow test results to be applied to

full-scale models. Based on these studies, backbone relations are fit to all test results and

modified to represent the behavior of the beam at full-scale. These backbone relations can be

used directly in computer software, and the load-deformation results of one specific modeling

effort are presented.

4.1 EFFECTIVE STIFFNESS

Elastic analysis approaches require estimation of the effective elastic bending and shear stiffness

values. In FEMA 35614, stiffness values of 0.5 c gE I and 0.4 c cwE A are recommended for bending

and shear, respectively. ASCE 41-06 including Supplement #115 incorporates a lower value for

effective stiffness of 0.3 c gE I , with a mean value obtained from tests of 0.2 c gE I .17 The New

Zealand Code (NZS-3101 1995)18 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×

=+ ×

(Eq. 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]. For beams with aspect ratio ln/h = 2.4, Equation 1 yields a beam with

effective elastic stiffness of around fifty percent of the gross section stiffness, 0.5 c gE I , whereas

48

for a ductility ratio of 6, the effective (secant) stiffness drops to eighteen percent of the gross

section properties, 0.18 c gE I . All of these values are summarized and compared with the test

results in Figure 4.1.

Figure 4.2 plots the secant stiffness normalized with respect to the concrete gross section

stiffness versus the chord rotation. Secant stiffness is calculated assuming fixed end conditions

according to: 3

12n

c effV lE I ×

=×Δ

. The initial stiffness of each residential beam is

approximately 0.25 c gE I , with an effective stiffness at the yield rotation (~1.0% rotation)

of 0.12 c gE I . Effective secant stiffness values corresponding to ASCE 41-06 limit states are

approximately 0.15 c gE I at Immediate Occupancy (~0.6% rotation), 0.075 c gE I at Life Safety

(~1.8% rotation), and 0.05 c gE I at Collapse Prevention (~3% rotation). The effective stiffness

ratio ( eff gI I ) does not vary significantly for the three different configurations (Fig. 4.2), i.e.

beam without slab (CB24F, CB24D), beam with RC slab (CB24F-RC), and beam with PT slab

(CB24F-PT, CB24F-1/2-PT). The initial stiffness ratio for the beams with slabs is moderately

higher (~25%) for rotations up to about 2%; however, after significant flexural cracks form at the

slab-wall interface, generally at ~3% rotation, the stiffness ratio is nearly the same for all three

test configurations.

The low secant stiffness ratios ( eff gI I ) relative to recommended values (Table 4.1)

might imply that significant damage (cracking, concrete spalling) is required to achieve these

ratios. However, photos of beam damage, Figures 3.10-13 for the beams without slabs, and

Figures 3.14-17 for the beams with slabs, do not show significant spalling and diagonal crack

widths are limited to 1/32” even at 6% total rotation (Table 3.2); damage is concentrated at the

beam-wall interface in the form of 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 as10.5 to 13.8 'cf . The larger diagonal crack widths observed for CB24F-1/2-

PT, with only one-half the required transverse reinforcement, indicate that the quantity of

transverse reinforcement provided in CB24F, CB24F-RC, and CB24F-PT could likely be

reduced moderately without compromising deformation capacity. Current modeling of the load-

deformation response of coupling beams tends to focus on shear behavior19; however, for the 2.4

49

and 3.33 aspect ratio beams tested, flexural and slip/extension deformations at and adjacent to

the beam-wall interface generally accounted for more than 85% of the total rotation.

Of the various approaches noted above for estimating the effective stiffness at yield, i.e.

FEMA 356 ( )0.5 c gE I , ASCE 41-06 ( )0.3 c gE I , and NZS-3101 1995 for low ductility

( )0.5 c gE I , 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 (2007)17 is

actually 0.2 c gE I at low axial load, the value of 0.3 c gE I is used as a compromise to address

issues associated with deformation compatibility checks for gravity columns].

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

using the approach recommended by Alsiwat and Saatcioglu20, where the crack width that

develops at the beam-wall interface depends on bar slip and bar extension (strain). Using a

coupling beam effective stiffness derived from a moment-curvature analysis of the beam cross-

section at the beam-wall interface ( )~ 0.5 c gE I and the slip/extension model noted above, the

effective stiffness at yield reduces to 0.12 c gE I , which is consistent with the effective stiffness at

the yield rotation (approximately 1.0% for all beams) derived for the tests (Fig. 4.2). Additional

details of the slip/extension calculations are included in Appendix B.

Table 4.1 provides a summary of the effective stiffness and yield rotation for each of the

different models discussed above. Based on these results, use of the model detailed in ASCE 41-

06 Supplement #1 is recommended, i.e., use a moment-curvature analysis to define the secant

stiffness 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 (~ 0.12 c gE I for the test beams).

4.2 EFFECT OF SCALE

As previously stated, the tests were conducted at one-half 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 contribution of flexural deformations (curvature) and

slip/extension to the yield rotation of the test beams at full scale (i.e. prototype beams) is

assessed using the same approach as noted in the previous paragraph for the one-half scale

50

beams. The study is extended to consider coupling beam aspect ratios beyond those tested, by

varying the beam length. Results are reported in Figure 4.3, where the effective yield rotation 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 moderate impact on the slip rotation, producing roughly a

15 to 20% increase from aspect ratios of 1.0 to 3.0. However, for a given aspect ratio, slip

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 one-half scale tests of

0.12 c gE I increases to 0.14 c gE I for the full-scale prototypes due to the reduction in the relative

contribution of slip rotation. Based on these results, we recommend use of an effective yield

stiffness value of 0.15 c gE I for full-scale coupling beams. Figure 4.4 provides a summary of

calculated values of effective yield stiffness for coupling beams with aspect ratios

2.0 4.0nl h≤ ≤ , for both full-scale and half-scale beams (for comparison purposes). The

specifics of these calculations are provided in Appendix C.

Table 4.1 Effective Stiffness Values

EIeff [% EIg] θy [% drift]

Test Results 14.0 (12.5)1

0.70 (1.00)1

FEMA 356 50.0 0.23

ASCE 41 30.0 0.39

ASCE 41 S1, w/slip hinge

16.5 (13.0)1

0.75 (0.95)1

NZS-3101 95 (μ=1) 50.0 0.23

1Modifications for 1/2-scale

51

1 2 3 4Ln/h

0

0.2

0.4

0.6

I eff

/Ig

m=1.25

m=3.0

m=4.5

m=6.0

Figure 4.1 Effective stiffness plotted as a function of aspect ratio for various levels of

displacement ductility (NZS 3101-1995). Included on the plot are test results at the

corresponding ductility levels.

0 2 4 6Beam Chord Rotation [%]

0

0.1

0.2

0.3

I eff/

I g

CB24F-PTCB24F-RCCB24F

Figure 4.2 Effective stiffness vs. rotation: ln/h = 2.4

52

1 2 3 4ln/h

0.002

0.003

0.004

0.005

0.006

Slip

Rot

atio

n [ra

d]

1/2-scale2/3-scale3/4-scaleFull-scale

Figure 4.3 Yield rotation due to slip/extension vs. aspect ratio for various testing scales

2 2.4 2.8 3.2 3.6 4ln/h

0.05

0.1

0.15

0.2

0.25

Effe

ctiv

e St

iffne

ss [I

eff/

I g]

Full-scale1/2-scale

Figure 4.4 Effective elastic stiffness as a function of gross section stiffness calculated for

various aspect ratios and testing scales

53

4.3 BACKBONE RELATIONS

Linearized backbone relations for normalized shear strength versus rotation are plotted in Figure

4.6 as dotted lines for the three configurations of beams tested, i.e. 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 Figure 4.5, which

plots the peaks of the load-deformation curves for CB24F and CB24D. The backbone relations

that are modified to represent full-scale beams are also plotted in Figure 4.6, as discussed in the

prior subsection. 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 Figure 4.6, with the ratios of ave nV V being approximately 1.1 (no slab), 1.3 (RC

slab), and 1.4 (PT slab), where Vave is defined as the average shear force resisted by the beam

between the yield point and the onset of significant lateral strength degradation.

ASCE 41-06 with Supplement #1 modeling parameters also are plotted on Figure 4.6 and

indicate that the test beams are more flexible at yield and that they attain substantially higher

deformation capacity prior to lateral strength degradation than the standard backbone relation.

The elastic stiffness of the ASCE 41 relation is based on a bending stiffness of 0.3 c gE I , or about

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 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

(0.14EcIg/0.3EcIg = 0.47) 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 maintained to very large rotations

(10 to 12%) compared to the ASCE 41-06 residual strength ratio of 0.8 at a plastic rotation value

of 5.0%. Therefore, it is reasonable to use a plastic rotation value of 5.0% with no strength

54

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. 4.6 apply for the beam aspect ratios tested (2.4 and 3.33), as

well as to beams between these ratios. It is reasonable to assume these values can be extrapolated

modestly to apply to beams with 2.0 4.0nl h≤ ≤ .

55

0 2 4 6 8 10 12 14Beam Chord Rotation [%]

0

0.2

0.4

0.6

0.8

1

1.2

1.4

V/V

ncod

e

CB24FCB24DLinear Backbone

Figure 4.5 Determination of backbone relation from test data

0 2 4 6 8 10 12 14Rotation [% drift]

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

V/V

ncod

e

PT SlabRC SlabNo SlabASCE 41-06

Figure 4.6 Backbone load-deformation for full-scale beam models and ASCE 41-06 model

(1/2-scale test results are dotted lines)

56

4.4 MODEL VS. TEST RESULTS

Based on the backbone and effective stiffness relations discussed above, 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 utilizing a

rotational spring at the ends of the beam to account for both nonlinear flexural and slip/extension

deformations (Mn hinge) and one utilizing a nonlinear shear spring at beam mid-span to account

for both shear and slip/extension deformations (Vn hinge). Both models were subjected to the

same loading protocol used in the tests (Fig. 2.8).

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 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 were defined using

the Alsiwat and Saatcioglu20 model discussed above, whereas the nonlinear flexural hinges are

modeled using the backbone relations derived from test results (Fig. 4.6, excluding the elastic

portion). 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 the beam mid-span to simulate the effects of nonlinear

deformations. The shear hinge properties are defined using the backbone relations derived from

the test results (Fig. 4.6).

Figures 4.7 and 4.8 shows cyclic load-deformation plots for the two models and the test

results for CB24F. Both models accurately capture the overall load-displacement response of the

member; however, the Mn-hinge model (Fig. 4.7) captures the unloading characteristics better

than the Vn-hinge model (Fig. 4.8), 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. As noted previously, for the

beam test aspect ratios (2.4 and 3.33), flexural and slip/extension deformations account for

approximately 80-85% of total deformation whereas shear deformations generally account for

only l5-20% of total deformation. Therefore, in both models, the flexural and shear hinges are

used to represent flexural deformations, whereas shear deformations are not considered.

57

Therefore, depending on the computer program used, modeling studies similar to those presented

here should be conducted to calibrate available model parameters with test results.

Specifically, these models were created using Perform 3D, as it is the common program

used by design engineers in nonlinear modeling of structural systems. The parameters used in

each model are summarized in detail in Appendix D.

58

-0.12 -0.06 0 0.06 0.12Beam Chord Rotation [rad]

-200

-100

0

100

200

Late

ral L

oad

[k]

-890

-445

0

445

890

Late

ral L

oad

[kN

]

Test (CB24F)Mn Hinge

Figure 4.7 Cyclic load-deformation: CB24F vs. moment hinge model

-0.12 -0.06 0 0.06 0.12Beam Chord Rotation [rad]

-200

-100

0

100

200

Late

ral L

oad

[k]

-890

-445

0

445

890

Late

ral L

oad

[kN

]

Test (CB24F)Vn Hinge

Figure 4.8 Cyclic load-deformation: CB24F vs. shear hinge model

59

5 Conclusions

Eight coupling beam specimens with ln/h ratios of 2.4 and 3.33, and varying geometries and

reinforcement layouts, were tested under reversed cyclic loading and double curvature bending.

The following conclusions can be drawn from the test results.

1) Beams detailed according to the new provision in ACI 318-08, which allows for full

section confinement, have performance, in terms of strength and ductility, that is better

than beams detailed according to the old provision in ACI 318-05, which requires

confinement of the diagonal bar groups.

2) Including a reinforced concrete slab increases the beam shear strength approximately 15-

20%, whereas adding post-tensioning increases the beam shear strength an additional

10%. However, the strength increase was directly related to the increase in beam moment

strength, as the beam shear force was limited by flexural yielding.

3) Beams detailed to satisfy 1/2*Ash perform well at chord rotations θ < 3.0%. However, at

very large rotations (θ > 6.0%), the beams experienced greater levels of damage (i.e.

more spalling of cover concrete and substantially larger shear cracks > 1/4”) compared

with beams detailed to satisfy Ash. The results indicate that the amount of transverse

reinforcement required could be modestly reduced for the beam aspect ratios tested,

especially for beams with lower ductility requirements (θ < 3.0%.). However, further

study is necessary to determine if less transverse reinforcement could be used for

rotations exceeding 3%, or for beams with lower aspect ratios (< 2).

60

4) Effective elastic stiffness values for test beams are determined to be ~15% of the gross

section stiffness, values that are much less than FEMA and ASCE 41 prescribed values of

50% and 30%, respectively. Designers should therefore utilize the slip/extension hinge

model detailed in Supplement 1 to ASCE 41 to better approximate the elastic stiffness of

the coupling beam.

5) Most damage experienced by coupling beams with aspect ratio ranging from 2.4 to 3.33

is concentrated at the beam-wall interface in the form of slip/extension of diagonal

reinforcement, even when axial load is applied to the beam via post-tensioning. Beams

not detailed with full section confinement experience more damage at large rotations (θ >

6.0%).

6) ACI 318-08 implies equivalence between diagonally-reinforced and “frame beams” for

aspect ratios between 2.0 and 4.0. However, frame beams typically achieve maximum

plastic chord rotations of 3.5 to 4.0%, for cases where the expected shear stresses are

4.0 to 5.0 'cf , or about one-half the values for diagonally-reinforced coupling beams

tested. Changes to ACI 318 code should be considered to reduce the shear stress allowed

for frame beams ( )e.g., 5.0 'cf , or to the ACI commentary to identify this significant

difference in performance.

7) 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. Therefore, 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.

61

REFERENCES

1. Paulay, T., and Binney, J. R., 1974, “Diagonally Reinforced Coupling Beams of Shear

Walls,” Shear in Reinforced Concrete, SP-42, American Concrete Institute, Farmington

Hills, Mich., pp. 579-598.

2. Tassios, T. P.; Moretti, M.; and Bezas, A., 1996, “On the Coupling Behavior and Ductility

of Reinforced Concrete Coupling Beams of Shear Walls,” ACI Structural Journal, V. 93,

No. 6, Nov.-Dec., pp. 711-720.

3. Kwan, A. K. H. and Zhao, Z. Z., 2001, “Testing of coupling beams with equal end

rotations maintained and local joint deformation allowed,” Structures and Buildings,

Thomas Telford, London, 152, No. 1, 67–78.

4. Galano, L., and Vignoli, A., 2000, “Seismic Behavior of Short Coupling Beams with

Different Reinforcement Layouts,” ACI Structural Journal, V. 97, No. 6, Nov.-Dec., pp.

876-885.

5. Xiao, Y.; Esmaeily-Ghasemabadi, A.; and H. Wu, "High-Strength Concrete Beams

Subjected to Cyclic Shear," ACI Structural Journal Vol. 96 No.3, May-June 1999,

pp.392-399.

6. Klemencic, R., Fry, J.A., Hurtado, G., and Moehle, J.P., 2006, “Performance of Post-

tensioned slab-core wall connections,” PTI Journal, 4(2), 7-23.

7. Kang, T. H.-K., and Wallace, J. W., “Dynamic Responses of Flat Plate Systems with Shear

Reinforcement,” ACI Structural Journal, V. 102, No. 5, Sept.-Oct. 2005, pp. 763-773.

8. Kang, T. H.-K., and Wallace, J. W., “Punching of Reinforced and Post-Tensioned

Concrete Slab-Column Connections,” ACI Structural Journal, V. 103, No. 4, July-August

2006, pp. 531-540.

9. ACI Committee 318, 1995, “Building Code Requirements for Structural Concrete (ACI

318-95) and Commentary (318R-95),” American Concrete Institute, Farmington Hills,

Mich., 430 pp.



Mich., 430 pp.

62



Mich., 430 pp.

12. ENR, 2007. “Good News for Tall, Concrete Cores,” story by Nadine Post, Engineering

News Record, 16 May 2007, pp. 10-11.

13. Wallace, J. W., 2007, “Modeling issues for tall reinforced core wall buildings,” The

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

14. Federal Emergency Management Agency, 2000, “Prestandard and Commentary for the

Seismic Rehabilitation of Buildings (FEMA-356),” Washington DC.

15. American Society of Civil Engineers, 2007, “ASCE/SEI Standard 41-06, Seismic

Rehabilitation of Existing Buildings,” Reston, VA.

16. Coull, A., 1974, “Stiffening of Coupled Shear Walls against Foundation Movement,”

Structural Engineer, V. 52, Issue 1, pp. 23-26.

17. Elwood, K.J., et al. (2007), “Update to ASCE/SEI 41 Concrete Provisions,” Earthquake

Spectra, EERI, Vol. 23, Issue 3, pp. 493-523.

18. New Zealand Standards Association (NZS), 1995, “NZS 3101:1995 Concrete Structures

Standard,” Wellington, New Zealand, 256 pp.

19. New Zealand Standards Association (NZS), 2006, “NZS 3101:2006 Concrete Structures

Standard,” Wellington, New Zealand, 256 pp.

20. Alsiwat, J., and Saatcioglu, M. (1992), “Reinforcement Anchorage Slip under

Monotonic Loading,” Journal of Structural Engineering, ASCE, Vol. 118, No. 9, pp.

2421-2438.

63

Appendix A – Summary of test results

64

CB24F


-200

-100

0

100

200La

tera

l Loa

d [k

]-12 -6 0 6 12


0 0.01 0.02 0.03 0.04Rotation [% drift]

0

20

40

60

80

100

% C

ontri

butio

n

FlexureSlipShear

65

Residual (zero displacement) damage photos

After Rotation = 0.01




66




67

CB24D


-200

-100

0

100

200La

tera

l Loa

d [k

]-12 -6 0 6 12


0 0.01 0.02 0.03 0.04Rotation [% drift]

0

20

40

60

80

100

% C

ontri

butio

n

FlexureSlipShear

68






69




70

CB24F-RC


-200

-100

0

100

200La

tera

l Loa

d [k

]-12 -6 0 6 12


0 0.01 0.02 0.03 0.04Rotation [% drift]

0

20

40

60

80

100

% C

ontri

butio

n

FlexureSlipShear

71

CB24F-PT


-200

-100

0

100

200La

tera

l Loa

d [k

]-12 -6 0 6 12


0 0.01 0.02 0.03 0.04Rotation [% drift]

0

20

40

60

80

100

% C

ontri

butio

n

FlexureSlipShear

72






73




74

CB24F-1/2-PT


-200

-100

0

100

200La

tera

l Loa

d [k

]-12 -6 0 6 12


0 0.01 0.02 0.03 0.04Rotation [% drift]

0

20

40

60

80

100

% C

ontri

butio

n

FlexureSlipShear

75






76




77

CB33F


-150

-100

-50

0

50

100

150La

tera

l Loa

d [k

]-10 -5 0 5 10


78

Damage photos at peak deformation

Rotation = 0.0075

Rotation = 0.01

Rotation = 0.015

Rotation = 0.02

79

Rotation = 0.03

Rotation = 0.04

Rotation = 0.06

80






81




82

CB33D


-150

-100

-50

0

50

100

150La

tera

l Loa

d [k

]-10 -5 0 5 10


83


Rotation = 0.01

Rotation = 0.015

Rotation = 0.02

Rotation = 0.03

84

Rotation = 0.04

Rotation = 0.06

85






86



87

FB33


-80

-40

0

40

80La

tera

l Loa

d [k

]-8 -4 0 4 8


88


Rotation = 0.0075

Rotation = 0.01

Rotation = 0.015

Rotation = 0.02

89

Rotation = 0.03

Rotation = 0.04

Rotation = 0.05

Rotation = 0.06

90






91



92

Appendix B – Slip/extension calculations

Problem Statement

For the cross-section of CB24F, determine the rotation, θ, due to slip/extension of flexural

reinforcement at the beam-wall interface prior to yielding.

θd

x

δtot

θd

x

δtot

Given:

2 2

'

7 / 8" 22.23

0.6 387

6.8 46.970 482.7

33" 83812.625"

@

2200 ( )

5"

b

b

c

y

d

s y

y

d mm

A in mm

f ksi MPaf ksi MPa

L mmd

f f

M in k from M analysis

x

φ

= =

= =

= == =

= ==

=

= − −

=

93

Calculations: Determine @ totM δθ−

(482.7 ) (22.23 ) 3.24 4 (838 )y b

ed

f d MPa mmu Mpal mm× ×

= = =× ×

22.23 1.744 4 3.2

s b se s

e

f d f mmL fu MPa× ×

= = = ×× ×

' 46.89(20 ) (20 22.23/ 4) 18.14 30 30

b cu

d fu MPa= − × = − × =

1 '

30 0.80sc

mmf

δ = =

@ fs = fy

'

''

'2.5

1

@

1.74 840

3.194

( ) 0.0104

1.25 / 2 1.05

0.0104 1.05 0.0417"

0.0417" 0.0054712.625" 5"

2200

/ 2200 / 0.0054 40

tot

e e y

y be

e

es s

u

ext y e

tot s ext

tot

y

y

L L f mm

f du MPa

L

u mmu

L mm

mm mm

d xM in k

K M

δ

δ δ

δ ε

δ δ δ

δθ

θ

= = × =

×= =

×

= × =

= × × =

= + = + =

= = =− −

= −

= = = 2200

Result

This M-θ relation represents the deformation characteristics of the beam in the elastic region due

to slip/extension of the flexural reinforcement. It can be implemented as an M-θ hinge in a

model to modify the elastic stiffness of the member.

94

-0.008 -0.004 0 0.004 0.008Rotation (rad)

-4000

-2000

0

2000

4000

Mom

ent [

in-k

]

Slip/Extension springsSlip/Extension springs

95

APPENDIX C – Procedure to Estimate EIeff

Problem Statement

Determine an estimate for the effective elastic stiffness (EIeff) as a function of the gross section

stiffness (EIg), considering the influence of slip/extension deformations.

Calculations

1) Calculate θslip@yield, by following the procedure in Appendix B.

2) Calculate θflex@yield, by the following:

a. @

miny

ACIy

M

VV V

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠

b. Use EIeff for flexure = 0.5EIg

c. 2 2

@12 ( .) 6

y n y nflex

eff g

V l V lyield

EI for flex EIθ

× ×= =

× ×

3) Calculate θtot:

a. tot slip flexθ θ θ≅ +

4) Calculate EIeff:

a. 2

12eff y n

g g tot

EI V lEI EI θ

×=

× ×

Results

This value of EIeff as a function of EIg can be directly input to a model (modifying the flexural

stiffness) in place of a slip/extension hinge model. This has the same impact as the slip hinge,

but would ease computation time.

96

APPENDIX D – Modeling Parameters

Summarized below are the parameters used in modeling of diagonally reinforced coupling beams

using CSI Perform 3D. Specifically, these parameters are for CB24F.

Mn-Hinge Model

The Moment-hinge model consists of an elastic RC cross-section, Mn-θ hinges, and

Slip/Extension hinges. The properties listed are for CB24F.

Mn-Rotation Springs

Slip/Ext. Springs

The elastic RC cross-section has the following properties:

Cross Section: Beam, Reinforced Concrete Section

Shape and Dimensions

Section Shape: Rectangle

B: 12 [in] D: 15 [in]

Section Stiffness

Axial Area: 180 [in2]

Shear Area: 0 (Shear area = 0 means no shear deformation)

Material Stiffness

Young’s Modulus: 1800 [ksi] Poisson’s Ratio: 0.2 Shear Modulus: 692

97

The Slip/Extension Hinges have the following properties:

Inelastic: Semi-Rigid Moment Connection

Basic F-D Relationship

K0: 402200 [k-in/rad2]

FU: 3100 [in-k]

DX: 0.14 [rad]

The Mn-θ hinges have the following properties:

Inelastic: Moment Hinge, Rotation Type


FY: 2350 [in-k] DU: 0.075 [rad]

FU: 2500 [in-k] DX: 0.130 [rad]

Strength Loss

DL: 0.08 FR/FU: 0.3

DR: 0.1 Interaction Factor: 0.25

Cyclic Degradation

Point Energy Factor

Y 0.5

U 0.45

L 0.4

R 0.35

X 0.35

Unloading Stiffness Factor: 0.5

Alternatively, similar results can be obtained by modifying the cross-section properties such that

Young’s Modulus = 0.15EcIg = 540 [ksi], and eliminating the slip/ext hinge altogether.

98

Vn-Hinge Model

The Shear-hinge model consists of an elastic RC cross-section, Vn-δ hinges, and Slip/Extension

hinges. The properties listed are for CB24F.

Vn-Displacement Hinge

Slip/Ext. Springs

The elastic RC cross-section has the following properties:

Cross Section: Beam, Reinforced Concrete Section

Shape and Dimensions

Section Shape: Rectangle

B: 12 [in] D: 15 [in]

Section Stiffness

Axial Area: 180 [in2]

Shear Area: 0 (Shear area = 0 means no shear deformation)

Material Stiffness

Young’s Modulus: 1800 [ksi] Poisson’s Ratio: 0.2 Shear Modulus: 692

The Slip/Extension Hinges have the following properties:

Inelastic: Semi-Rigid Moment Connection


K0: 402200 [k-in/rad2]

FU: 3100 [in-k]

DX: 0.14 [rad]

99

The Vn-δ hinges have the following properties:

Inelastic: Shear Hinge, Displacement Type


FY: 130 [k] DU: 2.7 [in]

FU: 136 [k] DX: 4.7 [in]

Strength Loss

DL: 2.88 [in] FR/FU: 0.3

DR: 3.31 [in] Interaction Factor: 0.25

Cyclic Degradation

Point Energy Factor

Y 0.5

U 0.45

L 0.4

R 0.35

X 0.35

Unloading Stiffness Factor: 0.5

Alternatively, similar results can be obtained by modifying the cross-section properties such that

Young’s Modulus = 0.15EcIg = 540 [ksi], and eliminating the slip/ext hinge altogether.

100

APPENDIX E – Material Testing

Diagonal #7 bars; tested by twining laboratories; based on given fy, fu, and % elongation

0 0.04 0.08 0.12 0.16 0.2

ε [in/in]

0

20

40

60

80

100

σ [k

si]

bar1bar2bar3

101

Concrete cylinders CB24F, CB24D, CB33F, CB33D; 6”x12” tested by twining laboratories; curve fit based on f’c

0 0.002 0.004 0.006 0.008 0.01

ε [in/in]

0

2

4

6

8

σ [k

si]cyl1cyl2cyl3cyl4cyl5cyl6

Concrete Cylinders CB24F-RC; 6”x12” tested by twining laboratories; 4”x8” tested by ucla; curve fit based on f’c

0 0.002 0.004 0.006 0.008 0.01

ε [in/in]

0

2

4

6

8

10

σ [k

si]

twining1twining2twining3twining4ucla1ucla2ucla3

102

Concrete Cylinders CB24F-PT; 6”x12” tested by twining laboratories; 4”x8” tested by ucla; curve fit based on f’c

0 0.002 0.004 0.006 0.008 0.01

ε [in/in]

0

2

4

6

8

10

σ [k

si]twining1twining2twining3twining4ucla1ucla2ucla3

Concrete cylinders CB24F-1/2-PT; 6”x12” tested by twining laboratories; 4”x8” tested by ucla; curve fit based on f’c

0 0.002 0.004 0.006 0.008 0.01

ε [in/in]

0

2

4

6

8

σ [k

si]

twining1twining2twining3twining4ucla1ucla2ucla3

103

Concrete cylinder tests FB33; 6”x12” tested by twining laboratories; 4”x8” tested by ucla; curve fit based on f’c

0 0.002 0.004 0.006 0.008 0.01

ε [in/in]

0

2

4

6

8

σ [k

si]twining1twining2twining3twining4ucla1ucla2ucla3

pankow reinforced concrete link beams

Documents

pt damage

tall building

environmental

steel reaction

transverse

pt cross section

pt beam construction

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