design of rectangular openings in unbonded post-tensioned...

Design of Rectangular Openings

in Unbonded Post-Tensioned

Precast Concrete Walls

Structural Engineering Research Report

Department of Civil Engineering and Geological Sciences

University of Notre Dame

Notre Dame, Indiana

April 2001

Michael Allen and Yahya C. Kurama Report #NDSE-01-01

Design of Rectangular Openings

in Unbonded Post-Tensioned

Precast Concrete Walls

Structural Engineering Research Report

Department of Civil Engineering and Geological Sciences

University of Notre Dame

Notre Dame, Indiana

April 2001

by

Michael AllenFormer Graduate

Research Assistant

and

Yahya C. KuramaAssistant Professor

Report #NDSE-01-01

ABSTRACT

Recent research has shown that unbonded post-tensioned precast walls can be used as primary gravityand lateral load resisting systems in seismic regions. This report investigates the use of rectangularopenings in the walls to accommodate architectural, mechanical, and/or safety requirements. Theopenings can cause large tensile stresses, and thus, cracking in the wall panels. Under lateral loads,cracking in the panels can also occur due to gap opening along the horizontal joints between thepanels and at the base.

Bonded mild steel reinforcement is needed in the wall panels to limit the size of the cracks. The reportdescribes an analytical investigation on the behavior and design of walls with openings for twoloading stages: (1) vertical loads due to gravity and post-tensioning; and (2) combined vertical andlateral loads. For each loading stage, critical regions in the wall panels where reinforcement is neededare identified and design approaches are proposed to determine the amount of the required panelreinforcement. The effect of opening length, opening height, wall length, and initial stress in the wallsdue to post-tensioning and gravity on the behavior and design of the walls is investigated.

This report may be downloaded from http://www.nd.edu/~concrete/

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TABLE OF CONTENTS

LIST OF TABLES ............................................................................................................. viii

LIST OF FIGURES .......................................................................................................... x

LIST OF SYMBOLS ......................................................................................................... xvi

ACKNOWLEDGMENTS ............................................................................................... xxiv

CHAPTER 1 INTRODUCTION .................................................................................... 1

1.1 Overview .............................................................................................................. 1

1.2 Unbonded Post-Tensioned Precast Concrete Walls ........................................... 2

1.3 Walls with Openings ............................................................................................. 3

1.4 Objectives ............................................................................................................ 5

1.5 Scope and Approach ............................................................................................. 5

1.6 Organization of Report .......................................................................................... 7

CHAPTER 2 BACKGROUND ....................................................................................... 8

2.1 Unbonded Post-Tensioned Precast Concrete Walls .............................................. 8

2.1.1 Behavior along the horizontal joints ............................................................. 10

2.1.2 Structural design parameters ........................................................................ 11

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2.1.3 Base-shear-roof-drift relationship ................................................................ 11

2.1.3.1 Decompression state ........................................................................... 11

2.1.3.2 Softening state .................................................................................... 12

2.1.3.3 Yielding state ...................................................................................... 13

2.1.3.4 Failure state ......................................................................................... 13

2.2 Prestressed Beams with Openings ......................................................................... 13

2.3 Monolithic Cast-in-Place Concrete Walls with Openings ......................................... 13

2.4 Precast Concrete Walls with Openings .................................................................. 14

2.5 Fiber Element Model ............................................................................................ 14

2.6 Closed-Form Solutions for Infinite Elastic Panels with Openings ........................... 15

CHAPTER 3 PARAMETRIC INVESTIGATION ........................................................... 19

3.1 Opening Dimensions, lo and ho ............................................................................ 19

3.2 Panel Dimensions, lp, hp, and tp ............................................................................. 20

3.3 Initial Concrete Stress, fci ...................................................................................... 22

3.4 Post-Tensioning Loads ........................................................................................ 23

3.5 Gravity Loads ....................................................................................................... 23

3.6 Spiral Confinement ............................................................................................... 23

3.7 Analysis of the Walls ............................................................................................. 24

CHAPTER 4 ANALYTICAL MODEL .......................................................................... 25

4.1 Modeling of the Post-Tensioning Bars ................................................................. 25

4.2 Modeling of the Wall Panels ................................................................................. 27

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4.2.1 Refinement of the finite element mesh ......................................................... 29

4.3 Modeling of Concrete in Compression ................................................................. 29

4.4 Concrete Failure Envelope under Biaxial Loading ................................................ 30

4.4.1 ABAQUS biaxial failure ratios ...................................................................... 32

4.4.2 ABAQUS failure ratio 1 .............................................................................. 33



4.4.5 ABAQUS failure ratio 4 ............................................................................... 35

4.4.6 Failure ratios used to model the walls ........................................................... 36

4.5 Modeling of Bonded Mild Steel Reinforcement ..................................................... 37

4.6 Modeling of Behavior Along the Horizontal Joints .............................................. 37

4.7 Modeling of Gravity Loads ................................................................................... 39

4.8 Modeling of Lateral Loads ................................................................................... 39

4.9 Verification of the Analytical Model ...................................................................... 39

4.9.1 Comparisons with the fiber element model ................................................. 39

4.9.2 Comparisons with closed-form solutions ...................................................... 40

CHAPTER 5. DESIGN OF PANEL REINFORCEMENT UNDER VERTICAL LOADS ONLY ...................................................................... 42

5.1 Critical Panel Regions under Vertical Loads .......................................................... 42

5.2 Design of Panel Reinforcement: Overview .............................................................. 45

5.3 Truss Model ........................................................................................................ 45

5.3.1 Estimation of Panel Top Stresses ....................................................................... 47

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5.3.2 Estimation of Cr and xGp ...................................................................................... 51

5.3.3 Estimation of Side Chord Stresses and xGs ........................................................... 51

5.4 Placement of the Panel Reinforcement ............................................................... 52

5.5 Design of Upper Story Panels .............................................................................. 55

5.6 Base Panel Reinforcement Results ......................................................................... 59

5.6.1 Result averages ............................................................................................. 63

CHAPTER 6 DESIGN OF PANEL REINFORCEMENT UNDER COMBINED VERTICAL AND LATERAL LOADS ................................. 64

6.1 Effect of Openings on the Lateral Load Behavior of the Walls ................................ 64

6.2 Critical Panel Regions ........................................................................................... 65

6.3 Determining the Critical Panel Sections for Design: Overview ............................... 69

6.4 Walls Without Openings ....................................................................................... 69

6.4.1 Contact length at the bottom of the base panel ............................................ 70

6.4.2 Contact length at the top of the base panel ................................................... 71

6.5 Walls With Openings ............................................................................................ 72

6.5.1 Contact length at the bottom of the base panel ............................................ 72

6.5.2 “Large” openings versus “small” openings .................................................... 74

6.5.3 Contact length at the top of the base panel ................................................... 76

6.6 Critical Section at the Bottom of the Base Panel .................................................. 77

6.7 Critical Section at the Top of the Base Panel ....................................................... 77

6.8 Overview of the Proposed Design Approach ......................................................... 78

6.9 Design of Panel Top Reinforcement ..................................................................... 80

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6.9.1 Forces at the top of the panel ....................................................................... 81

6.9.1.1 Axial force, Nptl ................................................................................... 82

6.9.1.2 Shear force, Vptl ................................................................................... 85

6.9.2 Forces in the tension side chord .................................................................... 88

6.9.2.1 Axial force, N2 ................................................................................... 89

6.9.2.2 Shear force, V2 .................................................................................... 92

6.9.2.3 Moment, M2m ..................................................................................... 95

6.9.3 Design tension force and required reinforcement .......................................... 99

6.10 Design of Panel Side Reinforcement .................................................................. 100

6.11 Design of Panel Bottom Reinforcement .............................................................. 100

6.11.1 Forces in the compression side chord ........................................................... 100

6.11.2 Forces in Section 4 .................................................................................... 101

6.11.3 Design tension force and required reinforcement ...................................... 102

6.12 Design of Reinforcement Around Openings ........................................................ 104

6.13 Placement of the Panel Reinforcement ................................................................. 104

6.14 Design of Shear Reinforcement .......................................................................... 105

6.15 Roof-Drift at the Failure State ........................................................................... 106

6.16 Design of the Upper Story Panels ....................................................................... 106

6.16.1 Required reinforcement in the upper story panels ........................................ 108

6.17 Base Panel Reinforcement Results ........................................................................ 112

6.17.1 Panel top reinforcement, Dc1b ....................................................................... 113

6.17.2 Panel side reinforcement, Dc2b ...................................................................... 117

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6.17.3 Panel bottom reinforcement, Dc3b ................................................................ 121

6.17.4 Result averages ........................................................................................... 125

CHAPTER 7 CONCLUSIONS AND RECOMMENDATIONS ....................................... 126

REFERENCES ............................................................................................................... 129

APPENDIX A DESIGN EXAMPLE, VERTICAL LOADS ONLY ................................ 132

APPENDIX B DESIGN EXAMPLE, COMBINED VERTICAL AND LATERAL LOADS ........................................................................ 136

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LIST OF TABLES

TABLE 2.1 VALUES OF k ................................................................................................ 17

TABLE 3.1. PARAMETRIC INVESTIGATION ............................................................... 20

TABLE 3.2 POST-TENSIONING AND GRAVITY LOADS .......................................... 22

TABLE 3.3 CONCRETE PARAMETERS ........................................................................ 24

TABLE 5.1 REQUIRED REINFORCEMENT AT TOP AND BOTTOM OF OPENING ................................................................. 59

TABLE 5.2 AVERAGE OF PREDICTED/FEM REINFORCEMENT RATIO ........................................................................... 61

TABLE 6.1 CONTACT LENGTH AT THE BOTTOM OF THE BASE PANEL ................................................................................................ 71

TABLE 6.2 CONTACT LENGTH AT THE TOP OF THE BASE PANEL ............................................................................................. 73

TABLE 6.3 CRITICAL SECTION AT THE BOTTOM OF THE BASE PANEL ............................................................................................. 78

TABLE 6.4 CRITICAL SECTION AT THE TOP OF THE BASE PANEL ............................................................................................ 79

TABLE 6.5 REQUIRED REINFORCEMENT AT THE TOP OF THE BASE PANEL ............................................................................... 113

TABLE 6.6 REQUIRED REINFORCEMENT AT THE SIDE OF THE BASE PANEL .............................................................................. 117

TABLE 6.7 REQUIRED REINFORCEMENT AT THE BOTTOM OF THE BASE PANEL ............................................................................... 121

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TABLE 6.8 AVERAGE OF PREDICTED / FEM REINFORCEMENT RATIO ....................................................................... 125

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LIST OF FIGURES

Figure 1.1 Unbonded post-tensioned precast concrete wall: (a) elevation; (b) cross-section near base ......................................................... 2

Figure 1.2 Wall under lateral loads: (a) displaced shape; (b) stresses on the base panel .............................................................................. 3

Figure 1.3 Wall with openings ......................................................................................... 4

Figure 1.4 Cracking in a wall panel ................................................................................... 5

Figure 2.1 Wall WH1M: (a) elevation; (b) cross-section near base (half the wall length) ............................................................................................ 9

Figure 2.2 Prototype building plan layout ......................................................................... 10

Figure 2.3 Base-shear versus roof-drift relationship of Wall WH1M .................................. 12

Figure 2.4 Infinite elastic plane with an opening ............................................................... 16

Figure 3.1 Elevation and cross-section of the parametric walls: (a) lp=20 ft, (f=0.29; (b) lp=20 ft, (f=0.18; (c) lp=20 ft, (f=0.11; (d) lp=20 ft, (f=0.057; (e) lp=15 ft, (f=0.18; (f) lp=12 ft, (f=0.18 .......................... 21

Figure 4.1 Finite element mesh: (a) refined mesh with openings; (b) coarse mesh ........... 26

Figure 4.2 Stress-strain relationship of the post-tensioning steel ....................................... 27

Figure 4.3 Concrete stress-strain relationships .................................................................. 30

Figure 4.4 Concrete biaxial failure envelope ........................................................................ 31

Figure 4.5 The 9x9 finite element model ............................................................................... 32

Figure 4.6 Failure ratio 1: (a) biaxial failure envelope (b) stress-strain relationships ......... 33

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Figure 4.7 Effect of failure ratio 2 on the concrete biaxial failure envelope ....................... 34

Figure 4.8 Failure ratio 3: (a) biaxial failure envelope; (b) stress-strain relationships ........ 35

Figure 4.9 Effect of failure ratio 4 on the concrete biaxial failure envelope ...................... 35

Figure 4.10 ABAQUS versus Mander concrete stress-strain relationships .......................... 36

Figure 4.11 Deformed finite element model: (a) entire wall; (b) base panel ......................... 38

Figure 4.12 Verification of the finite element model: (a) base-shear-roof-drift relationship; (b) contact length .................................. 40

Figure 4.13 Finite element model versus closed-form solutions ......................................... 41

Figure 5.1 Stress contours in the base panel: (a) maximum principle stresses (b) minimum principle stresses ....................................................................... 43

Figure 5.2 Stress contours in the base panel: (a) horizontal stresses; (b) vertical stresses ......................................................................................... 44

Figure 5.3 Panel stresses .................................................................................................. 46

Figure 5.4 Design of panel reinforcement: (a) truss model; (b) truss model enlarged ........ 47

Figure 5.5 Estimation of fp0: (a) bottom two panels; (b) fp0 versus 2c ................................. 48

Figure 5.6 Panel top stresses: (a) (h=0.25, (f=0.29, lp=20 ft; (b) (l=0.20, (f=0.29, lp=20 ft; (c) (l=0.20, (h=0.25, (f=0.18; (d) (l=0.20, (h=0.25, lp=20 ft .............................................................................. 50

Figure 5.7 Side chord stresses: (a) (h=0.25, (f=0.29, lp=20 ft; (b) (l=0.20, (f=0.29, lp=20 ft;(c) (l=0.20, (h=0.25, (f=0.18; (d) (l=0.20, (h=0.25, lp=20 ft .. .......................................................................... 53

Figure 5.8 Panel reinforcement example ........................................................................... 54

Figure 5.9 Principle stress contours in Wall WH1M ((f=0.29, lp=20 ft): (a) without opening; (b) with opening ((l=0.40, (h=0.38) ............................... 55

Figure 5.10 Horizontal stress contours in Wall WH1M ((f=0.29, lp=20 ft): (a) without opening; (b) with opening ((l=0.40, (h=0.38) .................................. 57

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Figure 5.11 Vertical stress contours in Wall WH1M ((f=0.29, lp=20 ft): (a) without opening; (b) with opening ((l=0.40, (h=0.38) ................................ 58

Figure 5.12 Effect of (l on Dv: (a) (h=0.25, lp=20 ft; (b) (h=0.25, (f=0.18; (c) (f=0.057, lp=20 ft; (d) (f=0.29, lp=20 ft ......................................................... 60

Figure 5.13 Effect of (h on Dv: (a) (l=0.30, lp=20 ft; (b) (l=0.30, (f=0.18; (c) (f=0.057, lp=20 ft; (d) (f=0.29, lp=20 ft ........................................................... 61

Figure 5.14 Effect of lp on Dv: (a) (l=0.30, (f=0.18; (b) (h=0.25, (f=0.18 .............................. 62

Figure 5.15 Effect of (f on Dv: (a) (l=0.30, lp=20 ft; (b) (h=0.25, lp=20ft ............................ 62

Figure 6.1 Base-shear-roof-drift relationship of walls without and with openings .............. 65

Figure 6.2 Stress contours under lateral loads: (a) maximum principle stresses (b) minimum principle stresses ............................................................................ 66

Figure 6.3 Stress contours in the base panel: (a) horizontal stresses; vertical stresses ......... 68

Figure 6.4 Panel reinforcement ......................................................................................... 69

Figure 6.5 Estimation of cb ................................................................................................. 70

Figure 6.6 Estimation of c tN ................................................................................................ 72

Figure 6.7 Contact length at the base of walls without and with openings .......................... 74

Figure 6.8 Opening size factor: (a) small opening; (b) large opening ................................. 75

Figure 6.9 Contact length at top of the base panel of a wall with and without openings .......................................................................................... 76

Figure 6.10 Design forces: (a) critical sections; (b) free body diagram for Sections 1 and 2; (c) free body diagram for Sections 2 and 3 ................... 80

Figure 6.11 Effect of (l on Nptl: (a) (h=0.25, lp=20 ft; (b) (h=0.25, (f=0.18; (c) (f=0.29, lp=20 ft .......................................................................................... 82

Figure 6.12 Effect of (h on Nptl: (a) (l=0.30, lp=20 ft; (b) (l=0.30, (f=0.18; (c) (f=0.29, lp=20 ft ........................................................................................ 83

Figure 6.13 Effect of lp on Nptl: (a) (l=0.30, (f=0.18, (b) (h=0.25, (f=0.18 ........................... 84

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Figure 6.14 Effect of (f on Nptl: (a) (l=0.30, lp=20 ft; (b) (h=0.25, lp=20 ft ........................ 84

Figure 6.15 Effect of (l on Vptl: (a) (h=0.25, lp=20 ft; (b) (h=0.25, (f=0.18; (c) (f=0.29, lp=20 ft ......................................................................................... 85

Figure 6.16 Effect of (h on Vptl: (a) (l=0.30, lp=20 ft; (b) (l=0.30, (f=0.18; (c) (f=0.29, lp=20 ft ....................................................................................... 86

Figure 6.17 Effect of lp on Vptl: (a) (l=0.30, (f=0.18, (b) (h=0.25, (f=0.18 ......................... 87

Figure 6.18 Effect of (f on Vptl: (a) (l=0.30, lp=20 ft; (b) (h=0.25, lp=20 ft ........................ 87

Figure 6.19 Effect of (l on N2: (a) (h=0.25, lp=20 ft; (b) (h=0.25, (f=0.18; (c) (f=0.29, lp=20 ft ........................................................................................... 89

Figure 6.20 Effect of (h on N2: (a) (l=0.30, lp=20 ft; (b) (l=0.30, (f=0.18; (c) (f=0.29, lp=20 ft ......................................................................................... 90

Figure 6.21 Effect of lp on N2: (a) (l=0.30, (f=0.18, (b) (h=0.25, (f=0.18 ........................... 91

Figure 6.22 Effect of (f on N2: (a) (l=0.30, lp=20 ft; (b) (h=0.25, lp=20 ft .......................... 91

Figure 6.23 Effect of (l on V2: (a) (h=0.25, lp=20 ft; (b) (h=0.25, (f=0.18; (c) (f=0.29, lp=20 ft ......................................................................................... 92

Figure 6.24 Effect of (h on V2: (a) (l=0.30, lp=20 ft; (b) (l=0.30, (f=0.18; (c) (f=0.29, lp=20 ft ......................................................................................... 93

Figure 6.25 Effect of lp on V2: (a) (l=0.30, (f=0.18, (b) (h=0.25, (f=0.18 ........................... 94

Figure 6.26 Effect of (f on V2: (a) (l=0.30, lp=20 ft; (b) (h=0.25, lp=20 ft .......................... 94

Figure 6.27 Frame structure ............................................................................................. 95

Figure 6.28 Effect of (l on M2m: (a) (h=0.25, lp=20 ft; (b) (h=0.25, (f=0.18; (c) (f=0.29, lp=20 ft ......................................................................................... 96

Figure 6.29 Effect of (h on M2m: (a) (l=0.30, lp=20 ft; (b) (l=0.30, (f=0.18; (c) (f=0.29, lp=20 ft ......................................................................................... 97

Figure 6.30 Effect of lp on M2m: (a) (l=0.30, (f=0.18, (b) (h=0.25, (f=0.18 ......................... 98

Figure 6.31 Effect of (f on M2m: (a) (l=0.30, lp=20 ft; (b) (h=0.25, lp=20 ft ........................ 98

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Figure 6.32 Estimation of Tl ............................................................................................. 99

Figure 6.33 Estimation of T2 ........................................................................................... 100

Figure 6.34 Axial stress distribution in the compression chord ............................................. 101

Figure 6.35 Shear stress distribution in the compression chord .......................................... 102

Figure 6.36 Estimation of T3 ............................................................................................ 103

Figure 6.37 Principle stress contours in Wall WH1M ((f=0.29, lp=20 ft); (a) without openings; (b) with openings ((l=0.40, (h=0.38) ................................ 107

Figure 6.38 Horizontal stress contours in Wall WH1M ((f=0.29, lp=20 ft); (a) without openings; (b) with openings ((l=0.40, (h=0.38) ................................ 109

Figure 6.39 Vertical stress contours in Wall WH1M ((f=0.29, lp=20 ft); (a) without openings; (b) with openings ((l=0.40, (h=0.38) ................................ 110

Figure 6.40 Required reinforcement in the upper story panels ........................................ 112

Figure 6.41 Effect of (l on Dc1b: (a) (h=0.25, lp=20 ft; (b) (h=0.25,(f=0.18; (c) (f=0.29, lp=20 ft ....................................................................................... 114

Figure 6.42 Effect of (h on Dc1b: (a) (l=0.30, lp=20 ft; (b) (l=0.30,(f=0.18; (c) (f=0.29, lp=20 ft ........................................................................................ 115

Figure 6.43 Effect of lp on Dc1b: (a) (l=0.30, (f=0.18; (b) (h=0.25, (f=0.18 ......................... 116

Figure 6.44 Effect of (f on Dc1b: (a) (l=0.30, lp=20 ft; (b) (h=0.25, lp=20 ft ......................... 116

Figure 6.45 Effect of (l on Dc2b: (a) (h=0.25, lp=20 ft; (b) (h=0.25,(f=0.18; (c) (f=0.29, lp=20 ft ........................................................................................ 118

Figure 6.46 Effect of (h on Dc2b: (a) (l=0.30, lp=20 ft; (b) (l=0.30,(f=0.18; (c) (f=0.29, lp=20 ft ......................................................................................... 119



Figure 6.49 Effect of (l on Dc3b: (a) (h=0.25, lp=20 ft; (b) (h=0.25,(f=0.18; (c) (f=0.29, lp=20 ft ........................................................................................ 122

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Figure 6.50 Effect of (h on Dc3b: (a) (l=0.30, lp=20 ft; (b) (l=0.30,(f=0.18; (c) (f=0.29, lp=20 ft ............................................................................................ 123



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LIST OF SYMBOLS

a = variable in closed-form solution

aG = variable in closed-form solution

abN = length of rectangular stress block at bottom of base panel without opening

Ac1b = area of reinforcing steel at top of base panel, combined loading

Ac2b = area of reinforcing steel at sides of base panel, combined loading

Ac3b = area of reinforcing steel at bottom of base panel, combined loading

Ac4b = area of reinforcing steel at top and bottom of opening in base panel, combined loading

Ac5b = area of reinforcing steel at sides of opening in base panel, combined loading

Ag = gross cross-section area of wall

Amin = area of minimum reinforcement (2 #5 bars)

ap = area of an individual post-tensioning bar

apj = first coefficient for panel top stress distribution in region j

Av = area of reinforcing steel at top and bottom of opening, vertical loads only

bpj = second coefficient for panel top stress distribution in region j

C1 = compression force in Section 1



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cb = contact length at bottom of base panel with opening

cbN = contact length at bottom of base panel without opening

cbe = distance from end of wall to edge of contact region at base of wall

cpj = third coefficient for panel top stress distribution in region j

Cr = compression force in truss model

ct = contact length at top of base panel with opening

c tN = contact length at top of base panel without opening

Cv = compression force at top of opening under vertical loads only

dc = effective depth of horizontal chord measured from compression edge to longitudinal reinforcement in tension

f11 = normal stress along 1-1 axis

f22 = normal stress along 2-2 axis

fc = stress in concrete

fcN = compressive strength of unconfined concrete

fcc = compressive strength of spiral confined concrete

fci = initial stress in concrete due to gravity and post-tensioning

fp = compression stress applied to top of panel

fp0 = compression stress at top of panel at centerline

fp1 = compression stress at top of panel at x1

fp2 = compression stress at top of panel at x2

fpa = uniformly distributed stress transferred to panel from panel above

fpe = compression stress at top of panel at edge based on a linear stress distribution

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fpi = initial stress in post-tensioning bar

fpp = uniformly distributed stress due to gravity loads applied on floor level above panel

fpu = ultimate strength of post-tensioning steel

fpy = yield strength of post-tensioning steel

fse = compression stress in side chord at edge of panel

fsr = compression stress in side chord at xr

ftm = maximum tension stress at top of opening

fy = yield strength of mild steel

Ga = sum of gravity loads applied on upper story panels

Gb = total gravity load acting at base of wall

Gp = gravity load applied on floor level above panel

Gptl = resultant of uniformly distributed gravity load applied left of critical section at panel top

hc = height of horizontal chord in base panel

hcu = height of horizontal chord in upper story panels

ho = height of opening

hp = height of base panel

hpu = height of upper story panel

ht1 = height of tension zone in Section 1

ht3 = height of tension zone in Section 3

htv = height of tension zone at top of opening, vertical loads only

k = characteristic of opening aspect ratio

lc = length of side chord

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lo = length of opening

lp = length of panel

lt2 = length of tension zone in Section 2

M2 = moment of axial force in Section 2 about middle of Section 2

M2m = moment of axial force in Section 2 about panel centerline

M3 = moment of axial force in Section 3 about center of Section 3

M4 = moment of axial force in Section 4 about center of Section 4

Mp2 = moment acting in base panel at bottom of opening

Mpb = moment acting at base of wall due to lateral loads and post-tensioning

Mpt = moment acting at top of base panel due to lateral loads and post-tensioning

Mr = moment of axial force in compression side chord about middle of chord

n = story number

N2 = axial force in Section 2



nmax = total number of stories

Npb = total axial force at base of wall due to gravity and post-tensioning

Npt = axial force transferred to top of base panel from panel above

Nptl = axial force transferred to base panel at left of critical section at panel top

Nr = axial force in compression side chord

Pf = sum of forces in post-tensioning bars at failure

Pi = total initial post-tensioning force

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tc = thickness of spiral confined concrete

tp = thickness of panel

s = opening size factor

T1 = tension force in Section 1



Tv = tension force occurring at top of opening under vertical loads only

U = constant in closed-form solution

V = shear force applied at top of frame

V2 = shear force in Section 2

V4 = shear force in Section 4

Vp = shear force applied at top of base panel

Vpb = shear force at base of wall

Vpt = shear force transferred to top of base panel from panel above

Vptl = shear force transferred to base panel at left of critical section at panel top

Vr = shear force in compression side chord

x = horizontal distance measured from panel centerline

x1 = location where Fp1 changes to Fp2

x2 = location where Fp2 changes to Fp3

xcb = distance of Section 3 from compression end of panel

xct = distance of Section 1 from compression end of panel

xGp = resultant location of Cr at top of panel

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xr = distance over which stresses are summed to determine Cr

xs = distance used to define a “large” opening

xGs = resultant location of Cr at top of side chord

yc3 = location of compression stress resultant in Section 3

" = factor for magnitude of equivalent rectangular stress block

$1 = factor for depth of equivalent rectangular stress block

(f = ratio of fci to fcN

(h = ratio of ho to hp

(l = ratio of lo to lp

,c = strain in concrete

,cc = strain in spiral confined concrete when maximum compressive stress is reached

,cu = ultimate strain capacity of spiral confined concrete

2 = angle (in radians) around edge of opening for closed-form solution

2c = angle to determine fp0 (in degrees)

2t = angle of truss model (in degrees)

>a = height to length ratio of base panel

>h = horizontal chord height factor

>l = vertical chord length factor

Dc1b = reinforcement ratio at top of base panel, combined loading

Dc1n = reinforcement ratio at top of nth panel, combined loading

Dc2b = reinforcement ratio at sides of base panel, combined loading

Dc2n = reinforcement ratio at sides of nth panel, combined loading

xxii

Dc3b = reinforcement ratio at bottom of base panel, combined loading

Dc3n = reinforcement ratio at bottom of nth panel, combined loading

Dc4b = reinforcement ratio at top and bottom of opening in base panel, combined loading

Dc4n = reinforcement ratio at top and bottom of opening in nth panel, combined loading

Dc5b = reinforcement ratio at sides of opening in base panel, combined loading

Dc5n =reinforcement ratio at sides of opening in nth panel, combined loading

Dmin,h = minimum reinforcement ratio in horizontal chords

Dmin,v = minimum reinforcement ratio in vertical chords

Dsp = spiral reinforcement ratio

Dv = reinforcement ratio at top and bottom of opening, vertical loads only

Fp = stress distribution at top of panel, vertical loads

Fp1 = second order panel top stress distribution in region 1



Fpj = second order panel top stress distribution in region j

Fs = stress distribution in side chord, vertical loads

Fs1 = second order side chord stress distribution in region 1

Fs2 = first order side chord stress distribution in region 2

F2 = stress around edge of opening

H = function of stress applied to infinite plane and location around opening edge

S = stress distribution factor

T = stresses function around edge of opening

xxiii

n = stress function around edge of opening

no = stress function for boundary conditions around edge of opening

xxiv

ACKNOWLEDGMENTS

The research was funded by a PCI Daniel P. Jenny Research Fellowship and by the University ofNotre Dame. The support of the Precast/Prestressed Concrete Institute and the University of NotreDame is gratefully acknowledged.

The support of the PCI Research Director P. Johal and Past Chairman of the Research andDevelopment Committee H. Wilden, H. Wilden & Associates, Inc. is gratefully acknowledged. Thesupport and advice provided by the PCI Ad Hoc Advisory Group on the Fellowship is acknowledged:N. Cleland, Blue Ridge Design, Inc.; T. D'Arcy, The Consulting Engineers Group, Inc.; J. Hoffman,Sturm Engineering Company; and S. Pessiki, Lehigh University. The researchers also wish to thanktwo other individuals for their contributions to the project: K. Baur, High Concrete Structures, Inc.and R. Sause, Lehigh University.

M. Nietfeld helped in the development of the finite element model during the summer of 1998 throughthe Research Experiences for Undergraduates (REU) Program funded by the National ScienceFoundation. The support of the National Science Foundation for the REU Program is acknowledged.

The opinions, findings, and conclusions expressed in this report are those of the authors and do notnecessarily reflect the views of the individuals and organizations acknowledged above.

1

CHAPTER 1

INTRODUCTION

1.1 Overview

Precast concrete is a widely used type of construction throughout the world. Its cost effectivenessand fast production enable innovation in both design and construction. Due to the high technologyand low labor cost involved with precast concrete structures, this method matches the strengths andconditions of the construction industry in the United States.

The development of precast concrete structural systems in the United States has not been asextensive as in some other countries (Priestley 1991). One of the major factors inhibiting thedevelopment of precast construction in the U.S. is the uncertainty in the behavior of precast concretestructures in seismic regions. A lack of experience on the seismic response of multi-story structuralsystems to strong earthquakes and little experimental data have resulted in precast systems being lesscommon in seismic regions in the U.S.

Current U.S. model building codes (e.g., NEHRP-97, IBC 2000) also restrict the innovation ofprecast concrete systems in seismic regions. In the absence of experimental data, the codes requirethat precast concrete systems emulate the behavior of monolithic cast-in-place reinforced concretesystems. Extensive testing is required for the design of “non-emulative” precast systems. Thisrequirement for testing, because it is expensive and time consuming, inhibits the development of newand innovative systems. This has caused most previous research on precast concrete to focus on theemulation of monolithic cast-in-place concrete. However, in many cases, precast concrete systemsthat emulate monolithic concrete systems perform poorly in seismic regions because of problems inthe joint regions between the precast members (Kurama et al. 1996). Furthermore, emulation reducesthe economic advantages of precast concrete and limits the realization of the true potential of thesesystems.

In order to develop recommendations for the use of precast concrete systems in seismic regions, thePREcast Seismic Structural Systems (PRESSS) Research Program was initiated (Priestley 1991)with funding from the National Science Foundation (NSF), the Precast/Prestressed Concrete Institute(PCI), and the Precast/Prestressed Concrete Manufacturers Association of California, Inc. (PCMAC).

2

The main objectives of the PRESSS Program were to develop recommendations for the seismicdesign of precast concrete structural systems and to develop new systems. The research program wasconducted in three phases. Phase I focused on identifying and evaluating the most promising seismicsystems. Phase II involved analytical and experimental studies of the systems selected in Phase I.Finally, Phase III investigated the design, erection, testing, and analysis of a five-story precastconcrete building (Priestley et al. 1999).

1.2 Unbonded Post-Tensioned Precast Concrete Walls

Previous research conducted as a part of Phase II and Phase III of the PRESSS Program has shownthat unbonded post-tensioned precast concrete walls may offer significant advantages as primarylateral load resisting systems in seismic regions (Kurama et al. 1999a, 1999b, Priestley et al. 1999).These walls do not emulate the behavior of monolithic cast-in-place reinforced concrete walls. Asan example, Figure 1.1 shows the elevation and cross-section near the base of a six-story unbondedpost-tensioned precast concrete wall. The wall is constructed by joining precast wall panels acrosshorizontal joints using post-tensioning bars which are anchored to the wall only at the roof and atthe foundation.

PT bar(unbonded)

PT anchoragelateral load

gravityload

horizontaljoint

foundation

wall panel

spiral reinforcement

PT bar wire mesh

spiral reinforcement

(a)

(b)

Figure 1.1 Unbonded post-tensioned precast wall: (a) elevation; (b) cross-section near base

3

The post-tensioning bars are placed inside oversized ducts to allow the wall to displace laterallywithout resulting in the kinking of the bars. No grout is placed inside the ducts, and thus the post-tensioning bars are not bonded to the wall panels. Dry-pack or grout may be used between the panelsfor construction tolerances and for alignment purposes. Spiral reinforcing steel is used to confine theconcrete near the base of the wall. Wire mesh is used as bonded reinforcement in the panels.

Unbonded post-tensioned precast walls resist gravity loads as well as lateral loads. During a seismicevent, the precompression forces from post-tensioning and gravity may be overcome, leading to theformation of gaps along the horizontal joints between the panels and between the wall and thefoundation. Figure 1.2(a) shows an example of this behavior for a wall subjected to vertical (due topost-tensioning and gravity) and lateral loads. The presence of the gaps along the horizontal jointscauses the compression stresses due to post-tensioning and gravity and the shear stresses due tolateral loads to be distributed over the length of each panel that is in contact with the foundation orwith the adjacent panels. Figure 1.2(b) gives an example of these contact stresses at the top andbottom of the base panel.

(b)

compression stressesdue to PT and gravity

shear stressesdue to lateral loads

discretegap

(a)

Figure 1.2 Wall under lateral loads: (a) displaced shape; (b) stresses on the base panel

1.3 Walls with Openings

The previous research on the seismic behavior and design of unbonded post-tensioned walls islimited to walls without openings. However, the use of openings in the wall panels as shown inFigure 1.3 may be desirable. For example, openings may be needed to accommodate windows,

4

doors, and mechanical penetrations due to functional and/or architectural requirements. Largeopenings may also be necessary in precast concrete parking structures for security reasons.

lateral load

gravityload

opening

Figure 1.3 Wall with openings

The seismic design of unbonded post-tensioned precast concrete walls can be done using a procedurewhich was developed previously for walls without openings. This design procedure, which isdescribed in detail by Kurama et al. (1999a, 1999b), can be used to determine the wall length andthickness as well as the required amount of post-tensioning and spiral reinforcement.

The use of openings in the wall panels is not addressed by the previous design approach. Theseopenings can result in large tension stresses, and thus, cracking in the panels under vertical andlateral loads. The gaps which form along the horizontal joints under lateral loads may also causecracking in the panels. These cracks can limit the vertical and lateral load carrying capacity of thewalls by causing premature failure of the wall panels. Thus, bonded mild steel reinforcement maybe needed in the panels to limit the size of the cracks.

Critical locations in a precast wall panel where cracks may form under vertical (due to post-tensioning and gravity) and lateral loads are shown in Figure 1.4. The lateral loads are assumed tobe acting from left to right. The opening can cause cracking in the panel under the presence ofvertical loads only. This cracking occurs at the center of the top and bottom edges of the opening.The design of the panel reinforcement required to limit the size of these cracks is discussed inChapter 5.

5

Figure 1.4 Cracking in a wall panel

During a seismic event, cracking may occur at the sides of the panel and near the corners of theopening. Cracking may also occur at the top and bottom of the panel due to the formation of gapsalong the horizontal joints. The determination of the critical panel regions where cracking may occurunder combined vertical and lateral loads and the design of the reinforcement required to limit thesize of these cracks are discussed in Chapter 6.

1.4 Objectives

The objectives of the research described in this report are:

1) To determine the effect of the openings on the behavior of the walls and the wall panels;

2) To determine the critical locations in the wall panels and the required amount of panelreinforcement;

3) To develop design methods for the panel reinforcement.

1.5 Scope and Approach

The research is conducted as a part of a Daniel P. Jenny Research Fellowship funded by thePrecast/Prestressed Concrete Institute. The project focuses on two stages of loading for the walls.In the first stage, the walls are subjected to vertical forces only, such as due to gravity loads and post-tensioning. This is the loading stage which a typical wall would be subjected to during most of itsservice life. The second stage of loading is the combination of vertical loads with lateral loads, suchas due to earthquakes. The behavior and design of the walls in this stage are investigated based ona series of nonlinear static push-over analyses. The finite element program, ABAQUS (Hibbitt et al.1998) is used to model the walls.

6

For each loading stage, the critical regions in the wall panels are identified and a design approachto determine the required amount of bonded panel reinforcement to control the cracks and preventpremature failure of the walls is proposed. The effect of opening length and height, wall length, andinitial stress in the concrete due to post-tensioning and gravity loads on the behavior and design ofthe walls is investigated.

The number of stories is kept constant at six stories because the design of the panel reinforcementis not expected to be influenced significantly by the number of stories. It is assumed that horizontaljoints exist at each floor level and at the foundation since larger panel sizes may cause difficultiesin the transportation of the panels to the construction site due to increased weight and/or size,particularly for longer walls. Only rectangular openings which are located at the center of the wallpanels are considered. It is assumed that each wall panel contains an opening and that the openingsize in a wall is constant.

To achieve the research objectives, the following approach was taken:

1) A finite element model was developed for the walls using the ABAQUS Program (Hibbitt et al.1998). The model was verified by comparing the results with a second analytical model developedduring previous research and with closed-form analytical solutions for elastic panels with openings.

2) A series of parametric walls were determined. The parameters which were varied are: (1) lengthof the openings; (2) height of the openings; (3) length of the panels; and (4) initial stress in theconcrete due to post-tensioning.

3) Finite element analyses were performed on the parametric walls to investigate the behavior anddesign of the panels under vertical loads (due to gravity and post-tensioning) only.

4) A design approach was developed to determine the required bonded mild steel reinforcement inthe panels under vertical loads.

5) Finite element analyses were performed on the parametric walls to investigate the behavior anddesign of the panels under combined vertical and lateral loads.

6) A design approach was developed to determine the required bonded mild steel reinforcement inthe panels under combined vertical and lateral loads.

7

1.6 Organization of Report

The remainder of this report is organized as follows:

Chapter 2 discusses the previous research on unbonded post-tensioned precast concrete walls,concrete members with openings, and closed-form analytical solutions for stresses aroundrectangular openings in infinite elastic panels.

Chapter 3 talks about the parameters studied by the research and the parametric walls.

Chapter 4 discusses the development and verification of the finite element model used to investigatethe behavior and design of the walls.

Chapter 5 discusses the behavior and design of the walls under vertical loads only and proposes amethod for the design of the required reinforcement in the panels.

Chapter 6 discusses the behavior and design of the walls under combined vertical and lateral loadsand proposes a method for the design of the required reinforcement in the panels.

Chapter 7 presents a summary and conclusions of the research.

8

CHAPTER 2

BACKGROUND

This chapter provides background information for the report. An overview of the previous researchon the seismic behavior and design of unbonded post-tensioned precast concrete walls is presented.Then, previous research on concrete members and structures, both cast-in-place and precast, withopenings is discussed. Finally, available closed-form analytical solutions which can be used todetermine the stresses around rectangular openings in infinite elastic planes are summarized.

2.1 Unbonded Post-Tensioned Precast Concrete Walls

Previous research by Kurama et al. (1996, 1999a, 1999b) and Priestley et al. (1999) have shown theadvantages of unbonded post-tensioned precast concrete walls for seismic regions. As an example,Figure 2.1 shows the elevation and cross-section near base of a six-story prototype wall referred toas Wall WH1M which was designed for a six-story office building (with a plan layout as shown inFigure 2.2) in a region with high seismicity (e.g., Western U.S.) and a site with a medium soil profile(Kurama et al. 1999a). The wall consists of six precast concrete panels with horizontal joints at thefloor levels and at the foundation.

The wall panels are connected using post-tensioning bars which are placed inside oversized ductsas shown in Figure 2.1. The ducts allow the post-tensioning bars to be placed in the wall after thepanels are erected. The post-tensioning bars are prestressed to fpi=0.60fpu where fpu=160 ksi is theultimate strength of the post-tensioning steel. The ducts are not grouted after post-tensioning in orderto prevent bond from forming between the bars and the concrete.

The panels do not have any bonded reinforcement connecting them along the horizontal joints.During a seismic event, the lack of bonded reinforcement across the horizontal joints allows thepanels to form gaps as shown in Figure 1.2. The unbonded post-tensioning bars resist the lateralloads and control the size of the gaps during the earthquake. After the earthquake, the post-tensioning bars provide a restoring force that pulls the wall back towards its original undeformedposition.

9

8288

94100 in.

f =0.60fpi pu

CL

a =1.49 in.2p

#3 Spiralsρ = 7.3%sp

12 in

.

120 in.

7670

64

20 ft

81 ft

gravity loadslateral loads

(a)

(b)

spiralreinforcement

Figure 2.1 Wall WH1M: (a) elevation; (b) cross-section near base (half the wall length)

10

8 x 24 ft. = 192 ft.

gravity loadframe

lateral loadframe

wall

columnL-beam

invertedT-beam

N

S

hollo

w-

core

pane

ls

40 ft

.

30 ft

.

40 ft

.

Figure 2.2 Prototype building plan layout

Unbonding of the post-tensioning steel has two important advantages: (1) it results in a uniformstrain distribution in the steel, and thus, delays the nonlinear straining (i.e., yielding) of the post-tensioning bars; (2) it significantly reduces the amount of tensile stresses in the concrete, thusreducing damage due to cracking.

The strength of the unconfined concrete, fc1 in Wall WH1M is assumed to be 6 ksi. During a seismicevent, large compression stresses form in the bottom corners of the base (i.e., first story) panel dueto the formation of a gap along the base panel to foundation joint. To increase the compressivestrength and ductility of the concrete in these regions, spiral reinforcement is used to confine theconcrete along the height of the base panel as shown in Figure 2.1. The spiral reinforcement ratio(i.e., volume of spiral reinforcing steel divided by volume of confined concrete core) near the baseof Wall WH1M is !sp=7.3%. The effect of the spiral reinforcement on the concrete is discussed inChapter 4.

2.1.1 Behavior along the horizontal joints

During a seismic event, two modes of displacement can occur along the horizontal joints of anunbonded post-tensioned precast wall. The first mode is the formation of gaps as shown in Figure1.2. The second mode is shear slip. Previous research (Kurama et al. 1996, 1999a, 1999b) has shownthat shear slip along the horizontal joints is not desired. Thus, the walls are designed not to have slipalong the joints (Kurama et al. 1996, 1999a, 1999b).

11

2.1.2 Structural design parameters

The seismic response characteristics of unbonded post-tensioned precast concrete walls depend ontheir design. Kurama et al. (1996) identified and studied the effect of eleven structural designparameters on the lateral load behavior of the walls. A design approach for the walls was developedusing these parameters which are: (1) eccentricity of the post-tensioning bars; (2) unbonded lengthof the post-tensioning bars; (3) initial stress in the post-tensioning bars; (4) area of the post-tensioning bars; (5) initial stress in the concrete due to post-tensioning; (6) initial stress in theconcrete due to gravity loads; (7) strength of unconfined concrete; (8) amount of spiralreinforcement; (9) length of the wall; (10) thickness of the wall; and (11) aspect ratio of the wallcross-section (i.e. wall length divided by wall thickness).

2.1.3 Base-shear-roof-drift relationship

During a seismic event, an unbonded post-tensioned precast wall experiences lateral displacementsas shown in Figure 1.2. Figure 2.3 shows the nonlinear base-shear-roof-drift relationship of WallWH1M under combined lateral and vertical loads. The base-shear is equal to the sum of the lateralloads applied to the wall. The roof-drift is determined by dividing the horizontal displacement of theroof with the wall height. The lateral loads are applied at the floor and roof levels and are distributedin a triangular pattern over the height of the wall as shown in Figure 2.1.

Kurama et al. (1996) identified four important states on the base-shear-roof-drift relationship of aproperly designed wall. These states are: (1) decompression state; (2) softening state; (3) yieldingstate; and (4) failure state.

2.1.3.1 Decompression state

The decompression state identifies the point on the base-shear-roof-drift relationship (Figure 2.3)when the initial stress in the concrete, fci due to post-tensioning and gravity is lost under the actionof lateral loads. Decompression begins at the extreme tension corner at the base of the wall andindicates the initiation of a gap between the base panel and the foundation. Decompression alsoidentifies the end of the linear response of the wall.

12

0 0.5 1 1.5 2 2.5

200

400

600

800

1000

yielding state

decompression state

failure state

softening state

roof-drift (%)

base

-she

ar (

kips

)

Figure 2.3 Base-shear versus roof-drift relationship of Wall WH1M

2.1.3.2 Softening state

The softening state identifies the point on the base-shear-roof-drift relationship when there is asignificant reduction in the lateral stiffness of the wall (Figure 2.3). The reduction in the lateralstiffness of an unbonded post-tensioned wall usually occurs in a smooth and continuous manner.Softening can occur because of three reasons: (1) decrease in the contact length along the horizontaljoints as a result of the formation of gaps; (2) nonlinear behavior of the concrete in compression; and(3) nonlinear behavior of the post-tensioning bars in tension. In a properly designed wall, nonlinearbehavior of the post-tensioning bars occurs after significant softening of the wall due to theformation of gaps along the horizontal joints and/or nonlinear behavior of the concrete incompression. If the initial stress in the concrete, fci is large, the formation of the gaps is delayed andnonlinear behavior of the concrete governs the softening state. If the initial stress in the concrete issmall, the nonlinear behavior of the concrete is delayed and formation of the gaps governs thesoftening state (Kurama et al. 1999b).

13

2.1.3.3 Yielding state

The yielding state identifies the point on the base-shear-roof-drift relationship when the strain in thepost-tensioning bars first reaches the linear limit strain (i.e., the limit of proportionality). Prior to theyielding state, the post-tensioning bars behave in a linear-elastic manner. In a properly designed wall,the yielding state is reached after large nonlinear lateral displacements which occur primarily dueto the formation of gaps along the horizontal joints and nonlinear behavior of the concrete incompression. Prior to the yielding state, noticeable damage to the concrete in compression is smallas a result of the use of spiral reinforcement in the base panel. The cover concrete may experiencesome damage over a small region near the base of the wall (Kurama et al. 1999b).

2.1.3.4 Failure state

The failure state identifies the axial-flexural failure of the wall as a result of crushing of the spiralconfined concrete. The confined concrete crushes when the spiral reinforcement fractures. Asdescribed in Kurama et al. (1999b), this is the desired mode of failure in a properly designed wall.Sufficient spiral reinforcement is provided in the base panel to ensure that the failure state is reachedat a roof-drift significantly larger than the roof-drift at the yielding state.

2.2 Prestressed Beams with Openings

The behavior and design of post-tensioned precast concrete beams with openings have beeninvestigated by Abdalla and Kennedy (1995, 1995b, 1995c), Barney et al. (1977), Kennedy andAbdalla (1992), and Kennedy and El-Laithy (1982). These studies indicate that the largest tensionstresses around an opening occur near the centerline of the opening on the edges perpendicular tothe post-tensioning forces. Kennedy and El-Laithy (1982) used a truss analogy to design the beamreinforcement around the openings. A similar approach is used in this report for the design of thepanel reinforcement in unbonded post-tensioned precast walls under vertical loads (Chapter 5).

2.3 Monolithic Cast-in-Place Concrete Walls with Openings

The seismic behavior and design of monolithic cast-in-place reinforced concrete walls with openingshave been investigated by many researchers including Ali and Wight (1991), Kato et al. (1995),Kobayashi et al. (1995), Paulay and Priestley (1992), Powers and Wallace (1998), Taylor et al.(1998), Wallace (1998), and Yanez et al. (1992). Several different methods were proposed for thedesign of the walls including: (1) the strength reduction method; and (2) the truss analogy. In thestrength reduction method (Kobayashi et al. 1995), the lateral strength of a wall with openings isreduced using an equivalent wall without openings. The truss analogy method (Taylor et al. 1998)uses struts and ties along predicted directions of the principle compression and tension stresses in

14

the wall to design horizontal reinforcement to ensure that the lateral forces have an adequate loadpath to the base of the wall.

As the openings become large, the walls begin to behave like coupled shear wall systems. Aktan andBertero (1987), Park and Paulay (1974), Paulay and Priestley (1992), and Paulay and Santhakumar(1976) are a few of those who have investigated the seismic behavior and design of coupledreinforced concrete shear walls.

The behavior and design of unbonded post-tensioned precast walls with openings is significantlydifferent than monolithic cast-in-place concrete walls with openings as a result of the formation ofgaps along the horizontal joints. Thus, the previous research on monolithic cast-in-place walls withopenings and coupled shear walls is not investigated further in the report.

2.4 Precast Concrete Walls with Openings

The lateral load behavior and design of precast concrete walls with openings have been investigatedby Mackertich and Aswad (1997). A method for designing the walls as solid walls with an equivalentreduced thickness was proposed.

The previous research on precast concrete walls with openings has focused on walls that emulate thebehavior of monolithic cast-in-place reinforced concrete walls. The use of openings in non-emulativeprecast walls in which gaps form along the horizontal joints under lateral loads has not beeninvestigated. The research described in this report shows that these gaps have a significant effect onthe behavior and design of the walls.

2.5 Fiber Element Model

The previous research on unbonded post-tensioned precast concrete walls used an analytical modelbased on fiber elements as described by Kurama et al. (1996, 1999a, 1999b). This model, which isreferred to as the fiber element model, was developed using the DRAIN-2DX Program (Prakash etal. 1993). The fiber element model was used to verify the finite element model described in Chapter4 of this report.

In the previous wall model, fiber elements are used to represent the axial-flexural behavior of theconcrete wall panels. Each fiber element consists of a number of parallel concrete fibers along thedirection of the height of the panel. Each concrete fiber has a location in the panel cross-section, across-sectional area, and a uniaxial concrete stress-strain relationship (Kurama et al. 1999b). Thestress-strain relationship of the concrete fibers is a multi-linear idealization of the smooth stress-strain relationship for the unconfined concrete or the spiral confined concrete based on a modeldeveloped by Mander et al. (1988).

15

The post-tensioning bars are modeled using truss elements. The post-tensioning of the wall issimulated by initial tensile forces in the truss elements which are equilibrated by compressionstresses in the fiber elements. The stress-strain relationship of the truss elements is a bilinearidealization of the smooth stress-strain relationship of the post-tensioning steel (Kurama et al.1999b).

In the fiber element model of an unbonded post-tensioned precast wall, the gaps that form along thehorizontal joints are represented as distributed tensile deformation in the wall panels. The tensilestrength and stiffness of the concrete fibers modeling the panels are set to zero and the bonded panelreinforcement (i.e., wire mesh, which is not continuous across the horizontal joints) is neglected toallow the displacements from the gaps to be distributed in the wall panels. The reduction in thelateral stiffness of the wall due to the formation of gaps is represented by the zero stiffness of thefibers that go into tension.

The fiber elements modeling the wall panels are based on the assumption that plane sections remainplane. However, in an unbonded post-tensioned precast wall, the formation of gaps violates theassumption that plane sections remain plane in the regions of the wall panels immediately adjacentto the horizontal joints. Therefore, the local stresses and strains in the wall panels near the horizontaljoints are not accurately captured in the fiber element model. However, Kurama et al. (1996, 1999b)determined that the effect of the gaps on the overall lateral load behavior of the walls is accuratelycaptured.

The fiber element model cannot accurately represent the stress distributions around the openings inunbonded post-tensioned precast walls, and thus, cannot be used to investigate the local behavior anddesign around the openings in the wall panels. Thus, a finite element model is developed for wallswith openings as described in Chapter 4.

2.6 Closed-Form Solutions for Infinite Elastic Panels with Openings

Closed-form analytical solutions are available (Savin 1961) for the maximum tensile stresses aroundrectangular openings in infinite elastic panels for a limited number of load and openingconfigurations. A brief overview of the solutions presented by Savin is given in this section. Thesesolutions are used to verify the finite element model described in Chapter 4.

As an example, Figure 2.4 shows a linear-elastic panel loaded with a uniform compression stress,fp. The panel is assumed to be infinitely long in both the horizontal and vertical directions. Thepresence of the opening results in the development of axial stresses (in the horizontal, X-direction)above and below the opening as shown in Figure 2.4. The stresses near the opening are tensile, withthe maximum tension stress, ftm occurring at the center of the opening edge.

16

X

Y

fp

loho

ftm

ftm

fp

Figure 2.4 Infinite elastic plane with an opening

The stresses around the opening are represented using a function &() as:

(2.1)

( )

( ) ( )( )

( ) ( )

( ) ( ) ( )

( ) ( ) ( )

ω ζ

ζζ ζ ζ

ζ

ζ

ζ

ζ

π

π

θ

=

++

+−

+− −

+

+ − − −+

+ − + − ++

+ − + − + −

=

=

=

−

U

a a

a e

a e

f e

k i

k i

pi

1

2

a a

24

a a a a

80

5 a a 4 a a 2

896

7 a a 5 a a 2 a a

2304

21 a a 14 a a 5 a a 4

11264

2

32 2

5

4 4 2 27

5 5 3 39

6 6 4 4 2 211

2

2

where U is a constant, k is a value characterizing the dimensions (length, lo and height, ho as shownin Figure 2.4) and orientation of the opening, and fp is the stress applied to the panel. The angle

identifies the location around the opening at which the stress is being calculated, with =0 at thecenter of the opening edge parallel to the surface where fp is applied. In Figure 2.4, ftm occurs at =0.The values of k for three different opening aspect ratios of lo/ho are given in Table 2.1 (Savin 1961).

17

TABLE 2.1 VALUES OF k

lo / ho k

1.0 1/4

3.2 1/6

5.0 5/36

It is noted that fp is considered positive when applied as a tension stress. Thus, the compressionstresses applied on the panel in Figure 2.4 are negative. It can be shown that U is canceled duringthe solution, and thus will not be given a value. Eq. (2.1) is an infinite series used to define thestresses around a rectangular opening as a continuous function. As more terms are used in Eq. (2.1),the accuracy of the solution increases. However, the effect of the higher order terms on the solutiondecreases rapidly as seen in Eq. (2.1).

The function &(") is used to determine the stress functions around the opening. These stressfunctions are given by:

(2.2)( ) ( )[ ] ( )ϕ ζ ω ζ ϕ ζ= +f

4p

o

where the determination of Qo(") is described in detail by Savin (1961).

The functions &(") and Q(") are solved for aspect ratios of lo/ho=1, 3.2, and 5 with four, four, and fiveterms used in Eq. (2.1), respectively. The values of k in Eq. (2.1) are 1/4, 1/6, and 5/36 for lo/ho=1,3.2, and 5, respectively as shown in Table 2.1. The equations for &(") and Q(") for these cases are(Savin 1961):

(2.3)

( )

( )

ω ζζ

ζ ζ ζ

ϕ ζ ζζ ζ ζ

ζ ζ ζ

= − + −

=− + − −

+ +

U

f Up

1 1

6

1

56

1

176

0 .250 .42 5 0 .04 76 0 .00 86

0 .00 60 0 .00 24 0 .00 14

3 7 11

3 5

7 9 11

(2.4)

( )

( )

ω ζζ

ζ ζ ζ ζ

ϕ ζζ

ζ ζ ζ

= + − − −

= − + +

U

f Up

10 .5 0 .12 5 0 .03 8 0 .01 1

0 .250 .33 8 0 .02 3 0 .00 95

3 5 7

3 5

18

(2.5)

( )

( )

ω ζζ

ζ ζ ζ ζ

ϕ ζζ

ζ ζ ζ ζ

= + − − −

= − + + +

U

f Up

10 .643 0 .098 0 .038 0 .011

0 .250 .323 0 .016 0 .008 0 .003

3 5 7

3 5 7

where Eqs. (2.3), (2.4), and (2.5) are for lo/ho=1, 3.2, and 5, respectively.

The stresses along the boundary of the opening, 1 can be calculated by:

(2.6)( )( )σ

ϕ ζω ζθ =

′′

4R e

where Re signifies the real part of the differentials.

The results obtained from the closed-form solutions above are compared with the results obtainedfrom the finite element model in Chapter 4.

19

CHAPTER 3

PARAMETRIC INVESTIGATION

A parametric investigation was conducted on unbonded post-tensioned precast walls with rectangularopenings using a finite element model which is described in Chapter 4. Table 3.1 shows the differentparameters that were investigated which include: (1) the opening length, lo; (2) the opening height,ho; (3) the panel (wall) length, lp; and (4) the initial stress in the concrete, fci due to post-tensioningand gravity loads.

In Table 3.1, the opening length, lo is normalized with respect to the panel length, lp. The openingheight, ho is normalized with respect to the height of the base panel, hp=16 ft. The resultingnondimensionalized ratios are referred to as l=lo/lp and h=ho/hp, respectively. Similarly, the initialconcrete stress at the base of the walls, fci is normalized with respect to the assumed unconfinedconcrete strength, fc'=6 ksi, and is referred to as f=fci/fc1.

The parametric wall with lp=20 ft and f=0.29 is the same as Wall WH1M described in Chapter 2.It is noted that the findings and conclusions presented in the report are limited to the ranges of theparameters that were considered. The ranges of these parameters are described below.

3.1 Opening Dimensions, lo and ho

The effect of the opening dimensions on the behavior and design of the walls was investigated byvarying the opening length and height as shown in Table 3.1. The opening length ratio, l=lo/lp waslimited to 0.10l0.40 and the opening height ratio, h=ho/hp (where hp=16 ft for the base panel) waslimited to 0.13h0.38.

The vertical chords on both sides of the openings must be able to accommodate the placement of thepost-tensioning bars inside the chords since the placement of the bars inside the openings is notrecommended due to safety reasons. In addition, the maximum length and height of the openings arelimited to ensure that the wall panels remain stable under combined vertical and lateral loads.

20

Table 3.1. PARAMETRIC INVESTIGATION

l h

lp = 20 ft lp= 15 ft lp = 12 ft

f = 0.29 f = 0.18 f = 0.11 f = 0.057 f = 0.18 f = 0.18

0.10 0.13 x — — x --- ---

0.20 0.13 x --- --- x --- ---

0.30 0.13 x x x x x x

0.40 0.13 x --- --- x --- ---

0.10 0.25 x x x x x x

0.20 0.25 x x x x x x

0.30 0.25 x x x x x x

0.40 0.25 x x x x x x

0.10 0.38 x --- --- x --- ---

0.20 0.38 x --- --- x --- ---

0.30 0.38 x x x x x x

0.40 0.38 x --- --- x --- —

Note: l = lo/lp, h = ho/hp, and f = fci/f c1, where hp = 16 ft and fc1 = 6.0 ksi.

Under lateral loads, all or most of the axial forces in the wall (due to gravity and post-tensioning)are transferred to the foundation through the vertical chord on the compression side of the base panelas a result of the gaps that form along the horizontal joints. If the openings are too large, thecompression chord in the base panel could become unstable under these concentrated forces. Theformation of gaps could also result in large shear forces in the horizontal chords as the vertical forcesare transferred to the compression chord at the side.

For openings that are larger than those studied in this investigation, it may be necessary to usebracing in the openings to stabilize the panels under lateral loads. The presence of larger openingsmay also necessitate that the walls be designed as either coupled wall systems or as frame systems.

3.2 Panel Dimensions, lp, hp, and tp

The effect of the panel length, lp on the behavior and design of the walls was studied as shown inTable 3.1. Figure 3.1 shows the elevation and cross-section near base of the parametric walls studied.The panel height, hp was kept the same throughout the research. The base panel (i.e., first story)height was assumed to be equal to 16 ft and the upper story panel heights were assumed to be equalto 13 ft. The thickness of the panels, tp was kept constant at 12 in.

21

f =0.60fpi pu

CL

a =1.49 in.2p

#3 Spiralsρ = 7.3%

sp

12 in

.

120 in.

100 in.

8894

7064

8276

20 ft

81 ft

20 ft

81 ft

f =0.60fpi pu

CL

a =1.49 in.2p

#3 Spiralsρ = 4.9%

sp

12 in

.

120 in.

8591 in.

7973

20 ft

81 ft

CL

#3 Spiralsρ = 2.9%sp

12 in

.

120 in.

f =0.60fpi pu

a =1.49 in.2p

85 in.79

20 ft

81 ft

CL

12 in

.

120 in.

12 ft

81 ft

CL

#3 Spiralsρ = 2.1%

sp

12 in

.

72 in.

f =0.60fpi pu

a =1.15 in.2p 46

52 in.

15 ft

81 ft

f =0.60fpi pu

CL

#3 Spiralsρ = 2.9%

sp

12 in

.

90 in.

a =1.83 in.2p

65 in.59

(a) (b) (c)

(d) (e) (f)

Figure 3.1 Elevation and cross-section of the parametric walls: (a) lp=20 ft, f=0.29; (b) lp=20 ft, f=0.18; (c) lp=20 ft, f=0.11; (d) lp=20 ft, f=0.057; (e) lp=15 ft, f=0.18; (f) lp=12 ft, f=0.18

22

The parametric investigation of the panel length was limited to 12 ftlp20 ft. The maximum panellength is limited by the maximum weight and size requirements for the transportation of the panelsto the construction site. Shorter panel lengths can be used in wall systems with vertical joints similarto the walls investigated by Priestley et al. (1999), Kurama (2001), and Perez (1998).

3.3 Initial Concrete Stress, fci

The parametric investigation of the initial stress in the concrete, fci due to post-tensioning and gravityloads was limited to 0.057f0.29. The variation in the initial concrete stress was achieved byvarying the area of the post-tensioning steel. The gravity load was kept constant as described later.Table 3.2 shows the initial concrete stress at the base of the parametric walls which is determinedas the total initial post-tensioning force, Pi plus the gravity force at the base, Gb divided by the grosscross-sectional area of the walls, Ag (Ag=lptp).

The variation in the initial concrete stress is intended to represent walls designed for different levelsof seismicity. The initial concrete stress ratios of f=fci/fc1=0.29, 0.18, and 0.11 represent walls forregions with high, moderate, and low seismicity, respectively. Figures 3.1(a)-(c) show the elevationand cross-section of the parametric walls with lp=20 ft and f=0.29, 0.18, and 0.11, respectively. Thecase with f=0.057 (Figure 3.1(d)) represents a wall with gravity loads only, without any post-tensioning. Figures 3.1(e) and (f) show the elevation and cross-section of two walls with f=0.18 forlp=15 ft and 12 ft, respectively.

TABLE 3.2 POST-TENSIONING AND GRAVITY LOADS

lp=20 ft lp=15 ft lp=12 ft

f=0.29 f=0.18 f=0.11 f=0.057 f=0.18 f=0.18

fpi/fpu 0.60 0.60 0.60 --- 0.60 0.60

ap (in2) 1.49 1.49 1.49 --- 1.83 1.15

Number of bars 28 16 8 --- 8 8

Pi (kips) 4010 2290 1140 --- 1410 883

Gb (kips) 983 983 983 983 983 983

Gb/(Ag fc1) 0.341 0.341 0.341 0.341 0.455 0.569

fci=(Gb+Pi)/Ag (ksi) 1.72 1.08 0.68 0.341 1.08 1.08

23

3.4 Post-Tensioning Loads

The post-tensioning bars were prestressed to fpi=0.60fpu, where fpu=160 ksi is the assumed ultimatestrength of the post-tensioning steel. The centroid of the post-tensioning bars on each side of the wallcenterline was located at a constant proportion (68.3%) of the distance from the wall centerline tothe wall end (i.e., half the wall length). This location was selected such that the bars could be placedoutside the opening (i.e., inside the vertical side chords) for the largest opening length that wasstudied (i.e., l=0.40).

For the parametric walls with lp=20 ft, the area of each post-tensioning bar, ap was kept at 1.49 in2

as shown in Figure 3.1 and Table 3.2. For the walls with lp=15 ft and 12 ft, the ap values were 1.83in2 and 1.15 in2, respectively. The number of post-tensioning bars used in each wall was determinedto achieve the desired initial concrete stress, fci. For lp=20 ft, the number of bars used was 14, 8, and4 on each side of the wall centerline with f=0.29, 0.18, and 0.11, respectively. The walls with lp=15ft and 12 ft each used 4 bars on each side of the wall centerline resulting in f=0.18. The total initialpost-tensioning force, Pi in the walls is shown in Table 3.2.

3.5 Gravity Loads

The amount of gravity load on the parametric walls was determined from the tributary area basedon the building layout shown in Figure 2.2 as described in Kurama et al. (1996). The resulting totalgravity force acting at the base of the walls, Gb is equal to 983 kips as shown in Table 3.2. The valuesof the gravity force, Gb normalized with Agfc1 are also shown in Table 3.2.

3.6 Spiral Confinement

As discussed in Chapter 2, the desired failure state for a properly designed unbonded post-tensionedprecast concrete wall is reached when the spiral confined concrete at the base of the wall crushes.Thus, the seismic design of the walls requires that the design roof-drift capacity corresponding to thedesired failure state is greater than the expected design roof-drift demand under a survival levelearthquake as described by Kurama et al. (1999a, 1999b). The spiral reinforcement for Wall WH1Mwas designed based on this requirement resulting in a design roof-drift capacity at the desired failurestate of approximately 2.35%.

The spiral reinforcement used to confine the concrete in the base panel of the other parametric walls(except for the wall with f=0.057) was designed such that the desired failure state (correspondingto the crushing of the spiral confined concrete) is expected to be reached at a roof-drift ofapproximately 2.35%. Thus, all parametric walls are expected to fail at the same roof-drift value of2.35%.

24

Table 3.3 shows the spiral reinforcement ratio, !sp (i.e., the volume of spiral reinforcing steel dividedby the volume of the confined concrete core) in each parametric wall. The stress-strain behavior ofthe spiral confined concrete was determined using a concrete confinement model developed byMander et al. (1988). Based on the Mander model, crushing of the spiral confined concrete occursat an ultimate compressive strain of 0cu which is reached when the spiral reinforcement fractures. Thecompressive strength, fcc and the ultimate strain capacity, 0cu of the spiral confined concrete for eachwall are given in Table 3.3.

TABLE 3.3 CONCRETE PARAMETERS


f=0.29 f=0.18 f=0.11 f=0.057 f=0.18 f=0.18

!sp (%) 7.3 4.9 2.9 --- 2.9 2.1

fc1 (ksi) 6.00 6.00 6.00 6.00 6.00 6.00

fcc (ksi) 14.8 12.7 10.6 --- 10.6 9.22

0cu 0.0538 0.0428 0.0319 --- 0.0319 0.0270

3.7 Analysis of the Walls

Each parametric wall was first subjected to gravity and post-tensioning forces using the finiteelement model described in Chapter 4. Then, lateral loads were applied on the walls (except for thewall with f=0.057) using the finite element model. A nonlinear static push over analysis withequivalent lateral loads applied at the floor and roof levels was used. The lateral loads weredistributed in a triangular pattern over the height of the walls as shown in Figure 2.1. The results ofthe analyses under vertical loads only (due to gravity and post-tensioning) are described in Chapter5. The results of the analyses under combined vertical and lateral loads are described in Chapter 6.

25

CHAPTER 4

ANALYTICAL MODEL

This chapter describes an analytical model to investigate the behavior and design of unbonded post-tensioned precast walls with rectangular panel openings. As an example, Figure 4.1(a) shows theanalytical model for Wall WH1M with l=0.40 and h=0.38 which was developed using the FiniteElement Program ABAQUS (Hibbitt et al. 1998). The model, which is referred to as the finiteelement model, can be used to conduct nonlinear analyses of walls with and without openings undervertical and lateral loads. Modeling of the post-tensioning bars, wall panels, behavior along thehorizontal joints, and loading as well as verification of the model are described below.

4.1 Modeling of the Post-Tensioning Bars

The unbonded post-tensioning bars are modeled using truss elements. The post-tensioning loads aresimulated by initial tensile forces in the truss elements which are equilibrated with compressionstresses in the wall panels. Figure 4.2 shows the uniaxial stress-strain relationship of the post-tensioning steel in tension which is modeled as a tri-linear (dashed line) approximation of the smoothstress-strain relationship (solid line). The yield stress of the tri-linear relationship is determined fromthe linear limit (i.e., limit of proportionality) stress of the smooth stress-strain relationship as fpy=120ksi. The strain-hardening stiffness is determined to minimize the difference between the smooth andtri-linear stress-strain relationships. It is noted that the strains in the post-tensioning bars remainsmall as a result of unbonding.

As discussed in Chapter 1, the post-tensioning bars are anchored to the wall only at the foundationand at the roof. In the finite element model, the post-tensioning anchors at the foundation arerepresented by restraining the vertical and horizontal translational degrees of freedom of the trusselement nodes located at the foundation level. The post-tensioning anchors at the roof level aremodeled using rigid elements. The rigid elements are connected to the truss elements nodes locatedat the roof level and share nodes with the elements used to model the wall panels, thus constrainingthe displacements of the post-tensioning anchors to the displacements of the wall.

26

(a) (b)

gap/contactsurface

nonlinearplane stresselement

trusselement

Figure 4.1 Finite element model: (a) refined mesh with openings; (b) coarse mesh

27

f = 160pu

stress (ksi)

strain

tri-linear approximationsmooth stress-strain relationship

f = 120py

0.00414 0.03510

Figure 4.2 Stress-strain relationship of the post-tensioning steel

4.2 Modeling of the Wall Panels

The precast concrete wall panels are modeled using nonlinear rectangular plane stress elements. Itis assumed that the panels remain stable under the applied post-tensioning, gravity, and lateral loads.As shown in Figure 4.1, a large number of elements are used to model the panels. The number ofelements is increased at the top of the wall near the post-tensioning anchors and in the base panelwhich is the most critical panel. The size of the elements in the base panel is 2x2 in., uniform. Thesecond story panel is modeled using 4x4 in. elements, except for the bottom 12 in. and the top 24in. of the panel as described below.

To improve the modeling of the behavior along the horizontal joints, the elements adjacent to eachjoint are modeled using the same element size. Thus, the bottom 12 in. of the second story panel ismodeled using 2x2 in. elements (same as the base panel). The top 24 in. of the second story panelis modeled with 8x8 in. elements to provide a transition to the third story panel. The third, fourth,and fifth story panels are modeled using 8x8 in. elements. The bottom half of the sixth (top) storypanel is modeled using 8x8 in. elements. The upper half of the sixth story panel is modeled using4x4 in. elements due to the presence of the post-tensioning anchors.

The openings are modeled by regions in the wall panels without any elements. To accurately modelthe size of the openings, it may be necessary to use smaller elements around the openings. Figure

28

4.1(a) shows the use of smaller elements around the openings in Wall WH1M. A row of 4x4 in.elements is used at the top and bottom of the openings in the third, fourth, and fifth story panels andat the bottom of the opening in the sixth story panel to accommodate for the panel and openingheights.

The spiral reinforcement in the wall panels was not modeled explicitly. Instead, the nonlinearuniaxial compressive stress-strain relationship of the spiral confined concrete in the panels wasdetermined using a confinement model developed by Mander et al. (1988). This stress-strainrelationship was used in the plane stress elements modeling the panels as described further inSections 4.3 and 4.4.

Variations in the concrete properties across the thickness of the wall panels (such as cover concrete)were ignored to reduce the size of the model and to prevent numerical problems due to the crushingof the cover concrete. Crushing of the cover concrete at the two ends of the walls near the base canalso cause numerical problems. These numerical problems were prevented by modeling the coverconcrete at the wall ends using spiral confined concrete properties. Previous investigations of wallswithout openings (Kurama et al. 1996) have shown that crushing of the cover concrete occurs overa small region near the bottom corners of the base panel and does not have a significant effect on thebehavior of the walls.

As shown in Figure 2.1, the spiral reinforcement is placed along the entire height of the base paneland is used over a sufficient length of the panel such that crushing of the unconfined concrete awayfrom the ends of the panel does not occur. The interior portions of the base panel are unconfined andare reinforced with wire mesh only. Kurama et al. (1996) showed that the strains in these regionsremain mostly in the linear-elastic range. In the linear-elastic range, the stress-strain relationshipsof the unconfined concrete and spiral confined concrete are similar. Because of this similarity, thestress-strain relationship for the spiral confined concrete was used throughout the entire base panel.

Preliminary finite element analysis results showed that the stresses in the upper story (i.e., secondstory and up) panels remained in the linear-elastic range. To simplify the model, the concrete in theupper story panels was modeled using a linear-elastic stress-strain relationship. The use of a linear-elastic stress-strain relationship also improved the behavior of the roof panel in the regions near thepost-tensioning anchors. It is noted that the local behavior of the walls near the post-tensioninganchors is not a focus of this research, and thus, is not investigated.

As described in Chapter 2, each post-tensioning bar is placed inside an oversized ungrouted duct toallow the wall to displace laterally without coming into contact with the post-tensioning bars. Theseducts cause a decrease in the area of the wall at the locations of the ducts. Kurama et al. (1996) haveshown that the effect of the ducts on the behavior of the walls is small. Thus, the reduction in theconcrete area due to the ducts was not modeled.

29

4.2.1 Refinement of the finite element mesh

Figure 4.1 shows two finite element models that were developed for the walls. The refined modelin Figure 4.1(a) is described above. The coarse model (Figure 4.1(b)) consists of 10x10 in. elementsfor most of the wall, with 5x5 in. elements at the right side of the bottom one and a half panels, alongthe horizontal joint between the bottom two panels, at the location of the lateral loads, and at the topof the sixth story panel. Openings were not used in this earlier wall model.

Preliminary finite element analyses conducted using the model in Figure 4.1(b) showed that the meshhad to be refined. The refinement of the mesh was done in a number of stages until furtherrefinement did not have a significant effect on the results and the analyses could be completed in areasonable amount of computational time. The model in Figure 4.1(a) shows the final mesh whichwas used to investigate the behavior and design of the walls as described in Chapters 5 and 6.

4.3 Modeling of Concrete in Compression

As stated above, the nonlinear uniaxial compressive stress-strain relationship of concrete wasdetermined using a model developed by Mander et al. (1988) and used in the plane stress elementsmodeling the wall panels. As an example, the solid lines in Figure 4.3 show the Mander stress-strainrelationships for the unconfined concrete and spiral confined concrete in Wall WH1M (with!sp=7.3%). The concrete stresses are normalized with the strength of the unconfined concrete, fc1=6ksi. The compressive strength of the spiral confined concrete, fcc is reached at a strain of 0cc. Thespiral confined concrete crushes (desired failure state) at a strain of 0cu when the spiral reinforcementfractures.

The dashed line in Figure 4.3 shows a multi-linear approximation of the spiral confined concretestress-strain relationship. This relationship is similar to the multi-linear relationship used in the fiberelement model (with the DRAIN-2DX Program) which is described in Chapter 2.

Numerical problems were encountered during the push-over analyses of the walls (using the finiteelement model) up to the desired failure state using the multi-linear concrete stress-strain relationshiprepresented by the dashed line in Figure 4.3. As the elements at the bottom corner of a wall reachedthe compressive strength of the spiral confined concrete, fcc and began to resist smaller stresses withincreased lateral displacements of the wall, the finite element program was unable to continue withthe analysis. As shown in Figure 4.3, the stress-strain relationship for the concrete has a negativestiffness after reaching the compressive strength, fcc. This negative stiffness caused the numericalproblems that prohibited the analyses from continuing.

30

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

0.5

1

1.5

2

2.5

3

ε

unconfined concrete

fc′

(f , ε )cc cc

(f , ε )cu cu

c′f

/ fc

c

modified multi-linear approximation (ABAQUS)

multi-linear approximation (DRAIN-2DX and ABAQUS)

Mander model

crushing of spiral confined concrete (desired failure state)

spiral confined concrete (ρ = 7.3%)sp

Figure 4.3 Concrete stress-strain relationships

To overcome the numerical problems, a modified multi-linear concrete stress-strain relationship wasused in the finite element model as shown by the dotted line in Figure 4.3. In order to reach the roof-drift corresponding to the desired failure state of a wall, it was necessary to maintain a positivestiffness in the concrete stress-strain relationship throughout the analysis. It was also necessary toensure that the modified relationship would not add strength to the system. This required the use oflower stresses in the modified stress-strain relationship as shown in Figure 4.3.

The numerical problems described above are associated with elements which are subjected to largecompression stresses. The largest compression stresses occur at the bottom of the base panel, andthus it was necessary to modify the stress-strain relationship of these elements only. Thus, themodified stress-strain relationship was used only in the bottom 20 in. of the walls. This modificationallowed the analyses to reach the roof-drift of approximately 2.35% corresponding to the desiredfailure state of the parametric walls as described in Sections 2.1.3.4 and 3.6.

4.4 Concrete Failure Envelope under Biaxial Loading

The concrete stress-strain relationship discussed above is for uniaxial compression with or withoutlateral confinement. However, the plane stress elements used for the wall panels model the behaviorof concrete under biaxial loading. To model the behavior of concrete under biaxial stress conditions,a failure envelope is used by the ABAQUS Program. The thick line in Figure 4.4 shows the defaultfailure envelope used by ABAQUS to model the failure of concrete under biaxial loading. The

31

stresses are normalized with the strength of the spiral confined concrete, fcc. Compression stressesare shown using negative values.

The default biaxial failure envelope in ABAQUS can cause an increase in the strength of theconcrete under the condition of compressive stresses in one direction and compressive stresses inthe transverse direction that act as confining stresses as shown by the shaded area in Figure 4.4. Theconcrete model by Mander et al. (1988) already accounts for the confinement provided by the spiralreinforcement as discussed earlier. Thus, it is necessary to modify the biaxial failure envelope inABAQUS such that additional concrete confinement does not occur due to the stresses in thetransverse direction.

.

−1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

11 cc f / f

22cc

f /

f

range of transversecompression stresses inelements with largecompression stresses

increasein strength

2

1

−0.27

ABAQUS default envelope (1.16, 0.09, 1.28, 0.33)modified envelope (0.80, 1.00, 0.75, 0.33)

Figure 4.4 Concrete biaxial failure envelope

The ABAQUS biaxial failure envelope is determined based on four parameters which are referredto as failure ratios (Hibbitt et al. 1998). These ratios are:

1) The ratio of the ultimate biaxial compressive stress to the ultimate uniaxial compressive stress.

2) The absolute value of the ratio of uniaxial tensile stress at failure to the ultimate uniaxialcompressive stress.

3) The ratio of the magnitude of a principal component of plastic strain at ultimate stress in biaxialcompression to the plastic strain at ultimate stress in uniaxial compression.

32

4) The ratio of the tensile principal stress at cracking, in plane stress, when the other principle stressis at the ultimate compressive value, to the tensile cracking stress under uniaxial tension.

By default, the ABAQUS Program uses 1.16, 0.09, 1.28, and 0.33 for failure ratios 1, 2, 3, and 4,respectively. To model the behavior of the spiral confined concrete in the parametric walls, theABAQUS default biaxial failure envelope was modified by using 0.80, 1.00, 0.75, and 0.33 for thefailure ratios as shown by the thin line in Figure 4.4. The determination of the modified biaxialfailure envelope for the spiral confined concrete in the parametric walls is described below.

4.4.1 ABAQUS biaxial failure ratios

In order to determine the effect of the ABAQUS biaxial failure ratios on the concrete stress-strainrelationship, a small finite element model was created using ABAQUS. This model consisted of a9x9 mesh, with 1x1 in. nonlinear plane stress elements as shown in Figure 4.5. The bottom edge ofthe model was restrained for vertical (Y-axis) displacements only and one side of the model wasrestrained for horizontal (X-axis) displacements only. The model was then used to determine thefailure ratios for the spiral confined concrete as follows.

X

Y

Figure 4.5 The 9x9 finite element model

The top edge of the model was loaded with a constant uniform stress. Then, a uniform stress wasapplied to the unrestrained side of the model. This stress was increased until the concrete failed. Forexample, to determine the failure envelope for biaxial compression, a constant uniform compressionstress was first applied to the top edge, and then the unrestrained side was compressed until theconcrete failed. The procedure was repeated for different values of the compression stress appliedat the top of the model, starting from a stress of zero (i.e., uniaxial compression) to the condition inwhich the compression stress applied at the top and side edges of the model were the same. A similarmethod was used to determine the biaxial compression-tension regions and the biaxial tension regionof the failure envelope.

33

The 9x9 model was used to study the effect of each failure ratio on the concrete stress-strainrelationship and the biaxial failure envelope. The multi-linear approximation of the Mander stress-strain relationship (shown by the dashed line in Figure 4.3) was used in the plane stress elements inthe model. The investigation of each failure ratio is described below. The results of the investigationwere used to determine the failure ratios to model the behavior of the spiral confined concrete in theparametric walls.

4.4.2 ABAQUS failure ratio 1

The effect of failure ratio 1 on the biaxial failure envelope and on the stress-strain relationship of theconcrete is studied using values of 1.16 (default), 1.0, 0.8, and 0.6 as shown in Figures 4.6(a) and(b), respectively. The ABAQUS default values are used for the other failure ratios.

f / f = -0.2722 cc(f = -4.0 ksi)22

0 0.02 0.04 0.06 0.08 0.10

0.25

0.5

0.75

1

1.25

1.16 (default)1.0 0.8 0.6 Mander model

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

f11

/ fcc

f 22 /

f cc

1.16 (default)1.0 0.8 0.6

2

1

εc

f /

f cc

11

(a) (b)

ρ = 7.3%sp

Figure 4.6 Failure ratio 1: (a) biaxial failure envelope; (b) stress-strain relationships

Figure 4.6(b) compares the ABAQUS stress-strain relationships (solid lines) with the Mander stress-strain relationship (dashed line) for the spiral confined concrete in Wall WH1M (with !sp=7.3%)under a constant transverse compressive stress of f22=-4.0 ksi (corresponding to f22/fcc=-0.27 whichis the maximum normalized transverse compression stress expected in the parametric walls). Theresults indicate that the strength of concrete under biaxial compression decreases significantly asfailure ratio 1 decreases. The larger ultimate strain, 0cu of the ABAQUS stress-strain relationshipsin Figure 4.6(b) is due to failure ratio 3, which is discussed below.


The effect of failure ratio 2 on the concrete biaxial failure envelope is studied using values of 0.09(default), 0.5, and 1.0 as shown in Figure 4.7. The ABAQUS default values are used for the other

34

failure ratios. The results indicate that the strength of concrete in tension increases as failure ratio2 increases.

−1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

0.09 (default)0.5 1.0

11 cc f / f

22cc

f /

f

2

1

Figure 4.7 Effect of failure ratio 2 on the concrete biaxial failure envelope


The effect of failure ratio 3 on the biaxial failure envelope and on the stress-strain relationship of theconcrete is studied using values of 0.5, 1.28 (default), and 2.0 as shown in Figures 4.8(a) and (b),respectively. The ABAQUS default values are used for the other failure ratios.

Figure 4.8(b) compares the ABAQUS stress-strain relationships (solid lines) with the Mander stress-strain relationship (dashed line) for the spiral confined concrete in Wall WH1M (with !sp=7.3%)under a constant transverse compressive stress of f22=-4.0 ksi (corresponding to f22/fcc=-0.27). Thelarger compressive strength, fcc of the ABAQUS stress-strain relationships as compared with theMander stress-strain relationship is due to failure ratio 1, which is described earlier.

The results indicate that failure ratio 3 does not affect the biaxial failure envelope, however, it hasa significant effect on the stress-strain relationship of the concrete. Figure 4.8(b) shows that the strainat which the spiral confined concrete crushes, 0cu increases significantly as failure ratio 3 increases.

35

0 0.02 0.04 0.06 0.08 0.10

0.25

0.5

0.75

1

1.25

2.01.28 (default)0.5Mander model

f / f = -0.2722 cc(f = -4.0 ksi)22

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

f11

/ fcc

f 22 /

f cc

2.01.28 (default) 0.5

f /

fcc

11

2

1

εc

(a) (b)

ρ = 7.3%sp

Figure 4.8 Failure ratio 3: (a) biaxial failure envelope; (b) stress-strain relationships


The effect of failure ratio 4 on the concrete biaxial failure envelope is studied using values of 0.0,0.33 (default), and 1.0 as shown in Figure 4.9. The ABAQUS default values are used for the otherfailure ratios. The results indicate that failure ratio 4 has a small effect on the biaxial compression-tension regions of the failure envelope.

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

1.00.33 (default) 0.0

2

1

11 cc f / f

22cc

f /

f

Figure 4.9 Effect of failure ratio 4 on the concrete biaxial failure envelope

36

4.4.6 Failure ratios used for the parametric walls

Through the use of the simple 9x9 element model, the value of each failure ratio to be used in themodeling of the spiral confined concrete in the parametric walls was determined. Values of 0.80 and0.75 were used for failure ratios 1 and 3, respectively, such that the spiral confined concrete does notexperience an increase in strength or ductility under biaxial compression.

In the finite element model of the walls, large compressive stresses occur in elements with onlyrelatively small compressive stresses in the transverse direction. The range of the compressionstresses in these elements is shown in Figure 4.4 (corresponding to f22/fcc=-0.27). Additional concreteconfinement due to biaxial compression was prevented only in this range of the transverse stresses.As shown in Figure 4.4, this results in a reduction in the compressive strength of the concrete withlarger transverse stresses. However, the finite element analysis results indicate that these stressconditions do not exist in the parametric walls.

As an example, Figure 4.10 compares the ABAQUS and Mander stress-strain relationships for thespiral confined concrete in Wall WH1M (with !sp=7.3%) under a constant transverse compressionstress of f22=-4.0 ksi corresponding to f22/fcc=-0.27. The results indicate that the ABAQUS stress-strain relationship using values of 0.80 and 0.75 for failure ratios 1 and 3, respectively compares wellwith the Mander stress-strain relationship for the spiral confined concrete.

εc

11cc

f /

f

0 0.02 0.04 0.06 0.08 0.10

0.25

0.5

0.75

1

1.25

ABAQUS (0.80, 1.00, 0.75, 0.33)Mander model

ρ = 7.3%sp

f / f = -0.2722 cc

(f = -4.0 ksi)22

Figure 4.10 ABAQUS versus Mander concrete stress-strain relationships

37

Failure ratio 4 was left at the ABAQUS default value of 0.33 since it has a small effect on theconcrete biaxial failure envelope. The selection of failure ratio 2 is discussed below.

4.5 Modeling of Bonded Mild Steel Reinforcement

The bonded mild steel reinforcement and the wire mesh in the wall panels was not modeledexplicitly. Since the actual amount and location of the bonded panel reinforcement in a wall is notknown in advance, an explicit modeling of the steel would necessitate an iterative analysis procedureto determine the required area, number, and location of the bars using an accurate representation ofeach bar in the wall panels. This iterative procedure would significantly limit the number of casesthat could be investigated by the research because of an increased number of analyses necessary foreach case, difficulties in the modeling of individual bars and cracking of concrete, numericalproblems, and execution time.

These difficulties were overcome by modeling the effect of the bonded steel reinforcement usingelastic tension properties in the plane stress elements for the wall panels. This was achieved by usinga value of 1.0 for the concrete failure ratio 2 described above. It is assumed that: (1) the panels arereinforced with a sufficient amount of bonded mild steel to limit the size of the cracks; (2) the panelreinforcement does not yield; and (3) the reinforcement is distributed uniformly in the tension zone.Based on these assumptions, the required area of reinforcement can be determined from the tensionstresses in the plane stress elements modeling the wall panels as described in Chapters 5 and 6.

It is noted that as a result of using elastic tension properties for the wall panels, the redistribution ofthe stresses due to concrete cracking and the actual placement of the panel reinforcement cannot bemodeled. However, in a properly designed panel with a sufficient amount of well-distributedreinforcement, the cracks remain small, and thus, are not expected to significantly affect thebehavior. Furthermore, the actual placement of the panel reinforcement is likely to be similar to theassumed uniform distribution in the tension regions.

It is noted that the biggest “cracks” in an unbonded post-tensioned precast wall are the gaps that formalong the horizontal joints under lateral loads. The bonded panel reinforcement does not cross thehorizontal joints, and thus, does not restrict the formation of the gaps. This results in relatively smalltensile stresses, and thus less cracking in the walls as compared with monolithic cast-in-placereinforced concrete walls. The gaps are accurately represented in the finite element model by usingnonlinear gap/contact surfaces along the horizontal joints as described below.

4.6 Modeling of Behavior Along the Horizontal Joints

As discussed in Chapter 2, two types of behavior are possible along the horizontal joints of anunbonded post-tensioned precast wall under lateral loads. The possible behaviors are: (1) formationof gaps; and (2) slip. Previous research (Kurama et al. 1996, 1999a, 1999b) has shown that the slip

38

behavior is not desired, and thus, the walls are designed not to have slip along the horizontal joints.In the finite element model, this is represented by providing friction along the joints to ensure thatslip does not occur. The formation of gaps along the horizontal joints is modeled as described below.

As shown in Figures 1.2(a) and 4.11, discrete gaps form along the horizontal joints between the wallpanels and between the wall and the foundation if the precompression from the post-tensioning andgravity is overcome due to lateral loads. The formation of these gaps is modeled using nonlineargap/contact surfaces at the top and bottom of the wall panels. To allow for the formation of the gaps,the gap/contact surfaces in adjacent panels do not share the same nodes. Figure 4.11(a) shows thedisplaced shape of the entire wall model and Figure 4.11(b) shows the deformed shape of the basepanel. The displacements are enlarged to show the gaps along the horizontal joints. The largest gapforms between the base panel and the foundation.

(a) (b)

Figure 4.11 Deformed finite element model: (a) entire wall; (b) base panel

39

As lateral loads are applied to the wall, the compression stresses (due to gravity and post-tensioning)acting on the gap/contact surfaces decrease near the tension side of the wall and increase near thecompression side of the wall. When the contact stress becomes zero, the surfaces have separated, anda gap has initiated on the tension side of the wall. When one node of a gap/contact surface hasseparated, the entire length of the surface is modeled as being opened. The length of each gap/contactsurface is the same as the length of the plane stress elements along the horizontal joints in the panels.Thus, a large number of elements are used along the horizontal joints at the top and bottom of thepanels to get an accurate estimate of the gap length. The gap/contact surfaces along the horizontaljoints of the base panel were modeled using 2x2 in. elements. The gaps along the horizontal jointsin the upper floor levels are expected to be either very small or zero. Thus, larger elements (i.e., 8x8in. elements) were used along the upper floor joints.

4.7 Modeling of Gravity Loads

The gravity loads were modeled as uniformly distributed loads applied to the panels at each floor androof level.

4.8 Modeling of Lateral Loads

The lateral loads were applied at the floor and roof levels and were distributed in a triangular patternalong the height of the walls (with the maximum load at the roof) as shown in Figure 2.1. Eachlateral load was distributed uniformly over the top two feet of the wall panel in that story.

4.9 Verification of the Analytical Model

The finite element model was verified using two methods: (1) the results obtained using the finiteelement model of walls without openings were compared with the results obtained using the fiberelement model discussed in Chapter 2; and (2) the stresses around rectangular openings in infiniteelastic panels were determined using the finite element model and were compared with the stressesdetermined using the closed-form analytical solutions described in Chapter 2.

4.9.1 Comparisons with the fiber element model

The results obtained for walls using the finite element model were compared with results obtainedusing a second analytical model, which is referred to as the fiber element model. As described inChapter 2, the fiber element model cannot be used to investigate the behavior and design of wallswith openings. Thus, the comparison between the two models was done on walls without openings.

40

The comparison between the finite element model and the fiber element model was done using WallWH1M, with the openings removed. Figure 4.12(a) shows the nonlinear base-shear-roof-driftrelationship of the wall under vertical and lateral loads obtained using the two models. Figure 4.12(b)shows the length of the base panel that is in contact with the foundation (normalized with respectto the wall length) as a function of the roof-drift. The results indicate that both the global (i.e., base-shear-roof-drift) and the local (i.e., contact length) behaviors of the wall predicted by the two modelsare similar.

yieldingstate

decompressionstate

2.52.01.51.00.50

200

400

600

800

1000

roof-drift (%)

base

-she

ar (

kips

)

Finite Element Model (ABAQUS)

Fiber Element Model (DRAIN-2DX)

2.52.01.51.00.50

0.2

0.4

0.6

0.8

1.0

roof-drift (%)

cont

act l

engt

h at

bas

e / w

all l

engt

hFinite Element Model (ABAQUS)

Fiber Element Model (DRAIN-2DX)

(a) (b)Figure 4.12 Verification of the finite element model: (a) base-shear-roof-drift relationship;

(b) contact length

4.9.2 Comparisons with closed-form solutions

The finite element model was used to estimate the stresses around rectangular openings in infiniteelastic panels. The results were compared with the closed-form analytical solutions described inChapter 2. As an example, Figure 4.13 shows comparisons between the maximum tension stress, ftm

(at the middle of the top and bottom edges of the opening) calculated using the closed-form solutions(with Eqs. (2.3)-(2.6)) and the maximum tension stress predicted using the finite element model ofthree panels with opening aspect ratios of lo/ho=1.0 (with lo=60 in., ho=60 in.), 3.2 (with lo=192 in.,ho=60 in.), and 5.0 (with lo=300 in., ho=60 in.). The panels were loaded with a uniform compressionstress of fp=1.0 ksi.

The closed-form solutions in Chapter 2 are for infinite panels. A large panel with finite dimensionswas used in the finite element model. The panel length was 3000 in. and the panel height was 600in. (ten times the dimensions of the largest opening studied). The panel was modeled using 2.5x2.5in. elements around the opening. The element size was increased away from the opening to amaximum size of 10x10 in. The results in Figure 4.13 indicate that the finite element model iscapable of accurately predicting the maximum tension stresses adjacent to the opening for differentvalues of opening length, lo and height, ho.

41

0 3.0 6.0

0.6

1.2

closed-form solutionfinite element model

f = 1.0 ksip

l / ho o

f (

ksi)

tm

XY

1.0 2.0 4.0 5.0

0.8

0.4

0.2

1.0

fp

lo

ho

ftm

ftm

fp

Figure 4.13 Finite element model versus closed-form solutions

42

CHAPTER 5

DESIGN OF PANEL REINFORCEMENT UNDER VERTICAL LOADS ONLY

This chapter talks about the behavior and design of the parametric walls under vertical loads (dueto post-tensioning and gravity) only. A design approach is proposed to estimate the required bondedmild steel reinforcement in the wall panels. For each wall, the base panel is investigated in detailsince the gravity loads are maximum at the base while the post-tensioning loads are constant alongthe height of the wall, and thus, the most critical panel is the base panel. An example is included inAppendix A to demonstrate the proposed design approach.

5.1 Critical Panel Regions under Vertical Loads

As described earlier, a typical unbonded post-tensioned precast wall would primarily be subjectedto vertical loads due to gravity and post-tensioning during most of its service life. Under theseconditions, cracks may form in the wall panels due to the presence of openings. In order to limit thesize of these cracks, the most critical locations in the wall panels need to be identified and reinforcedwith a sufficient amount of bonded mild steel.

As an example, Figure 5.1 shows the principle stress contours in the base panel of Wall WH1M(lp=20 ft and f=0.29) using the finite element model. Figures 5.1(a) and (b) show the maximum andminimum principle stress contours, respectively in the panel without and with an opening (l=0.40,h=0.38). The opening in Figure 5.1 is the largest opening considered by the research.

The red shaded areas in Figure 5.1(a) show the critical regions of the panel where the maximum

principle stresses exceed in tension. The red shaded areas in Figure 5.1 (b) indicate regions7 5. ′f c

of the panel where the minimum principle stresses are tensile (i.e., greater than zero). The signconvention for the stresses is tension positive. Cracking is expected to occur in the critical regions

where the maximum principle stresses are larger than .7 5. ′f c

43

Figure 5.1 shows that the presence of an opening disrupts the flow of the compression stresses in thepanel, causing the formation of significant tensile stresses along the top and bottom edges of theopening. The maximum tensile stresses form at the middle of the opening edges.

(b)

(a)

Figure 5.1 Stress contours in the base panel: (a) maximum principle stresses; (b) minimum principle stresses

As described in Chapter 4, the panels were modeled using elastic tension properties assuming thatthe critical regions are reinforced with an adequate amount of well-distributed bonded mild steel.Thus, the tensile stresses in the panels are allowed to exceed the cracking stress. The required areaof the steel is determined by dividing the total force in the tension zone (by integrating the tensilestresses) with the assumed yield strength of the steel. The tensile strength of concrete is ignored inthe design of the reinforcement.

Figure 5.1 shows that the critical panel regions under vertical loads are at the top and bottom of theopening and that the maximum tensile stresses may be large enough to cause cracking. The finite

44

element analysis results indicate that the stresses above and below the opening are similar, and thus,similar amounts of reinforcement are needed in both locations.

Figures 5.2(a) and (b) show the horizontal and vertical stress contours, respectively, correspondingto the principle stress contours in Figures 5.1(a) and (b). The red shaded areas in Figure 5.2(a)

indicate regions of the panel where the horizontal stresses exceed in tension. Figures 5.1(a)7 5. ′f c

and 5.2(a) show that the critical principle tension stresses in the panel are mostly in the horizontaldirection, and thus, horizontal bonded mild steel reinforcement is required parallel to the edges atthe top and bottom of the opening.

(a)

(b)

Figure 5.2 Stress contours in the base panel: (a) horizontal stresses; (b) vertical stresses

45

It is noted that for the opening and panel sizes and loads considered for the parametric walls, shearreinforcement is not needed in the horizontal and vertical chords of the wall panels under verticalloads. Thus, the design of the chord shear reinforcement is not addressed in the report. A nominalamount of minimum shear reinforcement may be used in the horizontal chords.

5.2 Design of Panel Reinforcement: Overview

This section investigates the required panel reinforcement above and below the openings based onthe finite element analysis results of the parametric walls. A design approach is proposed to estimatethe amount of required reinforcement. For each wall, the design of the base panel is investigated indetail since the effect of the gravity loads is maximum at the base while the post-tensioning forceis constant over the height of the wall, and thus, the most critical panel is the base panel.

Figure 5.3 shows the compression stresses, 1p acting at the top of a typical base panel from thegravity loads applied at the first floor level and from the gravity and post-tensioning loads transferredto the panel from the panel above. These stresses are obtained from the finite element model and arereferred to as the panel top stresses. Only the stress distribution on the right half of the wall is showndue to symmetry about the wall centerline. The stress distribution above the panel is not uniformbecause of the presence of the openings. It is noted that the panel loading shown in Figure 5.3 issimilar to the loading in Figure 2.4 with the exception that the stress distribution in Figure 5.3 is notuniform.

The compression stresses, 1s at the top of the vertical chord on the right side of the opening are alsoshown in Figure 5.3. These stresses are in equilibrium with the panel top stresses and are referredto as the side chord stresses.

Similar to Figure 2.4, the presence of the opening results in axial stresses (in the horizontal, X-direction) in the top chord above the opening. The total design tension force, Tv in the top chord isdetermined by integrating the stresses in the tension zone over a height of htv as shown in Figure 5.3.Then, the required area of steel, Av is determined by dividing the design tension force, Tv with thesteel yield strength, fy. Since the stresses at the top and bottom of the opening are similar, a similaramount of reinforcement is needed below the opening.

5.3 Truss Model

The total design tension force, Tv in the top chord can be estimated using a “truss” model as shownin Figure 5.4(a) and enlarged in Figure 5.4(b). As shown in Figure 5.1, the opening disturbs the flowof the stresses from the top to the bottom of the panel. The truss model is used to represent the flowof the panel stresses around the opening.

46

.

Cv

Tvhtv

l o

ho

l p

hpσs

σp

panel top stresses

side chord stresses

CL

X

Yhc

l c

Figure 5.3 Panel stresses

The truss model uses only a portion of the panel top stresses and the side chord stresses. The portionof the total compression force used in the model is referred to as Cr, which is calculated from thepanel top stresses and side chord stresses over a distance of xr from the wall centerline as shown bythe shaded areas in Figure 5.4(a).

Let the location of the stress resultant Cr be equal to x*p and x*s at the top of the panel and at the topof the side chord, respectively. Based on the finite element analysis results, it is assumed that thelocation of Cr at a distance of lp/4hc (where hc=hp/2-ho/2 is the height of the top chord as shown inFigure 5.3) up from the top of the opening is the same as x*p. Thus, the total design tension forceabove the opening, Tv and the required area of steel, Av are estimated as:

(5.1)

T Cx x

l4

AT

f

v r

s p

p

vv

y

=−

=

The determination of Cr, x*p, and x*s for use in Eq. (5.1) is described below.

47

(a) (b)

fp2

CLCL

uniformdistributionl o

ho

l p

hp

x2

xr

x1

l / 4p

fpe

f p1fp0

f sr

fse

xp

xs

Cr

θt

Cr

Tv

l / 4p

C = Tv v

lineardistribution

Figure 5.4 Design of panel reinforcement: (a) truss model; (b) truss model enlarged

5.3.1 Estimation of panel top stresses

The panel top stresses are estimated by dividing the stress distribution from x=0 (panel centerline)to x=lp/2 (panel edge) into three regions between 0xx1, x1xx2, and x2xlp/2, where x is thedistance measured from the panel centerline. The stress distribution in each region, 1pj isapproximated using a second order curve as:

(5.2)σ p j p j p j p ja x b x c= + +2

where the subscript j indicates the stress distribution in the j’th region. The stress 1p1(0)=fp0 at x=0is estimated as:

(5.3)f cp0

pa

pp

f

4540) f= − +(θ

The stress fpp is equal to the uniformly distributed stress due to the gravity loads applied at the floorlevel above the panel, Gp and fpa is equal to the uniformly distributed stress due to the total initialpost-tensioning force, Pi (see Table 3.2) and the gravity loads applied at the upper floor levels, Ga

(e.g., second floor and up for the base panel):

48

(5.4)

fG

l t

fP G

l t

p p

p

p p

p ai a

p p

=

=+

where lp and tp are the panel length and thickness, respectively. The angle c (in degrees) is shownin Figure 5.5(a) and is calculated as:

(5.5)θc

pu o

o

h h

l=

−−tan ( )1

where hpu is the height of the upper story panel (i.e., the panel above). The angle c is measuredbetween the panel centerline at the top and the bottom corner of the opening in the upper story panel.Figure 5.5(b) shows how Eq. (5.3) (shown by the solid lines) compares with the fp0 values obtainedusing the finite element model (shown by the markers) for different values of l, h, f, and lp. Thestress fp0 is normalized with respect to the strength of the unconfined concrete, fc1=6 ksi. Thevariation in the angle c in Figure 5.5(b) is as a result of the variations in l and h.

0.30

0.10

0 30 60 90

f /

f′ c

p0

0.20

θ (degrees)c

θ

hpu

hp ho

lp

lo

hc

l c

c

γ = 0.29, l = 20 ftf p

γ = 0.18, l = 20 ftf p

γ = 0.11, l = 20 ftf p

γ = 0.057, l = 20 ftf p

γ = 0.18, l = 15 ftf p

γ = 0.18, l = 12 ftf p

(a) (b)Figure 5.5 Estimation of fp0: (a) bottom two panels; (b) fp0 versus c

The distances x1 and x2 are selected such that the stresses fp1 and fp2 (at x1 and x2) can be estimatedusing a linear stress distribution as shown by the dashed line in Figure 5.4(a). Based on the finiteelement analysis results, it is assumed that:

49

(5.6)x

l

2

x2 l l

4

1o

2

o p

=

=+

The stresses fpe, fp2, and fp1 for the linear stress distribution at x=lp/2, x2, and x1, respectively aredetermined as:

(5.7)

( )

( )

( )

f2 P G G

l tf

f f2 f f x

l

f f2 f f x

l

pe

i p a

p pp0

p2 p0

pe p0 2

p

p1 p0

pe p0 1

p

=+ +

−

= +−

= +−

It is assumed that the slope of 1p1 at x=0 is zero (i.e., 1p11(0)=0). Then, the three second order stressdistributions, 1p1 (0xx1), 1p2 (x1xx2), and 1p3 (x2xlp/2) can be determined using the followingboundary conditions with fp0, fp1, and fp2 as determined from Eqs. (5.3) and (5.7):

(5.8)

( )( )

( ) ( )( ) ( )( ) ( )( ) ( )

σ

σ

σ σ

σ σ

σ σ

σ σ

σ σ σ

p1 p0

p1

p2 1 p1 1 p1

p2 1 p1 1

p3 2 p2 2 p2

p3 2 p2 2

10 2 32

0 f

0 0

x x f

x x

x x f

x x

21

1

2

2

p

=

′ =

= =

′ = ′

= =

′ = ′

+ + =+ +

∫ ∫ ∫t dx t dx t dxP G G

p p

x

p px

x

p px

li p a

50

Figures 5.6(a)-(d) compare the estimated panel top stress distributions with stress distributionsdetermined using the finite element analysis results of a selected set of parametric walls for varyingvalues of l (with h=0.25, f=0.29, lp=20 ft), h (with l=0.20, f=0.29, lp=20 ft), lp (with l=0.20,h=0.25, f=0.18), and f (with l=0.20, h=0.25, lp=20 ft), respectively. The stresses are normalizedwith respect to fc'=6 ksi.

0 0.1 0.2 0.3 0.4 0.5

0.1

0.2

0.3

0.4

0.5

x/lp

γh = 0.25, γ

f = 0.29, l

p = 20 ft

γl = 0.40

γl = 0.30

γl = 0.20

γl = 0.10

0 0.1 0.2 0.3 0.4 0.5

0.1

0.2

0.3

0.4

0.5

x/lp

γl = 0.20, γ

f = 0.29, l

p = 20 ft

γh = 0.38

γh = 0.25

γh = 0.13

0 0.1 0.2 0.3 0.4 0.5

0.1

0.2

0.3

0.4

0.5

x/lp

γl = 0.20, γ

h = 0.25, γ

f = 0.18

lp = 20 ft

lp = 15 ft

lp = 12 ft

0 0.1 0.2 0.3 0.4 0.5

0.1

0.2

0.3

0.4

0.5

x/lp

γl = 0.20, γ

h = 0.25, l

p = 20 ft

γf = 0.29

γf = 0.18

γf = 0.11

γf = 0.057

(a) (b)

(c) (d)

σ p / f c′

σ p / f c′

σ p / f c′

σ p / f c′

FEMestimated

FEMestimatedFEM

estimated

FEMestimated

Figure 5.6 Panel top stresses: (a) h=0.25, f=0.29, lp=20 ft; (b) l=0.20, f=0.29, lp=20 ft; (c) l=0.20, h=0.25, f=0.18; (d) l=0.20, h=0.25, lp=20 ft

The results in Figure 5.6 indicate that the estimated stress distributions are close to the FEMdistributions (i.e., the distributions obtained using the finite element model). For the cases of l=0.30and 0.40 with h=0.25, f=0.29, and lp=20 ft (Figure 5.6(a)), which correspond to two of the longer

51

openings in Wall WH1M, the estimated stress distributions deviate from the FEM distributions nearthe end of the panels (i.e., near x=lp/2). However, this deviation does not significantly affect theestimation of the required panel reinforcement since only a portion of the stress distribution over adistance of xr from the panel centerline is used in the truss model.

5.3.2 Estimation of Cr and x*p

The truss model shown in Figure 5.4(a) uses the portion of the panel top stresses between 0xxr

with a compression stress resultant equal to Cr and a resultant location equal to x*p. Based on thefinite element analysis results, it is assumed that:

(5.9)x 0 .3 ll

2r co= +

where lc is the length of the side chord as shown in Figure 5.3. Thus, Cr and x*p can be determinedby integrating the estimated panel top stress distribution from x=0 to x=xr as:

(5.10)

C t dx dx

x txdx xdx

C

r p p

x

px

x

p p

p

x

px

x

r

r

r

= +

=+

∫ ∫

∫ ∫

σ σ

σ σ

1 2

1 2

1

1

1

0

0

1

Note that x1<xr<x2, thus both fp1 and fp2 need to be estimated to determine Cr and x*p.

5.3.3 Estimation of side chord stresses and x*s

The compression stress distribution at the top of the side chord is estimated as follows. The stressdistribution over x=lo/2 and x=lp/2 is divided into two regions, 1s1 and 1s2 between lo/2xxr andxrxlp/2, respectively where xr is given in Eq. (5.9). The compression stress resultant for 1s1 isassumed to be equal to Cr, and the compression stress resultant for 1s2 is assumed to be equal to(Pi+Gp+Ga)/2-Cr. The stress fsr at x=xr is estimated using a uniform stress distribution in the sidechord as shown by the dashed line in Figure 5.4(a) as:

(5.11)fP G G

l tsr

i p a

c p

=+ +

2

Then, the stress fse at x=lp/2 is determined assuming a linear stress distribution for 1s2 as:

52

(5.12)f fse sr=+ + −

−P G G 2C

0 .7 l ti p a r

c p

Finally, a second order stress distribution is assumed for 1s1 which can be determined using thefollowing boundary conditions:

(5.13)

( ) ( )( ) ( )

σ σ

σ σ

σ

s r s r sr

s r s rsr se

c

s r

x

x x f

x xf f

l

t dx Cr

1 2

1 2

p 1l2

0 .7

o

= =

′ = ′ =−

=∫

After finding 1s1, the location of the stress resultant Cr at the top of the side chord, x*s can bedetermined as follows:

(5.14)xt xdx

Cs

p sl

x

r

o

r

=∫ σ 1

2

Figures 5.7(a)-(d) compare the estimated side chord stress distributions with stress distributionsdetermined from the finite element analyses of a selected set of walls for varying values of l (withh=0.25, f=0.29, lp=20 ft), h (with l=0.20, f=0.29, lp=20 ft), lp (with l=0.20, h=0.25, f=0.18), andf (with l=0.20, h=0.25, lp=20 ft), respectively. The stresses are normalized with respect to fc'=6 ksi.The results indicate that the estimated stress distributions are close to the FEM distributions.

5.4 Placement of the Panel Reinforcement

The design approach described above can be used to determine the panel reinforcement needed tolimit the size of the cracks that can occur due to the presence of an opening. The finite elementresults indicate that the tension stresses above and below the opening are similar. Thus, thereinforcement determined for the top of the opening can and should also be used at the bottom of theopening. It is recommended that a minimum reinforcement of two No. 5 bars (with an area ofAmin=0.61 in2 ) is used along the top and bottom edges of each opening.

The reinforcing bars should be placed as close to the top and bottom edges of the opening as possibleon both sides (faces) of the panel. The reinforcement must be placed horizontally within the depthof the tension zone, htv (see Figure 5.3), above and below the opening. Based on the finite elementanalysis results, it may be assumed that:

(5.15)h htv l c= γ

53

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

x/lp

γh = 0.25, γ

f = 0.29, l

p = 20 ft

γl = 0.40

γl = 0.30

γl = 0.20

γl = 0.10

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

x/lp

γl = 0.20, γ

f = 0.29, l

p = 20 ft

γh = 0.38

γh = 0.25

γh = 0.13

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

x/lp

γl = 0.20, γ

h = 0.25, γ

f = 0.18

lp = 20 ft

lp = 15 ft

lp = 12 ft

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

x/lp

γl = 0.20, γ

h = 0.25, l

p = 20 ft

γf = 0.29

γf = 0.18

γf = 0.11

γf = 0.057

σ s / f c′

σ s / f c′

σ s / f c′

σ s / f c′

(a) (b)

(c) (d)

FEMestimated

FEMestimated

FEMestimated

FEMestimated

Figure 5.7 Side chord stresses: (a) h=0.25, f=0.29, lp=20 ft; (b) l=0.20, f=0.29, lp=20 ft; (c) l=0.20, h=0.25, f=0.18; (d)l=0.20, h=0.25, lp=20 ft

For the cases studied, the proposed design requirements can be met without the need for using verylarge bars since the amount of reinforcement needed in the panels is not excessive as shown later.As an example, Figure 5.8 shows the reinforcement required in the base panel of a wall with lp=20ft, f=0.18, l=0.30, and h=0.38.

54

l = 6 fto

h =

6 ft

o

l = 20 ftp

h =

16

ftp

2 #4 bars4 pairs @ 2.0 in.

t = 12 in.p

h =

16

ftp

1.5 in.

Figure 5.8 Panel reinforcement example

The reinforcing bars must be extended a sufficient distance on both sides of the middle of theopening in order to provide for full development of the yield strength of the steel. The length of thetension zones at the top and bottom of the opening extend the entire length of the opening (Figure5.1(a)). Thus, the length of each bar must be longer than the opening length to provide sufficientembedment beyond the opening corners.

5.5 Design of Upper Story Panels

The finite element analysis results of the parametric walls indicate that tension stresses form alongthe top and bottom edges of the openings in the upper story panels, similar to the base panel. Themagnitude of these tension stresses is most significant in walls with high initial stress conditions(e.g., f=0.29). For example, Figures 5.9(a) and (b) show the maximum principle stress contours overthe height of Wall WH1M (f=0.29 and lp=20 ft) without and with openings (l=0.40, h=0.38),respectively. The red shaded areas indicate regions of the wall panels where the maximum principle

stresses exceed in tension, and thus, where cracking is expected to occur.7 5. ′f c

Figure 5.9(b) shows that the size of the critical regions with tension stresses larger

than decrease in the upper story panels. Several of the upper story panels in Figure 5.9(b)7 5. ′f c

do not show any regions of tension stresses exceeding . However, the finite element analysis7 5. ′f c

results show that significantly large (but less than ) tensile stresses do form at the top and7 5. ′f c

bottom edges of the openings in these panels, and thus, bonded mild steel reinforcement isrecommended at the top and bottom of the openings.

55

(a) (b)

Figure 5.9 Principle stress contours in Wall WH1M (f=0.29, lp=20 ft): (a) without openings; (b) with openings (l=0.40, h=0.38)

56

Figures 5.10(a) and (b) show the horizontal stress contours corresponding to the principle stresscontours in Figures 5.9(a) and (b), respectively. Similarly, Figures 5.11(a) and (b) show the verticalstress contours corresponding to the principle stress contours in Figures 5.9(a) and (b), respectively.The red shaded areas in Figure 5.10 indicate regions of the panels where the horizontal stresses

exceed in tension. Figures 5.9(b) and 5.10(b) show that the critical principle tension stresses7 5. ′f c

in the panels are primarily in the horizontal direction, and thus, horizontal reinforcement is requiredat the top and bottom of the openings in the upper story panels similar to the base panel.

It is noted that the proposed design approach can be used for the design of the upper story panels aswell as the base panel which is the most critical panel. The only differences between the base paneland the upper story panels for the design of the panel reinforcement are the amount of gravity loadsand, possibly, the panel height. Thus, the design approach can be applied to the upper story panelsby using the appropriate panel height and the gravity loads acting on the panel.

Similar to the base panel, a minimum reinforcement of two No. 5 bars is recommended along thetop and bottom edges of the openings in the upper story panels. It is noted that the proposed designapproach cannot be used for the roof panel where there is no panel above and concentrated post-tensioning anchors forces are applied. The design of the roof panel for the post-tensioning anchorforces is not addressed by the research.

For walls where the post-tensioning forces are significantly larger than the gravity loads, it isrecommended that the reinforcement designed for the base panel is used in the upper story panelssince the stresses in the panels will be similar. For walls with little or no post-tensioning, it may bemore economical to determine the reinforcement needed in each panel.

57

(a) (b)Figure 5.10 Horizontal stress contours in Wall WH1M (f=0.29, lp=20 ft): (a) without openings; (b) with openings (l=0.40, h=0.38)

58

(a) (b)Figure 5.11 Vertical stress contours in Wall WH1M (f=0.29, lp=20 ft): (a) without openings; (b) with openings (l=0.40, h=0.38)

59

5.6 Base Panel Reinforcement Results

For each parametric wall, the required steel area, Av above and below the opening in the base panelwas predicted using the proposed design approach and compared with the steel area determinedusing the finite element model (which is referred to as the FEM steel area). In order to determine theFEM steel area, the tensile stresses above the opening were integrated over the height of the tensionzone, htv and then divided with the yield strength of steel. The steel yield strength, fy was assumedto be equal to 60 ksi.

Table 5.1 compares the predicted and FEM steel areas, Av for the base panels of the parametric wallsnormalized with respect to the chord area hctp resulting in the required reinforcement ratio, !v as:

(5.16)ρvv

c p

A

h t=

The shaded values in the table correspond to the cases where the predicted reinforcement ratio is lessthan the FEM reinforcement ratio.

TABLE 5.1 REQUIRED REINFORCEMENT AT TOP AND BOTTOM OF OPENING

l h

lp = 20 ft lp= 15 ft lp = 12 ft

f = 0.29 f = 0.18 f = 0.11 f = 0.057 f = 0.18 f = 0.18

!v (%)FEM

!v (%)pred.

!v (%)FEM

!v (%)pred.

!v (%)FEM

!v (%)pred.

!v (%)FEM

!v (%)pred.

!v (%)FEM

!v (%)pred.

!v (%)FEM

!v (%)pred.

0.10 0.13 0.11 0.11 --- --- --- --- 0.021 0.021 --- --- --- ---

0.20 0.13 0.20 0.22 --- --- --- --- 0.041 0.046 --- --- --- ---

0.30 0.13 0.28 0.32 0.18 0.21 0.12 0.14 0.058 0.069 0.17 0.18 0.15 0.16

0.40 0.13 0.34 0.40 --- --- --- --- 0.071 0.092 --- --- --- ---

0.10 0.25 0.12 0.12 0.076 0.078 0.049 0.051 0.023 0.024 0.070 0.066 0.062 0.055

0.20 0.25 0.22 0.23 0.14 0.15 0.092 0.10 0.044 0.049 0.12 0.12 0.11 0.10

0.30 0.25 0.28 0.32 0.18 0.21 0.012 0.14 0.059 0.071 0.18 0.19 0.17 0.17

0.40 0.25 0.31 0.37 0.21 0.25 0.14 0.17 0.070 0.091 0.22 0.24 0.22 0.23

0.10 0.38 0.13 0.14 --- --- --- --- 0.026 0.028 --- --- --- ---

0.20 0.38 0.22 0.24 --- --- --- --- 0.046 0.052 --- --- --- ---

0.30 0.38 0.26 0.29 0.17 0.19 0.11 0.13 0.058 0.070 0.18 0.19 0.18 0.18

0.40 0.38 0.26 0.30 --- --- --- --- 0.066 0.085 --- --- --- ---


60

Results from the observed trends and comparisons based on the parametric analyses are given inFigures 5.12-5.15 which show the predicted and FEM reinforcement ratios for different values ofl, h, lp, and f.

Figures 5.12(a)-(d) show the effect of l on !v for varying values of f (with h=0.25, lp=20 ft), lp (withh=0.25, f=0.18), h (with f=0.057 no post-tensioning, lp=20 ft), and h (with f=0.29, lp=20 ft),respectively. The results indicate that the required reinforcement ratio, !v increases as the openinglength, lo increases.

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

γl

ρ v (%

)

γh=0.25, lp=20 ft

γf=0.29

γf=0.18

γf=0.11

γf=0.057

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

γl

ρ v (%

)

γh=0.25, γ

f=0.18

lp=20 ftlp=15 ftlp=12 ft

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

γl

ρ v (%

)

γf=0.057, lp=20 ft

γh=0.38

γh=0.25

γh=0.13

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

γl

ρ v (%

)

γf=0.29, lp=20 ft

γh=0.38

γh=0.25

γh=0.13

(a) (b)

(c) (d)

FEMpredicted

FEMpredicted

FEMpredicted

FEMpredicted

Figure 5.12 Effect of l on !v: (a) h=0.25, lp=20 ft; (b) h=0.25, f=0.18; (c) f=0.057, lp=20 ft; (d) f=0.29, lp=20 ft

61

Figures 5.13(a)-(d) show the effect of h on !v for varying values of f (with l=0.30, lp=20 ft), lp (withl=0.30, f=0.18), l (with f=0.057 no post-tensioning, lp=20 ft), and l (with f=0.29, lp=20 ft),respectively. The results indicate that the opening height, ho has little or no effect on the requiredreinforcement ratio, !v.

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

γh

ρ v (%

)

γl=0.30, lp=20 ft

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

γh

ρ v (%

)

γl=0.30, γ

f=0.18


0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

γh

ρ v (%

)

γf=0.057, lp=20 ft

γl=0.40

γl=0.30

γl=0.20

γl=0.10

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

γh

ρ v (%

)

γf=0.29, lp=20 ft

γl=0.40

γl=0.30

γl=0.20

γl=0.10

FEMpredicted FEM

predicted

FEMpredicted

FEMpredicted

γf=0.29

γf=0.18

γf=0.11

γf=0.057

(a) (b)

(c) (d)Figure 5.13 Effect of h on !v: (a) l=0.30, lp=20 ft; (b) l=0.30, f=0.18; (c) f=0.057, lp=20 ft;

(d) f=0.29, lp=20 ft

62

Figures 5.14(a) and (b) show the effect of lp on !v for varying values of h (with l=0.30, f=0.18) andl (with h=0.25, f=0.18), respectively. The results indicate that the panel length, lp has little or noeffect on the required reinforcement ratio, !v as long as the opening length ratio l=lo/lp remainsconstant.

0.5 0.75 1 1.25 1.50

0.1

0.2

0.3

0.4

0.5

hp / lp

ρ v (%

)

γl = 0.30,γ

f= 0.18

γh=0.38

γh=0.25

γh=0.13

0.5 0.75 1 1.25 1.50

0.1

0.2

0.3

0.4

0.5

hp /lpρ v (

%)

γh= 0.25,γ

f= 0.18

γl=0.40

γl=0.30

γl=0.20

γl=0.10

FEMpredicted

FEMpredicted

(a) (b)

Figure 5.14 Effect of lp on !v: (a) l=0.30, f=0.18; (b) h=0.25, f=0.18

Finally, Figures 5.15(a) and (b) show the effect of f on !v for varying values of h (with l=0.30,lp=20 ft) and l (with h=0.25, lp=20 ft), respectively. The results indicate that the requiredreinforcement ratio, !v increases significantly as the initial concrete stress, fci increases.

0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

0.5

γf

ρ v (%

)

γl = 0.30, lp= 20 ft

γh=0.38

γh=0.25

γh=0.13

0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

0.5

γf

ρ v (%

)

γh= 0.25, lp = 20 ft

γl=0.40

γl=0.30

γl=0.20

γl=0.10

FEMpredicted

FEMpredicted

(a) (b)Figure 5.15 Effect of f on !v: (a) l=0.30, lp=20 ft; (b) h=0.25, lp=20ft

63

5.6.1 Result averages

Figures 5.12-5.15 show how that the predicted reinforcement ratios compare well with the FEMreinforcement ratios. Table 5.2 shows the averages of the predicted reinforcement ratio divided bythe FEM reinforcement ratio for each combination of lp and f. The results indicate that the designapproach, on average, provides a reasonably close and conservative (except for lp=12 ft) estimate forthe required reinforcement in the parametric walls.

TABLE 5.2 AVERAGE OF PREDICTED/FEM REINFORCEMENT RATIO


f=0.29 f=0.18 f=0.11 f=0.057 f=0.18 f=0.18

top of opening, !v 1.10 1.12 1.14 1.16 1.05 0.99

64

CHAPTER 6

DESIGN OF PANEL REINFORCEMENT UNDER COMBINED

VERTICAL AND LATERAL LOADS

This chapter investigates the required bonded panel reinforcement in unbonded post-tensionedprecast walls with rectangular openings under combined vertical loads and lateral loads due toearthquakes. A design approach is proposed to determine the critical regions in the panels and toestimate the required amount of reinforcement based on the finite element analyses of the parametricwalls. For each parametric wall, the design of the base panel is investigated in detail since the designforces (i.e., axial force, shear force, and bending moment) are maximum at the base, and thus, themost critical panel is the base panel. The design of the upper story panels is done using the designof the base panel. An example is included in Appendix B to demonstrate the proposed designapproach.

6.1 Effect of Openings on the Lateral Load Behavior of the Walls

The finite element model described in Chapter 4 was used to study the effect of rectangular openingsin the wall panels on the lateral load behavior of the walls. It was assumed that a sufficient amountof well distributed bonded steel reinforcement is used in the critical regions of the panels to controlthe cracking due to the openings and due to the gaps that form along the horizontal joints.

As an example, Figure 6.1 compares the base-shear-roof-drift relationship of Wall WH1M (lp=20ft, f=0.29) without and with openings (l=0.40, h=0.38). The opening size used in this wall is thelargest opening considered by the research. Figure 6.1 also compares the base-shear-roof-driftrelationship of the parametric wall with lp=12 ft and f=0.18, without and with openings (l=0.30,h=0.38).

The results in Figure 6.1 indicate that the base-shear-roof-drift relationship of the walls without andwith openings are similar. For the parametric walls which were studied by the research, the openingshave little or no effect on the lateral stiffness and strength of the walls or on the decompression,softening, yielding, and failure states as long as the panels are properly designed.

65

0 0.5 1.0 1.5 2.0 2.50

200

400

600

800

1000

roof-drift (%)

base

-she

ar (

kips

)

decompression state

softening state

failure state

yielding state

without openingswith openings (γ = 0.40, γ = 0.38)

l h

p f l = 20 ft, γ = 0.29


l h

p f l = 12 ft, γ = 0.18

decompression state

yielding statesoftening statefailure state

Figure 6.1 Base-shear-roof-drift relationship of walls without and with openings

Similarly, Figure 4.11(b) shows that the opening does not have a significant effect on the deformedshape of the panel as long as the cracks are controlled by using an adequate amount of well-distributed bonded panel reinforcement. This is because, the behavior of a wall with properlydesigned panels is governed by the formation of gaps along the horizontal joints. The post-tensioningbars are not bonded to the concrete and the bonded panel reinforcement does not cross the horizontaljoints. Thus, the post-tensioning bars and the panel reinforcement do not restrict the formation of thegaps. The design of the reinforcement in the panels to achieve this desired behavior is described inthe chapter.

6.2 Critical Panel Regions

As described in Chapter 1, large tension stresses may form in the wall panels under combinedvertical and lateral loads due to: (1) presence of openings; and (2) formation of gaps along thehorizontal joints. In order to limit the size of these cracks, the most critical regions in the panels needto be identified and reinforced.

The finite element analyses of the parametric walls show that the most critical panel is the base panelbecause of the presence of larger forces (i.e., axial force, shear force, and bending moment), and,

66

larger stresses near the base. Thus, the design of the base panel is investigated in detail. The designof the upper story panels is done using the design of the base panel.

Figure 6.2 shows the principle stress contours in the base panel of Wall WH1M (lp=20 ft, f=0.29)without and with an opening (l=0.40, h=0.38) when the desired mode of failure as described inChapter 2 is reached under lateral loads applied from left to right. This state, which corresponds tothe crushing of the spiral confined concrete at the base, is referred to as the desired failure state.

(a)

(b)Figure 6.2 Stress contours under lateral loads: (a) maximum principle stresses;

(b) minimum principle stresses

Figures 6.2(a) and (b) show the maximum and minimum principle stress contours in the base panels,respectively. The red shaded areas in Figure 6.2(a) indicate the critical regions in the panels wherethe maximum principle stresses exceed in tension and the red shaded areas in Figure 6.2(b)7 .5 fc′indicate regions where the minimum principle stresses are tensile (i.e., greater than zero). The sign

67

convention for the stresses is tension positive. Cracking is expected to occur in the critical regionswhere the maximum principle stresses are larger than . 7 .5 fc′

The results in Figure 6.2 show that, as a result of the formation of gaps along the horizontal joints,the amount of cracking that is expected to occur in the panels is significantly smaller than thecracking that would be expected in a comparable monolithic cast-in-place reinforced concrete wall.

As described in Chapter 4, the wall panels were modeled using elastic tension properties assumingthat the critical regions are reinforced with an adequate amount of well-distributed bonded mild steel.Thus, the tensile stresses in the panels are allowed to exceed the cracking stress. The required areaof the steel is determined by dividing the total force in the tension zone (by integrating the tensilestresses) with the assumed yield strength of the steel. The tensile strength of concrete is ignored inthe design of the reinforcement.

Figures 6.2(a) and (b) show that cracking can occur in walls without openings as well as walls withopenings. When no opening is present, the critical regions are at the top and bottom of the panelwhere large tensile stresses occur as a result of the gaps that form along the horizontal joints. Thepresence of an opening in the panel results in the following: (1) additional critical regions with largetensile stresses form near the corners of the opening and at the side of the panel; (2) the location ofthe critical region at the top of the panel moves to the right (for lateral loads applied from left toright); and (3) the flow of the compression stresses from the top contact region to the bottom contactregion in the panel is disrupted (Figure 6.2(b)).

Figure 6.2 shows that the most critical regions are at the top, bottom, and sides of the panel, and nearthe corners of the opening. Figures 6.3(a) and (b) show the horizontal and vertical stress contours,respectively, corresponding to the principle stress contours in Figure 6.2. The red shaded areasindicate the critical panel regions where the maximum tension stresses exceed in tension.7 .5 fc′Figures 6.2(a) and 6.3(a) indicate that the critical principle tension stresses at the top and bottom ofthe panel and near the corners of the opening are mostly in the horizontal direction, and thushorizontal reinforcement is needed at these locations. Similarly, Figures 6.2(b) and 6.3(b) show thatthe critical principle tension stresses at the side of the panel are mostly vertical, and thus verticalreinforcement is needed at the panel edges.

The panel reinforcement that is required in the critical regions is shown in Figure 6.4. Horizontalbars with steel areas of Ac1b and Ac3b are needed at the top and bottom of the base panel, respectively.Horizontal bars are also needed at the top and bottom of the opening. The area of the steel requiredin these regions is Ac4b. Vertical bars with a steel area of Ac2b are needed at the sides of the panel.

68

(a)

(b)

Figure 6.3 Stress contours in the base panel: (a) horizontal stresses; (b) vertical stresses

The finite element analysis results indicate that the tension forces at the sides of the opening are notlarge, and thus a nominal area of minimum vertical reinforcement, Ac5b=Amin as described later, issufficient. Transverse (i.e., shear) reinforcement (not shown in Fig. 6.4) may also be necessary in thetop, bottom, and side chords depending on the opening and panel dimensions and the forces in thechords. The required reinforcement must be placed symmetrically on both sides of the panelconsidering lateral loads applied in both directions (from left to right and from right to left).

69

hp hoA c2b A c2b

l p

l o

A c1b

A c3b

A c4b

A c4b

A c5bA c5b

Figure 6.4 Panel reinforcement

6.3 Determining the Critical Panel Sections for Design: Overview

A method is proposed for determining the location of the critical sections where maximum tensionforces develop at the top and bottom of the base panel of a wall with openings when the desiredfailure state of the wall is reached. This is necessary in order to design the panel reinforcement inthese regions. It is assumed that the lateral loads are applied from left to right.

As described in Section 6.2, other critical regions are located at the sides of the panel and near thecorners of the opening. The finite element analysis results show that the maximum tension forcesat the sides of the panel occur at the same height as the bottom of the opening, independent of theparameters that were studied by the research. The most critical locations adjacent to the opening areat the corners of the opening.

6.4 Walls Without Openings

The locations of the critical sections at the top and bottom of a base panel depend on the contactlength along the horizontal joints (i.e., length over which the panel is in contact with the panel aboveand with the foundation). First, the contact length along the horizontal joints of a panel without anopening is addressed as follows.

70

6.4.1 Contact length at the bottom of the base panel

For a wall without openings, the length of the base panel that is in contact with the foundation, cb1

is estimated by assuming a rectangular compression stress block in the contact zone as shown inFigure 6.5. The shear force at the base of the wall is not shown in Figure 6.5. The depth of therectangular stress block is ab1=1cb1. From equilibrium:

(6.1)′ =

= + +

cN

f t

N G G P

b

p b

cc c

p b p a f

αβ1

where fcc is the compressive strength of the spiral confined concrete, . and 1 are rectangular stressblock parameters for confined concrete (e.g., as given by Paulay and Priestley 1992), and tc is thewidth of the spiral confined concrete. It is assumed that the cover concrete is crushed and the spiralconfined concrete is at the verge of crushing corresponding to the desired failure state. For the casesstudied in this report, . = 0.9 and 1 = 1.0 may be used (Paulay and Priestley 1992).

hp

l p

M pb

CL

Npb

αfcc

cb′

a = β cb b1 ′′

assumed rectangularstress block

actual stress distribution

coverconcrete

Figure 6.5 Estimation of cb1

The force Npb is the total axial force at the base of the wall due to gravity and post-tensioning. Thegravity forces are divided into two parts: (1) Gp is the gravity force applied at the floor level abovethe panel (i.e., first floor); and (2) Ga is the sum of the gravity forces applied at the upper floor androof levels (i.e., second floor and up). The post tensioning force, Pf is equal to the sum of the forcesin the post-tensioning bars at the desired failure state as determined from a nonlinear push-overanalysis of the wall with no openings.

71

Eq. (6.1) was used to predict the contact length at the base of the parametric walls. Table 6.1compares the predicted contact lengths with the contact lengths determined from the finite elementanalysis results. The cases with l=0 and h=0 are for the walls without openings. The contact lengthcb1 is normalized by the panel length, lp. The results indicate that the contact lengths predicted usingEq. (6.1) compare well with the finite element analysis results.

It is noted that crushing of the cover concrete was not modeled in the finite element analyses of theparametric walls as described in Chapter 4. Thus, the thickness of the spiral confined concrete, tc wasreplaced with the panel thickness, tp in Eq. (6.1). Furthermore, the calculated contact lengths, cb1 forthe parametric walls are measured from the end of the wall panels.

TABLE 6.1 CONTACT LENGTH AT THE BOTTOM OF THE BASE PANEL

l h

lp = 20 ft lp= 15 ft lp = 12 ft

f = 0.29 f = 0.18 f = 0.11 f = 0.18 f = 0.18

cb11/lpor

cb/lp FEM

cb11/lpor

cb/lp pred.

cb11/lpor

cb/lp FEM

cb11/lpor

cb/lp pred.

cb11/lpor

cb/lp FEM

cb11/lpor

cb/lp pred.

cb11/lpor

cb/lp FEM

cb11/lpor

cb/lp pred.

cb11/lpor

cb/lp FEM

cb11/lpor

cb/lp pred.

0.00 0.00 0.14 0.14 0.11 0.11 0.083 0.084 0.13 0.12 0.15 0.14

0.10 0.13 0.14 0.14 --- --- --- --- --- --- --- ---

0.20 0.13 0.14 0.14 --- --- --- --- --- --- --- ---

0.30 0.13 0.14 0.14 0.11 0.11 0.083 0.084 0.13 0.12 0.15 0.14

0.40 0.13 0.14 0.14 --- --- --- --- --- --- --- ---

0.10 0.25 0.14 0.14 0.11 0.11 0.083 0.084 0.13 0.12 0.15 0.14

0.20 0.25 0.14 0.14 0.11 0.11 0.083 0.084 0.13 0.12 0.15 0.14

0.30 0.25 0.14 0.14 0.11 0.11 0.083 0.084 0.13 0.12 0.15 0.14

0.40 0.25 0.14 0.14 0.11 0.11 0.083 0.084 0.13 0.12 0.15 0.14

0.10 0.38 0.14 0.14 --- --- --- --- --- --- --- ---

0.20 0.38 0.14 0.14 --- --- --- --- --- --- --- ---

0.30 0.38 0.14 0.14 0.11 0.11 0.083 0.084 0.13 0.12 0.15 0.14

0.40 0.38 0.14 0.14 --- --- --- --- --- --- --- ---


6.4.2 Contact length at the top of the base panel

For a wall with no openings, the contact length at the top of the base panel, ct1 is determined fromequilibrium of the forces at the first floor level as shown Figure 6.6. The gravity force, Gp appliedat the first floor level and the story shear force are not shown in Figure 6.6. The contact length, ct1

can be estimated from moment equilibrium about the panel centerline as:

72

(6.2)′ = −

c

l M

Nt

p pt

p t

Ω 2

where Npt=Ga+Pf is the axial force transferred to the top of the base panel from the panel above. Themoment Mpt is the moment acting at the top of the panel as a result of the lateral forces applied atthe upper floor and roof levels and the forces in the post-tensioning bars at the desired failure stateas determined from a nonlinear push-over analysis of the wall.

In Eq. (6.2), is a stress distribution factor as follows. If the contact stress distribution along thehorizontal joint is linear, then is equal to 3.00. From the finite element analyses of the parametricwalls, it was determined that the distribution of the contact stresses at the top of the base panel isslightly nonlinear at the desired failure state. A factor of =2.75 was found to be appropriate for theestimation of ct1.

c thp

l p

M pt

Npt

CL

N = G + Pfapt

Figure 6.6 Estimation of ct1

73

Table 6.2 compares the predicted (using Eq. (6.2)) contact lengths for the parametric walls with thecontact lengths determined using the finite element analysis results. Similar to Table 6.1, the caseswith l=0 and h=0 are for the walls without openings and the contact length, ct1 is normalized withrespect to the panel length, lp. The results in Table 6.2 indicate that the predicted contact lengthscompare well with the finite element analysis results.

TABLE 6.2 CONTACT LENGTH AT THE TOP OF THE BASE PANEL

l h

lp = 20 ft lp= 15 ft lp = 12 ft

f = 0.29 f = 0.18 f = 0.11 f = 0.18 f = 0.18

ct11/lpor

ct/lpFEM

ct11/lpor

ct/lppred.

ct11/lpor

ct/lpFEM

ct11/lpor

ct/lppred.

ct11/lpor

ct/lpFEM

ct11/lpor

ct/lppred.

ct11/lpor

ct/lpFEM

ct11/lpor

ct/lppred.

ct11/lpor

ct/lpFEM

ct11/lpor

ct/lppred.

0.00 0.00 0.51 0.52 0.45 0.46 0.39 0.40 0.45 0.46 0.43 0.45

0.10 0.13 0.52 0.52 --- --- --- --- --- --- --- ---

0.20 0.13 0.53 0.52 --- --- --- --- --- --- --- ---

0.30 0.13 0.58 0.52 0.47 0.46 0.39 0.40 0.46 0.46 0.44 0.45

0.40 0.13 0.71 0.63 --- --- --- --- --- --- --- ---

0.10 0.25 0.53 0.52 0.46 0.46 0.39 0.40 0.45 0.46 0.44 0.45

0.20 0.25 0.55 0.52 0.47 0.46 0.39 0.40 0.46 0.46 0.44 0.45

0.30 0.25 0.63 0.56 0.49 0.46 0.40 0.40 0.47 0.46 0.44 0.45

0.40 0.25 0.77 0.67 0.64 0.50 0.42 0.40 0.50 0.50 0.46 0.49

0.10 0.38 0.53 0.52 --- --- --- --- --- --- --- ---

0.20 0.38 0.58 0.52 --- --- --- --- --- --- --- ---

0.30 0.38 0.68 0.60 0.58 0.46 0.41 0.40 0.49 0.46 0.46 0.45

0.40 0.38 0.77 0.72 --- --- --- --- --- --- --- ---


6.5 Walls With Openings

The contact length along the horizontal joints of a base panel with an opening is discussedbelow.

6.5.1 Contact length at the bottom of the base panel

The finite element analysis results show that, at the desired failure state, the presence of an openingdoes not have a significant effect on the contact length, cb along the base panel to foundation jointof a wall with openings (for the opening sizes considered by the research). Thus:

74

cb = cb1 (6.3)

where cb1 is estimated as described in Section 6.4.1.

As an example, Figure 6.7 compares the contact length (normalized with lp) at the base of WallWH1M (lp=20 ft, f=0.29) without and with openings (l=0.40, h=0.38). The contact length at thebase of a wall with lp=12 ft and f=0.18 without and with openings (l=0.30, h=0.38) is also shown.The normalized contact lengths at the base of the parametric walls at the desired failure state aregiven in Table 6.1. The results show that the openings do not have a significant effect on the contactlength along the base panel to foundation joint of the parametric walls for the opening sizes that werestudied.

0.5 1.0 1.5 2.0 2.50

0.2

0.4

0.6

0.8

1.0

roof-drift (%)

cont

act l

engt

h / l

p

0


l h

p f l = 20 ft, γ = 0.29


l h

p f l = 12 ft, γ = 0.18

Figure 6.7 Contact length at the base of walls without and with openings

6.5.2 “Large” opening versus “small” opening

For a wall with openings, the length of the contact region and the location of the critical section atthe top of the base panel depend on the size of the opening. For this purpose, an opening size factor,s is defined as follows:

75

(6.4)sc

xt

s

=′

≥ 1 .0

where ct1 is the contact length at the top of the panel with no opening as described in Section 6.4.2.If s>1.0, the opening is said to be a “large” opening. For s1.0, the opening is said to be a “small”opening, and s=1.0.

Figure 6.8 shows how the opening size factor is determined. The shaded panel regions in Figure 6.8represent the flow of the compression stresses from the top contact region to the bottom contactregion in a wall without openings. A large opening is considered to be an opening which disruptsthe flow of the compression stresses as shown in Figure 6.8(b).

The length xs in Eq. (6.4) is determined by drawing a straight line from the end of the contact region(i.e., gap tip) at the bottom of the panel to the upper right corner of the opening, and then extendingthe line until it intersects the top of the panel as shown by the dashed lines in Figure 6.8. Thus, xs canbe determined as:

(6.5)xl c

h hh cs

c b e

p op b e=

−+

+2

where cbe is the distance from the end of the panel to the end of the contact region at the base. Thedistance cbe is calculated as the sum of the contact length, cb and the length of the cover concrete atthe end of the wall.

(a) (b)

hp

l p

ho

l o

xs

l c

hc

ct′

hp

l p

ho

l o

xs

l c

hc

ct′

c = cbe be′ c = cbe be

′

Figure 6.8 Opening size factor: (a) small opening; (b) large opening

76

6.5.3 Contact length at the top of the base panel

As noted above, the contact length at the top of the base panel, ct depends on the size of the opening.Based on the finite element analysis results of the parametric walls, ct can be estimated as:

ct = ct1s (6.6)

For small openings (i.e., s=1.0), ct is equal to ct1, and thus the opening does not have an effect on thecontact length at the top of the panel. For large openings (i.e., s>1.0), ct is greater than ct1 anddepends on the size of the opening.

Figure 6.9 shows the contact length (normalized with lp) at the top of the base panel of Wall WH1M(lp=20 ft, f=0.29) without and with openings (l=0.40, h=0.38). As the size of the opening increases,the stiffness of the top chord above the opening decreases, and thus, the panel deforms more abovethe opening. This results in a larger contact length occurring along the horizontal joint as shown inFigure 6.9.

0 0.5 1 1.5 2 2.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

roof−drift (%)

l = 20 ft, γ = 0.29

without openings with openings (γ = 0.40, γ = 0.38)

p

cont

act l

engt

h / l

p

f

l h

Figure 6.9 Contact length at top of the base panel of a wall without and with openings

Table 6.2 compares the predicted (using Eq. (6.6)) contact lengths at the top of the base panels in theparametric walls at the desired failure state with the contact lengths determined using the finite

77

element analysis results. The shaded numbers correspond to cases for which the opening is classifiedas a “large” opening. The comparisons in Table 6.2 indicate that the predicted ct values arereasonably close to the finite element analysis results.

6.6 Critical Section at the Bottom of the Base Panel

The finite element analysis results indicate that the location of the critical section where themaximum tension force occurs at the bottom of the base panel, xcb can be estimated as:

(6.7)xc

cbt=′

2

where xcb is measured from the compression end of the panel (i.e., the right end for lateral loadsapplied from left to right) and ct1 is the contact length at the top of the panel with no opening asdescribed in Section 6.4.2. Eq. (6.7) indicates that, for the opening sizes studied by the research, xcb

does not depend on the presence of an opening in the panel. It is noted that, for the walls investigatedby the research, xcb>cb and thus, the critical section is outside the contact region at the base.

Table 6.3 compares the critical section location, xcb for the parametric walls as predicted using Eq.(6.7) with the location determined using the finite element analysis results. The xcb values arenormalized with respect to the panel length, lp. For each parametric wall, the location of the criticalsection from the finite element analysis results was determined by integrating the tensile stresses inthe tension zone at each vertical cross-section along the length of the panel at the bottom. Thelocation of the critical section was determined as the location along the length of the panel with themaximum resultant tension force. The comparisons in Table 6.3 indicate that the predicted xcb valuesare reasonably close to the finite element analysis results.

6.7 Critical Section at the Top of the Base Panel

The location of the critical section where the maximum tension force occurs at the top of the basepanel, xct can be estimated using the size factor, s as:

(6.8)xc

s

c

sctt t= =

′2

where xct is measured from the compression end of the panel (i.e., the right end for lateral loadsapplied from left to right). Eqs. (6.6) and (6.8) show that, for small openings, the critical section isat the end of the contact region (i.e., xct = ct=ct1). For large openings, the critical section is locatedinside the contact region (i.e., xct < ct).

78

TABLE 6.3 CRITICAL SECTION AT THE BOTTOM OF THE BASE PANEL

l h

lp = 20 ft lp= 15 ft lp = 12 ft

f = 0.29 f = 0.18 f = 0.11 f = 0.18 f = 0.18

xct/lpFEM

xct/lppred.

xct/lpFEM

xct/lppred.

xct/lpFEM

xct/lppred.

xct/lpFEM

xct/lppred.

xct/lpFEM

xct/lppred.

0.10 0.13 0.29 0.26 --- --- --- --- --- --- --- ---

0.20 0.13 0.28 0.26 --- --- --- --- --- --- --- ---

0.30 0.13 0.26 0.26 0.23 0.23 0.23 0.20 0.23 0.23 0.27 0.220.40 0.13 0.25 0.26 --- --- --- --- --- --- --- ---

0.10 0.25 0.28 0.26 0.25 0.23 0.25 0.20 0.25 0.23 0.27 0.22

0.20 0.25 0.27 0.26 0.23 0.23 0.23 0.20 0.25 0.23 0.27 0.22

0.30 0.25 0.27 0.26 0.22 0.23 0.22 0.20 0.23 0.23 0.27 0.22

0.40 0.25 0.23 0.26 0.21 0.23 0.21 0.20 0.23 0.23 0.24 0.22

0.10 0.38 0.27 0.26 --- --- --- --- --- --- --- ---

0.20 0.38 0.27 0.26 --- --- --- --- --- --- --- ---

0.30 0.38 0.25 0.26 0.20 0.23 0.20 0.20 0.23 0.23 0.24 0.22

0.40 0.38 0.23 0.26 --- --- --- --- --- --- --- ---


Table 6.4 compares the critical section location, xct for the parametric walls as predicted using Eq.(6.8) with the location determined using the finite element analysis results. The xct values arenormalized with respect to the panel length, lp. The shaded numbers correspond to cases for whichthe opening is classified as a “large” opening.

For each parametric wall, the location of the critical section from the finite element analysis resultswas determined by integrating the tensile stresses in the tension zone at each vertical cross-sectionalong the length of the panel at the top. The location of the critical section was determined as thelocation along the length of the panel with the maximum resultant tension force. The comparisonsin Table 6.4 indicate that the predicted xct values are reasonably close to the finite element analysisresults.

6.8 Overview of the Proposed Design Approach

This section provides an overview for the design of the required reinforcement in the critical regionsat the top, bottom, and sides of a base panel. A truss model approach similar to the approach usedin the design of the panels under vertical loads (Chapter 5) was found to be very sensitive to thegeometry of the truss. Thus, a different design approach based on two equilibrium free bodies of thepanel as shown in Figure 6.10 is proposed.

79

TABLE 6.4 CRITICAL SECTION AT THE TOP OF THE BASE PANEL

l h

lp = 20 ft lp= 15 ft lp = 12 ft

f = 0.29 f = 0.18 f = 0.11 f = 0.18 f = 0.18

xct/lpFEM

xct/lppred.

xct/lpFEM

xct/lppred.

xct/lpFEM

xct/lppred.

xct/lpFEM

xct/lppred.

xct/lpFEM

xct/lppred.

0.10 0.13 0.52 0.52 --- --- --- --- --- --- --- ---

0.20 0.13 0.53 0.52 --- --- --- --- --- --- --- ---

0.30 0.13 0.57 0.51 0.46 0.46 0.39 0.40 0.46 0.46 0.46 0.45

0.40 0.13 0.47 0.42 --- --- --- --- --- --- --- ---

0.10 0.25 0.53 0.52 0.45 0.46 0.39 0.40 0.45 0.46 0.44 0.45

0.20 0.25 0.55 0.52 0.46 0.46 0.39 0.40 0.45 0.46 0.44 0.45

0.30 0.25 0.47 0.48 0.47 0.46 0.39 0.40 0.46 0.46 0.46 0.45

0.40 0.25 0.42 0.40 0.43 0.42 0.40 0.40 0.48 0.41 0.46 0.41

0.10 0.38 0.53 0.52 --- --- --- --- --- --- --- ---

0.20 0.38 0.58 0.52 --- --- --- --- --- --- --- ---

0.30 0.38 0.42 0.45 0.43 0.46 0.39 0.40 0.47 0.45 0.46 0.44

0.40 0.38 0.33 0.37 --- --- --- --- --- --- --- ---


The finite element analysis results of the parametric walls show that the most critical state for thedesign of the reinforcement at the top, bottom, and sides of the panels is the desired failure state.Figure 6.10(a) shows the critical sections of a base panel under lateral loads acting from left to right.The locations of these critical sections are determined as described earlier.

Figure 6.10(b) shows the free body diagram used to determine the area of reinforcement required atthe top (Section 1) and side (Section 2) of the panel, and Figure 6.10(c) shows the free body diagramused to determine the area of reinforcement required at the bottom of the panel (Section 3). Section3 extends to the midheight of the panel at Section 4. It is noted that for the opening sizes that werestudied, the critical section at the bottom of the panel (i.e., Section 3) is located between the rightedge of the opening and the end of the panel (at a distance of xcb from the right end of the panel).

The area of the required reinforcement is determined by dividing the total tension force in the criticalregions (ignoring the tension strength of concrete) with the assumed yield strength of steel. In orderto estimate the design tension forces in the critical sections, the section and panel forces that areshown in Figures 6.10(b) and (c) need to be estimated as described below. The positive directionsassumed for the forces are shown in Figure 6.10. The axial forces in Sections 1, 2, 3, and 4 areassumed to act at the middle of each section.

80

2

3

4

1

1

2

1

2

3

4

critical section at panel top

critical section at panel side

critical section at panel bottom

midheight of panel

(a)

(b)

chph / 2

xct

xcb

N2

M 2

V2

N1

M 1

V1

Nptl

Gptl

Vptl

Vp

4

2

3(c)

N2

M 2

V2

N4

M 4

V4

V3 M 3

N3

Figure 6.10 Design forces: (a) critical sections; (b) free body diagram for Sections 1 and 2; (c) free body diagram for Sections 2 and 3

6.9 Design of Panel Top Reinforcement

The design of the reinforcement at the top of a base panel requires the estimation of the forcesapplied at the top of the panel and the forces that are carried by the tension (i.e., left for lateral loadsapplied from left to right) side chord based on the free body diagram in Figure 6.10(b).

81

6.9.1 Forces at the top of the panel

The free body diagram in Figure 6.10(b) shows three forces, Nptl, Vptl, and Gptl acting to the left ofthe critical section at the top of the panel. The forces Nptl and Vptl are the axial and shear forces thatare transferred to the base panel (from the upper story panel) at the left of the critical section. Theforce Gptl represents the resultant of the uniformly distributed gravity load applied at the first floorlevel to the left of the critical section. These forces are estimated as:

(6.9)

N ptl = −

= −

=

0 .2 11

sN

V 11

sV

Gl - x

lG

3 pt

p tl 2 p t

p tl

p c t

pp

The force Npt is the total axial force transferred to the base panel from the panel above asshown in Figure 6.6 and is equal to the sum of the gravity forces applied at the upper floor and rooflevels, Ga and the forces in the post-tensioning bars, Pf at the desired failure state. Similarly, the forceVpt is the total shear force transferred to the base panel from the panel above and is equal to the sumof the lateral forces applied at the upper floor and roof levels (i.e., second floor and up) when thedesired failure state is reached. Finally, Gp is the total gravity load applied at the first floor level.

As described before, the lateral load behavior of the walls is not significantly affected by theopenings as long as a sufficient amount of well-distributed bonded steel reinforcement is providedin the panels. Thus, Npt and Vpt can be estimated from a nonlinear push-over analysis of the wallignoring the presence of the openings.

The force Nptl is assumed to act at a distance of 2/3(ct - xct) to the left of the critical section (i.e.,Section 1). The gravity load is assumed to be uniformly distributed across the top of the panel,therefore the force Gptl is located at a distance of (lp-xct)/2 to the left of the critical section.

Eq. (6.9) shows that for small openings (i.e., s=1.0), Nptl=0 and Vptl=0. This is expected since forsmall openings, the critical section is at the end of the contact region. For large openings, s>1.0 andthus, Nptl and Vptl depend on the size of the opening. This is also expected since, for s>1.0, the criticalsection is located inside the contact region, and thus, some of the axial and shear stresses transferredto the base panel from the panel above are at the left of the critical section. The comparisons ofresults obtained using Eq. (6.9) with results obtained using the finite element analyses of theparametric walls are discussed below.

82

6.9.1.1 Axial force, Nptl

Figures 6.11-6.14 compare the values for Nptl predicted using Eq. (6.9) with values determined usingthe finite element analysis results of the parametric walls. The Nptl values are normalized with respectto Npt. The results indicate that the predicted forces follow the general trends obtained from the finiteelement analysis results.

Figures 6.11(a)-(c) show the effect of l on Nptl for varying values of f (with h=0.25, lp=20 ft), lp(with h=0.25, f=0.18), and h (with f=0.29, lp=20 ft), respectively. The results indicate that for largeopenings (i.e., s>1), Nptl/Npt increases as the opening length, lo increases.

0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

0.15

0.2

γl

Npt

l / N

pt

γh = 0.25, l

p = 20 ft

γf = 0.29

γf = 0.18

γf = 0.11

0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

0.15

0.2

γl

Npt

l / N

pt

γh = 0.25, γ

f = 0.18 ft

lp = 20 ft

lp = 15 ft

lp = 12 ft

0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

0.15

0.2

γl

Npt

l / N

pt

γf = 0.29, l

p = 20 ft

γh = 0.38

γh = 0.25

γh = 0.13

FEMpredicted

(a) (b)

(c)

FEMpredicted

FEMpredicted

Figure 6.11 Effect of l on Nptl: (a) h=0.25, lp=20 ft; (b) h=0.25, f=0.18; (c) f=0.29, lp=20 ft

83

Figures 6.12(a)-(c) show the effect of h on Nptl for varying values of f (with l=0.30, lp=20 ft), lp(with l=0.30, f=0.18), and l (with f=0.29, lp=20 ft), respectively. The results indicate that for largeopenings (i.e., s>1), Nptl/Npt increases as the opening height, ho increases.

0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

0.15

0.2

γh

Npt

l / N

pt

γl = 0.30, l

p = 20 ft

γf = 0.29

γf = 0.18

γf = 0.11

0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

0.15

0.2

γh

Npt

l / N

pt

γl = 0.30, γ

f = 0.18

lp = 20 ft

lp = 15 ft

lp = 12 ft

0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

0.15

0.2

γh

Npt

l / N

pt

γf = 0.29, l

p = 20 ft

γl = 0.40

γl = 0.30

γl = 0.20

γl = 0.10

FEMpredicted

(a) (b)

(c)

FEMpredicted

FEMpredicted

Figure 6.12 Effect of h on Nptl: (a) l=0.30, lp=20 ft: (b) l=0.30, f=0.18; (c) f=0.29, lp=20 ft

84

Figures 6.13(a) and (b) show the effect of lp on Nptl for varying values of h (with l=0.30, f=0.18)and l (with h=0.25, f=0.18), respectively. The results indicate that for large openings (i.e., s>1),Nptl/Npt decreases as the panel length, lp decreases.

0.5 0.75 1 1.25 1.50

0.05

0.1

0.15

0.2

hp / l

p

Npt

l / N

pt

γl= 0.30,γ

f= 0.18

γh=0.38

γh=0.25

γh=0.13

0.5 0.75 1 1.25 1.50

0.05

0.1

0.15

0.2

hp / lp

Npt

l / N

pt

γh= 0.25,γ

f= 0.18

γl=0.40

γl=0.30

γl=0.20

γl=0.10

FEMpredicted

(a) (b)

FEMpredicted

Figure 6.13 Effect of lp on Nptl: (a) l=0.30, f=0.18; (b) h=0.25, f=0.18

Finally, Figures 6.14(a) and (b) show the effect of f on Nptl for varying values of h (with l=0.30,lp=20 ft) and l (with h=0.25, lp=20 ft), respectively. The results indicate that for large openings (i.e.,s>1), Nptl/Npt increases as the initial concrete stress, fci increases.

0 0.1 0.2 0.3 0.40

0.05

0.1

0.15

0.2

γf

Npt

l / N

pt

γl

= 0.30, lp = 20 ft

γh=0.38

γh=0.25

γh=0.13

0 0.1 0.2 0.3 0.40

0.05

0.1

0.15

0.2

γf

Npt

l / N

pt

γh= 0.25, lp = 20 ft

γl=0.40

γl=0.30

γl=0.20

γl=0.10

FEMpredicted

(a) (b)

FEMpredicted

Figure 6.14 Effect of f on Nptl: (a) l=0.30, lp=20 ft; (b) h=0.25, lp=20 ft

85

6.9.1.2 Shear force, Vptl

Figures 6.15-6.18 compare the values for Vptl predicted using Eq. (6.9) with values determined usingthe finite element analysis results of the parametric walls. The Vptl values are normalized with respectto Vpt. The results indicate that the predicted forces usually follow the trends obtained from the finiteelement analysis results.

Figures 6.15(a)-(c) show the effect of l on Vptl for varying values of f (with h=0.25, lp=20 ft), lp(with h=0.25, f=0.18), and h (with f=0.29, lp=20 ft), respectively. The results indicate that for largeopenings (i.e., s>1), Vptl/Vpt usually increases as the opening length, lo increases.

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

γl

Vpt

l / V

pt

γh = 0.25, l

p = 20 ft

γf = 0.29

γf = 0.18

γf = 0.11

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

γl

Vpt

l / V

pt

γh = 0.25, γ

f = 0.18 ft

lp = 20 ft

lp = 15 ft

lp = 12 ft

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

γl

Vpt

l / V

pt

γf = 0.29, l

p = 20 ft

γh = 0.38

γh = 0.25

γh = 0.13

FEMpredicted

(a) (b)

(c)

FEMpredicted

FEMpredicted

Figure 6.15 Effect of l on Vptl: (a) h=0.25, lp=20 ft; (b) h=0.25, f=0.18; (c) f=0.29, lp=20 ft

86

Figures 6.16(a)-(c) show the effect of h on Vptl for varying values of f (with l=0.30, lp=20 ft), lp(with l=0.30, f=0.18), and l (with f=0.29, lp=20 ft), respectively. The results indicate that for largeopenings (i.e., s>1), Vptl/Vpt usually increases as the opening height, ho increases.

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

γh

Vpt

l / V

pt

γl = 0.30, l

p = 20 ft

γf = 0.29

γf = 0.18

γf = 0.11

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

γh

V ptl

/ V p

t

γl = 0.30, γ

f = 0.18

lp = 20 ft

lp = 15 ft

lp = 12 ft

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

γh

Vpt

l / V

pt

γf = 0.29, l

p = 20 ft

γl = 0.40

γl = 0.30

γl = 0.20

γl = 0.10

FEMpredicted

(a) (b)

(c)

FEMpredicted

FEMpredicted

Figure 6.16 Effect of h on Vptl: (a) l=0.30, lp=20 ft: (b) l=0.30, f=0.18; (c) f=0.29, lp=20 ft

87

Figures 6.17(a) and (b) show the effect of lp on Vptl for varying values of h (with l=0.30, f=0.18)and l (with h=0.25, f=0.18), respectively. The results indicate that for large openings (i.e., s>1),Vptl/Vpt decreases as the panel length, lp decreases.

0.5 0.75 1 1.25 1.50

0.2

0.4

0.6

0.8

1

hp / l

p

Vpt

l / V

pt

γl= 0.30,γf

= 0.18

γh=0.38

γh=0.25

γh=0.13

0.5 0.75 1 1.25 1.50

0.2

0.4

0.6

0.8

1

hp

/ lp

Vpt

l / V

pt

γh= 0.25,γf

= 0.18

γl=0.40

γl=0.30

γl=0.20

γl=0.10

FEMpredicted

(a) (b)

FEMpredicted

Figure 6.17 Effect of lp on Vptl: (a) l=0.30, f=0.18; (b) h=0.25, f=0.18

Finally, Figures 6.18(a) and (b) show the effect of f on Vptl for varying values of h (with l=0.30,lp=20 ft) and l (with h=0.25, lp=20 ft), respectively. The results indicate that for large openings (i.e.,s>1), Vptl/Vpt increases as the initial concrete stress, fci increases.

0 0.1 0.2 0.3 0.40

0.2

0.4

0.6

0.8

1

γf

Vpt

l / V

pt

γl =0.30, lp= 20 ft

γh=0.38

γh=0.25

γh=0.13

0 0.1 0.2 0.3 0.40

0.2

0.4

0.6

0.8

1

γf

Vpt

l / V

pt

γh=0.25, lp= 20 ft

γl=0.40

γl=0.30

γl=0.20

γl=0.10

FEMpredicted

(a) (b)

FEMpredicted

Figure 6.18 Effect of f on Vptl: (a) l=0.30, lp=20 ft; (b) h=0.25, lp=20 ft

88

6.9.2 Forces in the tension side chord

As shown in Figures 6.10(b) and (c), the design of the panel reinforcement requires the estimationof the forces that are carried by the tension (i.e., left for lateral loads applied from left to right) sidechord of the panel at Section 2. The finite element analysis results of the parametric walls indicatethat the distribution of the total story forces between the two side chords in the base panel issignificantly affected by the formation of gaps along the horizontal joints. The chord on thecompression side of the panel carries most of the story forces even if the opening is placed at thepanel centerline. Thus, the story forces cannot be distributed based on the relative stiffnesses of thechords.

The finite element analysis results were used to relate the forces in the tension side chord of a basepanel to the geometric properties of the panel as follows:

(6.10)

( )Nc

slN

V V

lt

ppb

a apb

2

3

2 l2 t

p

2 2p c

2

p p

2 2m 2

p c

0 .15 (1

4 .0c

sl1 .25

1 10 .75

G l

l h

M M Nl l

2

= −

=

+ −

+

= −−

ξ

γξ ξ

)

where

(6.11)

M M

l

l l

h

h h

h

l

m l h a p

lc

p ch

c

p ca

p

p

22

20 .15=

=−

=−

=

ξ ξ ξ

ξ ξ ξ, ,

In Eq. (6.10), N2 and V2 are the axial and shear forces carried by the tension chord, respectively, andNpb and Vpb are the total axial and shear forces in the base panel, respectively. It is noted that the signconvention for N2 is positive as shown in Figure 6.10(b) and Npb is the magnitude of the total axialcompression force in the wall.

The moment Mp2 is the total moment acting in the panel at the same height as Section 2 due to thelateral forces applied at the first floor level and above and the forces in the post-tensioning bars atthe desired failure state. The forces Npb, Vpb, and Mp2 can be estimated from a nonlinear push-overanalysis of the wall ignoring the presence of the openings. The force Gp is the total gravity loadacting at the first floor level.

The moment M2m represents the moment of the axial force in the tension chord, N2 about the panelcenterline and the moment M2 represents the moment of N2 about the chord centerline. Therelationship between M2 and M2m is given in Eq. (6.10).

89

The comparisons of results obtained using Eq. (6.10) with results obtained using the finite elementanalyses of the parametric walls are discussed below. It is noted that the expressions in Eq. (6.10)were verified only within the range of the parameters that were considered by the research, and thus,should be used with care.

6.9.2.1 Axial force, N2

Figures 6.19-6.22 compare the values for N2 predicted using Eq. (6.10) and (6.11) with valuesdetermined using the finite element analysis results. The N2 values are normalized with respect toNpb. The results indicate that Eq. (6.10) may need to be improved for the prediction of N2.

Figures 6.19(a)-(c) show the effect of l on N2 for varying values of f (with h=0.25, lp=20 ft), lp(with h=0.25, f=0.18), and h (with f=0.29, lp=20 ft), respectively. The results indicate that N2/Npb

decreases as the opening length, lo increases.

0 0.1 0.2 0.3 0.4 0.5−0.02

0

0.02

0.04

0.06

0.08

γl

N2 /

Npb

γh = 0.25, l

p = 20 ft

γf = 0.29

γf = 0.18

γf = 0.11

0 0.1 0.2 0.3 0.4 0.5−0.02

0

0.02

0.04

0.06

0.08

γl

N2 /

Npb

γh = 0.25, γ

f = 0.18

lp = 20 ft

lp = 15 ft

lp = 12 ft

0 0.1 0.2 0.3 0.4 0.5−0.02

0

0.02

0.04

0.06

0.08

γl

N2 /

Npb

γf = 0.29, l

p = 20 ft

γh = 0.38

γh = 0.25

γh = 0.13

FEMpredicted

FEMpredicted

FEMpredicted

(a) (b)

(c)Figure 6.19 Effect of l on N2: (a) h=0.25, lp=20 ft; (b) h=0.25, f=0.18; (c) f=0.29, lp=20 ft

90

Figures 6.20(a)-(c) show the effect of h on N2 for varying values of f (with l=0.30, lp=20 ft), lp(with l=0.30, f=0.18), and h (with f=0.29, lp=20 ft), respectively. The results indicate that thereis a small decrease in N2/Npb as the opening height, ho increases.

0 0.1 0.2 0.3 0.4 0.5−0.02

0

0.02

0.04

0.06

0.08

γh

N2 /

Npb

γl = 0.30, l

p = 20 ft

γf = 0.29

γf = 0.18

γf = 0.11

0 0.1 0.2 0.3 0.4 0.5−0.02

0

0.02

0.04

0.06

0.08

γh

N2 /

Npb

γl = 0.30, γ

f = 0.18

lp = 20 ft

lp = 15 ft

lp = 12 ft

0 0.1 0.2 0.3 0.4 0.5−0.02

0

0.02

0.04

0.06

0.08

γh

N2 /

Npb

γf = 0.29, l

p = 20 ft

γl = 0.40

γl = 0.30

γl = 0.20

γl = 0.10

FEMpredicted

FEMpredicted

(a) (b)

(c)

FEMpredicted

Figure 6.20 Effect of h on N2: (a) l=0.30, lp=20 ft; (b) l=0.30, f=0.18; (c) f=0.29, lp=20 ft

91

Figures 6.21(a) and (b) show the effect of lp on N2 for varying values of h (with l=0.30, f=0.18)and l (with h=0.25, f=0.18), respectively. The results indicate that the panel length, lp has littleeffect on N2/Npb.

0.5 0.75 1 1.25 1.50

0.02

0.04

0.06

0.08

hp / l

p

N2 /

Npb

γl= 0.30,γf = 0.18

γh=0.38

γh=0.25

γh=0.13

0.5 0.75 1 1.25 1.50

0.02

0.04

0.06

0.08

N2 /

Npb

γh= 0.25,γf

= 0.18

γl=0.40

γl=0.30

γl=0.20

γl=0.10

FEMpredicted

FEMpredicted

(a) (b)

hp / l

p

Figure 6.21 Effect of lp on N2: (a) l=0.30, f=0.18; (b) h=0.25, f=0.18

Finally, Figures 6.22(a) and (b) show the effect of f on N2 for varying values of h (with l=0.30,lp=20 ft) and l (with h=0.25, lp=20 ft), respectively. The results indicate that N2/Npb decreases as theinitial concrete stress, fci increases.

0 0.1 0.2 0.3 0.4−0.02

0

0.02

0.04

0.06

0.08

γf

N2 /

Npb

γl= 0.30, lp = 20 ft

γh=0.38

γh=0.25

γh=0.13

0 0.1 0.2 0.3 0.4−0.02

0

0.02

0.04

0.06

0.08

γf

N2 /

Npb

γh= 0.25, lp= 20 ft

γl=0.40

γl=0.30

γl=0.20

γl=0.10

FEMpredicted

(a) (b)

FEMpredicted

Figure 6.22 Effect of f on N2: (a) l=0.30, lp=20 ft; (b) h=0.25, lp=20 ft

Figures 6.19-6.22 show that N2 is a relatively small force as compared to Npb, however, the finiteelement analysis results indicate that N2 has a considerable effect on the design of the panelreinforcement and should not be ignored. This is because, the magnitude of N2 is not negligible andit has a large moment arm with respect to Sections 1 and 3. It is noted that N2 is usually a tensionforce, however it becomes a small compression force for a limited number of walls with large l andf values as shown in Figures 6.19(a), 6.20(c), and 6.22(b). For the walls where N2 is a tension force,the axial force in the compression (i.e., right) chord, Nr is greater than Npb as discussed later.

92

6.9.2.2 Shear force, V2

The first term (in brackets) in the expression for V2 in Eq. (6.10) represents the portion of the totalstory shear force, Vpb carried by the tension chord. As described earlier, gravity loads were assumedto be uniformly distributed at the floor and roof levels. The finite element analyses show that the firstfloor gravity loads applied directly above the tension chord result in the development of additionalshear forces in the chord. The second term in the expression for V2 represents this additional shearforce, where Gp is the total gravity load applied at the first floor level.

Figures 6.23-6.26 compare the values for V2 predicted using Eqs. (6.10) and (6.11) with valuesdetermined using the finite element analysis results. The V2 values are normalized with respect toVpb. The results indicate that the predicted forces are reasonably close to the forces from the finiteelement analyses. Figures 6.23(a)-(c) show the effect of l on V2 for varying values of f (withh=0.25, lp=20 ft), lp (with h=0.25, f=0.18), and h (with f=0.29, lp=20 ft), respectively. The resultsindicate thatV2/Vpb increases as the opening length, lo increases.

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

γl

V2 /

Vpb

γh = 0.25, l

p = 20 ft

γf = 0.29

γf = 0.18

γf = 0.11

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

γl

V2 /

Vpb

γh = 0.25, γ

f = 0.18

lp = 20 ft

lp = 15 ft

lp = 12 ft

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

γl

V2 /

Vpb

γf = 0.29, l

p = 20 ft

γh = 0.38

γh = 0.25

γh = 0.13

FEMpredicted

FEMpredicted

FEMpredicted

(a) (b)

(c)Figure 6.23 Effect of l on V2: (a) h=0.25, lp=20 ft; (b) h=0.25, f=0.18; (c) f=0.29, lp=20 ft

93

Figures 6.24(a)-(c) show the effect of h on V2 for varying values of f (with l=0.30, lp=20 ft), lp(with l=0.30, f=0.18), and h (with f=0.29, lp=20 ft), respectively. The results indicate that theopening height, ho has little effect on V2/Vpb.

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

γh

V2 /

Vpb

γl = 0.30, l

p = 20 ft

γf = 0.29

γf = 0.18

γf = 0.11

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

γh

V2 /

Vpb

γl = 0.30, γ

f = 0.18

lp = 20 ft

lp = 15 ft

lp = 12 ft

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

γh

V2 /

Vpb

γf = 0.29, l

p = 20 ft

γl = 0.40

γl = 0.30

γl = 0.20

γl = 0.10

FEMpredicted

FEMpredicted

(c)

(a) (b)

FEMpredicted

Figure 6.24 Effect of h on V2: (a) l=0.30, lp=20 ft; (b) l=0.30, f=0.18;(c) f=0.29, lp=20 ft

94

Figures 6.25(a) and (b) show the effect of lp on V2 for varying values of h (with l=0.30, f=0.18)and l (with h=0.25, f=0.18), respectively. The results indicate that V2/Vpb increases significantlyas the panel length, lp decreases.

0.5 0.75 1 1.25 1.50

0.1

0.2

0.3

0.4

hp / l

p

V2 /

Vpb

γl= 0.30,γ

f= 0.18

γh=0.38

γh=0.25

γh=0.13

0.5 0.75 1 1.25 1.50

0.1

0.2

0.3

0.4

V2 /

Vpb

γh= 0.25,γ

f= 0.18

γl=0.40

γl=0.30

γl=0.20

γl=0.10

(a) (b)

FEMpredicted

FEMpredicted

hp / l

p

Figure 6.25 Effect of lp on V2: (a) l=0.30, f=0.18; (b) h=0.25, f=0.18

Finally, Figures 6.26(a) and (b) show the effect of f on V2 for varying values of h (with l=0.30,lp=20 ft) and l (with h=0.25, lp=20 ft), respectively.

0 0.1 0.2 0.3 0.40

0.05

0.1

0.15

0.2

0.25

γf

V2 /

Vpb

γl =0.30, lp= 20 ft

γh=0.38

γh=0.25

γh=0.13

0 0.1 0.2 0.3 0.40

0.05

0.1

0.15

0.2

0.25

γf

V2 /

Vpb

γh= 0.25, lp= 20 ft

γl=0.40

γl=0.30

γl=0.20

γl=0.10

FEMpredicted

FEMpredicted

(a) (b)

Figure 6.26 Effect of f on V2: (a) l=0.30, lp=20 ft; (b) h=0.25, lp=20 ft

As shown above, the normalized shear force in the tension chord, V2/Vpb increases significantly asthe panel length, lp decreases. Figure 6.25 shows that for lp=12 ft, about 30% or more of the totalstory shear force, Vpb is carried by the tension chord. This increase in the chord shear force is dueto a reduction in the flexural stiffness of the side chords relative to the stiffness of the horizontalchords as follows.

95

As lp decreases, the length of the side chords, lc and thus the flexural stiffness of the side chordsdecreases while there is an increase in the flexural stiffness of the horizontal chord. This results inthe panel behaving more like a frame structure under lateral loads applied at the top, with the sidechords behaving like the columns in the frame as shown in Figure 6.27. The columns in the frameeach carry 50% of the total story shear force. Thus, the amount of shear force carried in each sidechord in a panel will approach 0.5Vpb as lp decreases. This trend can be seen in Figure 6.25.

column

V

lc

Figure 6.27 Frame structure

6.9.2.3 Moment, M2m

Figures 6.28-6.31 compare the values for M2m predicted using Eqs. (6.10) and (6.11) with valuesdetermined using the finite element analysis results. The M2m values are normalized with respect toMp2. The results indicate that the predicted moments are reasonably close to the moments from thefinite element analyses.

96

Figures 6.28(a)-(c) show the effect of l on M2m for varying values of f (with h=0.25, lp=20 ft), lp(with h=0.25, f=0.18), and h (with f=0.29, lp=20 ft), respectively. The results indicate that thereis a decrease in M2m/Mp2 as the opening length, lo increases.

0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

0.15

0.2

γl

M2m

/ M

p2

γh = 0.25, l

p = 20 ft

γf = 0.29

γf = 0.18

γf = 0.11

0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

0.15

0.2

γl

M2m

/ M

p2

γh = 0.25, γ

f = 0.18

lp = 20 ft

lp = 15 ft

lp = 12 ft

0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

0.15

0.2

γl

M2m

/ M

p2

γf = 0.29, l

p = 20 ft

γh = 0.38

γh = 0.25

γh = 0.13

FEMpredicted

FEMpredicted

FEMpredicted

(a) (b)

(c)Figure 6.28 Effect of l on M2m: (a) h=0.25, lp=20 ft; (b) h=0.25, f=0.18; (c) f=0.29, lp=20 ft

97

Figures 6.29(a)-(c) show the effect of h on M2m for varying values of f (with l=0.30, lp=20 ft), lp(with l=0.30, f=0.18), and h (with f=0.29, lp=20 ft), respectively. The results indicate that M2m/Mp2

decreases slightly as the opening height, ho increases.

0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

0.15

γh

M2m

/ M

p2

γl = 0.30, l

p = 20 ft

γf = 0.29

γf = 0.18

γf = 0.11

0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

0.15

γh

M2m

/ M

p2

γl = 0.30, γ

f = 0.18

lp = 20 ft

lp = 15 ft

lp = 12 ft

0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

0.15

γh

M2m

/ M

p2

γf = 0.29, l

p = 20 ft

γl = 0.40

γl = 0.30

γl = 0.20

γl = 0.10

FEMpredicted

(a) (b)

(c)

FEMpredicted

FEMpredicted

Figure 6.29 Effect of h on M2m: (a) l=0.30, lp=20 ft; (b) l=0.30, f=0.18; (c) f=0.29, lp=20 ft

98

Figures 6.30(a) and (b) show the effect of lp on M2m for varying values of h (with l=0.30, f=0.18)and l (with h=0.25, f=0.18), respectively. The results indicate that M2m/Mp2 increases as the panellength, lp decreases. This increase in M2m/Mp2 is similar to the increase in V2/Vpb as lp decreases asdescribed in the previous section.

0.5 0.75 1 1.25 1.50

0.05

0.1

0.15

hp / l

p

M2m

/ M

p2

γl=0.30,γ

f= 0.18

γh=0.38

γh=0.25

γh=0.13

0.5 0.75 1 1.25 1.50

0.05

0.1

0.15

M2m

/ M

p2

γh= 0.25,γ

f= 0.18

γl=0.40

γl=0.30

γl=0.20

γl=0.10

FEMpredicted

FEMpredicted

(a) (b)

hp / l

p

Figure 6.30 Effect of lp on M2m: (a) l=0.30, f=0.18; (b) h=0.25, f=0.18

Finally, Figures 6.31(a) and (b) show the effect of f on M2m for varying values of h (with l=0.30,lp=20 ft) and l (with h=0.25, lp=20 ft), respectively. The results indicate that the initial concretestress, fci has little effect on M2m/Mp2.

0 0.1 0.2 0.3 0.40

0.02

0.04

0.06

0.08

0.1

γf

M2m

/ M

p2

γl=0.30, lp= 20 ft

γh=0.38

γh=0.25

γh=0.13

0 0.1 0.2 0.3 0.40

0.02

0.04

0.06

0.08

0.1

γf

M2m

/ M

p2

γh= 0.25, lp=20 ft

γl=0.40

γl=0.30

γl=0.20

γl=0.10

FEMpredicted

(a) (b)

FEMpredicted

Figure 6.31 Effect of f on M2m: (a) l=0.30, lp=20 ft; (b) h=0.25, lp=20 ft

99

6.9.3 Design tension force and required reinforcement

Once the tension chord forces (N2, V2, and M2) and the forces at the top of the panel to the left ofSection 1 (Gptl, Nptl, and Vptl) are predicted as described above, the design forces for the top of thepanel (N1, V1, M1) can be determined from equilibrium of the free body diagram in Figure 6.10(b).The force Vp is the lateral force applied to the wall at the first floor level at the desired failure state.

The design tension force, T1 at the top of the panel is determined from the design axial force, N1 anddesign moment, M1 by assuming a linear distribution of axial stresses in Section 1 as shown inFigure 6.32. The estimated linear stress distribution is used to determine the height of the tensionzone, ht1 at the top of the panel. The design tension force, T1 is determined by integrating the stressesin the tension zone over the height ht1. The area of the required reinforcement in Section 1, Ac1b

(Figure 6.4) is equal to the design tension force divided by the assumed yield strength of steel, fy.Then, the reinforcement ratio for Section 1, !c1b is calculated by normalizing Ac1b with the cross-sectional area of the top chord, hctp as:

(6.12)

AT

f

A

h t

c1b1

y

c1bc1b

c p

=

=ρ

C1

T1

assumed linearstress distribution

1

xct

h t1

Figure 6.32 Estimation of T1

100

6.10 Design of Panel Side Reinforcement

The design tension force, T2 at the side of the panel is determined from the design axial force, N2 anddesign moment, M2 by assuming a linear distribution of axial stresses in Section 2 (similar to theestimation of T1 in Section 1) as shown in Figure 6.33. The estimated linear stress distribution isused to determine the length of the tension zone, lt2 at the side of the panel. The design tension force,T2 is determined by integrating the stresses in the tension zone over the length lt2. The requiredreinforcement area, Ac2b (Figure 6.4) and ratio, !c2b for Section 2 are calculated as:

(6.13)

AT

f

A

l t

c2b2

y

c2bc2b

c p

=

=ρ

C2

T2 assumed linearstress distribution

2

l t2

hc


6.11 Design of Panel Bottom Reinforcement

The design of the reinforcement at the bottom of a base panel uses the free body diagram in Figure6.10(c) and requires the estimation of the forces in Section 4.

6.11.1 Forces in the compression side chord

Once the forces in the tension chord of the panel are predicted as described above, the forces in thecompression side chord can be found using equilibrium as follows:

101

(6.14)

N N N

V V V

M M M Nl l

2V

h

2

r pb 2

r pb 2

r p2 2m r

p c

ro

= += −

= − −−

−

where Nr, Vr, and Mr are the axial force (applied at the chord centerline), shear force, and moment(about the chord centerline) carried by the compression chord at midheight of the panel, respectively.

6.11.2 Forces in Section 4

For the parametric walls that were studied, the maximum tension force at the bottom of the paneloccurs between the right edge of the opening and the right end of the panel (at a distance of xcb fromthe compression end of the panel). Thus, only a portion of the axial and shear stresses in thecompression chord between xcb and lc as shown by the shaded areas in Figures 6.34 and 6.35,respectively are used in the design of the panel reinforcement. The resultant of these stresses arereferred to as N4, V4, and M4 as shown in the free body diagram in Figure 6.10(c).

The forces N4, V4, and M4 are determined based on assumed distributions of the axial and shearstresses in the compression chord as follows. The finite element analyses indicate that thedistribution of the axial stresses in the compression chord at Section 4 is very close to a lineardistribution. As an example, Figure 6.34 shows a comparison between the estimated axial stressesusing a linear distribution (based on Nr and Mr) and the stresses determined using the finite elementanalysis results of Wall WH1M (f = 0.29, lp=20 ft) with l = 0.20 and h = 0.25. The force N4 isdetermined by integrating the estimated linear stress distribution between xcb and lc. The moment M4

is determined by taking the moment of the axial stresses acting in the chord region between xcb andlc about the middle of this chord region.

ph / 2

xcb

l c

finite element distribution

estimated distribution

Figure 6.34 Axial stress distribution in the compression chord

102

Similarly, the distribution of the shear stresses in the compression chord is very close to a parabolicdistribution with zero shear stresses adjacent to the opening and the panel edge. The finite elementanalysis results show that the maximum shear stress occurs at a distance of xcb from the right end ofthe panel. As an example, Figure 6.35 compares the estimated shear stresses with the stressesdetermined using the finite element analysis results of Wall WH1M (f = 0.29, lp=20 ft) with l = 0.20and h = 0.25.

ph / 2

xcb

l c

finite element distribution

estimated distribution

Figure 6.35 Shear stress distribution in the compression chord

Assuming that the shear stress distribution is parabolic on either side of xcb and is continuous at xcb,the shear force to the left of xcb can be found as:

(6.15)V (l x

l)V4

c cb

cr=

−

Eq. (6.15) shows that V4 is proportional to the distance between xcb and lc. The ratio of this distanceto the chord length, lc directly determines the chord shear force to the left of xcb.

6.11.3 Design tension force and required reinforcement

Once the tension chord forces (N2, V2, and M2) and the compression chord forces (N4, V4, and M4)are estimated, the design forces at the bottom of the panel (N3, V3, M3) can be determined fromequilibrium of the free body diagram in Figure 6.10(c). The axial force, N3 is assumed to be actingat midheight of Section 3 (i.e., at a height of hp/4).

Figure 6.36 shows the distribution of the axial stresses in Section 3 as determined from the finiteelement analysis of Wall WH1M (f = 0.29, lp=20 ft) with l = 0.20 and h = 0.25. The tensile stressesat the bottom of the panel are assumed to be distributed linearly over a height, ht3 of:

103

(6.16)h 0.10 l (1 0 .55 0 .55t3 p= − −γ γl h )

Thus, the location of the design tension force, T3 from the bottom of the panel is equal to yt3=ht3/3.

l c

ph / 2

xcb

yc3ht3

T3

C 3

yt3

3


Figure 6.36 shows that the distribution of the compression stresses in Section 3 is difficult toapproximate directly. However, the location, yc3 of the compression stress resultant, C3 from thebottom of the panel can be approximated as:

(6.17)y hca

t3 33 .751

= −

ξ

Then, the design tension force, T3 can be determined using moment equilibrium about thecompression stress resultant, C3 (at a height of yc3 above the base) as:

(6.18)TM N (y h / 4 )

y y3

3 3 c3 p

c3 t3

=− −

−

The required reinforcement area, Ac3b (Figure 6.4) and ratio, !c3b for Section 3 are calculated as:

(6.19)

AT

f

A

h t

c3b3

y

c3bc3b

c p

=

=ρ

104

6.12 Design of Reinforcement Around Openings

This section talks about the design of the reinforcement around the opening in a base panel. Thefinite element analysis results of the parametric walls show that the most critical state for the designof the reinforcement around the opening may not be the desired failure state. Earlier states whichcorrespond to smaller lateral displacements with larger contact lengths at the top and bottom of thepanel may result in larger tension stresses around the opening.

The finite element analysis results indicate that the amount of reinforcement needed at the top andbottom of the opening in a base panel under combined vertical and lateral loads is similar to thereinforcement required at the top of the panel. Thus, the reinforcement required at the top of thepanel, !c1b should also be used at the top and bottom of the opening as:

(6.20)A Ac4b c1b

c4 c1b

==ρ ρb

where Ac4b and !c4b are the reinforcement area and ratio, respectively required at the top and bottomof the opening (Figure 6.4). The reinforcement must not be less than the required reinforcement area,Av and ratio, !v under vertical loads only (due to post-tensioning and gravity loads). Thedetermination of Av and !v is described in Chapter 5.

The finite element analysis results indicate that the tensile stresses adjacent to the sides of an openingare not large, and thus, a nominal amount of minimum reinforcement, Amin can be used. It isrecommended that at least two No. 5 bars are placed along the sides of each opening and thus,Amin=0.61 in2. The resulting reinforcement ratio is !min,s. Thus:

(6.21)

A A 0 .61 inc5b m in2

5b

= =

= =ρ ρc sc p

A

l tm in ,m in

where Ac5b and !c5b are the reinforcement area and ratio, respectively required at the sides of theopening (Figure 6.4).

6.13 Placement of the Panel Reinforcement

The proposed design approach can be used to determine the area of reinforcement needed to controlcracking in the critical sections of the base panel as shown in Figure 6.4. The depth of the tensionzone at the critical sections can also be estimated. The required reinforcement must be placed withinthe tension zone for each critical section, considering lateral loads applied from left to right and fromright to left. It is recommended that the steel is placed in two layers, near each face of the panel, and

105

as close to the panel and opening edges as possible. It is also recommended that a minimumreinforcement of two No. 5 bars (with an area of Amin=0.61 in2) is used in each critical region.

The panel top and bottom reinforcement, Ac1b and Ac3b must be extended over the entire panel length,lp. This is because, even though the most critical state for the design of the reinforcement is thedesired failure state, the location of the critical sections change during the seismic response of thewall, depending on the contact length along the horizontal joints. The reinforcement must beextended a sufficient distance on both sides of the critical section at the desired failure state in orderto provide for full development of the yield strength of the steel. Since the critical section at thebottom is very close to the end of the panel, the bottom reinforcement may have to be bent verticallyinto the sides to ensure adequate development length.

The tension stresses at the sides of the panel are significantly smaller than those at the top andbottom of the panel. The required panel side reinforcement, Ac2b must be placed near the ends of thepanel on both sides. The reinforcement must be extended a sufficient distance on both sides of thecritical section (at the same height as the bottom of the opening) in order to provide for fulldevelopment of the yield strength of the steel. The side reinforcement may have to be benthorizontally at the bottom of the panel if the height of the horizontal chord is not sufficient for thedevelopment of the steel.

The reinforcement needed around the openings must be placed as close to the edges of the openingsas possible. The reinforcement must be extended a sufficient distance on both sides of the openingcorners in order to provide for full development of the yield strength of the steel.

6.14 Design of Shear Reinforcement

The free body diagrams in Figure 6.10 can be used to determine the design shear forces in the top,bottom, and side chords of a base panel. The design of the chord shear reinforcement can be doneusing these forces and is not addressed further in the report.

As described earlier, the largest shear forces are carried by the vertical chord on the compression sideof the panel. For the walls that were studied, only nominal shear reinforcement may be needed in thecompression side chord because of the presence of large compression forces. A nominal amount ofshear reinforcement may also be sufficient for the tension side chord, because of significantly smallerdesign shear forces.

In the horizontal chords of the panel above and below the opening, it may be necessary to design forshear if the opening reaches a length such that lo/2dc, where dc is the effective depth of the chordmeasured from the compression edge to the longitudinal reinforcement in tension. For the openingsand panel sizes studied in this research, this condition never occurred, and thus a nominal amountof shear reinforcement may be sufficient in the horizontal chords.

106

6.15 Roof-Drift at the Desired Failure State

As discussed in Chapters 2 and 3, the roof-drift used in the design of the panel reinforcementcorresponds to the desired failure state when the spiral confined concrete crushes. The design roof-drift of the parametric walls was assumed to be equal to 2.35%, which is the roof-drift at the desiredfailure state of Wall WH1M.

The design roof-drift at the desired failure state of a wall depends on a number of factors includingbuilding and wall characteristics, seismicity of the region, and soil conditions of the site. Thus, thedesign roof-drift of the parametric walls with different initial stress, f (representing walls designedfor regions with different seismicity) and length, lp can be different.

In order to investigate the effect of the design roof-drift on the panel reinforcement, lateral loadswere applied to a parametric wall with lp=20 ft, f=0.18, l=0.20, and h=0.25 until a roof-drift of1.70% was reached. This roof-drift value represents a 28% reduction from the design roof-drift of2.35%. The finite element analysis results showed that the tension forces at the critical panel sectionscorresponding to the roof-drift of 1.70% were similar to the tension forces corresponding to the roof-drift of 2.35%. The locations of the critical panel sections were also similar.

As discussed in Chapter 2, the roof-drift capacity of an unbonded post-tensioned precast walldepends on the amount of spiral reinforcement provided in the base panel. Walls designed for thesame base-shear capacity but different roof-drift levels experience similar forces and moments atfailure. Figure 2.3 shows that once the base-shear capacity of a wall is reached, the roof-driftincreases without a significant change in base-shear, and thus the forces and moments in the panelsremain almost the same. Similarly, Figures 4.12(b) and 6.7 show that the contact length between thewall and the foundation does not change significantly once the base-shear capacity is reached.

Based on the finite element analysis results, it is concluded that the proposed design approach forthe panel reinforcement can be used for different levels of design roof-drift as long as the base-shearcapacity of the wall is reached.

6.16 Design of the Upper Story Panels

The research described in this report focuses on the detailed design of the base (i.e., first story) panel.The design of the upper story panels is done using the design of the base panel as follows. Thereinforcement required in the upper story panels of the parametric walls was compared with thereinforcement required in the base panel. As an example, Figures 6.37(a) and (b) show the maximumprinciple stress contours over the height of Wall WH1M (f=0.29, lp=20 ft) without and withopenings (l=0.40 and h=0.38), respectively when the desired failure state is reached. The red shadedareas indicate the critical regions of the panels where the maximum principle stresses exceed

in tension, and thus, where cracking is expected to occur.7 .5 fc′

107

(a) (b)Figure 6.37 Principle stress contours in Wall WH1M (f=0.29, lp=20 ft):

(a) without openings; (b) with openings (l=0.40, h=0.38)

108

For the wall with no openings, Figure 6.37(a) shows that the critical regions are located at the topand bottom of the base panel only. It is noted that the top of the sixth story (i.e., roof) panel may alsobe critical due to the post-tensioning anchors, however, the design of these regions is not a focus ofthis research.

Figure 6.37(b) shows that the openings cause critical regions to form in the upper story panels. Thetop and bottom of each panel and the top and bottom of each opening experience significant tensionstresses. Large tension stresses can also form at the sides of the lower story panels.

Figures 6.38(a) and (b) show the horizontal stress contours corresponding to the principle stresscontours in Figures 6.37(a) and (b), respectively. Similar to the stresses in the base panel as describedin more detail earlier in the chapter, the principle tension stresses that occur at the top and bottomof the upper story panels and at the top and bottom of the openings are governed by horizontaltension stresses. These regions require horizontal reinforcement to control the cracks that may form.

Similarly, Figures 6.39(a) and (b) show the vertical stress contours corresponding to the principlestress contours in Figures 6.37(a) and (b), respectively. The results indicate that the principle tensionstresses that occur at the sides of the panels are governed by vertical tension stresses. These regionsrequire vertical reinforcement to control the cracks that may form.

6.16.1 Required reinforcement in the upper story panels

The finite element analysis results show that the reinforcement required at the top, bottom, and sidesof the panels is maximum in the base panel. The required reinforcement decreases significantly forthe upper story panels.

The reinforcement required at the top and bottom of the openings decreases gradually for the upperstory panels. It is noted that in Figures 6.37(b) and 6.38(b), the opening in the second story panelresults in larger tension stresses than the opening in the base panel. However, as described in Section6.12, larger tension stresses develop at the top and bottom of the opening in the base panel at earlierstates corresponding to smaller lateral displacements with larger contact lengths along the horizontaljoints.

It is recommended that the reinforcement at the top, bottom, and sides of the wall panels and aroundthe openings be determined by:

109

(a) (b)

Figure 6.38 Horizontal stress contours in Wall WH1M (f=0.29, lp=20 ft): (a) without openings; (b) with openings (l=0.40, h=0.38)

110

(a) (b)Figure 6.39 Vertical stress contours in Wall WH1M (f=0.29, lp=20 ft):

(a) without openings; (b) with openings (l=0.40, h=0.38)

111

( )

ρ ρ ρ

ρ ρ ρ

ρ ρ ρ

ρ ρ ρ ρ ρ

ρ

c1n c1bm ax

m ax

2

c2n c2bm ax

m ax

2

c3n c3bm ax

m ax

2

c4n 4m ax

2

5n

n n

n 1pane l top re in fo rcem ent

n n

n 1pane l s ide re in fo rcem en t

n n

n 1pane l bo ttom re in fo rcem ent

1n 1

n 1open ing top and bo ttom re in fo rcem ent

=−−

≥

=−−

≥

=−−

≥

= + − −−

−

≥

m in ,

m in ,

m in ,

m in ,

h

s

h

v c b v h

c

,

,

, (6 .22 )

,

= =ρ ρc5b open ing side re in fo rcem entm in ,s ,

where n is the story number of the panel being designed (e.g., n=1 for the base panel) and nmax is thetotal number of stories in the wall. It is noted that an in-depth analysis of the upper story panels wasnot performed, and thus the reinforcement ratios given by Eq. (6.22) are recommendations only andshould be used with caution.

As an example, Figure 6.40 shows the variation in the required reinforcement in the panels of a sixstory (i.e., nmax=6) wall based on Eq. (6.22). The reinforcement ratios !c1b, !c2b, and !c3b correspondto the reinforcement ratios at the top, sides, and bottom of the base panel, respectively. Thereinforcement ratios !c4b and !c5b correspond to the reinforcement ratios at the top and bottom of theopening and the sides of the opening in the base panel, respectively.

For the top, bottom, and sides of the panels, the largest decrease in reinforcement occurs betweenthe base panel and the second story panel. The decrease in reinforcement becomes smaller for theother panels (i.e., second and third stories, third and fourth stories, etc.). For the top and bottom ofthe openings, the decrease in the required reinforcement is smallest between the base panel and thesecond story panel. In the upper story panels, the top and bottom of the openings require the largestamount of reinforcement as compared with the reinforcement required in the other critical regions.

The reinforcement ratio !v corresponds to the reinforcement required at the top and bottom of theopenings under vertical loads only (due to gravity and post-tensioning) as described in Chapter 5.The reinforcement required at the top and bottom of the opening in the top story (i.e., roof) can beestimated as the reinforcement required under gravity and post-tensioning loads only.

It is recommended that a minimum reinforcement of two No. 5 bars is placed in the critical regionsof each panel. The minimum reinforcement ratios, !min,s and !min,h are determined by dividing the areaof the recommended two No. 5 bars with the cross-sectional area of the vertical and horizontalchords, respectively. The reinforcement ratios determined using Eq. (6.22) should be limited by theminimum reinforcement ratios as shown in Figure 6.40.

112

0

1

2

3

4

5

6st

ory

ρ (%)

ρc2b

ρc3b

min,sρ = ρ

c5b c4bρ = ρ

c1b

ρc1n (panel top reinforcement)

ρc3n (panel bottom reinforcement)

ρc2n (panel side reinforcement)

ρc4n (opening top and bottom reinforcement)

ρv

ρc5n (opening side reinforcement)

ρmin,h

Figure 6.40 Required reinforcement in the upper story panels

6.17 Base Panel Reinforcement Results

This section presents the results from the design of the base panel in the parametric walls. For eachwall, the required reinforcement areas at the top, bottom, and sides of the base panel were predictedusing the proposed design approach. The predicted steel areas were then compared with the steelareas determined using the finite element analysis results. The steel areas from the finite elementanalyses were determined by integrating the tension stresses with the largest resultant force in eachcritical region. The yield strength of the steel, fy was assumed to be 60 ksi.

The comparison between the predicted and finite element analysis results is presented below basedon the required reinforcement ratio in each critical section (i.e., !c1b, !c2b, and !c3b). As describedearlier, the design of the reinforcement at the top and bottom of the openings and in the upper storypanels depends on !c1b, !c2b, and !c3b and is not discussed further in this section.

113

6.17.1 Panel top reinforcement, !c1b

Table 6.5 compares the predicted and FEM (i.e., determined using the finite element model)reinforcement ratios for Section 1, !c1b at the top of the base panels of the parametric walls. Theshaded numbers correspond to cases in which the predicted reinforcement ratio is less than the FEMratio.

TABLE 6.5 REQUIRED REINFORCEMENT AT THE TOP OF THE BASE PANEL

l h

lp = 20 ft lp= 15 ft lp = 12 ft

f = 0.29 f = 0.18 f = 0.11 f = 0.18 f = 0.18

!c1b (%)FEM

!c1b (%)pred.

!c1b (%)FEM

!c1b (%)pred.

!cb (%)FEM

!c1b (%)pred.

!c1b (%)FEM

!c1b (%)pred.

!c1b (%)FEM

!c1b (%)pred.

0.10 0.13 0.61 0.86 --- --- --- --- --- --- --- ---

0.20 0.13 0.54 0.69 --- --- --- --- --- --- --- ---

0.30 0.13 0.38 0.50 0.45 0.47 0.44 0.43 0.37 0.40 0.29 0.38

0.40 0.13 0.10 0.31 --- --- --- --- --- --- --- ---

0.10 0.25 0.67 0.91 0.61 0.81 0.52 0.73 0.54 0.64 0.44 0.58

0.20 0.25 0.48 0.70 0.57 0.64 0.52 0.59 0.50 0.53 0.40 0.49

0.30 0.25 0.22 0.34 0.46 0.48 0.50 0.48 0.40 0.40 0.32 0.37

0.40 0.25 0.14 0.54 0.26 0.22 0.42 0.38 0.27 0.26 0.24 0.27

0.10 0.38 0.69 0.97 --- --- --- --- --- --- --- ---

0.20 0.38 0.42 0.70 --- --- --- --- --- --- --- ---

0.30 0.38 0.27 0.43 0.42 0.49 0.57 0.56 0.43 0.36 0.36 0.34

0.40 0.38 0.96 1.11 --- --- --- --- --- --- --- ---


114

Figures 6.41(a)-(c) show the effect of l on !c1b for varying values of f (with h=0.25, lp=20 ft), lp(with h=0.25, f=0.18), and h (with f=0.29, lp=20 ft), respectively. The results indicate that, formost cases, !c1b decreases as the opening length, lo increases. Figure 6.41(c) shows that for large f

and h values, !c1b can increase if lo gets very long. The proposed design approach seems to be ableto capture this trend.

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

γl

ρ c1b (

%)

γh=0.25, lp=20 ft

γf=0.29

γf=0.18

γf=0.11

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

γl

γh=0.25, γ

f=0.18


0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

γl

γf=0.29, lp=20 ft

γh=0.38

γh=0.25

γh=0.13

(a) (b)

(c)

FEMpredicted

FEMpredicted

FEMpredicted

ρ c1b (

%)

ρ c1b (

%)

Figure 6.41 Effect of l on !c1b: (a) h=0.25, lp=20 ft; (b) h=0.25,f=0.18; (c) f=0.29, lp=20 ft

115

Figures 6.42(a)-(c) show the effect of h on !c1b for varying values of f (with l=0.30, lp=20 ft), lp(with l=0.30, f=0.18), and l (with f=0.29, lp=20 ft), respectively. The results indicate that, for mostcases, the opening height, ho has little effect on !c1b. Figure 6.42(c) shows that for large f and l

values, !c1b can increase significantly if ho gets very tall. The proposed design approach seems to beable to capture this trend.

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

γh

γl=0.30, lp=20 ft

γf=0.29

γf=0.18

γf=0.11

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

γh

γl=0.30, γ

f=0.18

lp =20 ftlp =15 ftlp =12 ft

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

γh

γf=0.29, lp=20 ft

γl=0.40

γl=0.30

γl=0.20

γl=0.10

(a) (b)

(c)

FEMpredicted

FEMpredicted

FEMpredicted

ρ c1b (

%)

ρ c1b (

%)

ρ c1b (

%)

Figure 6.42 Effect of h on !c1b: (a) l=0.30, lp=20 ft; (b) l=0.30,f=0.18; (c) f=0.29, lp=20 ft

116

Figures 6.43(a) and (b) show the effect of lp on !c1b for varying values of h (with l=0.30, f=0.18)and l (with h=0.25, f=0.18), respectively. The results indicate that, for most cases, !c1b decreasesslightly as the panel length, lp decreases.

0.5 0.75 1 1.25 1.50

0.2

0.4

0.6

0.8

1

γl= 0.30,γ

f= 0.18

γh=0.38

γh=0.25

γh=0.13

0.5 0.75 1 1.25 1.50

0.2

0.4

0.6

0.8

1

hp / lp

γh= 0.25,γ

f= 0.18

γl=0.40

γl=0.30

γl=0.20

γl=0.10

hp / lp

(a) (b)

FEMpredicted

FEMpredicted

ρ c1b (

%)

ρ c1b (

%)

Figure 6.43 Effect of lp on !c1b: (a) l=0.30, f=0.18; (b) h=0.25, f=0.18

Finally, Figures 6.44(a) and (b) show the effect of f on !c1b for varying values of h (with l=0.30,lp=20 ft) and l (with h=0.25, lp=20 ft), respectively. The results indicate that, for most cases, !c1b

decreases as the initial concrete stress, fci increases. Figure 6.44(b) shows that !c1b increases as fci

increases if lo get very short.

0 0.1 0.2 0.3 0.40

0.2

0.4

0.6

0.8

1

1.2

γf

γl= 0.30, lp= 20 ft

γh=0.38

γh=0.25

γh=0.13

0 0.1 0.2 0.3 0.40

0.2

0.4

0.6

0.8

1

1.2

γf

γh= 0.25, lp= 20 ft

γl=0.40

γl=0.30

γl=0.20

γl=0.10

(a) (b)

FEMpredicted

FEMpredicted

ρ c1b (

%)

ρ c1b (

%)

Figure 6.44 Effect of f on !c1b: (a) l=0.30, lp=20 ft; (b) h=0.25, lp=20 ft

117

6.17.2 Panel side reinforcement, !c2b

Table 6.6 compares the predicted and FEM (i.e., determined using the finite element model)reinforcing ratios for Section 2, !c2b at the side of the base panels of the parametric walls. The shadednumbers correspond to cases in which the predicted reinforcement ratio is less than the FEM ratio.

TABLE 6.6 REQUIRED REINFORCEMENT AT THE SIDE OF THE BASE PANEL

l h

lp = 20 ft lp= 15 ft lp = 12 ft

f = 0.29 f = 0.18 f = 0.11 f = 0.18 f = 0.18

!c2b (%)FEM

!c2b (%)pred.

!c2b (%)FEM

!c2b (%)pred.

!c2 (%)FEM

!c2b (%)pred.

!c2b (%)FEM

!c2b (%)pred.

!c2b (%)FEM

!c2b (%)pred.

0.10 0.13 0.33 0.43 --- --- --- --- --- --- --- ---0.20 0.13 0.32 0.45 --- --- --- --- --- --- --- ---

0.30 0.13 0.33 0.49 0.22 0.33 0.16 0.22 0.36 0.55 0.51 0.91

0.40 0.13 0.40 0.55 --- --- --- --- --- --- --- ---

0.10 0.25 0.30 0.33 0.22 0.23 0.16 0.15 0.32 0.37 0.46 0.63

0.20 0.25 0.28 0.33 0.21 0.23 0.16 0.15 0.32 0.39 0.47 0.65

0.30 0.25 0.29 0.37 0.19 0.25 0.15 0.16 0.33 0.42 0.49 0.71

0.40 0.25 0.39 0.42 0.21 0.29 0.14 0.19 0.37 0.47 0.51 0.77

0.10 0.38 0.26 0.27 --- --- --- --- --- --- --- ---

0.20 0.38 0.24 0.24 --- --- --- --- --- --- --- ---

0.30 0.38 0.25 0.26 0.16 0.18 0.12 0.12 0.29 0.31 0.46 0.53

0.40 0.38 0.37 0.31 --- --- --- --- --- --- --- ---


118

Figures 6.45(a)-(c) show the effect of l on !c2b for varying values of f (with h=0.25, lp=20 ft), lp(with h=0.25, f=0.18), and h (with f=0.29, lp=20 ft), respectively. The results indicate that, formost cases, !c2b increases slightly as the opening length, lo increases.

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

γl

γh=0.25, lp=20 ft

γf=0.29

γf=0.18

γf=0.11

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

γl

γh=0.25, γ

f=0.18


0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

γl

γf=0.29, lp=20 ft

γh=0.38

γh=0.25

γh=0.13

(a) (b)

(c)

FEMpredicted

FEMpredicted

FEMpredicted

ρ c2b (

%)

ρ c2b (

%)

ρ c2b (

%)


119

Figures 6.46(a)-(c) show the effect of h on !c2b for varying values of f (with l=0.30, lp=20 ft), lp(with l=0.30, f=0.18), and l (with f=0.29, lp=20 ft), respectively. The results indicate that !c2b

decreases slightly as the opening height, ho increases.

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

γh

γl=0.30, lp=20 ft

γf=0.29

γf=0.18

γf=0.11

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

γh

γl=0.30, γ

f=0.18


0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

γh

γf=0.29, lp=20 ft

γl=0.40

γl=0.30

γl=0.20

γl=0.10

(a) (b)

(c)

FEMpredicted

FEMpredicted

FEMpredicted

ρ c2b (

%)

ρ c2b (

%)

ρ c2b (

%)


120

Figures 6.47(a) and (b) show the effect of lp on !c2b for varying values of h (with l=0.30, f=0.18)and l (with h=0.25, f=0.18), respectively. The results indicate that !c2b increases significantly asthe panel length, lp decreases.

0.5 0.75 1 1.25 1.50

0.2

0.4

0.6

0.8

1

hp / l p

γl= 0.30,γ

f= 0.18

γh=0.38

γh=0.25

γh=0.13

0.5 0.75 1 1.25 1.50

0.2

0.4

0.6

0.8

1

hp / lp

γh= 0.25,γ

f= 0.18

γl=0.40

γl=0.30

γl=0.20

γl=0.10

(a) (b)

FEMpredicted

FEMpredicted

ρ c2b (

%)

ρ c2b (

%)


Finally, Figures 6.48(a) and (b) show the effect of f on !c2b for varying values of h (with l=0.30,lp=20 ft) and l (with h=0.25, lp=20 ft), respectively. The results indicate that !c2b increasessignificantly as the initial concrete stress, fci increases.

(a) (b)

FEMpredicted

FEMpredicted

0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

0.5

0.6

γf

γl= 0.30 , lp= 20 ft

γh=0.38

γh=0.25

γh=0.13

0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

0.5

0.6

γf

γh= 0.25, l p= 20 ft

γl=0.40

γl=0.30

γl=0.20

γl=0.10

ρ c2b (

%)

ρ c2b (

%)


121

6.17.3 Panel bottom reinforcement, !c3b

Table 6.7 compares the predicted and FEM (i.e., determined using the finite element model)reinforcing ratios for Section 3, !c3b at the bottom of the base panels of the parametric walls. Theshaded numbers correspond to cases in which the predicted reinforcement ratio is less than the FEMratio.

TABLE 6.7 REQUIRED REINFORCEMENT AT THE BOTTOM OF THE BASE PANEL

l h

lp = 20 ft lp= 15 ft lp = 12 ft

f = 0.29 f = 0.18 f = 0.11 f = 0.18 f = 0.18

!c3b (%)FEM

!c3b (%)pred.

!c3b (%)FEM

!c3b (%)pred.

!c3b (%)FEM

!c3b (%)pred.

!c3b (%)FEM

!c3b (%)pred.

!c3b (%)FEM

!c3b (%)pred.

0.10 0.13 0.96 1.22 --- --- --- --- --- --- --- ---

0.20 0.13 0.94 1.13 --- --- --- --- --- --- --- ---

0.30 0.13 0.89 1.11 0.67 0.79 0.54 0.53 0.64 0.68 0.56 0.69

0.40 0.13 0.81 1.16 --- --- --- --- --- --- --- ---

0.10 0.25 1.09 1.43 0.79 1.12 0.63 0.77 0.77 0.82 0.67 0.75

0.20 0.25 1.05 1.31 0.78 1.01 0.63 0.71 0.76 0.78 0.66 0.73

0.30 0.25 0.99 1.26 0.76 0.93 0.62 0.64 0.73 0.76 0.64 0.74

0.40 0.25 0.88 1.31 0.71 0.89 0.60 0.60 0.71 0.77 0.62 0.77

0.10 0.38 1.25 1.76 --- --- --- --- --- --- --- ---

0.20 0.38 1.20 1.59 --- --- --- --- --- --- --- ---

0.30 0.38 1.11 1.49 0.87 1.15 0.72 0.82 0.86 0.88 0.75 0.82

0.40 0.38 0.95 1.45 --- --- --- --- --- --- --- ---


122

Figures 6.49(a)-(c) show the effect of l on !c3b for varying values of f (with h=0.25, lp=20 ft), lp(with h=0.25, f=0.18), and h (with f=0.29, lp=20 ft), respectively. The results indicate that !c3b

decreases slightly as the opening length, lo increases.

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

γl

γh=0.25, lp=20 ft

γf=0.29

γf=0.18

γf=0.11

γl

γh=0.25, γ

f=0.18


(a) (b)

FEMpredicted

FEMpredicted

ρ c3b (

%)

ρ c3b (

%)

γf=0.29, lp=20 ft

γh=0.38

γh=0.25

γh=0.13

γl

(c)

FEMpredicted

ρ c3b (

%)


123

Figures 6.50(a)-(c) show the effect of h on !c3b for varying values of f (with l=0.30, lp=20 ft), lp(with l=0.30, f=0.18), and l (with f=0.29, lp=20 ft), respectively. The results indicate that !c3b

increases as the opening height, ho increases.

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

γl=0.30, l

p=20 ft

γf=0.29

γf=0.18

γf=0.11

γl=0.30, γ

f=0.18


γf=0.29, lp=20 ft

γl=0.40

γl=0.30

γl=0.20

γl=0.10

γh

γh

(a) (b)

γh

(c)

FEMpredicted FEM

predicted

FEMpredicted

ρ c3b (

%)

ρ c3b (

%)

ρ c3b (

%)


124

Figures 6.51(a) and (b) show the effect of lp on !c3b for varying values of h (with l=0.30, f=0.18)and l (with h=0.25, f=0.18), respectively. The results indicate that !c3b decreases slightly as thepanel length, lp decreases.

hp / l p hp / lp

(a) (b)

FEMpredicted

FEMpredicted

0.5 0.75 1 1.25 1.50

0.2

0.4

0.6

0.8

1

1.2

γl= 0.30,γ

f= 0.18

γh=0.38

γh=0.25

γh=0.13

0.5 0.75 1 1.25 1.50

0.2

0.4

0.6

0.8

1

1.2

γh= 0.25,γ

f= 0.18

γl=0.40

γl=0.30

γl=0.20

γl=0.10

ρ c3b (

%)

ρ c3b (

%)


Finally, Figures 6.52(a) and (b) show the effect of f on !c1b for varying values of h (with l=0.30,lp=20 ft) and l (with h=0.25, lp=20 ft), respectively. The results indicate that !c3b increasessignificantly as the initial concrete stress, fci increases.

0 0.1 0.2 0.3 0.40

0.5

1

1.5

γf

γl= 0.30, lp= 20 ft

γh=0.38

γh=0.25

γh=0.13

0 0.1 0.2 0.3 0.40

0.5

1

1.5

γf

γh= 0.25, lp= 20 ft

γl=0.40

γl=0.30

γl=0.20

γl=0.10

(a) (b)

FEMpredicted

FEMpredicted

ρ c3b (

%)

ρ c3b (

%)


125

6.17.4 Result averages

Table 6.8 shows the averages of the predicted reinforcement ratio divided by the FEM reinforcementratio at the top, sides, and bottom of the base panels of the parametric walls for each combinationof lp and f. On average, the proposed design approach is conservative. For the largest initial stresscondition with f=0.29, the design approach is over conservative for the reinforcement required atthe top of the panels. This is due mostly to the two cases shown in Figure 6.41(c) with h=0.13 andh=0.25 for lp=20 ft, f=0.29, and l=0.40, where the predicted reinforcement ratio is significantlylarger than the FEM ratio as shown in Table 6.5. However, in both of these cases, the FEMreinforcement ratio is small, and thus the predicted steel area is not overly large. Table 6.8 shows thatthe proposed design approach works reasonably well for the lower initial stress conditions.

TABLE 6.8 AVERAGE OF PREDICTED / FEM REINFORCEMENT RATIOSlp = 20 ft lp= 15 ft lp = 12 ft

f = 0.29 f = 0.18 f = 0.11 f = 0.18 f = 0.18

panel top, !c1b 1.78 1.10 1.06 1.03 1.19

panel side, !c2b 1.18 1.24 1.11 1.26 1.44

panel bottom, !c3b 1.34 1.28 1.08 1.05 1.16

126

CHAPTER 7

SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS

The research described in this report investigates the behavior and design of unbonded post-tensionedprecast walls with rectangular panel openings. The research is divided into two main parts: (1) wallsunder vertical loads only, due to gravity and post-tensioning; and (2) walls under combined verticaland lateral loads. A summary and main conclusions from the research are presented below.

1. Openings in unbonded post-tensioned precast walls may be needed to accommodate architectural,mechanical, and/or safety requirements. Previous research on these walls is limited to walls withoutopenings. Thus, it is necessary to study the effect of the openings on the behavior and design of thewalls.

2. A finite element model was developed using the ABAQUS Program to investigate the behavior anddesign of walls with openings. The finite element model was verified by comparing the results witha second analytical model which uses fiber elements for the panels. In addition, the stresses aroundthe openings were compared with available closed-form analytical solutions.

3. The finite element model was used to conduct a parametric investigation on the behavior anddesign of a series of walls with openings. The parameters studied were: (1) opening length; (2)opening height; (3) panel length; and (4) initial stress in the concrete due to post-tensioning andgravity loads.

4. The parametric investigation was used to identify the critical regions in the panels where largetensile stresses, and thus, cracking can occur under post-tensioning, gravity, and lateral loads.

5. Cracking in the panels can cause premature failure of the walls under post-tensioning, gravity, andlateral loads. Bonded mild steel reinforcement is used to limit the cracking in the critical regions ofthe panels.

127

Walls Under Vertical Loads Only

1. The presence of the openings in the wall panels can cause cracking at the top and bottom of theopenings under vertical loads only, due to post-tensioning and gravity loads. This is the loading stagethat a typical wall would be subjected to during most of its service life.

2. A design approach is proposed to determine the required reinforcement at the top and bottom ofthe openings. The design approach can be used for walls with and without post-tensioning.

3. The required panel reinforcement predicted by the proposed design approach compares well withthe reinforcement determined using the finite element analysis results. On average, the proposeddesign approach is conservative.

Walls Under Combined Vertical and Lateral Loads

1. Unbonded post-tensioned precast walls are used as primary lateral load resisting systems in seismicregions. The lateral load behavior of the walls is governed by the formation of gaps along thehorizontal joints between the wall panels and between the base panel and the foundation.

2. The post-tensioning bars are not bonded to the concrete and the bonded panel reinforcement is notcontinuous across the horizontal joints. Thus, the post-tensioning bars and the bonded panelreinforcement do not restrict the formation of the gaps along the joints. This results in significantlyless cracking in the walls as compared with monolithic cast-in-place reinforced concrete walls.

3. During a seismic event, cracks can form at the top and bottom of the panels due to the formationof gaps along the horizontal joints. Cracking due to the gaps can occur in panels without openingsas well as with openings and is most significant in the base panel where the largest gap forms betweenthe wall and the foundation.

4. Under combined vertical and lateral loads, the presence of openings in the wall panels can affectthe location and magnitude of the maximum tensile forces at the top and bottom of the panels and cancause additional cracks to form at the sides of the panels and at the top and bottom of the openings.

5. The finite element analysis results indicate that the most critical panel is the base panel. In eachpanel, bonded mild steel reinforcement is required at the top, bottom and sides of the panel, andaround the opening.

6. The effect of the openings is largest in the upper story panels where the top and bottom of theopenings require the largest amount of reinforcement as compared with the reinforcement requiredin the other critical regions.

128

7. The size of the openings has an important effect on the behavior and design of the walls. Anopening size factor is proposed to differentiate between “small” and “large” openings.

8. Under combined vertical and lateral loads, the panels begin to behave more like frame structuresas the panel length decreases. This occurs as a result of a decrease in the flexural stiffness of the sidechords with respect to the stiffness of the horizontal chords.

9. The overall behavior of the walls under combined vertical and lateral loads is not significantlyaffected by the presence of the openings as long as a sufficient amount of well-distributed bondedmild steel reinforcement is used in the panels to control the cracks. This is because, the lateral loadbehavior of the walls is governed by the gaps that form along the horizontal joints.

10. A design approach is proposed to determine the required reinforcement in the critical regions ofwall panels to prevent premature failure of the walls due to cracking. The design of the base panelis investigated in detail. The design of the upper story panels is done using the design of the basepanel.

11. The required panel reinforcement predicted by the proposed design approach compares well withthe reinforcement determined using the finite element analysis results. On average, the proposeddesign approach is conservative.

Placement of the Panel Reinforcement

1. The required bonded panel reinforcement should be placed in two layers (near the faces of thepanels) within the tension zone in each critical region. The reinforcement must be placed as close tothe panel and opening edges as possible. The reinforcement must be extended a sufficient distanceon both sides of the critical sections for full development of the yield strength of the steel.

129

REFERENCES

Abdalla, H. and Kennedy, J., “Design of Prestressed Concrete Beams with Openings,” ASCE Journalof Structural Engineering, V. 121, No.5, May 1995, pp. 890-898.

Abdalla, H. and Kennedy, J., “Dynamic Analysis of Prestressed Concrete Beams,” ASCE Journal ofStructural Engineering, V. 121, No.7, July 1995, pp. 1058-1068.

Abdalla, H. and Kennedy, J., “Design Against Cracking at Openings in Prestressed Concrete Beams,”PCI Journal, V. 40, No.6, November-December 1995, pp. 60-75.

Aktan, A. and Bertero, V., “Evaluation of Seismic Response of RC Buildings Loaded to Failure,”ASCE Journal of Structural Engineering, V. 113, No. 5, May 1987, pp. 1092-1108.

Ali, A. and Wight, J., “RC Structural Walls with Staggered Door Openings,” ASCE Journal ofStructural Engineering, V. 117, No. 5, May 1991, pp. 1514-1531.

Allen, M. and Kurama, Y., “Design of Openings in Precast Walls under Vertical Loads,” submittedto the PCI Journal, 2001.

Allen, M. and Kurama, Y., “Design of Openings in Precast Walls under Vertical and Lateral Loads,”submitted to the PCI Journal, 2001.

Barney, G., Corley, W.m Hanson, J., and Parmalee, R., “Behavior and Design of PrestressedConcrete Beams with Large Web Openings,” PCI Journal, V. 22, No. 6, November-December 1977,pp. 32-61.

Hibbitt, Karlsson, and Sorenson, ABAQUS User’s Manual, Hibbitt, Karlsson, & Sorenson,Inc.,Version 5.8, 1998.

IBC 2000, International Building Code 2000, International Building Code Council, Inc., Falls Church,VA, 2000.

Kato, D., Kabeyasawa, T., Otani, S., and Aoyama, H., “Earthquake-Resistant Design of Shearwallswith one Opening,” ACI Structural Journal, V. 92, No. 4, July-August 1995, pp. 495-500.

130

Kennedy, J. and El-Laithy, A. M., “Cracking at Openings in Prestressed Beams at Transfer,” ASCEJournal of the Structural Division, V. 108, No. ST6, June 1982, pp. 1250-1265.

Kennedy, J. and Abdalla, H., “Static Response of Prestressed Girders with Openings,” ASCE Journalof Structural Engineering, V. 118, No. 2, February 1992, pp. 488-504.

Kobayashi, J., Korenaga, T., Shibata, A., Akino, K., and Taira, T., “Effect of Small Openings onStrength and Stiffness of Shear Walls in Reactor Buildings,” Nuclear Engineering and Design, V.156, 1995, pp. 17-27.

Kurama, Y., Pessiki, S., Sause, R., and Lu, L-W., and El-Sheikh, M., “Analytical Modeling andLateral Load Behavior of Unbonded Post-Tensioned Precast Concrete Walls,” Research Report No.EQ-96-02, Department of Civil and Environmental Engineering, Lehigh University, Bethlehem, PA,November 1996, 191 pp.

Kurama, Y., Pessiki, S., Sause, R., and Lu, L.-W., “Seismic Behavior and Design of Unbonded Post-Tensioned Precast Concrete Walls,” PCI Journal, V. 44, No. 3, May-June 1999, pp. 72-89.

Kurama, Y., Sause, R., Pessiki, S., and Lu, L.-W., “Lateral Load Behavior and Seismic Design ofUnbonded Post-Tensioned Precast Concrete Walls,” ACI Structural Journal, V. 96, No. 4, July-August 1999, pp. 622-632.

Kurama, Y., “Simplified Seismic Design Approach for Friction-Damped Unbonded Post-TensionedPrecast Concrete Walls,” accepted to the ACI Structural Journal, 2001.

Mackertich, S. and Aswad, A., “Lateral Deformations of Perforated Shear Walls for Low and Mid-Rise Buildings,” PCI Journal, V. 42, No. 1, January-February 1997, pp. 30-41.

Mander, J., Priestley, M., and Park, R., “Theoretical Stress-Strain Model for Confined Concrete,”ASCE Journal of Structural Engineering, V. 114, No. 8, August 1988.

NEHRP-97, NEHRP Recommended Provisions for Development of Seismic Regulations for NewBuildings, Building Structural Safety Council, 1997 Edition, Federal Emergency ManagementAgency, Washington, D.C., 1998.

Park, R. and Paulay, T., Reinforced Concrete Structures, John Wiley & Sons, New York, 1975.

Paulay, T., and Priestley, M., Seismic Design of Reinforced Concrete and Masonry Buildings, JohnWiley and Sons, New York, 1992.

Paulay, T. and Santhakumar, A., “Ductile Behavior of Coupled Shear Walls,” ASCE Journal of theStructural Division, V. 102, No. ST1, January 1976, pp. 93-108.

131

Perez, F., “Lateral Load Behavior of Precast Concrete Walls with Ductile Vertical Joint Connectors,”M.S. Thesis, Department of Civil and Environmental Engineering, Lehigh University, Bethlehem, PA,December 1998.

Powers, K. and Wallace, J., “Evaluation of Older RC Perforated Walls Using Data from theNorthridge Earthquake,” 6th U.S. National Conference on Earthquake Engineering, EarthquakeEngineering Research Institute, Seattle, WA, May 31-June 4, 1998.

Prakash, V., Powell, G., and Campbell, S., “DRAIN-2DX Base Program Description and User Guide;Version 1.10,” Report No. UCB/SEMM-93/17, Department of Civil Engineering, University ofCalifornia, Berkeley, CA, 1993

Priestley, M., “Overview of PRESSS Research Program,” PCI Journal, V. 36, No. 4, July-August1991, pp.50-57.

Priestley, M., Sritharan, S., Conley, R., and Pampanin, S., “Preliminary Results and Conclusions Fromthe PRESSS Five-Story Precast Concrete Test Building,” PCI Journal, V. 44, No. 6, November-December 1999, pp. 42-67.

Savin, G. N., Stress Concentrations Around Holes, Pergamon Press, New York, 1961.

Taylor, C., Cote, P., and Wallace, J., “Design of Slender Reinforced Concrete Walls with Openings,”ACI Structural Journal, V. 95, No. 4, July-August 1998, pp. 420-433.

Wallace, J., “Reinforced Concrete Walls: Recent Research & ACI 318-2001,” 6th U.S. NationalConference on Earthquake Engineering, Earthquake Engineering Research Institute, Seattle, WA,May31-June 4, 1998.

Yanez, F., Park, R., and Paulay, T., “Seismic Behavior of Walls with Irregular Openings,”Proceedings of the Tenth World Conference on Earthquake Engineering, 1992, pp. 3303-3308.

132

APPENDIX A

DESIGN EXAMPLE, VERTICAL LOADS ONLY

The following example demonstrates the implementation of the design approach proposed in Chapter5 to determine the required panel reinforcement in a six story wall with lp = 20 ft, (f = 0.18, (l = 0.30,and (h = 0.38. The dimensions of the wall and the placement of the post-tensioning bars are shownin Figure 3.1(b). The applied loads, wall and opening dimensions, and material properties are givenbelow.

A.1 Data

Vertical loads:

Gb = 983 kips (total gravity load at the base of the wall)Gp = 172 kipsGa = 983-172 = 811 kipsPi = 2280 kips

Wall and panel dimensions:

lp = 240 in.hp = 192 in.hpu = 160 in.tp = 12 in.

Opening and chord dimensions:

lo = 72 in.lc = (240-72)/2 = 84 in.ho = 72 in.hc = (192-72)/2 = 60 in.hcu = (160-72)/2 = 44 in. (horizontal chord height for the upper story panels)

133

Material properties:

fc! = 6 ksify = 60 ksi

A.2 Design Overview

The design of the base panel, which is the most critical panel, is presented below. The post-tensioningforces for the wall are significantly larger than the gravity loads. Thus, the reinforcement determinedfor the base panel will be used in the upper story panels as described at the end of the example.

A.3 Estimation of Panel Top Stresses

In order to determine the amount of reinforcement required at the top and bottom of the opening inthe base panel, it is necessary to solve Eq. (5.1) for Tv and Av. The first step is to estimate the stressesat the top of the panel by using Eq. (5.2) for the regions between 0#x#x1 and x1#x#x2, where x ismeasured from the centerline of the panel. Using Eq. (5.6):

x1 = 72/2 = 36 in.x2 = (2(72+240)/4 = 96 in.

The stresses fp1 and fp2 (corresponding to x1 and x2) are estimated using a linear stress-distribution asshown by the dashed line in Figure 5.4(a). From Eq. (5.4), fpp and fpa can be determined as:

fpp = 172/(240(12) = 0.0597 ksifpa = (2280+811)/(240(12) = 1.07 ksi

The angle 2c is calculated using Eq. (5.5):

2c = tan-1((160-72)/72) = 50.7/

The stresses fpp and fpa, and the angle 2c are then used in Eq. (5.3):

fp0 = 1.07(50.7-40)/45+0.0597 = 0.315 ksi

The stresses fpe, fp2, and fp1 are determined using Eq. (5.7):

fpe = 2(2280+172+811)/(240(12)-0.315 = 1.96 ksifp2 = 0.315+2(1.96-0.315)96/240 = 1.63 ksifp1 = 0.315+2(1.96-0.315)36/240 = 0.809 ksi

134

The stresses fp1 and fp2 are then used in Eq. (5.8) to determine the boundary conditions for Fp1 and Fp2as:

Fp1(0) = 0.315 ksiFpN1(0) = 0Fp2(36) = Fp1(36) = 0.809 ksiFpN2(36) = FpN1(36)Fp2(96) = 1.63 ksi

These boundary conditions can be used to determine the functions Fp1 and Fp2 for the panel topstresses as:

Fp1 = 3.74(10-4x2+0.315 ksiFp2 = -2.24(10-4x2+0.0431x-0.463 ksi

A.4 Estimation of Cr and xGp

The functions Fp1 and Fp2 are then used to estimate Cr and xGp using Eqs. (5.9) and (5.10) as follows:

xr = 0.3(84+72/2 = 61.2 in.Cr = 533 kipsxGp = 38.7 in.

A.5 Estimation of Side Chord Stresses and xGs

After determining Cr and xGp, the stresses in the side chord can be estimated. The stress fsr is foundusing Eq. (5.11):

fsr = (2280+172+811)/(2(84(12) = 1.62 ksi

The stress fse is calculated using Eq. (5.12):

fse = (2280+172+811-2(533)/(0.7(84(12)-1.62 = 1.49 ksi

The boundary conditions for Fs1 are determined using Eq. (5.13):

Fs1 (61.2) = Fs2(61.2) = 1.49 ksiFsN1(61.2) = FsN2(61.2) = (1.62-1.49)/(0.7(84) = 0.00221

12 dx = 533 kipss1

61.2σ

36∫

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These boundary conditions are then used to solve for Fs1:

Fs1 = 6.37(10-4x2-0.0803x+4.13 ksi

From Eq. (5.14), xGs can be calculated as:

xGs = 48.0 in.

A.6 Estimation of Tv and Av

The design tension force Tv, the required reinforcement area Av, and the reinforcement, ratio Dv in thebase panel are estimated using Eqs. (5.1) and (5.16):

Tv = 533(48.0-38.7)/60 = 82.6 kipsAv = 82.6/60 = 1.38 in2 > Amin = 0.61 in2 O.K. Thus,Dv = 1.38(100/(60(12) = 0.19% as shown in Table 5.1.

A.7 Reinforcement Selection and Placement

The required reinforcement in the base panel can be provided by using 8-#4 bars, four bars near eachside (face) of the panel, placed at both the top and the bottom of the opening as shown in Figure 5.8.Using Eq. (5.15), the bars must be placed within a distance of:

htv = 0.30(60 = 18 in.

from the opening edges to ensure that the steel is within the tension zone. Using a center-to-centerspacing of 2.0 in. and allowing 1.5 in. of cover concrete to the center of the first layer of bars, thesteel can be placed within 7.5 in. from the edges of the opening. The bars must be extended asufficient distance past the opening corners for full development.

A.8 Design of Upper Story Panels

As described earlier, the reinforcement ratio of Dv = 0.19% for the base panel is also used in the upperstory panels. Since the height of the horizontal chord in the upper story panels is less than the chordheight in the base panel, a smaller reinforcement area can be used in the upper story panels as:

Av = 0.19(12(44/100 = 1.00 in2 > Amin = 0.61 in2 O.K.

The design of the roof panel for the post-tensioning anchor forces is not addressed in the example.

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APPENDIX B

DESIGN EXAMPLE, COMBINED VERTICAL AND LATERAL LOADS

The following example demonstrates the implementation of the design approach proposed in Chapter6 to determine the required panel reinforcement in a six story wall with lp = 20 ft, (f = 0.18, (l = 0.30,and (h = 0.38 The dimensions of the wall and the placement of the post-tensioning bars are shownin Figure 3.1(b). The applied loads, wall and opening dimensions, and material properties are givenbelow.

B.1 Data

Gravity and post-tensioning loads:

Gb = 983 kips (total gravity load the base of the wall)Gp = 172 kipsGa = 983-172 = 811 kipsPi = 2280 kips

The following load effects are determined from a nonlinear push-over analysis of the wall ignoringthe presence of the openings:

Pf = 2580 kipsVpb = 587 kipsVp = 30.5 kipsVpt = 557 kipsMpt = 273000 [email protected] = 350000 kip@in.

Wall and panel dimensions:

lp = 240 in.hp = 192 in.

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hpu = 160 in. (height of the upper story panels)tp = 12 in.tc = 10 in.

Opening and chord dimensions:lo = 72 in.lc = (240-72)/2 = 84 in.ho = 72 in.hc = (192-72)/2 = 60 in.hcu = (160-72)/2 = 44 in. (horizontal chord height for the upper story panels)

Material properties:

fc! = 6 ksifcc = 12.7 ksify = 60 ksi

B.2 Design Overview

The design of the base panel, which is the most critical panel under combined lateral and verticalloads, is presented below. The design of the upper story panels is done using the design of the basepanel as described at the end of the example. The design of the wall under vertical loads is describedin Appendix A.

B.3 Determination of The Critical Panel Sections

In order to determine the critical panel sections, it is necessary to estimate the contact length at thebottom of the base panel, cb and at the top of the base panel, ct. The contact length at the bottom ofthe base panel ignoring the presence of the opening, cbN, can be determined using Eq. (6.1) as:

Npb = 172+811+2580 = 3560 kipscbN = 3560/(0.9(1.0(12.7(10) = 31.1 in.

The contact length at the bottom of the base panel with the opening, cb can be determined using Eq.(6.3) as:

cb = cbN = 31.1 in.

138

Assuming a concrete cover of 1.0 in., the distance from the end of the panel to the end of the contactregion at the base, cbe is calculated as:

cbe = 31.1+1.0 = 32.1 in.

The contact length at the top of the base panel ignoring the presence of the opening, c tN is estimatedusing Eq. (6.2) as:

Npt = 811+2580 = 3390 kipsc tN = 2.75(240/2-273000/3390) = 109 in.

The classification of the opening as a “large” or a “small” opening is done based on the length xs,which is determined using Eq. (6.5) as:

xs = 2(84-32.1)/(192+72)(192+32.1 = 108 in.

The opening size factor, s is calculated using Eq. (6.4) as:

s = 109/108 = 1.0

Since s = 1.0, the opening is classified as “small.”

The contact length at the top of the base panel with the opening is determined using Eq. (6.6) as:

ct = 109(1.0 = 109 in.

The critical section at the bottom of the base panel, xcb is estimated using Eq. (6.7) as:

xcb = 109/2 = 54.5 in.

The critical section at the top of the base panel, xct is estimated using Eq. (6.8) as:

xct = 109/1.02 = 109 in.

B.4 Design of Panel Top Reinforcement

The forces at the top of the base panel as shown in Figure 6.10(b) are estimated using Eq. (6.9) as:

Nptl = 0.2(1-1/1.03)3390 = 0 kipsVptl = (1-1/1.02)557 = 0 kipsGptl = (240-109)/240(172 = 93.9 kips

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The forces in the tension side chord are estimated using Eqs. (6.10) and (6.11). From Eq. (6.11):

>l = 84/(240-84) = 0.538>h = 60/(192-60) = 0.455>a = 192/240 = 0.800M2m = 0.15(0.538(0.455(0.8002(350000 = 8220 kip@in.

From Eq. (6.10), the forces in the tension side chord are estimated as:

N2 = 0.15(0.5383[1-109/(1.0(240)]3560 = 45.5 kipsV2 = 4.0(0.302[109/(1.0(240)]2+(1.25-1/0.800)2/0.800587+0.75(172(842/(240(192) = 63.3kipsM2 = 8220 - 45.5(240-84)/2 = 4670 kip@in.

The design axial force N1, shear force V1, and moment M1, are determined using the free bodydiagram in Figure 6.10(b) as:

N1 = 32.8 kipsV1 = 139 kipsM1 = 7790 kip@in.

The design tension force, T1 is estimated based on a linear axial stress distribution with N1 = 32.8 kipsand M1 = 7790 kip@in as:

T1 = 212 kips

Then, the required reinforcement area, Ac1b and ratio, Dc1b at the top of the base panel are calculatedusing Eq. (6.12) as:

Ac1b = 212/60 = 3.53 in2 > Amin = 0.61 in2 O.K. Thus,Dc1b = 3.53/(60(12) = 0.49% as shown in Table 6.5.

B.5 Design of Panel Side Reinforcement

The design tension force, T2 is estimated based on a linear axial stress distribution with N2 = 45.5 kipsand M2 = 4670 kip@in. as:

T2 = 108 kips

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Then, the required reinforcement area, Ac2b and ratio, Dc2b at the sides of the base panel are calculatedusing Eq. (6.13) as:


B.6 Design of Panel Bottom Reinforcement

The forces in the compression side chord are determined using Eq. (6.14) as:

Nr = 3560+45.5 = 3610 kipsVr = 587-63.3 = 524 kipsMr = 350000-8220-3610(240-84)/2-524(72/2 = 41300 kip@in.

The axial force, N4 and moment, M4 in Section 4 are determined based on a linear axial stressdistribution with Nr = 3610 kips and Mr = 41300 kip@in. as:

N4 = 600 kipsM4 = 1740 kip@in.

The shear force, V4 in Section 4 is determined using Eq. (6.15) as:

V4 = (84-54.5)/84(524 = 184 kips

The forces in Section 3 are determined using the free body diagram in Figure 6.10(c) as:

N3 = 246 kipsV3 = 646 kipsM3 = 13700 kip@in.

The height of the tension zone in Section 3, ht3 is determined using Eq. (6.16) as:

ht3 = 0.10(240(1-0.55(0.30-0.55(0.38) = 15.0 in.

The location of the tension stress resultant, T3 from the bottom of the panel, yt3 is:

yt3 = 15.0/3 = 5.00 in.

The location of the compression stress resultant, C3 from the bottom of the panel, yc3 is approximatedusing Eq. (6.17) as:

yc3 = (3.75-1/0.800)15.0 = 37.5 in.

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The design tension force, T3 is determined using Eq. (6.18) as:

T3 = [13700-246(37.5-192/4)]/(37.5-5.00) = 501 kips

The required reinforcement area, Ac3b and ratio, Dc3b at the bottom of the base panel are calculatedusing Eq. (6.19) as:


B.7 Design of Reinforcement Around The Opening

The reinforcement at the top and bottom of the opening in the base panel is determined using Eq.(6.20) as:

Ac4b = Ac1b = 3.53 in2 > Av = 1.38 in2 and Ac4b = Ac1b = 3.53 in2 > Amin = 0.61 in2 O.K. Thus,Dc4b = Dc1b = 0.49%

where Av = 1.38 in2 is the reinforcement required at the top and bottom of the opening under verticalloads only (due to gravity and post-tensioning) as determined in Appendix A.

Based on Eq. (6.21), the required reinforcement at the sides of the opening is equal to the minimumreinforcement as:

Ac5b = Amin = 0.61 in2

Dc5b = Dmin,s = 0.61/(84(12) = 0.061%

B.8 Design of Reinforcement in Upper Story Panels

The design of the reinforcement in the upper story panels is done using Eq. (6.22). The reinforcementratio at the top and bottom of the openings under vertical loads only is:

Dv = 0.19% (as determined in Appendix A)

The minimum reinforcement ratios for the upper story panels are:

Dmin,h = 0.61/(44(12) = 0.12%Dmin,s = 0.61/(84(12) = 0.061%

142

The required reinforcement ratios for the second story panel are:

Dc12 = 0.49[(6-2)/(6-1)]2 = 0.31% > 0.12% O.K.Dc22 = 0.18[(6-2)/(6-1)]2 = 0.12% > 0.061% O.K.Dc32 = 1.04[(6-2)/(6-1)]2 = 0.67% > 0.12% O.K.Dc42 = 0.19+(0.49-0.19)1-[(2-1)/(6-1)]2 = 0.48% > 0.12% O.K.Dc52 = 0.061%

The required reinforcement ratios for the third story panel are:

Dc13 = 0.49[(6-3)/(6-1)]2 = 0.18% > 0.12% O.K.Dc23 = 0.18[(6-3)/(6-1)]2 = 0.065% > 0.061% O.K.Dc33 = 1.04[(6-3)/(6-1)]2 = 0.37% > 0.12% O.K.Dc43 = 0.19+(0.49-0.19)1-[(3-1)/(6-1)]2 = 0.44% > 0.12% O.K.Dc53 = 0.061%

The required reinforcement ratios for the fourth story panel are:

Dc14 = 0.49[(6-4)/(6-1)]2 = 0.078%< 0.12% Y Dc14 = 0.12%Dc24 = 0.18[(6-4)/(6-1)]2 = 0.029% < 0.061% Y Dc24 = 0.061%Dc34 = 1.04[(6-4)/(6-1)]2 = 0.17% > 0.12% O.K.Dc44 = 0.19+(0.49-0.19)1-[(4-1)/(6-1)]2 = 0.38% > 0.12% O.K.Dc54 = 0.061%

The required reinforcement ratios for the fifth story panel are:

Dc15 = 0.49[(6-5)/(6-1)]2 = 0.020%< 0.12% Y Dc15 = 0.12%Dc25 = 0.18[(6-5)/(6-1)]2 = 0.0072% < 0.061% Y Dc25 = 0.061%Dc35 = 1.04[(6-5)/(6-1)]2 = 0.042% < 0.12% Y Dc35 = 0.12%Dc45 = 0.19+(0.49-0.19)1-[(5-1)/(6-1)]2 = 0.30% > 0.12% O.K.Dc55 = 0.061%

The design of the sixth (top) story panel is governed by the post-tensioning anchor forces and is notaddressed in the example.

STRUCTURAL ENGINEERING RESEARCH REPORT SERIESLIST OF TECHNICAL REPORTS

NDSE-01-01 “Design of Rectangular Openings in Unbonded Post-Tensioned Precast ConcreteWalls,” by M. Allen and Y. Kurama, April 2001, 142 pp. (this report may bedownloaded from http://www.nd.edu/~concrete/).

design of rectangular openings in unbonded post-tensioned...

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