Analysis of a Reinforced Concrete Shear Wall

118 Pages·2007·3.44 MB·English

Analysis of a Reinforced Concrete Shear Wall
M.Sc Thesis
Bj(cid:246)rk Hauksd(cid:243)ttir
s053069
Instructors
Bjarni Bessason
Per Golterman
February 2007

Abstract
InJune2000twomajorearthquakeswithmomentmagnitude6.6occurred,after88years
of rest, in the central part of the South Iceland Seismic Zone (SIZS). Earthquakes in this
region have several times since the settlement of Iceland caused collapse of the majority
of houses and number of casualties. It has been estimated that no more than one fourth
of the strain energy in the SIZS was released in the two June 2000 earthquakes resulting
in that large earthquakes may occur in the zone during the next few decades.
Themainobjectiveoftheresearchworkpresentedinthisthesisistostudythenonlin-
earbehaviorofareinforcedconcreteshearwallwithdi(cid:27)erentreinforcementarrangements
inanidealizedthreestorybuildinglocatedintheSISZsubjectedtoastep-wiseincreasing
lateral earthquake load.
Four di(cid:27)erent reinforcement arrangements of the shear wall are considered. Firstly,
a reinforcement in which the design is based on the Stringer method. Secondly, a rein-
forcement in which the design is based on linear elastic (cid:28)nite element method analysis
using general purpose FE-program (SAP2000). Thirdly, a reinforcement again based on
linear elastic FEM but here using a building specialized FE-program (ETABS), which
has a special post-processor to present section forces. Fourthly, a reinforcement based on
minimum reinforcement requirements from Eurocode 2.
The nonlinear behavior of the four di(cid:27)erent reinforced shear walls is then tested by
non-linear pushoveranalysis using the general purpose FE-program ANSYS. An attempt
is made to evaluate crack width calculations as a function of load to re(cid:29)ect the damage.
The study show that di(cid:27)erent reinforcement layouts a(cid:27)ect the response of the wall
andthedi(cid:27)erenceincrackwidthismainlyduetotheboundaryreinforcement.Thecrack
widthscalculatedbyusingtheinformationfromANSYSseemtobepromisinganduseful
when designing and analysing structures in seismic zones.
i

Symbols
Q = Set of generalized stresses
D = Distribution
q = strains
W = work per unit volume
(cid:178)¯ = strains distribution
σ¯ = stress distribution
λ = indeterminate factor
Pi = external forces
ui = displacements
dV = volume element
σx = stresses in x direction (horizontal)
σy = stresses in y direction (vertical)
τxy = strains
ftx = Tensile strength of reinforcement in x direction (horizontal)
fty = Tensile strength of reinforcement in y direction (vertical)
fY = Yield strength of reinforcement
Asx = Tensile reinforcement area in x direction (horizontal)
Asy = Tensile reinforcement area in y direction (vertical)
σc = concretes strength
ν = e(cid:27)ectiveness factor
t = thickness
F = calculated compression/tension force
fyd = Design yield point of steel
As,t = Reinforcement area for tension stringer
Ac,needed = Needed concrete area to take up compression
fcd = Design concrete strength
C = Total force that concrete can uptake
As,c = Reinforcement area for compression stringer
As = Reinforcement are for rectangle mesh area
ft = Tensile strength of steel
(cid:178)u = maximum strain in steel
fc = compressive strength of concrete
(cid:178)c1 = concrete strain at peak stress
(cid:178)cu = ultimate strain in concrete
∆ = structural displacement
µ = ductility
SE = strength to resist earthquake-induced force
iii
Abstract
wk = the design crack width
srm = the average (cid:28)nal crack spacing
εsm = the mean strain allowing under the relevant load
β = coe(cid:30)cient relating the average crack width to the design value
σs = the stress in the tension reinforcement at cracked section
σsr = the stress in the tension reinforcement at the (cid:28)rst crack
β1 = Coe(cid:30)cient which takes account of the bond properties
β2 = Coe(cid:30)cient which takes account of the loading
φ = bar size
k1 = Coe(cid:30)cient which takes account of the bond properties
k2 = Coe(cid:30)cient which takes account of the form of the strain distribution
H = height of the analyzed building
W = width of the analyzed building
L = Length of the analyzed shear wall
h = story height
tw = the shear wall thickness
ts = the slab/roof thickness
ρc = density of concrete
ρg = density of glass
tg = thickness of double glass
Ec = Young’s modulus for concrete
T1 = the fundamental period of vibration
Fb = the seismic base shear force
Sd = Design spectrum
Ac = total a(cid:27)ective area of shear wall
ag = ground acceleration
q = behavior factor
Fi = horizontal forces acting on the shear wall
T = vibrating period
S = soil factor
mi,j = storey masses
zi,j = heights of the masses
fct = tensile strength of concrete
f1 = Ultimate compressive strength for state of biaxial compression
f2 = Ultimate compressive strength for state of uniaxial compression
σha = ultimate biaxial compressive strength
Ec = secant modulus of elasticity
βt = shear coe(cid:30)cient for open crack
βc = shear coe(cid:30)cient for closed crack
Tc = multiplier for amount of tensile stress relaxation
Es = modulus of elasticity for steel
iv
Contents
Abstract i
Symbols iii
Contents v
List of Figures vii
List of Tables ix
Preface xi
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Theory 5
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Analysis Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.1 Linear Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.2 Plastic Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.2.1 The Lower Bound Theorem . . . . . . . . . . . . . . . . . 8
2.3 Design Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3.1 Disks with Orthogonal Reinforcement . . . . . . . . . . . . . . . . 9
2.3.2 Stringer Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Finite Element Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.5 Nonlinear Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5.1 Concrete and Steel . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5.2 Reinforced Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5.3 Mathematical Modeling . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5.3.1 Elastic Based Model - Before Yielding Point . . . . . . . 17
2.5.3.2 Elastic-Strain Hardening Plastic Model - After Yielding
Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5.3.3 The Shape of an Initial Yield Surface . . . . . . . . . . . 18
2.5.3.4 The evolution of Subsequent Loading Surface . . . . . . . 19
2.5.3.5 The Flow Rule . . . . . . . . . . . . . . . . . . . . . . . . 19
2.5.4 Finite Element Modeling of Cracks . . . . . . . . . . . . . . . . . . 19
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Contents
2.6 Ductility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.7 Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.8 Methods to Calculate Cracks . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.8.1 Calculation of design crack widths . . . . . . . . . . . . . . . . . . 22
2.9 Shear Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3 The Building and the Load 25
3.1 The Building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1.1 The Mass of the Building . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 Pushover Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2.0.1 Lateral Force Patterns. . . . . . . . . . . . . . . . . . . . 28
3.2.0.2 Capacity Curve . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Load - Lateral Force Method of Analysis . . . . . . . . . . . . . . . . . . . 29
3.3.1 Can the Lateral Force Method be used? . . . . . . . . . . . . . . . 29
3.3.2 The Design Response Spectra . . . . . . . . . . . . . . . . . . . . . 30
3.3.3 Vertical Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4 Reinforcement Design 35
4.1 The Stringer Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.1.1 The Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.1.2 Calculation of Shear Stresses and Stringer Forces . . . . . . . . . . 37
4.2 Linear Elastic FE-analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2.1 Modeling in SAP2000 and ETABS . . . . . . . . . . . . . . . . . . 43
4.2.2 ETABS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2.3 SAP2000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3 Minimum Reinforcement according to EC2 . . . . . . . . . . . . . . . . . 53
4.3.1 Vertical Reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3.2 Horizontal Reinforcement . . . . . . . . . . . . . . . . . . . . . . . 53
5 Nonlinear Pushover Analysis 55
5.1 Calculation Process in ANSYS . . . . . . . . . . . . . . . . . . . . . . . . 55
5.2 Element Type - Reinforced Concrete Solid . . . . . . . . . . . . . . . . . . 55
5.3 Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.4 Analytical Nonlinear Model . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.5 Analytical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.5.1 Cracks and Reinforcement Yielding . . . . . . . . . . . . . . . . . . 63
5.5.2 Calculations of Crack width . . . . . . . . . . . . . . . . . . . . . . 68
6 Summary and Conclusion 75
Appendices 77
A MATLAB script for Design Response spectra 77
B Calculations for Stringer method 79
C Modeling in ETABS 95
References 103
vi
List of Figures
1.1 Iceland lies on the Mid Atlantic Ridge . . . . . . . . . . . . . . . . . . . . 1
1.2 Damage because of the earthquakes. . . . . . . . . . . . . . . . . . . . . . 2
2.1 Uniaxial stress-strain relation for rigid-plastic material [18] . . . . . . . . 6
2.2 Maximum work hypothesis [18] . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Disk element with stress in the concrete [18] . . . . . . . . . . . . . . . . . 9
2.4 Disk divided into nodes, stringer and mesh rectangle areas [13] . . . . . . 11
2.5 Stress-strain diagram for concrete [9] . . . . . . . . . . . . . . . . . . . . . 13
2.6 Typical stress-strain diagram of reinforcing steel [9] . . . . . . . . . . . . . 13
2.7 Typical load-displacement relationship for reinforced concrete element [21] 14
2.8 Biaxial strength Envelope for Plain Concrete [19] . . . . . . . . . . . . . . 15
2.9 Triaxial strength surface in principal stress space [19] . . . . . . . . . . . . 16
2.10 Typical load-displacement relationship for reinforced concrete element [21] 16
2.11 Loading surfaces of concrete in biaxial stress plane for a work-hardening-
plasticity model [4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.12 Kinematic hardening rule [1] . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.13 Relationship between strength and ductility [21] . . . . . . . . . . . . . . 21
2.14 The Shear Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.15 Structural wall [21] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.1 Plan View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 The Shear Wall Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 The Longitudinal Wall Dimensions . . . . . . . . . . . . . . . . . . . . . . 27
3.4 Horizontal ground acceleration for Iceland . . . . . . . . . . . . . . . . . . 30
3.5 Horizontal design spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.6 Forces applied on the shear wall. . . . . . . . . . . . . . . . . . . . . . . . 34
4.1 The wall divided into nodes, stringers and areas . . . . . . . . . . . . . . . 36
4.2 The forces acting on the wall for Stringer Method. . . . . . . . . . . . . . 36
4.3 Sign of the shear stresses. . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.4 Horizontal Stringer Forces for stringerline 1 to 3 . . . . . . . . . . . . . . 40
4.5 Horizontal Stringer Forces for stringerline 4 to 6 . . . . . . . . . . . . . . 40
4.6 Horizontal Stringer Forces for stringeline 7 to 10 . . . . . . . . . . . . . . 41
4.7 Vertical Stringer Forces for stringerline 11 to 14 . . . . . . . . . . . . . . . 41
4.8 Vertical Stringer Forces for stringerline 15 to 18 . . . . . . . . . . . . . . . 42
4.9 Reinforcement of the wall based on Stringer method . . . . . . . . . . . . 42
4.10 Shell Element [26] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.11 Deformations of a shell element in ETABS [8] . . . . . . . . . . . . . . . . 44
4.12 Pier and spandrel forces in ETABS . . . . . . . . . . . . . . . . . . . . . . 45
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Contents
4.13 Pier labeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.14 Moment, M3, in spandrels . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.15 Moment, M3, in piers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.16 Reinforcement of the wall based on analysis in ETABS . . . . . . . . . . . 47
4.17 The basic types of shell stresses [22] . . . . . . . . . . . . . . . . . . . . . 48
4.18 Normal stresses, σx, from the SAP2000 analysis . . . . . . . . . . . . . . . 49
4.19 Normal stresses, σy, from the SAP2000 analysis . . . . . . . . . . . . . . . 49
4.20 Shear stresses, τxy, from analysis in SAP2000 . . . . . . . . . . . . . . . . 50
4.21 Reinforcement arrangement of the wall based on analysis in SAP2000 . . 52
4.22 Minimum reinforcement according to EC2 . . . . . . . . . . . . . . . . . . 53
5.1 SOLID65 element in ANSYS [1] . . . . . . . . . . . . . . . . . . . . . . . . 56
5.2 Bilinear Hardening Concrete Model . . . . . . . . . . . . . . . . . . . . . . 57
5.3 Normal distribution of compressive strength results [20] . . . . . . . . . . 57
5.4 Steel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.5 Modeling of the wall in Ansys . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.6 Element numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.7 Load de(cid:29)ection curves for di(cid:27)erent analysis . . . . . . . . . . . . . . . . . 62
5.8 Ductility curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.9 Ductility curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.10 Cracking signs in ANSYS, NL=1 . . . . . . . . . . . . . . . . . . . . . . . 64
5.11 Cracks at design earthquake load (NL = 1) for in the wall designed with
Stringer method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.12 Cracksatdesignearthquakeload(NL=1)inthewalldesignedfromETABS 65
5.13 Cracks at design earthquake load (NL = 1) in the wall designed from
SAP2000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.14 Cracks at design earthquake load (NL = 1) in the wall with minimum
reinforcement, EC2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.15 Steel stresses in element no 787 above middle window . . . . . . . . . . . 67
5.16 Steel stresses in element no 670 below middle window . . . . . . . . . . . 67
5.17 Computed crack width in element 787 . . . . . . . . . . . . . . . . . . . . 69
5.18 Computed crack width in element 670 . . . . . . . . . . . . . . . . . . . . 69
5.19 Design crack width in element 1026 . . . . . . . . . . . . . . . . . . . . . . 70
5.20 Cracks at middle window for Stringer . . . . . . . . . . . . . . . . . . . . 71
5.21 Cracks width for Stringer . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.22 Cracks at middle window for ETABS . . . . . . . . . . . . . . . . . . . . . 72
5.23 Cracks width for ETABS . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.24 Cracks at middle window for SAP2000 . . . . . . . . . . . . . . . . . . . . 73
5.25 Cracks width for SAP2000 . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.26 Cracks at middle window for EC2. . . . . . . . . . . . . . . . . . . . . . . 74
5.27 Cracks width for EC2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
C.1 Spandrel labeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
C.2 Axial forces in spandrels, P . . . . . . . . . . . . . . . . . . . . . . . . . . 96
C.3 Shear forces in spandrels, V2 . . . . . . . . . . . . . . . . . . . . . . . . . 96
C.4 Moment forces in spandrels, M3 . . . . . . . . . . . . . . . . . . . . . . . . 97
C.5 Pier labeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
C.6 Axial forces in Piers, P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
C.7 Shear forces in piers, V2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
C.8 Moment forces in piers, M3 . . . . . . . . . . . . . . . . . . . . . . . . . . 100
viii