Unbonded post-tensioned shear walls of calcium silicate
element masonry
Citation for published version (APA):
Meer, van der, L. J. (2013). Unbonded post-tensioned shear walls of calcium silicate element masonry. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR753875
DOI:
10.6100/IR753875
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of calcium silicate element masonry
of calcium silicate element masonry
PROEFSCHRIFT
ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de rector magni cus, prof.dr.ir. C.J. van Duijn, voor een
commissie aangewezen door het College voor Promoties in het openbaar te verdedigen op donderdag 23 mei 2013 om 16.00 uur
door
Lex Jasper van der Meer
prof.ir.-arch. D.R.W. Martens
Copromotor:
dr.ir. A.T. Vermeltfoort
Bouwstenen 181
A catalogue record is available from the Eindhoven University of Technology Library ISBN: 978-90-386-3374-9
is dissertation was prepared in X E LATEX by the author and printed by Eindhoven
Uni-versity of Technology Printservice
Cover design: Sonja van der Meer (SvdM architecture & engineering) © Copyright 2013, L.J. van der Meer
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopy-ing, recording or otherwise, without the prior written permission from the copyright owner.
prof.dr.ir. B. de Vries (chair)
Eindhoven University of Technology, e Netherlands prof.ir.-arch. D.R.W. Martens
Eindhoven University of Technology, e Netherlands dr.ir. A.T. Vermeltfoort
Eindhoven University of Technology, e Netherlands Professor J. Ingham PhD MBA
e University of Auckland, New Zealand Prof.Dr.-Ing. E. Gunkler
Hochschule Ostwestfalen-Lippe, Germany prof.dr.ir. D.A. Hordijk
Del University of Technology, the Netherlands prof.dr.ir. A.S.J. Suiker
is dissertation would not have been completed without the support of family, friends, colleagues and other people involved. I would like to thank all of you for your support, assistance, guidance, advice and encouragement.
First of all, I would like to thank my supervisors, Prof. Dirk Martens and Dr. Ad Vermelt-foort. Aer I obtained my MSc degree, Dirk Martens offered me a temporary position as a research assistant for the group of Masonry Structures, which allowed us to write the PhD research proposal. Ad Vermeltfoort supervised one of my MSc research projects and captured my interest in experimental research. I would like to thank you both for the valuable and numerous discussions that gave direction to the project.
e effort of the other members of the doctoral committee is also highly appreciated. Many thanks for your feedback, which helped to improve the quality of my work. In particular I would like to thank Prof. Jason Ingham, who was willing to spend the time and effort to travel to Eindhoven, literally from the other side of the world. Furthermore I owe gratitude to Prof. Akke Suiker, who provided invaluable advice related to nite element modelling.
To VNK and IKOB BKB, I would like to express my sincere gratitude for generously fund-ing our research project for four and a half years. A special word of thanks goes to Tijn Coppens, former director of VNK, for his enthusiastic involvement in the project from day one, which continued aer his retirement.
Our research was discussed from a practical viewpoint during several meetings with a committee of industrial partners. I would like to thank Hans Verkleij (Calduran Kalk-zandsteen), Arthur Claessens (Xella Nederland) and Andrew Zielinski (Dywidag Sys-tems International) for their participation in these meetings and also for arranging the materials for the experimental investigations as well as some very insightful construction site visits. Furthermore I would like to thank Simon Wijte (Adviesbureau Hageman), Erik
Kampman and Ton van Beek (IKOB BKB) for their participation in the meetings. I would like to acknowledge Erik Kampman also for conducting related experimental research at IKOB BKB.
e experimental research would have been impossible without the dedicated support of the staff of the Pieter van Musschenbroek laboratory: Eric Wijen, eo van de Loo, Rien Canters, Cor Naninck, Johan van den Oever, Martien Ceelen, Toon van Alen and Hans Lamers. I enjoyed working in the lab together with all of you a lot and I would like to thank you for the good working atmosphere. Furthermore I owe a word of thanks to the staff of the Building Physics laboratory for facilitating the outdoor prestress loss experiments, especially to “weatherman” Wout van Bommel for sending me the climate data. Further thanks goes to MSc students Jop Courage, Gerwin Paters and Jeroen Hendriks for their assistance with some of the experimental work, to Gerard Nabuurs (Xella Nederland) for constructing the shear wall specimens, and last but not least to VEBO and Cugla for providing materials for the shear wall specimens free of charge.
On the computational side, the soware DIANA developed by TNO DIANA was a great asset. I would like to thank Professor Paulo Lourenço (University of Minho) for his ma-sonry macro-model, for putting its source code online and for answering some of my questions. At TNO DIANA I would like to thank Gerd-Jan Schreppers for his interest in my work with Lourenço’s model and Wijtze Pieter Kikstra for his attempts to incorporate a ‘3D’-version of Lourenço’s model in DIANA and to solve my issues with the model. Of course I would also like to thank all my (former) PhD colleagues of the9th oor: Sander Zegers, Paul Teeuwen, Dennis Schoenmakers, Roel Spoorenberg, Frank Huijben, Ronald van der Meulen, Sarme Silitonga, Juan Manuel Davila Delgado and Saleh Mohammadi. ank you for the “gezelligheid” during the lunch- and coffee-breaks, aer-work beers and all-you-can-eat sushi dinners. I would also like to thank Bram van Gessel from the Department of Applied Physics for our regular lunch-time run-training, which helped to clear the mind. Furthermore I am grateful for the support of the secretaries of the 9th oor, Nathaly Rombley, Karin Nisselrooij-Steenbergen, Bianca Magielse and Marlyn Aretz, may she rest in peace.
I would like to thank my parents, Ruurd and Gerri, my sister Sonja, her boyfriend David and my girlfriend’s parents Frans and Gerry for the pleasant times together, which dis-tracted my attention from PhD matters. To Sonja I owe a special thanks for designing the cover of this dissertation. Last but certainly not least I wish to thank my girlfriend Emmy. anks for your never-ending support and for our high-quality time together! Lex van der Meer
CASIEL masonry consists of prefabricated CAlcium SIlicate ELements (CASIELs) that are bonded by thin-layer mortar. In Northwest Europe, CASIEL masonry is popular for application in load-bearing walls and shear walls in low-rise buildings. Shear walls have to bear vertical loads and ensure the overall lateral stability of a building for horizontal loads such as wind and seismic loading. e recent development of CASIELs with a compress-ive strength of 36 MPa and above is bene cial for the application of CASIEL masonry in load-bearing walls and shear walls in medium-rise buildings. However, due to the mod-est tensile strength, the horizontal load capacity of unreinforced CASIEL masonry shear walls is determined by the overturning moment about the base of the wall. e over-turning moment and hence the horizontal load capacity can be increased by vertically post-tensioning a CASIEL masonry shear wall.
A CASIEL masonry shear wall can be post-tensioned by means of prestressing tendons inserted in cavities in the wall, which are anchored and prestressed aer construction of the wall. Despite the availability of high-strength CASIEL masonry and suitable post-tensioning systems, post-tensioned CASIEL masonry has rarely been applied in construc-tion practice. Apart from practical aspects, the absence of post-tensioned CASIEL ma-sonry in construction practice is due to a lack of experimental evidence and a lack of design rules for such masonry in building codes. erefore, the research described in this dissertation was aimed at experimental, numerical and analytical investigation of the mechanical behaviour of post-tensioned CASIEL masonry shear walls and the eval-uation and development of design rules. e investigation was con ned to unbonded post-tensioning (UPT) by means of high-strength steel tendons, shear walls subjected to monotonic horizontal loading and two types of CASIELs with a high compressive strength.
In international scienti c literature, attention has been paid to UPT shear walls of precast
concrete, concrete masonry, clay masonry and calcium silicate block masonry. CASIEL masonry differs from other types of masonry by the larger size of the masonry units (ele-ments), the use of thin-layer mortar and the presence of a kicker course in each wall– oor interface. e kicker course, which consists of a mortar joint of variable thickness and one course of calcium silicate blocks, is used to create a smooth and level surface for the CASIELs on top of a oor in a building.
An important aspect of post-tensioned masonry is the prestress loss due to time-depen-dent deformations such as creep and shrinkage of the masonry and relaxation of the prestressing steel, which should be accounted for in the design. Chapter 2 is dedicated to the investigation of prestress loss due to creep and shrinkage of UPT CASIEL masonry. Creep, shrinkage and prestress loss were measured on 38 large-scale high-strength CA-SIEL specimens, including a thin-layer mortar bed joint or a wall– oor connection with a kicker course, for a period of at least 300 days. Initial moisture content of the specimens and temperature and relative humidity of the environment were carefully controlled. Measured creep and shrinkage data were curve- tted and used to predict prestress loss by means of visco-elastic nite element (FE) simulations as well as a simple analytical expres-sion found in Eurocode 2 [74]. ese predictions were validated by prestress loss experi-ments, conducted simultaneously with the creep and shrinkage experiments. Predictions for building practice indicated acceptable amounts of ultimate prestress loss ranging from 10 to 30%. Future improvements could be made for the wall– oor connection including the kicker course, which has a relatively high contribution to the overall prestress loss. An investigation of the mechanical behaviour of UPT CASIEL masonry shear walls sub-jected to monotonic horizontal loading is described in Chapter 3. is behaviour is to a large extent similar to the mechanical behaviour of unreinforced shear walls where the prestressing force is treated as an additional axial load. e bene cial effect of post-tensioning is mainly due to this increase of axial load. However, for shear walls acting as a cantilever, upli of the shear wall aer cracking of the wall at the base activates the UPT tendons, which results in a further increase of the horizontal load capacity. Eight single-storey UPT CASIEL masonry shear walls were tested to gather experimental evid-ence for their mechanical behaviour including activation of the UPT tendons. Two unit types, two prestress levels and rectangular as well as T-shaped cross-sections were con-sidered. e tests on rectangular shear walls were consistent in their behaviour: rst base separation was observed, then the UPT tendons were activated and nally toe crushing occurred in the compressed zone at the base of the wall. e shear walls with T-shaped cross-section all failed prematurely due to shear failure at the web– ange interface. is type of failure is characterised by opening or cracking of the head joints of the web– ange interface and diagonal cracking of the interlocking CASIELs of the web.
ree models were developed to reproduce the mechanical behaviour as observed ex-perimentally. e rst two models were based on several assumptions, including the assumptions that plane sections remain plane and that shear deformations are negligible. Both models were based on global compatibility of the masonry deformations and the UPT tendons between their anchorages, which required an iterative procedure. In the rst model, closed-form solutions for the deformations of the cracked masonry shear wall were used, while numerical integration using MATLAB was employed in the second model. e third model was a FE macro-model for orthotropic plasticity in a state of plane stress, originally developed by Lourenço [55], but modi ed to overcome some de cien-cies of the original model. e kicker course was included in all models as a no-tension linear-elastic layer with an adjusted stiffness.
Results for all three models were compared to the experimental results for rectangular walls. e non-linear stiffness as observed experimentally was reproduced quite well by all three models, despite the assumptions for the rst two models. e maximum ho-rizontal displacement was underestimated by all models, which was attributed to non-linear behaviour of the kicker course as well as a non-non-linear strain distribution in the compressed zone. e latter strain distribution was also observed in the FE simulations, which underestimated the experimental displacements to a lesser extent. For all models, it was particularly difficult to accurately estimate the material properties to be used, be-cause of the heterogeneity at the shear wall base due to the kicker course. e non-linear behaviour of the shear walls with T-shaped cross-section could be approximated reason-ably well by the FE model, but the web– ange failure, which is related to the local failure of joints in the web– ange interface, could not be adequately captured by the FE macro-model. In spite of the aforementioned shortcomings of the developed models, these mod-els offer a sound basis for the development of design rules for rectangular shear walls, particularly related to the activation of UPT tendons. Additional research is needed on the non-linear behaviour of the kicker course and the web– ange shear capacity of shear walls with non-rectangular sections.
While Chapter 3 was focused on the mechanical behaviour of individual shear walls, the behaviour of multiple shear walls in a building was studied in Chapter 4, in particular the redistribution of the horizontal load among shear walls aer cracking. In CASIEL masonry shear wall buildings it is generally assumed that the reinforced concrete storey oor slabs act as rigid diaphragms. e horizontal load acting on the building facade is then distributed among the shear walls based on the linear-elastic in-plane stiffness of the individual shear walls. However, the horizontal load capacity of an individual CASIEL masonry shear wall is based on a cracked cross-section and non-linear material beha-viour. is discrepancy is accounted for in Eurocode 6 [75], by allowing the redistribu-tion of a maximum of 15% of the horizontal load on one shear wall, based on the initial
linear-elastic horizontal load distribution, to the other shear walls in the same loading direction. As little information related to horizontal load redistribution is available in in-ternational literature, a FE study was conducted, using the previously developed macro-model. One half of a symmetric six-storey office building with symmetric horizontal loading was modelled, containing two shear walls in the considered direction of load-ing. Each of these shear walls was either unreinforced or UPT and three cross-sectional shapes were considered. Horizontal load redistributions up to 30% were found, especially when one shear wall had a high initial stiffness and a low level of axial load compared to the other shear wall. To draw general conclusions with respect to horizontal load redis-tribution, it is required that more complex building geometries, including asymmetric building layout and/or asymmetric loading, are studied. Particular attention should be paid to horizontal load distribution in buildings where a brittle type of shear failure such as cracking of the web– ange interface is decisive, as horizontal load redistribution is not self-evident in this case.
Chapter 5 is dedicated to the development of design rules. Based on a comparison of ex-isting design rules in various (inter)national building codes, the research presented in this dissertation and additional literature, design rules for UPT CASIEL masonry shear walls were developed. Short-term and long-term material properties for the investigated types of CASIEL masonry were extracted from the experimental research. Particular attention was paid to a design method for the activation of UPT tendons based on design charts, which were created by using the analytical model from Chapter 3. Furthermore, it was noted that insufficient design rules are available for the shear capacity of the web– ange interface. e average shear stress at the web– ange interface at crack initiation could be correlated reasonably well to the splitting tensile strength of the CASIELs. However, addi-tional experimental, numerical and analytical research in this area is required to develop adequate design rules. Chapter 5 is concluded by proposing construction guidelines for UPT CASIEL masonry.
Acknowledgements vii
Summary ix
List of Abbreviations xix
List of Symbols xxi
1 Introduction 1
1.1 Motivation of the research program . . . 1
1.2 Objectives of the research program . . . 4
1.2.1 Creep, shrinkage and prestress loss . . . 4
1.2.2 Post-tensioned shear wall behaviour . . . 5
1.2.3 Post-tensioned shear wall assemblies . . . 6
1.2.4 Design and construction guidelines . . . 6
1.3 Research signi cance . . . 6
1.3.1 Scienti c contribution . . . 6
1.3.2 Utilisation . . . 7
1.4 Research methodology . . . 8
1.5 esis outline . . . 9
2 Creep, shrinkage and prestress loss 11 2.1 Literature review . . . 12
2.1.1 Creep and shrinkage . . . 12
2.1.2 Prestress loss . . . 14
2.2 Experiments . . . 15 2.2.1 Design of a test-setup . . . 15 2.2.2 Test scheme . . . 15 2.2.3 Preparation of specimens . . . 17 2.2.4 Measurements . . . 20 2.3 Modelling . . . 22 2.3.1 Analytical model . . . 22
2.3.2 Viscoelastic nite element model . . . 24
2.4 Results and discussion . . . 25
2.4.1 Short-term material properties . . . 25
2.4.2 Axial shrinkage . . . 28
2.4.3 Axial creep . . . 29
2.4.4 Prestress loss . . . 31
2.5 Parametric study . . . 34
3 Post-tensioned shear wall behaviour 37 3.1 Literature review . . . 38
3.2 Experimental research . . . 40
3.2.1 Material properties of CASIEL masonry . . . 40
3.2.2 Material properties of the prestressing steel . . . 45
3.2.3 Test scheme . . . 46
3.2.4 Construction of test specimens . . . 46
3.2.5 Test set-up . . . 49
3.2.6 Instrumentation scheme . . . 50
3.2.7 Post-tensioning and test procedure . . . 53
3.2.8 Test observations . . . 54
3.2.9 Processing of experimental results . . . 54
3.2.10 Experimental results and discussion . . . 54
3.3 Semi-analytical model . . . 61
3.3.1 Stress-strain diagram . . . 62
3.3.2 Dimensionless variables and parameters . . . 62
3.3.3 Illustration of the semi-analytical model . . . 63
3.3.4 Uncracked, linear-elastic behaviour . . . 63
3.3.5 Cracked, linear-elastic behaviour . . . 67
3.3.6 Cracked, elasto-plastic behaviour . . . 69
3.3.7 In uence of the kicker course . . . 72
3.3.8 Total displacements and rotation . . . 73
3.3.9 Activation of prestressing tendons . . . 74
3.4 MATLAB numerical model . . . 79
3.5 DIANA nite element model . . . 84
3.5.1 Introduction to numerical modelling of masonry . . . 85
3.5.2 Rankine-Hill model by Lourenço [55] . . . 86
3.5.3 Implementation as a user-supplied subroutine in DIANA . . . 92
3.5.4 Drawbacks and suggested improvements . . . 92
3.5.5 Consistent tangent for the apex . . . 94
3.5.6 e super-hyperbolic Rankine-type yield criterion . . . 95
3.5.7 A single-surface Rankine-Hill yield criterion . . . 97
3.5.8 Implementation of a substepping scheme . . . 103
3.5.9 Performance of the single-surface Rankine-Hill model . . . 105
3.6 Comparison of results . . . 110
3.6.1 FE model of UPT shear walls . . . 110
3.6.2 Material parameters . . . 111
3.6.3 Results for rectangular shear walls . . . 112
3.6.4 Results for T-shaped shear walls . . . 117
3.6.5 Conclusions . . . 118
4 Post-tensioned shear wall assemblies 121 4.1 Introduction to shear wall interaction . . . 121
4.2 Description of numerical study . . . 122
4.2.1 Building layout and shear wall con gurations . . . 122
4.2.2 FE model . . . 123
4.3 Results of linear-elastic analysis . . . 127
4.4 Results of non-linear analysis . . . 129
4.4.1 Load-displacement diagram . . . 129
4.4.2 Non-linear distribution of horizontal and vertical load . . . 129
4.4.3 Cracking and plasticity . . . 132
4.5 Discussion . . . 136
5 Design and construction guidelines 139 5.1 Design rules: state-of-the-art . . . 139
5.1.1 General design principles for prestressed masonry . . . 140
5.1.2 Intersecting walls as anges . . . 142
5.1.3 Load distribution among shear walls . . . 143
5.1.4 Permissible tendon stresses . . . 144
5.1.5 Introduction of prestress . . . 144
5.1.6 Prestress loss . . . 145
5.1.8 Combined axial compression, exure and shear . . . 152
5.1.9 Shear at the web- ange interface . . . 155
5.1.10 Serviceability Limit State (cracking and de ection) . . . 156
5.1.11 Construction guidelines, including corrosion and re protection . 157 5.2 Proposed design rules and guidelines . . . 159
5.2.1 General design principles . . . 159
5.2.2 Material properties of CASIEL masonry . . . 159
5.2.3 Effective ange width and shear wall interaction . . . 164
5.2.4 Permissible tendon stress . . . 164
5.2.5 Introduction of prestress . . . 164
5.2.6 Prestress loss . . . 166
5.2.7 Combined axial compression and bending . . . 167
5.2.8 Combined axial compression, exure and shear . . . 176
5.2.9 Shear at the web– ange interface . . . 180
5.2.10 Serviceability Limit State (SLS) . . . 182
5.3 Construction guidelines . . . 183
6 Conclusions and Recommendations 187 6.1 Creep, shrinkage and prestress loss . . . 187
6.2 Post-tensioned shear wall behaviour . . . 189
6.3 Post-tensioned shear wall assemblies . . . 190
6.4 Design rules and construction guidelines . . . 191
6.5 Outlook . . . 193
References 195 A Short-term material properties 205 A.1 CASIELs (masonry units) . . . 205
A.2 Mortar prisms . . . 212
A.3 CASIEL masonry . . . 214
A.4 Concrete . . . 216
A.5 Prestressing bars . . . 216
B Creep, shrinkage and prestress loss 219 B.1 Determination of moisture content of specimens . . . 219
B.2 Relation between weight loss and shrinkage . . . 221
B.3 Analytical derivation of aging coefficient . . . 221
B.5 Prestress loss: outdoor vs climate room . . . 235
C Post-tensioned shear wall behaviour 237
D Auxiliary components of the FE model 243
D.1 Gradients of the yield surface and plastic potential . . . 243 D.2 Jacobian for the return mapping . . . 245
Curriculum Vitae 253
BPT Bonded Post-Tensioning
C28/35 Strength class of concrete indicating cylindric compressive strength (28 MPa) and cube compressive strength (35 MPa)
CAD Computer Aided Design CASIEL CAlcium SIlicate ELement
CFRP Carbon Fibre Reinforced Polymers CoV Coefficient of Variance
CS44 Strength class of calcium silicate (CS) element indicating normalised unit compressive strength (44 MPa)
DeMec Demountable Mechanical strain gauge
DTI Direct Tension Indicator, a single use mechanical load cell that can be applied between the anchor nut and the anchor plate to indicate a speci ed level of tension in a prestressing bar
ESECMaSE Enhanced Safety and Efficient Construction of Masonry Structures in Europe ESG Electrical Strain Gauges
ETA European Technical Approval FE Finite Element
FEM Finite Element Method
FeP1100 Strength class of prestressing steel indicating characteristic ultimate tensile strength (1100 MPa)
GFRP Glass Fibre Reinforced Polymers GPM General Purpose Mortar
HE600B HEB denotes a European wide- ange steel beam, 600 indicates a height of the cross-section of 600 mm
KCM Kicker Course Mortar
LVDT Linear Variable Differential Transformer, used for measurement of relative deformations
PE PolyEthylene
PRESSS PREcast Seismic Structural Systems PT Post-Tensioning
RC Reinforced Concrete RH Relative Humidity SLS Serviceability Limit State TLM in-Layer Mortar ULS Ultimate Limit State
UPT Unbonded Post-Tensioning URM UnReinforced Masonry USSR User-Supplied SubRoutines V/S ratio Volume to exposed Surface ratio
Symbols that are used throughout this dissertation are listed here, in the following order:
𝑎 − 𝒂 − 𝐴 − 𝑨 − 𝑏 − 𝒃 − 𝐵 − 𝑩 − … − 𝛼 − 𝜶 − 𝛽 − 𝜷 − …. Bold symbols represent vectors or
matrices. In mathematical equations,(…)is used for scalars,{…}for vectors and[…]for matrices. An attempt was made to maintain the common meaning of symbols for speci c subjects, which resulted in a few ambiguous symbols. In case of ambiguity the Chapter or Section, in which the symbol is used, is indicated. When the same symbol is used for a variable and for its dimensionless or normalised equivalent,𝑎denotes the variable with real dimensions and𝑎̄denotes the dimensionless or normalised equivalent of the variable
𝑎.
𝑎 Auxiliary parameter for elliptic hardening, de ned in equation (3.3) 𝐴 In general: Cross-sectional area
𝐴 In Chapter 2: A constant in equation (2.2)
𝐴 In Appendix D: An auxiliary scalar, de ned in equation (D.4c) 𝐴ac Contact area between anchor plate and concrete
𝐴am Area of the masonry that is indirectly loaded by the prestressing load, which distributes through the anchor plate and oor slab
𝐴e Effective cross-sectional area 𝐴g Gross cross-sectional area
𝐴m Cross-sectional area of the masonry
𝐴mv Area of the part of the masonry cross-section that offers resistance to shear 𝐴n Net cross-sectional area
𝐴p Cross-sectional area of the prestressing tendons 𝐴pq Cross-sectional area of the 𝑞thprestressing tendon 𝐴wf Gross area of the web– ange interface
𝐴wfe Effective area of the web– ange interface
𝑨 Auxiliary matrix for recursive update of the elastoplastic consistent tangent with substepping, de ned in equation (3.109d)
𝑏 In Chapter 3: An auxiliary parameter for elliptic hardening, de ned in equation (3.3)
𝑏 In Chapter 5: ickness of the shear wall in code equations, corresponding to 𝑡w
𝑏c Width of the compressed ange, excluding 𝑡w
̄𝑏
c Dimensionless width of the compressed ange ̄𝑏c= 𝑏c 𝑙w
𝑏t Width of the ange in tension, excluding 𝑡w
̄𝑏
t Dimensionless width of the ange in tension ̄𝑏t = 𝑏t 𝑙w
𝐵 In Chapter 2: A constant in equation (2.3)
𝐵 In Appendix D: An auxiliary scalar, de ned in equation (D.6d) 𝑐 Length of the compressed zone in code equations, corresponding to 𝛽 𝑐 In equation (5.38): 𝑐 = 1.0 ≤ 0.5 + 𝜆v ≤ 1.5
𝐶 An auxiliary scalar, de ned in equation (D.11h)
𝐶1 A constant in equation (5.14), related to the longitudinal reinforcement ratio 𝐶2 A constant in equation (5.14), related to the aspect ratio of the shear wall 𝑑H Denominator of 𝑓H, de ned in equation (3.97c)
𝑑R Denominator of 𝑓R, de ned in equation (3.96c) 𝐷 In Chapter 2: A constant in equation (2.3)
𝐷 In Appendix D: An auxiliary scalar, de ned in equation (D.13h) 𝑫 Elasticity matrix
𝑫−1 Inverted elasticity matrix
𝑫ep Elastoplastic consistent tangent with respect to 𝝈
𝑫𝒙ep Elastoplastic consistent tangent with respect to {𝝈, ∆𝜿, ∆𝜆}T
𝑒 Eccentricity of the vertical axial load resultant in a horizontal section of the shear wall
̄
𝑒 Normalised eccentricity of the vertical axial load resultant in a horizontal section of the shear wall ̄𝑒 = 𝑒 𝑙w
𝑒pq Eccentricity of the 𝑞thprestressing tendon
𝑒p0 Initial eccentricity of the axial load due to prestress 𝐸 Young’s modulus
𝐸ctest Young’s modulus derived from compression tests on column-shaped specimens
𝐸kc Young’s modulus of the kicker course
𝐸kcm Young’s modulus of the kicker course mortar 𝐸m Young’s modulus of the masonry
𝐸m(𝜏) Young’s modulus of the masonry as a function of age at loading 𝜏 𝐸p Young’s modulus of the prestressing tendons
𝐸pq Young’s modulus of the 𝑞thprestressing tendon
𝐸x, 𝐸y, 𝐸z Young’s modulus in the respective direction for orthotropic (FE) models 𝐸α Linear-elastic spring stiffness of the 𝛼thKelvin group, see Figure 2.7 𝑓1 Rankine-type yield criterion [55], de ned by equation (3.74) 𝑓2 Hill-type yield criterion [55], de ned by equation (3.81)
𝑓b Normalised compressive strength of CASIELs according to NEN-EN 772-1 [77]
𝑓cd Concrete design compressive strength
𝑓b,dry Oven-dry compressive strength of CASIELs according to NEN-EN 772-1 [77] 𝑓ctest Compressive strength derived from compression tests on column-shaped
specimens
𝑓d Design compressive strength according to NEN-EN 1996-1-1 [75]
𝑓H Hill-type yield criterion (as a rational expression), de ned by equation (3.97a) 𝑓k Characteristic compressive strength according to NEN-EN 1996-1-1 [75] 𝑓m Mean compressive strength of the masonry
𝑓md Design compressive strength of the masonry 𝑓mk Characteristic compressive strength of the masonry 𝑓mp Mean compressive strength of mortar prisms 𝑓mpt Mean exural strength of mortar prisms
𝑓mx, 𝑓my Compressive strength in the respective direction for orthotropic (FE) models 𝑓p Tensile strength of the prestressing tendons
𝑓p0.1k Characteristic 0.1% proof stress of the prestressing tendons 𝑓pk Characteristic tensile strength of the prestressing tendons 𝑓pu Ultimate tensile strength of the prestressing tendons 𝑓py 0.2% proof stress of the prestressing tendons
𝑓R Superhyberbolic Rankine-type yield criterion (as a rational expression), de ned in equation (3.96a)
𝑓RH Superquadric single-surface Rankine-Hill yield criterion, de ned in equation (3.99)
𝑓t Masonry tensile strength
̄
𝑓t Dimensionless masonry tensile strength ̄𝑓t = 𝑓t 𝑓mk 𝑓ts Splitting tensile strength of the masonry units
𝑓tsd Design splitting tensile strength of the masonry units 𝑓tsk Characteristic splitting tensile strength of the masonry units
𝑓tx, 𝑓ty Tensile strength in the respective direction for orthotropic (FE) models 𝑓v0 Initial shear strength according to ACI 530-08 [4]
𝑓vd Design shear strength according to NEN-EN 1996-1-1 [75] 𝑓vk Characteristic shear strength according to NEN-EN 1996-1-1 [75] 𝑓vk0 Characteristic initial shear strength according to NEN-EN 1996-1-1 [75]
̄
𝑓vk0 Dimensionless masonry characteristic initial shear strength ̄𝑓vk0 = 𝑓vk0 𝑓mk 𝑓vk0,kc Characteristic initial shear strength of the kicker course
𝑓vlt Shear strength limit related to diagonal tension failure according to NEN-EN 1996-1-1 [75]
̄
𝑓vlt Dimensionless masonry shear strength limit related to diagonal tension failure ̄𝑓vlt = 𝑓vlt 𝑓mk
𝑓vvd Design shear strength of the web– ange interface based on interlocking of the masonry units
𝑓vvk Characteristic shear strength of the web– ange interface based on interlocking of the masonry units
𝑓xk1 Tensile strength of the masonry perpendicular to the bed joints 𝑓xk2 Tensile strength of the masonry parallel to the bed joints
𝐹∗ Factor in equation (5.38) related to the tensile strength of the masonry units 𝑔1 Plastic potential of the Rankine-type yield criterion 𝑓1, equal to 𝑓1with
𝛼 = 𝛼g= 1
𝑔H Plastic potential of 𝑓H, equal to 𝑓H
𝑔R Plastic potential of 𝑓R, equal to 𝑓Rwith 𝛼 = 𝛼g = 1 𝑔RH Plastic potential of 𝑓RH, equal to 𝑓RH with 𝛼 = 𝛼g = 1
𝐺fcx, 𝐺fcy Compressive fracture energy in the respective direction for orthotropic (FE) models
𝐺ftx, 𝐺fcy Tensile fracture energy in the respective direction for orthotropic (FE) models 𝐺xy, 𝐺yz, 𝐺xz Shear modulus in the respective plane for orthotropic (FE) models
ℎCASx Height of the 𝑥thCASIEL from the base of the wall, see Table 3.6 and Figures 3.4 and 3.5
ℎf ickness of the oor slab, see Table 3.6 and Figures 3.4 and 3.5
ℎH Plastic modulus for the Hill part of the single-surface Rankine-Hill yield criterion, de ned in equation (3.103)
ℎkc Height of the kicker course
ℎkcb Height of the kicker course block, see Table 3.6 and Figures 3.4 and 3.5 ℎkcj Height of the kicker course joint, see Table 3.6 and Figures 3.4 and 3.5 ℎR Plastic modulus for the Rankine part of the single-surface Rankine-Hill yield
criterion, de ned in equation (3.102) ℎu Height of the masonry unit (CASIEL)
ℎw Height of the shear wall (in Section 3.3 the height of the shear wall excluding the kicker course)
𝒉 Plastic moduli vector ℎR, ℎH T
𝐻0 Initial slope 𝜕 ̄𝜎ci 𝜅c 𝜕𝜅c of the hardening law ̄𝜎cai 𝜅c , used in equation (3.105)
𝑖 Iteration number in Section 3.3.10
𝑖 Horizontal segment number in Section 3.4
𝐼m Second moment of area of the (uncracked) masonry cross-section 𝐼n Second moment of area of the net masonry cross-section
𝑰 Identitity matrix
𝑗 Parameter in equation (5.1)
𝐽 (𝑡) Creep compliance 𝐽 (𝑡) = 1 𝐸0 (1 + 𝜑(𝑡))
𝐽 𝑡, 𝑡0 Creep compliance for creep caused by a load applied at 𝑡 = 𝑡0 𝑱 Jacobian for the return mapping
𝑱−1 Inverted Jacobian
𝑘 A factor describing the amount of non-linearity in a hardening stress–strain diagram, de ned in equation (3.2)
𝑘 Substep counter in Section 3.5.8 𝑘1 A constant in equation (5.2) 𝑘2 A constant in equation (5.2) 𝑘c Speci c creep 𝑘c= 𝜀cr 𝜎m0
𝑘c(𝑡) Speci c creep 𝑘c(𝑡) = 𝜀cr(𝑡) 𝜎m0
𝑘c0 Speci c creep at time 𝑡 = 0, only for curve tting of 95% upper bound, see Table 2.10
𝑘c 𝑡, 𝑡0 Speci c creep 𝑘c 𝑡, 𝑡0 = 𝜀cr 𝑡, 𝑡0 𝜎m0 for a load applied at 𝑡 = 𝑡0
𝑘E Empirical ratio of the masonry Young’s modulus to the masonry characteristic compressive strength 𝑘E= 𝐸m 𝑓mk
𝑘Ekc Empirical ratio of the Young’s modulus of the kicker course to the masonry characteristic compressive strength 𝑘Ekc = 𝐸kc 𝑓mk
𝑘Ep Empirical ratio of the Young’s modulus of the prestressing tendons to the tendon characteristic tensile strength 𝑘Ep = 𝐸p 𝑓pk
𝑘f e part of the total axial load that is carried by the ange
𝑘h Ratio of masonry wall height and prestressing tendon length between its anchorages 𝑘h = ℎw 𝑙p
𝑘i Part of the area of the web– ange interface that is intersected by interlocking units
𝐾 A constant in equation (3.1)
𝐾m Axial stiffness of the masonry 𝐾m= 𝐸m𝐴m 𝑙m (for a shear wall 𝑙m= ℎw) 𝐾p Axial stiffness of the prestressing tendons 𝐾p = 𝐸p𝐴p 𝑙p
𝑙c Length of the compressed zone based on a linear-elastic no-tension stress–strain characteristic
𝑙m Height of the masonry between the prestressing tendon anchorages 𝑙p Length of the prestressing tendons between their anchorages 𝑙pq Length of the 𝑞thprestressing tendon between its anchorages
∆𝑙pq Length change of the 𝑞thprestressing tendon between its anchorages
𝑙t Minimum spacing of the prestressing tendons, see Table 3.6 and Figures 3.4 and 3.5
𝑙u Length of the masonry unit (CASIEL) 𝑙w Length of the shear wall
𝑚i Auxiliary component for ̄𝜎ci 𝜅c , de ned in equation (3.83) 𝑚 In Section 3.4: iteration number
𝑚 In Section 3.5: Power (𝑚 = 2𝑝 ∈ ℕ) that modi es the superquadric single-surface Rankine-Hill criterion
̄
𝑚 Dimensionless moment ̄𝑚 = 𝑀 𝑓m𝐴m𝑙w
∆ ̄𝑚 Dimensionless moment increment due to activation of prestressing tendons ∆ ̄𝑚 = ∆𝑀 𝑓m𝐴m𝑙w
̄
𝑚act Dimensionless moment capacity including activation of the prestressing tendons
∆ ̄𝑚tot,rel[%] Percentage of increase of the moment capacity due to activation of the prestressing tendons
∆ ̄𝑚tot,rel,∞[%] Percentage of increase of the moment capacity due to activation of the prestressing tendons with respect to the moment capacity without activation based on a rectangular stress block
𝑀0 Bending moment due to eccentric prestress
∆𝑀 Counteracting bending moment due to activation of the prestressing tendons ∆𝑀max Maximum counteracting bending moment due to activation of the
prestressing tendons, as observed experimentally, see Table 3.7 𝑛 In Section 3.4: Load step number
𝑛 In Section 3.5: Power (𝑛 = 2𝑝 ∈ ℕ) that modi es the superhyperbolic Rankine-type yield criterion
𝑛H Numerator of 𝑓H, de ned in equation (3.97b) 𝑛i Number of segments in the cross-section 𝑛N Number of elements in the array 𝑵
𝑛R Numerator of 𝑓R, de ned in equation (3.96b) 𝑛q Number of prestressing tendons
𝑛κ Number of elements in the array 𝜿
̄
𝑛 Axial load ratio of the masonry ̄𝑛 = 𝑁 𝑓m𝐴m
∆ ̄𝑛 Axial load ratio increment due to activation of the prestressing tendons ∆ ̄𝑛 = ∆𝑃 𝑓m𝑡w𝑙w
̄
𝑛a Axial load ratio of a shear wall due to axial loads other than prestress
̄
𝑛ad Design axial load ratio of a shear wall due to axial loads other than prestress
̄
𝑛ak Characteristic axial load ratio of a shear wall due to axial loads other than prestress
̄
𝑛c Axial load ratio of a shear wall with anges, when the neutral axis is at the web– ange interface of the compressed ange
̄
𝑛p Axial load ratio of the prestressing tendons ̄𝑛p = 𝑃 𝐴p𝑓p
̄
𝑛pd Design axial load ratio of the prestressing tendons
̄
𝑛pe Axial load ratio of the prestressing tendons aer all losses have occurred (i.e. working prestress)
̄
𝑛pk Characteristic axial load ratio of the prestressing tendons
̄
𝑛t Axial load ratio of a shear wall with anges, when the neutral axis is at the web– ange interface of the ange in tension
𝑁 Axial load
𝑁i Axial load in segment 𝑖
𝑁v e compressive force acting normal to the shear surface 𝑁∗ e design axial load according to NZS 4230:2004 [80]
𝑵 An array that contains the segment axial loads 𝑁i for each segment 𝑖 𝑜 Overlap of the masonry units (CASIELs)
𝑃 Axial load due to prestress
∆𝑃 Increase of axial load due to activation of the prestressing tendons
∆𝑃c+s+r Time-dependent prestress loss due to creep and shrinkage of the masonry and relaxation of the prestressing tendons
𝑃d e compressive force in the masonry acting normal to the head joint 𝑃dl e axial dead load
𝑃e e effective prestress, i.e. aer all losses have occurred
∆𝑃e Immediate prestress loss due to elastic shortening of the masonry ∆𝑃max Maximum increase of axial load in the prestressing tendons, as observed
experimentally, see Table 3.7
∆𝑃q Increase of axial load in the 𝑞thprestressing tendon due to activation of the tendons
𝑃t Axial load due to prestress at transfer
𝑷 Projection matrix for the elastoplastic consistent tangent for a substepping scheme, de ned in equation (3.109a)
𝑷c Projection matrix for the Hill-type yield criterion, de ned in equation (3.82) 𝑷t Projection matrix for the Rankine-type yield criterion, de ned in
equation (3.75)
𝑞 Prestressing tendon number
𝑞 In Section 3.5.8: Total number of substeps in current loadstep 𝑄 First moment of area of the masonry cross-section
𝑸 Projection matrix for the strain soening hypothesis (Rankine-type yield criterion), de ned in equation (3.78)
𝑟(𝑡) Relaxation–time function
𝑅2 Coefficient of determination (R-squared) 𝑠 Standard deviation
𝑡 Time
𝑡c ickness of the compressed ange of the shear wall
̄𝑡
c Dimensionless thickness of the compressed ange ̄𝑡c = 𝑡c 𝑙w
𝑡i ickness of the 𝑖thsegment of the cross-section
𝑡t ickness of the ange on the tension side of the shear wall
̄𝑡
t Dimensionless thickness of the ange in tension ̄𝑡t = 𝑡t 𝑙w
̄𝑡
w Dimensionless thickness of the web ̄𝑡w= 𝑡w 𝑙w
𝑇 Temperature
∆𝑇 Change in temperature
𝑢c0 Vertical displacement of the top of the shear wall due to prestressing at the initial neutral axis
𝑢r Vertical displacement of the top of the shear wall at the compressed face of the wall
̄
𝑢r Dimensionless vertical displacement of the top of the shear wall at the compressed face of the wall ̄𝑢r = 𝑢r ℎw𝜀y
𝑣bm Basic nominal shear strength provided by the masonry according to NZS 4230:2004 [80]
𝑣g Maximum allowable total shear stress to avoid critical shear related failures according to NZS 4230:2004 [80]
𝑣m Contribution of the masonry to the total shear strength according to NZS 4230:2004 [80]
𝑣n Total shear strength according to NZS 4230:2004 [80]
𝑣p Contribution of the axial load to the total shear strength according to NZS 4230:2004 [80]
𝑣s Contribution of shear reinforcement to the total shear strength according to NZS 4230:2004 [80]
̄
𝑣 Dimensionless horizontal load ̄𝑣 = 𝑉 𝑓m𝑡w𝑙w
̄
𝑣bending Dimensionless horizontal load capacity for bending according to Kranzler [45]
̄
𝑣cr Dimensionless horizontal load at crack initiation at the base (decompression)
̄
𝑣friction Dimensionless horizontal load capacity for friction according to Kranzler [45]
̄
𝑣gaping Dimensionless horizontal load capacity for gaping according to Kranzler [45]
̄
𝑣tension Dimensionless horizontal load capacity for diagonal tension according to Kranzler [45]
̄
𝑣u Dimensionless horizontal load corresponding to 𝜀r = 𝜀uat the base of the wall
̄
𝑣u∞ Dimensionless horizontal load corresponding to 𝜀r → ∞ at the base of the wall
̄
𝑣y Dimensionless horizontal load corresponding to 𝜀r = 𝜀yat the base of the wall 𝑉 Horizontal load
𝑉cr Horizontal load at crack initiation at the base (decompression) 𝑉Ed Design shear load according to NEN-EN 1996-1-1 [75]
𝑉max Maximum horizontal load, i.e. horizontal load capacity, as observed experimentally, see Table 3.7
𝑉n Nominal shear capacity according to NZS 4230:2004 [80] 𝑉r Shear resistance according to CSA S304.1-04 [27]
𝑉Rd Design shear resistance according to NEN-EN 1996-1-1 [75]
𝑉sc/y Horizontal load at which shear cracking of the masonry (sc) or yield of the prestressing tendons (y) occurs, as observed experimentally, see Table 3.7 𝑉u Horizontal load corresponding to 𝜀r = 𝜀uat the base of the wall
𝑉y Horizontal load corresponding to 𝜀r = 𝜀yat the base of the wall 𝑉∗ Design shear force according to NZS 4230:2004 [80]
𝑊m Section modulus of the (uncracked) masonry cross-section
𝑥c0 Distance from the centroid of the cross-section to the compressed face of the wall
̄
𝑥c0 Normalised distance from the centroid of the cross-section to the compressed face of the wall
𝑥cc Distance from the centroid of the compressed part of the cross-section to the compressed face of the wall
̄
𝑥cc Normalised distance from the centroid of the compressed part of the cross-section to the compressed face of the wall
𝑥cg Distance from the compressed face of the shear wall to its center of gravity 𝑥i Distance from the centroid of the 𝑖thsegment to the compressed face of the
wall
𝑥pq Distance from the compressed face of the shear wall to the 𝑞thprestressing tendon
𝑧 Vertical coordinate from the top to the base of the shear wall, see Figure 3.19a 𝑧cr 𝑧-coordinate for the boundary between zone A and zone B in the shear wall
model, see Figure 3.19a
𝑧mp Distance from the centroid of the cross-section to the centroid of the prestressing tendons (corresponds to 𝑒p0)
𝑧u 𝑧-coordinate for the boundary between zone C and the kicker course in the shear wall model, see Figure 3.19a
𝑧y 𝑧-coordinate for the boundary between zone B and zone C in the shear wall model, see Figure 3.19a
𝛼 In Chapter 2: A counter for Kelvin groups 𝛼 In Section 3.2: A constant in equation (3.1)
𝛼 In Section 3.2: Proportionality factor in equation (3.3)
𝛼 In Section 3.5: Shear stress contribution to failure for the Rankine-type yield criterion
𝛼g Replaces 𝛼 in 𝑓1to obtain 𝑔1 𝛼g = 1
𝛼k Relative substep size ∆𝜺k ∆𝜺
𝛼m ermal expansion coefficient of the masonry
𝛼p ermal expansion coefficient of the prestressing tendons 𝛽 A constant in equation (3.1)
𝛽 In Sections 3.3 and 3.4: Length of the compressed zone
𝛽 In Section 3.5: A parameter that couples the normal stress values for biaxial compression (Hill-type yield criterion)
𝛽 In equation (5.24): A compressive strength enhancement factor for concentrated loads
𝛽cr Compressed zone length between zone A and zone B in the shear wall model, see Figure 3.19a
𝛽u Compressed zone length between zone C and the kicker course in the shear wall model, see Figure 3.19a
𝛽y Compressed zone length between zone B and zone C in the shear wall model, see Figure 3.19a
̄
𝛽 Normalised compressed zone length ̄𝛽 = 𝛽 𝑙w
̄
𝛽cr Normalised compressed zone length between zone A and zone B in the shear wall model, see Figure 3.19b
̄
𝛽u Normalised compressed zone length between zone C and the kicker course in the shear wall model, see Figure 3.19b
̄
𝛽y Normalised compressed zone length between zone B and zone C in the shear wall model, see Figure 3.19b
𝛾 Shear stress contribution to failure for the Hill-type yield criterion 𝛾G,inf Partial load factor for permanent loads, which are favourable for the
mechanical behaviour under consideration
𝛾G,sup Partial load factor for permanent loads, which are infavourable for the mechanical behaviour under consideration
𝛾M Partial material factor for masonry
𝛾P Partial load factor for the prestressing load
𝛾P,sup Partial load factor for the prestressing load, which is infavourable for the mechanical behaviour under consideration
𝛾Q Partial load factor for variable loads 𝛾S Partial material factor for steel
̄
𝛿 Dimensionless horizontal displacement at the top of the shear wall
̄
𝛿 = 𝛿𝑙w ℎw2𝜀y
𝛿max Maximum horizontal displacement as observed experimentally 𝛿[%] Prestress loss in %
𝛿dm[%] Prestress loss in % as measured by demountable mechanical (DeMec) strain gauge at the (masonry) specimen surface
𝛿esg[%] Prestress loss in % as measured by electrical strain gauges (ESG) at the prestressing bar surface
𝛿∆T[%] Prestress loss in % due to a change in temperature 𝜀cr 𝑡, 𝑡0 Creep strain at time 𝑡 caused by a load applied at 𝑡 = 𝑡0 𝜀ctest Strain corresponding to 𝑓ctest
𝜀dm(𝑡) Strain in the (masonry) specimen at time 𝑡 as measured by demountable mechanical (DeMec) strain gauge at the specimen surface
𝜀dm0 Instantaneous strain in the (masonry) specimen due to prestressing as measured by demountable mechanical (DeMec) strain gauge at the specimen surface
𝜀esg(𝑡) Strain in the prestressing bars at time 𝑡 as measured by electrical strain gauges (ESG) at the prestressing bar surface
𝜀esg0 Instantaneous strain in the prestressing bars due to prestressing as measured by electrical strain gauges (ESG) at the prestressing bar surface
𝜀i Average strain in the 𝑖thsegment of the cross-section 𝜀m Compressive strain in the masonry corresponding to 𝑓m
𝜀m0 Instantaneous linear-elastic strain in the masonry due to prestressing 𝜀shr(𝑡) Shrinkage–time function
𝜀shr0 Shrinkage at time 𝑡 = 0, only for curve tting of 95% upper bound, see Table 2.9
𝜀shr∞ Ultimate shrinkage
𝜀p Strain in the prestressing tendons
𝜀pq0 Strain in the 𝑞thprestressing tendon due to prestressing
∆𝜀pq Strain change in the 𝑞thprestressing tendon due to activation of the tendons 𝜀pu Strain in the prestressing tendon corresponding to 𝑓pu
𝜀py Strain in the prestressing tendon corresponding to 𝑓py
𝜀r Vertical strain in the masonry at the compressed face of the shear wall
̄
𝜀r Normalised vertical strain in the masonry at the compressed face of the shear wall ̄𝜀r = 𝜀r 𝜀y
𝜀y Strain in the masonry at initiation of plasticity (yield) based on a bilinear stress–strain diagram 𝜀y = 𝑓m 𝐸m
𝜀u Ultimate strain in the masonry based on a bilinear stress–strain diagram
̄
𝜀u Ductility ratio ̄𝜀u = 𝜀u 𝜀y 𝜺p Plastic strain vector
∆𝜺p Plastic strain increment vector
𝜖∆ ̄m Convergence norm for ∆ ̄𝑚, de ned in equation (3.62) 𝜖∆ ̄n Convergence norm for ∆ ̄𝑛, de ned in equation (3.62) 𝜁 Normalised 𝑧-coordinate from the top 𝜁 = 𝑧 ℎw
𝜁cr 𝜁-coordinate for the boundary between zone A and zone B in the shear wall model, see Figure 3.19b
𝜁u 𝜁-coordinate for the boundary between zone C and the kicker course in the shear wall model, see Figure 3.19b
𝜁y 𝜁-coordinate for the boundary between zone B and zone C in the shear wall model, see Figure 3.19b
𝜂 Ratio of kicker course height to wall height 𝜂 = ℎkc ℎw, see Figure 3.19b 𝜂α Dashpot viscosity of the 𝛼thKelvin group, see Figure 2.7
𝜼 Back stress vector 𝜎̄tx 𝜅t , ̄𝜎ty 𝜅t , 𝟎 T
𝜃 Average squared eccentricity of prestressing tendons 𝜃 = 1 𝑛q ∑𝑛1q 𝑒pq 𝑙w 2 𝜅 In Sections 3.3 and 3.4: Curvature
̄
𝜅 Dimensionless curvature ̄𝜅 = 𝜅𝑙w 𝜀y
𝜅c In Section 3.5: Equivalent plastic strain for the Hill-type yield criterion ∆𝜅c Equivalent plastic strain increment (Hill-type yield criterion)
𝜅mi Equivalent plastic strain corresponding to ̄𝜎mi, see Figure 3.28b 𝜅p Equivalent plastic strain corresponding to ̄𝜎pi, see Figure 3.28b
𝜅t In Section 3.5: Equivalent plastic strain for the Rankine-type yield criterion ∆𝜅t Equivalent plastic strain increment (Rankine-type yield criterion)
𝜿 In Section 3.4: An array containing all elements 𝜅i 𝜿 In Section 3.5: Equivalent plastic strain vector 𝜅t, 𝜅c T
∆𝜿 Equivalent plastic strain increment vector ∆𝜅t, ∆𝜅c T
𝜆 Aspect ratio of the shear wall 𝜆 = ℎw 𝑙w
𝜆v Shear slenderness according to Kranzler [45] 𝜆v = 𝜓ℎw 𝑙w
∆𝜆 Incremental plastic multiplier for the superquadric single-surface Rankine-Hill yield criterion
∆𝜆c Incremental plastic multiplier for the Hill-type yield criterion ∆𝜆t Incremental plastic multiplier for the Rankine-type yield criterion 𝜇 In general: Coefficient of friction
𝜇 In Section 3.5: Distance from the vertex of the (super)hyperbola to either asymptote of the (super)hyperbolic approximation of the Rankine-type yield criterion, see Figure 3.30a
𝜇 In equation (5.2): axial load ratio of the prestressing tendons at transfer 𝜇kc Coefficient of friction of the kicker course
𝜇red Reduced value of the coefficient of friction according to NEN-EN 1996-1-1 [75]
𝜇tol Ratio of residual and initial tensile strength in the weakest direction 𝜈 Poisson factor
𝜈xy, 𝜈yz, 𝜈xz Poisson factor in the respective plane for orthotropic (FE) models 𝜉x, 𝜉y Reduced stress in the respective direction
𝝃 Reduced stress vector 𝝃 = 𝝈 − 𝜼
𝝅 Projection vector for the Rankine-type yield criterion 𝝅 = {1, 1, 𝟎}T
𝜌 Ratio of the axial stiffness of the prestressing tendons 𝐾p = 𝐸p𝐴p 𝑙p and the axial stiffness of the masonry 𝐾m = 𝐸m𝐴m 𝑙m (for a shear wall 𝑙m = ℎw) 𝜌1000 Relaxation loss in % aer 1000 h for at an average temperature of 20∘C, see
equation (5.2)
𝜌dry Dry density of CASIELs according to NEN-EN 772-10 [78] 𝜎i Average stress in the 𝑖thsegment of the cross-section 𝜎m(𝑡) Stress in the masonry at time 𝑡
∆𝜎m(𝑡) Stress decrement in the masonry at the centroid of the prestressing tendon(s) due to prestressing, in equation (5.1)
𝜎m0 Instananeous stress in the masonry due to prestressing
𝜎m∞ Stress in the masonry aer all losses have occurred (i.e. working prestress) 𝜎pe Effective stress in the prestressing tendon(s), i.e. aer losses have occured,
used in equations (5.6)–(5.9)
𝜎pq0 Stress in the 𝑞thprestressing tendon due to prestressing
∆𝜎pq Stress change in the 𝑞thprestressing tendon due to activation of the tendons ∆𝜎pr Loss of stress due to relaxation of the prestressing tendons, see equation (5.2) 𝜎pt Stress in the prestressing tendons at transfer
𝜎pu Ultimate stress in the prestressing tendon(s), i.e. including activation of the tendons, used in equations (5.6)–(5.9)
𝜎p∞ Stress in the prestressing tendons aer all losses have occurred (i.e. working prestress)
𝜎x, 𝜎y, 𝜎z Stress in the respective direction
̄
𝜎 In Section 3.3: Normalised stress ̄𝜎 = 𝜎 𝑓m
̄
𝜎c Measure for equivalent stress in both directions (𝑥 and 𝑦), equal to 𝑑H, de ned in equation (3.85)
̄
𝜎ci 𝜅c Equivalent stress for the Hill-type yield criterion in direction 𝑖, see Figure 3.28b
̄
𝜎cai 𝜅c Parabolic or elliptic hardening branch of ̄𝜎ci 𝜅c , de ned in equations (3.83) and (3.105) respectively
̄
𝜎cbi 𝜅c Parabolic soening branch of ̄𝜎ci 𝜅c , de ned in equation (3.83b)
̄
𝜎cci 𝜅c Exponential soening branch of ̄𝜎ci 𝜅c , de ned in equation (3.83c)
̄
𝜎ii Equivalent stress corresponding to proportionality limit for ̄𝜎ci 𝜅c , see Figure 3.28b
̄
𝜎mi Equivalent stress corresponding to the boundary between parabolic and exponential soening for ̄𝜎ci 𝜅c , see Figure 3.28b
̄
𝜎pi Equivalent stress corresponding to the peak compressive stress 𝑓mifor
̄
𝜎ci 𝜅c , see Figure 3.28b
̄
𝜎ri Equivalent stress corresponding to the residual compressive strength for
̄
𝜎ci 𝜅c , see Figure 3.28b
̄
𝜎ti 𝜅t Equivalent stress for the Rankine-type yield criterion in direction 𝑖, see Figure 3.28a and equation (3.76)
𝝈 Stress vector
𝝈n Stress vector at the beginning of the current step 𝝈n+1 Updated stress vector at the end of the current step 𝝈trial Trial stress 𝝈n+ 𝑫∆𝜺
𝜏cu Pure shear stress for the Hill-type yield criterion (for 𝜎x = 𝜎y = 0) 𝜏tu Pure shear stress for the Rankine-type yield criterion (for 𝜎x = 𝜎y = 0)
𝜏wf Average shear stress in the web– ange interface, see equation (5.40) 𝜏xy, 𝜏yz, 𝜏xz Shear stress in the respective plane
𝜑 In Sections 3.3 and 3.4: Total angle of rotation at the top of the shear wall 𝜑0 Angle of rotation at the top of the shear wall due to eccentric prestressing
̄
𝜑bending Angle of rotation at the top of the shear wall due to bending about the initial neutral axis
̄
𝜑bending Dimensionless angle of rotation at the top of the shear wall due to bending about the initial neutral axis
𝜑uplift Angle of rotation at the top of the shear wall due to upli of the shear wall
̄
𝜑uplift Dimensionless angle of rotation at the top of the shear wall due to upli of the shear wall
𝜑(𝑡) In Chapter 2: Creep coefficient as a function of time 𝜑(𝑡) = 𝜀cr(𝑡) 𝜀m0 𝜑 𝑡, 𝑡0 Creep coefficient as a function of time 𝑡, where 𝑡0is the time at application of
the load that initiated creep
𝜑(𝑡, 𝜏) Creep coefficient as a function of time 𝑡, where 𝜏 is the age at loading 𝛷 Strength reduction factor in NZS 4230:2004 [80]
𝛷m Strength reduction factor in CSA S304.1-04 [27] 𝜒(𝑡) In Chapter 2: Aging coefficient as a function of time 𝑡
𝜒(𝑡, 𝜏) Aging coefficient as a function of time 𝑡, where 𝜏 is the age at loading [37] 𝜒(𝜁) In Section 3.3: Ratio of the distance from the axial load resultant to the
compressed face of the wall to the compressed zone length, see Figure 3.21 𝜓 A constant for consideration of boundary conditions in determination of the
shear slenderness, 𝜓 = 0.5 for a xed– xed shear wall and 𝜓 = 1.0 for a cantilevered ( xed–free) shear wall
𝜓(𝜁) In Section 3.3: Ratio of the area of the bilinear stress block to the area of the enclosing rectangle, see Figure 3.21
Introduction
In this chapter, the research program of this dissertation is introduced. Motivation and objectives are described in Sections 1.1 and 1.2 respectively. e research signi cance is underlined in Section 1.3, taking into account both the scienti c contribution and util-isation aspects. Methodology of the research is presented in Section 1.4 and the outline of this dissertation is illustrated in Section 1.5.
1.1 Motivation of the research program
Research should be driven by social and scienti c needs. Nowadays the urgent need for skilled workers and the scarcity of raw materials and cheap energy sources have a negative impact on the construction industry. New construction techniques have to be developed in order to cope with these problems. In this dissertation a new technique is introduced, which can efficiently provide the overall stability of CASIEL masonry buildings by post-tensioning CASIEL masonry shear walls.
CASIEL masonry consists of prefabricated CAlcium SIlicate ELements (CASIELs) that are bonded by thin-layer mortar (TLM). A typical CASIEL has a length of 900 or 1000 mm, a height between 540 and 650 mm and a thickness ranging from 100 to 300 mm, while the TLM joints have a thickness of only 2 to 3 mm. CASIELs are categorised by their normalised compressive strength, which ranges from 12 to 44 MPa. With a speci c mass between 1725 and 2350 kg/m3, a CASIEL can weigh up to 350 kg. e construction of a CASIEL masonry wall starts with a kicker course, which consists of a general purpose
mortar (GPM) joint of 10 to 50 mm and one course of calcium silicate blocks with a height of approximately 100 mm. e kicker course is necessary to create a smooth and level surface as a starting point for the CASIELs, and to achieve the desired storey height. More information on CASIEL masonry is given by e.g. Berkers [14], Vermeltfoort [104]. e Dutch calcium silicate industry introduced CASIELs in the 1980s, as an alternative for bricks and blocks, and developed a complete system for the efficient construction of CASIEL masonry walls [14]. Aer its introduction, CASIEL masonry construction be-came popular in the Netherlands, with a market share of 60% in 1997 for housing, com-mercial and industrial buildings [88]. e use of small cranes to li the heavy CASIELs to their position in the wall dramatically reduced construction time and labour costs on site compared to traditional bricklaying. Construction waste on site was also reduced, be-cause the CASIELs were cut to size by the manufacturer based on architectural drawings and delivered at the construction site in packages per wall.
Due to its excellent compressive strength, CASIEL masonry is especially suitable for ap-plication in load-bearing walls in low- to medium-rise buildings. If load-bearing walls are constructed with CASIEL masonry, it is convenient to construct the shear walls, which have to ensure lateral stability of the building, with CASIEL masonry as well. For slender unreinforced CASIEL masonry shear walls, the overturning moment due to lateral load-ing is most critical. Because the exural strength of CASIEL masonry is quite modest, es-pecially at the wall– oor interface, the lateral load capacity is governed by the active axial load on the shear wall. For axial forces smaller than half the axial load capacity for centric compressive loading (axial load ratio< 0.5), the bending moment capacity increases with increasing axial load, see Figure 1.1, which depends on the distribution of the building weight. As the building layout is mainly determined by architectural requirements, it is oen not possible to guarantee sufficient axial force in a CASIEL masonry shear wall to achieve the required horizontal load capacity. In this case, the horizontal load capacity of a CASIEL masonry shear wall can be increased by applying vertical prestress.
Since the bene cial effect of prestressing is limited to half the axial load capacity for centric compressive loading (axial load ratio < 0.5), as shown in Figure 1.1, a relatively high compressive strength of the masonry is needed. In the last decade, new production techniques and the use of new aggregates have resulted in CASIELs with a compressive strength of 36 MPa and above. Combined with TLM, the resulting compressive strength of the masonry is comparable to that of medium-strength concrete. anks to this de-velopment, prestressing of CASIEL masonry shear walls in multi-storey buildings has become feasible. In this context, the research described in this dissertation considers CASIELs with a compressive strength exceeding 36 MPa, from two Dutch manufactur-ers: Calduran (C) and Silka (S).
... Pres tres sing . 0 . 0.5 . 1 .
Axial load ratio . B en din g m om en t ca p aci ty
Figure 1.1: Relation between axial load ratio and bending moment capacity for an unreinforced
shear wall and illustration of the bene cial effect of prestressing
e prestressing of structures may generally be achieved by two different construction techniques: pre-tensioning and post-tensioning. e rst method is used for prefabric-ated structural elements such as oor slabs, while the latter method is appropriate for prestressing on site. In the research described in this dissertation, only post-tensioning was considered. Different materials may be used for the post-tensioning of CASIEL ma-sonry, e.g. high-strength steel, glass bre reinforced polymers (GFRP) or carbon bre reinforced polymers (CFRP). e research described in this dissertation was focused on high-strength prestressing steel, which is used for the majority of post-tensioning systems available on the market. A post-tensioning system consists of prestressing tendons (bars or strands), anchorages, anchorage plates and some kind of corrosion protection.
A distinction can be made between unbonded tensioning (UPT) and bonded post-tensioning (BPT). In the case of BPT masonry, prestressing tendons in cavities in the ma-sonry are anchored and prestressed, aer which bond between mama-sonry and prestressing tendons is achieved by grouting the cavities. In the case of UPT masonry, prestressing tendons are either applied in cavities in the masonry or externally, and bond between masonry and prestressing tendons is not obtained. BPT and UPT shear walls exhibit fundamentally different mechanical behaviour. While the mechanical behaviour of BPT shear walls is based on strain compatibility of the shear wall material and the prestressing tendons, the mechanical behaviour of UPT shear walls relies on global compatibility of wall and tendon deformations between the anchorages. BPT shear walls generally have a higher stiffness aer cracking than UPT shear walls [47], but stiffness was not the primary concern of the investigation described in this dissertation. Moreover, grouting is an ad-ditional, time-consuming and expensive operation, which contains the risk of grout leak-age. For these reasons, the research described in this dissertation was con ned to UPT CASIEL masonry.
