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University of Calgary
PRISM: University of Calgary's Digital Repository
Graduate Studies The Vault: Electronic Theses and Dissertations
2017
Reliability Assessment of Historical Masonry
Structures
Seyedain Boroujeni, Setare
Seyedain Boroujeni, S. (2017). Reliability Assessment of Historical Masonry Structures
(Unpublished master's thesis). University of Calgary, Calgary, AB. doi:10.11575/PRISM/27613
http://hdl.handle.net/11023/3855
master thesis
University of Calgary graduate students retain copyright ownership and moral rights for their
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UNIVERSITY OF CALGARY
Reliability Assessment of Historical Masonry Structures
by
Setare Seyedain Boroujeni
A THESIS
SUBMITTED TO THE FACULTY OF GRADUATE STUDIES
IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE
DEGREE OF MASTER OF SCIENCE
GRADUATE PROGRAM IN CIVIL ENGINEERING
CALGARY, ALBERTA
MAY, 2017
© Setare Seyedain Boroujeni 2017
ii
Abstract
Despite the high level of vulnerability of unreinforced masonry structures under applied loads and
the importance of their reliability evaluation, there is no formal methodology to assess the
reliability of historic masonry structures. To develop an appropriate methodology, estimations of
probabilistic models of structural resistance and load effects are required to formulate a limit state
function. The stochastic characteristics of materials play key roles in the determination of
probabilistic models of structural resistance. Therefore, in current study, methodologies for
estimating the statistical characteristics of historic masonry materials through non-destructive tests
are described. Best fit probabilistic models for load effects are also presented. Target reliability
index and different approaches for calculating suitable targets for historic structures are described
as well. Evaluation of the reliability level of historical structures through the recommended
procedure would lead to more realistic and accurate levels of reliability estimation without
requiring degradation of the historic structure.
iii
Acknowledgements
First and foremost, I owe my sincere gratitude to Dr. Nigel Shrive, my supervisor, for his
continuous and unconditional support and encouragement during my master program. He has
always been a constant source of inspiration. This work would not have been possible without his
tireless guidance, patience, scholarly inputs, and insightful comments.
My parents, my mentors Mr. Fakhroddin Seyedain and Mrs. Ahraf Seyedain, I cannot thank you
enough for your continuous love, support and encouragement throughout every single step of my
academic and personal life.
I would like to thank my husband, Danial Arab who has always loved me unconditionally and
whose good example has taught me to work hard for the things that I aspire to achieve.
Last but not least, this research study was carried out at the Civil Engineering Department of the
University of Calgary with funding provided by the Natural Science and Engineering Research
Council (NSERC) and Canadian Masonry Design Center, whose support is appreciatively
acknowledged.
iv
Dedication
This thesis is dedicated with respect and love to:
My wonderful parents who never stop sacrificing themselves for us,
My love and my life, Danial who has made the happiest moments of my life,
My lovely brothers, Sahand and Sepehr who never left their little sister side,
I could not have done it without your faith, support and constant encouragement.
v
Table of Content
Abstract ........................................................................................................................................... ii
Acknowledgements ........................................................................................................................ iii
Dedication ...................................................................................................................................... iv
Table of Content ............................................................................................................................. v
List of Tables ............................................................................................................................... viii
List of Figures ................................................................................................................................. x
List of Symbols .............................................................................................................................. xi
Chapter I: Overview ........................................................................................................................ 1
1.1 Introduction ...................................................................................................................... 1
1.2 Motivation ........................................................................................................................ 2
1.3 Contributions .................................................................................................................... 3
1.4 Thesis Summery ............................................................................................................... 5
Chapter II: Previous Research Work .............................................................................................. 8
2.1 Introduction ...................................................................................................................... 8
2.2 Literature Review ............................................................................................................. 8
Chapter III: Structural Reliability Analysis .................................................................................. 15
3.1 Introduction .................................................................................................................... 15
3.1.1 Uncertainties ........................................................................................................... 15
3.1.2 Heritage ................................................................................................................... 17
3.1.3 Past Performance .................................................................................................... 17
3.1.4 Economics ............................................................................................................... 17
3.1.5 Change in Design Practice ...................................................................................... 18
3.1.6 Service Life ............................................................................................................. 18
3.2 Limit States .................................................................................................................... 18
3.3 Limit State Function ....................................................................................................... 19
3.4 Basic Theory of Reliability Assessment ........................................................................ 20
3.5 Structural Reliability Techniques ................................................................................... 26
3.5.1 Asymptotic Techniques .......................................................................................... 26
First Order Reliability Method (FORM) ......................................................... 27
Second Order Reliability Method (SORM) ..................................................... 30
First Order/Second Order Reliability in Original Space (FOROS/SOROS) ... 30
vi
3.5.2 Simulation Based Techniques ................................................................................. 31
3.5.3 Response Surface Method....................................................................................... 34
3.5.4 Artificial Neural Networks (ANNs)........................................................................ 34
3.6 Reliability Analysis and Software .................................................................................. 34
3.7 Deterioration................................................................................................................... 35
3.7.1 Modeling of Deterioration ...................................................................................... 37
3.8 Bayes’ Theorem ............................................................................................................. 38
3.9 Framework for Reliability Assessment .......................................................................... 39
3.10 Reliability Assessment under Seismic Loading ......................................................... 41
Chapter IV: Stochastic Modeling of Applied Load ...................................................................... 44
4.1 Introduction .................................................................................................................... 44
4.2 Dead Load ...................................................................................................................... 45
4.3 Live Load ....................................................................................................................... 47
4.3.1 Total Live Load....................................................................................................... 49
4.3.2 Point-in-time Live Load .......................................................................................... 55
4.3.3 Distribution ............................................................................................................. 55
4.4 Wind Load ...................................................................................................................... 57
4.4.1 Point-in-time Wind Load ........................................................................................ 63
4.4.2 Distribution ............................................................................................................. 64
4.5 Snow Load...................................................................................................................... 65
4.5.1 Ground Snow Load ................................................................................................. 66
4.5.2 Snow Related Loads ............................................................................................... 73
4.5.3 Point-in-time Snow Load ........................................................................................ 74
4.5.4 Distribution ............................................................................................................. 74
Chapter V: Stochastic Modeling of Historic Masonry Materials ................................................. 76
5.1 Introduction .................................................................................................................... 76
5.2 Evaluation of Structural Performance ............................................................................ 77
5.3 Modulus of Elasticity ..................................................................................................... 83
5.4 Compressive Strength .................................................................................................... 84
5.5 Shear Strength ................................................................................................................ 98
5.6 Tensile Strength............................................................................................................ 100
5.7 Cohesion and Friction Coefficient ............................................................................... 104
vii
Chapter VI: Reliability Assessment Methodology ..................................................................... 106
6.1 Introduction .................................................................................................................. 106
6.2 Reliability Assessment Procedure ................................................................................ 106
6.3 Limitations ................................................................................................................... 112
Chapter VII: Conclusion ............................................................................................................. 114
7.1 Thesis Contributions .................................................................................................... 114
7.2 Future Work ................................................................................................................. 116
References ................................................................................................................................... 118
Appendix ..................................................................................................................................... 132
viii
List of Tables
Table 3-1 Influential factors on the target probability of failure (Schueremans, 2001) 24
Table 3-2 Target reliability and consequences class according to GruSiBau (1981) 25
Table 3-3 Tentative target reliability values 𝛽𝑇 (𝑃𝑓𝑇) (Diamantidis, 1999, 2001) 25
Table 3-4 Required target reliability according to JCSS (2001) for a 50-year observation period
26
Table 3-5 Integration of software for structural reliability analysis 35
Table 3-6 Deterioration mechanisms 36
Table 3-7 Deterioration modeling (Maes et al., 1999) 37
Table 3-8 Degradation functions (Mori et al., 1993) 37
Table 4-1 Summary of statistical parameters for dead load 47
Table 4-2 Parameters related to probabilistic modeling of live load (JCSS, 2001) 52
Table 4-3 Statistical parameters for live load recommended by Bartlett et al. (2003) 54
Table 4-4 Summary of recommended stochastic parameters provided by different authors 54
Table 4-5 Stochastic characteristics of point-in-time live load 56
Table 4-6 Summary of recommendations for distribution type 55
Table 4-7 Stochastic characteristics for maximum wind velocity in 50-year (Bartlett et al., 2003)
59
Table 4-8 Summery of statistical parameters transformation factors reported by different authors
60
Table 4-9 Statistical characteristics of variables being influential on wind load (JCSS, 2001) 63
Table 4-10 Statistical parameters of maximum 3-hour wind velocity in Canada 64
Table 4-11 Summery of the best fit distribution for parameters involved in wind load estimation
65
Table 4-12 Characteristics of the annual maximum depth in Canada (Bartlett et al., 2003) 67
Table 4-13 Characteristics of the 50-year maximum depth in Canada (Bartlett et al., 2003) 67
Table 4-14 Statistical characteristics of snow density (Kariyawasam et al., 1997) 68
Table 4-15 Statistical characteristics of transformation factor (Taylor & Allen, 2000) 70
Table 4-16 The exposure coefficient (𝐶𝑒) and shape factor (ƞ𝑎) (JCSS, 2001) 71
Table 4-17 Summary of stochastic characteristics of snow load variables (JCSS, 2001) 73
Table 4-18 Statistical characteristics of point-in-time snow (Bartlett et al. 2003) 74
ix
Table 4-19 Summery of recommended distributions for the snow load random variables 75
Table 5-1 Nondestructive testing techniques and in-situ evaluation approaches 79
Table 5-2 Summary of standards and codes including various non-destructive methods 81
Table 5-3 Classification of testing techniques appropriate for evaluation of masonry (Harvey JR.
& Schuller, 2010) 82
Table 5-4 Values of 𝑐1 (Schubert, 2010) 83
Table 5-5 Masonry compressive strength determination from the strengths of its components 86
Table 5-6 Statistical characteristics of 𝑌 involved in probablistic model of compressive strength
JCSS (2011) 86
Table 5-7 Summary of relations between 𝐸 and 𝑓𝑚′ found in literatures 93
Table 5-8 Values of the CoV of compressive strength reported by different researchers 95
Table 5-9 Recommended distributions found in the literature 95
Table 5-10 Values of compressive strength found in the literature (TLM is Thin-Layer Mortar) 96
Table 5-11 Flexural tensile strength of masonry (Schubert, 2010) 101
Table 5-12 Statistical characteristics of 𝑌 involved in probablistic model of tensile strength
(JCSS, 2011) 103
Table 5-13 Statistical characteristics of 𝑌 involved in probablistic model of cohesion
(JCSS, 2011) 104
Table 5-14 Statistical characteristics of 𝑌 involved in probablistic model of friction coefficient
(JCSS, 2011) 105
Table A-1 Probabilistic model of dead load 136
Table A-2 Probabilistic model of live load 137
Table A-3 Probabilistic model of wind load 138
Table A-4 Considered load combination 138
Table A-5 Probabilistic model of modulus of elasticity 139
Table A-6 Probabilistic model of compressive strength 139
x
List of Figures
Figure 3-1 Flowchart for general evaluation procedure of existing structures. ............................ 16
Figure 3-2 Load effect and resistance probability density functions. ........................................... 22
Figure 3-3 Limit state function and probability of failure. ........................................................... 28
Figure 3-4 Standard normal space related to FORM and SORM. ................................................ 33
Figure 3-5 FOROS illustration...................................................................................................... 33
Figure 3-6 Example of the updating compressive strength of a masonry unit using Bayes’
theorem (Glowienka, 2007) .......................................................................................................... 38
Figure 3-7 Reliability assessment framework............................................................................... 40
Figure 4-1 Representation of live load. ......................................................................................... 48
Figure 4-2 Redistribution of snow load applied on a duopitch roof (JCSS, 2001). ...................... 71
Figure 4-3 𝐶𝑟𝑜 as a function of the roof angle. ............................................................................. 72
Figure 5-1 Ultrasonic Pulse Method ............................................................................................. 84
Figure 5-2 Flatjack technique ....................................................................................................... 90
Figure 5-3 Schematic of flatjack technique. ................................................................................. 91
Figure 5-4 Distribution of the coefficient k for clay and concrete. .............................................. 92
Figure 5-5 Cumulative probability of coefficient k. ..................................................................... 94
Figure 6-1 Flowchart for reliability assessment of structures. .................................................... 110
Figure 6-2 Recommended approaches for estimation of the probabilistic models of historic
masonry materials ....................................................................................................................... 111
Figure A-1 Clay brick masonry shear wall ................................................................................. 132
Figure A-2 General compression failure ..................................................................................... 133
Figure A-3 Flexural failure ......................................................................................................... 134
Figure A-4 Sliding failure .......................................................................................................... 134
Figure A-5 Diagonal shear failure .............................................................................................. 135
xi
List of Symbols
Symbol Definition
Chapter III
𝐿
Load effect
𝑅 Structural resistance
𝑓 Probability density function
𝐹 Cumulative distribution function
𝑃𝑓 Probability of failure
𝑋 Random variable
𝜃 Limit state model uncertainty
CoV Coefficient of Variation
𝑔(𝑥) Limit state function
β Reliability index
𝛽𝑇 Target reliability index
𝑝𝑓𝑇 Target failure probability
𝑡𝐿 Preset reference period
𝑅𝑓 Probability of survival
A Analysis model uncertainty
A Effective area
A Desired peak acceleration
𝐴𝐼 Influence area
𝐴𝑐 Activity factor
𝐴0 Reference area
A1 Independent area 1
A2 Independent area 2
Z Safety margin
𝑚 Mean value
𝜎 Standard deviation
ф Cumulative distribution function of the variable with a standard normal
distribution
∅ Probability density function of a standard normal variable
𝛼 Sensitivity factor
𝑡𝐿 Residual service life
𝑛𝑝 Number of endangered lives
𝑆𝑐 Social criterion factor
W Warning factor
𝐶𝑓 Cost factor
𝑥𝑖∗ Design point on the limit state function
xii
𝑥′∗ Vector of the transformed random variables involved in design point
∇𝑙(𝑥∗) Gradient vector of the log-likelihood
𝑃(𝑥∗) Projection matrix
𝑃∗(𝑥∗) Density matrix minus the projection matrix
𝐻𝑙(𝑥∗) Hessian matrix of likelihood
𝑇 Transpose and matrix determinant
det Transpose and matrix determinant
𝑋 Vector of basic random variables
𝑁 Total number of samples
𝑁𝑓 Total number of failures
𝑓𝑑𝑒𝑔 Degradation function
𝑡
t
Time
Envelope function
Sf Unit power spectrum
fp Predominant frequency
TD Duration
𝑎𝑘(t) K time histories
𝐹𝑋𝑖𝑗(𝑥𝑖𝑗/𝑎)
Conditional cumulative distribution function for 𝑖th critical variable and the
𝑗th member (𝑋𝑖𝑗)
𝑅𝑖𝑗 Reliability conditional on 𝑖th critical value for the 𝑗th member
𝑦𝑖𝑗 Capacity of 𝑗th member for 𝑖th critical value
𝑓𝐴(𝑎) Probability distribution function of A during any one year
𝐹𝑋𝑖,𝑛 Cumulative demand distribution in n-year life time
𝑓𝑌𝑖(𝑧)
Chapter IV
Density function of the capacity of a critical variable
𝑊 Load intensity (psf)
𝑌 Random variable modeling the mean of load on the floor
є (𝑥, 𝑦) Stochastic process with zero mean associated with the deviations from the
average
𝑉 Random variable (with mean zero) accounting for the variation of the load
𝑈(𝑥, 𝑦) Random field concerning the spatial variation related to the load
𝑞 Equivalent load with uniform distribution
𝑈 Unit load
𝐸 Expected value operator
Var Variance
µ Mean of all unit floor loads associated with office buildings
𝜎2 Variance in individual floor
𝜎𝑠 An experimental constant
I(x,y) Influence function
xiii
𝐿𝑚𝑎𝑥 Maximum sustained uniformly distributed load
λ Occurrence rate of sustained load changes in [1/year]
�̅�𝑚𝑎𝑥 Maximum equivalent uniformly distributed live load during a 50-year
reference period
𝐿 Nominal uniformly distributed live load
𝐵 Tributary area
𝜎𝐿𝑚𝑎𝑥
2 Variance of maximum live load during 50 years
𝑝 Wind pressure against the surface of structures
𝑞 Reference velocity pressure
𝐶𝑒 Exposure factor
𝐶𝑝 External pressure factor
𝐶𝑔 Gust factor
𝑞𝑇 1 in 𝑇 year reference wind pressure
𝑉𝑇 1 in 𝑇 year wind velocity
𝜌 Density of air
�̅�50 Mean maximum velocity likely in a 50-year design life
𝑉50 1 in 50-year velocity
𝐶𝑜𝑉𝑎 CoV of the maximum annual wind velocity
𝐶𝑎 Aerodynamic shape factor
𝐶𝑔 Gust factor
𝐶𝑟 Roughness factor
𝐶𝑑 Dynamic factor
𝑆 Snow load on a roof of a structure
𝐼𝑠 Importance factor for snow load
𝑆𝑠 Ground snow load with the probability of exceedance of 1 in 50 per year
𝑆𝑟 Associated rain load
𝑆𝑔 Ground snow load
𝐶𝑏 Basic roof snow load
𝐶𝑤 Wind exposure factor
𝐶𝑠 Slope factor
𝐶𝑎 Accumulation factor
𝐶𝑔𝑟 Overall combination of transformation factors
𝐶𝑒 Exposure coefficient
𝐶𝑡 Thermal coefficient
𝐶 Ground to roof transformation factor
𝑑 Snow depth
𝛾 Unit weight of snow
𝜆𝑔 Mean of 𝑙𝑛 𝑆𝑔
𝜉𝑔2 Variance of 𝑙𝑛 𝑆𝑔
xiv
𝛾(𝑑) Average weight density of the snow
𝛾∞ Unit weight at 𝑡 = ∞
𝛾0 Unit weight at 𝑡 = 0
𝐸 Wind exposure factor
T Thermal characteristic factor
𝜀 Error term
𝑟 Transformation factor of snow load on ground to snow load on roofs
ℎ Altitude of the site of the building
ℎ𝑟 Reference altitude
𝑘 Coefficient accounts for the type of region in which structure is located
ƞ𝑎 Shape coefficient
𝐶𝑟 Redistribution coefficient
𝑢(𝐻) Averaged wind speed during one-week period at roof level of H
𝑚𝑔 Sample mean
𝑠𝑔
Chapter V
Sample standard deviation
𝐸𝑚 Mean of modulus elasticity of masonry
𝑐1 Coefficient involved in probabilistic model of modulus of elasticity
𝑐2 Coefficient involved in tensile strength perpendicular to the units
𝑐3 Coefficient involved in tensile strength parallel to the units
𝑓𝑚′ Compressive strength of masonry
𝑌2 Log-normal variable
𝐸𝑚,𝑗 Stochastic modeling of modulus elasticity of masonry
𝑓𝑏 Mean value of the unit compressive strengths
𝑓𝑚𝑜 Mean value of mortar compressive strengths
𝑘 Coefficient
𝑌 Variable related to uncertainties
𝑓𝑚,𝑗 Stochastic modeling of compressive strength
𝛼 Coefficient
𝛽 Coefficient
𝐸𝐿 Modulus of elasticity under Long-term loading
𝐸𝑚 Modulus of elasticity under short-term loading
𝑓𝑏𝑡,𝑙 Mean of tensile strength perpendicular to units
𝑓𝑏𝑡,𝑠 Mean of tensile strength parallel to the units
𝑓𝑏𝑡,𝑙,𝑗 Stochastic modeling of tensile strength perpendicular to units
𝑓𝑏𝑡,𝑠,𝑗 Stochastic modeling of tensile strength parallel to the units
𝑓𝑣,𝑗 Stochastic modeling of cohesion
𝑓𝑣,𝑚 Mean of cohesion
𝜇𝑚 Mean of friction coefficient
xv
𝜇𝑗 Stochastic modeling of Friction coefficient
Abbreviations
ANNs Artificial Neural Networks
ULS Ultimate Limit State
SLS Serviceability Limit State
PLS Phenomenological Limit State
MVFOSM Mean Value First Order Moment
FORM First Order Reliability Method
SORM Second Order Reliability Method
FOROS/SOROS First Order/Second Order Reliability in Original Space
MCS Monte Carlo Simulation
IS Importance Sampling
FEM Finite Element Model
RSM Response Surface Method
PGA Peak Ground Acceleration
GPM General Purpose Mortar
TLM Thin Layer Mortar
UPM Ultrasonic Pulse Method
CS Calcium Silicate
AAC Autoclave Aerated Concrete
LC Lightweight Concrete
CB Clay Brick
1
Chapter I: Overview
There is an undeniable fact that uncertainty is an inseparable part of every scientific field. This
fact highlights the need for probabilistic treatment of problems in different fields of study. In order
to avoid complex solutions and calculations, deterministic approaches are commonly prevalent in
structural engineering. Semi-probabilistic methods are also used as simplified methods in order to
combine deterministic and probabilistic approaches. However, the most accurate and efficient
solutions can be achieved through fully-probabilistic approaches which guarantee comprehensive
reliability assessment of structures.
Historic structures are of great importance as they reflect the history and culture of a people and
their society. The conservation and maintenance of heritage structures requires considerable care.
Evaluation of the reliability of historic masonry structures has gained significant importance, since
the consequences of structural failure could be serious and irreparable. Heritage masonry structures
are typically unreinforced and have low and limited load bearing capacity. Therefore, assessment
of the reliability of these structures is of special importance (D’Ayala and Speranza, 2003).
Modern codes have been developed mostly for the design of new buildings based on modern
materials, whereas historical structures were designed according to the rules and experiences
available at the time. The expected service life of a historical structure was more than that of a new
structure designed by current codes, because historical structures were expected to be preserved
for future generations. These buildings have withstood all the applied loads during their service
life to date which provides some evidence as to the serviceability and safety of these structures.
Based on the age of a structure and the type of the applied loads, different information can be
obtained from its past performance. As an example, in the case of a 30-100 year-old building which
has experienced earthquakes with ground motions less than the design event during its service-life
to date, one may conclude that the structure has presented satisfactory performance regarding the
dead and other variable loads but not potential seismic ones. In contrast, a historical structure
2
dating back a number of centuries, has demonstrated satisfactory performance under a much bigger
variety of loads and hazards.
As these heritage structures were constructed before modern codes and standards were written,
current codes will not necessarily provide satisfactorily accurate estimates of load or resistance
when it comes to evaluation of the historical materials or the structure. This inaccuracy may lead
to inefficient and even destructive strengthening modifications. Due to the uncertain nature of the
applied loads as well as unexpected structural responses under different loading conditions,
deterministic evaluations may result in inaccurate evaluations and unnecessary strengthening.
Probabilistic approaches are used to get more accurate evaluations and consequently efficient
modifications. Through probabilistic reliability assessment of masonry structures, the necessity
and level of upgrading and rehabilitation can be defined, which should result in more appropriate
and efficient interventions (Lagomarsino and Cattari, 2015).
When “upgrading” or strengthening an existing structure, it could be more expensive to intervene
to meet a certain criterion compared to providing structural safety in the design stage of a new
structure. Therefore, determining which criteria should be satisfied and subsequently, the level of
upgrading is of special importance. Preservation of the values of a historical building plays a key
role in upgrading. Preservation should minimize the destruction of the originality of a historical
structure in terms of both materials and architecture.
Identification of vulnerable historical structures and determination of a reasonable criterion for
upgrading and strengthening them are major challenges in structural engineering and cultural
heritage preservation. To date, considerable research has been allocated to structural steel,
concrete and timber but only a few studies have been focused on the reliability assessment of
masonry structures. Despite the high level of vulnerability of unreinforced masonry structures
under applied loads and the importance of the reliability evaluation of historic masonry structures,
there is no specified methodology to assess the reliability of them.
3
Moreover, there are difficulties to apply the methodologies recommended by the National Building
Code of Canada (NBCC, 2010) and Canadian Standard Association (CSA, 2014) standards with
respect to the structural evaluation of existing structures (Allen,1991a), including:
In the case of historical structures, there are structural systems, components and materials which
are not addressed by the NBCC (2010) or CSA (2014). Structural evaluation faces difficulties due
to the lack of information on their structural properties.
Although historical structures do not mostly follow the instructions of the code, they have
performed satisfactorily over the years. Also, a number of structural characteristics such as dead
load, strength, deterioration, fatigue and creep can be measured directly. These constitute some of
the information not considered by the NBCC’s and CSA’s design criteria.
Most of the requirements are focused on specifying the required and economical percentage of
strengthening materials and their related arrangement during construction. But, it is uneconomical
to apply these strengthening modifications to an existing structure. Therefore, alternative
requirements need to be defined.
Consequently, as an initial step towards satisfying this goal, the main focus of the research
described in this thesis is to develop a determinate, step-by-step methodology for assessing the
reliability level of historical masonry structures under applied loads. The presented methodology
can be used in a code to estimate the reliability level with some modifications and generalization.
To achieve the goal, the relevant literature is reviewed, followed by a description of reliability
analysis. Methods to obtain the necessary structural information for reliability analysis are
proposed allowing a logical assessment method to be described.
Despite the high level of vulnerability of historic structures to the applied loads and the necessity
of reliability assessment of them, there is still no formal approach for reliability assessment of
historical masonry structures. The current thesis contributes to the development of a step-by-step
reliability-based assessment procedure being appropriate and applicable for historical masonry
structures. The main contributions of this study are:
4
1) A comprehensive literature review was done to understand various aspects of reliability
assessment of masonry structures specifically historical masonry ones. The theory of reliability
assessments of structures, reliability assessment techniques and suitable combination among finite
element structural analysis, reliability analysis methods and reliability software are reviewed
precisely.
2) Values for the target reliability index or target probability of failure (as decision criteria) being
appropriate for historic masonry structures are determined. Different values of target probability
of failure or target reliability index are recommended by different codes and standards. The
necessity of meeting target probability of failure values or the target reliability index recommended
by different codes and standards is still a controversial issue in the case of historic masonry
structures. This is due to that fact that the recommended values are determined for new masonry
materials, structural configurations and construction methodologies and may not be appropriate
for historical masonry structures having specific criteria and requirements. A number of
calculation approaches and target reliability index and failure probability values showed to be
appropriate for historic structures are discussed here.
3) Deterioration as a prevalent destructive process happening in historic structures may result in a
change in the structural resistance of masonry, the loading conditions or analysis model. In this
thesis, approaches for integrating deterioration into reliability assessment are investigated and
degradation functions are reported.
4) Reliability assessment of structures subjected to earthquakes is still challenging and complex.
This is due to the fact that different uncertainties including those in the nature of ground motions
and those in nonlinear behaviour of structures are involved in reliability assessment under seismic
loading. Reliability assessment of existing structures under seismic loading is studied and a
specific methodology is described in this regard.
5) To formulate the limit state function as the base of reliability assessment, probabilistic models
of applied loads have to be determined. In this study, the best stochastic models of applied loads
are presented.
6) The statistical characteristics of materials play key roles in the estimation of the structural
resistance probabilistic model. Various techniques can be used to determine stochastic
5
characteristics of historic masonry materials and to get more realistic information about the
material properties. However, destructive testing methods are not recommended as such tests may
result in irreparable damage to these valuable structures. Moreover, some techniques are not
capable of determining the variability of material properties (over an element or structural system).
As estimation of the variability is essential for reliability assessment, these techniques may be
considered inappropriate with respect to structural reliability assessment. Therefore, the
applicability and capabilities of current testing techniques in order to estimate the statistical
characteristics of historic masonry materials are studied in this research.
7) Compressive strength is the most important material property influencing the structural
resistance of historic masonry structures and consequently, influences the reliability assessment of
these structures. Codes are necessarily conservative and are also generally aimed at design or
assessment with modern masonry materials. Therefore, the use of code values for historical
structures may result in inaccurate structural reliability assessment. Various techniques can
estimate the mean value of compressive strength. However, testing techniques which damage the
structure (for example, by removing samples of the masonry) are not recommended for historical
buildings because of their destructive nature. Moreover, there are restrictions in the application of
some destructive techniques for estimation of the variability of compressive strength (over an
element or structural system). Here, a procedure is proposed to determine probabilistic model of
historic masonry compressive strength using non-destructive testing techniques.
Chapter I introduces the necessities of reliability assessment of historical masonry structures. to
do so, the most important difficulties, concerns and restrictions in the application of the new
reliability assessment methods for historic masonry structures are presented and the research
motivations are expressed.
A brief review of the literature focusing on the reliability assessment of historic masonry structures
were discussed in Chapter II.
Chapter III explains structural reliability analysis. Reliability assessment and the concerns
associated with the reliability assessment of existing structures are discussed. Basic theory and the
6
required procedure of reliability assessment are explained. Target reliability index and failure
probability values and different approaches for calculating suitable targets for historic structures
are described. Different limit state functions are introduced. Available techniques for reliability
assessment of structures are briefly discussed. Deterioration, as a common physicochemical
process resulting in gradual alteration in characteristics of historical masonry, is introduced and
the integration of deterioration into reliability analysis is studied. Finally, as the reliability
assessment of structures under seismic loading is complicated compared to the reliability
assessment under other loading conditions, the appropriate methodology for reliability assessment
of masonry structures subjected to earthquake is investigated.
An overview on probabilistic modeling of applied loads as one of the essential parameters in the
determination of the limit state function is provided in Chapter IV. The stochastic models of dead
load, live load, wind load, snow load are described. The recommendations of different researchers
and codes with respect to probabilistic models and the stochastic characteristics of applied loads
are studied and compared together. Point-in-time components of live load, wind load and snow
load were also studied.
Chapter V covers the stochastic characteristics and probabilistic models of historic masonry
materials. Masonry materials and their associated stochastic characteristics play key roles in the
determination of probabilistic models of structural resistance. A summary of the techniques which
are used to evaluate the characteristics of masonry materials is presented. The advantages and
disadvantages of these testing techniques as well as their feasibilities and capabilities in the
determination of the stochastic characteristics of historic masonry materials are studied. As codes
of practice mostly recommend calculation procedures, values and the best fit distributions for new
masonry materials, the accuracy, applicability and accordance of these recommendations to
historical masonry materials are investigated and discussed. Compressive strength is usually key
information in the determination of structural resistance of historic masonry structures. Therefore,
special attention is given in this thesis to the determination of the stochastic characteristics and
probabilistic model of historical masonry compressive strength using non-destructive techniques.
An approach is proposed here to estimate the compressive strength of ancient masonry structures
and its associated variability without any degradation to the historical value.
7
Chapter VI provides a summary of the proposed procedure for reliability assessment of historic
masonry structures as well as the limitations.
Chapter VII offers concluding remarks of this research including thesis contributions and
recommendations for future work.
8
Chapter II: Previous Research Work
Structural evaluation plays a key role in the preservation and use of structural and cultural
heritages. Traditionally, variables used in structural evaluation approaches are treated as
deterministic values. The reality that applied loads and structural resistances are uncertain and
parameters are random necessitates the application of probabilistic approaches for structural
evaluation leading to structural reliability assessment. Historical structures are among the most
complicated structures in terms of reliability assessment. Different material properties with various
stochastic characteristics, complicated configurations, shortage of available construction
documents and limitations in the application of destructive testing techniques are the main
concerns and difficulties in reliability assessment of historic structures. A brief review of previous
research work focusing on different aspects of reliability analysis of historic structure is presented
here.
Reliability assessment of structures is a developing field of civil engineering. In 1926, Mayer
suggested considering deviations in design procedure through stochastic techniques to lead to safer
and more reliable designs (reported by Brehm, 2011). Streletzki (1974) and Wierzbicki (1936) also
introduced load and resistance parameters as random variables and consequently, a probability of
failure for each structure. Although probabilistic approaches are obviously beneficial, they have
not been the focus of much research due to the computational complexity. The theory of reliability
assessment of structures was developed significantly by Freudenthal (1947: 1956). As the
formulations involving convolution functions were difficult to be solved by hand, it was not
possible to apply reliability analysis in practice until the work of Cornell and Lind during the late
1960s and early 1970s. A second moment reliability index was proposed by Cornell (1967). A
description of a format-invariant reliability index was provided by Hasofer and Lind (1974). Then,
Rackwitz and Fiessler formulated an efficient numerical procedure to calculate the reliability index
9
in 1978. To do reliability analysis, Melchers (1999) and Van Dyck (1995) proposed the use of
basic reliability problems to outline the generalized reliability problem.
The Joint Committee on Structural Safety (JCSS) was founded in 1976. Since then, this committee
has played a considerable role in support of the development and improvement of probabilistic
approaches and reliability methods. Moreover, the JCSS has provided and enlarged the available
database. The “Probabilistic model code” (JCSS, 2003) is the outcome of all the committee’s
efforts and findings which provides users with probabilistic models related to different variables
in the field of civil engineering. The more recent version (JCSS, 2011) covers stochastic models
of random variables of different structures categorized based on their construction materials,
including concrete structures, steel structures, and masonry structures. However, this version
(JCSS, 2011) includes statistical characteristics of new construction materials in the case of
masonry structures which is not the focus of this research. Therefore, there is still a lack of
knowledge in the guidelines recommended by JCSS (2011) regarding the statistical characteristics
of historic masonry materials.
Several researchers have developed general procedures and guidelines for reliability assessment
of structures. Ellingwood (1996) investigated the condition assessment of existing structures using
reliability assessment. His research mostly focused on general and somehow qualitative
explanations regarding the procedures for probabilistic assessment of existing buildings. Faber
(2000) reviewed the general philosophy, theoretical concepts and tools being important to be able
to do reliability assessment of existing structures. General comments regarding the application of
reliability analysis in the assessment of steel structures and concrete structures were presented.
Rackwitz (2001) also reviewed theory and methods of structural reliability. The cases for which
these methods are not applicable were explained and new fields having potential application were
discussed. The importance of sampling schemes to update probability estimates were outlined.
A report prepared by BOMEL Limited for the Health and Safety Executive (2001) reviewed the
developments of structural design methods including the earliest building methods, limit state
design methods and probabilistic analysis. The report focused on the application of probabilistic
methods in the design and assessment of pressure systems. Guidelines regarding risk analysis in
conjunction with reliability analysis were also prepared. The report was mainly for industry use.
The Federal Institute of Materials Research and Testing (BAM) (2006) prepared a guideline for
10
the assessment of existing structures (F08a). The report contains a chapter titled “probabilistic
assessment routine”. However, the chapter only includes general explanations and
recommendations regarding structural reliability assessment and procedures rather than more
specific guidelines. ISO 13822 (2010) provided general guidelines regarding the assessment of
existing structures.
Ditlevsen and Madsen (1996) published a book regarding the structural reliability. This book
includes both structural reliability philosophy and methods in details. Reliability of existing
structures and system reliability analysis are also included in this book. This book mostly covers
the reliability assessment of new structures rather than historical ones.
Reliability assessment of an existing building subjected to earthquakes has not been the focus of
many researchers due to the complexity. Among the research which has been done in this field,
the work of Shah and Dong (1984) is the one of the more comprehensive. They (Shah & Dong,
1984) presented a methodology to assess the structural reliability of existing buildings under
probabilistic seismic loadings. Through the presented methodology, the intensity and frequency
content of the ground motion are represented using a peak power spectrum obtained through
seismic hazard evaluation for the site. A set of time histories are then generated by Monte-Carlo
simulation (MCS) with the assumption of a Gaussian model for seismic time histories. Regarding
the response parameter being considered, a statistical analysis is done. A Gumbel distribution
(Generalized Extreme Value distribution Type 1) is fitted to the data. A probability distribution is
attributed to the ultimate ductility capacity and finally, the reliability level of the structure is
assessed through comparison of the distributions of demand and capacity.
Damage and deterioration are unavoidable aspects of existing structures. An analytical
investigation was carried out to examine the influence of damage and redundancy deterioration on
structural reliability (with a focus on trusses and bridges) by Frangopol et al. (1987). They showed
that probabilistic assessment of damaged structures should be done considering system reliability
methods rather than a single element one. Moreover, it was found that deterministic assessment
methods (rather than probabilistic approaches) can be reasonably accurate to estimate the influence
of deterioration on structural reliability.
Some researchers worked on different aspects of reliability assessment of masonry structures.
Schueremans and Van Gemert (1998) developed a probabilistic methodology to evaluate the
11
reliability of masonry structures with a focus on theory and calculations of probability of local
failure in the case of masonry shear walls. The study was part of a program involving structural
strengthening of ancient masonry considering risk analysis, grouting and non-destructive tests.
Two case studies also were investigated and the reliability of each structure was assessed including
a masonry sewer system and the facade of the St.-Amandus chapel at Erembodegem in Belgium.
Maes et al. (1999) developed a general limit state framework concerning loads, resistances and
stresses. Different structural limit states were introduced including structural limit states (ultimate
limit states and serviceability limit states) and phenomenological limit states (cultural value limit
state, appearance limit state and sustainability limit state). Besides performance requirements
being common between new buildings and old buildings, other requirements which are influential
on reliability-based assessment of existing buildings were presented. Economics, heritage value,
uncertainties, past performance, changing design practice and service life are among the most
important ones. Different deterioration mechanisms and their incorporation into reliability
assessment were discussed.
A procedure to calculate structural reliability of common masonry walls under vertical bending
was presented by Stewart and Lawrence (2002). The procedure was applied to a masonry column
and the probability of failure was calculated according to an ultimate limit state (crushing of the
brick masonry) and an appearance limit state (first-cracking). Structural reliability was found to
be sensitive to wall width, workmanship and the difference between the designed and the
constructed thickness.
Lawrence and Stewart (2009) also examined the reliability of masonry walls which were designed
to be under compression according to AS3700. The probabilistic models of behaviour were
obtained using the results of many tests done on full-scale masonry walls. Then, the structural
reliability of unreinforced masonry walls under concentric vertical loading was determined
considering different variants including unit compressive strength, mortar type, tributary area and
live-to-dead load ratios. They (Lawrence & Stewart, 2009) compared the calculated reliabilities to
a target reliability index specified in AS5104 and suggested an increment in the capacity reduction
factor (∅) from 0.45 to 0.75 for 𝐻/𝑡 < 30 where 𝐻 stands for height and 𝑡 is thickness. Lawrence
and Stewart (2011) also worked on the development of a methodology for reliability assessment
of unreinforced masonry walls under vertical bending. Using reliability analysis, they calibrated
12
the reduction factor (partial safety factor) for bending. Model error, wall height and discretization
of wall thickness were investigated as influential parameters on the reliability of masonry walls
under vertical bending. The probabilistic model of model error was determined through the
stochastic analysis of results obtained from testing of 118 walls. Statistical characteristics of
flexural tensile strength were derived based on the stochastic analysis of a database containing
7332 individual measurements of bond strength. Through the comparison of calculated reliabilities
with their associated target reliability index, the capacity reduction factor (∅) reported by AS3700
was recommended to decrease from 0.6 to 0.47. However, they (Lawrence & Stewart, 2011)
suggested further study be done on the influence of wall length and mortar type to see if these
perameters are influential on the structural reliability and if the behaviour model needs to be
improved.
Graubner and Glowienka (2008) provided the main statistical parameters required for the
reliability assessment of big size masonry. They also quantified model uncertainties which have
to be considered in the probabilistic calculations. They (Graubner and Glowienka, 2008) believed
that the statistical characteristics of big sized masonry showed small scatter compared to other
kinds of masonry.
An investigation was done on reliability assessment of masonry arch bridges considering both
serviceability and ultimate limit state functions by Casas (2010). Various failure modes as well as
the structural behaviour of the masonry arch bridges under low and high load levels were studied.
A methodology for reliability assessment of masonry arch bridges was developed. This
methodology was then applied to an existing arch bridge. However, due to the lack of experimental
data, the experimental application of the proposed methodology should be considered as
preliminary. Moreover, there is a need to update the preliminary proposed models with more
laboratory and full scale testing results. It was mentioned that probabilistic assessment of masonry
arch bridges is not common as the determination of their failure criteria is difficult. This is due to
that fact that these kinds of structures usually show a global nature of failure rather than based on
the failure of components.
Schueremans and Verstrynge (2008) worked on the use of reliability assessment techniques to
evaluate the safety of the Romanesque city wall of Leuven. Reliability analysis was done
considering three limit states including rotation equilibrium, stress in the masonry and stress in the
13
subsoil. The geometry and material properties were estimated through survey. The procedures of
reliability analysis and safety assessment are briefly explained. Brehm and Lissel (2012) studied
the reliability of unreinforced masonry bracing wall subjected to wind loading. In this study, the
walls were designed according to the German code to satisfy the provided level of reliability. The
shear capacity of the walls was determined by both analytical models and test data, and were
compared to determine the most realistic model.
Al-habahbeh & Stewart (2015) investigated the stochastic reliability of unreinforced masonry
(URM) walls under blast loading. Finite element modeling (FEM) was combined with MCS in this
study. They concluded that URM walls are highly reliable in the case of moderate blast. Wall
thickness, blast pressure, bond strength and material strength are the most influential parameters
on the reliability of URM walls subjected to blast. Mojsilovic´ and Stewart (2015) studied the
reliability of mortar joint thickness in load bearing masonry walls. Four different sites in
Switzerland were chosen to collect data on the thickness of the mortar joints in clay block masonry
walls. Using reliability analysis of the masonry under compression, it was concluded that
probabilistic modeling of bed joint thickness leads to a higher reliability index than that calculated
from a deterministic value of bed joint thickness. Different levels of reliability were observed at
the different sites. Moreover, thicker joints resulted in lower reliability indices.
The realistic reliability analysis of complex structural systems involving different materials and
different structural elements and systems has become an important challenge in the field of
structural reliability assessment and analysis. The finite-element method is a powerful tool being
used in the analysis of complicated structures, and has been combined with reliability analysis in
attempts to capture the desirable aspects of these two methods (reliability analysis and the finite
element method). The outcome of these attempts was the stochastic finite element method (SFEM)
developed by several researchers (Haldar and Nee, 1989; Haldar and Gao, 1997; Haldar and
Mahadevan, 2000).
Quite a few researchers have studied the structural reliability of historical masonry structures.
Typically, the research studies were specifically aimed at performing a reliability assessment of a
masonry element under determined loading conditions and did not propose a step-by-step
methodology for reliability assessment of historic masonry structures using appropriate
approaches and formulae.
14
The literature review which contributes to the content of individual thesis chapters, is mentioned
in the relevant sections.
15
Chapter III: Structural Reliability Analysis
Reliability assessment is a procedure for evaluating the reliability of a historic building during a
performance period, based on information gathered about the building, from material properties to
structural layout. In other words, reliability assessment includes estimation of the reliability level
of a structure considering certain limit states during its service life. Figure 3-1 shows the flowchart
for a general evaluation procedure of existing structures (ISO 13822, 2003). Human safety, human
comfort, building function, damage and economics are the fundamental structural safety and
performance requirements. The basics of safety and performance requirements are similar in cases
of both new and historic structures. However, there are some concerns regarding the assessment
of historic structures: these concerns include the six factors discussed below (Maes, 1999).
3.1.1 Uncertainties
Uncertainty refers to a situation associated with imperfect and/or unknown information.
Uncertainty is categorized as aleatory uncertainty (inherent natural variability) and epistemic
uncertainty (model uncertainty, statistical uncertainty and measurement error). There are many
sources of uncertainty, including material heterogeneity and lack of construction details. At the
design stage, load and resistance factors are representative of the uncertainties in loads and
resistance. At the evaluation stage, existing structures may experience either an increase (e.g. due
to deterioration) or a decrease (e.g. measuring properties by test) in load and resistance
uncertainties in comparison with the design stage. Therefore, the range of uncertainties is broader
in the evaluation stage which necessitates identification of key components and details of the
structure (Allen, 1991a). Monitoring structures during the evaluation procedure can provide
information to update the uncertainties.
16
Figure 3-1 Flowchart for general evaluation procedure of existing structures (ISO 13822, 2010)
Requests/Needs
Identification of the evaluation
Scenarios
Preliminary evaluation
Investigation of documents and other evidence
Initial inspection
Initial check
Decisions on immediate plans
Suggestions for detailed evaluation
Doubt about safety?
Detailed evaluation?
Detailed review and search for documents
Detailed inspection and material examination
Determination of plans
Determination of structural properties
Analysis of structures
Verification
Additional inspection?
Reporting evaluation outcomes
Decision
Reliability sufficient?
Intervention
Construction
Strengthening
Demolish, build new
Operation
Monitoring
Change or restriction
in use
Maintenance
Periodical inspection
Yes
Yes
No
No
No
No
Yes
Yes
17
3.1.2 Heritage
In the case of heritage structures, preserving existing material is of high value, implying the
application of less intervention. Also, it is important to use materials which are already there
instead of a new structure as a supporter. In brief, heritage criteria require the minimum destruction
be applied to the heritage value of existing materials and systems in any renovation and
strengthening process (Allen, 1991a).
3.1.3 Past Performance
Historical buildings have withstood all the loads applied during their service life to date which
provides some evidence as to the serviceability and safety of these structures. Based on different
factors including the age of a structure and the type of the applied loads, different information can
be obtained from its past performance. As an example, in the case of a building aging less than
100 years which has experienced earthquakes with ground motions less than the design event
during its service-life to date, one may conclude that the structure has presented satisfactory
performance regarding the dead and other variable loads but not potential seismic ones. In contrast,
a historical structure dating back a number of centuries, has demonstrated satisfactory performance
under a much bigger variety of loads and hazards (Allen,1991a). For example, the possible gradual
loss in load bearing capacity during past earthquakes may lead to alteration of dominant failure
modes and therefore difficulties in accurate structural evaluation. Structural integrity and the
ability to absorb local failure without extensive structural collapse are key properties of existing
masonry structures. However, these properties cannot easily be quantified and are mostly based
on engineering judgment.
3.1.4 Economics
In the design stage, the cost of providing a high degree of safety is low, so that reducing safety
factors in specific situations is not beneficial in terms of saving money. Therefore, it is more
suitable practically to use general criteria, presented by CSA/NBCC, which are conservatively
applicable to all situations (Diamantidis, 1999; Ellingwood, 1996). In the evaluation stage, the cost
18
of not meeting a criterion and, consequently, the requirement of upgrading can be large. Therefore,
due to economic considerations, determination of criteria for each situation according to the
fundamental requirements of life safety, comfort, function, and economics is necessary for
upgrading (Allen, 1991a). Moreover, Maes et al. (1999) proved that implementation of materials
and arrangements specified by some current requirements are economical and feasible during the
primary construction phase but uneconomical and infeasible after construction.
3.1.5 Change in Design Practice
Historical structures were designed and constructed many years ago based on the engineering
principles, from rules of thumb and tradition in engineering experience, which were available that
time. Therefore, they are mostly designed for compression leading to vulnerability when subject
to the lateral loads or eccentricity of vertical loads.
3.1.6 Service Life
Historic structures need to be preserved for future generation. Therefore, a new level of
performance may be considered for protected structures by the preservation authority, which may
lead to a new service life higher than that specified in modern design codes. The new expected
service life has to be satisfied with minimum level of upgrading and disturbance. Therefore, there
is a need to evaluate historic structures accurately before upgrading.
The calculation of a probability of failure necessitates the definition of failure. To do so, limit
states have to be defined. Limit states can be applied to all parts of a structure from the entire
structure to a single member. Commonly, several limit states are considered for a single structure.
There are several known limit state categories being used in structural engineering (Maes et al,
1999):
19
Ultimate limit state (ULS)
The ultimate limit state refers to load bearing capacity, overturning and overall stability, i.e., limit
state exceedance denotes failure of structure or member probably leading to serious injury or
human fatality. Therefore, ULS is based on the concept of safety.
Serviceability limit state (SLS)
Human life is not in danger through exceedance of a serviceability limit state. In general,
serviceability limit states refer to gradual deterioration, residents’ comfort or maintenance cost.
Regarding masonry structures, SLS refers to the control of cracking, the control of deflections and
inclinations, local damage, inappropriate function of non-bearing structural elements.
Phenomenological limit state (PLS)
PLS refers to different limit states including the value limit state (the influence of
engineering/architectural changes on historical/architectural value), the appearance limit state
(improper discoloration, corrosion, local crumbling and graffiti) and the sustainability limit state
(waste and recycling assessment). Regarding the limit state value, it should be noted that it may
be based on the summation of different values, including antiquity value, historical value, symbolic
value, heuristic value and so on (Baldioli, 1998).
The limit state function is determined based on the input variables and failure modes. Codes of practice
present formulae to calculate the load effects on a member as well as its capacity (using material
properties) under different failure modes. Therefore, when the desired limit state (e.g. ULS) is selected
and the applied loads on each structural element are determined, the probable failure modes of each
member should be estimated. The formulae of the structural resistance and the load effects associated
with each structural member under the determined probable failure modes can be derived from codes
of practice. Limit state functions associated with different failure modes of a member can be calculated
20
through the subtraction of the recommended formulae for load effect from the formulae of the
resistance.
The term reliability refers to the complement of the probability of failure. Structural safety is a
function of the applied loads and the structural capacity of the members. Reliability analysis of a
structure involves the calculation of its probability of failure under the applied loads. The general
reliability problem can be determined starting from the primary reliability problem. In the primary
reliability problem, only load effects (𝐿) and structural resistance (𝑅) are considered, both as
random variables with known probability density functions (𝑓𝐿(𝑙) and 𝑓𝑅(𝑟)). The probability of
𝑅 − 𝐿 <0 equals the probability of failure (𝑃𝑓). In order to do reliability analysis, a procedure
should be followed:
I) The hazard and failure modes under the considered hazard have to be determined.
II) A limit state function involving the variables from stage I have to be determined in order to
calculate the probability of failure. In structural engineering, the formulation of limit state function
is commonly in the basic form as follows:
𝑔(𝑅, 𝐿) = 𝑅 − 𝐿 (3-1)
where 𝑅 is the resistance and 𝐿 is a load effect. The safe and failure performances can be
conventionally defined as:
𝑔(𝑅, 𝐿) < 0 → 𝐹𝑎𝑖𝑙𝑢𝑟𝑒 (3-2)
𝑔(𝑅, 𝐿) = 0 → 𝐶𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑆𝑖𝑡𝑢𝑎𝑡𝑖𝑜𝑛 (𝐶𝑜𝑛𝑠𝑖𝑑𝑒𝑟𝑑 𝑆𝑎𝑓𝑒) (3-3)
𝑔(𝑅, 𝐿) > 0 → 𝑆𝑎𝑓𝑒 (3-4)
21
III) The random variables (X) which influence the structural behaviour including structural
resistance (𝑅) and load effect (𝐿) have to be identified. Then, the probabilistic characteristics of
these random variables should be determined.
IV) The probability of failure (𝑃𝑓) or reliability index (𝛽) as a reliability measure are calculated.
In general, 𝑅 and 𝐿 are a function of time and as a result 𝑃𝑓 is also a function of time. But due to
the complexity of the mathematical calculation of 𝑃𝑓, a preset reference period (𝑡𝐿) (design service
life) can be considered for reliability analysis, leading to transformation of the probability density
functions into time-invariant probability density functions. However, considering 𝑡𝐿 is possible
but it is not necessary. The generalized reliability analysis is as follows (Schueremans &
Verstrynge, 2008):
𝑃𝑓 = 𝑃(𝑔(𝑅, 𝐿) < 0) = 𝑃(𝑅 < 𝐿) = ∬ 𝑓𝑅,𝐿(𝑟, 𝑙)𝑑𝑟𝑑𝑙
𝑟<𝑙
(3-5)
Consequently, the probability of survival (𝑅𝑓) is
𝑅𝑓 = 1 − 𝑃𝑓 (3-6)
In most cases 𝑅 (resistance) and 𝐿 (load effect) are independent having continuous densities
(𝑓𝑅(𝑟), 𝑓𝐿(𝑙)). Therefore, the probability of failure can be numerically computed where Equation
3-5 can be rewritten as a convolution integral (Brehm, 2011), see Figure 3-2.
𝑃𝑓 = ∬ 𝑓𝑅(𝑟). 𝑓𝐿(𝑙)𝑑𝑟𝑑𝑙 = ∫ 𝐹𝑅(𝑙)𝑓𝐿(𝑙)𝑑𝑙
+∞
−∞
+∞
−∞
(3-7)
where 𝐹𝑅 denotes the cumulative distribution function of the resistance 𝑅, 𝑓𝐿 denotes the
probability density function of load effects 𝐿.
22
Figure 3-2 Load effect and resistance probability density functions.
Equation 3-7 can only be solved in closed form for a small number of cases. Special analysis
methods are required to solve the integral of Equation 3-7 in most cases. The most common
reliability analysis methods are briefly discussed in section 3-5.
V) Target failure probabilities (𝑝𝑓𝑇) or a target reliability index (𝛽𝑇) are determined. The safety
assessment of structures necessitates definition of target safety levels as decision criteria. Codes
of practice present different values as the target probability of failure, e.g. 𝑃𝑓𝑇 = 5. 10−4,
according to Eurocode (EN1990, 2002). Regarding historical structures, the necessity of meeting
the target probability of failure values reported by codes is still a controversial issue. This is due
to that fact that these values are reported for new structures, not historical ones with specific criteria
and requirements. Several studies have advocated widening the discussion to develop a more
accurate target probability of failure (Ditlevsen, 1982; EC1, 1994; ISO2394, 1998; Melchers,
1999; NEN 6700, 1991; Van Dyck, 1995). Other performance criteria are also considered in these
studies in addition to those mentioned in Section 3-1 regarding the assessment of historical
buildings. Some of these criteria and parameters include possible damage (fatality, environmental
damage, economic damage and socio-cultural damage), risk level (public buildings, bridges, off-
shore structures and so on) and warning level (steady failure with a visible clue and sudden failure
without a clue).
Target probability of failure values can be calculated according to proposed experimental formulas
(Melchers 1999; Van Dyck 1995). The following formula was proposed by the Construction
23
Industry Research and Information Association (CIRIA, 1977) in order to calculate the probability
of failure considering a preset design service life (𝑡𝐿):
𝑃𝑓𝑇 = 10−4𝑆𝑐
𝑡𝐿
𝑛𝑝 (3-8)
where 𝑛𝑝 denotes the average number of building residents or the building’s immediate
neighbourhood residents. 𝑆𝑐 is the social criterion factor, with values shown in Table 3-1. Allen
(1991b) developed another empirical formula:
𝑃𝑓𝑇 = 10−5𝑡𝐿
√𝑛𝑝
𝐴𝑐
𝑊 (3-9)
in which 𝐴𝑐 is activity factor and W is warning factor, see Table 3-1. Both formula have
advantages. The first one includes 𝑆𝑐 accounting the importance of historical buildings or their
preservation value. The second one accounts for more differentiation. But none of them considered
the number of injuries and the economic cost of failure. A formula has been proposed to calculate
the nominal target failure probabilities for historical structures accounting for the cost factor and
reassigning the social factor (Schueremans, 2001):
𝑃𝑓𝑇 =10−4𝑆𝑐𝑡𝐿𝐴𝑐𝐶𝑓
𝑛𝑝𝑊 (3-10)
where 𝐶𝑓 denotes the cost factor according to (CEB, 1976, 1978), see Table 3-1. As this formula
includes all the above mentioned factors in addition to the 𝑆𝑐 which specifically accounts the level
of importance of historical structures, it seems to be more compatible with historic structures.
Therefore, it is recommended to use Equation 3-10 to estimate the target probability failure in case
of historic structures.
GruSiBau (1981) categorized failure consequences based on defined consequence classes. The
reliability index was determined for each class and consequence for a 50-year observation period,
see Table 3-2. Diamantidis (1999, 2001) also presented final tentative target reliability values as
24
listed in Table 3-3. JCSS (2001) developed a different approach to account for risk to both human
life and investment. Through this approach, the target reliability links to the relative cost of
reliability enhancement of structures. Table 3-4 presents the required target reliability according
to JCSS (2001).
Table 3-1 Influential factors on the target probability of failure (Schueremans, 2001).
Residual Service Life/years (𝑡𝐿)
Number of Endangered Lives (𝑛𝑝)
Economic Factor (𝐶𝑓)
Not significant 10
Significant 1
Very significant 0.1
Warning Factor (W)
Fail-Safe condition 0.01
Steady failure with some warning clue 0.1
Steady failure hidden from view 0.3
Unexpected failure with no warning 1.0
Activity Factor (𝐴𝑐)
Post-disaster activity 0.3
Normal activities 1.0
- Building 1.0
- Bridges 3.0
- High exposure structures (offshore structures) 10.0
Social Criterion Factor (𝑆𝑐)
Historical structures of great importance (e.g. listed by UNESCO) 0.005
Historical structures listed as nationally important 0.05
Historical structures listed as regionally important 0.5
Not-listed historical structures 5.0
25
Table 3-2 Target reliability and consequences class according to GruSiBau (1981).
Consequence Class Ultimate Limit State Serviceability Limit State
1
Safe for human life
Small economic influence
Small economic impact
Small interference with use
𝛽 = 4.2 𝛽 = 2.5
2
Dangerous for human life
Significant economic influence
Significant economic influence
Considerable interference with use
𝛽 = 4.7 𝛽 = 3.0
3
Very dangerous for human life
Large economic influence
Large economic influence
Large interference with use
𝛽 = 5.2 𝛽 = 3.5
Table 3-3 Tentative target reliability values 𝛽𝑇 (𝑃𝑓𝑇) (Diamantidis, 1999, 2001).
Costs of Safety
Measures
SLS
(Permanent)
ULS - Failure Consequences
Low Moderate Significant
High 1.0 (0.2) 2.8 (3. 10−3) 3.3 (5. 10−4) 3.8 (7. 10−5)
Moderate 1.5 (7. 10−2) 3.3 (5. 10−4) 3.8 (7. 10−5) 4.3 (8. 10−6)
Low 2.0 (2. 10−2) 3.8 (7. 10−5) 4.3 (8. 10−6) 4.8 (8. 10−7)
26
Table 3-4 Required target reliability according to JCSS (2001) for a 50-year observation period.
Relative Cost
of Reliability
Enhancement
Failure Consequences
Minor Average Major
e.g. agricultural building e.g. residential building e.g. high-rise building
Large
𝛽 = 1.7 𝛽 = 2.0 𝛽 = 2.6
𝑃𝑓 ≈ 5. 10−2 𝑃𝑓 ≈ 3. 10−2 𝑃𝑓 ≈ 5. 10−3
Medium
𝛽 = 2.6 𝛽 = 3.2 𝛽 = 3.5
𝑃𝑓 ≈ 5. 10−3 𝑃𝑓 ≈ 7. 10−4 𝑃𝑓 ≈ 3. 10−4
Small
𝛽 = 3.2 𝛽 = 3.5 𝛽 = 3.8
𝑃𝑓 ≈ 7. 10−4 𝑃𝑓 ≈ 3. 10−4 𝑃𝑓 ≈ 10−5
Structural reliability techniques that have been developed can be classified into two general
categories including asymptotic techniques and simulation based approaches. A discussion on the
applicability of each technique in the estimation of structural reliability, as well as a brief overview
of their related limit state functions (explicit or implicit) is presented in the following sections.
3.5.1 Asymptotic Techniques
Early developments regarding asymptotic techniques were only focused on the functions of the
first moment (mean) and the second moment (variance). Numerous methods have been developed
based on second moment formulation using the reliability index (𝛽) proposed by Cornell (1969)
and Ang et al. (1975) including Mean Value First Order Moment (MVFOSM) (Cornell, 1969),
Generalized Safety Index (Hasofer & Lind, 1974), First Order Reliability Method (FORM)
(Rackwitz & Flessler, 1978) and Second Order Reliability Method (SORM) (Breitung, 1991).
27
FORM and SORM are the most developed structural reliability techniques and are discussed
further in the following sections. First Order/Second Order Reliability in Original Space
(FOROS/SOROS) as a modified version of FORM and SORM are explained as well.
First Order Reliability Method (FORM)
The first order reliability method (FORM) is one of the reliable analytical methods which make
use of first and second moments of random variables. The name comes from the fact that FORM
linearizes the limit state function using the first-order of the Taylor series approximation.
Cornell (1969) proposed the original formulation of the FORM in which the limit state function is
linearized at the mean values of the random variables using the first-order Taylor series
approximation. In this formulation both 𝑅 and 𝐿 are assumed to be independent and normally
distributed. The limit state function (𝑔(𝑅, 𝐿)) is consequently normally distributed and therefore,
the moments can be calculated according to the following expressions:
𝑔(𝑅, 𝐿) = 𝑅 − 𝐿 (3-11)
𝑚𝑔 = 𝑚𝑅 − 𝑚𝐿 (3-12)
𝜎𝑔 = √𝜎𝑅2 + 𝜎𝐿
2 (3-13)
where 𝑚𝑔 is mean value of 𝑔(𝑅, 𝐿) and 𝜎𝑔 is standard deviation of 𝑔(𝑅, 𝐿). As 𝑔(𝑅, 𝐿) is normally
distributed, the probability of failure can be calculated using the formula of cumulative distribution
function of normal distribution (𝐹𝑔) as follows
𝑃𝑓 = 𝑃 (𝑔(𝑅, 𝐿) < 0) = 𝐹𝑔(0) = ɸ (𝑥 − 𝑚𝑔
𝜎𝑔) = ɸ (
0 − 𝑚𝑔
𝜎𝑔) = ɸ (
−𝑚𝑔
𝜎𝑔) (3-14)
28
Cornell (1969) used the term in the parenthesis (ɸ (−𝑚𝑔
𝜎𝑔)) to define the reliability index as follows,
see Figure 3-3.
𝛽 =𝑚𝑔
𝜎𝑔= −ф−1(𝑃𝑓) (3-15)
𝛽 =1
𝐶𝑜𝑉𝑔 (3-16)
where ф−1 is the inverse of the cumulative distribution function of the variable with a standard
(Gaussian) normal distribution.
Figure 3-3 Limit state function and probability of failure (Glowienka, 2007).
The definition of reliability index by Cornell (1969) has several advantages. 𝛽 is independent from
the distribution type of 𝑔. The stochastic moments of the variables (mean 𝑚𝑖 and standard
deviation 𝜎𝑖) are only required to calculate the approximate structural reliability. However, the
29
missing parameter in the calculation of reliability index, i.e. distribution type, can influence the
estimate of reliability significantly especially in the case of structural engineering problems
(Brehm, 2011). Madsen et al. (1986) reported the formulation of the limit state as the more
important shortcoming related to this formulation and also discussed the mathematical treatment
of the limit state function as the reason for the shortcoming.
A modified formulation was developed by Hasofer & Lind (1974) so that basic variables are
transformed into standard normal space at the design point on the limit state function. The
reliability index refers to the minimum distance between the origin of the transformed random
variable space to the most likely failure point (design point). Distribution type of 𝑔 is considered
using this advanced FORM.
In this method, an initial design point (𝑥𝑖∗) is determined by assuming values for n-1 of random
variables (𝑋𝑖) (usually mean values). By solving the limit state function (𝑔 = 0), the remaining
random variable is obtained to ensure the design point is located on the failure boundary. Reduced
variables are obtained as follows
𝑥𝑖′∗ =
𝑥𝑖∗ − 𝜇𝑋𝑖
𝜎𝑋𝑖
(3-17)
where 𝑥′∗ is the vector of the transformed random variable. The reliability index is the shortest
distance between the design point and the centre of the transformed random variables. An iterative
procedure is used to estimate the reliability index which is discussed in more detail in Nowak &
Collins (2000). The reliability index and probability of failure are obtained by:
𝛽 = √(𝑥′∗)𝑇(𝑥′∗) (3-18)
𝑃𝑓 = ɸ(−𝛽) (3-19)
As shown in Figure 3-4, in the computation of 𝑃𝑓, the limit state approximation is done linearly at
the design point. However, the original and/or transformed limit state has been proven to be
30
nonlinear in the case of practical applications. Therefore, no exact reliability index value can be
calculated by FORM. SORM was introduced to overcome this imperfection of FORM.
Second Order Reliability Method (SORM)
Second order approximations are introduced by SORM regarding the nonlinear limit state, leading
to a more accurate reliability index, see Figure 3-4. In SORM, a quadric surface at the design point
is considered to approximate the failure surface (Breitung, 1984; Der Kiureghian et al., 1987). The
failure surface is also approximated by a set of tangent hyperplanes by Ditlevsen (1984) and
Madsen et al. (1986) during the same period. Analysis utilized asymptotic approximation as
proposed by Breitung (1984):
𝑃𝑓𝑆𝑂𝑅𝑀≈ ɸ(−𝛽) ∏(1 + 𝛽к𝑖)
−12⁄
𝑛−1
𝑖=1
(3-20)
where к𝒊 is the principal curvature of the limit state related to the minimum distance point, and 𝛽
denotes the reliability index obtained by FORM.
The reliability index determined through the procedure of SORM is expected to be more accurate
compared to that obtained by FORM. As the formulation of SORM is based on second-order partial
derivatives of the limit state function on the transformed space, SORM is more complicated than
FORM.
First Order/Second Order Reliability in Original Space (FOROS/SOROS)
Breitung (1994) and Geyikens (1993) proposed other asymptotic techniques, namely First
Order/Second Order Reliability in Original Space (FOROS/SOROS). Through these methods,
there is no need to transform random variables to a standard normal space. The analysis is done in
the original space of the random variables. In other words, in order to evaluate the probability of
failure in the original space, the full model does not need to be transformed to the normal space,
see Figure 3-5. There are some advantages with this method compared to both FORM and SORM.
31
As an example, through application of either FORM or SORM, a simple linear limit state is
changed to nonlinear, probably leading to complexity and lack of accuracy. However, it remains
linear in either FOROS or SOROS. The probability of failure in these methods can be determined
as:
𝑃𝑓 =(2𝜋)
(𝑛−1)2⁄ 𝑓(𝑥∗)
‖𝛻𝑙(𝑥∗)‖√‖𝑑 𝑒𝑡 (𝑃∗𝑇(𝑥∗)𝐻𝑙(𝑥∗)𝑃∗(𝑥∗) − 𝑃(𝑥∗))‖
(3-21)
where 𝑥∗ and ∇𝑙(𝑥∗) denote the PLM (Point of Maximum Likelihood) and the gradient vector of
the log-likelihood, respectively. 𝑃(𝑥∗) is the projection matrix and 𝑃∗(𝑥∗) is density matrix minus
the projection matrix. 𝐻𝑙(𝑥∗) refers to the Hessian matrix of likelihood. T is the transpose and
matrix determinant is shown by det.
3.5.2 Simulation Based Techniques
Enrico Fermi (1930’s) used Monte Carlo Simulation (MCS) in order to calculate neutron diffusion
but he did not publish any studies related to it. MCS is executed in a way in which joint probability
density functions are used to generate a number of independent samples (N) of the vector of
random variables (X). The limit state function (g(X)) of each sample (𝑥𝑖) is then evaluated, and
the probability of failure is calculated as:
𝑃𝑓𝑠𝑖𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛≈
𝑁𝑓
𝑁 (3-22)
where 𝑁 and 𝑁𝑓 denote the total number of samples and the total number of failures (performance
function less than zero), respectively. The major deficiency of MCS is related to the large number
of analyses required to obtain accurate results; i.e. even in the case of simple problems, a large
number of simulations have to be executed. Importance Sampling (IS) (Harbitz, 1986) and
Adaptive Sampling (Karamchandani et al., 1989) are examples of other approaches that have been
32
proposed based on simulation. In the case of both explicit and implicit limit state functions,
simulation-based approaches can be utilized. However, a particular algorithm needs to be
employed to set the simulation method and the FEM regarding the implicit limit state functions.
Simulation-based approaches may lose their practicality in cases of time consuming structural
analysis such as nonlinear analysis, dynamic analysis and complex systems of structures.
33
Figure 3-4 Standard normal space related to FORM and SORM (Hassanien, 2006).
Figure 3-5 FOROS illustration (Hassanien, 2006).
34
3.5.3 Response Surface Method
A relationship between a collection of input variables and the output response of a system is
designed to be found by a set of statistical and mathematical approaches, defined as the Response
Surface Method (RSM) (Myers & Montgomery, 1995; Khuri & Cornell, 1996). The main goal of
the application of RSM in structural reliability analysis is defined as the approximation of the
original implicit limit state function (Faravelli, 1989; Rajashekhar et. al., 1993; Abdelatif et al.,
1999). The Lagrange interpolation method and the regression method, developed by Abdelatif et
al. (1998) are among the most important response surface methods utilized for solving reliability
problems. The collection process of input data in addition to the design of experiments are factors
which RSM mainly depends on. The main concern regarding the design of experiments is the level
of accuracy in locating the points in the failure point vicinity, given that the failure point is not yet
determined.
3.5.4 Artificial Neural Networks (ANNs)
In structural reliability, limit state functions can be approximated by Artificial Neural Networks.
The application of ANNs in structural reliability has only been investigated slightly (Hurdato et
al., 2001; Gomes, et al., 2004; Deng et al., 2005). An ANN is commonly established based on
groups called layers. There are two typical layers in this model: an input layer and an output layer.
The input layer is responsible for providing data to the network (Murotsu, 1993) and the output
layer is responsible for obtaining the network response to a specific input. To estimate the weights
of the input variables related to the output response, a learning process is considered.
Structural analysis and reliability methods need to be appropriately integrated in order to enable
reliability assessment in real situations. Integrated software developments have been achieved
through several research programs. Table 3-5 presents the list of suitable combinations among
finite element structural analysis, reliability analysis methods and reliability software
(Schueremans and Verstrynge, 2008).
35
Table 3-5 Integration of software for structural reliability analysis.
Software Tool Structural
Analysis (FEM)
Reliability Analysis Method
MC FORM/SORM Use of RS
NASREL NASCOM
COMREL -
SYSREL -
PERMAS-RA PERMAS
PROBAB DIANA
NESSUS ABAQUS/ANSYS
SBRA Included
OPTIMUS NASTRAN
Deterioration refers to physicochemical processes having environmental sources and introduces
the concept of time dependency into reliability analysis due to its gradual influence on the
behaviour and the appearance of buildings. Deterioration results in a reduction in the defined
standard level of performance (Franke & Schuman, 1998).
One or both groups of basic variables (resistance (𝑅) and load effect (𝐿)) can be influenced by
deterioration as well as the analysis model. In other words, deterioration may change masonry
resistance, but also may alter the loading conditions or might affect the analysis model. To be able
to describe the influence of deterioration on reliability assessment fully, all these influences on one
or more of the basic variables in addition to the analysis model need to be examined. Deterioration
processes impacting material strength (and consequently, structural limit states) and those not
directly impacting structural limit states (but influencing the phenomenological limit states) cannot
be clearly differentiated. For example, as residual strength is influenced by freeze-thaw cycles,
36
they can be placed in the first category. On the other hand, building appearance is also influenced
by spalling (Maes, 1999).
Existing masonry structures, especially the ones with a service life of more than 50 years, which
is proposed as the standard design service-life, experience many deterioration processes (Maes,
1999). The most well-known deterioration processes related to masonry structures are presented
in Table 3-6. Frank & Schuman (1998) provided a more detailed overview in this regard.
Table 3-6 Deterioration mechanisms (Maes et al., 1999).
Deterioration Type Cause Consequences Imposed to
Freeze-thaw cycles -Temperature change
- Moisture content - Spalling
- Section resistance
- Appearance
Chemicals
(e.g. sulfates,
chemical spill,
acids, road salt)
- Diffusion
- Absorption,
- Capillary suction
- Moisture content
- Concentration difference
- Chemical reaction
- Disintegration
- Disfiguration
- Efflorescence
- Salt crystallization
- Discoloration
- Material resistance
- Appearance
Abrasion - Particulate - Material Erosion - Material resistance
- Appearance
Pollution
(𝐶𝑂2, 𝑂2, 𝑁𝑂𝑥) - Chemical agents
- Disintegration
- Discoloration
- Material resistance
- Appearance
Solar radiation - UV, IR - Disintegration
- Discoloration
- Material resistance
- Appearance
Biological and
microbiological
agents
- Fungi
- Bird droppings
- Biological deposits
- Vegetation or plants
- Disintegration
- Spalling
- Material resistance
- Appearance
37
3.7.1 Modeling of Deterioration
In order to integrate deterioration into reliability analysis, one or more of the basic variables related
to either resistance or load effects, analysis model uncertainty (𝐴) or limit state model uncertainty
(𝜃) have to account for their time-dependency, see Table 3-7. Mori et al. (1993) proposed several
possible degradation functions presented in Table 3-8.
Table 3-7 Deterioration modeling (Maes et al., 1999).
Basic Variables Time Dependency
Load effects (𝐿) 𝐿(𝑡) = 𝑆0. 𝑓𝑑𝑒𝑔,𝐿(𝑡)
Resistance (𝑅) 𝑅(𝑡) = 𝑅0. 𝑓𝑑𝑒𝑔,𝑅(𝑡)
Analysis model parameters (A) 𝐴(𝑡) = 𝐴0. 𝑓𝑑𝑒𝑔,𝐴(𝑡)
LS model uncertainty (𝜃) 𝜃(𝑡) = 𝜃0. 𝑓𝑑𝑒𝑔,𝜃(𝑡)
𝐿0, 𝑅0, 𝐴0, 𝜃0 are random variables reported at a reference time
Table 3-8 Degradation functions (Mori et al., 1993).
Model Formula
Linear 𝑓𝑑𝑒𝑔(𝑡) = 1 − 𝑐. 𝑡
Parabolic 𝑓𝑑𝑒𝑔(𝑡) = 1 − 𝑐. 𝑡2
Square Root 𝑓𝑑𝑒𝑔(𝑡) = 1 − 𝑐. √𝑡
c is a constant value
38
Lack of data is one of the most important problems in assessing historical structures. This
necessitates finding methods to overcome this shortcoming. Available prior information obtained
by either expert opinion or testing results can be used to increase the sample size. This enlargement
can consequently result in better estimation of the probabilistic models of materials. Bayes (1763)
developed Bayes’ theorem as a data updating method which is discussed briefly as follows.
In this method, the stochastic distribution of the prior parameters can be updated using new data
(e.g. test data). The same type of distribution is usually considered for both the prior and posterior
distributions. The distribution of the compressive strength of a masonry unit before and after the
updates besides the associated likelihood distribution was illustrated by Glowienka (2007), see
Figure 3-6.
Figure 3-6 Example of the updating compressive strength of a masonry unit using Bayes’
theorem (Glowienka, 2007).
Bayes’ theorem is expressed as the following relationship
𝑓𝜃(𝜃|𝑥) =𝑓𝑋(𝑥|𝜃). 𝑓𝜃(𝜃)
𝑓𝑋(𝑥)=
𝑓𝑋(𝑥|𝜃). 𝑓𝜃(𝜃)
∫ 𝑓𝑋(𝑥|𝜃). 𝑓𝜃(𝜃)𝑑𝜃 (3-23)
39
where 𝑓𝑋(𝑥|𝜃) is the probability density function of a random variable dependent on 𝜃. 𝑓𝜃(𝜃) is
the prior probability distribution function of the corresponding vector of parameters and 𝑓𝜃(𝜃|𝑥)
is the posterior probability distribution function of the vector of parameters. 𝑓𝑋(𝑥|𝜃) can be
estimated using the likelihood distribution of measured data. In the case of uncorrelated data,
𝑓𝑋(𝑥|𝜃) can be derived as follows
𝑓𝑋(𝑥|𝜃) = 𝑓𝑋(𝑥1|𝜃). … . 𝑓𝑋(𝑥𝑛|𝜃) = ∏ 𝑓𝑋(𝑥𝑖|𝜃) = 𝐿(𝜃|𝑥) 𝑛𝑖=1 (3-24)
The integral in Equation 3-23 sets the area under the probability distribution function to one, in
order to convert 𝑓𝜃(𝜃|𝑥) to a true probability of failure function. Therefore, the integral can be
considered as a constant coefficient. The posterior probability distribution function is commonly
formulated as
𝑓𝜃(𝜃|𝑥) = 𝑘 . 𝐿(𝜃|𝑥) . 𝑓𝜃(𝜃) (3-25)
The possibility of continuous application is one of the most important advantages of Bayes’
theorem as the updating can be done as often as required. For the next updating cycle, the estimated
posterior distribution serves as the prior one. A larger sample size would lead to more realistic
approximation of the distribution of 𝑋. However, updating will become inefficient with
enlargement of the sample size. Therefore, an evaluation is needed beforehand to determine if the
updating will be effective.
Considering all of the concepts mentioned above, the process of reliability analysis can be
summarized in a flowchart as shown in Figure 3-7.
40
Figure 3-7 Reliability assessment framework.
Limit State Function
- Resistance (R)
- Load Effect (L)
Random Variables
- Asymptotic Techniques
- Simulation-based Techniques
- Response Surface Method
- Artificial Neural Networks
(ANNs)
Reliability Analysis
𝛽 & 𝑃𝑓
𝛽 > 𝛽𝑇
𝑃𝑓 < 𝑃𝑓𝑇
Deterioration
𝛽 < 𝛽𝑇
𝑃𝑓 > 𝑃𝑓𝑇
Reliable Non-reliable
41
Reliability assessment of existing structures subjected to seismic loading is one of the most
challenging and complex assessments involving different aspects and concepts in the field of
structural engineering. In addition to the non-linear behaviour of the structure, there are
uncertainties in the nature of possible ground motions. Due to the complexity, new influential
parameters and almost different reliability assessment procedures, reliability assessment under
seismic loading is discussed separately in this section. Shah & Dong (1984) developed a method
to assess the reliability of existing structures subjected to probabilistic earthquake loadings. Based
on their studies, Monte-Carlo simulation can be used to obtain the distribution of the maximum
response, given the characteristics of an earthquake of strong motion. The procedure to do so is:
1) A unit power spectrum (Sf) and time envelope function (t) are constructed based on the
predominant frequency (fp) and duration (TD).
2) Monte-Carlo simulation is used to generate a set of K time histories (𝑎𝑘(t), k = 1, 2,…, K).
3) A desired peak acceleration (A = a) is selected and 𝑎𝑘(𝑡) is scaled using this acceleration.
4) A structural model is built and hysteretic characteristics selected.
5) An inelastic dynamic analysis is done for each sample 𝑎𝑘(𝑡) to estimate the maximum response
of the 𝑖th critical value for the 𝑗th member (𝑋𝑖𝑗,𝑘) for each time history (k).
6) Data (𝑋𝑖𝑗,𝑘(𝑘 = 1, 2, … , 𝐾)) are fitted to a probability distribution function. In other words, the
cumulative distribution function being conditional on 𝑎 (𝐹𝑋𝑖𝑗(𝑥𝑖𝑗/𝑎)) for 𝑖th critical variable and
the 𝑗th member (𝑋𝑖𝑗) is calculated in a way to fit the obtained data appropriately. The Gumbel
Extreme Value distribution is proved to fit the data well:
𝐹𝑋𝑖𝑗(𝑥𝑖𝑗|𝑎) = 𝑒𝑥𝑝 {−exp (−
𝑥𝑖𝑗 − 𝑏𝑖𝑗(𝑎)
𝑐𝑖𝑗(𝑎))} (3-26)
Given A = a, 𝑏𝑖𝑗(𝑎) and 𝑐𝑖𝑗(𝑎) are estimated by data fitting to the distribution. Given the capacity
of 𝑗th member for 𝑖th critical value (𝑦𝑖𝑗), the conditional reliability (𝑅𝑖𝑗) can be calculated by:
Rij = Reliability of the jth member regarding the ith critical value
42
𝑅𝑖𝑗 = 𝑃 [𝑋𝑖𝑗 < 𝑦𝑖𝑗] = 𝐹𝑋𝑖𝑗(𝑦𝑖𝑗) = 𝑒𝑥𝑝 {−𝑒𝑥𝑝 (−
𝑦𝑖𝑗 − 𝑏𝑖𝑗(𝑎)
𝑐𝑖𝑗(𝑎))} (3-27)
Considering the values obtained related to 𝑅𝑖𝑗, failure is anticipated to occur in members with low
reliability. The boundary of structural reliability is:
min(𝑅𝑗) ≥ 𝑅 ≥ ∏ 𝑅𝑗
𝑗
(3-28)
As there is only a slight difference between the two bounds, as the large inelastic deformation
generally concentrates on only weak members of a structure subjected to seismic loads, only the
lower bound is used as structural reliability:
𝑅𝑇𝑜𝑡𝑎𝑙 = ∏ 𝑅𝑖𝑗
𝑖𝑗
(3-29)
It should be considered that the reliability estimation given in the previous steps is conditional on
the PGA (A = a) and the capacity (𝑌𝑖𝑗 = 𝑦𝑖𝑗). The final probability of failure can be calculated by
knowing the probability distribution function of A and 𝑌𝑖, as follows. Two sequential events with
the same characteristics (A, fp and TD) show maximum load effect and demand independently.
Therefore, the cumulative demand distribution in n-year life time can be expressed as:
𝐹𝑋𝑖,𝑛= [𝐹𝑋𝑖
(𝑥𝑖)]𝑛 (3-30)
The seismic demand is considered only as a function of the peak ground acceleration A, in almost
all applied seismic hazard analysis. In other words, determinist estimation of 𝑓𝑝 and 𝑇𝐷 are
expressed by the local soil conditions and past experience. Therefore, in Equation 3-31 we have:
43
𝐹𝑋𝑖(𝑥𝑖) = ∫ 𝐹𝑋𝑖
(𝑥𝑖|𝑎) 𝑓𝐴(𝑎) 𝑑𝑎 (3-31)
where 𝑓𝐴(𝑎) denotes the probability distribution function of A during any one year. The reliability
of 𝑗th member in a lifetime of n years is:
𝑅𝑗 = ∫ 𝐹𝑋𝑖,𝑛(𝑧) 𝑓𝑌𝑖
(𝑧) 𝑑𝑧 (3-32)
in which 𝑓𝑌𝑖(𝑧) is the density function of the capacity of a critical variable.
It is obvious that identification of weak members by this method and their strengthening would
lead to improvement of the overall reliability of the structure. As either of the loading parameters
including intensity, peak acceleration, duration or frequency, cannot effectively be the
representative of total load effects or responses, estimation of structural responses or load effects
encounter important uncertainties. Uncertainties stem from three different sources including:
I) Seismic loads
II) Structural capacity
III) Load effects due to inadequate representation of load and consequently, calculation of
load effects
According to Shah & Dong (1984), the uncertainty which dominates the overall uncertainty in
final reliability analysis is the uncertainty in the ground motion. Therefore, to achieve more
accurate reliability analysis results in the case of structures subjected to seismic loads, estimation
of the loads or evaluation of the seismic hazard for the site should be done with great care.
44
Chapter IV: Stochastic Modeling of Applied Load
Structural members have to be designed to be able to carry their applied loads during their service
life. The term “load” denotes the actions leading to stress development in the members. Moreover,
effects that may be influential on the capacity of a member are also considered as loads (e.g.
corrosion). In this thesis, the term loads refer to the applied forces on members (e.g. dead load or
snow load). Effects like corrosion are not considered.
As explained in Chapter 3, in order to calculate reliability of a structure, limit state functions have
to be estimated. Limit state functions take into account both loads and resistance. Therefore, to be
able to calculate an accurate probability of failure and reliability index, stochastic modeling of the
applied loads needs to be as realistic as stochastic modeling of the resistance.
In this chapter, stochastic models of the most common applied loads on historical structures are
presented.
Generally, loads vary over time and space and, therefore, can be expressed as random variables. It
is complex and not efficient to assess the load stochastically for every design case. Therefore,
general stochastic models for different loads have been derived to be used in the estimation of the
probability of failure and the reliability index. Load actions have been modeled based on the kind
of the load. There are four categories commonly used to classify loads:
Dead Load: dead loads usually refer to the self-weight of the structure. They apply permanently
on a member and therefore, there is minor scatter around the mean - the variation is low.
Live Load: live loads refer to the loads from occupancy. These kinds of loads are time dependent
and vary significantly with time leading to higher coefficients of variation compared to other types
of loads.
Wind and Snow loads: these loads are also time dependent and change considerably with time.
45
Accidental loads: accidental loads refer to those with large quantity but short duration of
occurrence. The probability of occurrence of these kinds of loads is small during the observation
period. Earthquakes, explosions and avalanches are typical examples of accidental loads.
Dead load refers to the weight of structural members and their related components in addition to
the partitions and permanent equipment. Typically, dead load contributes about 70% of the total
load for ordinary residential structures and offices, which makes dead loads influential on the
evaluation of structural behaviours. Dead load has been shown to be less influential on the
probability of failure of concentrically compressed masonry members because of the small scatter
and sensitivity value. Shear failure occurs mostly under minimum axial load. Therefore, shear
walls are expected to be more sensitive to dead load and accordingly dead load is required to be
assessed in more detail. Dead loads are assumed to be constant over the service life. However, this
assumption is uncertain. Dimensional tolerances and uncertainty in the unit weight of materials
are the main causes of dead load uncertainty. In addition, the process of converting the load into
load effects leads to uncertainty. This process can be completed accurately in case of indeterminate
structures only if the sequence of construction is known.
Bartlett et al. (2003) presented a summary of the statistical parameters for loads which have been
used to calibrate the loads and load combination criteria for the 2005 National Building Code of
Canada (NBCC). Based on their literature review, the bias coefficient (the ratio of mean to nominal
values) and the coefficient of variation (CoV) (the ratio of standard deviation to mean value) of
dead load has been reported to be from 1.00 to 1.05 and 0.06 to 0.09, respectively. CoVs of
modeling and analysis which are usually assumed to be unbiased, are reported to be in the range
of 0.03 to 0.07. Therefore, the CoV of the dead load effect increases to a range of 0.05 to 0.10. A
bias of 1.05 and a CoV of 0.10 (reported by Ellingwood et al., 1980) were adopted by different
organizations (Standards Association of Australia, 1985, South African Bureau of Standards, 1989
and European Committee for Standardization, all reported by Kemp et al., 1998; Tabsh, 1997;
Ellingwood, 1999), see Table 4-1. In the case of counteraction of the dead load effects with the
effects of other loads, a normal distribution with a bias of 1.00 and a CoV of 0.10 was assumed
for dead load.
46
They (Bartlett et al., 2003) also expressed that there was a common assumption of normal
distribution of dead loads, perhaps due to the tendency of tolerances to be distributed normally.
However, there are no actual data available to verify this assumption. To some extent, the statistical
characteristics of dead loads depend on the type of construction materials and the size of the
structure. In other words, in the case of large structural components, weight is not very sensitive
to absolute dimensional tolerances which results in a reduction in dead load bias coefficients and
coefficients of variation. All the above-mentioned studies focused on steel and concrete structures.
Therefore, accordance of the proposed statistical parameters with masonry structures, especially
historic ones, is in doubt.
Brehm (2011) determined the stochastic model appropriate for the dead load of a masonry bracing
wall. He expressed that determination of dead load is based on the density of material and the
volume of the member. The density of a material varies over the member (e.g. concentration of
aggregate). However, members are usually considered to be homogeneous. Therefore, the
influence of the scatter of material density on dead load is considered to be negligible and
consequently dead load is assumed to distribute uniformly over a member. Mortar density is
assumed to be similar to the masonry unit density. Therefore, mortar joints, even the thicker one
(general purpose mortar, GPM), were expressed to have negligible effect on the scatter of dead
load. As the research was based on new masonry units produced in factories and plants, the scatter
of unit dimension was also considered to be negligible. However, the volume of a masonry wall
was assumed to be variable due to workmanship and provided a coefficient of variation regarding
geometry. Brehm (2011) also introduced the self-weight of the concrete slabs as the most
influential parameter on self-weight of the structure and reported influence of the coefficients of
variation of the concrete density and concrete slab thicknesses on the scatter of the slab self-weight.
He suggested a bias factor of 1.0 and coefficient of variation of 0.06 for dead load with normal
distribution (log-normal distribution in some cases), see Table 4-1.
The Joint Committee on Structural Safety (JCSS) probabilistic model code (2001) assumes self-
weight as a time independent variable with a probability of occurrence at an arbitrary point-in-
time almost equal to one. The uncertainties in the magnitude of self-weight are described as small
compared to other kinds of loads. A Gaussian distribution is suggested to be considered for the
weight density and dimensions of a structural member. To make the calculation of self-weight
47
simple, a Gaussian distribution is also assumed as the distribution type of self-weight. The JCSS
code presented the CoV of masonry weight density equal to 5% and mean values and standard
deviations for deviation of cross-section dimensions of masonry members from their nominal
values as 0.02 and 0.04 for unplastered and 0.02 and 0.02 for plastered, respectively. JCSS states
their reported values are derived from a large population from various sources.
Table 4-1 Summary of statistical parameters for dead load.
References Bias COV Distribution Type
Ellingwood et al. (1980) 1.05 0.1 -
Brehm (2011) 1.0 0.06 Normal/Log-normal
Bartlett et al. (2003) 1.0 0.10 Normal
The load associated with use and building occupancy, including the weight of people, equipment,
furniture and materials in storage, is referred to as live load. Live load varies over time and
location. Total live load is the sum of contributions categorized as permanent live load and
transient live load (Ferry Borges & Castanheta, 1971). Permanent live load is the component of
live load remaining constant for a period of time (usually the duration of the tenancy) but still
removable, e. g. furnishings, while transient live load is associated with a short or instantaneous
duration, e. g. group of people. The total live load is sum of these two components, see Figure 4-
1. Due to the large variation associated with live load, especially as a result of spatial variation, it
is complex to determine a stochastic model for the live load. Presentation of a detailed description
regarding the theory of live load modeling is beyond of scope of this thesis.
48
Figure 4-1 Representation of live load (Brehm, 2011).
A simplified approach to model live load probabilistically is (McGuire & Cornell, 1974; Pier &
Cornell, 1973)
𝑊(𝑥, 𝑦) = 𝑌 + є (𝑥, 𝑦) (4-1)
where W refers to load intensity (psf) applied on a floor, Y is a random variable modeling the
mean of load on the floor and є (𝑥, 𝑦) is a stochastic process with zero mean associated with the
deviations from the average. CIB (1989) outlined a similar model regarding live load modeling as
follows:
𝑊(𝑥, 𝑦) = 𝑚 + 𝑉 + 𝑈(𝑥, 𝑦)
(4-2)
where 𝑚 is the deterministic mean of live load depending on the usage type of building (e. g.
residential, office and so on). 𝑉 represents a random variable (with mean zero) accounting for the
49
variation of the load associated with two independent areas on the same floor or between floors
(A1 and A2). 𝑈(𝑥, 𝑦) stands for a random field concerning the spatial variation related to the load.
It is too sophisticated to determine the structural response by evaluating of the increment of every
single load based on Eq. 4-2. Therefore, a methodology was introduced by which an equivalent
load with uniform distribution (q) leading to the same load effect is determined as a preferable
alternative solution.
4.3.1 Total Live Load
Ellingwood & Culver (1977) presented mean and variance of unit load (U) as
𝐸 [𝑈] = µ (4-3)
Var [U] = 𝜎2 +1
𝐴∫ ∫ ∫ ∫ 𝐶𝑜𝑉 [є (𝑥, 𝑦), є (𝑢, 𝑣)] 𝑑𝑥𝑑𝑦𝑑𝑢𝑑𝑣 … (4-4)
where µ is mean of all unit floor loads associated with office buildings and 𝜎2 is variance in
individual floor.
They then simplified Eq. 4-4 as
Var [U] = 𝜎2 +𝜎𝑠
2
𝐴 (4-5)
in which 𝜎𝑠 denotes an experimental constant.
The mean and variance of the equivalent load with uniform distribution leading to same effect
(𝑞) are
E [q] = µ (4-6)
50
𝑉𝑎𝑟 [𝑞] = 𝜎2 +𝜎𝑠
2
𝐴𝜅 (4-7)
𝜅 =∬ 𝐼2(𝑥, 𝑦) 𝑑𝑥𝑑𝑦
𝐴
[∬ 𝐼(𝑥, 𝑦) 𝑑𝑥𝑑𝑦𝐴
]2 (4-8)
In which 𝐼(𝑥, 𝑦) is the influence function for the response quality sought. A is the influence area
over which I(x,y) assumes nonzero values. Considering multi-storey column loads, there is a slight
difference in the first term of Eq. 4-1 from that of floor loads (McGuire & Cornell, 1974). In
practice, the difference is negligible and may be ignored.
Ellingwood & Culver (1977) also determined the mean and variance of the maximum sustained
uniformly distributed load (𝐿𝑚𝑎𝑥 ) based on the National Bureau of Standards (NBS) load survey
results (office buildings) in the U.S.
𝐸[𝐿𝑚𝑎𝑥] = 0.924 + 1.853/√𝐴 𝑘𝑁/𝑚2 (4-9)
𝑉𝑎𝑟[𝐿𝑚𝑎𝑥] = 0.033 + 4.024/𝐴 𝑘𝑁/𝑚4 (4-10)
A similar analysis done earlier by McGuire & Cornell (1974) using U.K. data resulted in
𝐸[𝐿𝑚𝑎𝑥] = 0.867 + 2.101/√𝐴 𝑘𝑁/𝑚2 (4-11)
𝑉𝑎𝑟[𝐿𝑚𝑎𝑥] = 0.026 + 3.194/𝐴 𝑘𝑁/𝑚4 (4-12)
It has been shown that there is close agreement between 𝐸[𝐿𝑚𝑎𝑥] values based on the NBS and
U.K. survey results (Ellingwood & Culver, 1977). Regarding the maximum total equivalent
51
uniformly distributed load (q), the mean and variance were determined using the NBS survey
results (Ellingwood & Culver, 1977) as
𝐸[𝑞] = 0.895 + 7.588/√𝐴 𝑘𝑁/𝑚2 (4-13)
𝑉𝑎𝑟[𝑞] = 0.032 + 4.024/𝐴 𝑘𝑁/𝑚4 (4-14)
And the analysis of McGuire & Cornell (1974) using U.K. data resulted in
𝐸[𝑞] = 0.713 + 11.135/√𝐴 𝑘𝑁/𝑚2 (4-15)
𝑉𝑎𝑟[𝑞] = 0.026 + 3.194/𝐴 𝑘𝑁/𝑚4 (4-16)
Ellingwood & Culver (1977) also showed that there is a significant difference between the mean
values of the two expressions in terms of small influence areas. However, the difference is not
related to the difference in the NBS and U.K. load survey results. It is attributed to the parameters
which are used in the extraordinary load model.
JCSS (2001) based its suggested procedure regarding the determination of the stochastic
characteristics of permanent live load on the formula derived by Rackwitz (1995) who generalized
the stochastic moments of live load (similar to Ellingwood & Culver, 1977) as follows
𝐸[𝑞] = 𝐸[𝑊(𝑥, 𝑦)] = 𝑚𝑞 (4-17)
𝑉𝐴𝑅[𝑞] ≈ 𝜎𝑉2 + 𝜎𝑈
2 ∙ 𝐴0
𝐴 ∙ 𝜅 (4-18)
52
𝜅 ≈ ∫ 𝑖2(𝑥, 𝑦)𝑑𝐴
(∫ 𝑖(𝑥, 𝑦)𝑑𝐴)2 (4-19)
where 𝑚𝑞 refers to the mean of the live load. 𝜎𝑉 and 𝜎𝑈 are the standard deviations of V and U,
respectively. A refers to the effective area and 𝐴0 is the reference area of the load measure. 𝜅 is
defined based on the required stress resultants and the structural system. Values of 𝜅 are
recommended by Melchers (1999), JCSS (2001) and Hausmann (2007) (reported by Brehm
(2011)).
Table 4-2 Parameters related to probabilistic modeling of live load (JCSS, 2001).
Type of
Use
Ref.
area Permanent Load Transient Load
𝐴0
(𝑚2)
𝑚𝑝𝑒𝑟𝑚
(𝑘𝑁/𝑚2)
𝜎𝑉 (𝑘𝑁/𝑚2)
𝜎𝑈 (𝑘𝑁/𝑚2)
1/λ
(a) 𝑚𝑡𝑟𝑎𝑛
(𝑘𝑁/𝑚2) 𝜎𝑈
(𝑘𝑁/𝑚2)
1/v
(a)
dp
(d)
Office 20 0.5 0.30 0.60 5 0.20 0.40 0.3 1-3
Lobby 20 0.2 0.15 0.30 10 0.40 0.60 1.0 1-3
Residential 20 0.3 0.15 0.30 7 0.30 0.40 1.0 1-3
Hotel 20 0.3 0.05 0.10 10 0.20 0.40 0.1 1-3
Hospital 20 0.4 0.30 0.60 5-10 0.20 0.40 1.0 1-3
Laboratory 20 0.7 0.40 0.80 5-10 - - - -
Library 20 1.7 0.50 1.00 >10 - - - -
Classroom 100 0.6 0.15 0.40 >10 0.50 1.40 0.3 1-5
Sales room 100 0.9 0.60 1.60 1-5 0.40 1.10 1.1 1-14
Industrial
Light 100 1.0 1.00 2.8 5-10 - - - -
Heavy 100 3.0 1.50 4.10 5-10 - - - - aload fluctuation rate baverage load duration
JCSS (2001) suggested a similar method in order to calculate the transient live loads. As a result
of the large scatter of transient live loads, a stochastic field is used to model them. Therefore 𝜎𝑉 is
set to be zero. The importance of transient live load is more obvious in the case of small values of
the influence area (A) such as balconies or stairs. A summary of required parameters for the live
load model based on JCSS (2001) is presented in Table 4-2 where dp is duration of transient load
(year), λ is occurrence rate of permanent live load changes in [1/year] and v is occurrence rate of
transient live load changes in [1/year].
53
Bartlett et al. (2003) considered Eq. (4-20) derived by simulation results (Ellingwood and Culver,
1977) to calculate the expected value of maximum equivalent uniformly distributed live load
during a 50-year reference period (�̅�𝑚𝑎𝑥 (𝑘𝑁/𝑚2)).
�̅�𝑚𝑎𝑥 = 0.895 + 7.6/√𝐴𝐼 (4-20)
in which 𝐴𝐼 denotes the influence area (𝑚2). NBCC (1999) presented the corresponding nominal
uniformly distributed live load (𝐿 (𝑘𝑁/𝑚2)) as follows
𝐿 = 2.4 (0.3 + √9.8/𝐵) (4-21)
where 2.4 refers to the specified live load (𝑘𝑁/𝑚2) and B is the tributary area (𝑚2). ASCE7- 98
(ASCE 2000) considered the influence area to be equal to the tributary area regarding two-way
slabs and twice and four times of the tributary area for interior beams and columns, respectively.
Based on the above-mentioned formula, the bias can be calculated easily by dividing Eq. (4-20)
by Eq. (4-21) depending on the element type and its related influence area (Bartlett et al., 2003).
It was shown that with reduction of the nominal live load through increasing influence area, the
bias suggested by ASCE (2000) changes in accordance with ASCE-98 instead of Eq. (4-20)
(Bartlett et al., 2003).
The simulation results for office buildings done by Ellingwood and Culver (1977) was also used
to calculate the CoV of the maximum equivalent uniformly distributed live load regarding an office
floor over a 50-year reference period. The variance (𝜎𝐿𝑚𝑎𝑥
2 (𝑘𝑁/𝑚2)) of maximum live load during
50 years was reported to be
𝜎𝐿𝑚𝑎𝑥
2 = 0.033 + 4.025/√𝐴𝐼 (4-22)
By dividing the square root of Eq. (4-22) by Eq. (4-20), the CoV of the live load is calculated.
Bartlett et al. (2003) considered a bias of 0.9 and a CoV of 0.17 for a 50-year maximum live load.
54
Moreover, they found out that the CoV values related to beams reported by Kariyawasam et al.
(1997) are smaller than those reported for columns. Also, the CoVs are almost insensitive to the
influence area (Bartlett et al., 2003).
In order to transform the live load into a live load effect, two factors including modeling and
analysis were considered by Bartlett et al. (2003). The modeling factor was used to address the
simplification of the actual live load as an equivalent uniformly distributed load. This factor was
reported to be unbiased with a CoV of 0.20 (Ellingwood et al., 1980). These values were assumed
to be related to columns and were decreased to a CoV of 0.1 related to beams by Kariyawasam et
al. (1997). Analysis factors were used to transform the live load to load effects. Ellingwood et al.
(1980) suggested an assumption of unbiased with a CoV of 0.05 regarding this factor. Bartlet et
al. (2003) determined the combined effect of the modeling and analysis factors having a bias of
1.0 and a CoV of 0.206 and assumed them to be time-independent, see Table 4-3.
Table 4-3 Statistical parameters for live load recommended by Bartlett et al. (2003).
Type Bias CoV
50-year maximum live load 0.9 0.17
Transformation to load effect 1.0 0.206
Table 4-4 Summary of recommended stochastic parameters provided by different authors.
Reference Classification Mean CoV Bias q
CIB (1989)
Office 2.64 0.19 - 2.42
Residential 1.73 0.20 - 1.57
Classroom 1.63 0.12 - 1.63
Rackwitz (1995)
Office 1.81 0.20 - 1.54
Residential 1.52 0.29 - 1.32
Classroom 2.65 0.36 - 2.23
Glowienka (2007)
Office 2.51 0.37 - 2.09
Residential 1.81 0.28 - 1.59
Classroom 3.61 0.22 - 3.49
Brehm (2011) Residential - 0.2 1.1 -
55
There are other recommendations regarding the stochastic parameters of live load determined by
various researchers. A summary of recommended stochastic parameters provided by different
authors is presented in Table 4-4.
4.3.2 Point-in-time Live Load
Some studies have been directed to statistical analysis of point-in-time live load (Ellingwood &
Culver, 1977; Chalk & Corotis, 1980; Ellingwood et al., 1980). To model the load, a sustained
component (2-8 years duration) and an extraordinary component (one or twice per year) may be
considered. Bartlett et al. assumed a duration of 6 months for both components and derived point-
in-time live load parameters with a procedure presented by Wen (1990). The bias and the CoV
were reported to be 0.273 and 0.674, respectively. Transformation factors were assumed to be
same as those for a 50-year maximum live load. A Weibull distribution has been assumed for
point-in-time live load (Wen, 1990). The stochastic characteristics of point-in-time live load are
summarized in Table 4-5.
Table 4-5 Stochastic characteristics of point-in-time live load.
Load Type Bias CoV
Point-in-time Load 0.273 0.674
4.3.3 Distribution
It is recommended that the permanent component of live load be modeled by a Gamma distribution
(Pier & Cornel, 1973; Chalk & Corotis, 1980; Ellingwood, 1980; JCSS, 2001). Although there is
currently a lack of accurate data obtained from load measurements regarding transient live load,
the mean and standard deviation of the load was stated to be of almost equal quantity. Therefore,
an exponential distribution such as a Gamma distribution was also suggested to be used for the
transient component of live load (Pier & Cornel, 1973; Chalk & Corotis, 1980; Ellingwood, 1980;
Rackwitz, 1995). A Gamma distribution was recommended to be used to represent the Live load
56
effect (modeling and analysis factors) by Ellingwood et al. (1980) and Kariyawasam et al. (1997).
However, Bartlett et al. (2003) assumed a normal distribution for the overall effect of modeling
and analysis. The reason for this assumption was stated to be the dominance in the overall effect
uncertainty of a single factor (the modeling uncertainty) as well as the lack of data associated with
the modeling uncertainty.
Table 4-6 Summary of recommendations for distribution type.
Variable Type Distribution Type Reference
Permanent Live Load Gamma distribution
Pier & Cornel (1973); Chalk
& Corotis (1980); Ellingwood
(1980); JCSS (2001)
Transient Live Load Gamma distribution
(Pier & Cornel (1973); Chalk
& Corotis (1980); Ellingwood
(1980); Rackwitz (1995)
Modeling and Analysis Factors Gamma distribution
Ellingwood et al. (1980);
Kariyawasam et al. (1997)
Normal distribution Bartlett et al. (2003)
Total Live Load Gumbel distribution
Allen (1975); MacGregor
(1976); Ellingwood et al.
(1980); European Committee
for Standardization (1994);
Kariyawasam et al. (1997);
Tabsh (1997); Ellingwood
(1999); Brehm (2011)
Point-in-time Live Load Weibull distribution Wen (1990)
The maximum live load over the lifetime of a structure has been assumed to be represented by a
time-independent random variable using a Gumbel distribution (Allen, 1975; MacGregor, 1976;
Ellingwood et al., 1980; European Committee for Standardization, 1994; Kariyawasam et al.,
57
1997; Tabsh, 1997; Ellingwood, 1999; Brehm, 2011). Considering the live load as a Poisson
process may lead to obtaining different reliabilities in some cases in comparison with using an
equivalent Gumbel distribution. However, the assumption of representing the maximum live load
by a Gumbel distribution has been accepted in practice. A summary of the recommended
distribution types for different components and parameters as well as total live load is presented
in Table 4-6.
Wind is a natural phenomenon created by the differences in atmospheric temperature leading to
differences in air-pressure. Wind loads are one of the main horizontal loads on structures and
depend on many parameters. Among the parameters wind velocity (𝜐) and gust intensity,
estimated based on annual extremes, are the main influential ones. The shape of the
member/structure, the surrounding roughness, the dynamic behaviour of the structure and its
altitude above ground are other influential parameters. It is worth mentioning that to account for
the influence of altitude above ground, the wind load at an altitude of 10 m above ground is
considered as a reference value. In structural analysis, wind loads refer to the stresses or forces on
members due to the applied wind. Various studies have examined the uncertain nature of wind
load and estimated its related stochastic parameters, but as wind load parameters are strongly
dependent on the geographical region of the structure, the NBCC procedure for wind load
calculation as well as studies with respect to the regions of Canada are mostly presented here.
A simple gust factor approach was adopted by the 1995 NBCC to calculate wind load on structures.
This simplified procedure estimated the wind pressure (𝑝) against the surface of structures as
𝑝 = 𝑞𝐶𝑒𝐶𝑝𝐶𝑔 (4-23)
where 𝑞 is the reference velocity pressure, 𝐶𝑒 is the exposure factor, 𝐶𝑝 is the external pressure
factor and 𝐶𝑔 is the gust factor. The reference wind pressure is influenced by the climatic condition
of the region. With the assumption of constant climatic condition, the maximum distribution of the
wind pressure during the service life of a structure can be estimated by the annual maximum
58
pressure distribution. The reference wind pressure is normally derived using Bernoulli’s rule and
the wind velocity. The 1 in 𝑇 year reference wind pressure (𝑞𝑇) is estimated based on the 1 in 𝑇
year wind velocity (𝑉𝑇) (Appendix C of the 1995 NBCC) as follows
𝑞𝑇 = 1
2𝜌𝑉𝑇
2 (4-24)
where 𝜌 is the density of air which strongly depends on the air temperature (e.g. 1.2929 𝑘𝑔/𝑚3
considering dry air at 0 ºC and standard pressure of atmosphere). The CoV of air density was
estimated to be roughly 2.5%-5.0% (Ellingwood et al. 1980; MacGregor et al. 1997) and is
significantly less than the CoV of 𝑉𝑇2. Therefore, air density is assumed to be deterministic. NBCC
specifies values for each of the variables used in the calculation of wind load. However, they are
uncertain.
1 h wind velocity data related to over 100 sites (mostly airports) with 10-22 years of record are
used to derive the reference velocity pressures presented in Appendix C of the 1995 NBCC.
Locations which are listed in Appendix C are the centres of urban areas and, therefore, mostly
differ from the locations from where the data were collected. Interpolation was used to determine
the 1 in 30-year wind velocities and dispersion parameters related to these urban centres from the
collected data. Moreover, for each site, 1 in 10-year and 1 in 100-year velocities were derived as
well, and finally, 1 in 10-year, 1 in 30-year and 1 in 100-year reference pressures were calculated
by means of Eq. 4-24. It should be mentioned that there are no available documents including the
transformation factors used to calculate the velocities and dispersion parameters of the locations
listed in Appendix C from the sites where data were collected. The CoVs of maximum annual
wind velocities related to data collection sites were lower than the corresponding CoVs back-
calculated from the wind pressures of sites listed in Appendix C (Bartlett et al., 2003).
Bartlett et al. (2003) investigated the stochastic characteristics of wind load (1 in 50 year specified)
in three different regions of Canada (Regina, Rivière-du-Loup and Halifax). Those characteristics
have been accepted for calibration of the load and load combination criteria of the NBCC (2005).
The annual maximum wind velocity of 311 sites were provided by the Engineering Climatology
Section of the Canadian Meteorological Centre in Downsview, Ontario for this investigation. 223
59
out of the 311 sites had a data record of at least 10 years. Bartlett et al. (2003) calculated the bias
as the ratio of the mean maximum velocity likely in a 50-year design life (�̅�50) to the specified 1
in 50-year velocity (𝑉50) using
�̅�50
𝑉50=
1 + 3.050 𝐶𝑜𝑉𝑎
1 + 2.592 𝐶𝑜𝑉𝑎 (4-25)
where 𝐶𝑜𝑉𝑎 is the CoV of the maximum annual wind velocity. The stochastic characteristics
reported for the maximum wind velocity in 50-year period are summarized in Table 4-7.
Table 4-7 Stochastic characteristics for maximum wind velocity in 50-year (Bartlett et al., 2003).
Site 𝐶𝑜𝑉𝑎 Bias CoV
Regina 0.108 1.039 0.081
Rivière-du-Loup 0.170 1.054 0.112
Halifax 0.150 1.049 0.103
As mentioned above, in order to transfer the reference velocity pressure to the pressure developed
on the surface of structures, some coefficients including gust, exposure and pressure coefficients
must be considered. These coefficients are assumed to be time-independent and are considered to
fit a log-normal distribution (based on the Central Limit Theorem). Table 4-8 presents a summary
of the statistical parameters for these transformation factors reported by different authors.
60
Table 4-8 Summery of statistical parameters of transformation factors reported by different
authors (Bartlett et al., 2003).
Reference Bias CoV Comment
𝐶𝑒
Ellingwood et al. (1980) 1.0 0.16
Davenport (1981) 1.0 0.10 Reported by Kariyawasam et al. (1997)
Davenport (1982) 0.80 0.16 Low building
Ellingwood and Tekie (1999) 0.93 0.129 Low Building
Davenport (2000) 0.80 0.20 Tall building (simplified methods)
Bartlett et al. (2003) 0.80 0.16
𝐶𝑔
Ellingwood et al. (1980) 1.0 0.11
Davenport (1981) 1.0 0.05 Reported by Kariyawasam et al. (1997)
Ellingwood and Tekie (1999) 0.97 0.093
𝐶𝑝
Allen (1975) 1.0 0.10 Added a model error coefficient of 0.85
Ellingwood et al. (1980) 1.0 0.12
Davenport (1981) 1.0 0.10 Reported by Kariyawasam et al. (1997)
Ellingwood and Tekie (1999) 0.88 0.067
𝐶𝑒𝐶𝑔
Allen (1975) 0.85 0.13
𝐶𝑔𝐶𝑝
Ellingwood et al. (1980) 1.0 0.16
Davenport (1981) 1.0 0.11 Reported by Kariyawasam et al. (1997)
Davenport (1982) 0.80 0.15 Low buildings
Ellingwood and Tekie (1999) 0.85 0.115
Davenport (2000) 0.80 0.21 Tall building (simplified method)
Bartlett et al. 2003 0.85 0.15
𝐶𝑒𝐶𝑔𝐶𝑝
Allen 1975 0.72 0.174
Ellingwood et al. (1980) 0.85 0.239
Davenport (1981) 0.85 0.150 Reported by Kariyawasam et al. (1997)
Davenport (1982) 0.54 0.325 Low buildings, only external pressures
Rosowsky and Cheng (1999) 0.73 0.264 Low buildings, high wind regions
Ellingwood and Tekie (1999) 0.70 0.208 Normal distribution for all factors
Davenport (2000)
0.57
0.71
1.00
0.250
0.150
0.087
Simple method
Detailed method
Wind tunnel and meteorological investigation
Bartlett et al. (2003) 0.68 0.22 Log-normal distribution
61
The NBCC differs significantly from ASCE 2000 in terms of the transformation factors. For
example, the NBCC has reduced the gust and pressure coefficients for the design of low-rise
buildings to account for directionality and other factors while ASCE-98 (2000) suggests a separate
coefficient equal to 0.85 in this regard. The difference in pressure coefficients reported by the
NBCC and those reported by ASCE7-98 regarding taller buildings can be mentioned as another
example of the inconsistency between these two codes. This is due to the fact that the NBCC
pressure coefficients permit the designer to calculate the leeward suction at the mid-height of the
structure instead of at the top of the structure. Therefore, the pressure coefficients reported by the
NBCC are less severe than those reported by ASCE7-98.
JCSS (2001) estimated the wind load applied per unit area of a structure using the following
relations
i. Small rigid structures
𝑊 = 𝑞𝐶𝑎𝐶𝑔𝐶𝑟 (4-26)
ii. Large rigid structures sensitive to dynamic effects (natural frequency less than 1 Hz)
𝑊 = 𝑞𝐶𝑑𝐶𝑎𝐶𝑔𝐶𝑟 (4-27)
where q is the reference velocity pressure, 𝐶𝑎 is an aerodynamic shape factor, 𝐶𝑔 is the gust factor,
𝐶𝑟 is the roughness factor and 𝐶𝑑 is the dynamic factor. It is clear that the factors recommended
by JCSS are different from those recommended by the NBCC, except the gust coefficient and
therefore, their stochastic characteristics differ from each other. The mean and CoV of wind load
using the mean and CoV of the influential random variables assumed to be uncorrelated are as
follows
𝐸(𝑤) = 𝐸(𝐶𝑔)𝐸(𝐶𝑎)𝐸(𝐶𝑟)𝐸(𝑞) (4-28)
62
𝐶𝑜𝑉𝑊2 = 𝐶𝑜𝑉𝐶𝑔
2 + 𝐶𝑜𝑉𝐶𝑎
2 + 𝐶𝑜𝑉𝐶𝑟
2 + 𝐶𝑜𝑉𝑞2 (4-29)
𝐸(𝑤) = 𝐸(𝐶𝑑)𝐸(𝐶𝑔)𝐸(𝐶𝑎)𝐸(𝐶𝑟)𝐸(𝑞) (4-30)
𝐶𝑜𝑉𝑊2 = 𝐶𝑜𝑉𝐶𝑑
2 + 𝐶𝑜𝑉𝐶𝑔
2 + 𝐶𝑜𝑉𝐶𝑎
2 + 𝐶𝑜𝑉𝐶𝑟
2 + 𝐶𝑜𝑉𝑞2 (4-31)
A summary of the statistical characteristics of the above random variables as suggested by JCSS
(2001) is presented in Table 4-9. The coefficient of variation of the resulting wind pressure is
shown to be almost double the coefficient of variation of the wind velocity according to JCSS
(2001). Therefore, the CoV of wind pressure can be estimated directly from the CoV of the wind
velocity. However, it is clear that it would be an approximation and may lead to a less accurate
estimation compared with using the formula above and the stochastic characteristics of the
influential factors directly.
Generally, the characteristic value of the reference wind velocity and consequently the reference
wind pressure have been specified based on each code criteria and may differ from those of
another. In case of the NBCC, the reference wind velocity and reference wind pressure have been
determined based on the probability of being exceeded 1 in 50 years. In other words, these
reference values have commonly a 50-year return period. From a statistical point of view, there is
an obvious limitation regarding the number of extreme values that can be measured during an
observation period of 50 years. However, a relatively large database can be gained for an
observation period of 1 year. MCS can then be used to generate extreme values for large
observation periods like 50 year.
63
Table 4-9 Statistical characteristics of variables being influential on wind load (JCSS, 2001).
Reference Variable Bias CoV
Davenport (1987)
𝑞 0.8 0.2-0.3
𝐶𝑎
-Pressure coefficient
-Force coefficient
1.0
1.0
0.1-0.3
0.1-0.15
𝐶𝑔 1.0 0.1-0.15
𝐶𝑟 0.8 0.1-0.3
𝐶𝑑 1.0 0.1-0.2
Vanmarcke (1992)
Structure period
-Small amplitude
-Large amplitude
0.85
1.15
0.3-0.35
0.3-0.35
Structure damping
-Small amplitude
-Large amplitude
0.8
1.2
0.4-0.6
0.4-0.6
4.4.1 Point-in-time Wind Load
The statistical characteristics of point-in-time wind load which have been adopted for calibration
of load and load combination of the NBCC (2005) were estimated based on several assumptions.
Wind velocity and consequently, the resultant pressure are considered to be stochastic and the
transformation factors are time independent variables. 3 h is assumed to be the duration of each
wind pulse. Statistical parameters related to the maximum wind velocity in 3 h periods were
reported for three different regions of Canada (basis of calibration of load and load combination
of the NBCC (2005)) is presented in Table 4-10 (Bartlett et al., 2003). The transformation factors
which are used to transform wind velocity to wind pressure applied on the surface of structures are
the same for both maximum wind load over T years and the point-in-time wind load. Therefore,
64
the previously mentioned transformation factors and their related stochastic characteristics can be
used to estimate stochastic modeling of point-in-time wind load as well.
Table 4-10 Statistical parameters of maximum 3-hour wind velocity in Canada (Bartlett, 2003).
Site Bias CoV
Regina 0.156 0.716
Rivière-du-Loup 0.064 0.149
Halifax 0.084 1.001
One may note that as the bias represents the ratio of the mean to the characteristic value suggested
by the relative code, this ratio can be used a basis for understanding the appropriateness of the
characteristic values reported by the codes. As the bias factor was shown to be near to 1 for wind
velocity during the longer observation period (50 years) as presented in Table 4-7, this proves that
the characteristic values of the NBCC code that Bartlett et al. (2003) used were appropriate.
However, bias factors reported for point-in-time wind velocities of the three different regions in
Canada as represented in Table 4-10 show around 90% difference between the actual mean value
and the NBCC characteristic value. This may be due to the fact that Bartlett et al (2003) used the
characteristic values related to a 50-year return period to calculate the point-in-time wind load bias
ratio leading to this considerable difference.
4.4.2 Distribution
The best-fit distribution for a 50-year maximum wind velocity has been reported to be a Gumbel
distribution (JCSS, 2001; Bartlett et al, 2003) and for point-in-time velocity the best fit distribution
has been reported to be Weibull (Bartlett et al., 2003). The log-normal distribution is suggested for
each of the transformation factors, (JCSS, 2001) as well as their overall combination (Bartlett et
al., 2003). However, Ellingwood and Tekie (1999) recommended normal distribution for all
factors. The overall wind load pressure is suggested to be modeled by a Gumbel distribution (JCSS,
2001). However, this distribution usually misses the upper limit. Therefore, a Gumbel distribution
may not be the best choice although it is commonly used in different studies as its application is
65
simple and it needs only a few parameters. Kasperski (2000) suggested the Weibull distribution as
a better fit for wind load and to prevent ignorance of the upper limit. This is due to considering a
third parameter 𝜏, referred to as the shape parameter, besides the moments in the Weibull
distribution defining the upper limit. A summary of the recommended distributions for each
parameter as explained is presented in Table 4-11.
Table 4-11 Summery of the best fit distribution for parameters involved in wind load estimation.
Parameter Distribution Reference
Maximum velocity (50-year return period) Gumbel Bartlett et al. (2003); JCSS
(2003)
Point-in-time velocity Weibull Bartlett et al. (2003)
Each transformation factor
Normal Ellingwood and Tekie (1999)
Log-normal Bartlett et al. (2003); JCSS
(2003)
Overall transformation factor Log-normal Bartlett et al. (2003)
Wind pressure Gumbel JCSS (2001)
Weibull Kasperski (2000), Brehm (2011)
Snow loads are prevalent in mountainous and cold regions all over the word. There are
uncertainties in the nature of the occurrence of extreme snowfalls and consequently, the duration
and intensity of the resultant snow loads on structures. Therefore, snow loads should be treated in
a probabilistic manner. In order to calculate the actual snow load on a roof accurately, the
difference in the quantity of snow or rain being accumulated and that of the snow or rain being
revoked by the wind, melting or evaporation should be calculated (Ellingwood and O’Rourke,
1985). However, it is not possible to quantify snow in this manner as data are not available. As the
snow load and consequently its probabilistic parameters depend on the location of the structure
66
(like wind load), here only the snow load model and the stochastic parameters reported in the
NBCC and for Canada are presented. The NBCC presents a model to estimate the snow load on a
roof of a structure as follows
𝑆 = 𝐼𝑠[(𝐶𝑏𝐶𝑤𝐶𝑠𝐶𝑎)𝑆𝑠 + 𝑆𝑟] (4-32)
𝐶𝑔𝑟 = 𝐶𝑏𝐶𝑤𝐶𝑠𝐶𝑎 (4-33)
where 𝐼𝑠 is importance factor for snow load, 𝑆𝑠 is the ground snow load with the probability of
exceedance of 1 in 50 per year, 𝑆𝑟 is the associated rain load, 𝐶𝑏 refers to the basic roof snow load
of a value of 0.8, 𝐶𝑤, 𝐶𝑠 and 𝐶𝑎 are the wind exposure factor, the slope factor and the accumulation
factor, respectively. Based on the NBCC, 𝑆𝑟 has to be taken less than 𝐶𝑔𝑟𝑆𝑟.
4.5.1 Ground Snow Load
Ground snow loads are considered as the basis of estimation of roof snow load in Canada. The
snow depth (𝑑) and the unit weight of snow (𝛾) are the parameters used in the calculation of
ground snow load. In other words, data are usually reported as depth of snow by weather stations.
Snow depth then transfers to ground snow load by means of the relation between depth and density
of snow. Appendix C of the 2010 NBCC includes the values of ground snow load for selected
locations in Canada. In order to calculate these recommended values, Gumbel distributions were
fitted to the reported data of the maximum annual accumulated depth related to 1618 stations
having 7-38 years of record (Newark et al., 1989; NBCC, 1995). Then, the values for the 1 in 30
years were determined. Snow densities of various geographical regions having common climatic
characteristics (manifested by forest type) were estimated and site elevations were considered
through normalization of the loads. The next step was preparation of smoothed contour maps.
Through interpolation and rectification accounting for the elevation, the final loads related to the
different geographical regions listed in Appendix C were computed. It should be mentioned that
there are no documents available regarding the weighting factors used in interpolating the ground
67
snow values corresponding to the data collection sites to calculate the values concerning the sites
listed in Appendix C.
Bartlett et al. (2003) also studied the stochastic characteristics of snow load in Canada. They used
the data related to annual maximum snow depth collected from 1278 sites provided by the
Engineering Climatology Section of the Canadian Metrological Center. Table 4-12 presents the
characteristics of the annual maximum snow depth in Canada. Then, 50-year maximum snow
depths were calculated based on annual data, see Table 4-13.
Table 4-12 Characteristics of the annual maximum depth in Canada (Bartlett et al., 2003).
Type CoV Mean Median
Annual maximum snow depth 0.08-1.34* 0.491 0.47
* 95% of the CoV values are in the range of 0.22-0.95.
Table 4-13 Characteristics of the 50-year maximum depth in Canada (Bartlett et al., 2003).
Type Bias CoV
50-year maximum snow depth 1.1 0.2
Snow density is the other influential parameter on ground snow load. Therefore, determination of
stochastic characteristics of snow density is necessary to calculate the stochastic characteristics of
ground snow load and consequently those of the snow load on the roof of a structure. Statistical
characteristics of snow density have been summarized and reported by Kariyawasam et al. (1997)
based on the study of Newark (1984), see Table 4-14. The values were reported based on the forest
type in the region of the sites considered.
Taylor and Allen (2000) investigated the statistical characteristics the ground-to-roof
transformation factor in consistency with the definition of ground-to-roof snow load factor
recommended by 1995 NBCC. The database of 112 roofs in four Canadian cities during 13 years
68
were used to estimate the statistical values of the ground-to-roof transformation factor as
summarized in Table 4-15.
Table 4-14 Statistical characteristics of snow density (Kariyawasam et al., 1997).
Forest Mean
𝑘𝑔/𝑚3
Standard Deviation
𝑘𝑔/𝑚3 CoV Sites
Acadian 220 50 0.23 Fredericton, Halifax, St. John’s, Charlottetown
Aspen grove 220 40 0.18 Edmonton, Saskatoon, Winnipeg, Thunder Bay
Boreal 190 60 0.32 Northern Prairies, mid-northern Ontario and
Quebec
Coast 430 25 0.06 Coastal British Columbia
Columbia 360 35 0.10 Southeastern British Columbia
Great Lakes 220 60 0.27 Southern and central Ontario, southern Quebec
Montane 260 25 0.10 Interior British Columbia and Yukon
Prairie 210 40 0.19 Southern Prairies, Regina
Subalpine 360 30 0.08 Vancouver and Fraser Valley
Tundra 300 80 0.27 Arctic
Taiga 200 80 0.40 Subarctic, Yellowknife
Ellingwood and O’Rourke (1985) did a study regarding the calculation of snow loads applied to
structures using probabilistic models. They accepted the general formula of roof snow calculation
(NBCC, 1977; ISO 4355, 1981; ANSI A58.1, 1982) as a product of ground snow load and a
transformation factor as follows:
𝑆𝑟 = 𝐶𝑆𝑔 (4-34)
69
in which 𝑆𝑔 is the ground snow load and 𝐶 is the ground to roof transformation factor. Ellingwood
and O’Rourke (1985) estimated 𝑆𝑔 corresponding to the 98 percentile value over a 50-year return
period using Log-normal and Gumbel distribution functions based on the statistical study done by
Ellingwood and Redfield (1983), indicating that either the Log-normal or the Gumbel distribution
may be suitable for 𝑆𝑔
Log-normal distribution:
𝑆𝑔 = exp(𝜆𝑔 + 2.054𝜉𝑔) (4-35)
Gumbel distribution:
𝑆𝑔 = 𝜇𝑔 + 3.902/𝛼𝑔 (4-36)
𝜇𝑔 ≈ 𝑚𝑔 − 0.5772/𝛼𝑔 (4-37)
𝛼𝑔 ≈ 1.283/𝑠𝑔 (4-38)
where 𝜆𝑔 and 𝜉𝑔2 refer to the mean and variance of 𝑙𝑛 𝑆𝑔. 𝑚𝑔 and 𝑠𝑔 are the sample mean and
standard deviation.
O’Rourke et al. (1982) defined a relationship for determination of the overall combination of
transformation factors (𝐶𝑔𝑟) as follows
𝐶𝑔𝑟 = 0.47𝐸𝑇𝜀 (4-39)
in which 𝐸 is a wind exposure factor (from 0.9 to 1.3); T is thermal characteristic factor (from 0.1
to 1.2); and 𝜀 is the error term with a log-normal distribution, a mean value of 1.0 and a CoV of
70
0.44. It is worth mentioning that the O’Rourke et al. (1982) formula leads to transformation factor
values which are from 0.53 to 0.91 times the 1995 NBCC basic roof snow load factor equal to 08.
Table 4-15 Statistical characteristics of transformation factor (Taylor & Allen, 2000).
Location Bias CoV
Sheltered Location
Halifax
Chicoutimi
Saskatoon
Recommendation
0.61
0.71
0.30
0.60
0.45
0.44
0.11
0.42
Exposed locations
Halifax
Ottawa
Saskatoon
Recommendation
0.47
0.44
0.33
0.5
0.46
0.41
0.22
0.42
Drift locations
Chicoutimi
Ottawa
Saskatoon
Recommendation
0.86
0.60
0.61
0.60
0.30
0.38
0.43
0.42
JCSS (2001) also presents a probabilistic model for snow loads. According to JCSS (2001), the
snow load on a roof can be computed by the following relation
𝑆𝑟 = 𝑆𝑔𝑟𝑘ℎ/ℎ𝑟 (4-40)
𝑟 = ƞ𝑎𝐶𝑒𝐶𝑡 + 𝐶𝑟 (4-41)
71
in which 𝑆𝑔 accounts for the snow load on the ground at the weather station; 𝑟 is the transformation
factor of snow load on ground to snow load on roofs; ℎ is the altitude of the site of the building;
ℎ𝑟 is the reference altitude (300 m); 𝑘 is a coefficient accounts for the type of region in which
structure is located (𝑘 = 1.25 for coastal regions and 𝑘 = 1.5 for inland mountainous regions; ƞ𝑎
is a shape coefficient, (Table 4-16); 𝐶𝑒 is an exposure coefficient (Table 4-16); 𝐶𝑡 is a thermal
coefficient; and 𝐶𝑟 is a redistribution coefficient resulting from wind (neglected for monopitch
roofs, equal to ±𝐶𝑟𝑜 for symmetrical duopitch roofs, Figure 4-2, Figure 4-3).
Table 4-16 The exposure coefficient (𝐶𝑒) and shape factor (ƞ𝑎) (JCSS, 2001).
Slope of The Roof ƞ𝑎𝐶𝑒
𝛼 = 0° 0.4 + 0.6 exp (−0,1 𝑢(𝐻))*
𝛼 = 25° 0.7 + 0.3 exp (−0,1 𝑢(𝐻))
𝛼 = 60° 0
* 𝑢(𝐻) is the averaged wind speed during one-week period at roof level of H
Figure 4-2 Redistribution of snow load applied on a duopitch roof (JCSS, 2001).
72
Figure 4-3 𝐶𝑟𝑜 as a function of the roof angle (JCSS, 2001).
𝑆𝑔 is considered to be time dependant. However, it is space independent within a region with
almost same altitude and same climatic condition. 𝑆𝑔 is determined by either the water equivalent
of snow or snow depth. The reported values are used directly in the determination of ground snow
load in the first case. In the second case, snow depth values have to be transformed to snow load
using the following relation
𝑆𝑔 = 𝑑𝛾(𝑑) (4-42)
𝛾(𝑑) =𝜆𝛾∞
𝑑ln {1 +
𝛾0
𝛾∞[exp (
𝑑
𝜆) − 1]} (4-43)
where 𝑑 is the snow depth; 𝛾(𝑑) is the accounts for the average weight density of the snow; 𝛾∞ is
unit weight at 𝑡 = ∞ (equal to 5 𝐾𝑁/𝑚3); and 𝛾0 is unit weight at 𝑡 = 0 (equals to 1.70 𝐾𝑁/𝑚3).
The stochastic characteristics of snow load variables recommended by JCSS (2001) are
summarized in Table 4-17.
73
Table 4-17 Summary of stochastic characteristics of snow load variables (JCSS, 2001).
Symbol mean CoV
𝑆𝑔 Observation Observation
𝑑𝑔 Observation Observation
𝑘 1.25/1.5 Deterministic
ℎ𝑟 300 m Deterministic
𝛾0 1.7 Deterministic
𝛾∞ 5.0 Deterministic
𝜆 0.85 Deterministic
ƞ𝑎𝐶𝑒 Table 4-16 0.15
𝐶𝑡 0.8 – 1.0 Deterministic
𝐶𝑟𝑜 Figure 4-2 & Figure 4-3 1.0
4.5.2 Snow Related Loads
Associated rain load is one of the main snow-related loads accounting the weight of rain which
may fall over a snowpack. The values of associated rain load reported in the NBCC (Appendix A)
have been calculated based on the database related to winter rainfall according to the study of
Taylor and Allen (2000). The data being used in the determination of the probability models for
ground snow load are usually taken from daily measurements of the water equivalent ground load.
Therefore, as heavy rain can commonly penetrate down through the snowpack and drain away, it
is hard to claim the percentage of maximum rain on snow being captured by daily measurements
(Ellingwood & O’Rourke, 1985). Moreover, the available data of associated winter rainfall should
be filtered carefully to remove those on non-existent snowpacks. In other words, as there are no
available, reliable and clear data regarding the associated rain load, derivation of its statistical
parameters is difficult. Colbeck (1977) worked on roof loads resulting from rain-in-snow.
74
However, there is still a lack of a general and acceptable approach for analysis of associated rain
load.
Icing on eaves and the development of ice dams are two other associated snow loads which have
been shown occasionally to lead to structural distress (MacKinley, 1983). Roof overhangs and
moisture-resistant roofing members are substantially endangered with these effects. Critically,
warm structures being purely insulated are threatened with ice formation as well. However, there
is a of comprehensive and quantitative study in this regard.
4.5.3 Point-in-time Snow Load
The statistical characteristics of point-in-time snow load have not been investigated broadly and
comprehensively. Bartlett et al. (2003) just studied the point-in-time snow load in Canada. There
are no available documents regarding the details of their study and some general assumptions were
pointed out in their related publication. Snow accumulation was assumed to be a stochastic process
and transformation factors (converting depth to the load and the ground snow load the roof one)
were assumed to be time dependent. A 14-day period over 3 months of a year was considered as
the point-in-time pulse (i.e. 6 events per year). Table 18 presents the statistical characteristics of
point-in-time snow depth and transformation factor to load effect consistent with Canadian regions
and the 1995 NBCC criteria recommended by Bartlett et al. (2003).
Table 4-18 Statistical characteristics of point-in-time snow (Bartlett et al., 2003)
Parameter Bias CoV
Point-in-time depth 0.196 0.882
Transformation factor 0.600 0.420
4.5.4 Distribution
Different researchers have introduced almost separate coefficients and parameters as random
variables in the determination of snow loads. It is clear that each of these parameters have their
own distribution, and therefore differ from one reference to another. Here, the best fit distributions
for the random variables and parameters of the NBCC and JCSS, which are the two references
75
being explained above, are presented. Considering the NBCC criteria for the calculation of snow
load, snow densities were recommended to have a Normal distribution; 50 year-maximum depth
and point-in-time depth were fitted to Gumbel and Weibull distributions, respectively.
Transformation factors were allocated Log-normal distribution (Bartlett et al., 2003). JCSS (2003)
recommended a Gamma distribution for snow depth on the ground. Density has been considered
deterministic as well as the insulation parameter (one of the transformation factors). The shape and
redistribution coefficients (two other transformation factors) have been fitted to Beta distributions.
All other parameters were assumed to be deterministic, see Table 4-17. Ellingwood and O’Rourke
(1985) considered both Log-normal and Gumbel distributions for the ground snow load based on
the study done by Ellingwood and Redfield (1983) and recommended a Log-normal distribution
for the transformation factor. Ellingwood and Redfield (1983) proposed the Log-normal
distribution for regions where the snow cover shows intermittency over the snow season and the
annual extremes happen commonly with a severe winter storm. Moreover, large variabilities have
been shown to exist in annual extreme snow loads (Eurocode, 1994). The Log-normal distribution
has a longer upper tail compared to the Gumbel distribution to be able to represent these
variabilities. The recommended distributions regarding snow load random variables are
summarized in Table 4-19.
Table 4-19 Summery of recommended distributions for the snow load random variables.
Variable Distribution Comment Reference
Density Normal - Bartlett et al. (2003)
Snow depth
Gumbel 50-year maximum depth,
reported for 2005 NBCC Bartlett et al. (2003)
Gamma - JCSS (2001)
Weibull Point-in-time Bartlett et al. (2003)
Transformation factor
Log-normal Overall factor, based on
1995 NBCC Bartlett et al. (2003)
Beta Shape coefficient &
redistribution coefficient JCSS (2003)
Log-normal - Ellingwood &
O’Rouke (1985)
76
Chapter V: Stochastic Modeling of Historic Masonry Materials
Masonry has been used in construction for thousands of years, with many structures still standing.
In other words, masonry is among the most durable construction materials and masonry structures
usually outlive their design or expected service life. In the past, natural units like cut stones were
commonly used. They have been replaced with manufactured units as the primary building block
because of efficient industrial manufacturing. Hydraulic lime mortars have also been substituted
by Portland cement mortars. Therefore, a wide variety of materials in terms of units and mortar
have been used throughout construction history. Masonry is mostly identified as a homogenous
material in structural engineering. It leads to consideration of the characteristics of the units and
mortars after combination in order to estimate masonry properties (Brehm 2011).
The resistance of structures is mainly based on the strength of the materials from which the
structure is constructed. Therefore, determination of the material properties which are influential
on the structural resistance is a key step in the estimation of structural reliability. Historical
structures often contain multiple construction materials (different bricks or stones for example)
with various characteristics leading to more complexity in the identification and determination of
material properties. Additionally, there are usually no construction documents (even as-built
documents in modern construction often fail to provide enough information). In this situation,
there is a need for supplementary in-situ and laboratory tests to quantify material properties
(FEMA 365). As a result of possible damage to the structure’s fabric or even structural instability,
destructive techniques should not be used in order to get information about the type of materials
in the construction and their properties. All of these information shortages and concerns make
analysis and understanding of the behaviour of a historical structure more complicated but
necessary.
The main focus here is to present a possible procedure in order to get an estimate of the material
properties and their variability without applying destructive tests to the historical structure. There
are various formulas presented by standards and previous researchers in order to calculate the
77
resistance of a structure under its probable failure modes. The most frequently used and thus
influential material properties in these formulas are modulus of elasticity, compressive strength,
tensile strength, cohesion and coefficient of friction. Since compressive strength is most
fundamental for masonry, the main focus of this thesis is the estimation of the probabilistic model
related to compressive strength. Useful techniques in order to get information about other material
properties of historical structures and the stochastic modeling of them are also explained.
Evaluation of the performance of historical masonry structures and gathering information about
them would enhance understanding of their current conditions and capacity and would minimize
the modifications and additions applied to them. There is a wide variety of evaluation methods
available, showing different accuracy and feasibility. Some of these methods have been
specifically developed to evaluate masonry structures. However, some of them are borrowed from
other fields of science (e.g. archeology and aerospace). Investigation procedures usually include
several complementary evaluation methods. Different levels of damage or deconstruction can be
incurred by masonry by using masonry evaluation techniques. In case of historic structures, even
small damage can be lead to expensive and difficult repairs. Typically, investigation approaches
and relative evaluation methods should be collected carefully to have minimum disruption to
historical structures. A list of nondestructive testing techniques, in-situ evaluation approaches and
their related purposes is summarized in Table 5.1. Standards and codes which govern various
nondestructive testing methods are reported in Table 5-2. In order to choose an appropriate testing
technique, several criteria should be taken into account. The type of required information,
structural condition, cost and complexity are among the most important criteria in selection of
efficient and feasible testing techniques. Table 5-3 categorizes testing techniques being applicable
to evaluate or investigate different masonry conditions based on the above-mentioned criteria. An
effectiveness grade is given to each method based on the accuracy of the information that the
method can provide. This classification considering the cost, complexity and effectiveness grades
is based on the experiences of different authors who worked with these methods and is expressed
here only as a general guideline. In other words, all these parameters and consequently the
classification may differ based on various factors (e.g. location, accessibility and schedule). In the
78
case of load testing, a service level is usually considered so that no significant damage is applied
to the structure. If it is required to increase the load to reach a level near failure, considerable repair
may be needed. Moreover, in visual testing, slight repair may be required in the case of a borescope
and other small probe observations. However, larger openings for visual testing result in more
significant repairs.
Codes have not considered more detailed structural characteristics like cracks and fatigue in the
evaluation of structural resistance. However, in the case of more detailed reliability assessment,
there is a need to involve more detailed characteristics of structures in the estimation of resistance.
Cracks, spalls, efflorescence and surface erosion are among the most well-known signs of masonry
distress. Visual observation is a significant method in the evaluation of masonry distress. An
experienced investigator can conclude the reason of many crack patterns and surface erosion
patterns from only surface observation. Sometimes visual observation should be accompanied with
other subsurface investigations including non-destructive evaluation, probe openings and
borescope observations to estimate the subsurface conditions more accurately.
Cracks, spalls and surface erosion can be evaluated using a number of nondestructive techniques.
Ultrasonic Pulse Method (UPM) is used to estimate crack depth and extent in place as well as
surface condition. The impact echo method is also useful in getting information about cracks.
Petrography in the laboratory can be used to get information about distress mechanism where it is
possible to take samples from materials.
In masonry, the corrosion of embedded metal elements may lead to cracking. To investigate the
probability of active corrosion in subsurface layers, a non-destructive method namely half-cell
potential is used.
The appropriate non-destructive technique is selected based on the situation. However, it is
recommended to use visual observation (done by an expert) at the beginning of the performance
evaluation procedure. Generally, to do more accurate reliability evaluation, determination of more
exact limit state functions and consequently more detailed structural resistance evaluation is
necessary. Estimation of more exact limit state functions considering more detailed structural
characteristics is out of scope of current thesis. Here only the most important material properties
being used in code-based structural resistance evaluation and their stochastic characteristics are
discussed.
79
Table 5-1 Nondestructive testing techniques and in-situ evaluation approaches (Hussain &
Akhtar, 2017).
Testing Method Measured parameters
Stone masonry Brick masonry
Visual
Visual inspection Macroscopic flaws, cracks
and deformation
Macroscopic flaws, cracks
and deformation
Borescope Macroscopic flaws, cracks
and deformation
(underneath the surface)
Macroscopic flaws, cracks
and deformation
(underneath the surface)
Acoustic
Ultrasonic Pulse Method
(UPM)* Flaws, Crack, deterioration,
strength and modulus of
elasticity
Flaws, Crack, deterioration,
hardness of surface, strength
and modulus of elasticity Acoustic emission
Impact-Echo (IE)*
Mechanical Pulse Velocity
(MPV)
Voids and discontinuities of
masonry
Voids and discontinuities of
masonry
Physical methods
Rebound hammer*
Strength Hardness of surface and
strength Waitzmann hammer
Flatjack
Push or shove test _
Shear strength of brick
assembly
Bond wrench test _
Flexural bond strength
(Tensile strength)
Radar Method
Ground Penetrating Radar
(GPR)
Flaws, cracks, voids and
moisture content
Flaws, cracks, voids and
moisture content
80
Testing Method Measured parameters
Stone masonry Brick masonry
Penetrating radiation
methods
Radiography Surface voids, crack, path
density
_
Radiometry _
X-Ray Methods
X-Ray tomography Chemical composition
(stone), chemical attack,
inner structure
_
X-Ray diffraction _
X-Ray fluorescence _
Thermal imaging
Infrared thermography Crack, voids, delamination,
damage to thermal
insulation, air circulation
Crack, voids, delamination,
air circulation
Corrosion diagnose
Phenolphthalein indicator
test
Diagnosis of corrosion
Diagnosis of corrosion
Chloride penetration test
Half-cell potential test
Rapid chloride test
Quantab test
Volhard test
*intended to be used for concrete but can be applied or can be adopted for masonry as
well.
81
Table 5-2 Summary of standards and codes including various non-destructive methods.
Testing Methods Standards
Visual ASTM C823/C823M
NDIS 3418:1993
Ultrasonic Pulse Method (UPM) ASTM C597-97
NDIS 2416-1993
IS 13311 (Part 1):1992
BS 1881: Part 203: 1986
BS 4408: Part 5
BS 12504-4, Part 4 2004
ISO/DIS 8047, C-26-72
COST 17624
Impact-Echo (IE) ASTM C1383-15
Acoustic emission ASTM E2983
Rebound Hammer ASTM C805-97
ASTM D5873
IS 13311 (Part 2):1992
BS 1881: Part 202: 1986
EDIN EN 12398 (1996)
ISO/CD 8045
Flatjack ASTM C1196-14a
Ground Penetrating Radar (GPR) ASTM D6087-08
ASTM D6432-11
Radiography ASTM E1742/E1742M
NDIS 1401-1992
BS 1881: Part 205: 1970
BS 4408: Part 3
IR thermography ASTM D4788-88, D 4788-03 (2013)
Phenolphthalein indicator test ASTM C1202
ASTM C114
AASHTO T277
Half-cell potential test ASTM C876-91
Volhard test ASTM 1411-09
NT 208
BS 1881-Part 6
DS 423.28
NS 3671
82
Table 5-3 Classification of testing techniques appropriate for evaluation of masonry (Harvey JR.
& Schuller, 2010).
Destructive Non-destructive with
minor repairs Non-destructive
Sam
pli
ng (
core
tes
ting
& p
rism
tes
tin
g)
Bo
nd w
ren
ch t
est
In-s
itu
lo
ad t
esti
ng
Pet
rog
raph
y
Fla
tjac
k/S
hea
rjac
k
Gro
und
Pen
etra
tion
Rad
ar (
GP
R)
Ult
raso
nic
Pu
lse
Met
ho
d (
UP
M)
Imp
act
Ech
o (
IE)
Rad
iog
rap
hic
Infr
ared
Th
erm
og
raph
y (
IRT
)
Hal
f-ce
ll p
ote
nti
al
Reb
oun
d h
amm
er/S
urf
ace
test
s
Met
al d
etec
tio
n (
Ind
uct
ion
)
Vis
ual
Characteristics*
In-place strength G G E A E A A A
In-place stress E
In-place uniformity G G G G G G G G A
In-place deformability E E A A A
Location of crack E E G A
Movement of crack G
Rebar information E G E A E
Corrosion of rebar G A
Location of anchor and ties E E E
Voids in grout E A A E G A
Voids in masonry E A A E G A
Durability E E E G A
Performance under applied loads E
Cost**
M M H H M H H H H H M S S S
Complexity**
L M H H M H H H M M L L L L
* E = Excellent information, G= Good information, slightly variable or unclear results, A = Approximation, extremely variable
** L = Low, M = Medium, H = High
83
The modulus of elasticity is the most important material property for determining the deformation
of structures. JCSS (2011) as a probabilistic model code relates modulus of elasticity to the
compressive strength of masonry in order to estimate the stochastic modeling as follows
𝐸𝑚 = 𝑐1. 𝑓𝑚′ (5-1)
𝐸𝑚,𝑗 = 𝑌2 . 𝐸𝑚 (5-2)
where 𝐸𝑚 is mean of modulus elasticity of masonry, 𝑐1 is a coefficient according to Schubert
(2010) summarized in Table 5-4 and 𝑌2 is log-normal variable with a mean of 1.0 and coefficient
of 25% for all types of unit and mortar. It should be considered that 𝑐1 is claimed to be a prior
value needing updating using test data. As can be seen, the reported values of 𝑐1 are mostly
presented for new materials not historical ones. Therefore, using this formula to estimate the
modulus of elasticity of ancient building materials and understand their deformation behaviours
may lead to inaccuracy and unreliable estimation.
Table 5-4 Values of 𝑐1 involved in probabilistic model of modulus of elasticity (Schubert, 2010).
Unit Mortar 𝑐1
CS[1] GPM[4], TLM[5] 500
AAC[2] GPM 520
TLM 560
LC[3] GPM 1040
TLM 930
Perforated clay brick
GPM 1170
TLM 1190
Lightweight 1480
[1] Calcium Silicate, [2] Autoclave Aerated Concrete, [3] Lightweight concrete
[4] General Purpose Mortar, [5] Thin Layer Mortar
In order to prevent any damage and disturbance to historical structures, it is suggested to employ
non-destructive tests for these kinds of structures instead of loading techniques. The Ultrasonic
84
Pulse Method (UPM), Figure 5-1, is a nondestructive method which has proven to be efficient in
estimating the modulus of elasticity of masonry structures (Brozovsky 2013). Consequently, to
obtain a data set related to the modulus of elasticity of historical masonry structures, UPM is
recommended
Figure 5-1 Ultrasonic Pulse Method (n.d.).
The compressive strength of the masonry is usually the most important information being desired
to estimate for structural resistance evaluation. Compressive strength can be used to estimate the
load-bearing capacity of structures. There are difficulties in the estimation of compressive strength
of existing masonry structures including variable material properties, different construction
techniques, lack of knowledge about the existing damage made throughout the life of a masonry
structure and the absence of appropriate code. In the case of ancient structures, compressive
strength was probably not considered or specified in the design procedure and any documents
regarding the compressive strength of the masonry have been destroyed or lost throughout history.
Even if some documents regarding the compressive strength of the masonry in the design stage
could be found, they are not that reliable as the present compressive strength of the masonry
obviously is different from that hundreds of years ago. Weathering, deterioration and the effects
of long-term applied load are some of the causes of this difference in material strength. Moreover,
85
inspection, specimen removal and testing are restricted regarding heritage structures. All these
statements show the difficulties but necessities of the determination of the present compressive
strength of the historical masonry structure to be able to evaluate the structural resistance
accurately. Techniques used to estimate the compressive strength of existing structures
experimentally range from simulating the structure through testing specimens in laboratories to
removing a prism from the structure in order to do a test. Each technique has its own advantages
and disadvantages. However, an attempt is made here to find an optimum procedure that allows
estimation of the variability of the compressive strength of a structure in addition to the mean value
in order to be able to do reliability analysis. A brief review of the most well-known approaches
and testing methods for estimation of the compressive strength of masonry follows.
The compressive strength of a masonry structure is based on the compressive strength of its
components (units and mortar). Some of the methods are based on the determination of
compressive strength of whole masonry structure according to the estimation of compressive
strength of the unit and mortar individually. Table 5-5 illustrates some of the proposed expressions
for estimation of a deterministic value of compressive strength of brick masonry according to the
compressive strength of the brick and the mortar.
The JCSS (2011) probabilistic model code also proposed formulas regarding the estimation of a
stochastic model of masonry compressive strength using the compressive strengths of its
components to do reliability assessment
𝑓𝑚′ = 𝑘. 𝑓𝑏
𝛼. 𝑓𝑚𝑜𝛽 (5-3)
𝑓𝑚,𝑗 = 𝑌. 𝑓𝑚′ (5-4)
where , 𝑓𝑚′ , 𝑓𝑏, and 𝑓𝑚𝑜 represent the compressive strength of masonry, and the mean values of the
unit and mortar compressive strengths, respectively. K, 𝛼 and 𝛽 are coefficients. 𝑓𝑚,𝑗 represents
the probabilistic model for compressive strength and 𝑌 is a variable related to uncertainties
distributed lognormally, see Table 5-6.
86
Table 5-5 Masonry compressive strength determination from the strengths of its components.
Reference Expressions
Mann (1982) f'm=0.83fb0.49fmo
0.32
Hendry & Malek (1986) f'm=0.317fb0.531fmo
0.208
Dayaratnam (1987) f'm=0.275fb0.5fmo
0.5
Eurocode 6 (1996) f'm=0.5fb0.7fmo
0.3
Bennet et al. (1997) f'm=0.3fb
ACI 530.99 (1999) f'm =2.8+0.2fb
MSJC (2002) f'm=(400+0.25fb)/145
Dymiotis & Gutlederer (2002) f'm=0.3266 fb(1-0.0027fb+0.0147fmo)
Gumaste et al. (2006) f'm=0.317fb0.866fmo
0.134
Kushik et al. (2007) f'm=0.63fb0.49
fmo0.32
Garzón-Roca et al. (2013) f'm=0.53fb+0.93fmo-10.32
Table 5-6 Statistical characteristics of 𝑌 involved in probabilistic model of compressive strength
(JCSS, 2011).
Unit Type Mortar Mean CoV Distribution
CS[1]
TLM[4]
1.0 0.2 Log-normal
CS (Large sized units) 1.0 0.2 Log-normal
AAC[2] 1.0 0.16 Log-normal
AAC (Large sized units) 1.0 0.14 Log-normal
CB[3] GPM[5] 1.0 0.17 Log-normal
[1] Calcium Silicate, [2] Autoclave Aerated Concrete, [3] Clay Brick
[4] Thin Layer Mortar, [5] General Purpose Mortar
The procedure of determination of the compressive strength of a historical masonry structure based
on that of its components seems not to be effective due to the following reasons:
87
1) To get information about the compressive strengths of the unit and mortar, access to the
construction documents and/or application of material testing is required. In the case of
historical structures, typically construction documents are not available and removal of
units contravenes conservation guidelines.
2) There are often various types of mortars and units with different material properties in a
single historic structure. Therefore, it may not be appropriate to apply the expressions
above to the historical building.
3) The formulas above have mostly been developed for new masonry materials and therefore
may not be accurate enough for historical materials.
4) There is no standard correlation between strength of whole masonry and the strength of its
components. Different correlations have been suggested by different researchers leading to
different results.
5) Any error in measuring compressive strength of the unit and mortar would result in
propagation of the error and consequently, inaccuracy in estimation of the compressive
strength of the whole masonry (Sabri et al., 2015; Steil et al., 2001; Chunha et al., 2001)
6) As masonry constituents work together in a structure, the homogeneity of the components
is an influential factor in compressive strength of the whole masonry. Therefore, it may be
inaccurate to estimate the compressive strength of the masonry considering mortar and
units individually (Radovanović, 2015).
7) There is no standard approach to estimate the compressive strength of mortar in-situ. Some
codes (e.g. The American Standard ASTM C 780) propose testing samples which are
constructed with the same materials as those in the existing masonry structure. This method
seems not to be effective and feasible in the case of historical structures, as the strength of
the mortar has been changed over hundreds of years due to deterioration. Moreover, the
compressive strength of the mortar depends on several factors including water/cement
ratio, mix proportions, aggregate ratio and sand type (Nwofor, 2012; Appa Rao, 2001;
Neville, 1996) which give challenges to correct sampling. Workability is influential on the
strength of constructed masonry as well. Zejak (2015) showed an inconsistency between
the tests results of mortar compressive strength determined in the field with the structural
properties of the masonry. Overall, the properties of prepared mortar samples may differ
88
from the properties of mortar in historical structures leading to inaccuracy in the estimation
of the compressive strength of the whole masonry structure.
Core testing, categorized as small diameter core testing and large diameter core testing, is another
technique used for determining the compressive strength of masonry. Small diameter core testing
is used for units and mortar individually and large diameter core testing is appropriate for whole
masonry (unit and mortar together). There are some difficulties associated with these two methods.
In the case of small core testing techniques, mortar sampling is difficult and results are the
compressive strength of unit and mortar individually. The sampling direction is influential on the
estimated compressive strength. In other words, the compressive strengths of samples taken from
the same element differ by changing the sampling directions. In the case of large core testing, the
number of samples is restricted as the extraction of many large samples may influence the strength
of the structure. Moreover, it is not possible to do large core testing for elements with small cross
sections such as columns as the elements that are extracted from them might lead to deformation
or a decrease in the bearing capacity. The compressive strength of the extracted sample shows the
compressive strength of the location where sample is taken from. As a single historical structure
might be constructed by different masons from different materials with different strengths, this
testing method may lead to inaccuracy in terms of estimation of the compressive strength of the
whole masonry. The direction of sampling is also influential in the case of large core testing.
Masonry prism testing is another sampling method used for the estimation of compressive strength
of masonry. There are two ways of prism testing, including construction of prisms with the same
units and mortar as those by which the structure build up and extraction of prisms from existing
structures using a saw-cutting machine. The key advantage of this technique is its usability for all
kinds of masonry both in terms of the material (e. g. block, clay brick, and silicate brick) and
configuration (solid units and hollow units). Using each method of prism testing raises
controversial issues. In the case of the first way, it is challenging to be able to construct the prisms
in laboratories having exactly all detailed properties of the masonry in the existing structure. This
difference in detail may lead to inaccuracy in the estimation of compressive strength. Moreover, it
is hard and sometimes impossible to find the exact materials which had been used in the original
structure due to their uniqueness. The cost of the first way of prism testing is high as all materials
89
(unit and mortar) besides expert masons have to be provided. The problems of the second prism
testing method are the same as the large dimeter core testing method as mentioned above.
Both the core testing and prism testing are categorized as sampling techniques in Table 5-3.
Finding the appropriate sample size to remain intact during removal and transportation is very
difficult when extracting a sample from existing masonry. Any damage or fracture made to the
specimen throughout the procedure makes the sample useless. Overall, both core testing and prism
testing can easily lead to irrational and inaccurate estimation of the actual compressive strength
and methods involving sample extraction are considered as destructive testing methods which
makes them unacceptable for use on historic structures that are to be conserved.
Another testing method that has become popular in the evaluation of compressive strength is the
flat-jack testing method, See Figure 5-2. This method is an in-situ technique for evaluation of the
stress in the masonry for determining the masonry strength. The method is described as either non-
destructive with minor repairs or semi-destructive, based on different references. In this method,
thin stainless-steel bladders are placed into slots cut in masonry (in bed joints) some distance apart
– one above the other. The bladders are inflated to apply pressure and the deformation of masonry
located between them is monitored. Through the gradual increase of pressure, the stress-strain
relationship and loading cycles can be determined. The compressive strength is then estimated
based on the resultant experimental stress-stain curve together with a value of Young’s
compressive modulus. The compressive strength can be estimated more exactly if damage to the
masonry can be accepted: the maximum pressure applied by the flatjacks is more than the
compressive strength of masonry and the development of cracks is considered as a sign of failure.
Rossi (1987) and Noland et al. (1990) evaluated the accuracy of the flatjack technique. Both
showed that flatjack testing has an error ranging between 15%-20% which is reasonable and
acceptable for practical estimation of compressive strength. However, there are also some
difficulties regarding the flatjack technique besides the restrictions associated with its application
mentioned above. The section of masonry which is under testing has to be damaged and reach the
failure level to be able to estimate compressive strength. In the case of elements or construction
parts with small cross section (e.g. a narrow column), application of flatjack method may not be
possible due to the reduction of bearing capacity or total destruction of the element tested. Also, it
is difficult to interpolate results associated with unsuccessful cutting and recovered distances.
90
Reliable interpolation of results obtained from too weak or nonhomogeneous materials may be
challenging. In addition, there is a need to repair the masonry after using this technique which
involves repointing the joint with unloaded mortar in what was originally a loaded mortar joint
(thus altering the structure locally) and finding a compatible mortar may be expensive or even
impossible in the case of historical buildings. Flatjack technique can not also predict the
compressive strength of multi-wythe walls (common walls in the historic masonry structures)
accurately due to the short length of the bladder which can not reach to the inner wythe, see Figure
5-3. Moreover, in order to assess the reliability of a historical structure, it is not feasible to estimate
the variability in the compressive strength over an element or masonry structure by this method.
This is due to the fact that to obtain a CoV of compressive strength, flatjack testing should be
repeated in many locations leading to a high level of destruction and interference to a structure
with historical value.
Figure 5-2 Flatjack technique (n.d.).
91
Figure 5-3 Schematic of flatjack technique.
All the above statements regarding different methods of compressive strength estimation show that
there is a need for a new method to determine compressive strength which is compatible with
historical structures and able to provide the variability for reliability analysis. It is therefore
suggested that the compressive strength be estimated from the modulus of elasticity. Different
researchers and codes and standards recommend different relationships between the compressive
strength and the modulus of elasticity of masonry. A summary is presented in Table 5-7.
As with the expressions for estimation of whole masonry strength based on the strength of its
components, these formulas may also not lead to an exact value of the compressive strength
because they were developed for modern materials rather than historical ones. However, using a
non-destructive test multiple times and deriving therefrom a mean strength and the associated
variability will potentially provide more accurate values of that strength and variability than trying
to determine the means and CoVs of compressive strengths of the mortars and the units. The non-
destructive UPM can provide a measure of the variability of the materials, whist also not causing
damage to the structure under consideration. Obviously, the accuracy and applicability of this
suggested procedure should be checked experimentally in future work.
As can be seen, there is a wide range of suggestions that have been presented regarding the
relationship between E and f’m. The difference is only in the coefficients, here referred to as (k).
The distribution of the coefficients is illustrated in Figure 5-4. As can be observed, the coefficient
92
of f’m is most frequently in the range of 700 to 900 in the case of concrete masonry. This range is
also frequently reported in the case of clay masonry in addition to the coefficient being equal to
1000.
Figure 5-4 Distribution of the coefficient k for clay and concrete.
According to Figure 5-5, the fifty percent-probable coefficients are 800 and 890 for clay and
concrete masonry respectively. Therefore, it can be concluded that the reported coefficients which
are higher than 800 in the case of clay masonry and 890 in the case of concrete masonry lead to
more conservative compressive strength estimates, while those lower than these fifty-percent
probable coefficients result in higher values of compressive strength. Therefore, the compressive
strength of historical masonry structures could be estimated according to the values of E obtained
from the non-destructive tests and the relation between E and 𝑓𝑚′ . Coefficients for stone masonry
have not been reported, so this is clearly an area where work needs to be done to aid in the
estimation of the reliability and safety of historic structures. Given that stone units vary in size as
well as source material, it is likely that more than one coefficient will be needed to provide
reasonable estimates of strength from modulus measures. The best fit statistical distributions have
to be determined as well.
0
1
2
3
4
Fre
quen
cy
Relation between E and f'm
Concrete Masonry
Clay Masonry
93
Table 5-7 Summary of relations between 𝐸 and 𝑓𝑚′ found in literatures.
Reference Formula Type
BS 5628 (1978) 𝐸 = 900 𝑓𝑚′ Core filled concrete masonry
NZS (1990) 𝐸 = 800 𝑓𝑚′ Masonry
Eurocode 6 (1996) 𝐸 = 1000 𝑓𝑚′ Masonry
FEMA 306 (1999) 𝐸 = 550 𝑓𝑚′ Clay masonry
MSJC (2002) 𝐸 = 700 𝑓𝑚
′ Clay masonry
𝐸 = 900 𝑓𝑚′ Concrete masonry
IBC (2003) 𝐸 = 700 𝑓𝑚′ Masonry
CSA (2004) 𝐸 = 850 𝑓𝑚′ < 20000 MPa Clay and concrete masonry
NNI (2005) 𝐸 = (400 − 1000) 𝑓𝑚′ Masonry
AS3700 (1998)
𝐸𝑚∗∗ = 700 𝑓𝑚
′ 𝐸𝐿∗∗ = 450 𝑓𝑚
′ Clay units (5 MPa <f'uc<30 MPa, M2 and M3)
𝐸𝑚 = 1000𝑓𝑚′ 𝐸𝐿 = 660 𝑓𝑚
′ Clay units (f'uc>30 MPa)
𝐸𝑚 = 1000 𝑓𝑚′ 𝐸𝐿 = 500 𝑓𝑚
′ Concrete units (density>1800 kg/m3) and CS units
𝐸𝑚 = 750 𝑓𝑚′ 𝐸𝐿 = 500 𝑓𝑚
′ Concrete units (density<1800 kg/m3)
𝐸𝑚 = 1000 𝑓𝑚′ 𝐸𝐿 = 350 𝑓𝑚
′ Grouted concrete or clay masonry
𝐸𝑚 = 500 𝑓𝑚′ 𝐸𝐿 = 250 𝑓𝑚
′ AAC***
Paulay and Priestly (1992) 𝐸 = 1000 𝑓𝑚′ Masonry
Drysdale et al. (1994) 𝐸 = 210 − 1670 𝑓𝑚′ Masonry
Vermelfoort (2005) 𝐸 = 700 − 750 𝑓𝑚′ Clay brick masonry
Kaushik et al. (2007) 𝐸 = 550 𝑓𝑚′ Clay masonry
Mohamad et al. (2007) 𝐸 = 758 − 1021 𝑓𝑚′ Hollow concrete masonry
Budiwati (2009) 𝐸 = 1000 𝑓𝑚′ Clay and concrete masonry
Costigan (2015)
𝐸 = 230 𝑓𝑚′ PC-lime***
𝐸 = 130 𝑓𝑚′ Hydraulic lime***
𝐸 = 85 𝑓𝑚′ Feebly hydraulic lime***
*Eave is considered in this study **Em=Short-term loading, EL= Long-term loading, average of Em and EL is
considered in this study ***Modern material, therefore, not considered in this study and just for information and
comparison.
94
Figure 5-5 Cumulative probability of coefficient k.
Considering the coefficient of variation of the compressive strength of masonry obtained by
previous researchers, it can be concluded that the average coefficient of variation is 20% in the
case of both concrete masonry and clay masonry, Table 5-8. According to the research of
Galambos (1982), Schueremans & Van Gemert (1999), Schueremans (2001), Holicky & Markova
(2002), Park et al. (2009), Brehm (2011) and Mojsilovic & Stewart (2014), the best fit distribution
for the compressive strength of masonry (both concrete and clay) is reported to be the log normal
distribution, Table 5-9.
Some research has been done on the estimation of masonry compressive strength and its
variability. Schueremans (2001), Bojsiljkov et al. (2004), Löring (2005), Bojsiljkov & Tomazevic
(2005), Fehling & Stürz (2006a) and Kaushik et al. (2007) worked on clay brick which is common
in historic masonry structure. Calcium silicate and autoclave aerated concrete units are modern
masonry units. There has also been some research focused on the material characteristics of
masonry built with these kinds of modern materials (Jäger & Schöps (2004), Löring (2005),
Fehling & Stürz (2006a), Fehling & Stürz (2006b), Costa (2007), Graubner & Glowienka (2008),
Schermer (2007), Magenes (2007), Höveling et al. (2009) and Gunkler et al. (2009)). A summary
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
500 700 900 1100
Cu
mu
lati
ve
Pro
bab
ilit
y
Coefficient k
Concrete Masonry Clay Masonry
~800 ~890
95
of the results of these studies, obtained mostly through experimental tests, is presented in Table 5-
10. The data in that table give an idea about the expected range of compressive strength. The
compressive strength values obtained via other methods, such as the ones described above, can
also be compared to the values reported in Table 5-10
Table 5-8 Coefficient of variations of compressive strength reported by different researchers.
Reference CoVmean
Kirtschig & Kasten (1980) 0.17
Galambos et al. (1982) 0.18
Shah & Dong (1984) 0.25
Tschötschel (1989) 0.25
Schueremans (2001) 0.19
Holicky & Markova (2002) 0.20
Graubner & Glowienka (2008) 0.19
Table 5-9 Recommended distributions for compressive strength of masonry (both concrete and
clay) found in the literature.
Reference Distribution Comment
Galambos et al. (1982) LN -
Schueremans & Van Gemert (1999) LN Brick masonry
Schueremans (2001) LN Historical
masonry
Holicky & Markova (2002) LN -
Park et al. (2009) LN URM
Brehm (2011) LN CB, CS, AAC
Mojsilovic & Stewart (2015) LN -
96
Table 5-10 Values of compressive strength found in the literature (TLM is Thin-Layer Mortar).
Reference Unit Mean
(MPa) COV Description
Kaushik et al. (2007) Clay Brick
4.1 0.24 Weak mortar
7.5 0.18 Strong mortar
6.6 0.2 Intermediate
mortar
Schueremans (2001)
Clay Brick
(For Historic
Masonry)
4.53 0.17 Cores Ø150,
h=300mm
4.26 0.19 Pillars,
h=360mm
6.07 0.093 Wallettes,
h=570mm
4.3 0.137 Cores Ø113,
h=300mm
3.75 - wall,
h=2000mm
Fehling & Stürz (2006a) Clay Brick 6.7 - TLM, M5
Löring (2005) Clay Brick 5.6 - MG IIa
Bojsiljkov & Tomazevic
(2005) Clay Brick 4.6 -
M5, M6, M7,
M15, LM5
Bojsiljkov et al. (2004) Clay Brick 5.45 - M5, TLM
Fehling & Stürz (2006a) Calcium Silicate Wall 15 - TLM
Schermer (2007) Calcium Silicate Wall 15 - TLM
Magenes (2007) Calcium Silicate Wall 15 - TLM
Löring (2005) Calcium Silicate Wall 15 - TLM
Jäger & Schöps (2004) Calcium Silicate Wall 7.3 - TLM
Gunkler et al. (2009) Calcium Silicate Wall 20.6 - TLM
97
Reference Unit Mean
(MPa) COV Description
Fehling & Stürz (2006b)
Autoclave aerated
concrete
(unit-mortar
combination 4/TLM)
2.6 - TLM
Löring (2005) Autoclave aerated
concrete 2.3 - TLM
Costa (2007) Autoclave aerated
concrete 2.4 - TLM
Jäger & Schöps (2004) Autoclave aerated
concrete 3.3 - TLM
Höveling et al. (2009) Autoclave aerated
concrete 4.26 - TLM
Graubner & Glowienka (2008)
Calcium Silicate
13.5
0.16
Class 16
18.8 Class 20
27 Class 28
Autoclave aerated
concrete
2.4
0.14
Class 2
4.1 Class 4
5.5 Class 6
98
Understanding the shear failure of masonry structures can be important in order to evaluate
structural resistance to lateral loads (e.g. wind and earthquake). It is difficult to estimate the shear
strength without large scale testing. However, in-situ testing techniques applied to single masonry
units can provide comparable information to full scale ones. These in-situ testing methods are also
less destructive and more economical and consequently better than full scale testing in the case of
historic structures. Quite a few testing methods have been reported for estimation of the shear
strength of masonry. These methods are in situ testing techniques. An early one, named the “push
test”, was developed to be used in the evaluation of the seismic resistance of older unreinforced
masonry structures (ABK, 1984). In this method, a brick unit and its associated head joints (located
on opposite ends of the determined test unit) are removed. The next step is to place a hydraulic
ram in the wall and displace the wall laterally. The mortar bed joints located straight below and
above the designated test unit are sheared. This method is only appropriate for masonry structures
or elements having strong units and weak mortar. In such structural systems, shear cracks develop
in a stair-step pattern through mortar joints or bed joints slide over each other while the units
remain uncracked. In other words, bed joint sliding or stair-step cracking diagonally along mortar
joints dominate the shear failure. The shear strength of the mortar joints is then related to the shear
strength of the masonry system using empirical formulae. This method is not applicable for new,
modern masonry structures having strong mortar, but can be applied to historical structures as they
mostly have weak mortar. It should be noted however, that although application of the push test is
possible for historical structures from a scientific and theoretical point of view, it leads to
destruction and interference of historical systems which may be expensive to repair or even be
irreparable.
JCSS (2011) has not reported any direct approaches for estimation of the shear strength of
masonry. The committee introduced the tensile strength of the units (i.e. shear failure) and the
cohesion between units and mortar (i.e. sliding failure) as the two parameters determining the shear
strength of masonry walls. The presentation of a probabilistic model for estimation of both the
tensile strength and cohesion by JCSS (2011) may be the reason of not reporting direct approaches
associated with the estimation of shear strength of masonry.
99
A new American Society for Testing and Materials International (ASTM) standard presented three
semi- destructive test methods regarding the estimation of shear strength. The mortar joint shear
strength index is calculated by dividing the maximum recorded horizontal force by the gross area
of the upper and lower bed joints. One of these methods is the conventional approach with a
hydraulic ram as explained above. The second method involves flatjacks in order to apply
compression to the test location. The shear strength index and the coefficient of friction are the
outcomes of this method. The relationship between shear strength thickness and applied
compressive stress is obtained by doing the shear test while the compressive stress is increased.
Moreover, the coefficient of friction is the slope of the best-fit line passing through the measured
joint shear strength index and the applied compressive stress plotted versus each other. As the
friction coefficient is measured in this method instead of assumed, this method would lead to more
accurate results in terms of joint shear strength index. The other two testing procedures described
by ASTM assume a value of friction coefficient in their strength calculation procedures. The last
method is also an in-situ test which can be performed using special flatjacks. By pressurizing the
masonry horizontally between two flatjacks and monitoring the resultant deformation, the shear
strength of the masonry can be estimated. Using flatjacks rather than a hydraulic ram results in a
test that does not require the removal of masonry units.
Flatjack tests have a minor destructive effect on the masonry compared to the conventional
approach using a hydraulic hammer, so they may be more appropriate for masonry structures in
terms of destructivity. However, estimation of strength variability with flatjack tests is arguably
destructive as that would require making a large number of slots. Moreover, the flatjack method
has not shown good accuracy in cases of deep, multi-wythe and multi-layer structures as the
bladders cannot penetrate the masonry sufficiently to get the desired information (as mentioned
earlier). Therefore, there is a lack of an appropriate test method for estimation of the variability of
the shear strength of historic masonry structures required to be able to determine their reliability
under the vertical loading. Consequently, there is a potential area of research here. Here, shear
strength is proposed to be considered deterministic and to be estimated by few applications of
flatjack technique (as semi-destructive testing method) to restrict the potential destructions of
variability estimation.
100
Quite a few research studies have been directed to the estimation of the tensile strength of historic
masonry. This may be due to the fact that the tensile strength of masonry perpendicular to the bed
joints has been proved to be negligible. Therefore, it is usually considered to be zero in the
calculation of masonry resistance. However, in some cases (e.g. slender walls) where the flexural
tensile strength may significantly influence the structural resistance, there may be a necessity to
include a probabilistic model of tensile strength in reliability analysis.
The bond-wrench test can be used to estimate the tensile strength of historic masonry, although
the test is semi-destructive. In this method, the units are removed to get the instrument in place,
wrench off some units to get the bond strength and then mortar and all units are returned to their
original places. If the test does not lead to the damage to the prims under the test, it considers semi-
destructive. However, there are limitations in the repetitive application of the wrench test to
estimate the variability of tensile strength over a historic masonry system due to the destructivity.
In other words, in order to understand how the tensile strength changes over a historic masonry
system, the wrench test should be repeated many times which leads to destruction of the historic
structure. Therefore, it is recommended to neglect tensile strength of structural members which
their resistances are not sensitive to tensile strength. However, the member which its resistance is
sensitive to tensile strength, can be tested by wrench test as it is semi-destructive (minor
destruction).
Schubert (2010) worked on the flexural tensile strength of masonry perpendicular and parallel to
bed joints. As the cohesion of thin layer mortar (TLM) is reliable, masonry with TLM was
considered in this research. It was showed that there is considerable scatter in this property and the
scatter is mostly dependent on the characteristics of the mortar. The type of head joint (filled or
unfilled) was shown to have only a small influence on the flexural tensile strength of the masonry.
The flexural tensile strengths of masonry parallel and perpendicular to the bed joints are
summarized in Table 5-11.
101
Table 5-11 Flexural tensile strength of masonry (Schubert, 2010).
Unit Type Head Joint Number
of Tests
Mean of 𝑓𝑡
(𝑁/𝑚𝑚2)
Range of measured 𝑓𝑡
(𝑁/𝑚𝑚2)
Parallel to bed joints
CB[1] Un-filled 2 0.21 0.2, 0.22
CS[2]
non-perforated Filled 8 0.75 0.36 – 1.14
Un-filled 11 0.71 0.38 – 0.97
perforated Filled 4 0.48 0.45 – 0.51
Un-filled 4 0.25 0.29 – 0.35
ACC[3] Filled 10 0.43 0.22 – 0.64
Un-filled 6 0.2 0.16 – 0.24
Perpendicular to bed joints
CB Un-filled 3 0.28 0.26 – 0.3
CS
perforated Filled &
Un-filled 8 0.56 0.35 – 0.73
non-perforated Filled 4 0.34 0.23 – 0.48
ACC Filled &
Un-filled 23 0.4 0.25 – 0.81
Concrete Blocks Filled 5 0.33 0.22 – 0.44
[1] Clay Brick, [2] Calcium Silicate, [3] Autoclave Aerated Concrete
JCSS (2011) categorized tensile strength into longitudinal tensile strength and splitting tensile
strength and provided a probabilistic model for the tensile strength of the units as follows:
Tensile strength perpendicular to units (longitudinal tensile strength)
102
𝑓𝑏𝑡,𝑙 = 𝑐2. 𝑓𝑏 (5-5)
Tensile strength parallel to the units (splitting tensile strength)
𝑓𝑏𝑡,𝑠 = 𝑐3. 𝑓𝑏 (5-6)
where 𝑓𝑏𝑡,𝑙 and 𝑓𝑏𝑡,𝑠 are mean of tensile strength perpendicular to units and parallel to the units,
respectively. 𝑐2 and 𝑐3 are ratios with values for different materials and perforation being presented
in detail in JCSS (2011). JCSS (2011) does not describe shear strength individually. Therefore,
although the splitting tensile strength involves shear strength in concept, it is mentioned under the
category of tensile strength based on the JCSS (2011). The tensile strength of the units has a strong
effect on the shear capacity of the masonry walls. Therefore, JCSS (2011) just defined probabilistic
models for the tensile strength of different kinds of masonry units in the direction dominating the
shear strength of the masonry. In the case of calcium silicate units and autoclaved aerated concrete
units, the shear capacity is dominated by the tensile strength in the longitudinal direction whereas
the splitting tensile strength describes the shear capacity of clay brick units. Therefore,
probabilistic modeling of the tensile strength of calcium silicate units and autoclaved aerated
concrete units is suggested only in terms of the longitudinal direction and the tensile strength of
clay brick units is recommended for the perpendicular direction as follows
Tensile strength in the longitudinal direction of calcium silicate units and autoclaved aerated
concrete units
𝑓𝑏𝑡,𝑙,𝑗 = 𝑌. 𝑓𝑏𝑡,𝑙 (5-7)
Splitting tensile strength of clay brick units
𝑓𝑏𝑡,𝑠,𝑗 = 𝑌. 𝑓𝑏𝑡,𝑠 (5-8)
103
where 𝑌 is a lognormal variable, see Table 5-12. The large values of CoV reveal that the tensile
strength of the units shows large scatter.
Table 5-12 Statistical characteristics of 𝑌 involved in probabilistic model of tensile strength
(JCSS, 2011).
Unit Type CoV Mean Distribution
CS[1] 0.26 1.0 Log-normal
AAC[2] 0.16 1.0 Log-normal
CB[3] 0.24 1.0 Log-normal
[1] Calcium Silicate, [2] Autoclave Aerated Concrete, [3] Clay Brick
Li et al. (2016) reported mean values and the CoV for both the tensile strength of mortar and the
tensile strength of brick. The mean and CoV of the tensile strength of mortar were reported to be
1.21 and 0.30, respectively and the mean and CoV of brick tensile strength were reported to be
3.13 and 0.34, respectively. Moreover, a lognormal distribution was recommended to be the best-
fit distribution for the tensile strength of the mortar while the Weibull distribution was considered
to describe best the distribution of the tensile strength of brick. Li et al. (2016) did not specifically
define the type of unit and mortar that they used in their studies. Therefore, there may be
inaccuracy if the reported values are generalized for other brick masonry structures.
All of the studies mentioned above and the findings regarding the tensile strength of the masonry
(Schubert, 2010; JCSS, 2011, Li et al., 2016) are focused on new masonry materials (units and
mortar) and may not be accurate for historic masonry materials. Therefore, there is still a gap in
knowledge regarding the tensile strength of historic masonry and its variability in order to do
reliability analysis accurately, specifically in the case of elements in which tensile strength plays
a key role in the determination of load bearing capacity. In other historic masonry systems, where
tensile strength does not have significant influence and as mostly historical structures are
unreinforced, the tensile strength of the masonry and its variability (due to its brittle nature) can
be neglected in reliability analysis.
104
Cohesion and friction coefficient are the two parameters that influence the shear strength of the
masonry. However, no study could be found focusing on the evaluation of the stochastic
characteristics of the cohesion and friction coefficient of historic masonry structures.
JCSS (2011) only presents the probabilistic models related to cohesion and friction coefficient
considering new masonry materials which may not be accurate for reliability analysis of historic
masonry materials. The JCSS (2011) recommended probabilistic models for both cohesion and
friction coefficient are presented as follows:
- Cohesion
𝑓𝑣,𝑗 = 𝑌. 𝑓𝑣,𝑚 (5-9)
where 𝑓𝑣,𝑚 is the mean of cohesion and 𝑌 is a random variable according to Table 5-13.
- Friction coefficient
𝜇𝑗 = 𝑌. 𝜇𝑚
(5-10)
where a mean value of 0.8 is normally considered for the friction coefficient and 𝑌 is a random
variable given in Table 5-14.
Table 5-13 Statistical characteristics of 𝑌 involved in probabilistic model of cohesion (JCSS,
2011).
Unit Type Mean CoV Distribution
CS[1] 1.0 0.35 Log-normal
AAC[2] 1.0 0.35 Log-normal
CB[3] 1.0 0.40 Log-normal
[1] Calcium Silicate, [2] Autoclave Aerated Concrete, [3] Clay Brick
105
Table 5-14 Statistical characteristics of 𝑌 involved in probabilistic model of friction coefficient
(JCSS, 2011).
Unit Type Mean CoV Distribution
CS[1] 1.0 0.19 Log-normal
AAC[2] 1.0 0.19 Log-normal
[1] Calcium Silicate, [2] Autoclave Aerated Concrete,
Li et al. (2016) reported the mean, CoV and best-fit distribution associated with the cohesion of
unreinforced masonry constructed with an extruded brick. They described neither the type of brick
nor the type of mortar used in their studies. The Mean and CoV of the cohesion were reported to
be 1.81 and 0.27, respectively. Moreover, the lognormal distribution was considered for this
material property. However, as they did not mention exactly the characteristics of the materials
that were studied, it may not be accurate to generalize the reported values to other unreinforced
brick masonry structures, like historic ones.
Given the information above, there is a gap in knowledge regarding the probabilistic characteristics
of cohesion and friction coefficient of masonry structures especially historical masonry structures.
106
Chapter VI: Reliability Assessment Methodology
Historic masonry structures are constructed with various masonry materials and are mostly
unreinforced. Therefore, they have limited load bearing capacity, making them vulnerable to the
applied loading conditions. Assessing the reliability of their performance under applied loads is of
great importance. If there is a need to upgrade and strengthen historic structural systems to preserve
them, minimum interventions have to be applied to save their historical value. Reliability
assessment can therefore act as a useful tool to determine the level of required upgrading leading
to minimization of the intervention. Although there is need to analyze the reliability of a historic
masonry structure, no formal methodology has been proposed in this regard. In this chapter, an
attempt is made to initiate the development of a step-by-step methodology appropriate for
reliability assessment of historic masonry structural members considering the specific limitations
and criteria which exist in this regard. This is the first step before full structural reliability could
be determined. There is no doubt that there are still significant gaps in knowledge and limitations
in the development of a comprehensive and accurate reliability assessment methodology for
historical masonry structures. Investigation of all the limitations needs more time and research
beyond the scope of this thesis.
A general layout of the evaluation procedure of existing structures is presented as a flowchart by
Figure 3-1 in Chapter 3. To evaluate the structure, the process in the flowchart should be followed.
This evaluation involves different steps from the preliminary evaluation to the detailed one.
To start the structural evaluation, the first step is to gather information about the structure. This
includes investigation of documents, an initial inspection and initial check decisions on immediate
plans and suggestions for detailed evaluation. One of the main objectives of restoration of
historical structures is conservation. Therefore, a careful analysis of the structure is required.
107
Specific and comprehensive information about the structure’s materials and techniques as well as
its condition need to be gathered. Architectural surveying is the main step of direct investigation
which includes geometric and morphological measurements, materials survey, and
photogrammetry. The next step of data gathering is historical investigation. Critical reading and
understanding of bibliographies of iconographies and available documents are the main parts of
historical investigation. Historical documents can provide useful information about material
properties and construction methods. The integration of the knowledge and information obtained
leads to a more precise and comprehensive understanding of the structure under evaluation. The
more carefully the primary evaluation is performed, the more accurate is the assessment achieved.
In the case of historical structures, as there are typically almost no documents available, a field
inspection and direct investigation are of special importance.
When useful information is gathered from direct and historical investigation and study, a detailed
reliability assessment may be required. The flowchart (Figure 3-1) does not include the detailed
steps which need to be followed in order to do reliability assessment. Therefore, an attempt is made
here to review the process of reliability assessment (Figure 6-1) of a historic structure considering
the previously discussed parameters. Here the focus is on reliability assessment of each member
individually. System reliability assessment can provide more accurate evaluation since it considers
the probability of failure as the probability of failure of whole system including the individual
members. However, connection behaviour and strength are also considered in system reliability
which are influential in structural resistance of historic masonry structures. System reliability
assessment is thus highly complex in a new structure, but especially so in the case of historic
masonry structures because there is no knowledge and information about connections between the
different systems and materials typically found in such a structure: system (whole structure)
reliability is thus an area ripe for further research and is outside the scope of this thesis.
The basis of reliability assessment is the limit state function, being the resistance minus the load
effect, as discussed in Chapter 3. Codes of practice recommend formulas to estimate the required
resistance and load effects (defined as R and L in Chapter 3) for different structural members under
different failure scenarios. There is general acceptance in Canada that if historic structures can
satisfy 60% of the structural resistance required in the new NBCC, they are considered to be in a
108
safe condition. Therefore, the limit state function can be formulated for each historic structural
member to be 60% of required resistance minus the load effect recommended by NBCC code.
Material properties play a key role in the determination of structural resistance. CSA S304-14
recommends expressions for estimating the resistance of a masonry member with respect to
different failure scenarios, which include general material properties. Compressive strength, shear
strength, tensile strength, cohesion and friction coefficient are material properties which are
considered in determination of structural resistance. The stochastic characteristics of the material
properties of historic masonry structures are recommended to be estimated directly and specifically
for each structural element under evaluation due to their specific nature and characteristics using
field testing techniques (reasons were discussed in detail earlier). The available approaches,
concerns and recommendations associated with the estimation of the probabilistic models of these
material properties are discussed in detail in Chapter 5. As there is no non-destructive method to
estimate the shear strength, tensile strength, cohesion and friction coefficient more than one
approach is recommended for them here, see Figure 6-2. Shear strength is recommended to be
estimated using flatjack. The estimated value of shear strength can be considered either
deterministic to minimize the destruction caused by repeating the test or as a test data for Bayes’
theorem to update prior estimation. Tensile strength is proposed to be neglected in the case of an
ordinary member (non-sensitive to tensile strength). In the case of members which their resistance
is sensitive to their tensile strength, 4 approaches are recommended, including consideration of
tensile strength as deterministic; the use of the semi-destructive wrench test, considering the result
of wrench test as a test data for Bayes’ theorem to update available data and expand the database,
considering the results of the wrench test as a mean and using the probabilistic model of JCSS
(2011); and lastly, considering the probabilistic model of JCSS (2011) without modification.
Cohesion and the friction coefficient are also recommended to be modeled either deterministically
using recommended values, or by using the JCSS (2011) probabilistic model. It should be noted
that each approach adds its related error to the reliability assessment.
In order to determine the load effect, the probabilistic model of applied load needs to be known.
The stochastic characteristics of dead load, live load, wind load and snow load are described in
detail in Chapter 4.
109
Knowing the limit state function and the probabilistic models of the material properties and applied
loads, the reliability index needs to be defined in next step. There are several structural reliability
techniques which are briefly explained in Chapter 3. Asymptotic techniques and simulation based
approaches are two reliability methods that have been developed and are commonly used in
structural reliability assessment. The selection of the appropriate structural reliability technique
depends several factors like time concerns, required accuracy and the limit state function.
The final step is to compare the calculated reliability index or probability of failure with the target
reliability index as a decision criterion. As discussed in Chapter 3, the necessity of meeting the
target reliability index and target probability of failure as recommended by codes of practice (for
new structures) is a controversial issue in reliability assessment of historic masonry structures. The
target probability of failure and target reliability index values reported for masonry structures as
well as some formulae for the calculation of the targets considering specific criteria associated
with historic masonry structures are reported in Chapter 3.
The process of reliability assessment structures is summarized in Figure 6-1 (see the example of
the application of the method presented in Appendix). Although the process of reliability
assessment of structures is the same in general, there are some parameters which should be
considered in reliability assessment of historic masonry structures which make the reliability
assessment procedure more specific. The necessity of considering a specific target reliability index
and target probability of failure as decision criteria, the difficulty and restriction in estimating the
material properties and related variabilities for estimating their probabilistic models and the
likelihood of inaccuracy in the available limit state functions for describing the behaviour of
historic structures are the most challenging parameters necessitating the development of a specific
methodology to assess the reliability of historic masonry structures. The first two parameters are
discussed in this thesis. However, there is still plenty of room for development. Figure 6-2 shows
the recommended approaches for estimation of the probabilistic models of historic masonry
materials.
110
Figure 6-1 Flowchart for reliability assessment of structures.
Reliability Assessment of Historical Masonry Structures
Data Collection
(Direct and Historical Investigation)
Probabilistic Model of
Historic Masonry Material
Probabilistic Model of
Applied Loads Limit State
𝑔(𝑥) = 𝑅 − 𝐿
Structural Reliability Techniques
𝛽 & 𝑃𝑓
𝛽 > 𝛽𝑇
𝑃𝑓 < 𝑃𝑓𝑇
𝛽 < 𝛽𝑇
𝑃𝑓 > 𝑃𝑓𝑇
Reliable Non-reliable
111
Figure 6-2 Recommended approaches for estimation of the probabilistic models of historic
masonry materials.
Wrench Test, Deterministic
Wrench Test, Mean JCSS, Distribution and CoV
JCSS Probabilistic Model
Modulus of
Elasticity Compressive
Strength Shear
Strength Tensile
Strength
Cohesion & Friction Coefficient
Deterministic,
Recommended
values
JCSS Model
Non-sensitive to
Tensile Strength
Negligible, 0
Sensitive to
Tensile Strength
UPM E= k.f'm
Clay
Masonry Concrete
Masonry
k=700-900 Median: 890
k=700-900 Median: 800
Flatjack,
Deterministic
Probabilistic Model of
Historic Masonry Material
Flatjack, Bayes’ theorem
Wrench Test, Bayes’ theorem
112
The required resistance of historic masonry structures in the evaluation stage is thought to be 60%
of the required resistance in the design stage recommended by NBCC. This thought is unwritten
and no scientific base or detailed analysis and investigation has been reported for it. As formulation
of the limit state function is a key parameter in the outcome of reliability assessment, acceptance
and application of this general belief with respect to satisfactory structural resistance may result in
an inaccurate reliability index or probability of failure.
Moreover, the intent of formulae to calculate the resistance presented by codes of practice is
accepted here to satisfy safety. The appropriateness of these formulae for the evaluation of historic
masonry structures however, is a controversial issue as they are determined for new masonry
materials and constructions. Some research is recommended to be done to verify the
appropriateness of these formulae for historic masonry structures having specific characteristics.
Limit state functions formulated by these recommended expressions should be investigated in
terms of their accuracy in the estimation of reliability of historic structures.
Many researchers have investigated the relationship between compressive strength and modulus
of elasticity and have reported different relationships between these two material properties.
Nevertheless, this is the basis of the recommendation in this study regarding the calculation of
compressive strength, as discussed in Chapter 5. As different coefficients have been suggested for
this relationship, the median values of the coefficient are considered here as the final recommended
coefficient. However, the median value may not be the real representation of this coefficient which
may result in inaccurate estimation of compressive strength and any property derived from the
compressive strength. A study is therefore recommended to define the most accurate coefficient.
Moreover, no information was found regarding the relationship of compressive strength and
modulus of elasticity in the case of stone masonry. To be able to generalize the approach, this
relationship needs to be defined for stone masonry in its various forms.
The stochastic characteristics of some material properties of historic masonry structures cannot be
estimated directly due to the lack of non-destructive approaches. Therefore, deterministic values
of them or probabilistic models recommended for new masonry materials are used. To achieve a
113
more accurate reliability index or probability of failure, there is a need to develop non-destructive
testing approaches to estimate the probabilistic models of relevant properties accurately.
114
Chapter VII: Conclusion
Historical masonry structures should be preserved for future generations. Despite the high level of
vulnerability of unreinforced masonry structures under applied loads and the importance of their
reliability evaluation, there is no formal methodology to assess the reliability of historic masonry
structures. In this study, an attempt is made to initiate development of a step-by-step methodology
for assessing the reliability level of historic masonry structures. Reliability assessment requires the
limit state function to be formulated, being the structural resistance minus the load effect.
Therefore, in order to develop an appropriate determinate methodology, estimations of
probabilistic models of structural resistance and load effects are required to formulate a limit state
function. Moreover, a target reliability appropriate for historical structures has to be determined.
Currently, there is no formal method for reliability assessment of historical masonry structures.
This thesis contributes to the development of a reliability-based assessment methodology for
historical masonry structures as follows:
A comprehensive literature review was done on different aspects of reliability assessment of
masonry structures, specifically historical masonry ones, including the theory of reliability
assessment of structures, limit states, reliability assessment techniques and suitable combination
among finite element structural analysis, reliability analysis methods and reliability software.
Reliability assessment of historic masonry structures necessitate determination of target reliability
index or target probability of failure as decision criteria. Codes of practice recommended different
values as target probability of failure or target reliability index. Regarding historical structures, the
necessity of meeting the target probability of failure values or target reliability index presented by
different codes of practice is still a controversial issue. This is due to that fact that the
recommended values are reported considering new materials, structural configurations and
construction methodologies, not historical ones, with specific criteria and requirements. In this
thesis, different calculation approaches and values associated with target reliability index and
115
failure probability being appropriate for historic structures are described. Use of one formula is
recommended.
Deterioration is a common process in historic structures which may result in a change in masonry
resistance, the loading conditions or analysis model. Appropriate approaches for integration of the
deterioration into reliability assessment were studied and degradation functions were reported.
Reliability assessment of structures under seismic loading is a challenging and complex field of
science as involves different uncertainties including those in the nature of ground motions and
those in the nonlinear behaviour of structures. The specific methodology for reliability assessment
of existing structures under seismic loading was reviewed.
The stochastic modeling of applied loads as one of the key parameters in the determination of the
limit state function including dead load, live load, wind load and snow load were reviewed.
Different literature- and code-recommended probabilistic models and stochastic characteristics for
applied loads were studied and compared. Statistical characteristics of point-in-time components
of live load, wind load and snow load were also investigated. Finally, the best fit probabilistic
models for different load effects were presented.
The stochastic characteristics of construction materials play key roles in the determination of
probabilistic models of structural resistance. There are various testing techniques which can be
used to estimate stochastic characteristics of historical masonry materials. Destructive testing of a
historic masonry structure or its components to get more realistic information about the material
properties is not recommended as such tests may lead to irreparable damage to these valuable
structures. Moreover, some of testing techniques are not capable of estimating the variability of
material properties over an element or structural system, this being essential for reliability
assessment. A summary of testing techniques being used for the evaluation of masonry materials
properties was presented. The strengths and weaknesses of each technique as well as its
applicability in estimation of stochastic characteristics of historic masonry materials properties
were investigated. The calculation procedures, values and the best fit distributions recommended
by several codes of practice regarding the estimation of the statistical characteristics of masonry
material properties were presented. As code recommendations are mostly for new masonry
materials, the accuracy, applicability and accordance of them with historical masonry materials
were discussed.
116
Compressive strength is the most important material property influencing the structural resistance
of historic masonry systems and consequently affects the reliability assessment of these structures.
Codes are necessarily conservative and are also generally aimed at design or assessment with
modern masonry materials, so the use of code values for historical structures may lead to inaccurate
reliability assessment. There are various testing techniques which can be used in order to estimate
the mean value of compressive strength related to existing masonry structures. Testing techniques
which damage the structure (for example, by removing samples of the masonry) are not
recommended for historical buildings because of their destructive nature. Moreover, these
destructive techniques are also restricted in their ability to estimate the variability of compressive
strength over an element or structural system. Here, a proposal is made on how to determine a
probabilistic model of the masonry compressive strength from non-destructive tests.
This research focused on proposing a methodology for the estimation of a probabilistic model of
compressive strength of historic masonry structures, as the most important material characteristic
influencing the structural resistance. The idea was to get more reliable information directly without
any disturbance to the historical value of the structure. However, there are other material properties
which are influential on the structural resistance in some cases (e.g. flexural tensile strength in the
case of slender walls). There is still no procedure for estimating the stochastic characteristics of
these material properties without any disturbance to the historic masonry. Further work is needed
to assess the statistical characteristics of these material properties.
In this thesis, the procedure for estimating compressive strength was based on the relationship
between E as modulus of elasticity and 𝑓𝑚′ as compressive strength. No information was found
regarding this relationship for stone masonry. The focus has been on the development of
methodology for estimating of compressive strength of concrete masonry and clay masonry –
frequently used modern masonry materials. Therefore, determining the relationship between E and
𝑓𝑚′ for stone masonry is obviously an area where work needs to be done to aid in the estimation of
the reliability of historic masonry structures.
117
The accuracy and applicability of the proposed methodology regarding the estimation of the
stochastic characteristics of the compressive strength of historical masonry structures should be
evaluated through several experimental tests.
In order to do reliability assessment, a limit state function needs to be determined being structural
resistance minus load effect. Codes and standards recommend different formulas to calculate
structural resistance associated with different failure modes. The capability of these formulae to
be an accurate representation of the structural resistance of historical structures is still a
controversial issue. This is due to that fact that the recommended expressions are obtained
considering new materials, structural configurations and construction methodologies rather than
historical ones with specific criteria and requirements. Therefore, more research should be done in
this area.
The resistance and behaviour of connections play a key role in the resistance and behaviour of the
structural systems especially in case of historic masonry structures which may have weak
connections. This parameter should be considered in future research using system reliability
assessment.
118
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Appendix
This appendix includes a simple example on reliability assessment of an ordinary unreinforced
shear wall (see Figure A-1) using the recommended procedure. Different failure modes of
unreinforced shear wall are explained and their associated limit state functions are determined.
However, as the recommended procedure is the same for different elements under different failure
modes, the probabilities of failure of an unreinforced shear wall under two load combinations and
in the case of two failure modes are calculated as an example.
Figure A-1 Clay brick masonry shear wall.
System Definition
The system which is considered in this example is an unreinforced masonry shear wall. The wall
was constructed with clay units (190mm x 90mm x 57mm) and general purpose mortar. The wall
is considered to be an interior 3m high, 2m long and 0.09m thick wall. The wall is assumed to be
surrounded with a slab of 2 m width and 0.03 m thickness. Self-weight of the shear wall and the
slab is 2.5 𝑘𝑁/𝑚3.
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Failure Modes of Masonry Shear Wall
The combination of axial and shear stress, the material properties and slenderness ratio h/lw, can
determine the failure modes of shear walls. The failure modes of a shear wall include:
General compression failure
Overturning failure/Flexural failure
Sliding failure
Diagonal shear failure
General compression failure:
This failure occurs when the axial load applied on the shear wall is high and masonry compressive
strength is low, see Figure A-2.
Figure A-2 General compression failure.
Overturning failure/Flexural failure:
This failure is due to the exceedance of the vertical stress at the toe of the wall from the
compressive strength of the masonry. Therefore, The units located in the bottom corner of the wall
crush. This failure mode is more likely to occur for slender walls, Figure A-3.
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Figure A-3 Flexural failure.
Sliding failure:
This failure mode is more likely for low axial load and large horizontal force (mostly in an
unreinforced wall) which result in the exceedance of shear stress from the sliding strength of the
bed joints, Figure A-4.
Figure A-4 Sliding failure.
Diagonal shear failure:
This failure mode can occur when axial load is higher compared to sliding failure mode (potentially
through units), Figure A-5.
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Figure A-5 Diagonal shear failure.
Limit state
Ultimate limit state is considered in this example.
Limit State Function
Flexure (axial force and moment) limit state function:
𝑔(𝑥) = 𝑃𝑓
𝐴−
𝑀𝑓𝑦
𝐼 𝛼𝐷 = 0.9
(A-1)
𝑔(𝑥) = ∅𝑚𝑓𝑚′ −
𝑃𝑓
𝐴+
𝑀𝑓𝑦
𝐼 𝛼𝐷 = 1.25
(A-2)
where 𝑀𝑓 is factored moment and 𝑃𝑓 is factored axial load at the section under consideration.
Sliding Limit state Function:
𝑔(𝑥) = 𝑉𝑟 − 𝑊 (A-3)
𝑉𝑟 = ∅𝑚𝜇𝑃2 (A-4)
𝑃2 = 0.9 𝐷 (A-5)
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where 𝑊 is the applied wind load on the wall, 𝜇 is friction coefficient, 𝑃2 is compression force
perpendicular to the sliding face and 𝐷 is the applied dead load.
Diagonal Shear Limit State Function:
𝑔(𝑥) = 𝑉𝑟 − 𝑊 (A-6)
𝑉𝑟 = ∅𝑚(𝑉𝑚𝑏𝑤𝑑𝑣 + 0.25𝑃𝑑)𝛾𝑔 + (0.6∅𝑠𝐴𝑣𝑓𝑦𝑑𝑣
𝑆) ≤ 0.4∅𝑚√𝑓𝑚
′ 𝑏𝑤𝑑𝑣𝛾𝑔 (A-7)
𝑉𝑚 = 0.16 (2 −𝑀𝑓
𝑉𝑓𝑑) √𝑓𝑚
′ 0.25 ≤𝑀𝑓
𝑉𝑓𝑑≤ 1
(A-8)
where 𝑊 is the applied wind load on the wall, 𝑏𝑤 width of wall, 𝑑𝑣 is effective depth, 𝛾𝑔 is
grouting factor, 𝑀𝑓 is factored moment and 𝑉𝑓 is factored shear at the section under
consideration.
Probabilistic Model of Applied Load
Bartlett et al. (2003) presented a summary of the statistical parameters for loads which have been
used to calibrate the loads and load combination criteria for the 2005 National Building Code of
Canada (NBCC). Their recommended probabilistic model is considered in this example.
Dead load:
Dead load is considered to be the self-weight of the wall in addition to the self-weight of the slab.
Bias factor of dead load is considered to be equal to 1 (Bartlett et al., 2003). The probabilistic
model considered for dead load in this example can be found in Table A-1.
Table A-1 Probabilistic model of dead load.
Mean (𝐾𝑁) CoV Distribution Type
1.65 0.10 Normal
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Live load:
The nominal uniformly distributed live load is calculated as follows (NBCC, 1999):
𝐿 = 2.4 (0.3 + √9.8
𝐵) = 4.48 (A-9)
𝐵 = 2 × 2 = 4 𝑚2 (A-10)
where 𝐵 is tributary area. Having the bias value of 0.9, the mean value of live load is calculated to
be 4.03. The probabilistic model of 50-year maximum live load is considered in this example
(Bartlett et al., 2003). Table A-2 presents the probabilistic model of live load.
Table A-2 Probabilistic model of live load.
Mean (𝐾𝑁) CoV Distribution Type
4.03 0.17 Gumbel
Wind load:
The shear wall is assumed to resist the wind which is applied to a 6 m × 3 m wall locating
perpendicular to the shear wall. The reference velocity pressure is derived from Appendix C of the
2010 NBCC for Halifax (𝑞 = 0.58 𝐾𝑝𝑎). Knowing the reference velocity pressure, wind velocity
is calculated from the relationship between wind pressure and wind velocity (see Chapter 4). The
bias factor associated with wind velocity in 50-year is given to be equal to 1.049 for Halifax
(Bartlett et al., 2003). Therefore mean value of wind velocity is calculated to be equal to 1 𝑚/𝑠2.
Exposure factor (𝐶𝑒) is assumed to be 0.9 and the external pressure-gust coefficient (𝐶𝑝𝐶𝑔) is
assumed to be 1.15. The bias factor of the overall transformation factor, its CoV and distribution
type is considered based on the recommendation of Bartlett (2003) (see Chapter 4). The importance
factor is estimated to be 1.15.
138
Overall transformation factor = 𝐶𝑒𝐶𝑝𝐶𝑔 = 0.9 × 1.15 = 1.035 (A-11)
Table A-3 presents the considered probabilistic modeling for wind load parameters.
Table A-3 Probabilistic model of wind load.
Parameter Mean (𝑚/𝑠2) CoV Distribution Type
Wind velocity 1 0.17 Gumbel
Overall transformation factor 0.7 0.22 Lognormal
Considered Load combinations:
Two load combinations are selected to be considered in this example, see Table A-4.
Table A-4 Considered load combination.
Load Combination
Principal Load Companion Load
0.9D 1.4W
(0.9D or 1.25D) + 1.5L 0.4W
Probabilistic Model of Material Properties
Modulus of Elasticity:
The mean value and CoV of modulus of elasticity of clay brick masonry (with weak mortar)
reported by Kaushik et al. (2007) is considered in this example. The distribution of modulus of
elasticity is also considered to be log-normal as recommended by JCSS (2011). These assumptions
are made to be able to proceed this example. However, the modulus of elasticity and its statistical
characteristics are recommended to be estimated by UPM as a non-destructive test in the case of
real application of the procedure for reliability evaluation of historic masonry structures. Table A-
5 presents the probabilistic model of modulus of elasticity.
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Table A-5 Probabilistic model of modulus of elasticity.
Mean (𝑀𝑃𝑎) CoV Distribution type
2300 0.24 Log-normal
Compressive Strength:
The compressive strength of clay masonry is recommended to be estimated by modulus of
elasticity using following formula
𝐸 = 800 𝑓𝑚′ (A-12)
It should be noted that 800 is the median of the coefficient k values reported by different authors.
The probabilistic model of compressive strength is presented in Table A-6.
Table A-6 Probabilistic model of compressive strength.
Mean (𝑀𝑃𝑎) CoV Distribution type
2.88 0.24 Log-normal
Probability of Failure under Flexure
The probability of failure of the shear wall is estimated in both cases of the flexural failure mode
and diagonal shear failure mode under two load combinations. The Monte-Carlo Method is used
to calculate the probability of failure in this example. There are other reliability analysis methods
(mentioned in Chapter 3) which can be used.
Using the Mont-Carlo Method, the number of failures is estimated to be 1 out of 600 in the case
of flexural failure under the first load combination presented in Table A-4, which means
𝑃𝑓 =1
600= 0.0017 (A-13)
The target probability of failure is calculated as follows (see Chapter 3)
140
𝑃𝑓𝑇 =10−4𝑆𝑐𝑡𝐿𝐴𝑐𝐶𝑓
𝑛𝑝𝑊=
10−4 × 0.005 × 100 × 1.0 × 0.1
10 × 1.0= 5 × 10−7 (A-14)
where 𝑡𝐿 is assumed to be 100 years and 𝑛𝑝 is assumed to be equal to 10. The values of the other
parameters are derived from Table 3-1 in Chapter 3. Therefore probability of failure is more than
target probability of failure which means the shear wall is unreliable in case of flexural failure
under the first load combination.
However, no flexural failure occurs under the second load combination (𝑃𝑓 = 0). Diagonal shear
failure has not occurred under either load combination (𝑃𝑓 = 0) as well which means the
considered shear wall is probably not going to experience diagonal shear under these two load
combinations.
