CONCRETE BLOCK MASONRY CONSTRUCTION A DISSERTATION …
155
CONCRETE BLOCK MASONRY CONSTRUCTION TO RESIST SEVERE WINDS by YAHYA MOHAMMED AL-MENYAWI, B.Sc, M.Sc. A DISSERTATION IN CIVIL ENGINEERING Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Approved August, 2001
Transcript of CONCRETE BLOCK MASONRY CONSTRUCTION A DISSERTATION …
CONCRETE BLOCK MASONRY CONSTRUCTION
TO RESIST SEVERE WINDS
by
YAHYA MOHAMMED AL-MENYAWI, B.Sc, M.Sc.
A DISSERTATION
IN
CIVIL ENGINEERING
Submitted to the Graduate Faculty
of Texas Tech University in Partial Fulfillment of the Requirements for
the Degree of
DOCTOR OF PHILOSOPHY
Approved
August, 2001
ACKNOWLEDGEMENTS
I would like to express my deepest appreciation and gratitude to my advisory
committee chairman. Dr. Kishor C. Mehta, for his advice, objective supervision,
inspiration, encouragement, support, and fatherly guidance. I would also like to thank
him for the effort and time he spent orienting my steps and reviewing my work and for
his constructive suggestions throughout the preparation of this dissertation and
throughout my entire doctoral program.
I would also like to express my sincerest gratitude and thanks to Dr. Ernst W.
Kiesling for his professional and fatherly guidance throughout my stay at Texas Tech
University. I would like to thank him deeply for his encouragement and continuous
support and mentoring. I would like to express my thankfulness for his sincere inspiring
morals which will leave permanent prints in my life.
I would like to express my thanks and gratitude to Dr. James R. McDonald for his
support and guidance since I joined Texas Tech University and throughout my stay and
study. I would like also to thank Dr. C. V. G. Vallabhan for his mentoring and for the
knowledge he provided me with throughout the course of my study. I would also like to
express my sincere gratitude to Dr. James G. Surles, from the department of Mathematics
and Statistics at TTU for his help in the statistics involved in this dissertation as well as
throughout my course work.
I owe thanks to the Civil Engineering Department and the Wind Engineering
Research Center at Texas Tech University for helping to form me as a student and for
supporting and assisting me to build my life career. I would also like to thank April
MacDowel and Lynnetta Hibdon from the Wind Engineering Research Center for their
sincere help.
I wish to devote my entire life and this work to my mother and to the soul of my
father; to whom I owe my life and everything else. I would like to extend my
wholehearted appreciation, thanks and recognition to my family and friends for their care,
support, patience and love.
TABLE OF CONTENTS
ACKNOWLEDGEMENTS
ABSTRACT
LIST OF TABLES
LIST OF FIGURES
CHAPTER
11
vii
viii
ix
1. INTRODUCTION I
1.1 General 1
1.2 Current Practice of Masonry Design 2
1.2.1 Working Stress Design 2
1.2.2 Strength Design 3
1.2.3 Empirical Design Method 3
1.3 Problem Statement 3
1.4 Objectives 4
1.5 Scope 4
1.6 Dissertation Organization 5
2. FLEXURAL TENSILE STRENGTH OF MASONRY WALLS 7
2.1 Loading and Failure Mechanisms of Masonry Structures 7
2.2 Modulus of Rupture of Concrete Block Masonry Walls 10
2.2.1 Previous Research Values 13
2.2.2 Statistical Approach 15
2.2.2.1 Method I 16
2.2.2.2 Method II 16
2.2.2.3 Calculation of Variance 17
2.2.2.4 Confidence Limits 17
2.2.3 NCMA, CTL and UT-Austin Test Results 20
111
2.2.4 Other Code-specified Values 21
2.3 Flexural Tensile Strength Parallel to Bed Joints
(Normal to Head Joints) 21
2.4 Probabilistic Estimation of Flexural Strength 24
2.5 Monte Carlo Simulation Technique 25
2.6 Chapter Summary 29
3. PROBABILISTIC WIND LOADS ON LOW-RISE MASONRY
BUILDINGS 30
3.1 General 30
3.2 Estimation of Wind Loads 30
3.3 Variability in Factors for Wind Loads 34
3.4 Development of Wind Loads 39
3.4.1 Lateral Wind Pressure on Walls 41
3.4.2 Axial loads in Walls 41
3.4.3 Calculation of Stresses in Walls 46
3.5 Obtaining Probability Distribution of Wall Stresses - Methodology 47
3.6 Research Results 47
3.7 Quality of Results Obtained by The Monte Carlo Simulation 55
3.8 Chapter Summary 55
4. RESERVE STRENGTH AND PROBABILITY OF FAILURE 57
4.1 Introduction 57
4.2 Mathematical Approach 57
4.3 Parameters Affecting Reserve Strength 58
4.4 Extreme Value Distribution for Probability of Failure 58
4.5 Reserve Strength and Probability of Failure 60
4.6 Adequacy of the Monte Carlo Simulation Technique 66
4.7 Chapter Summary 67
IV
5. TWO-WAY INTERMITTENTLY REINFORCED WALL PANELS 68
5.1 Introduction 68
5.2 The Intermittent Reinforcement 68
5.3 Building Codes Addressing IRMW 70
5.4 Performance of IRMW During Famous Severe Wind Events 71
5.5 Analysis of Intermittently Reinforced Masonry Walls 71
5.5.1 Thin Plate Analysis 72
5.5.2 Strip Method 72
5.5.3 Yield-Line Analysis 73
5.6 Yield Line Analysis for Ultimate Lateral Strength Estimation 74
5.6.1 Adequacy of Yield-Line Method 75
5.6.2 Basic Fundamentals of Yield-Line Analysis 76
5.6.3 The Yield-Line Analysis 77
5.6.3.1 Analysis of Isotropic Plates Using Yield-line theory 77
5.6.3.2 Energy Method for Yield-Line Analysis 78
5.6.3.3 Analysis of Orthotropic Plates Using Yield-line Method 80
5.6.3.4 Effect of Elastic Support 85
5.6.3.5 Edge Support Conditions 85
5.6.4 Development of Yield-line Equations 88
5.6.4.1 Geometric Parameters of Fracture Pattern 89
5.6.4.2 Aspect Ratio 89
5.6.4.3 Edge Conditions of Wall Panels 89
5.6.4.3.1 Wall simply supported on four sides 90
5.6.4.3.2 Wall fixed at bottom and simply supported elsewhere 93
5.6.4.3.3 Wall fixed at sides, simply supported on top and bottom 96
5.6.4.3.4 Wall fixed at sides and bottom and simply supported on top 98
5.6.4.4 Moment Coefficients 100
5.7 Chapter Summary 101
6. TARGET PROBABILITY OF FAILURE 103
6.1 Introduction 103
6.2 Target Probability of Failure 104
6.3 Factor of MOR Increase (f) 107
6.4 Relationship Between Factor (f) and Intermittent Reinforcement Spacing 112
6.5 Estimation of Pilaster Spacing 114
6.6 Chapter Summary 117
7. SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS 119
7.1 Summeiry 119
7.2 Conclusions 120
7.3 Recommendations for Future Research 121
REFERENCES 124
APPENDIX
A. YIELD-LINE ANALYSIS 13 5
B. DESIGN OF PILASTERS 140
VI
ABSTRACT
Unreinforced masonry is a common construction system for low-rise commercial
buildings. It is often used as load-bearing wall system in large low-rise buildings such as
malls, warehouse and industrial buildings. Failure of such construction type is prevalent
during severe windstorm events such as tornadoes, hurricanes and severe thunderstorms.
Resistance of masonry construction to wind depends on the out-of-plane strength
of the exterior walls. Out-of-plane strength of unreinforced masonry walls depends
mainly on the wall modulus of rupture (MOR). The statistical analysis of previously
published investigations gives a mean value of the MOR of 55.5 psi with a COV of 26%.
Wind-induced load is probability-based and involves variables of wind speed, terrain
exposure, building enclosure type, and pressure direction. The Monte Carlo Simulation
using 1,000 observations of the MOR and the wind-induced loads is used to determine
the probability of failure of walls. For a wall of 10-in. thickness and 15-ft height of a
partially enclosed building located in terrain exposure C in Lubbock, Texas, the
probability of failure is 94% in 50-year life of the building.
A target probability of failure of 1.5% in a 50-year life is ascertained from the
current practice of masonry wall design. Failure strength of intermittently reinforced
walls is determined using yield-line theory analysis. A mathematical methodology to
relate the target probability of failure to the intermittent reinforcement spacing is
introduced. It is found that a wall with intermittent reinforcement placed at a spacing
equal to the wall height would survive in areas with design wind speeds up to 120 mph.
vu
LIST OF TABLES
2.1 Different Test Results for Obtaining Flexural Bond Tensile Strength 14
2.2 Comparison Between Different Methods of Finding Variance 17
2.3 MOR for Different Confidence Limits 18
2.4 Secfion Properties for Unreinforced, Ungrouted Concrete Block Masonry 24
3.1 Statistical Distributions and Parameters of Random Variables Involved
in the Wind Pressure Calculation 33
3.2 CompressionResuItingfrom Wall Self Weight 42
3.3 Compression Resulting from Roof Dead Load 42
4.1 Probability of Failure Percent for Walls (Partially Enclosed Building) 61
4.2 Probability of Failure Percent for Walls (Enclosed Building) 61
4.3 Lifetime Probability of Failure (Outward acting pressure) 64 4.4 Probability of Failure from 1,000 Observations versus 10,000
Observations Partially Enclosed Building - Exposure B 67
4.5 Probability of Failure from 1,000 Observations versus 10,000 Observations Partially Enclosed Building - Exposure C 67
5.1 Analytical to Experimental results Comparison 100
6.1 Factor (f) for Different Loading Conditions (90 mph) 109
6.2 Factor (f) for Different Loading Conditions (120 mph) 110
6.3 Limiting Aspect Ratio Corresponding to Certain Values of the Factor (f) 116
A. I Moment Coefficient (k) for Different Cases of Edge Condition 139
Vlll
LIST OF FIGURES
1.1 West Wall of Gymnasium Building (CND, 1981) 2
2.1 Wind Loading on Windward and Leeward Walls 9
2.2 Simple Beam and Cantilever Failure Modes of Masonry Walls 10
2.3 Joints and Associated Flexural Strength in Masonry Construcfion II
2.4 Flexural Stresses Across Bed Joints 12
2.5 MOR For Defined Confidence Limits 19
2.6 Relationship Between MOR and Flexural Tensile Strength Parallel to Bed Joint 23
2.7 Probability Density Distribution of MOR 26
2.8 Cumulafive Probability Distribution of MOR 26
2.9 ProbabilityDensityDistribufion of Flexural Strength 27
2.10 Cumulative Distribution of Flexural Strength 27
2.11 Effect of Number of Observations on Quality of Results -
MOR Probability Density Distribution 28
3.1 Pressure Coefficient For Roof Uplift 3 5
3.2 Gust-Pressure Coefficient (GCp) - Zones 4 and 5 Positive Pressure 38
3.3 Gust-Pressure Coefficient (GCp) - Zones 4 and 5 Negative Pressure 38
3.4 Probability Density Distribution of Wind Speeds - Lubbock, TX 40
3.5 Effect of Number of Observations on Quality of Results -
Wind Speed Probability Density Distribution 40
3.6 Wall Loading Mechanism 44
3.7 Probability Density Distribution of Uplift Loads Enclosed Building, Exposures B and C, 30ft Roof span 45
3.8 Probability Density Distribution of Uplift Loads Partially Enclosed Building, Exposures B and C, 30ft Roof span 45
3.9 Probability Distribution of Total Axial Stresses Uplift and Dead Load, Enclosed Building, Exp B 50
3.10 Probability Distribution of Total Axial Stresses Uplift and Dead Load, Enclosed Building, Exp C 50
IX
3.11 Probability Distribution of Total Axial Stresses Uplift and Dead Load, Partially Enclosed Building, Exp B 51
3.12 Probability Distribution of Total Axial Stresses Uplift and Dead Load, Partially Enclosed Building, Exp C 51
3.13 Probability Distribution of Tensile Stresses Enclosed Building 52
3.14 Probability Distribufion of Wall Stresses Partially Enclosed Building 52
3.15 Cumulative Distribution of Tensile Stresses Enclosed Building 53
3.16 Cumulative Distribution of Tensile Stresses Partially Enclosed Building 53
3.17 Probability Distribution of Wall Stresses Partially Enclosed Building, Exposure C 54
3.18 Effect of Number of Observations on Quality of Results -Partially Enclosed Building, Exp B 56
3.19 Effect of Number of Observations on Quality of Results -Partially Enclosed Building, Exp C 56
4.1 Probability Distribution of Reserve Strength OW walls, Partially Enclosed Building 62
4.2 Cumulative Distribution of Reserve Strength OW Walls, Partially Enclosed Building 62
4.3 Cumulative Distribution of Reserve Strength OW Walls, Enclosed Building 63
4.4 Cumulative Distribution of Reserve Strength
OW Walls, Partially Enclosed Building 63
4.5 Probability of Failure Within N Years 65
5.1 Standard Pilaster Blocks 69
5.2 Pilasters Formed Using Ordinary Blocks 69
5.3 Orthotropic Masonry Wall Panel 79
5.4 Equilibrium of a Wall Segment 79
5.5 Transformation of an Orthotropic Wall to Isotropic Wall 84
5.6 Explicit and Hidden Pilasters in Masonry Construction 87
5.7 Notations of Yield-Line Analysis 88
5.8 Failure Patterns and Geometric Parameters for Wall Symmetrically Supported on Four Edges 91
5.9 Failure Patterns and Geometric Parameters for Wall Fixed at the Bottom and Simple Elsewhere 94
5.10 Yield-Line Moment Coefficients Wall Simple at Top with Simple Sides 104
5.11 Yield-Line Moment Coefficients
Wall Simple at Top with Fixed Sides 104
6.1 Relationship Between MRI and Wind Speed Multiplier 106
6.2 Probability Distribution of Wind Induced Tensile Stresses Versus MOR 108
6.3 Probability Distribufion of MOR with the Application of the Factor (f) 108
6.4 Factor of MOR Increase (f) 111
6.5 Relationship Between Aspect Ratio (cp) and Factor (f) 115
6.6 Aspect Ratio Corresponding to Various Values of Factor (f) 118
A.l Different Cases of Edge Conditions 138
B.l Load Tributary Area of Pilasters 142
XI
CHAPTER I
INTRODUCTION
1.1 General
Unreinforced masonry is a common construction system for low-rise commercial
buildings. It is often used as load-bearing wall system in low-rise large structures such as
shopping malls, warehouse and industrial buildings. Masonry construcfion is very cost
effective in single-story, load-bearing buildings.
Collapse of masonry construction is not a common phenomenon under normal
loading conditions. However, failure is prevalent during severe windstorm events such as
tornadoes, hurricanes and severe thunderstorms.
Failure of unreinforced masonry construction is catastrophic in nature. The
implications and consequences of that nature of failure are potential fatalities, serious
injuries, and economic losses. The National Academy of Engineering report on the
damage in the Kalamazoo tornado of May 13, 1980, concluded that "Commercial
buildings with unreinforced masonry walls sustained significant structural damage.
Failures of load-bearing walls led to catastrophic collapse of the roof" (CND, 1981, pp.
49). One of the collapsed buildings in Kalamazoo was the gymnasium in St. Augustine
school. Figure 1.1 shows the collapsed wall of the building.
Resistance of masonry construction to wind depends on the out-of-plane strength
of the exterior walls. Out-of-plane strength of unreinforced masonry walls depends
mainly on the wall modulus of rupture (MOR). The MOR has large variability and cannot
be identified as an explicit value. MOR depends on the mortar ingredients, block
strength, texture, environmental conditions, history of loading, curing method and quality
of construction.
At the same time, wind pressures are also highly variable. Wind pressures depend
on wind speed, terrain exposure, gust effect factor and pressure coefficients, which are all
variable. Due to the variability of wind-induced loads and the MOR of unreinforced
masonry walls, masonry wall failure needs to be assessed on a probabilistic basis.
CHAPTER 1
INTRODUCTION
1.1 General
Unreinforced masonry is a common construction system for low-rise commercial
buildings. It is often used as load-bearing wall system in low-rise large structures such as
shopping malls, warehouse and industrial buildings. Masonry construction is very cost
effective in single-story, load-bearing buildings.
Collapse of masonry construction is not a common phenomenon under normal
loading conditions. However, failure is prevalent during severe windstorm events such as
tornadoes, hurricanes and severe thunderstorms.
Failure of unreinforced masonry construction is catastrophic in nature. The
implications and consequences of that nature of failure are potential fatalities, serious
injuries, and economic losses. The National Academy of Engineering report on the
damage in the Kalamazoo tornado of May 13, 1980, concluded that "Commercial
buildings with unreinforced masonry walls sustained significant structural damage.
Failures of load-bearing walls led to catastrophic collapse of the roof." (CND, 1981, pp.
49). One of the collapsed buildings in Kalamazoo was the gymnasium in St. Augustine
school. Figure 1.1 shows the collapsed wall of the building.
Resistance of masonry construction to wind depends on the out-of-plane strength
of the exterior walls. Out-of-plane strength of unreinforced masonry walls depends
mainly on the wall modulus of rupture (MOR). The MOR has large variability and cannot
be identified as an explicit value. MOR depends on the mortar ingredients, block
strength, texture, environmental conditions, history of loading, curing method and quality
of construction.
At the same time, wind pressures are also highly variable. Wind pressures depend
on wind speed, terrain exposure, gust effect factor and pressure coefficients, which are all
variable. Due to the variability of wind-induced loads and the MOR of unreinforced
masonry walls, masonry wall failure needs to be assessed on a probabilistic basis.
Fig. 1.1. West Wall of Gymnasium Building (CND, 1981)
1.2 Current Practice Of Masonry Design
One of the design documents for masonry construction in the United States is the
International Building Code (ICC, 2000). Basically, three design methods are introduced
in this code. Working Stress Design, Strength Design, and Empirical Design. In the
following, a brief overview on each method is given.
1.2.1 Working Stress Design
The international building code refers the design engineer to the ACI 530
document where working stress design is fully addressed (ACI, 1999). Some light is shed
in the IBC on special requirements for zones with high seismic activity. Basically the
method depends on specified allowable stress and the expected loads. The ACI 530
covers various cases of loading for unreinforced and reinforced masonry. For resistance
of lateral out-of-plane loads, values for the allowable flexural tensile stress are given for
different wall conditions. The engineer checks wind-induced stresses against allowable
stresses in this method.
1.2.2 Strength Design
The method depends on providing sufficient strength to resist factored loads.
Loads used in calculations are factored loads, where the load factors are given in the
ASCE 7-98. Nominal strength is reduced by a strength reduction factor (([)). Strength
reduction factor depends on the type of loading (axial, shear or bending). Equations
needed to evaluate different quantities are provided in the IBC (ICC, 2000). Some
restrictions on the material properties such as block strength and mortar types, as well as
quality control level, are specified. In this method, the engineer is responsible for
assuring that factored wind-induced stresses do not exceed specified tensile strength of
the wall.
1.2.3 Empirical Design Method
Empirical design method is convenient and easy to use. The height of the wall is
governed by wall thickness (see section 2109.4.1 in the IBC). The method is restricted for
use in non-seismic zones and when basic wind speed does not exceed 110 mph. It is also
not permitted for use when wall height exceeds 35 ft. The method does not require
calculating loads and resulting stresses and does not specify permissible tensile stresses.
The method accounts for slenderness of walls by providing limiting values for
spacing of lateral supports or wall height to thickness ratio. For ungrouted hollow block
walls, the wall height to wall thickness ratio should not exceed 18. For walls conforming
to the limitations specified in this method, allowable axial stresses based on the wall
gross cross-sectional area are provided. In this method, the engineer is not required to
calculate wind loads or resulting stresses.
1.3 Problem Statement
Masonry construction may fail due to insufficient flexural resistance to lateral
out-of-plane bending loads, or may fail due to excessive lateral loads. Both loads and
MOR are highly variable. Therefore, it is required to know at what loading level and
MOR an unreinforced masonry wall will fail. It is desirable to obtain the probability of
failure based on appropriate statistical analysis.
Providing walls with some vertical reinforcement would add to the out-of-plane
flexural strength and increase their ductility. It is understood that providing reinforcement
and grout in every cell makes the walls much more wind-resistant and provides them with
ductility; however, this may be expensive. The high cost of reinforced masoruy may
make owners and contractors reluctant to choose it as a construction system. It is
desirable to make the masonry system cost-effective by providing intermittent
reinforcement that is consistent with the target level of probability of failure.
1.4 Objectives
According to the problem statement, this research has three objectives. These
objectives are:
1. Ascertain the probability of failure of concrete masonry walls based on the
probability distribution of the MOR and the probability distribution of the wall
tensile stresses resulting from wind loading.
2. Estimate the ultimate strength of intermittently reinforced masonry walls to resist
out-of-plane lateral loads.
3. To develop a scheme of intermittent reinforcement spacing to match the target
probability of failure.
1.5 Scope
Unreinforced masonry walls can be used as infill walls in skeletal steel or
concrete buildings, can be used as shear walls, or can be used as load-bearing walls
supporting roofs. Exterior walls will experience out-of-plane bending resulting from wind
pressures. Load-bearing walls can be employed in low-rise or high-rise buildings. Low-
rise buildings are more common in the United States as well as in many other countries.
In addition, load-bearing walls subjected to out-of-plane loading are more critical in low-
rise buildings, as the axial compression in these walls is quite small. The scope of this
research focuses on single-story load-bearing walls subjected to out-of-plane loading
resulting from wind pressures.
Most commonly used concrete masonry units are of 8-in., 10-in., or 12-in.
thickness. This research focuses primarily on establishing a methodology for determining
the probability of failure of walls and a procedure to mitigate this failure. Thus, only 10-
in. block walls are studied. An extension of the methodology to 12-in. walls is available
(Al-Menyawi and Mehta, 2001).
Behavior of brick masonry is different from that of block masonry. Since block
masonry is far more commonly used worldwide, this research will be limited to concrete
block masonry.
Blocks are available in three types of concrete: (1) normal weight concrete, (2)
medium weight concrete, and (3) light weight concrete. Medium and lightweight concrete
blocks are not likely to be used for load-bearing walls in zones of severe winds; therefore,
only normal weight concrete blocks are considered in this research.
Out-of-plane strength of walls will be estimated using the Yield-Line theory for
plates. Masonry walls can be modeled as orthotropic plates, with different properties in
two orthogonal directions. Yield-line method for orthotropic plates is utilized, and
necessary developments are made to make it suitable for intermittently reinforced block
masonry walls.
1.6 Dissertation Organization
This dissertation is presented in seven chapters. The dissertation covers different
yet related subjects. Therefore, a review of previous work is not presented in an
independent chapter, rather is covered when relevant in the various chapters.
Chapters 2, 3 and 4 are concerned with establishing probability distribution of
unreinforced masonry flexural strength; probability distribution of wind-induced tensile
stresses in masonry walls, and obtaining probability of failure, respectively. Available
data is used to establish probability distribution for each involved variable. The Monte
Carlo Simulation technique is used to obtain probability of failure.
Chapter 5 proposes intermittently reinforced walls to reduce the probability of
failure. This proposal is an outgrovs^h of observing performance of intermittently
reinforced walls in severe wind events. Yield-line theory is used to estimate the ultimate
lateral load carrying capacity of intermittently reinforced walls. Different edge conditions
are examined and graphs are made available for the final solution. As an extension,
Appendix A shows the assessment of ultimate lateral load carrying capacity and gives
some useful results in a tabular form. Chapter 6 introduces a mathematical method to
relate the target probability of failure to the spacing of intermittent reinforcement. Based
on that methodology, intermittent reinforcement spacing schemes are suggested to
ascertain target probability of failure. Appendix B shows a general outline for the design
of pilaster.
Chapter 7 presents a summary of the research and conclusions that can be drawn
from it. Furthermore, ideas and recommendations for future research in the same topic
are proposed.
CHAPTER 2
FLEXURAL TENSILE STRENGTH OF MASONRY WALLS
2.1 Loading and Failure Mechanisms of Masonry Structures
In the last three decades, many researchers showed a great deal of interest in the
failure modes of masonry structures following severe wind events like tornadoes,
hurricanes and thunderstorms. In fact, understanding and categorizing failure modes is
the first step towards improving the design of such structures.
Researchers analyzed the loading mechanism of masonry structures during severe
winds as a step to analyze failures in such events. (Sparks et al., 1989) outiined the
loading mechanism as most masonry-walled structures carry wind loads in the following
manner. Wind forces on the windward and leeward walls of the building are transmitted
by the walls, either directly to the ground or to the floors or roof These elements then act
as diaphragms, which transmit the loads to the sidewalls, which in turn carry the loads in
shear to the ground. The sidewalls may also be subjected to out-of-plane bending by the
wind, as might the roof
Different failure modes should be studied to make it possible to provide necessary
design measures for improving the performance of this type of construction. Many
researchers spent tremendous efforts investigating and documenting failures of masonry
construction. Generally, four different modes of failure were found to be associated with
masonry walls subject to out-of-plane loading (Mehta and Minor, 1986). For masonry
walls subject to out-of-plane loading, the failure modes associated with different loading
mechanisms are classified as follows:
• Wall subject to external lateral pressure. In this mode, the roof loads are sufficient
to prevent any potential tension due to wind uplift. In this case, the roof is
supposed to withstand the wind and to provide reasonable diaphragm action.
Hence, walls act as simple beams. The external lateral pressure induces bending
moments in the wall section, which cause tensile stresses. When the tensile
stresses exceed the fiexural tensile strength of the wall section (modulus of
rupture or MOR), sudden collapse will take place.
7
• Lateral pressure and uplift. In this mode, the roof gravity loads are not sufficient
to cancel out the wind uplift. Therefore, walls are subject to lateral pressure as in
case [a] in addition to direct tensile forces due to roof uplift. Walls experience
tensile stresses resulting from combined direct tension and flexure. Again, failure
will take place when these stresses exceed the wall flexural tensile strength
(MOR).
• Walls subject to combined external and internal pressures. Due to high wind
pressure or debris impact on different building openings, such as garage doors,
windows and exterior doors, the structure envelope might be breached. In this
case, the walls will experience internal pressure in addition to the external
pressure. Excessive pressure acting laterally on the walls with or without roof
uplift leads to failure similar to that of the first two cases.
• Wall corner pressures. During severe wind events, corners experience very high
pressures due to turbulence and separation in the wind flow. It has been observed
that wall corners of masonry constructions fail and further failure follows due to
building envelope breach.
Recent research based on damage investigations of the Central Florida Tornadoes
of February 22-23, 1998, confirmed the modes of failure suggested earlier and reported
other modes associated basically with masonry shear walls (Pinelli and O'Neill, 2000).
Modes associated with shear walls are primarily shear failure due to in-plane loading.
According to the loading mechanism outlined above, the roof diaphragm transfers the
lateral load to the side (shear) walls. Lack of horizontal and vertical reinforcement makes
these walls susceptible to excessive in-plane load. The above type of failure associated
with in-plane loading of masonry shear walls is not included within the scope of this
dissertafion and will not be discussed.
Figure 2.1 shows wind loading mechanism on both the windward wall and the
leeward wall. It is noticeable that maximum flexural stresses in the windward wall are
associated with negative internal pressure. This will reduce the total uplift forces to some
extent, but will increase the bending moment in the wall significantly. On the other hand.
maximum flexural stresses in the leeward wall are associated with positive internal
pressure.
External Wind Load
c>
Uplift
Negative internal Pressure
Negative Internal Pressure
' = >
Windward Wall
Positive Internal Pressure
Positive Internal Pressure .^External Wind Load
Leeward Wall
/
External & Internal pressure
Fig. 2.1. Wind Loading on Windward and Leeward Walls
Figure 2.2 (a) demonstrates the simple beam failure mode, whereas Figure 2.2 (b)
demonstrates the cantilever failure mode of walls. Simple beam failure mode will prevail
as long as the roof is intact and can provide lateral support to the wall. Once the roof
support is lost, the wall will behave as a cantilever, which has significantly higher
flexural stresses than the simple beam.
Wind Load I [ ^
\ \ \ \ \ \ I t I \ I 1 1 I
I ( / / I / / / / / / / / / /
/
Wind Load I | > >
/ / / / / / / / / / / / / / / / / / / / / / / /
a. Masonry Wall in Simple Beam Action b. Masonry Wall in Cantilever Action
Fig. 2.2. Simple Beam and Cantilever Failure Modes of Masonry Walls
2.2 Modulus of Rupture of Concrete Block Masonry Walls
When brittle materials are loaded in flexure until failure, the tensile stress at
which failure occurs is called the modulus of rupture (MOR) of the material. As
explained earlier, unreinforced masonry is commonly used as load-bearing walls with
lightweight roof structures. During severe windstorms, external walls are exposed to high
pressures thus producing flexural stresses. These flexural stresses could be high enough
to fail the walls. In such cases, resistance of the load-bearing walls depends mainly on the
flexural tensile strength. It is well known that the bed joint is the weakest link in a
masonry wall assembly. Bond tensile strength of bed joints (i.e., MOR) becomes a major
factor in determining wall flexural strength.
Common practice defines the MOR as the bond flexural tensile strength of bed
joints. Other important terminology is illustrated in Figure 2.3.
10
BED .lOINT
M A S O N R Y B L O C K H E A D JOINT
Fracture along bed joints controlled by flexural tensile strength normal to bed joint (MOR)
Fracture along head joints controlled by flexural tensile strength parallel to bed joint (normal to head joints)
Fig. 2.3. Joints and Associated Flexural Strength in Masonry Construction
11
Typically, materials are assumed to have similar properties in tension and
compression. This is not true for block masonry (Anderson, 1980). A test performed on
thirteen specimens, with strain transducers fitted on the tension and compression sides.
Results showed that block masonry has different properties, and that, when it fails under
flexure, tensile stress at which it fails (MOR) is less than the compressive stress. Figure
2.4 represents graphically that phenomenon.
TENSION
COMPRESSION
iHIIillEIiEnmj^" COMPRESSION
SIMPLE THEORY OF BENDING
MATERIAL WITH DIFFERENT PROPERTIES IN TENSION & COMPRESSION
Fig. 2.4. Flexural Stresses Across Bed Joints
When MOR is determined, simple flexure formula is used. This is also done when
wind-induced flexural tensile stresses are obtained. This is a reasonable simplification for
brittle materials. In reality, all available research results are presented considering simple
formula of flexural stresses.
Many researchers have been concerned with establishing reliable values for MOR
of block masonry. Different sample shapes and testing methods have been used, as well
as different mortar ingredients. Therefore, values obtained by researchers varied between
43 and 225 psi. In this section, available test results are assembled, and the mean value
(p.) for MOR and associated variance (a^) are presented.
In review of results of different research programs, a large variation is found in
the values of MOR. It is the goal of research projects to determine values as close as
12
possible to what can be obtained in the field. Large variations in the test results confuse
researchers if they cannot account for discrepancies.
It is found in this study that different testing programs used different test methods,
sample curing procedures, and cement types. These factors led to much variation in the
results obtained. The goal of this section is to review previously published research
results and establish the mean value (p) for the MOR and an associated variance (a^). In
other words, the goal is to establish a probability distribution for the MOR.
This research does not discuss effect of material differences on MOR. ft is known
that mortar type (N, M, or S) affects MOR values. The results reported here do not
include tests using type N or type M mortars.
2.2.1 Previous Research Values
Table 2.1 summarizes test results obtained from previous research showing the
publication year, number of relevant samples, shape and size of samples, test method,
mean value for MOR, and coefficient of variation (COV). Tests were conducted between
1979 and 1992.
By investigating the different resuUs reported in Table 2.1, it is noticed that the
number of samples from different programs is generally small. In addition, sample
shapes and sizes are different, and testing methods were different as well. Even with
these variations, it can be seen that different testing programs led to values for the MOR
in a narrow range with a minimum of 43 psi and maximum of 65 psi.
It is known that tests cannot be duplicated due to the high heterogeneity of
materials and various environmental conditions such as workmanship and curing.
Considering the potential heterogeneity in testing, it can be said that, the effect of
different testing methods and different sample shapes or sizes is relatively small.
The COV values in each project ranged from a low of 8% to a high of 38%. This
large variation is the result of a small number of samples in each research project, i.e.,
sample size varied from 3 to 10. When the results shown in Table 2.1 are lumped
together, the total sample size is 33 samples. The mean value for the total sample size is
56 psi with COV of 24%.
13
Size Size (in)
No. of Relevant Samples
Testing Method
1979 Drysdale et al. Disks, (|)803
1988 Hamid & Drysdale
790 x 1590
1988 Drysdale & Essawy
10
1990 Matthys
390 X 990
1200X 2400
1990 Gazzolaetal.
1992 Hamid etal. 3x2
Bond Tensile Strength
(psi)
COV %
Disks were 48^ tested in splitting (Brazilian) tension at different angles.
Horizontal 62 simple beam with two concentrated loads applied at 400 mm from the supports, total span = 1400 mm
Single block 54 prisms were tested using bond wrench apparatus
Walls 43 constructed with half running bond, S mortar, and PCL.
Wallettes 54 loaded as simple beams with two concentrated loads.
1:3 scale units 65 are used to build walls. Walls are cut, then tested using bond wrench apparatus
19
38
24
27
31
Cumulative 33 56 24
14
2.2.2 Statisfical Approach
It is assumed that all testing programs represent the large population of masonry
walls. This assumption is supported by the fact that building codes define a single value
for the MOR of block masonry walls based on the mortar type. Also, since the testing
programs considered follow different technical rules and specifications, they can all be
considered representative samples. There is no tendency to accept one rather than
another. Therefore, all testing programs can be combined in one result of more
significance than the individual others. In this research, some assumptions are made and
some rules are established based on stafistical approaches. Basically, the following
assumptions and rules are observed:
• Different samples for different testing programs are drawn from the same
population.
• Since all samples represent the same population, the general mean value should
be influenced by number of observations in each sample.
• MOR obtained from different samples is normally distributed.
• Central Limit Theorem is valid, i.e., the mean value is normally distributed.
• The ratio between the variance of the population to the variance as obtained from
the different samples follows chi-square {y^) distribution.
The former assumptions are acceptable as long as the same or similar materials
are utilized. The available data as collected from previous research is mean values for the
MOR obtained from different testing programs and associated COV. Based on the
aforementioned assumptions, the unbiased expected general mean value ( i) could be
calculated from the following formula:
_^EML_^LJhl (2.1)
where ni = Number of data points in the i"' testing program and \i\ = the mean value
obtained from the i"" testing program.
15
There are actually two ways to obtain unbiased estimates for the general variance.
The estimated variance will have variance itself The factor governing which method
should be followed is the variance of the variance. To be able to decide, variance of the
variance will be obtained for the two methods, and the one that has smaller value will be
selected. The two methods are discussed below.
2.2.2.1 Method I
^ ^ = l | (2.2-A)
where a' is an estimate of the population variance, si' is the estimated variance for the
mean value of the MOR for i" testing program and C is the total number of testing
programs considered. Since only COV is available, the following equation is used to
obtain the quantity (Sj).
CO^- • A (2.3) 100
2.2.2.2 Method II
In this method, the sample variances are summed and all the samples are
considered as one large sample containing all the observations. To follow this approach,
the following equation is used:
a' = Z("--l)^/ (2.2-B)
where the terms are same as defined before.
16
2.2.2.3 Calculation of Variance
Since the variance obtained by either of the above described methods is just an
estimate, it would have its own distribution. Therefore, the estimated variance will have
a variance itself The variance of the variance is obtained by the following definition:
Var s'
k - 1 ) X ^ a'
Z' (2.4)
where % is the chi-square distribution. From which, the following expression can be
found:
.4
Var[sf] = 2 x - ^ (2.5) (",-1)
The better estimate is the one that would have the smaller variance. Table 2.2 shows the
outcome of the statistical analysis using the two methods.
Table 2.2: Comparison Between Different Methods of Finding Variance Method
Method I
Method II
Mean (p) 55.5
55.5
Variance (a^) 206.7
203.0
COV %
25.9
25.7
Var [Sil
0.0914 a^
0.0741 a'*
From Table 2.2, it is evident that both methods yield almost identical results, though
Method II gives a slightly better estimate.
2.2.2.4 Confidence Limits
Design is the process of establishing a relationship between strength and loading.
To establish this relationship, the statistical distribution of both loading and strength
should be defined. In this section, only the statistical distribution of strength in terms of
MOR is given. Since the required confidence limits are not well established at this point,
a range from 80% to 99% are defined. To establish confidence limits, student T-
distribution is used since the number of available observations is small. Table 2.3 17
presents mean values for the MOR based on different confidence limits. The following
formula is used to define the confidence limits:
MOR^, = // - [t^xcr) (2.6)
where CL is the target confidence limit, a = type I error (if 95% confidence is needed, a
= 0.05), t is the value obtained from the standard T-distribution corresponding to certain
probability (could be found in any standard statistics text book such as Milton and
Arnold, 1994) and a is an estimate of standard deviation (a = V203 , Table 2.2).
Table 2.3: MOR for Different Confidence Limits Confidence Level
80% 85% 90% 95% 99%
ta 1.32 1.49 1.72 2.07 2.82
MOR (psi) 36.8 34.4 31.2 26.2 16.0
The same values are shown in Figure 2.5 in graphical form.
18
Confidence Limit %
Fig. 2.5. MOR For Defined Confidence Limits
19
2.2.3 NCMA, CTL and UT-Austin Test Results
The National Concrete Masonry Association (NCMA), Concrete Technology
Laboratories (CTL), and the University of Texas at Austin (UT-Austin) conducted an
extensive research program on the flexural bond strength (MOR) of concrete masonry for
type M, S and N Portland cement/lime mortars at the request of the International
Conference of Building Officials (Headstrom et al., 1991; Melander et al., 1993).
Tests were performed for various mortar types, and for each type of mortar,
various cement types and curing methods were used. Thus, for each mortar type, data for
a number of test groups were available. For mortar type S, the mean values of the MOR
varied between 124 and 225 psi with an associated COV between 10.1% and 23.5%, as
obtained by NCMA. The same mean values varied between 140 and 195 psi with an
associated COV between 10% and 21%, as obtained by CTL. Mean values of MOR
obtained from the UT-Austin testing program varied between 163 and 220 psi with an
associated COV between 14.0% and 20.0%. In addition, NCMA conducted another
research project to evaluate flexural bond strength of masonry walls (MOR) and to
compare between different curing methods and various masonry materials. The program
results showed MOR variation between 134 and 169 psi for mortar type S with an
associated COV between 26.0% and 33.0%.
These values of MOR are significantly higher than those reported previously in
Table 2.1. By investigating the associated research, it was found that the testing was done
under very strict levels of quality control of materials and curing, which could not be met
in reality. Headstrom et al. (1991) and Melander et al. (1993) concluded that the flexural
bond strengths obtained in that test program are intended only to provide a means of
evaluating and comparing materials and should not be used as design values or compared
directly with allowable flexural bond stresses prescribed by building codes. Therefore, it
is decided to exclude these values from the present study since they do not represent
reality in the field.
20
2.2.4 Other Code-specified Values
Building codes provide allowable and ultimate values of MOR for different types
of mortars and/or blocks. One of these codes is the International Building Code (ICC,
2000), which specified the MOR to be 50 psi for non-grouted hollow block masonry
walls constructed with mortar type M or S. The British Standards BS 5628: Part 1
specified different values for the MOR depending on the block strength, thickness and
mortar type. For block sizes from 6-in. to 12-in., and for mortar types (i) and (ii), MOR
values varied between 21.5 psi and 35.7 psi. It is understood that the code-defined values
for MOR imply some confidence limits; however, these limits are not explicitly declared.
Also, these limits depend on the design method and other parameters involved in each
code.
2.3 Flexural Tensile Strength Parallel to Bed Joints (Normal to Head Joints)
Normally, in large structures like shopping malls or industrial plants, walls are
supported at the roof and on the foundation. Therefore, the walls act as a laterally loaded
one-way plate. In such cases, the MOR is of the most interest since it determines the
flexural capacity of the wall assembly when loaded in out-of-plane bending. However, in
some cases, where cross walls or pilasters are provided, the wall acts as a laterally loaded
two-way plate. In such cases, the flexural tensile strength parallel to bed joints (see
Figure 2.3) becomes important. Actually, lateral load distribution depends on both
flexural tensile strength normal to bed joints (MOR) and flexural tensile strength parallel
to bed joints (normal to head joints). Therefore, it is important to determine flexural
tensile strength parallel to bed joints (fracture along head joints. Figure 2.3).
Unfortimately, there is a dearth of data on flexural tensile strength parallel to bed joints.
The few sources of available data were collected to determine the ratio between flexural
tensile strength parallel to bed joints and flexural tensile strength normal to bed joints
(MOR, see Figure 2.3). Head joint might be stronger than the blocks and fracture occurs
within the blocks. Actually, the flexural tensile strength parallel to the bed joint is much
higher than that normal to the bed joints. Hamid and Drysdale (1988) reported that tensile
strength parallel to the bed joint is two to four times higher than tensile strength normal to
21
the bed joint (MOR). In this dissertation, some values for tensile strength parallel to bed
joints are reported with associated COV. Values are obtained exttacted from three
different testing programs conducted between 1979 and 1990.
Drysdale et al. (1979) performed tests to obtain tensile strength of concrete
masonry in all directions. Disks were used for that purpose, and they were loaded in
splitting tension mode. For that test, reported value of splitting tensile stress was 116 psi
with an associated COV of 5.2%. For the same test the MOR was 48 psi. Later, another
experimental program yielded a value of 191 psi for the flexural tensile strength parallel
to the bed joints with a COV of 8.5% for a group of 20 samples. The flexural tensile
strength normal to the bed joint (MOR) obtained from the same experimental program
was 54 psi with a COV of 24% (Drysdale and Essawy, 1988).
Gazzola et al. (1990) reported a value of 139 psi for tensile flexural strength
parallel to bed joints with an associated COV of 10.5%). The test program covered
flexural tensile strength in all directions, and the aforementioned value was obtained by
testing 5 samples shaped as wallettes and tested as simple beams.
Figure 2.6 shows the relationship between flexural tensile strength parallel to bed
joints and flexural tensile strength normal to bed joints (MOR) for the available data.
Figure 2.6 suggests some ratio between the tensile strengths in two perpendicular
directions. A ratio of 2.85 is suggested, and used further in subsequent analysis. Flexural
tensile strength parallel to bed joints is assumed to have the same variability of that
normal to bed joints (MOR), which is an approximation due to the lack of experimental
data. It should be noted that horizontal section modulus of walls is larger than the vertical
section modulus by about 13% (NCMA TEK 14-1, 1993). Both section modulus and
tensile strength will define the orthogonal ratio. From the available data, an orthogonal
ratio of 2.5 is considered reasonable. Orthogonal ratio (y) is defined as the ratio of the
flexural strength of masonry when failure plane is normal to bed joints to that when
failure plane is parallel to the bed joints. The orthogonal ratio is used later to assess the
ultimate lateral load carrying capacity of masonry walls with pilasters using the yield-line
theory. British Standards BS 5628: part I recommends an orthogonal ratio around 3.0.
22
240
47 49 51 MOR (psi)
53 55
Fig. 2.6. Relationship Between MOR and Flexural Tensile Strength Parallel to Bed Joint
(Drysdale et al., 1979, Drysdale and Essawy, 1988, and Gazzola et al, 1990)
23
2.4 Probabilistic Estimation of Flexural Strength
While estimating the probability distribution of MOR is an important step, the
probabilistic distribution of moment resistance, (MR), is still required to be estimated.
Since unreinforced masonry is quite brittle and does not have much stiength beyond first
cracking, the elastic relationships can be used to determine the moment resistance given
the MOR and the section properties. For this research, section properties provided by
National Concrete Masonry Association (NCMA TEK 14-1, 1993) are used. Table 2.4
summarizes section properties for 10-in. ungrouted block masonry walls.
Table 2.4: Section Nominal Size
10-in.
Properties for Unreinforced, Ungrouted Concrete Block Masonry An (inVft)
50.4
Ix (in7ft)
635.3
Sx (inVft)
132.0
R(in)
3.55
Table 2.4 shows A, the area of the cross section per linear foot of wall; L, the moment of
inertial about the X-axis; Sx, the section modulus about the X-axis; and R, the radius of
gyration. The moment resistance is estimated from the following equation:
MR = MOR X S^. (2.7)
It is known that section properties have some variability that will contribute to the overall
variability of moment resistance. However, this depends on the manufacturer and overall
quality control, and it is generally difficult to account for. In addition to that, it is
expected that such variability should be quite small and therefore it is neglected in this
research. Thus, MOR is the only considered random variable.
The MOR is assumed normally distributed, and its parameters are estimated. The
mean value (p) is 55.5 psi and the standard deviation (a) is 14.25 psi (COV is 25.7%). It
is possible to find a closed form distribution based on this information. Figure 2.7 shows
the probability density distiibution of the MOR. The cumulative distiibution of the MOR
is obtained by integrating the density function numerically. Figure 2.8 shows the MOR
cumulative distribution. These graphs are for type S mortar, made using Portland cement
24
and lime. Furthennore, the probability density distribution of moment resistance normal
to bed joints is found. Figure 2.9 shows the probability density distribution of the moment
resistance normal to bed joints, and Figure 2.10 shows its cumulative distribution.
Figures 2.9 and 2.10 are plotted for 10-in. walls.
2.5 Monte Carlo Simulation Technique
In this research, different random variables are involved, either from the strength
side or from the loading side. Probability distributions of both strength and loading need
to be merged together to obtain probability of failure. In such cases, closed form solutions
are not available and some numerical techniques must be employed. The Monte Carlo
Simulation is used within this research. This technique generates artificial random
observations for a known distribution with known parameters. The idea is to generate
random numbers between zero and 1.0 and consider them as probability of occurrence,
and hence, to obtain the corresponding random variable value. The accuracy of results
depends on the number of generated observations. Hence, it is desirable to generate a
large number of observations. At certain point, the whole process becomes impossible to
complete. Thus, it is desirable to study the accuracy of the Monte Carlo Simulation
technique to determine the least number of observations that will yield satisfactory
results.
For that purpose, a comparison is made when 100, 500, 1,000, and 10,000 random
observations are used. Figure 2.11 shows the probability density distribution of MOR for
different number of observations as compared to the closed form solution.
In review of Figure 2.11, 100 observations will gives a poor representation
compared to 500 observations or more. Also, the graph reveals that with a minimum of
500 observations, a fairly good representation can be achieved. The difference between
the distributions obtained from 1,000 observations and 10,000 observations is not
significant. Using 1,000 observations or more yields very comparable results to the
closed form solution. For this research, 1,000 observations on the strength side is
sufficient.
25
3.0
2.0
^ ^ XI ta
Si o
1.0
0.0
20 40 60 80 100 120
MOR (psi)
Fig. 2.7. Probability Density Distribution of MOR
100
^ ^ j2
o
U
Fig 2 8 Cumulative Probability Distribution of MOR 26
0.03
0.02
o
0.01
0.00
4000 8000 12000
Flexural Strength (in.lb/ft)
Fig. 2.9. Probability Density Distribution of Flexural Strength
16000
100
80
a Si o
60
« 40 3 B 3
u 20
4000 12000 8000
Strength (in.lb/ft)
Fig. 2.10. Cumulative Distribution of Flexural Strength
27
16000
^ 3
XI
XI
o
—*—lOOobs
—e—500obs
—B—10000 obs
Closed Form
20 40 60 80 100 120
MOR (psi)
Fig. 2.11. Effect of Number of Observations on Quality of Results - MOR Probability Density Distribution
28
2.6 Chapter Summary
This chapter focused on obtaining probability distribution of flexural tensile
strength of unreinforced masonry walls. To do that, modulus of rupture (MOR) data for
concrete block masonry from six previously conducted projects was assembled and
analyzed statistically. Statistical analysis of data shows that a reasonable mean value and
COV of MOR can be obtained even though the tests were conducted using different types
of specimens and in different laboratories. A mean value for the MOR of 55.5 psi and an
associated COV of 25.7% are obtained. In addition, values of MOR for different
confidence levels are established. Results of other tests performed by NCMA, CTL, and
UTA are not used in this research since they were meant to compare effects of different
materials and curing methods on the MOR and they were performed imder strict quality
control that cannot be reproduced in the site.
In further development, probability distribution of flexural strength of walls is
established based on the available outcome. The results obtained in this chapter will be
used along with probability distribution of wind induced tensile stresses in walls to
establish probability of failure of unreinforced masonry walls given the design wind
speed.
MOR probability density distribution can be numerically represented using the
Monte Carlo Simulation technique. One thousand artificially generated random
observations are found to be adequate.
An orthogonal ratio (y, ratio of tensile strength of head joints to bed joints) of
unreinforced masonry wall panels of 2.5 is suggested and will be used in subsequent
chapters. This value is based on very limited experimental observations.
29
CHAPTER 3
PROBABILISTIC WIND LOADS ON LOW RISE
MASONRY BUILDINGS
3.1 General
Wind loads along with other natural loads, are considered of utmost importance.
However, wind loads are not easy to estimate. This is due to the fact that they depend on
vagaries of nature, and thus, they are somewhat associated with random nature in many
aspects such as direction, magnitude, fluctuations, etc. There is high variability in wind
characteristics and wind-structure interaction. At present, the most reliable analytical
procedure to assess wind loads on buildings is outlined in the ASCE 7-98 document
(ASCE, 1999). The procedure outiined in the ASCE 7-98 involves many factors that are
classified into three categories of variables: (1) geometric, (2) wind characteristics, and
(3) structure-related. To establish probability of failure or survival for buildings subjected
to wind loads, the variability of wind loads as well as variability in resistance, should be
taken into account. This chapter is concemed with assessing wind loads on exterior
masonry walls, accoimting for inherent variability resulting from different parameters.
The target of this chapter is to obtain a probability distribution of wind-induced tensile
stresses in masonry walls.
3.2 Estimation of Wind Loads
Modem design codes are based on accepted probability of failure calculated from
probability distribution of strength and probability distribution of loads. Modem design
methods are typically called limit state design, or load and resistance factor design
(LRFD). Compared to classical allowable stress design (ASD), LRFD provides the
following advantages: stmctural behavior is addressed at or near the limit state(s) that are
essential for adequate safety and rational and quantitative treatment for design
uncertainties arising from randomness and modeling errors (EUingwood and Tekie,
1997). Establishing probability of failure needs both the probability distributions of wind-
induced stresses and the strength to be well known. Since this is a difficult process, some
30
assumptions and simplifications are made. In early versions of LRFD design methods,
only variability in wind speed was considered. This was a reasonable assumption since
the wind speed is squared in the wind load equations, and its influence is much more
significant than all other parameters. Advancement in wind load simulation and
modeling made it possible to obtain statistical models for other parameters involved in
the wind load equations. In some cases, this was based on experimental work supported
by analytical models, and in other cases, it was based on what wind experts know and
believe. This is understood through the fact that Delphi method was used to establish the
statistical distribution of most of the wind load parameters (EUingwood and Tekie, 1997).
Previous research was concerned with the wind pressure on main wind-force
resisting system to establish load factors for different cases of loading. In this research,
the author is interested in studying the effect of wind loads acting directly on exterior
masonry walls. Thus, it is required to establish a probabilistic model for the wind load on
a low-rise building, considering all effects, such as wind pressure on walls, wind uplift on
roofs, and dead loads resulting from roof as well as wall self weight. This is done using
the probability distribution of involved variables.
The reference document of use in this research is the ASCE 7-98, according to
which, the wind pressure (W) is calculated from the following equation:
W = q GC^ - q, (GC^^). (3.1)
In which, q, or qn, is the velocity pressure, calculated from the following equation:
q = 0.00256 K^ K._, K, V' I. (3.2)
A thorough look will reveal that variability in wind pressure depends on the following
parameters:
• Wind speed (V),
• Importance factor (I),
• Gust effect factor (G),
31
• Pressure coefficient (Cp),
• Terrain and exposure coefficient (K^),
• Topographic factor (KzO,
• Wind Directionality factor (Kd),
• Combined Gust-Pressure coefficient (GCpi) for internal pressure,
• Combined Gust-Pressure coefficient (GCp) for components and cladding
(C&C).
As mentioned earlier, among these parameters, the most important factor is the
wind speed, since it is squared in the wind pressure equation and it has relatively large
variability. However, all of the parameters are considered random variables and will
contribute to the wind pressure probability distribution.
A vital step in the analysis is to know the distribution of different parameters
contributing to the wind pressure. This has been the scope of many research programs
over the last 30 years. It is not the scope of this research to find the distribution that
represents any of these parameters the best, but rather to use the best available
information.
Importance factor (I) converts the wind speed from 50-year MRI to any other
MRI. In this research, the wind speed is represented on a probabilistic basis (maximum
armual wind speed). The importance factor, therefore, is taken as 1.0.
For this research, the wind directionality factor (Kd) and the topographic effect
factor (Kzt) are considered constants and do not contribute to the variability of wind
pressure. Wind directionality factor (Kd) is taken as 0.85 since it is unlikely that wind
would hit the building in the worst direction with the highest wind speed. Topographic
effect factor (Kzt) is taken as 1.0 (assuming flat terrain condition).
For this research, the probability distribution of wind pressures needs to be
established. To do that, the observations at specific station are considered. The wind
speed observations should be fit to a proper statistical model. The ASCE 7-98 currently
uses Extreme Value Type I distribution for the development of wind speed map. In this
research. Extreme Value Type I distribution is used.
32
It is known that the exposure coefficient (Kz), pressure coefficient (Cp), and gust
effect factor (G) are likely to be correlated; however, the correlation model is not known.
Thus, it is almost impossible to consider the variability of the parameters together in one
statistical model. Researchers and wind engineering experts have assumed that the
parameters are independent and normally distributed. The distributions of these
parameters (Kz, Cp, and G) along with the distribution parameters (means and standard
deviations) are taken from the available research documentation. Some sources reported
experimental values that can be used for Delphi process (EUingwood and Tekie, 1997).
The following table presents the values used in this research (Table 3.1).
Table 3.1: Statistical Distributions and Parameters of Random Variables Involved in the Wind Pressure Calculation (EUingwood and Tekie, 1997)
Variable Structural
Element
Exp. C z<20ft
Exp. B z<20ft
Exp. B Exp. C
Nominal
Value
0.90
0.62
0.80 0.85
Mean Standard
Deviation
0.12
0.12
0.09
Statistical
Distribution
Normal
Normal
Normal
Kz
G
GC„
WW
WR
LW
LR
0.84
0.63
0.80 0.85 0.85
0.80 0.80
-1.04
-0.50
-0.70
0.18 0.18 -0.18
0.77 0.82 0.83
0.69 0.71
-0.92
-0.46
-0.61
0.15 0.13 -0.16
0.08 0.10
O.IO 0.10
0.15
0.07
0.09
0.05 0.06 0.05
Normal
Normal
Wind pressures acting on a single wall constitutes pressures on C&C. The values
in Table 3.1 do not include the combined gust-pressure coefficient (GCp) for C&C.
Therefore; reasonable estimation should be made for that variable. Also, for internal
pressure, the combined gust-pressure coefficient (GCpj) is only reported for enclosed 33
buildings. In severe wind events, the building envelope is likely to be breached, and this
breach will develop high internal pressures. When the building envelope is breached, it
should be treated as partially enclosed building.
In the following section, a comprehensive statistical approach to develop the
probability distribution for wind loads acting on low-rise buildings is presented and
applied. During the course of presenting that approach, any required but not available
details are appropriately assumed or estimated based on available knowledge.
3.3 Variability in Factors for Wind Loads
The loads acting on masonry wall on a low-rise building are generally produced
from the following components:
• Roof dead load,
• Roof uplift due to wind,
• Wall self weight,
• Wind pressure on the wall.
Among these components, dead loads have the least variability. There are two
dead load components involved. The wall self weight, which is quite consistent and has
the least variability. Thus, it is assumed dependent only on the wall thickness and specific
weight of blocks. Masonry load bearing walls are often associated with lightweight roof
systems. Roof dead load depends basically on the span; in other words, a longer span
produces a heavier roof system. The range of variation is not that large; however, it is
assumed in this research that the roof dead load varies from 10.0 lb/ft to 14.0 lb/ft when
the roof span varies from 20 ft to 40 ft. Between these two limits, the roof dead load is
assumed linearly varying with the span.
The other two components, which are the roof uplift and the wind pressure on
walls, are highly variable since they are related to wind characteristics. Since roof uplift
is dependent on the windward and leeward values for the pressure coefficient (Cp), an
assumption is made to make the calculations possible. According to the information
provided by previous research (EUingwood and Tekie, 1997) and listed in Table 3.1, the
coefficient of pressure (Cp) has a windward roof mean value of -0.92 associated with a
34
standard deviation of 0.15, and it has a leeward roof mean value of-0.61 associated with
a standard deviation of 0.09. In this research, it is assumed that the windward roof uplift
pressure coefficient is represented by a mean value of -0.92 and a standard deviation of
0.15, and that, the leeward roof uplift pressure coefficient is always higher by 0.31 than
that of the windward side. Between windward and leeward sides, uplift pressure varies
linearly. Figure 3.1 shows the uplift pressure coefficient distribution on the roof
ROOF UPLIFT PRESSURE COEFFICIENT
Uplift
Windward Wall Leeward Wall
Fig. 3.1. Pressure Coefficient For Roof Uplift
Mean values for coefficient of pressure at both windward and leeward walls are
given in Table 3.1, and can be readily used. This is useful for main wind force resisting
System (MWFRS). As far as individual walls are concemed, values for components and
cladding should be used. In this case, the statistical distribution for the combined gust-
pressure coefficient (GCp) is needed. This is not available in previous research; hence, it
has to be assumed.
In review of the values presented in Table 3.1, nominal values of the pressure
coefficient are seen as mean values plus a standard deviation. Also, the standard
deviation is about \4% of the mean value. Nominal values for (GCp) are given in Figure
35
6-5 A in the ASCE 7-98. If the nominal values are denoted by (n), the following equation
is used to obtain the disfribution parameters for the (GCp):
GC^„ = 1.14 X GC^. (3.3)
The other assumption that needs to be done is the distribution type. Actually, both the
individual gust effect factor and pressure coefficient are normally distributed; therefore, it
is reasonable assumption to consider the combined gust effect factor (GCp) as normally
distributed as well. Now, the (GCp) can be readily known completely, from the statistical
point of view. Figures 3.2 and 3.3 show the mean value and standard deviation of
combined gust-pressure coefficient (GCp) as a function of wall area.
The values of combined pressure-gust effect factor in the variable range, along
with the associated values for standard deviation, are expressed by the following
equations:
Positive Pressure - Zone 4 and 5
GCp, = 1.032 - 0.1551og(^) (3.4-A)
a p. = 0.145 - 0.022 log(.4) (3.4-B)
Negative Pressure - Zone 4
GC^, = -1.12 + O.I551og(^) (3.5-A)
cr . = 0.157 - 0.022 log(^) (3.5-B)
36
Negative Pressure - Zone 5
GCp, = -1.538 + 0.3101og(yi) (3.6-A)
(Tp, = 0.216 - 0.044 log(^) (3.6-B)
where (A) is the effective wind area (ASCE 7-98 section 6.2).
The combined gust-pressure coefficient for intemal pressure (GCpi) for partially
enclosed stmctures is required to be established; statistical parameters for enclosed
buildings are given in Table 3.1. In review of the available values of (GCpi) for enclosed
buildings, it is seen that the nominal value is almost the mean value plus a half standard
deviation. Further, the standard deviation is about 33% of the mean value. The following
equation for internal combined gust-pressure coefficient for partially enclosed building is
assumed:
GCp„, = 1.16 GCp,. (3.7)
Given that the nominal value for the combined gust-pressure coefficient for
intemal pressure of partially enclosed buildings is 0.55, the mean value is estimated as
±0.47 and the standard deviation is estimated as 0.16. Further, the combined intemal
gust-pressure factor for partially enclosed buildings is assumed normally distributed
similar to that of enclosed buildings. This establishes GCpi statistics of partially enclosed
buildings.
37
1.5
1.0
0.88 u o
0.5
0.12
0.0
— — St. Dev. GCp
10 100 Area (ft')
0.61
0.09
1000
Fig. 3.2. Gust-Pressure Coefficient (GCp) - Zones 4 and 5 Positive Pressure
0.5
0.17 0.14
0.0
^ -0-5
-0.96
-1.0
-1.23
-1.5
•Mean GCp - Zone 4
•St. Dev. GCp-Zone 4
Mean GCp - Zone 5
St. Dev. GCp - Zone 5
0.10
-0.70
' ° Area (ft') 100 1000
Fig. 3.3. Gust-Pressure Coefficient (GCp) for Zones 4 and 5 Negative Pressure
38
The available and assumed data make it possible to estimate the probability
distribution of wind loads on a low-rise building with lightweight roof In the following
section, the methodology for wind loads is discussed.
3.4 Development of Wind Loads
As mentioned earlier, the most important parameter in the wind load calculation is
the wind speed. In this research, maximum armual wind speeds are assumed to follow
Exfreme Value Type I distribution. This is the same distribution used in the ASCE 7-98
docimient. Records for maximum annual wind speeds are available at 400 stations all
over the United States territories for different numbers of years. The wind speed records
of Lubbock, Texas, are used to develop the probability distribution of maximum armual
wind speeds. Records are available from 1973 to 1990, with a maximum of 87.6 mph and
minimtim of 51.5 mph (CPP, 2001). Based on these records, the mean maximum annual
wind speed is 68.4 mph, and the standard deviation is 9.62 mph. The 50-year MRI for
this city is 93.4 mph. According to ASCE 7-98, Lubbock falls in the 90 mph wind speed
zone. Since the distribution of maximum annual wind speeds is following Extreme Value
Type I, a closed form distribution can be plotted for wind speed probability density
function. This is shown in Figure 3.4.
Since the Monte Carlo Simulation is used to develop the probability distribution
of wind-induced tensile stresses in walls, it is desirable to check the adequacy of obtained
results against the number of artificially generated observations. For that purpose, 100,
500, 1,000 and 10,000 observations are generated and the resulting density functions are
compared to the closed form solution. Figure 3.5 shows the plots of wind speed
probability density fianction generated by different number of observations versus that of
the closed form. The comparison shows that both 1,000 and 10,000 observations
produces fairly good plot for the probability density function of wind speed. In this
research, 1,000 observations are used to represent probability distribution of wind speed.
Wind loads and wind-induced stresses in masonry walls are developed for a
building located in Lubbock, Texas (basic wind speed of 90 mph), with wall height of 15
ft and wall thickness of 10-in.
39
4.00
3.00
2 2.00 <a
XI o
1.00
0.00
r
40 60 80 100 120 Wind Speed (mph)
140 160
5.00
4.00
^ 3.00
XI O T .00
1.00
Fig. 3.4. Probability Density Distribution of Wind Speeds
Lubbock, TX
0.00 1 • w
- * - 1 0 0 o b s
-0—SOOobs
- a - 1 0 0 0 obs
—•—10000 obs
closed Form
40 60 80 100 120 140 160 180
Wind Speed (mph) Fig. 3.5. Effect of Number of Observations on Quality of ResuUs
- Wind Speed Probability Density Distribution
40
3.4.1 Lateral Wind Pressure on Walls
The probability distribution of the lateral wind pressure is an outcome involving
many factors in addition to wind speed. Therefore, the same process followed to obtain
the probability distribution of wind speed is followed for all the factors involved in the
wind load calculation. The factors are terrain-exposure coefficient (Kz), pressure
coefficient (Cp), gust effect factor (G), and combined gust-pressure coefficient (GCp),
each with its associated probability distribution. The combined gust-pressure coefficient
for components and cladding depends on the effective wind area (ASCE, 1999). This is to
account for the fact that smaller areas have more correlated wind pressure. According to
the ASCE 7-98, the minimum effective wind area that can be considered is equal to the
wind span times one third the span or 10 ft^, whichever is larger. The combined gust-
pressure coefficient varies with the effective wind area until it reaches 500 ft^, where it
becomes constant. In this research, 10-in. block size is considered. The maximum
slendemess ratio allowed by IBC is 18, which results in a wall height of 15 ft (ICC,
2000). The effective width is taken as one third of that height. This produces an effective
wind area of 75 ft .
In previous research, it has been reported that 1,000 generated observations
suffices the objective of obtaining the wind pressure distribution (McAnuIty, 1998). Even
though the previous research was done to obtain load factors for hurricane winds, the
process in this research is similar. Nevertheless, the adequacy of the number of
observations used is checked throughout the development of this research.
3.4.2 Axial Loads in Walls
As mentioned before, variability in wall self weight is negligible. The wall height
and the wall thickness, along with the concrete block unit weight control the weight of
the wall. In some cases, walls are covered or decorated by wall veneers; however, wall
veneers are supported on nips from the foundation beams and are coimected to the wall
by flexible ties, which do not transfer vertical loads to the stmctural walls. Therefore,
only the wall self weight is considered.
41
According to the IBC, wall thickness is controlled by wall height. The IBC
limited slenderness ratio of hollow block walls to 18 (ICC, 2000). Since 10-in. walls are
studied in this research, wall height is limited to 15 ft. Considering normal weight
concrete blocks, a typical value of 135 lb/ft3 is used for the unit weight of concrete. This
value is based on many experiments and accumulated experience and the variation within
it is negligible. Compression at mid-height is calculated since maximum bending stresses
are expected at that level. Table 3.2 summarizes the process of estimating the wall self
weight.
Table 3.2: Compression Resuhing from Wall Self Weight
Height Area Weight Weight @ mid- Compression®
(ft) (in^) (Ib/ft ) height (lb) mid-height (psi)
15 50.4 47.25 354.38 7.03
It is assumed in this research that roof dead load is proportional to the roof span,
which coincides with the practice. For that reason, spans from 20 ft to 40 ft, with 5 ft
increments are be considered. Table 3.3 presents the compressive stress in the wall due to
the roof dead load.
Table 3.3: Compression Resulting from Roof Dead Load Span (ft)
Roof D.L. (Ib/ft^)
Wall Compression (psi)
20
lO.O
1.98
25
11.0
2.73
30
12.0
3.57
35
13.0
4.51
40
14.0
5.56
Compression resulting from wall self weight and roof dead load is small. Thus,
assuming constant values will not make any significant difference in the obtained results.
Uplift loads are handled in a similar way to wind loads. The exception is that the
pressure coefficient (Cp) is not considered as an independent random variable at the
leeward side, but rather dependent on that on the windward side (Cp (leeward roof) = Cp
(windward roof) + 0.31).
42
Figure 3.6 shows the loading mechanism of walls including different components
of wind loads and dead loads. Since external pressures under consideration are applied to
one surface (the wall under consideration), pressures for components and cladding (C&C)
are considered. Main wind force resisting system (MWFRS) is considered for the roof
uplift. Figure 3.6 shows two different loading scenarios.
The first loading scenario refers to walls subjected to inward (IW) acting pressure.
In this case, negative internal pressure is considered since it produces higher flexural
tensile stresses in the wall. Negative internal pressure reduces the total uplift forces.
The second loading scenario refers to walls subjected to outward (OW) acting
pressure. In this case, positive intemal pressure is considered since it produces higher
flexural tensile stresses in the wall. Positive intemal pressure increases the total uplift
forces.
Combined gust-pressure coefficient of components and cladding (C&C) is higher
for walls subjected to outward acting pressure than that for walls subjected to inward
acting pressure. Furthermore, due to the loading scenarios, tensile stresses resulting from
uplift forces are higher for walls subjected to outward acting pressure. Therefore, the
second loading scenario is likely to produce higher tensile stresses than the first one.
Previous research assessed stmctural adequacy of walls subjected to outward acting
presstires and pointed out that they are more critical (Al-Menyawi and Mehta, 2001).
Figures 3.7 and 3.8 show the probability distribution of uplift loads including
intemal pressure for walls subjected to both inward (IW) and outward (OW) acting
pressures, for a roof span of 30 ft. Both terrain exposures B and C are represented. As
expected, the figures show that total uplift forces in case of a wall subjected to outward
acting pressure are higher than those in the case of a wall subjected to inward acting
pressure.
43
< = I
>i > 0 0
•a
lA
A -a o
x : o
bx: • t-H
CS O
i-J
CO
oi
44
^ QN
XI ca
Si o
- A - O W - B
' -A-OW-C
- • - I W - B
- B - I W - C
150 450 300 Uplift Load (lb/ft)
Fig. 3.7. Probability Density Distribution of UpUft Loads Enclosed Building, Exposures B and C, 30ft Roof span
600
2.5
^ 1.5
XI o
0.5
-150
— • • • • I i
—A-OW-B
- • - I W - B
—13~ 1W - L.
150 300 Uplift Load (lb/ft)
450 600
Fig. 3.8. Probability Distribution of Uplift Loads Partially Enclosed Building, Exposures B and C, 30ft Roof span
45
3.4.3 Calculation of Stresses in Walls
Basically, two kinds of straining actions act on the wall section: (1) bending
resulting from the lateral pressure and (2) axial load resulting from the wall self weight,
roof dead load, and roof uplift. According to the basic mechanics of solids, stresses (t)
resulting from axial loads (F) acting on an area (A) are expressed as follows:
' = f <'-'^ The above equation has two conditions to be accurate and correct:
• Loads should be applied axially, and
• Section of consideration should be far enough from the load point of
application.
For axial loads applied to the wall, it is assumed that loads are axially applied. For
stresses resulting from bending, the wall is assumed to be acting as a simple beam
spanning vertically. This concept has been adopted by British Standards (BS 5628: Part
1, 1992). Stress in the wall due to lateral loads thus is expressed by the following
equation:
/ = ^ , (3.9)
where M is the bending moment resulting from the lateral wind pressure, and S is the
wall section modulus. Since the wall is modeled as simple beam, it is possible to express
the bending moment by the following formula:
M = , (3-10) 8
where q is the wind pressure on the wall and h is the wall height, expressed in consistent
units.
46
3.5 Obtaining Probabilitv Distribution of Wall Stresses: Methodology
To check flexural adequacy of walls, stresses resulting from wind pressure need
to be checked against MOR. Tensile stresses result from direct flexure in addition to roof
uplift. For instance, the tensile stresses resulting from wind pressure depend on the
following parameters:
• Exposure category, B or C,
• Type of structure enclosure, enclosed or partially enclosed,
• Roof span, 20 ft through 40 ft.
The above parameters will generate 20 cases. As mentioned earlier, 1,000
observations are generated involving all random variables contributing to the wind
pressure. During the course of generating random variables, two different approaches are
considered: (1) generating one random probability per observation, which is used for
generating values for all variables, such as wind speed, pressure coefficient (Cp), gust
effect factor (G), terrain exposure coefficient (Kz), and intemal gust-pressure coefficient
(GCpi); and (2) generating a random probability for each related group of variables, such
as (Cp) for walls and roof or gust effect factor (G). The two approaches yielded almost
the same results; this is attributed to the large number of generated observations. For the
20 cases, 1,000 results exist for each one. From the 1,000 results, the probability
distribution of tensile stresses in the wall is obtained.
3.6 Research Results
It is the target of this chapter to present the probability distribution of wind-
induced tensile stresses in masonry walls including extemal pressure on wall, roof uplift,
roof dead loads, and wall self weight. Results are obtained in two stages: (1) calculating
total stresses resulting from axial load, and (2) calculating flexural tensile stresses.
According to Figure 3.6, axial loads include wall self weight, roof dead load, roof
uplift and wind intemal pressure on the roof Figures 3.9 and 3.10 show the probability
distribution of wall stresses resulting from total axial loads for an enclosed building with
30 ft span. Figures 3.9 and 3.10 show no likelihood of tensUe stresses in the wall due to
axial loads in case of enclosed buildings, in other words, stresses are always compressive.
47
Figures 3.11 and 3.12 show the probability distribution of wall stresses resulting from
total axial loads for a partially enclosed building with 30 ft span. Tensile stresses are only
likely to occur in walls of partially enclosed buildings located in terrain exposure C;
however, the values of such tensile stresses are very small.
Since walls subjected to outward acting pressure are likely to have higher tensile
stresses, results for walls subjected to inward acting pressure are not shown in this
research. Figures 3.9 through 3.12 show that stresses resulting from axial loads are small,
which complies with the insignificant effect of roof span on the obtained results. Thus,
results are only shown for 30 ft roof spans. Effect of roof span on the obtained results is
shown later for verification of the above statement.
Figure 3.13 show the probability distribution of total tensile stresses in walls
subjected to outward acting pressures, in an enclosed building located in terrain
exposures B and C, and Figure 3.14 show the same probability distribution for a similar
wall in a partially enclosed building. Figures 3.15 and 3.16 show the cumulative
probability distribution of tensile stress in walls for the same cases.
The probability distribution of tensile stresses resulting from the wind pressure
does not follow any known distribution. Generally, the distribution is tailed one with low
probabilities at the high end, which is similar to that of the wind speed. It is important to
notice that, all parameters involved in the wind pressure equations are normally
distributed except the wind speed, but the resulting tensile stress is more influenced by
the wind speed distribution. This is attributed to the fact that, wind speed is squared in the
velocity pressure equation, and therefore, it has much more influence on the resulting
stress.
In review of Figures 3.13 to 3.16, it is noticeable that walls of buildings built in
exposure C experience significantly higher levels of stresses as compared to walls built in
exposure B. This is expected since mean values for the terrain exposure coefficient (Kz)
are higher for terrain exposure C. Furthermore; wind-induced tensile stresses for partially
enclosed buildings are significantiy higher than those of enclosed buildings. This is
attributed to the higher combined gust-pressure coefficient for intemal pressure of
partially enclosed buildings.
48
From the cumulative probability distribution of tensile stresses in walls of
enclosed buildings (Figure 3.15), it is seen that, for exposure B, there is a negligible
probability for the tensile stresses to exceed 55.5 psi (mean value of the MOR). For
exposure C, the probability of exceeding 55.5 psi is about 1%. This is based on 90 mph
basic design wind speed. From the cumulative probability distribution of tensile stresses
in walls of partially enclosed buildings (Figure 3.16), it is seen that, for exposure B, there
is a probability of about 1.7% for the stresses to exceed 55.5 psi. For exposure C, the
probability of exceeding 55.5 psi is more than 4%. These probabilities are annual. Annual
probability of failure is a reciprocal of the Mean Recurrence Interval (MRI). In other
words, 4% annual probability corresponds to 25-year MRI.
In this research, results are obtained for roof span varying from 20 ft to 40 ft with
5 ft increments. For the case of a wall subjected to outward acting pressure, in a partially
enclosed building located in exposure C, the results for different roof spans are presented
in Figure 3.17. Figure 3.17 confirms that the roof span is an insignificant parameter. This
is attributed to two reasons: (1) stresses resulting from axial loads are small, (2) uplift
loads are counteracted by dead loads, and both of them increase with the roof span,
therefore, they tend to neutralize the roof span.
49
60
^
Pro
babi
lity
o
20
0 i i a^-'A : ^>-^^^^ A
[
—A— OW Pressure
0 IW Pressure
A A A A •12 -9
Stress (psi)
Fig. 3.9. Probability Distribution of Total Axial Stresses Uplift and Dead Load, Enclosed Building, Exp B
XI C3
X2
o a.
80
60
40
20
0 k
• O W Pressure
• IW Pressure
•12 Stress (psi)
Fig. 3.10. Probability Distribution of Total Axial Stresses Uplift and Dead Load, Enclosed Building, Exp C
50
XI la
X) o
100
80
60
40
20
t B tf-^ ^
A 0 W Pressure
0 IW Pressure
^""''^ 0 0—0 0 0 0 # o 4 ^ h. k
-Yl -9 -6 -3 Stress (psi)
Fig. 3.11. Probability Distribution of Total Axial Stresses Uplift and Dead Load, Partially Enclosed Building, Exp B
80
60
% 40 la
Si
o
20
-A— OW Pressure
-d— IW Pressure
0 A A A A-
-15 -10 -5 0 5
Stress (psi)
Fig. 3.12. Probability Distribution of Total Axial Stresses Uplift and Dead Load, Partially Enclosed Building, Exp C
10
51
40
Stress (psi)
Fig. 3.13. Probability Distribution of Tensile Stresses
Enclosed Building
100
XI o
40
Stress (psi)
Fig. 3.14. probability Distribution of Wall Stresses
Partially Enclosed Building
100
52
X) la
XI o
100
80
60
^ 40 3
u
20
1—H—n -a—'•
^^^^—'iy » _ j fl a ta—H—H—H—a—a
—e—EXPB
- B - E X P C
-20
XI ta
XI o
100
80
60
ta 40 3
u 20
0 A-
-20
20 40 Stress (psi)
60 80
Fig. 3.15. Cumulative Distribution of Tensile Stresses Enclosed Building
20 40 Stress (psi)
60 80
Fig. 3.16. Cumulative Distribution of Tensile Stresses Partially Enclosed Building
53
X3 ta
X) o
-20 0 20 40 60 80 100
Stress (psi)
Fig. 3.17. Probability Distribution of Wall Stresses Partially Enclosed Building, Exposure C
54
3.7 Quality of Results Obtained by the Monte Carlo Simulation
It is important to check the quality of results obtained using the Monte Carlo
Simulation technique since they are used further in this research. For that purpose, total
tensile stresses in a wall subjected to outward acting pressure of a partially enclosed
building are calculated using 100, 500, 1,000 and 10,000 observations. Both ten-ain
exposures B and C are considered. The probability density distribution is compared for
the different number of observations. This comparison is shown in Figures 3.18 and 3.19.
It is noticeable that 1,000 and 10,000 observations give fair results, and the results are
quite comparable. This supports the decision of using 1,000 observations further in this
research. No closed form solution for the probability distribution of wind-induced tensile
stresses in walls is available to use it in this comparison.
3.8 Chapter Summary
Probability distribution and distribution parameters of combined gust-pressure
coefficients for components and cladding and for intemal pressure in partially enclosed
buildings are logically assumed. Assumptions were made by emulating probability
distribution of similar wind load parameters.
Probability distribution of wall flexural tensile stresses is determined considering
variability in wind speed, terrain exposure coefficient, gust effect factor, and pressure
coefficient based on the best available data. Tensile stresses include those resulting from
axial loads and flexural stresses (flexural stresses are significantly higher). The Monte
Carlo Simulation technique is used successfully for that purpose. The Monte Carlo
Simulation technique produces adequately good results using 1,000 artificially generated
random observations. It is shown that walls subjected to outward acting pressure are more
critical than those subjected to inward acting pressure. Through analysis, the roof span
proved to be insignificant parameter.
55
100 obs
500 obs
000 obs
10000 obs
-20 20 40 60 80
Stress (psi)
Fig. 3.18. Effect of Number of Observations on Quality of Results - Partially Enclosed Building, Exp B
X Q^
X2 O
-a—100 obs
—•—500 obs
—A—1000 obs
—e—10000 obs
-20 0 20 40 60 80 100
Stress (psi)
Fig. 3.19. Effect of Number of Observations on Quality of Results - Partially Enclosed Building, Exp C
56
CHAPTER 4
RESERVE STRENGTH AND PROBABILITY OF FAILURE
4.1 Introduction
Reserve strength of masonry constmction depends on lateral (out-of-plane)
flexural strength and wind-induced tensile stresses. Both strength and wind-induced
sfresses are variables. In Chapter 2, the probability disfribution of masonry walls flexural
tensile strength is established. In Chapter 3, the probability disfribution of wind-induced
tensile stresses is established, taking into account dead loads, wind pressure on walls, and
roof uplift, and considering the variability of involved parameters. The following step is
to use both distributions to obtain the probability of failure or survival of a masonry wall
subjected to wind loads. It is reported in Chapter 3 that the loading distribution does not
follow any known distribution, which makes closed form solution unavailable. Therefore,
numerical mathematical techniques should be used to obtain the target probability. The
probability of failure or survival depends on the 3-second gust wind speed of the zone,
terrain exposure category B or C, and type of building enclosure, either enclosed or
partially enclosed. It is demonstrated in Chapter 3 that the roof span is not a significant
parameter and does not influence the results.
4.2 Mathematical Approach
If the symbol (Q) represents the tensile stress resulting from the wind load and the
symbol (R ) represents the flexural tensile strength of the masonry wall (MOR), and each
one is following its own probabilistic distribution, failure would happen if
Q - R > 0 (4.1)
and probability of failure is defined as P[(Q-R) > 0]. To obtain such probability, a new
random variable is introduced, which is used for reserve strength (RS). RS is defined as
follows:
57
RS = R - Q. (4.2)
Numerically, if the 1,000 random observations for Q, the tensile stresses in the
wall resulting from the wind pressure, are available, along with the 1,000 random
observations of the flexural tensile strength, 1,000 random observations can be obtained
for the newly introduced random variable, RS. Using the generated observations, the
probability distribution of reserve strength (RS) can be found, and probability of failure
or survival can be calculated using numerical integration. The probability of failure is
defined as P(RS < 0), and the probability of survival is obtained from the following
equation:
P{RS > 0) = 1 - P(RS < 0). (4.3)
4.3 Parameters Affecting Reserve Strength
As mentioned earlier, reserve strength, RS, is the difference between the
resistance, R, and the wind-induced stresses, including the dead load effects, Q.
Resistance is only dependent on MOR. Other parameters affecting the resistance are not
considered in this research, such as mortar type, quality control, and strength of blocks.
On the other hand, wind-induced tensile stresses in the walls are affected mainly by wind
speed, exposure category B or C, and enclosure type of the building, enclosed or partially
enclosed. The above parameters are studied, and their effect on reserve strength and
probability of failure is reported in this research.
4.4 Extreme Value Distribution of the Reserve Strength
Wind speed distribution used in the analysis is an extreme value probability
distribution. Thus, it represents the maximum wind speed likely to happen in any
individual year. Other parameters are not represented as extreme value since they don't
depend on time. Since wind speed is squared in the equation of velocity pressure, which
is the goveming formula, and since all pressures are directly proportional to the square of
the wind speed, it is very likely, from the statistical standpoint, that the maximum tensile
58
stress in the wall in any year will occur with the maximum wind speed during that same
year. Nevertheless, this is not very accurate. The reason is that all parameters involved in
the velocity pressure, or pressure on the wall component are random variables, so at the
time the wind speed is maximum, the other parameters or some of them may be minimum
or shifted from the maximum. On the other hand, the maximum tensile stress in the wall
component in any year may occur at wind speed slightly less than the maximum annual
one because the other parameters may be all maximum at some time. Thus, in statistics
terminology, it can be said that the maximum annual tensile stress in the wall is very
likely to occur when the wind speed is maximum, but it may happen also at a different
time with a small likelihood. Therefore, the term Q used in the equations above is an
approximation to the maximum annual tensile stress in the wall, and fiirther, it can be
said that it is a lower bound estimate to the true maximum. Actually, since the wind speed
is squared in the equation of velocity pressure, it has an overwhelming effect on the
extreme value distribution of the tensile stresses in the wall, and it makes the
approximation valid and close enough to the tme values.
Some researchers have stated that most of the variability in the velocity pressure
equation comes from the wind speed since it is squared (McAnulaty, 1998). In different
research, the effect of various probability distributions of other parameters was studied
and was found ineffective (EUingwood, 1980). As a comment on the latter research, it has
been concluded that the results were virtually identical since the statistical characteristics
of the wind load are determined primarily by those of velocity squared (EUingwood,
1981).
Since one term in the reserve strength equation is represented as maximum annual
probability distribution, this will make the resulting reserve strength distribution of
annual type. Therefore, it can be noticed that probabilities of failure presented later in this
chapter are of annual nature as well. These probabilities of failure are actually lower
bound to the tme ones. However, it is not very likely to see higher annual probabilities of
failure. It is not the scope of this research to develop a statistical model for the tme
extreme value of tensile stresses in the wall, especially since the approximation is
59
reasonable and agrees with the basic fundamentals of statistics and findings of previous
research.
4.5 Reserve Strength and Probability of Failure
For each of the cases mentioned above (terrain exposure, and building enclosure
type), the reserve strength is calculated, and its probability distribution is obtained. To
make the comparison easier, wind-induced tensile stresses in different walls are
compared directly to modulus of rupture (MOR) values. Thus, wind-induced stresses are
calculated for enclosed and partially enclosed buildings, exposure B and C, for walls
subjected to outward acting pressures. Resulting values are subtracted from MOR values
to obtain reserve strength values, then probability distribution of reserve strength is
obtained.
Figure 4.1 shows the probability distribution of reserve strength of a wall
subjected to outward acting pressure, for a partially enclosed building located in terrain
exposure C. The resulting probability distribution is not represented by a smooth graph.
This is attributed to the nature of results obtained from the Monte Carlo Simulation
technique. Figure 4.2 shows the cumulative probability distribution of reserve strength of
a wall subjected to outward acting pressure, for a partially enclosed building located in
terrain exposure C. This figure is an integrated form of Figure 4.1. Normally, numerical
fluctuations disappear by integration, which is why the graph in Figure 4.2 looks
smoother. The cumulative distribution is more important and will be shown for different
cases. The annual probability of failure is the intercept value on the probability axis at
reserve strength equal to zero. Thus, it is not necessary to show the entire cumulative
distribution, but rather a partial one showing the sought probability of failure.
Figures 4.3 and 4.4 show the partial cumulative distribution of the reserve
strength for walls subjected to outward acting pressure in enclosed and partially enclosed
buildings. Terrain exposure B and C are considered. It is noticeable that the probability of
failure is higher for buildings located in terrain exposure C than for buildings located in
terrain exposure B. This is an expected result. Furthermore, probability of failure is
60
higher for partially enclosed buildings than enclosed buildings. This is attributed to the
higher combined internal gust-pressure coefficients.
The probability P (RS < 0) can be obtained from the cumulative distribution, or
by numerical integration of the reserve strength probability distribution. Since the whole
process is based on Monte Carlo Simulation, the annual probability of failure will not
always be a constant number, but will rather change in some narrow range. This is
attributed to the randomness of the artificially generated observations. To have the best
estimate, the process should be repeated a number of times, and an average value should
be considered. Fortunately, the range is quite narrow, which makes the reliability of the
obtained results quite high. For that sake, twenty runs were performed to obtain an
average value for the armual probability of failure. Wind speeds are generated as before,
for Lubbock, Texas, which has a 50-year MRI wind speed of 93.4 mph.
Tables 4.1 and 4.2 present annual probability of failures of different walls
subjected to inward and outward acting pressure. Table 4.1 summarizes the obtained
results for partially enclosed buildings, whereas Table 4.2 summarizes the obtained
results for enclosed buildings.
Table 4.1: Probability Wall
OW - (B)
OW-(C)
of Failtire Percent for Walls (Partially Enclosed Building) Min P(RS < 0)
1.00
4.50
Average P(RS < 0)
1.56
5.39
Max P(RS < 0)
2.10
6.60
Table 4.2: Probability of FaUure Percent for Walls (Enclosed Building) Wall
OW - (B)
OW-(C)
Min P(RS < 0)
0.10
0.80
Average P(RS < 0)
0.36
1.40
Max P(RS < 0)
0.70
1.90
61
XI ea
XI o
-50 -25 0 25 50
Flexural Tensile Reserve Strength (psi)
75 100
Fig. 4.1. Probability Distribution of Reserve Strength OW walls. Partially Enclosed Building
100
80
3
e 3
u
60
40
20
- • - E X P B
- e - E X P C
-50 -25 0 25 50
Flexural Tensile Reserve Strength (psi)
75 100
Fig. 4.2. Cumulative Distribution of Reserve Strength OW Walls, Partially Enclosed Building
62
10
^ ^
ta XI o
> •^ 4
u
- • - E X P B ^
1 -*-EXPC
1 • — — = 4 i 8 1 i "? -1 1 • '
-50 -40 -30 -20
Flexural Tensile Reserve Strength (psi)
-10
Fig. 4.3. Cumulative Distribution of Reserve Strength OW Walls, Enclosed Building
Flexural Tensile Reserve Strength (psi)
Fig. 4.4. Cumulative Distribution of Reserve Strength OW Walls, Partially Enclosed Building
63
In review of the average values of annual probability of failure for different types
of enclosed buildings walls, it can be noticed that annual probabilities of failure are as
high as 0.36% for buildings located in ten-ain exposure B. If the same constmction is
located in open terrain (Exposure C), the annual probabilities of failure will be as high as
1.40%.
For partially enclosed buildings, the annual probability of failure is as high as
1.56% for buildings located in ten-ain exposure B. For the same constmction located in
ten-ain exposure C, the probability of failure is as high as 5.39%.
It is important to assess the probability of failure during the building's lifetime.
To obtain the probability of failure in any number of years using the annual probability of
failure the following equation can be used.
P„ = 1 - (1 - PiRS<0)y (4.4)
where n is the number of years and Pn is the probability of failure in n years.
Fig. 4.5 presents the corresponding probability of failure in any number of years
obtained from the annual probability of failure. The figure shows that for an annual
probability of failure of 5.39%, the probability of failure in 10 years is about 45%, and it
is about 94%) in 50 years. Typically, buildings are designed for 50 years. Table 4.3
presents lifetime probability of failure for the different cases covered in this research.
Table 4.3: Lifetime Probability of Failure (Outward Acting Pressure) Wall
P. End. - (B)
P. End. - (C)
End. - (B)
End. - (C)
Annual
Probability
1.56
5.39
0.36
1.40
10 year
Lifetime
14.6
42.5
3.5
13.2
30 year
Lifetime
37.6
81.0
10.3
34.5
50 year
Lifetime
54.4
93.7
16.5
50.6
64
1)
I '3 O
XI ta
X) o
(U
100
75
50
25
- - e - l O Years
- B - 2 0 Years
- * - 3 0 Years
—»—40 Years
—•—50 Years
1.5 3 4.5
Annual Probability of Failure %
Fig. 4.5. Probability of Failure Within N Years
65
Design codes accounted for the aforementioned fact through the following
parameters: (1) load factor for wind load, which accounts for the probability of exceeding
the design wind speed during the structure lifetime, and, (2) strength reduction factor (^),
which accounts for variability in material strength and cross section dimensions.
Tables 4.1 and 4.2 are constmcted for a zone with 90 mph basic design wind
speed (Lubbock, Texas) according to the ASCE 7-98. Design wind speeds in the interior
of the United States are almost constant at that level, and it increases as we move toward
the coastal areas. It is understood that hurricane wind speeds do not follow Extreme
Value Type I distribution, however, we expect to see higher probabilities of failure as we
move toward coastal areas.
4.6 Adequacy of the Monte Carlo Simulation Technique
The Monte Carlo Simulation technique has been verified and proved to be
adequate to represent MOR, wind speed and wind-induced tensile stresses in walls.
Probability of failure is an interaction between MOR and tensile stresses resulting from
wind pressure; hence, the process needs to be verified for adequacy of probability of
failure results. To do that, 1,000 and 10,000 artificial random observations for both MOR
and wind-induced tensile stresses in walls are used to obtain aimual probability of failure.
This is done for walls subjected to outward acting pressure, in partially enclosed building
located in exposure B and C. The whole process is repeated 20 times to eliminate the
effect of randomness in obtained results.
Tables 4.3 and 4.4 summarize the comparison outcome. It is noticeable that using
10,000 observations increased the overall accuracy. This is shown through the narrower
range in which the probability of failure varies. Nevertheless, the average of the 20 runs
is almost the same as obtained from 1,000 observations. This proves that the averaging
technique is a successful one and that resuhs obtained from 1,000 observations are
adequate.
66
Table 4.4: Probability of Failure from 1,000 Observations vs. 10,000 Observations Partially Enclosed Building - Exposure B
No. of Observations Min P(RS < 0) Average P(RS < 0) Max P(RS < 0)
1,000 1.00 1.56 2.10
10,000 1.43 1.60 1.72
Table 4.5 : Probability of Failure from 1,000 Observations vs. Partially Enclosed Building - Exposure C
No. of Observations Min P(RS <
1,000
10,000
4.50
5.02
0) Average P(RS
5.39
5.39
10,000 Observations
<0) Max P(RS < 0)
6.60
5.78
4.7 Chapter Summary
Aimual probabilities of failure for unreinforced masonry walls (10-in. thick and
15 ft high) are established for different wind load conditions (terrain exposure, and
building enclosure type). Probability of failure over any number of years is calculated
from the established annual probability of failure. For a wall in an enclosed building
located in terrain exposure C, the probability of failure in a 50-year lifetime is 51%. For
the same case, if the building envelope is breached (partially enclosed), the lifetime
probability of failure becomes 94%.
Based on the obtained probabilities of failures, it is concluded that one-way wall
panels have high probability of failure during the building's lifetime. It is believed that
two-way wall panels would have higher lateral strength. Two-way action can be
employed to control the potential failure of walls.
67
CHAPTER 5
TWO-WAY ACTION OF MASONRY WALL PANELS
5.1 Introduction
It is common to have load-bearing walls acting as a one-way plate in the vertical
direction, especially, in large stmctures, like commercial malls, churches, gymnasiums,
factories, etc. For such cases, it is demonstrated in Chapter 4 that masonry walls have
probability of failure during the building lifetime. High probability of failure of masonry
stmctures has two basic adverse effects: (1) potential failure itself and (2) catastrophic
nature of failure. It is vital to bring masonry walls to a target level of probability of
failure. For that reason, it is suggested in this research to reinforce masonry walls,
intermittently, to create either hidden or explicit pilasters at certain spacing. This will
change the structtiral behavior of walls, from one-way plate action to two-way plate
action, since a portion of the load can go to the pilasters. The problem will be how to
estimate the collapse load (limit state resistance) for walls with two-way action. Thus, it
is required to employ an analytical technique to assess the ultimate lateral load carrying
capacity of masonry walls. This chapter evaluates the performance of two-way wall
panels for wind-induced loads and suggests an analytical method for the assessment of
wall lateral capacity.
5.2 The Intermittent Reinforcement
As the name suggests, the intermittent reinforcement is a concentrated vertical
reinforcement in particular cells in the block wall to form a hidden reinforced pilaster or
an explicit thick pilaster formed using standard pilaster blocks. This reinforcement is
intermittent because it is targeted to space the reinforcement as far apart as possible to
make it cost effective.
Figure 5.1 shows the standard pilaster units used in block masonry constmction. It
is an important feature that they have grooves that wall blocks may fit in for better
connection. This type of connection is not supposed to prevent rotation unless special
68
steel ties are provided. Figure 5.2 shows hidden pilasters fomied by reinforcing and
grouting cells within the block masonry wall itself
^ • r " "
/
\
RFT
^ ^ ^
U^ X^^^S^
PILASTER BLOCK
Fig. 5.1. Standard Pilaster Blocks
RFT
HIDDEN BLOCK Fig. 5.2. Pilasters Formed Using Ordinary Blocks
69
By providing such intermittent reinforcement and forming these pilasters, the
hidden or explicit pilaster will have much higher strength than the unreinforced wall, and
accordingly, will not deform in the same way. In other words, they are assumed to act
more like a lateral support for the horizontal direction of the wall. This will change the
wall behavior from a one-way plate spatming vertically to a two-way plate, with lateral
loads transferred in both vertical and horizontal directions.
It is known that two-way plates have much higher capacity than one-way plates.
The increased capacity will depend on the ratio of flexural strength of the direction
parallel and normal to the bed joints (orthogonal ratio y), and the wall aspect ratio (cp),
defined as the wall height divided by the pilaster spacing. Since pilasters are stiffer than
other parts of the wall, they will absorb significantly higher load. For ease of reference in
this research, intermittently reinforced masonry walls will be referred to as IRMW. Also,
it is assumed that pilasters are strong enough to act as support for wall and that roof
diaphragm is able to provide support for wall and pilasters.
5.3 Building Codes Addressing IRMW
South Florida Building Code (SFBC) is considered a pioneer in adopting the idea
of intermittently reinforced walls more than 40 years ago, realizing the nature of masonry
walls performance and brittleness of failure (Saffir, 1983). SFBC mandates providing tie
beams and tie columns all around the exterior wall panels such that the area of any panel
should not exceed 24.0 m^ (256 ft^). Considering the 15 ft high walls, the tie column
spacing should not exceed 17 ft. In other words, the aspect ratio (cp) of the wall panel
shall be limited to 0.88. Based on a long history of aftermath damage investigations,
researchers reported that SFBC is both economical and practical (Saffir, 1983). Recent
damage investigations reported that failure of masonry constmctions during severe wind
storms and hurricanes in the region of South Florida was only reported for buildings
constmcted before the development of that code, or where code requirements are violated
(ZoUo, 1993; FEMA, 1992).
70
5.4 Perfomiance of IRMW during Famous Severe Wind Events
It is useful to check masonry construction damage investigation reports and to
evaluate performance of intermittently reinforced walls. Fortunately, Hurricanes Andrew
and Opal provided a wealth of such information.
The FEMA damage report following Hurricane Andrew reported that the main
cause of failure of masonry buildings was the lack of vertical wall reinforcement. Further,
It has been reported that, where failures of the buildings did occur, poor mortar joints,
lack of tie beams, horizontal reinforcement, tie columns, tie anchors, and misplaced or
missing hurricane straps between walls and roof stmcture were observed (FEMA, 1992).
Other damage investigation reports following Hurricane Andrew reported that one
of the main causes of failure of one particular masonry building was the lack of grouting
in the reinforced cells. Otherwise, it was reported that masonry wall systems meeting
SFBC requirements performed well and that their mass also contributes to stability
against uplift and overtuming. Also, it was reported that failures did occur due to the loss
of lateral support when the roof diaphragm is gone for any reason (ZoUo, 1993).
After Hurricane Opal, masonry stmctures were observed, and some failures were
fotmd. Mainly, failures were attributed to poor constmction or to loss of connection
between the wall and the roof diaphragm. For residential stmctures, no collapse of well-
supported walls was reported. For industrial and institutional structures, there is only one
incident were the windward wall was damaged. The wall was multiwythe, with 4-in.
brick outer and 8-in. block irmer. The report attributed the failure to breaching of the wall
by a missile, which initiated the failure through the increase of intemal pressure
(McGinley et al., 1996).
5.5 Analysis of Intermittently Reinforced Masonry Walls
By providing the intermittent reinforcement, the wall acts as a two-way laterally
loaded plate. It should be noted that masonry walls are anisotropic and highly
heterogeneous. For modeling, they are asstimed as orthotropic plates having different
flexural strength and section properties in two perpendicular directions. Normally,
directions parallel to bed joints and normal to bed joints are considered. Methods adopted
from classical mechanics of solids and numerical solutions are utilized to solve plate
71
problems. In the following section, different available analytical methods for masonry
walls analysis are discussed, and an analytical method will be selected and developed for
further use in this research.
5.5.1 Thin Plate Analysis
Masonry walls are typically 8-in., 10-in., or 12-in. thick and span vertically about
18 times the thickness, or slightly less, as required by building codes. In the horizontal
direction, the span can be longer. Thus, thin plate analysis is considered applicable for
masonry wall panels.
Tests have shown that, at failure, deflection did not exceed the wall thickness
(Abboud et al., 1996). This means that membrane forces can be neglected, and simple
thin plate theory with shear deformation (Mindlin plate theory) can be used.
Due to the orthotropic nature (different strength in two orthogonal directions) of
masonry wall panels, classical plate analysis may become tedious and time consuming.
Therefore, nimierical techniques such as Finite Element Method and the method of Finite
Differences may be used. These methods are quite powerful; however. Finite Element
Method is more powerful since it can handle different boundary conditions more easily.
Thin plate analysis can be extended to account for material non-linearity resulting from
cracking. In such case, non-linear finite element analysis needs to be employed. This is
useful when the full behavior during loading is required.
5.5.2 Strip Method
As a simplification to the plate analysis, the strip method is sometimes utilized. It
can assume that the plate has different stiffness in two perpendicular directions, based on
the boundary conditions and the span in each direction and based on the compatibility of
displacements. The share of lateral load transmitted in each direction can be determined.
This method is the basis for the analysis method provided in the National
Concrete Masonry Association (NCMA TEK 14-3, 1995). In that solution, an isotropic
plate was assumed. It is known that solutions obtained by strip method can deviate from
the exact solution by about 20% - 30%. However, it is a good and reliable method of
72
analysis and gives fast yet conservative results. Both classical plate theory and strip
method are based on the theory of elasticity and cannot therefore give an idea about the
ultimate lateral load carrying capacity of the wall.
5.5.3 Yield-Line Analysis
Yield-line analysis has been under development since the 1920s. It evolved
basically from the theory of plasticity, and it studies the state of the plate at failure.
The method is based on assuming failure mechanism and either studying the
equilibrium of wall segments at failure or equating the energy dissipated along yield lines
with the work done by loads undergoing virtual displacement.
This method has been used and investigated by many researchers for masonry
walls (Drysdale and Essawy, 1988; Candy, 1988; Fried et al., 1988; Mann and Tonn,
1988) and has been adopted for analysis and design purposes by some building codes
(British Standards, BS 5628: Part 1, 1992). The method has proven to be reliable and
adequate for analysis. The tme advantage of this method is that it deals with the wall
panel at failure, therefore, it gives the failure load, which is of interest in this research.
The yield-line theory displays adequate accuracy and can handle orthotropic
plates and different boundary conditions or walls with openings. The only material
properties to pursue yield-line analysis of masonry wall panels are flexural tensile
strength in two perpendicular directions. In this research, plate analysis of wall panels is
performed for the sole purpose of determining the collapse load (limit state resistance).
Yield-line theory is a direct and reliable method to obtain this load. Also, yield-line is an
easy and fast method compared to other methods.
For the aforementioned reasons, this method is the one that will be utilized further
in this research. Necessary developments are formulated to make it completely tailored
for handling concrete block masonry wall panels with different edge conditions under
lateral loads resulting from wind pressure. This is addressed in more detail further in this
chapter.
73
5.6 Yield Line Analysis for Ultimate Lateral Strength Estimation
Masonry walls subjected to lateral (out-of plane) wind pressure act like laterally
loaded plates. Unreinforced masonry walls spanning vertically are easy and
straightforward to analyze since they act as one-way plates or simple beams. Once wall
panels become intermittently reinforced, they act as two-way plates. Since wind loads on
walls are quite important, many engineering authorities have been concemed with
providing methods of analysis for two-way loaded wall panels. Some of these methods
are based on elastic analysis. However, elastic analysis in such cases does not give a good
idea about wall ultimate lateral load carrying capacity, or failure load. In reality, once
intermittently reinforced wall panels start cracking and yield, elastic linear behavior does
not hold valid. For a partially reinforced wall with a small amount of reinforcement,
inherent reserve strength is quite significant as compared to linear elastic limit. When
design for severe winds is pursued, this reserve strength plays an important role.
Actually, failure load may not be estimated with a high level of accuracy. However, it is
possible to estimate upper and lower bounds for the failure load. Methods used to
estimate failure loads are generally derived from the theory of plasticity, which states that
the ultimate collapse load of a structure lies between two limits, an upper bound and
lower bound of the true collapse load (Nilson, 1997). These limits can be found by well-
established methods, and they can be forced to converge to the true value through trial
and error processes with numerical techniques. It is appropriate to introduce the upper
bound and lower bound theorems at this point (Nilson, 1997, pp. 484).
Lower bound theorem: If, for a given extemal load, it is possible to find a
distribution of moments that satisfies equilibrium requirements, with the moment
not exceeding the yield moment at any location, and if the boundary conditions
are satisfied, then the given load is a lower bound of the tme carrying capacity.
Upper bound theorem: If, for a small increment of displacement, the intemal work
done by the plate, assuming that the moment at every plastic hinge is equal to the
yield moment, and that boundary conditions are satisfied, is equal to the extemal
work done by the given load for that same increment of displacement, then that
load is an upper bound of the tme carrying capacity.
74
5.6.1 Adequacy of Yield-Line Method
The yield-line method of analysis gives an upper bound to the ultimate load
capacity of a plate by a study of assumed mechanisms of collapse. Yield-line being an
upper bound method, gives failure or ultimate loads higher than the tme capacity. On the
other hand, plastic behavior and real support conditions are difficult to model for
masonry wall panels. Thus, obtained collapse loads may not be overestimated.. Many
researchers spent tremendous efforts comparing collapse load obtained from yield-line
analysis to that obtained from lab tests. Drysdale and Essawy (1988) conducted intensive
research to evaluate the yield-line method as an analytical approach for laterally loaded
unreinforced masonry walls. For five different wall cases, the ratio of collapse load as
obtained from yield-line to that obtained from laboratory test was 0.98, 0.96, 1.20, 0.89,
and 0.97. These values display high consistency and compliance with tme fracture load.
The general conclusion of the conducted research was that yield-line analysis provides
good predictions of failure pressure, and it is a suggested approach as an adequate
analytical method. Also, yield-line analysis is adopted by the British Standards (BS 5628,
Part 1, 1992) as an analytical method to obtain moments and stresses in laterally loaded
masonry walls.
Accuracy of results obtained from yield-line analysis depends greatly on the
correctness of assumed fracture lines, and actual flexural strengths in the vertical and
horizontal directions of masonry wall. Modeling of support conditions is equally
important and can contribute to deviation from the true collapse load if not approached
with care. Yield-line analysis can give fairly good estimation for failure loads if proper
fracture lines pattern is ascertained. If the assumed fracture line pattem is different from
the tme one, an upper bound solution will result. The fracture line pattem depends greatly
on the wall aspect ratio (cp), i.e., height to horizontal span ratio. It depends also on the
orthogonal ratio (y) of wall flexural strength.
Drysdale and Essawy (1988) were successful in predicting fracture lines for walls
with different aspect ratios. The study was concerned with estimating the failure load of
75
laterally loaded wall panels using yield-line method, and comparison of the results with
experiments.
To develop an accurate yield-line solution, it is necessary to estimate the ultimate
flexural strength of a wall panel in two orthogonal directions. Normally, the directions of
interest are the direction normal to the bed joints and the direction normal to the head
joints. Flexural strength of masonry constmction in two directions is presented in Chapter
2 of this dissertation. Once an algorithm for failure pattem is established, fracture load
can be obtained for various aspect ratios (cp) and edge conditions.
5.6.2 Basic Fundamentals of Yield-Line Analysis
Yield-line analysis is always performed at the collapse position of a plate. It assumes that
at the ultimate (collapse) load, the wall panel will fracture into segments intersecting at
yield lines. The whole assembly is known as fracture pattern. When the plate is on the
verge of collapse as a mechanism, axes of rotation will be fully developed over supports.
Yield-line analysis for wall panels will be based on the following basic
fundamentals (Ghali and Neville, 1998):
1. At fracture, the bending moment per unit length along all fracture lines, along
non-reinforced area is constant and equal to fracture moment.
2. The wall parts rotate about axes along the supported edges.
3. At fracture, elastic deformations are small compared to the plastic deformations
and are therefore ignored. From this assumption and the previous one it follows
that fractured wall parts are plane and therefore they intersect in straight lines. In
other words, yield lines are straight.
4. The lines of fracture on the sides of two adjacent wall parts pass through the point
of intersection of their axes of rotation.
Yield lines can be positive or negative. Positive lines are those taking place in the field
zone, and negative lines are those taking place on the support lines.
The first and most important step to establish a yield-line solution is to assume a
reasonable fracture pattem and axes of rotation. Nilson (1997) provided guidelines to
76
establish proper fracture lines and axes of rotations. The following guidelines are relevant
to masonry wall panels:
1. Yield lines are straight lines because they represent the intersection of two planes.
2. Yield lines represent axes of rotation.
3. The supported edges of the wall will also establish axes of rotation. If the edge is
fixed, a negative yield line may form providing constant resistance to rotation. If
the edge is simply supported, the axis of rotation provides zero resistance.
4. A yield line between two slab segments must pass through the point of
intersection of the axes of rotation of the adjacent wall segments.
5.6.3 The Yield-Line Analysis
There are generally two possible approaches in the yield-line theory. The first one
is an energy method in which the external work done by the loads during a small virtual
movement of the collapse mechanism is equated to the intemal work. It is also called the
method of Virtual work (Ghali and Neville, 1998). The altemative approach is by the
study of the equilibrium of the various parts of the wall into which the wall is divided by
the yield lines. The energy method is adopted in this research and will be discussed in
detail.
5.6.3.1 Analysis of Isotropic Plates Using Yield-line theory
It is known that masonry walls are anisotropic and heterogeneous. For ease and
practicality of analysis and design, they are modeled as orthotropic plates with different
strengths in vertical and horizontal directions. However, analysis of isotropic plates is a
cmcial step for the understanding and development of solutions for orthotropic plates.
If a plate, with resistance Mi and M2 in two perpendicular directions as shown in
Figure 5.3, is loaded laterally up to collapse, it will collapse at inclined yield lines. It is
required to know the yield moment (Ma) along the yield line, so that it would be possible
to analyze the plate further. If a wedge of the plate such as that shown in Figure 5.4 is
considered, the moment along the yield line can be written in the following form:
77
M„ = M, cos^or + Mjsin^a (5.1)
Having inclined yield lines will produce torsion moments as well. The torsion moment
(M,) can be found from the following equation:
M, = (M, -M2) sin a cos« (5.2)
In the case of isotropic plate, where Mi and M2 are equal to M, it can be seen that, at any
inclined surface, the yield moment will be equal to M, no matter what the inclination
angle is. Also, it can be noticed, that the torsion moment along the yield line (Mt) is equal
to zero.
5.6.3.2 Energy Method for Yield-Line Analysis
In this method, the fracture pattem will be assumed, based on the guidelines mentioned
earlier. Wall segments will deflect a virtual disteince A. Due to that deflection, each wall
segment will rotate an angle 9. At fracture, and due to the virtual rotation, energy will be
dissipated along yield lines. At the same time, loads on different plate segments will do
virtual work. The virtual energy dissipated along the yield lines is set equal to the virtual
work done. From this equation, the value of the ultimate moments along yield lines and
ultimate load are obtained.
78
._x
Ml (N
>
Fig. 5.3. Orthotropic Masonry Wall Panel
Fig. 5.4. Equilibrium of a Wall Segment (Reproduced from Ghali and Neville, 1998)
79
Since the yield-line method is an upper bound solution, all assumed fracture
patterns would give ultimate load higher than the tme load-canying capacity of the wall.
Only if the correct fracture pattern is assumed, answers will be close enough to the tme
collapse load. For this purpose, energy and virtual work equations can be written as
functions of the fracture geometry, and using differentiation techniques, the correct
fracture pattern can be found. The correct fracture pattem can be defined as that one
which results in the maximum moment along yield lines, and consequently, least ultimate
load. In other words, it is that one that makes the moment as a fianction of fracture pattem
geometry maximum. Thus, the technique that will be used here is such that the energy
and virtual work equations are written as functions of the fracture geometry parameters,
and moment derivatives with respect to these parameters will be set equal to zero. Hence,
an appropriate fracture pattern geometry can be obtained. The following equation is used
to estimate the energy dissipated along the yield line:
U = Y^M. G (5.3)
where U is the dissipated energy, M is the yield moment vector, and 0 is the rotation
vector at the yield line. The above equation is in a vector form, and it can be expanded to
the following scalar form:
U = X ^ ^ • ^ . + H^y • ^y (5.4)
where x and y denote the components of the moment vector and rotation vector in two
perpendicular directions X and Y.
5.6.3.3 Analysis of Orthotropic Plates Using Yield-line Method
Compared to isotropic plates, moment at any yield line is a function of the
moments in the two main perpendicular directions and the inclination angle of the yield
line. Also, torsion along yield lines is not equal to zero. Since masonry walls dealt with in
80
this research are always rectangular in shape, some fiirther simplification based on the
fundamental method can be done.
The basic idea behind the simplified method is to deal with a transformed plate.
The orthotropic properties are completely represented in the transformed plate. The
method depends on considering one dimension as the original one. Then, depending on
the orthogonal ratio (y), the other dimension will be altered to have an equivalent
isotropic plate with new dimensions.
To be able to do that, the orthogonal ratio should be well established. In this
research, a value of 2.5 is justified based on available test results (limited number of
tests) and previous research (see Chapter 2).
A transformation process following that developed earlier (Johansen, 1962) will
be followed. The method is outlined according to available references (Johansen, 1962;
GhaU and Neville, 1998).
A part of a plate ABCDEF, as shown in Figure 5.5, is limited by positive and
negative yield lines and a free edge and is assumed to rotate an angle 9 about an axis R-
R. It is assumed that the ultimate positive and negative moment capacity of the wall panel
is M in the direction of the bed joint, and yM normal to bed joint (fracture occurring
along head joints). The vectors u and C represent the resultant positive and negative
moments respectively. Using the energy method, the intemal virtual energy dissipated
along any yield line U can be expressed as:
U = {mc^ + mb^ )G^ + y • {mc^ + mb^, )-0y (5.5)
where x and y denote vector components in the X and Y directions (Cx is the X-
component of vector C and Cy is Y-component of vector C) and G is the rotation vector
as shown. Assuming that the virtual deflection at a point n, at a distance r from the axis of
rotation R-R, is unity, the rotation 9 and its components can be written as:
G = - (5.6-A) r
81
0,-G X cosa =— (5.6-B) r
0y=G X sina = - (5.6-C) r
where a is the angle between the X axis and the axis of rotation R-R. Rewriting the
above equation, the internal energy dissipated along any yield line can be expressed as:
U = G [ [mc^ + mbj cosa + y-(mCy+mby) sina J. (5.7)
The derivation can be completed for a general case, where uniform, line, and
concentrated loads are applied to the plates. However, for masonry walls, the only likely
applied load is uniform load (q), resulting from wind pressure. If the deflection at any
point (x,y) is w, the virtual work done by the applied load on any plate part is expressed
by the following formula:
W = ^^q w dx dy. (5.8)
The basic principle is that intemal energy dissipated along all yield lines is set equal to
work done on all plate parts. This is expressed by the following equation:
{mc^ + mb^ )— + y • \mCy + mb^) ^
yy = Y^ \\q w dxdy. (5.9)
If an isotropic plate is considered, loaded by a uniform load q*, and have all its
dimensions in the X-direction multiplied by a factor X, the plate is assumed to have the
same fracture pattem and the same deflections at all corresponding points. The intemal
energy U* can be written as follows:
82
1 1 U* = (mAc^ + mAb^) — + (mc+mb) — . (5.10)
Similarly, the virtual work done by the uniform loading q* can be written as follows:
W*= ^jq* w A dx dy . (5.11)
The following virtual work equation represents all plate parts:
(wc, + mb^ )y + Y ^^^y ^ ^^y ^~ = YJ 11^*^ ^^y- (5-12)
If the two plates are said to be equivalent, then the following constraint can be deducted:
A' 7- (5.13-A)
Or, in other form.
A = Ir'
(5.13-B)
The above concludes that if the orthotropic plate dimensions are (b) in the X-
direction and (a) in the Y-direction, the moment capacity in the X-direction is M, and the
moment capacity in the Y-direction is yM, then the X-direction is known as the strong
direction. In other words, its span can be ahered to account for the high moment capacity.
The alteration ratio is proved to be -?=. In this case, the plate can be analyzed as if its
moment capacity is M in both directions. It should be noted that this is tme only for
orthotropic plates. Since, the horizontal direction of the wall is always stronger, it will be
83
muUiplied by the above defined ratio. Thus, the equivalent isotropic plate will be shorter
horizontally than the original orthotropic one.
R \
NEGATIVE YIELD LINES . - i
D y-.rr-
• X
AXES OF ROTATION
POSITIVE YIELD LINES
Fig. 5.5. Transformation of an Orthotropic WaU to Isotropic Wall (Reproduced from Ghali and Neville, 1998)
84
5.6.3.4 Effect of Elastic Support
Pilasters, hidden or explicit, are not like real rigid supports since they deform
depending on how much load they take and depending on their flexural stiffness.
However, they can support the wall laterally. If the pilaster can resist the load transfen-ed
to it at the time the wall collapses, then, for the yield-line analysis, it will be like rigid
support. Also, elastic deformations in the pilaster should not have any significant effects
on the results, since they are much smaller than wall deflection at collapse. Point (3) in
the yield-line basic fundamentals refer to this fact (see section 5.6.2). The only concem is
that pilasters should be adequately designed to support their share of the load at collapse
with adequate level of safety. Appendix B may be referred to for guidelines of pilaster
design according to building codes.
5.6.3.5 Edge Support Conditions
The scope of this research is concemed with masonry stmctures with light
roofing, typically made of steel or timber. Actually, this category of constmction is
common. Light roofing is normally supported on bond beam, on top of walls. Bond
beams are secured to walls by ties, spaced at somewhat large distances, and don't have
any significant flexural strength. Therefore, it is assumed that the roof diaphragm can
only prevent translation of walls at the support line, but is not supposed to provide any
flexural resistance to walls. Thus, the edge support condition at the top of the wall is
assumed to be simply supported.
At the bottom, the wall rests on the wall footing. Specifications do not mandate
providing any reinforcement dowels between the wall and the footing. In addition, it is
not required to have any groove in the foundation for the wall. Such foundation type
carmot provide any flexural resistance, either. Therefore, as long as no dowels are
provided between the wall and the footing, it is reasonable to assume the wall is simply
supported (hinged) at the bottom. If dowels and grouting are provided to connect the wall
to the foundation, the bottom edge support may be considered fixed.
When hidden or explicit pilasters are provided, wall behavior changes from one
way to two-way plate. According to Figure 5.6, pilasters constmcted using special
85
pilaster blocks will form a hinged lateral support, since it can only prevent translation,
whereas walls can rotate in their grooves. On the other hand, hidden or implicit pilasters
are integral part of the wall. They provide lateral support by being reinforced and
grouted, with sufficient strength and stiffness to resist lateral loads. Since these hidden
pilasters are integral parts of the walls, they can prevent rotation as well as translation.
Therefore, it can be concluded that explicit pilasters will provide hinged supports,
whereas hidden pilasters will provide fixed supports.
In construction, many details can be developed to constmct a fixed support, or
generally to enhance the performance of supports. This is not the scope of this research,
and it is left to the discretion of designers and contractors concerned with constmction
details.
Some notations are used commonly to describe the edge conditions for yield-line
analysis. These notations are shown in Figure 5.7.
86
I
L
<
w Q Q
s
i •
u o m z Q 9 EC
Z o P CJ UJ Z Z O o i < o z o O o
s
N
^ x$
:
i" :-S$
g
3 B i Tcnr
CJ
o CO
oi
u
o ^-» o )-l +-•
a o O
o t/2
C/3
T3
o "B. w
87
5.6.4 Development of Yield-line Equations
To analyze the wall panel for ultimate canying capacity, it should be first
transformed into equivalent isofropic plate. This is done by multiplying the horizontal
dimension (b) by the ratio - ^ . Where y is the orthogonal ratio. This will result in an
equivalent isofropic plate having shorter horizontal dimension. By doing that, the
calculated ultimate moment will correspond to that normal to the bed joint (causing
fracture along head joints). The calculated moment will be presented in the following
form:
yM = k-q-b^ (5.14)
where k is a moment coefficient, q is the ultimate load carrying capacity of the wall
(collapse load), and b is the original horizontal dimension of the wall, before the
transformation.
The analysis process is concerned with obtaining the values for the moment
coefficient (k) for the different cases considered in this research. Actually, such values
exist for some cases in the British standards (BS 5628 Part 1). The British standards
cover a range of aspect ratio (9) starting from 0.3 and some cases of edge conditions. The
purpose of pursing this analysis is to cover new cases and to outline the method.
5<v>^yyyvyyyv
FREE EDGE
HINGED EDGE
FIXED EDGE
Fig. 5.7. Notafions of Yield-Line Analysis
88
5.6.4.1 Geometric Parameters of Fracture Pattern
As mentioned earlier, the moment along the yield lines can be expressed in temis
of some parameters related to the geometry of the fracture lines. In other words, moment
along yield lines is expressed as a ftmction of some geometric parameters. By minimizing
that ftmction, the best estimate of fracture pattem can be found, and therefore, collapse
load can be detemiined with a good level of accuracy. In some cases, fracture pattern can
be described by one geomefric parameter, and in other cases, more than one parameter
will be required. In such cases, partial derivatives should be used for the establishment of
these parameters. The number of required geomefric parameters depends on the edge
conditions. In the following, fracture pattern parameters used in this research will be
outlined for different edge conditions.
5.6.4.2 Aspect Ratio
One of the useful features of yield-line analysis for rectangular plates is that, it is
controlled mainly by the aspect ratio of the wall panel. Aspect ratio (cp) is defined as the
vertical dimension of the wall (a) divided by the horizontal dimension (b). In this
research, all the moment coefficients obtained analytically will be presented in
dimensionless form as a function of the aspect ratio.
5.6.4.3 Edge Conditions of Wall Panels
Edge support conditions of walls differ depending on the location of the wall and
its use. Another parameter is the stmcttiral design and construction details that are to be
followed. The support at the roof diaphragm is always simple. Other supports can be
either simple or fixed. It is believed that special pilaster imits (Figure 5.1) are often
preferred. Also, for simplicity, a limited number of cases for edge conditions will be
considered. The following four cases are believed common and are more often used in
constmction, and therefore, will be emphasized in this research.
In the yield-line analysis, the displacement (A) appears on both the virtual energy
side and the virtual work side. Therefore, it is convenient to assume it as unity (1.0) from
the begitming. Thus, equations show virtual energy and work for a unit displacement.
89
5.6.4.3.1 Wall simply supported on four sides. Such walls are the typical ones,
and they exist almost in all locations. Simple supports at sides are resulting from the use
of special pilaster units. Simple edge conditions will result in a fracture pattern that can
be determined by one geometric parameter. Even though this case of edge support
conditions is simple and can be described by one geometric parameter, it should be
noticed that there are two potential fracture patterns depending on the wall aspect ratio.
Figure 5.8 shows the fracture pattems (A) and (B) and the used geometric parameters.
For fracture pattern (A), the virtual energy dissipated along yield lines is expressed by the
following equation:
U = 2M a 2b — + —
(5.15-A)
where b* is the transformed isotropic horizontal dimension. The virtual work done by the
loads on different wall segments is expressed by the following equation:
W q a 'b' X
3J (5.15-B)
From Equations 5.15-A and 5.15-B, the following expression for the moment as a
function of the geometric parameter (x) can be obtained:
M = f IL.' 2\ q-a ( 3b X - 2x
12 [a^+ 2b'x ^' (5.15-C)
90
X X
\
\ "J'-.n^WWMWm
\
PATERRN (A)
PATTERN (B)
Fig. 5.8. Failure Pattems and Geometric Parameters for Wall Symmefrically Supported on Four Edges
91
By maximizing the moment expression with respect to the geometric parameter (x), the
following expression for (x) can be found:
a X =
b') + 3 - a (5.15-D)
For fracture pattem (B), the virtual energy dissipated along yield lines is expressed by the
following equation:
U = 2M ^b' _ 2a
X b* (5.15-E)
And the virtual work done by the loads on different wall segments is expressed by the
following equation:
W = qa a X
2 ~ 3", (5.15-F)
From Equations 5.15-E and 5.15-F, the following expression for the moment as a
function of the geometrical parameter can be obtained:
M = q a ~[2
3ax - 2x
b* -f 2ax (5.15-G)
By maximizing the moment expression with respect to the geometric parameter (x), the
following expression for (x) can be foimd:
92
X =
2 ^ Ti'V
v « y . 3 - * :
a I
(5.15-H)
where a and b* are the isotropic wall dimensions in the vertical and horizontal directions,
respectively.
The minimum ultimate load of the two pattems should be used. It is found that for
aspect ratio ((p) up to 0.632, Pattern (A) predominates; afterwards, Pattem (B)
predominates (for higher aspect ratios, i.e., closer pilaster spacing).
5.6.4.3.2 Wall fixed at bottom and simply supported elsewhere. This case is a
further development of the previous one. One of the ways overall lateral strength of walls
can be increased is to provide fixed support at the bottom. When this is done, two
geometric parameters for the fracture pattems will be necessary. Figure 5.9 shows the
expected fracture pattems (A) and (B) and the chosen geometric parameters.
The intemal energy (U) can be obtained by adding that dissipated by the
additional moment at the fixed support to the intemal energy of case 5.6.4.3.1. The
difference from the previous case will be the use of two geometric parameters. Virtual
work (W) will not change from the previous case. Knowing (U) and (W), moment (M)
expressions for this case can be obtained.
93
\
\ /
7
\
/
\
1
PATERRN (A)
':^
y y
„ ^ ,^-,^'
.^'
1
y"
,^ .y^. X
V 'X
PATTERN (B)
Fig. 5.9. Failure Pattems and Geometric Parameters for Wall Fixed at the Bottom and Simple Elsewhere
94
For fracture pattern (A), the virtual energy dissipated along yield lines is
expressed by the following equation:
U = M ^ b' 2b' 2a^ + -I- —
\a-y y Xj (5.16-A)
and the virtual work done by the loads on different wall segments is expressed by the
following equation:
W = qa ^b'
(5.16-B)
From Equations 5.16-A and 5.16-B, the following expression for the moment as a
function of the geometrical parameter can be obtained:
q ( \3ab* - lax)- xy • {a - y) M =
b'xy + 2b'x(a - y) + 2ay{a - y) (5.16-C)
The above equation is to be partially differentiated with respect to both x and y, and
solved for the derivative set equal to zero. By maximizing the moment expression with
respect to the geometric parameter (y), the following expression for (y) can be found:
y = (2 - V2) a. (5.16-D)
By maximizing the moment expression with respect to the geometric parameter (x), an
expression for (x) can be found. No easy expression can be obtained for the geometric
parameter x; therefore, the equations are solved numerically.
For fracture pattem (B), the virtual energy dissipated along yield lines is
expressed by the following equation:
95
U ^ M 4a b' 2b' ]
b y X (5.16-E)
The virtual work done by the loads on different wall segments is expressed by the
following equation:
W = ^ {3a - y)- (5.16-F)
From Equations 5.16-E and 5.16-F, the following expression for the moment as a
function of the geometrical parameter can be obtained:
M = q-b xy • (3a - X y)
b' X + 2b* y Aaxy (5.16-G)
The above equation is to be partially differentiated with respect to both x and y,
and solved for the derivative set equal to zero. It is found that for aspect ratio (cp) up to
0.762, Pattem (A) predominates; afterwards, Pattem (B) predominates (for higher aspect
ratios).
5.6.4.3.3 Wall fixed at sides, simply supported on top and bottom. This case is
similar to case 5.6.4.3.1, with fixed sides instead of the simple supports. Symmetric edge
conditions will result in a fracture pattem that can be determined by one geometric
parameter. As in case 5.6.4.3.1, it should be noticed that, there are two potential fracttire
pattems depending on the wall aspect ratio. Figure 5.8 still can be referred to for fracture
pattems (A) and (B) and the used geometric parameters.
For fracture pattem (A), the virtual energy dissipated along yield lines is
expressed by the following equation:
U = AM a 2b — ^- —
V- a (5.17-A)
96
The virtual work done by the loads on different wall segments is expressed by the
following equation:
W = qa 'b'
(5.17-B)
From Equations 5.17-A and 5.17-B, the following expression for the moment as a
function of the geometrical parameter can be obtained:
M = a^ f3b'x - 2x 2 \
24 2 7 '
a + b X (5.17-C)
By differentiating the moment expression with respect to the geometric parameter (x) and
equating the derivative to zero, the geometric parameter (x) can be obtained. This can be
done numerically for different values of aspect ratio (cp).
For fracture pattem (B), the virtual energy dissipated along yield lines is
expressed by the following equation:
U = M 2b* 8fl
X b (5.17-D)
The virtual work done by the loads on different wall segments is expressed by the
following equation:
w - r»- If - I (5.17-E)
97
From Equations 5.17-D and 5.17-F, the following expression for the moment as a
function of the geometrical parameter can be obtained:
M = q^
12 3ax - 2x^
b* + Aax (5.17-F)
The geometric parameter (x) can be estimated by maximizing the moment expression.
The minimum ultimate load of the two pattems should be used. It is found that for aspect
ratio (cp) up to - ^ , Pattem (A) predominates; afterwards, Pattem (B) predominates (for
higher aspect ratios).
5.6.4.3.4 Wall fixed at sides and bottom and simply supported on top. As a further
development of the previous case, a fixed support can be provided between the wall and
the foundation. When this is done, two geometric parameters for the fracture pattems will
be necessary. Figure 5.9 is still representing the expected fracture pattems (A) and (B)
and the chosen geometric parameters.
For fracture pattem (A), the virtual energy dissipated along yield lines is
expressed by the following equation:
U = M ^ b* 2b' Aa
-f -I- —
^a-y y x
(5.18-A)
The virtual work done by the loads on different wall segments is expressed by the
following equation:
W = q-a (b- \
(5.18-B)
98
From Equations 5.18-A and 5.18-B, the following expression for the moment as a
function of the geometric parameter can be obtained:
M = qa (36* - 2x)-xy-{a - y)
b'xy -f 2b'x(a - y) + Aay{a - y) (5.18-C)
The above equation is to be partially differentiated with respect to both x and y, and
solved for the derivative set equal to zero. By maximizing the moment expression with
respect to the geometric parameter (y), the following expression for (y) can be found:
;; = (2 - V2) a. (5.18-D)
By maximizing the moment expression with respect to the geometric parameter (x), an
expression for (x) can be fotmd, which may be solved numerically.
For fracture pattem (B), the virtual energy dissipated along yield lines is
expressed by the following equation:
U = M 8fl b' 2b
+ - + — b y X
*\ (5.18-E)
The virtual work done by the loads on different wall segments is expressed by the
following equation:
W = q-b'
{3a - X - y). (5.18-F)
From the Equations 5.18-E and 5.18-F, the following expression for the moment as a
function of the geometrical parameter can be obtained:
99
M = i l ^ ( ^ ^ • ( 3 Q - X - y) ^
6 16* X + Ib'^y + Saxy^ (5.18-G)
The above equation is to be partially differentiated with respect to both x and y, and
solved for the derivative set equal to zero. Values of x and y are estimated numerically. It
is found that, for aspect ratio (cp) up to 0.853, Pattern (A) predominates; afterwards,
Pattem (B) predominates (for higher aspect ratios).
5.6.4.4 Moment Coefficients
The obtained results are compared to those tabulated in the British Standards (BS
5628: Part 1) and found identical. The values of moment coefficient (k, see Equation
5.14) for cases (a) and (b) are presented graphically in Figure 5.10. The values of moment
coefficient (k) for cases (c) and (d) are presented in Figure 5.11.
Some experimental results are published (Drysdale and Essawy, 1988) to evaluate
the yield-line as an analytical method. Table 5.1 presents a comparison between resuhs
obtained in this research and available experimental results. It should be noted that,
experimentally, it is difficult to model all support conditions.
Table 5.1: Comparison of Analytical Results to Experimental Results Support
Condition Wall dimensions
b x a (m)
Aspect ratio
(cp)
qe KPa
qA KPa
Ratio qA/qE
Case (a) 2.80x3.40 0.82 10.21 9.05 0.89
Case (a) 2.80x5.00 0.56 6.82 6.16 0.90
Case (a) 2.80x5.80 0.48 4.77 5.50 1.15
The above comparison shows that, yield-line analysis does not really overestimate
the failure load. Some differences to the conservative side are attributed to the difference
in orthogonal ratio (y). Also, wall own weight will produce compressive stresses, which
would enhance the wall behavior under flexure, even by a small amount, and this is not
100
taken into account in the analytical results. Generally, the analytical results are in good
agreement with available experimental results.
5.7 Chapter Summary
It IS proposed to increase the sfrength of wall panels by providing them with
intermittent vertical reinforcement. This will change the wall panels' structural behavior
to two-way action. Two-way action of walls will provide them with additional strength
due to the two-way action itself and due to the high orthogonal ratio of walls. High
orthogonal ratio is attributed to the fact that horizontal direction of the wall is much
stronger than the vertical direction.
Review of tiie performance of intermittentiy reinforced walls showed that IRMW
generally performed well in severe wind events. When damage was reported, evidence
was there that standards were not met properly or constmction was of poor quality.
Two-way masonry wall panels can be modeled as orthotropic plates having
different strength properties in two directions, parallel to bed joints, and normal to bed
joints (orthogonal ratio y is 2.5). Among the different analytical methods to solve the
two-way wall panels, the yield-line method is employed. Yield-line method is a reliable
analysis and design method for unreinforced concrete block masonry walls. It has been
used by design codes and has been subject to intensive testing which confirmed its
reliability and adequacy for analysis and design purposes.
Through some fracture mechanism geometric parameters, the best estimate of
fracture pattems has been established. The governing fracture pattem depends on the wall
panel aspect ratio (cp). By changing the wall panel dimensions, the break down aspect
ratio for each combination of edge support conditions is obtained.
By the virtue of Yield-line method, the ultimate lateral load carrying capacity of
masonry wall panels can be estimated for different combinations of edge support
conditions. Results are presented in the form of moment coefficient (k), which is plotted
in Figures 5.10 and 5.11. Similar values are tabulated in the British Standards (BS 5628:
Part 1).
101
0.10
0.08
0.06
0.04
0.02
0.00
—B— Bottom Simple
—»— Bottom Fixed
0.00 0.50 1.00 1.50
Aspect Ratio ((p=a/b)
Fig. 5.10. Yield-Line Moment Coefficients Wall Simple at Top with Simple Sides
2.00
0.05
0.04
0.03
0.02
0.01
0.00
0.00
-B— Bottom Simple
-e—Bottom Fixed
0.50 1.50 1.00
Aspect Ratio (a/b)
Fig. 5.11. Yield-Line Moment Coefficients Wall Simple at Top with Fixed Sides
2.00
102
CHAPTER 6
TARGET PROBABILITY OF FAILURE
6.1 Introduction
In practical design, a low probability of failure is targeted. This is the spirit of the
standard of practice. Based on the importance of the stmcture and intended lifetime, this
target value can be defined.
In Chapter 4, armual probabilities of failure of unreinforced masonry wall (10-in.
thick and 15 ft high) are established (Tables 4.1 and 4.2). Based on the intended lifetime
of the stmcttore, lifetime probability of failure can be obtained (Table 4.5). In some cases,
lifetime probability of failtire seems high. It is targeted to reduce the probability of failure
to some limiting value.
Probability of failure is an outcome of both the probability distribution of wall
ultimate lateral load carrying capacity (expressed in terms of MOR) and probability
distribution of wind induced tensile stresses. Probability of failure can be reduced by
increasing the wall ultimate lateral load carrying capacity or by reducing the wind
induced loads. Practically, only ultimate lateral load carrying capacity can be increased.
Increasing the ultimate lateral load carrying capacity can be done by using higher
strength mortars, by using bed joint ties, or by providing two-way action of wall through
vertical intermittent reinforcement. In this research, increasing the wall ultimate lateral
load carrying capacity is proposed to be done by changing the wall to intermittently
reinforced masonry wall (IRMW). Intermittent reinforcement will create pilasters in the
walls that will result into two-way action for wall. IRMW has significantly higher
strength than unreinforced masonry walls because of the two-way action and because of
the high orthogonal ratio of flexural strength.
Ultimate lateral load carrying capacity of IRMW is estimated by the virtue of
yield-line theory. The orthotropic wall panel is transformed into an equivalent isotropic
wall with modified (shorter) horizontal dimension. Ultimate lateral load carrying capacity
is estimated for different wall edge conditions.
103
A relationship between the ultimate lateral load carrying capacity and probability
of failure needs to be established. The wall aspect ratio is not known a priory; therefore,
ultimate lateral load carrying capacity cannot be estimated on a probabilistic basis. Thus,
an intermediate parameter is necessary at this point. The intermediate parameter is related
to the probability of failure on one side and is related to the ultimate lateral load carrying
capacity on the other side. The intennediate parameter will be referred to as factor (f). If
the target probability of failure is known, by iteration, the factor (f) can be obtained.
It is required to relate the probability of failure to the spacing of the intermittent
reinforcement. This is done by multiplying the MOR by a factor (f). This factor maybe
called "MOR increase factor". The proposed factor expresses the required increase in
lateral flexural strength to meet certain probability of failure. The proposed factor should
be always greater than 1.0. By repeating the work done in Chapters 2, 3 and 4, with the
use of the factor (f), resulting probabilities of failures can be obtained. Probability of
failure is to be obtained for the same cases of walls, with different wind load conditions
(exposure B, exposure C, enclosed and partially enclosed buildings).
It is also required to relate the factor (f) to the ultimate lateral load carrying
capacity. This is done by converting the additional increase in lateral strength provided
by intermittent reinforcement to an equivalent increase in the MOR as mentioned before.
By comparing the ultimate lateral load carrying capacity of walls with one-way behavior
to that of walls with two-way behavior, an equation for the factor (f) can be obtained in
terms of the wall aspect ratio, which represents the spacing of pilasters or intermittent
reinforcement.
6.2 Target Probability of Failure
The first step towards mitigating undesirable failure of masonry stmctures is to set
forth a target probability of failure. This needs to be done in the light of available codes
of practices. In this research, standards will be emulated to obtain the target probability of
failure. However, this research establishes a methodology rather than obtaining design
value; therefore, target probability of failure can be changed.
104
Defining the target of probability of failure depends on the intended design
lifetime of the stmcture. This is shown in Chapter 4 (Table 4.5). In this research, only
normal structures are studied; therefore, lifetime is assumed to be 50 years. For other
lifetime periods, a different value for the target probability of failure can be calculated.
Some analysis of the ASCE 7-98 wind load factor and other strength reduction
factors in building codes needs to be done. Wind load factor as defined by the ASCE 7-
98 is 1.60 (ASCE, 1999). Also, UBC (ICBO, 1997) and IBC (ICC, 2000) specify on a
strength reduction factor (([)) of 0.80 for masonry walls under flexure. The two combined
factors will result in a total factor of 2.0. To translate this into a probability of failure, it
needs to be taken at one side, either loading or sfrength. For instance, it can be all taken
to the wind side. Since wind loads are proportional to the square of wind speed, then the
factor that should be applied to wind speed is V2 .
The ASCE 7-98 load factors for wind loads are based on limit state wind speed of
500-year mean recun-ence interval (MRI) (ASCE 7-98, section C6.5.4). The MRI is the
reciprocal of the annual probability of occurrence, which is 0.2% for 500-year MRI.
According to ASCE 7-98 Table C6-3, 500 year MRI means that the wind speed
multiplier is 1.23. In this research, to accotmt for strength reduction factor (([>), the
multiplier used is v2 . MRI corresponding to V2 can be obtained by extrapolation since
the relation between the MRI and the multiplier is log linear. Figure 6.1 shows the
relationship between MRI and wind speed multiplier. The multiplier of v2 corresponds
to about 3200-year MRI. Such MRI will suggest a target armual probability of failure of
0.0315%). For 50-year lifetime, the total probability of failure will be 1.5%. This level of
probability of failure is targeted in this research.
105
1.60
1.40
•^ 1.20 a. 3 •S 1.00 a •o B 0.80
0.60
0.40
y = 0.099Ln(x) +0.616
10 100
MRI (years)
1000 10000
Fig. 6.1. Relationship Between MRI and Wind Speed Multiplier
106
6.3 MOR Increase Factor (f)
After a target probability of failure is defined, the next step is to answer the
question, "what MOR can give this probability of failure?" This can be better seen from
the following equation:
RS = R - Q (4.2)
where R is the MOR, Q is the wind induced tensile stresses, and RS is the reserve
strength. When MOR increases, the reserve strength will increase; hence, the probability
of failure will decrease. Figure 6.2 shows the distribution of both R and Q. Probability of
failure is the intersection area of both R and Q graphs. Increasing R implies a right shift
of the entire R graph; hence, a decrease of the intersection area. If MOR is multiplied by
a factor (f), a new distribution of R will result. This idea is shown schematically in Figure
6.3.
107
Si
0 B a n -20 0 20
e - P r o b - Q
• - P r o b - R I
fl B BS P a
40 60 80 100 120
Stress (psi) Fig. 6.2. Probability Distribution of Wind Induced Tensile
Stresses Versus MOR
-B—Prob-Q
•O • -Prob-R
-A—Prob-fR
» H H
120
Stress (psi) Fig. 6.3. Probability Distribution of MOR with the Application
of the Factor (f) 108
The probability of failure is obtained by using the Monte Carlo Simulation.
Therefore, the results are in numerical form. Strength obtained from the yield-line
analysis cannot be represented numerically since the intermittent reinforcement spacing is
unknown. This shows the necessity of the factor (f) to relate between numerical values of
probability of failure and yield line equations. If the MOR is multiplied by an increase
factor (f) during the process of obtaining probability of failure, it will reduce the
probability of failure. Iteratively, the factor (f) can be increased, and the probability of
failure is to be calculated, until it reaches the target level. The probability of failure
depends on loading conditions, such as terrain exposure and building enclosure type.
Therefore, for each case, the factor (f) has a specific value. Table 6.1 gives the values of
the factor (f) for walls with outward acting pressure, in zones with 90 mph basic wind
speeds.
Table 6.1: Factor (f) for Different Wind Load Conditions (90 mph) Wind Load Condition f
Exposure B, Enclosed building 1.45
Exposure C, Enclosed building 1.90
Exposure B, Partially Enclosed building 1.95
Exposure C, Partially Enclosed building 2.50
The annual probability of failure for unreinforced masonry walls for the
conditions listed in Table 6.1 ranges from 0.36% for a building located in terrain
exposure B up to 5.39% for a building located in terrain exposure C. This is for enclosed
and partially enclosed buildings (Chapter 4, Tables 4.1 and 4.2). The relationship
between the factor (f) and the probability of failure is not linear.
109
Table 6.1 is valid only for inland zones of the United States with basic wind
speeds of 90 mph. For higher wind speeds, required factor (f) will be higher. Table 6.2
gives the values of the factor (f) for 120 mph wind speed zones. This is based on the
approximation that 120 mph wind speed has the same variability and probability
distribution type (Extreme Value Type I) as the 90 mph wind speed.
Table 6.2: Factor (f) for Different Loading Conditions (120 mph) Wall description
Exposure B, Enclosed building 2.40
Exposure C, Enclosed building 3.20
Exposure B, Partially Enclosed building 2.90
Exposure C, Partially Enclosed building 4.60
For different wind speeds between 90 and 120 mph, Figure 6.4 can be used to
obtain the required factor (f). The relationship between the wind speed and the factor (f)
is represented by a concave curve of mild slope, which suggests possible use of linear
interpolation.
110
o 3.0
80 90 100 110
Basic Design Wind Speed (mph)
Fig. 6.4. Factor of MOR Increase (f)
120 130
111
6.4 Relationship Between Factor (f) and Intermittent Reinforcement Spacing
By establishing a target level of probability of failure, the required factor (f),
which represents the increase in the MOR to meet that probability of failure, is obtained.
It is possible to relate the factor (f) to the spacing of the intermittent reinforcement;
hence, intermittent reinforcement spacing that will correspond to specific probability of
failure can be determined. This section derives this relationship mathematically.
As mentioned before, reserve strength is defined as the difference between
strength and applied loads. In mathematical form, it can be defined as follows:
RS = R - Q (4.2)
where RS denotes the target reserve strength, R denotes the available strength, and Q
denotes the stresses resulting from wind and dead loads.
At the failure point, the resistance R should be equal to the stress Q, and the
reserve strength will be equal to zero. This can be used to calculate the ultimate lateral
load at failure. R can be calculated for mean values or nominal values in a
straightforward manner. For walls spanned vertically, this can be written as follows:
R = ' ^ . (6.1) S
In the above equation, R is the resistance expressed in terms of MOR, h is the wall
height, S is the section modulus, ks is the simple beam moment coefficient (ks = 0.125),
and q is the total ultimate load intensity at which failure is anticipated.
To increase the reserve strength to a target limit, the resistance R (MOR) is
multiplied by an increase factor (f). This can be expressed in the following form:
fR = ^ ^ ^ (6.2)
112
where q is the increased lateral load carrying capacity of the two-way wall. What the
intermittent reinforcement does is change the wall behavior to a two-way plate, as
mentioned in Chapter 5. In such a case, the total ultimate load at failure will increase
from q to q . The ultimate load at failure q* in the new case can be defined by the
following formula:
Ji qb' R = - ^ (6.3)
yS
where y is the orthogonal ratio (assumed 2.5 in this research, see Chapter 2) and b is the
intermittent reinforcement spacing. This is the same formula outlined in Chapter 6. The
above formula does not tell in an easy way how much additional strength is gained.
Therefore, it may be rewritten in slightly different form. The new form will make use the
factor (f). To do that, the additional lateral strength of the wall gained by providing
intermittent reinforcement is converted into equivalent fictitious increase in the MOR, via
multiplying it by the factor (f). The modified MOR (R*) is obtained as follows:
R' = /R (6.4)
where the factor (f) is obtained as:
f = i- = ^ . cp' - y (6.5) a k
where cp is the wall aspect ratio. In the above equation, both k and cp are variables, but
they are interdependent.
113
6.5 Estimation of Pilaster Spacing
In the yield-line analysis, the aspect ratio (cp) of the wall panel is a given
parameter. This is practical when the wall height and pilaster spacing are known. In this
research, the pilaster spacing is the target variable, and it needs to be estimated. Wall
height is assumed to be 15 ft for 10-in. walls. It is understood that the moment coefficient
(k) and the pilaster spacing (b) are interdependent; therefore, a mathematical procedure
needs to be adopted. An easy way for doing that is to put the solutions of the yield-line
2
problem in the fomi of — . The aspect ratio (cp) can be determined in a straightforward k
marmer once the factor (f) is determined and the proper edge conditions are chosen.
Figure 6.5 shows a set of graphs necessary to solve the equations of the yield-line
analysis for given values of the factor (f). This technique is used to obtain the aspect ratio
that results in specific value for the factor (f).
Tables 6.1 and 6.2 show that values of the factor (f) vary between 1.45 and 4.60.
The above-mentioned procedure is used to obtain the pilaster spacing corresponding to
the previously calculated values of the factor (f). A range for the factor (f) between 1.05
and 5.00 is considered, and for any required values in the middle, linear interpolation can
be used. Results are expressed in terms of aspect ratio (cp). For pilaster spacing, wall
height is to be divided by the tabulated values. Table 6.3 presents minimum aspect ratio
for different values of factor (f) for various wall panel support conditions.
114
o
o u &
1.80
1.40
1.00
0.60
0.20
0.00 2.00 .00 4.00 6.00
Factor of MOR Increase (f)
Fig. 6.5. Relationship Between Aspect Ratio (tp) and Factor (f)
10.00
Case (a): Simply supported at four sides Case (b): Fixed at the bottom and simply supported elsewhere Case (c): Simply supported at top and bottom, and fixed elsewhere Case (d): Simply supported at top and fixed elsewhere
115
Table 6.3: Aspect Ratio Corresponding to Certain Values of the Factor (f) f
1.05
1.50
1.75
2.00
2.50
3.00
3.50
4.00
4.50
5.00
Case (a)
<0.10
0.22
0.31
0.39
0.52
0.63
0.73
0.82
0.91
0.98
Case (b)
<0.10
<0.10
0.12
0.21
0.36
0.48
0.60
0.70
0.78
0.87
Case (c)
<0.10
0.16
0.22
0.27
0.37
0.45
0.52
0.58
0.64
0.70
Case (d)
<0.10
<0.10
<0.10
0.14
0.26
0.35
0.42
0.50
0.56
0.62
Case (a): Simply supported at four sides
Case (b): Fixed at bottom and simply supported elsewhere
Case (c): Simply supported at top and the bottom, and fixed elsewhere
Case (d): Simply supported at top and fixed elsewhere
For better use and benefit, the above resuhs are represented graphically in Figure 6.6.
Results presented in Table 6.3 show that pilaster spacing can be as large as three
times the wall height, depending on the wall edge conditions and wind load conditions.
Moreover, it is seen that providing pilasters at spacing equal to the wall height provides
adequate lateral strength to meet the target probability of failure for any wind load
conditions and wind speed up to 120 mph.
Table 6.2 presents values for the MOR increase factor (f) for 120 mph wind speed
and different wind load conditions (f varies from 2.40 to 4.60). Aspect ratios (pilaster
spacing) are calculated for these cases and the values are presented in Table 6.3 (the last
six rows). By reviewing the values in Table 6.3 for the cases associated with 120 mph
wind speed, it can be seen that pilasters are spaced at closer distances. Also, providing
fixed support between the wall and the foundation increases the pilaster spacing (about
23%) for an f value of 3.00). In some cases (values above the line in Table 6.3), required
aspect ratio is very small (i.e., pilaster spacing is very large). It is appropriate to limit the 116
minimum aspect ratio (maximum spacing). This limit is better established using
experimental methods. British Standards (BS 5628: Part 1, 1992) limit the minimum
aspect ratio to 0.3 and recommends analyzing the wall as a one-way plate spanning
vertically for values less than that limit. In previous experimental research, the two-way
behavior was verified for an aspect ratio as low as 0.48 (Drysdale and Essawy, 1988). In
this research, it is recommended to limit the minimum aspect ratio to 0.30.
Obtained values of aspect ratio (cp) seem low. This is attributed to the orthotropic
nature of block masonry walls. It is shown in Chapter 2 that the orthogonal ratio of
concrete block walls is about 2.5. This value is based on limited number of experiments;
however, it is supported by British Standards.
6.6 Chapter Summary
In this research, a lifetime of 50 years is considered since this length represents
common structures. Based on that lifetime, an annual probability of failure of 0.03% is
suggested. The suggested annual probability of failure is estimated from current standards
of practice (ASCE 7-98, 1999; ICC, 2000). This annual probability of failure con-esponds
to 1.5%) lifetime probability of failure.
To develop a mathematical basis for achieving target probability of failure, an
MOR increase factor (f) is introduced. It represents the gain in ultimate lateral carrying
capacity resulting from the two-way action.
Values in Table 6.3 suggests that pilaster spacing can be as large as three times
the wall height, depending on the wall edge conditions and wind load conditions.
Moreover, it can be concluded that providing pilasters at spacing equal to the wall height
will provide adequate strength to bring the probability of failure below the target value
for any wind load condition for wind speed up to 120 mph.
British Standards limit the aspect ratio to 0.30 for two-way action. Experimental
data on the behavior of two-way wall panels is available for an aspect ratio of 0.48. It is
recommended in this research to limit the aspect ratio to 0.30.
117
1.00
0.80
0.60 CO
D, 0.40
0.20
0.00
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
Factor of MOR Increase (f) Fig. 6.6. Aspect Ratio Corresponding to Various Values of
Factor (f)
Case (a): Simply supported at four sides Case (b): Fixed at the bottom and simply supported elsewhere Case (c): Simply supported at top and bottom, and fixed elsewhere Case (d): Simply supported at top and fixed elsewhere
118
CHAPTER 7
SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS
7.1 Summary
Unreinforced masonry constmction may experience failures in severe wind
events. Unreinforced masonry failure is catastrophic in nature. This research focuses on
providing masonry walls with additional lateral strength to meet a target probability of
failure. The additional strength is achieved by intermittently reinforcing the walls.
This research has three main objectives. These objectives are to assess the
probability of failure of unreinforced masonry walls, to estimate the ultimate lateral load
carrying capacity of intermittently reinforced masonry walls, and to determine the
intermittent reinforcement spacing necessary to meet the target probability of failure.
The research scope focuses on unreinforced low-rise masonry constmction with
lightweight roofs. Construction under consideration is commonly used with normal
weight blocks and mortar type S. Walls in the study are subjected to out-of-plane bending
resulting from direct wind pressures.
This research is based on statistical analysis of available data for MOR of
masonry walls. Experimental values of MOR from different investigations are assumed to
be representative and from the same population. MOR is assumed to be normally
distributed. On the loading side, probability distributions of combined gust-pressure
coefficients (GCp) for components and cladding (C&C) and for intemal pressure in
partially enclosed buildings are obtained using logical extrapolation of available data.
Probability distribution of other variables involved in the calculation of wind-induced
loads are obtained from literature. Roof dead load is assumed to be dependent on the roof
span, varying from 10 Ib/ft to 14 Ib/ft when roof span varies from 20ft to 40ft.
Monte Carlo Simulation is a convenient method for determining a wind load
probability distribution since wind load parameters such as wind speed, pressure
coefficients, gust effect factors, terrain exposures, and building enclosure types have
random variability. A total of 1,000 sample points gives a reasonable cumulative
distribution of wind loads.
119
Annual failure probability of unreinforced masonry walls is established for
different loading conditions (type of terrain exposure and building enclosure type). The
probability of failure in any number of years is calculated from the annual probability of
failure. A 10-in, wall of 15 ft height is used to ascertain probability of failure.
By providing walls with intermittent reinforcement, wall stmctural behavior
changes to two-way plate action. Two-way masonry wall panels are modeled as
orthotropic plates having different strength properties parallel to bed joints, and normal to
bed joints. The yield-line method is used to analyze Intermittently Reinforced Masonry
Walls (IRMW). Yield-line method gives reliable estimates of the ultimate lateral load
carrying capacity of masonry walls, provided the assumed edge conditions are met.
It is possible to relate intermittent reinforcement spacing to target probability of
failure through the MOR increase factor (f). This factor expresses the increase in lateral
strength gained from the two-way action in terms of a virtual increase in the MOR.
7.2 Conclusions
Throughout this research, various studies have been conducted, and the following
conclusions are drawn:
1. For mortar type S made with Portland cement and lime, bed joint MOR
(Flexural tensile strength normal to bed joint) has a mean value of 55.5 psi
with a COV of 25.7% (standard deviation a is 14.25 psi). Flexural tensile
strength parallel to bed joints (normal to head joints) is considered 2.5 times
the strength normal to the bed joints. These values are based on extremely
limited experimental data available in literature.
2. Monte Carlo Simulation of wind speed, terrain exposure, building enclosure
type, pressure coefficients, and gust effect factor provides wind loads on a
probabilistic basis. A Monte Carlo Simulation sample of 1,000 observations
gives a reasonable and smooth cumulative distribution for the probability
disfribution of wind loads. Results obtained using 1,000 observations are
comparable to those obtained using 10,000 observations.
120
3. Wind-induced tensile stresses are more critical for outward acting pressures.
Roof uplift-induced pressures, which vary with roof span, have small effects
on tensile stresses in the walls. Roof span proved to be generally an
insignificant parameter.
4. Armual probability of failure for unreinforced masonry walls is established
using a probability distribution of both the MOR and wind-induced tensile
stresses. For a partially enclosed building located in terrain exposure C, the
probability of failure of unreinforced masonry wall (10-in. thick and 15 ft
high) in a 50-year building life is 94%.
5. Two-way plate action significantly increases out-of-plane bending resistance
of masonry walls against failure. This conclusion is based on yield-line theory
analysis for orthotropic plates. Intermittent reinforcement (pilasters) and grout
provides two-way action in masonry walls. This conclusion assumes that edge
conditions are met as assumed.
6. Current practice of factored loads in ASCE 7-98 and strength reduction
factors in IBC for masonry walls translates into 1.5% probability of failure in
a 50-year life. On an aimual basis, the target probability of failure is 0.03%,
which is equivalent to a 3,200-year Mean Recurrence Interval (MRI).
7. Intermittent reinforcement (pilaster) spacing equal to wall height prevents
failure of walls (meet the target probability of failure) for design wind speeds
up to 120 mph, even when the building is located in an open terrain (terrain
exposure C) and it is partially enclosed.
A caveat for conclusion 5, states that increase in failure resistance needs to be verified
with physical experiments.
7.3 Recommendations for Future Research
This research is a step toward improving masonry constmction to make it more
resistant to severe winds. However, the subject of masonry constmction is diverse and
still needs tremendous research efforts in proportion to its importance as a constmction
material. Thus, further developments and research need to be done for the completeness
121
and better understanding of physical phenomena. In this section, some recommendations
for future research are presented.
1. Data from experimental programs are available for MOR of type S mortar,
this data is not extensive. It is recommended to design a testing program to
obtain the probability distributions for various mortar types. The probability
distributions along, with their parameters, are necessary to be obtained and
verified. This effort will help extend this research for more coverage and
generality.
2. During the course of assessing wind loads on low-rise masonry buildings, the
probability distribution and distribution parameters of the combined gust-
pressure coefficient for components and cladding (C&C) and the combined
gust-pressure coefficient for intemal pressure in partially enclosed buildings
were assumed. It is recommended that these assumptions be verified through
experimental measurements.
3. In this research, intermittent reinforcement is proposed to increase the lateral
strength of wall panels and to reduce the probability of failure to a target level
by changing the wall behavior to two-way action. It is believed that
intermittent reinforcement will also increase the wall ductility. Ductility is not
assessed in this research; therefore, it should be investigated in a separate
study.
4. This research establishes an analytical methodology to assess the probability
of failure of unreinforced masonry walls and shows how to reduce the
probability of failure by adding intermittent reinforcement at specified
spacing. It is required to verify the suggested spacing experimentally and to
verify the lateral strength increase when two-way action is utilized.
Experimental work is meant to verify the two-way action of intermittently
reinforced wall panels when small aspect ratios (less than 0.4) are targeted and
to verify the ratio between the ultimate lateral load carrying capacity of the
two-way wall to that of the one-way wall. Scale models may be used for this
purpose (1:3 scale models have been used successfully in similar cases).
122
5. The same research is required to be done for more cases. Among these cases
are masonry construction with various mortar types (mortar type N and mortar
type S made with masonry cement) and roofing types (heavy roofing). The
research needs also to be extended to cover the cases of walls with openings
(doors or windows) and the effect of different kinds of loading on the roof
(concentrated load on the roof diaphragm).
123
REFERENCES
Abboud, B., Hamid, A., and Harris, H. "Flexural Behavior of Reinforced Concrete Masonry Walls under Out-of-Plane Monotonic Loads." ACI Stmctural Joumal May-June 1996.
Abrams D. P. "Diagnosing Lateral Strength of Existing Concrete Masonry Buildings." Structural engineering in natural hazards mitigation. Proceedings, Stmctures Congress, ASCE, Irvine, California, 1993.
ACI, American Concrete Institute "Building Code Requirements for Masonry Structures," ACI 530-99/ASCE 5-99/TMS 402-99, 1999.
Al-Menyawi, Y., and Mehta, K. C. "Performance Of Unreinforced Masonry Walls During Wind Loading." Proceedings, ASCE Texas Section, Fall meeting. El Paso, Oct., 2000, 243-250.
Al-Menyawi, Y. M. and Mehta, K. C. "Reserve Strength and Failure Probabilities for Unreinforced Masonry Walls." Americas Conference on Wind Engineering, Clemson University, June 3-6, 2001.
Al-Menyawi, Y. M., Mehta, K. C, and Kiesling, E. W. "What is the Real MOR for Unreinforced Concrete Block Walls?." Proceedings, ASCE Texas Section, Spring meeting, San Antonio, Mar, 2001.
Amrhein, J. E. "1988 Provisions for Masonry Design." Seismic Engineering, Research and practice: Proceedings, Stmctures Congress, ASCE, 1989, 729-732.
Anderson, C. "Tensile Bond Tests With Concrete Blocks." Intemational Joumal of Masonry Constmction, Vol. 1 No. 4, 1980, 134-148.
Andreaus, U. and Paolo, A. D. "3-D Analysis of Masonry Columns with Grouted Reinforcement Bars." Brick and Block Masonry, University College of Dublin, Ireland, 1988, 1507-1518.
Antrobus, N. A., Leiva G., Men-yman, M. and Klingner, R. E. "In-Plane Behavior of Concrete Masonry Coupled Walls." Structural Design, Analysis and Testing, Proceedings, Stmctures Congress, ASCE, 1989, 969-978.
Arya, S. K. and Hegemier, G. A. "Finite Element method for Interface Problems." Joumal of the Stmctural Division, Vol. 108, no. 2, 1982, 327-342.
124
ASCE 7-98 "Minimum Loads on Building and Other Stmctures." American Society of Civil Engineers, 1999.
ASTM, C140-99b. "Standard Test Methods for Sampling and Testing Concrete Masonry Units and Related Units." American Society for Testing and Materials, Vol. 04-05, 2000.
ASTM, C270-97a. "Standard Specification for Mortar for Unit Masonry." American Society for Testing and Materials, Vol. 04-05, 2000.
ASTM, C90-97. "Standard Specification for Load bearing Concrete Masonry Units." American Society for Testing and Materials, Vol. 04-05, 2000.
Atkinson, R. H., Amadei, B. P., Saeb, S. and Sture, S. "Response of Masonry Bed Joints in Direct Shear." Joumal of Structural Engineering, Vol 115, No. 9, 1989, 2276-2296.
Baqi, A., Bhandari N. M. and Trikha, D. N. "Experimental Study of Prestressed Masonry Flexural Elements." Journal of Structural Engineering, Vol. 1, No. 3, 1999, 245-254.
Baumaim, H. U. "Ductile Masonry Construction In California." Worldwide Advances in Structural Concrete and Masonry, Proceedings of the CCMS Symposium held in Conjunction with Structures Congress XIV, ASCE, 1996, 93-100.
BSI, British Standards Institution. "Code of Practice for Use of Masonry, Part 1. Stmctural Use of Unreinforced Masonry." BS 5628, Part 1, London, 1992.
Candy, C. C. E. "The Energy Line Method for Masonry Panels Under Lateral Loading." Brick and Block Masonry, University College of Dublin, Ireland, 1988, 1158-1170.
Cement and Concrete Research Association. "Recent Developments in Yield-line Theory." Magazine of Concrete Research; Special Publications, London, 1965.
Colville, J. "Overview of Masonry Design and Construction." Restmcturing: America and Beyond, Proceedings, Stmctures Congress, ASCE, 1995, 1685-1688.
Committee on Natural Disasters/Commission on Sociotechnical Systems. "The Kalamazoo Tomado of May 13, 1980." National Academy Press, Washington, D.C., 1981.
125
Copeland, R. E., and Saxer, E. L. "Tests of Structural Bond of Masonry Mortars to Concrete Block." Joumal of the American Concrete Institute, November, 1964, 1411-1451.
CPP "Data Base of Peak Gust Wind Speeds," A final report prepared for Wind Engineering Research Center, Texas Tech University, by CPP, Inc., Fort Collins, Colorado, 2001.
Drysdale, R. G., Heidebrecht, A. C. and Hamid, A. A. "Tensile Strength of Concrete Masonry." Journal of the Stmctural Division, Vol. 105, No. 7, 1979, 1261-1276.
Drysdale, R., and Essawy, A. "Out-of-Plane Bending of Concrete Block Walls." Journal of Structural Engineering, Vol. 114, No. 1, 1988.
Drysdale, R., and Khattab M. "In-Plane Behavior of Grouted Concrete Masonry under Biaxial Tension-Compression." ACI Structural Journal, Nov.-Dec, 1995.
Edgeil, G. J. "The Use of Bed Joint Reinforcement to Enhance the Lateral Load Resistance of a Calcium Silicate Brickwork Wall." Brick and Block Masonry, University College of Dublin, Ireland, 1988, 617-630.
EUingwood, B. "Wind and Snow Load Statistics for probabilistic Design." Joumal of Stmctural Division, ASCE, Vol. 107, No. 7, 1981, 1345-1349.
EUingwood, B., and Tekie, P.B. "Wind Load Statistics for probability-Based Structural Design." NAHB Research Center, Upper Marlboro, MD, 1997.
EUingwood, B., Galambos, T., MacGregor, J., and Cornell, C. "Development of a Probability Based Load Criterion for American National Standard A58." NBS Special Publication 577, U.S. Department of Commerce, 1980.
EUingwood, B., Galambos, T., MacGregor, J., and Comdl, C. "Probability Based Load Criteria: Load Factors and Load Combinations." Journal of the Structural Division, ASCE, Vol. 108, No. 5, May, 1980, 978-997.
Essawy, A. S. "Strength of Hollow Concrete Block Masonry Walls Subject to Lateral (out-of-plane) Loading." PhD dissertation, McMaster University, Canada, November, 1986.
Essawy, A. S., Drysdale, R. G., and Mirza, F. F. "Nonlinear Macroscopic Finite Element Model For Masonry Walls." Proceedings, New Analysis Techniques for Stmctural Masonry, ASCE, Subhash C. Anand, ed., 1985, 19-45.
126
Ewing, R. D. and Kariotis, J. C. "Nonlinear Finite Element Analysis of Experiments on Two-Story Reinforced Masonry Shear Walls." Structural Design, Analysis and Testing, Proceedings, Structures Congress, ASCE, 1989, 979-988.
Ewing, R.D., Kariotis, J. C. and El-Mustafa, A. M. "Con-elation of Finite Element Analysis and Experiments on Reinforced Masonry Walls." Brick and Block Masonry, University College of Dublin, Ireland, 1988, 631-641.
FEMA, Building Performance Assessment Report: Midwest Tornadoes of May 3, 1999, Oklahoma and Kansas. Observations, Recommendations and Technical Guidance, Federal Emergency Management Agency, Aug. 1999.
FEMA, Building performance: Hurticane Andrew in Florida, Federal Emergency Management Agency and Federal Insurance Administration, Dec. 1992.
FEMA, Building performance: Hurricane Iniki in Hawaii, Federal Emergency Management Agency and Federal Insurance Administration, Jan. 1993.
Fishburn, Curus C. "Effect of Mortar Properties on Strength of Masonry." Monograph 36, National Bureau of Standards, 1961.
Fried, A., Anderson, C. and David, S. "Predicting the Transverse Lateral Strength of Masonry Walls." Brick and Block Masonry, University College of Dublin, Ireland, 1988, 1171-1194.
Fudge. C , Bright, N. and Ohler, A. "Flexural Strength of Masonry with Reference to a New Code of practice for Etirope." The Masonry Society Joumal, Volume 10, No. 1, August, 1991,69-74.
Gabriel, C. and Wattz, E. "How long will it last? Condition Assessment of the Building Envelope." Restmcturing: America and Beyond, Proceedings, Stmctures Congress, ASCE, 1995, 642-657.
Galambos, T., EUingwood, B., MacGregor, J., and Comell, C. "Probability Based Load Criteria: Assessment of Current Design Practice." Joumal of the structural division, ASCE, Vol. 108, No. 5, 1980, 959-977.
Gambarotta, L. and Lagomarsino, S. "A finite Element Damage Model for the Evaluation and Rehabilitation of Brick Masonry Shear Walls." Worldwide Advances in Stmctural Concrete and Masonry: Proceedings of the CCMS Symposium held in Conjunction with Structures Congress XIV, ASCE, 1996, 72-81.
127
Ganesan, T. P. and Ramamurthy, K. "Behavior of Concrete Hollow-Block Masonry Pnsms Under Axial Compression." Journal of Stmctural Engineering, Vol. 118, No.7, 1992, 1751-1769.
Ganesan, T. P. and Ramamurthy, K. "A Simplified Finite Element Method for Design of Hollow Block Masonry Walls." Brick and Block Masonry, University College of Dublin, Ireland, 1988, 1437-1446.
Gazzola, E. A., Drysdale, R. G. and Essawy, A. S. "Bending of Concrete Masonry Wallettes at Different Angles to Bed Joints." Proceedings of the third North American Masonry Conference, 1985, 27-1 - 27-14.
Gazzola, E., Bagnariol, D. Toneff, J., and Drysdale, R. "Influence of Mortar Materials on the Flexural Tensile Bond Strength of Block and Brick Masonry." ASTM STP 871, Masonry: Research, Application, and Problems, 1985.
Geoffrey W. B. "Performance of Commercial Masonry Stmctures in Hun-icane Andrew." Conference proceedings, ASCE, Hun-icanes of 1992: Lessons Learned and Implications for the Future, 1993, 456-464.
Ghali, A., Neville, A.M., Garas, F.K. and Virdi, K.S. (Editor) Stmctural Analysis : A Unified Classical and Matrix Approach. 4"' Edition, Routledge, E&FN SPON, 1998.
Ghosh, S. K. "Flexural Bond Strength of Masonry." Proceedings of the Fifth Canadian Masonry Symposium, Volume 2, University of British Columbia, 1989, 735-744.
Ghosh, S. K. "Flexural Bond Strength of Masonry: An Experimental Overview." Proceedings of the Fifth North American Conference, The Masonry Society, Volumell, 1990, 701-712.
Graber, D. W. "Concrete Masonry Design and Constmction Considerations in Florida." Conference proceedings, ASCE, Buildings, Donal R. Sherman, ed., 1987, 1-44.
Grimm, C. T. "Procedure for Standardizing Masonry Flexural Strength." Brick and Block Masonry, University College of Dublin, Ireland, 1988, 1040-1048.
Grimm, C. T. and Tucker, R. L. "Flexural Strength of Masonry Prisms Versus Wall Panels." Joumal of Stmctural Engineering, Vol. 111, No. 9, 1985, 2021-2032.
Hamid, A. A., and Drysdale, R. "Flexural Tensile Strength of Concrete Block Masonry." Journal of Stmctural Engineering, 1988, Vol. 114, No. 1.
128
Hamid, A. A., and Drysdale, R. "Proposed Failure Criteria for Concrete Block Masonry under Biaxial Stresses." Journal of tiie Structtiral Division, Vol. 107, No. 8, 1981, 1675-1687.
Hamid, A. A., Assis, G. F. and Han-is, H. G. "Towards Developing a Flexural Strength Design Methodology for Concrete Masonry." Masonry: Components to Assemblages: ASTM STP 1063, J. H. Matthys, Ed., 1990, 350-365.
Hamid, A. A., Elnawawy, O. A. and Chandrakeerthy, S. R. "Flexural Tensile Strength of Partially Grouted Concrete Masonry." Journal of Stmctural Engineering, Vol. 118, No. 12, 1992,3377-3393.
Hamid, A.A., Abboud, B.E., Farah, M. and Han-is, H.G. "Flexural Behavior of Vertically Spanned Reinforced Concrete Block Masonry Walls." Proceedings, Fifth Canadian Masonry Symposium, University of British Columbia, Vancouver, Canada, 1989,209-218.
Hamid, A.A., Harris, G.H. and Catherine, C. "Flexural Strength of Joint Reinforced Block Masonry Walls." Brick and Block Masonry, University College of Dublin, Ireland, 1988,653-664.
Headstrom, E.G., Tahrini, K. M., Thomas, R. D., Dubovoy, V. S., Klingner, R. E., and Cook, R. A. "Flexural Bond Strength of Concrete Masonry Prisms Using Portland Cement and Hydrated Lime Mortars." The Masonry Society Joumal, Vol. 9, No. 2,1991,8-23.
Huffmgton, H. J. "Theoretical Determination of Rigidity properties of Orthotropic Stiffened Plates." Joumal of Applied Mechanics, Vol. 23, 1956.
ICBO: Uniform Building Code, Intemational Conference of Building Officials, Whittier, CA. 1997.
ICC: Intemational Building Code, Intemational Code Council, VA, 2000.
Inga, A. T. "Wind Induced Damage on Single Story Non-Reinforced Concrete Masonry Walls." M.S. report, Department of Civil Engineering, Texas Tech University, Lubbock, TX, 1984.
Johansen, K. W. "Yield-line Theory." English Translation, Cement and Concrete Association, London, 1962.
Johnson, D. "Masonry Bond Strength." M.S. thesis. Department of Civil Engineering, Queensland Institute of Technology, 1983.
129
Ju, S. H. and Lin, M. C. "Comparison of Building Analysis Assuming Rigid or Flexible Floors." Journal of Structural Engineering, Vol. 1, No. 1, 1999, 25-31.
Kariotis, J. "Dynamic Testing of Unreinforced Masonry Walls for Loading Normal to the Wall Plane." Restmcturing: America and Beyond, Proceedings, Stmctures Congress, ASCE, 1995, 972-975.
Kriebel, D.L., Mehta, K.C., and Smith, D.A. "An Investigation of Load Factors for Flood and Combined Wind and Flood - Phase I Project Report." Submitted to American Society of Civil Engineers, Texas Tech University, Lubbock, Texas, 1996.
Kriebel, D.L., Mehta, K.C., and Smith, D.A. "An Investigation of Load Factors for Flood and Combined Wind and Flood - Phase II Project Report." Submitted to American Society of Civil Engineers, Texas Tech University, Lubbock, Texas, 1998.
Leland, K. B. "The Strength of Roof Uplift in Unreinforced Concrete Masonry." M.S. thesis. Department of Civil Engineering, Clemson University, 1988.
Lournco, P. B., Rots, J. G. and Blaauwendraad, J. "Continuum Model for Masonry: Parameter Estimation and Validation." ASCE, Joumal of Stmctural Engineering, Vol. 124,No. 6, 1998.
Mann, W. and Tonn, V. "The Load-Bearing Behavior of Biaxially Sparmed Masonry Walls Subjected Simultaneously to Horizontal and Vertical Loading." Brick and Block Masonry, University College of Dublin, Irdand, 1988, 1195-1203.
Manzouri, T. Shing, B. and Amadei, B. "Analysis of Masonry Structures with ElasticA^iscoplastic Models." Worldwide Advances in Structural Concrete and Masonry: Proceedings of the CCMS Symposium held in Conjunction with Stmctures Congress XIV, ASCE, 1996, 61-71.
Marquis, E. L., and Borcheh, J. G. "Bond Strength Comparison of Laboratory and Field Tests." Volume 1, Proceedings, Fourth Canadian Masonry Symposium, 1986, 94-106.
Matthys, J. H. "Concrete Masonry Prism and Wall Flexural Bond Strength Using Conventional Masonry Mortars." Masonry: Components to Assemblages: ASTM STP 1063, J. H. Matthys, Ed., American Society for Testing and Materials, Philadelphia, 1990, 350-365.
Matthys, J. H. "Flexural Bond Strengths of Portland Cement Lime and Masonry Cement Mortars." Brick and Block Masonry, University College of Dublin, Ireland, 1988, 284-291.
130
McAnulty, J. N. "Wind Load Factors For Atlantic And Gulf Coast Humcane Winds." M.S. Thesis, Department of Civil Engineering, Texas Tech University, Lubbock, TX, 1998.
McGinley, W. M. "Bond Wrench Testing-Calibration Procedures and Proposed Apparatus and Testing Procedures Modifications." The Sixth North American Masonry Conference, 1993, 159-172.
McGinley, W. M. "Flexural Bond Strength Testing - An Evaluation of Bond Wrench Testing Procedures." Masonry: Design and Constmction, Problems and Repair, ASTM STP 1180, 1993,213-227.
McGinley, W. M., Samblanet, P. J. and Subasic, C. A. "An Investigation of the Effects of Hurricane Opal on Masonry." A report by the the Masonry Society disaster Investigation Team, The Masonry Society, 1996.
Mehta, K. C. and Minor, J. E. "Wind Loading Mechanism on Masonry Constmction." The Masonry Society Joumal, 1986.
Mehta, K. C , Cheshire, R. H. and McDonald, J. R. "Wind Resistance Categorization of Buildings for Insurance." Joumal of Wind Engineering and Industrial Aerodynamics, 41-44, 1992, 2617-2628.
Melander, J. M., Ghosh, S. K., Dubovy, V. S., Hedstrom, E. G., and Klinger, R. E. "Flexural Bond Strength of Concrete Masonry Prisms Using Masonry Cement Mortars." Masonry: Design and Construction, Problems and Repair, ASTM STP 1180,1993, 152-164.
Miltenberger, M. A., Colville, J., and Wolde-Tinsae, A. M. "A Proposed Flexural Bond Strength Test Method." The Sixth North American Masonry Conference, 1993, 137-148.
Milton, J. S. and Arnold, J. C. "Introduction to Probability and Statistics: Principles and Applications for Engineering and Computing Sciences." McGraw-Hill, December, 1994.
Minor, J. E. "Investigation of Engineered Building Wind Failure." Stmctural Engineering in The 21" Century, Proceedings, Stmctures Congress, ASCE, 1999, 1011-1014.
NCMA TEK 3-3A "Reinforced Concrete Masonry Constmction." National Concrete Masonry Association, 1997.
131
NCMA TEK 14-1 "Section Properties of Concrete Masonry Walls." National Concrete Masonry Association, 1993.
NCMA TEK 14-3 "Designing Concrete Masonry Walls for Wind Loads." National Concrete Masonry Association, 1995.
NCMA TEK 14-lOA "Lateral Support of Concrete Masonry Walls." National Concrete Masonry Association, 1994.
NCMA "Research Evaluation of the Flexural Tensile Strength of Concrete Masonry." National Concrete Masonry Association, Research And Development Laboratory, Project No. 93-172, Order No. MR 10, 1994.
Neis, V. v., and Chow, D. Y. T. "Tensile Bond Testing of Stmctural Masonry Units." Proceedings, Second Canadian Masonry Symposium, Ottawa, Canada, 1980, 381-395.
Nelson, J. K. and Morgan J. R. "Hurricane Damage on Galveston's West Beach." Joumal of Stmctural Engineering, Vol. 111, No. 9, 1985, 1993-2007.
Nilson, A. H. Design of Concrete Stmctures. 12"' ed., the McGraw-Hill, Inc, New York, 1997.
Noland, J. L. "An Overview of the U.S. Coordinated Program for Masonry Building Research." Stmctural Design, Analysis and Testing, Proceedings, Stmctures Congress, ASCE, 1989, 949-958.
Okada, T., Hiraaishi, H. Kaminosono, T. and Teshigawara M. "Flexural Behavior of Reinforced Masonry Walls and Beams." Conference proceedings. Third Conference on Dynamic Response of Structures, Gary C. Hart and Richard B. Nelson, eds., ASCE, 1986, 481-488.
Page, A.W., Kleeman, P.W., and Manicka, D. "An In-plane Finite Element Model for Brick Masonry." New Analysis Techniques for Stmctural Masonry, Subhash C. Anand,ed., 1995, 1-17.
Phang, M. "Wind Damage Investigation of Low-Rise Buildings." Stmctural Engineering in The 21" Century, Proceedings, Stmctures Congress, 1999, 1015-1021.
Pinelli, J.-P., O'Neill, S. "Effect of tornadoes on residential masonry structures." Joumal of Wind and Structures, Vol. 3, No. 1, 2000, 23-40.
132
Pinelli, J.-P., O'Neill, S., Subramanian, C. S., and Leonard, M. "Effect of tomadoes on commercial stmctures." Proceedings, 10* International Conference on Wind Engineering, Copenhagen, Denmark, 1999, 1451-1456.
Porter, M. L. and Borcheh, J. G. "Update of Masonry Building Code and Specification, 1995 to 1998 Editions." Structural Engineering in The 21" Century, Proceedings, Structures Congress, 1999, 113-116.
Rodriguez, R., Hamid, A., and Lan-alde, J. "Flexural Behavior of Post-Tensioned Concrete Masonry Walls Subjected to Out-of-Plane Loads." ACI Sfructural Joumal, January-February, 1998.
Romano, F., Ganduscio S. and Zingone, G. "Cracked Nonlinear Masonry Stability under Vertical and Lateral Loads." Journal of Stmctural Engineering, Vol 119, No. 1, 1993,69-87.
Rosowsky, D. V. and Checg, N. "Reliability of Light-Frame Roofs in High-Wind Regions, I: Wind Loads." Joumal of Stmctural Engineering, Vol. 3, No. 7, 1999, 725-733.
Rosowsky, D. V. and Checg, N. "Reliability of Light-Frame Roofs in High-Wind Regions, II: Reliability Analysis." Journal of Stmctural Engineering, Vol. 3, No. 7,1999,734-739.
Saffir, H. S. "Low-Rise Building For Hurricane Wind Loads in South Florida." Joumal of the Wind Engineering and Industrial Aerodynamics, Vol 14, 1983, 79-90.
Saffir, H. S. "Upgrading Building Code Requirements in Hurricane-Prone Areas." Stmctural Engineering in The 21" Century, Proceedings, Stmctures Congress, ASCE, 1999, 1036-1042.
Sayed-Ahmed, E. Y. and Shrive, N. G. "Nonlinear Finite-Element model of Hollow Masonry." Joumal of Stmctural Engineering, Vol. 122, No. 6, 1996, 683-690.
Schubert, P., and Metzemacher, H. "On the Flexural Strength of Masonry." Masonry Intemational, Volume 6, No. 1, 1992, 21-28.
Shing, P. B. "Inelastic Performance of Single-Story Masonry Shear Wall." Stmctural Design, Analysis and Testing, Proceedings, Stmctures Congress , ASCE, 1989, 959-968.
Shoemaker, W. L. and Womak, A. S. "Masonry Walls Subjected to Wind-Induced Lateral Loads and Uplift." Journal of Wind Engineering and Industrial Aerodynamics, Vol. 36, 1990, 709-716.
133
Sparks, P. S., Liu, H. and Saffir H. S. "Wind Damage to Masonry Buildings." Journal of Aerospace Engineering, Vol. 2, No. 4, 1989, 186-198.
Thurlimann, B. and Guggisberg, R. "Failure Criterion for Laterally Loaded Masonry Walls: Experimental Investigations." Brick and Block Masonry, University College of Dublin, Ireland, 1988, 699-706.
Timoshenko S. and Woinowsky-Krieger S. Theory of Plates and Shells. (2e) McGraw-Hill, New York, 1959.
Ugural, A. C. Stresses in Plates and Shells. (2e) McGraw-Hill, New York, 1999.
Wacker, J. "Local Wind Pressures for Rectangular Buildings in Turbulent Boundary Layers." Proceedings, Wind Climate in Cities, J. E. Cermak et al. (eds), 1995, 185-207.
Zhuge, Y. Thambiratnam, D. and Corderoy, J. "Nonlinear Dynamic Analysis of Unreinforced Masonry." Journal of Structural Engineering, Vol. 124, No. 3, 1998,270-277.
ZoUo, R. F. "Hurricane Andrew: August 24, 1992 - Structural Performance of Buildings in Dade County, Florida." University of Miami, Coral Gables, Florida, 1993.
134
APPENDIX A
YIELD-LINE ANALYSIS
135
A.l General
Yield-line analysis is a method of plastic analysis. In this research it is used to
estimate the ultimate lateral load carrying capacity of unreinforced wall panels. To obtain
good estimates, the true ideal fracture pattem should be found. In this appendix, an
example of obtaining fracture pattems will be presented. In addition, moments
coefficients presented before in graphical form will be represented in Tabular form for
more convenience of use.
A.2 Obtaining Fracture Pattem
Reference is made to Figure 5.8, and case (a) is considered where the four sides of
the wall panel are simply supported. For pattem A, the energy dissipated along fracture
lines, work done by loads and moment as a function of fracture pattern geometry are
expressed by the following three equations:
U = 2M
W = q-a
a — + X
2 ^
a
M = q -a ~V2
'b'
3bx
\
2x 2A
a 2b X
(5.15-A)
(5.15-B)
(5.15-C)
The moment should be maximum along fracttire lines. Therefore, the moment expression
is to be differentiated and set equal to zero. An expression for the geometric parameter
(x) can be obtained by solving the resulting equation.
X can be expressed by the following equation:
X = (A.l)
136
For different values of aspect ratio (cp), X can be found and substittited in the moment
expression. Moment coefficients can be obtained therefore. It should be noted that
moment coefficients in this research are expressed for the horizontal dimension (b).
A. 3 Moment Coefficients
There are different cases for the edge conditions. In this research, only four cases
are considered. Figure A.l shows the different cases covered. Table A.l presents the
moment coefficient in Tabular form. To obtain the moment in the horizontal direction
(the strong direction), the following equation is used:
yM = k-q-b' (A.2)
Where y is the orthogonal ratio, M is the maximum moment the can be carried by the
weak direction of the wall (normal to bed joints), and q is the ultimate lateral load
carrying capacity. Figure A.l shows the different cases of edge conditions presented in
the table.
137
A\\ \ \ ' s \ \ ' \ \ \ \ \ \ \ \ \ \ \ ' v \ \ \ \ \ \ \ \ \ \ \ \V
.\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\' fWSfSWWVWW.V.WSWSWSMVWVVSiW.
sWWWWVW-.V'sV^W'sWWWWS.VWW"
(U
CO
<4-(
o CO
a
CI o O (U 01
w
•--s ^ s s s ^
^ s s \ > \ s \ \
//////
/
\ \ > ^ \ \ \ \ \ \ \ \ \ N
•^^
Mi lA
< u
c
and
fix
xt
A
R
O
D or
ted
on
= t
s:
^ tG Q
< ox
icwwwww<M)v<xw.»xwwvwv<\^
138
Jable A.1: Moment Coefficient (k) for Different Cases of Edge Condition ^ ^ p e c t Ratio (cp)
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
1.10
1.20
1.25
1.30
1.40
1.50
1.60
1.70
1.75
Case (a)
0.003
0.009
0.016
0.024
0.032
0.039
0.046
0.052
0.057
0.061
0.065
0.069
0.070
0.072
0.075
0.077
0.080
0.082
0.083
Case (b)
0.002
0.006
0.012
0.019
0.026
0.032
0.038
0.044
0.049
0.053
0.057
0.061
0.063
0.064
0.067
0.070
0.073
0.075
0.076
Case (c)
0.002
0.008
0.013
0.019
0.023
0.027
0.030
0.033
0.036
0.038
0.039
0.041
0.042
0.042
0.043
0.044
0.045
0.046
0.047
Case (d) V /
0.002
0.006
0.010
0.015
0.019
0.023
0.026
0.029
0.032
0.034
0.036
0.037
0.038
0.039
0.040
0.041
0.042
0.043
0.044
139
APPENDIX B
DESIGN OF PILASTERS
140
B.l General
Providing unreinforced masonry walls with pilasters or intermittent reinforcement
will significantly increase the wall ultimate lateral carrying capacity. If the walls can
safely support the load produced by the wind pressure, then, the entire load path should
be properly designed to ascertain overall structural integrity and adequacy. The purpose
of this appendix is to provide some guidelines for the design of Pilasters, since they are
will become the critical link in the overall masonry wall assembly. There are many cases
of edge conditions, which will affect the amount of the load transmitted to the pilaster.
Also, aspect ratio will affect the fracture pattem, which will affect the load transmitted to
the pilaster in return. This appendix serves as recommended guidelines for the design of
pilasters and does not cover all cases. It is the responsibility of the engineer to verify the
information outlined herein, check it against the current version of building codes and to
extend it to further cases.
B.2 Load Estimation on Pilasters
Load on the pilaster are generated by direct wind pressure on the wall, as well as
tension from the roof Tension from the roof will go directly to the wall.
Notwithstanding, this part of the load is small. Therefore, this analysis will be based on
the ultimate lateral carrying capacity of the wall panel. The following steps outline the
procedure of finding the straining actions in a pilaster.
• Knowing the panel edge conditions and the aspect ratio, the moment coefficient
(k) can be found. This is done by the virtue of Figures 6.10 through 6.11.
• Using ASCE 7-98 (Eq. 6-18), the lateral load (q) on the wall can be estimated. It
should be noted that this equation is for Components and Cladding (C&C).
• According to Figure B.l, and the load q, the bending moment in the pilaster can
be found. Figure B.l represents edge conditions for case (a) only. For other cases,
the wind load on the pilaster should be altered accordingly.
• Wind uplift on the roof is calculated on the pilaster.
• The pilaster takes a share from the dead load, dead load on the pilaster comprise
its own weight at the level of the maximum moment and roof dead load
141
transmitted to the pilaster. The latter part is calculated based on the roof load and
roof structural system.
PATERRN (A)
b + C
PATTERN (B)
Fig. B.l. Pilaster Load Tributary Area 142
B.3 Load Factors
Load factors shall be calculated according to ASCE 7-98 section 2.3.2 for strength
design, or according to the strictest of ACI 530 section 2.1.1.1 or ASCE 7-98 section
2.4.1 for allowable stress design. Different combinations should be checked and the one
that gives the most critical straining actions should be considered.
B.4 Design of Pilaster
Pilasters are meant to prevent the catastrophic collapse of wall panels. It is
important to design each pilaster to support its load safely. Pilasters are typically
collecting their loads from one surface. Therefore, pilasters are designed as components
and cladding (C&C) rather than main wind force resisting system (MWFRS).
Allowable stress design shall be done according to ACI 530-99. Strength design
shall be done according to the Intemational Building Code, ICC 2000. For each load
combination, the amount of reinforcement should be obtained and the most critical case
should be considered.
B.5 Detailing of PUaster
Standard details for reinforced masonry are recommended here. The design
engineer should consider the fact that wind can blow from any direction, and therefore,
reinforcement should be provided in a way that can provide strength to the section
irrespective the wind direction.
143
