Static Resistance Functions for Unbonded Retrofit Unreinforced Concrete Masonry 
Walls with Plate Connection for Blast Design Applications 
 
by 
 
Eric Gagnet 
 
 
 
 
A thesis submitted to the Graduate Faculty of 
Auburn University 
in partial fulfillment of the 
requirements for the Degree of 
Master of Science 
 
Auburn, Alabama 
August 3, 2013 
 
 
 
 
 
 
 
 
 
 
Approved by 
 
James S. Davidson, Chair, Professor 
Robert W. Barnes, James J. Mallett Associate Professor 
Justin D. Marshall, Assistant Professor 
 
 
  
ii 
 
Abstract 
 Over the past two decades the use of explosives to attack both commercial and 
government buildings has increased. A substantial amount of research has been 
conducted to be able to protect not only the occupants but the structure also. New 
construction practices have been developed to mitigate the damages, but retrofit of 
existing buildings does not have a single answer. The resistance functions of all the 
components of the wall including a retrofit system are needed to be able to model an 
accurate simulation to ensure safe design of the retrofit.  
 For design purposes the amount of energy dissipation is important; however, the 
design codes, such as UFC 3-340-02, do not use energy dissipation as a criterion for safe 
design. Rather deflection and rotation limits are set for different levels of protection. In 
order to determine if a wall meets these limits, a dynamic analysis is performed to 
determine the maximum deflection a wall will exhibit during a loading. Typically a 
single-degree-of-freedom (SDOF) model is used to solve for the midspan deflection. This 
simplified method requires a definition of the relationship between resistance and 
deflection.  
Unreinforced, ungrouted masonry (URM) walls are commonly found throughout 
the world. Masonry buildings have been around for thousands of years and comprise over 
70% of buildings worldwide. Because of their popularity and susceptibility to 
fragmentation, it is imperative to ensure masonry walls are capable of withstanding a 
iii 
blast load. The two main categories of URM walls examined are non-arching walls and  
arching walls. Several methods have been developed to simulate how these walls will 
resist a lateral load and an in depth comparison is presented. 
 Many methods have also been developed for the resistance definition of an 
unbonded retrofit system. These methods have also been heavily examined. An analytical 
method was developed to incorporate the effects of the retrofit connection on the 
resistance function. This method was validated using finite element modeling. Many 
different geometric models were used to assess the validity of the algorithm.  
Lastly, all of the components of the walls resistance definitions were combined to 
form the total resistance of the wall. This resistance was used in a single-degree-of-
freedom model to generate a retrofit system for a specific design example.  
 
  
iv 
 
Table of Contents 
Abstract ............................................................................................................................... ii 
Table of Contents ............................................................................................................... iv 
List of Figures .................................................................................................................... vi 
List of Tables ..................................................................................................................... xi 
Notation............................................................................................................................. xii 
Chapter 1: Introduction ....................................................................................................... 1 
1.1 Background ............................................................................................................... 1 
1.2 Objective ................................................................................................................... 5 
1.3 Methodology and Scope ........................................................................................... 5 
Chapter 2: Literature Review .............................................................................................. 7 
2.1 Overview ................................................................................................................... 7 
2.2 Masonry Wall............................................................................................................ 8 
2.2.1 Non-Arching walls ........................................................................................... 10 
2.2.2 Arching Wall Resistance Functions ................................................................. 15 
2.3 Unbonded Retrofit System ...................................................................................... 19 
2.4 Unbonded Retrofit System Connection .................................................................. 23 
2.5 Combination of Systems ......................................................................................... 28 
2.6 Material Selection ................................................................................................... 30 
2.7 Response Limits ...................................................................................................... 31 
Chapter 3: Review of Design Resistance Functions for Unreinforced Masonry .............. 36 
3.1 Overview ................................................................................................................. 36 
3.2 Non-Arching Masonry Resistance Functions ......................................................... 36 
3.2.1 Wiehle Resistance Function ............................................................................. 37 
3.2.2 Moradi Method ................................................................................................ 39 
3.2.4 Unified Facilities Criteria (UFC) 3-340-02 Methodology ............................... 44 
3.2.5 Comparison of Non-Arching Resistance Functions ........................................ 46 
3.3 Arching Masonry Resistance Functions ................................................................. 64 
3.3.1 Wiehle Arching Resistance Curve ................................................................... 64 
v 
3.3.2 Moradi Arching Resistance Curve ................................................................... 68 
3.3.3 SBEDS Arching Resistance Curve .................................................................. 74 
3.3.4 Comparison of Arching Resistance Functions ................................................. 76 
3.4 SDOF Implications ................................................................................................. 88 
3.5 Dynamic Testing ..................................................................................................... 92 
3.5.1 Testing Overview ............................................................................................. 92 
3.5.2 Dynamic Test Results ...................................................................................... 94 
3.6 Conclusion ............................................................................................................ 101 
Chapter 4: Comparison of Unbonded Membrane System Analytical Models ............... 102 
4.1 Overview ............................................................................................................... 102 
4.2 Cable Analogy Method ......................................................................................... 103 
4.3 Seide Resistance Function .................................................................................... 107 
4.4 Parabolic Deflection Method ................................................................................ 108 
4.5 Stress Strain Conversion ....................................................................................... 112 
4.6 Comparison of Resistance Functions .................................................................... 115 
4.7 Finite Element Validation of Membrane Methods ............................................... 121 
4.7.1 Linear Material Definition Validation ........................................................... 121 
4.7.2 Nonlinear Material Definition Validation ...................................................... 124 
Chapter 5: Resistance Function Incorporating Connections ........................................... 128 
5.1 Overview ............................................................................................................... 128 
5.2 Unbonded Retrofit Connection Resistance Function Development ..................... 128 
5.3 Finite Element Validation of the Plate Algorithm ................................................ 137 
Chapter 6: Dynamic Modeling and Comparison ............................................................ 148 
6.1 Overview ............................................................................................................... 148 
6.2 Single-Degree-of-Freedom Development ............................................................. 148 
6.3 Blast Loading ........................................................................................................ 154 
6.5 Dynamic Testing ................................................................................................... 155 
6.5.1 Test 1 Using Polypropylene ........................................................................... 155 
6.5.2 Test 2 using PVC Membrane ......................................................................... 159 
Chapter 7: Design Examples ........................................................................................... 164 
7.1 Overview ............................................................................................................... 164 
7.2 Problem Statement ................................................................................................ 164 
7.3 Wall #1 .................................................................................................................. 167 
7.4 Wall #2 .................................................................................................................. 170 
vi 
Chapter 8: Conclusion..................................................................................................... 172 
8.1 Masonry Walls ...................................................................................................... 172 
8.2 Membrane Resistance Functions .......................................................................... 173 
8.3 Full Resistance Curve Including Plate Connections Algorithm ........................... 175 
8.4 Single-Degree of Freedom Analysis ..................................................................... 176 
References ....................................................................................................................... 177 
Appendix A ..................................................................................................................... 182 
Appendix B ..................................................................................................................... 189 
Appendix C ..................................................................................................................... 196 
Appendix D ..................................................................................................................... 201 
 
  
vii 
 
List of Figures 
Figure 2.1 Idealized Resistance Curves ............................................................................ 11 
Figure 2.2 Compressive Resultant Force (Paulay and Priestley 1992) ............................. 12 
Figure 2.3 Idealized Arching Wall Forces ........................................................................ 15 
Figure 2.4 Arching Mechanism for Gapped Wall ............................................................ 17 
Figure 2.5 Membrane Retrofit System Drawing ............................................................... 20 
Figure 2.6 Installation of Membrane Retrofit System ...................................................... 20 
Figure 2.7 Free Body Diagram of Cable Analogy ............................................................ 21 
Figure 2.8 Suggested Catcher System Design (Department of the Air Force 2000) ........ 23 
Figure 2.9: Tensile Test using Angles for Connection ..................................................... 26 
Figure 2.10: Idealized FBD of Mechanical Connections (Hoemann et al. 2010) ............. 28 
Figure 2.11: Idealized Resistance Curve Combination ..................................................... 29 
Figure 6.4: Component Damage Levels (PDC-TR 06-08 2008) ...................................... 34 
Figure 3.1: Decaying Phases Assumed by Wiehle et al. (1969) ....................................... 38 
Figure 3.2: Free Body Diagram of Non-Arching Wall (Moradi et al. 2010) .................... 40 
Figure 3.3: Moradi Method for a Non-Arching Resistance Curve Flow Chart ................ 42 
Figure 3.4: Idealized Resistance-Deflection Curve for SBEDS ....................................... 44 
Figure 3.5: UFC Simply Supported Wall ......................................................................... 45 
Figure 3.6: Comparison of Non-Arching Methods ........................................................... 47 
Figure 3.7: Wiehle Non-Arching Variable Wall Thickness ............................................. 49 
Figure 3.8: SBEDS Non-Arching Variable Wall Thickness ............................................ 50 
viii 
Figure 3.9: Moradi Non-Arching Variable Wall Thickness ............................................. 51 
Figure 3.10: Wiehle Non-Arching Variable Brick Type .................................................. 53 
Figure 3.11: SBEDS Non-Arching Variable Brick Type ................................................. 54 
Figure 3.12: Moradi Non-Arching Variable Brick Type .................................................. 56 
Figure 3.13: Wiehle Non-Arching Variable Axial Loads ................................................. 57 
Figure 3.14: SBEDS Non-Arching Variable Axial Loads ................................................ 58 
Figure 3.15: Moradi Non-Arching Variable Axial Loads ................................................ 59 
Figure 3.16: Comparison of All Non-Arching Walls ....................................................... 60 
Figure 3.17: Deflection at Peak Pressure for Non-Arching Walls .................................... 62 
Figure 3.18: Energy Dissipation for Non-Arching Walls ................................................. 63 
Figure 3.19: UFC (DoD 2008) Gapped Wall Schematic .................................................. 66 
Figure 3.20: UFC Gapped Wall Comparison ................................................................... 68 
Figure 3.21: FBD of Moradi Arching Wall (Moradi et al. 2010) ..................................... 69 
Figure 3.22: Shortening of URM Wall ............................................................................. 71 
Figure 3.23: Flow Chart of Moradi Arching Resistance Curve Development ................. 73 
Figure 3.24: Wiehle Arching Variable Wall Thickness.................................................... 77 
Figure 3.25: SBEDS Arching Variable Wall Thickness ................................................... 78 
Figure 3.26: Moradi Arching Variable Wall Thickness ................................................... 79 
Figure 3.27: Wiehle Arching Variable Brick Type .......................................................... 81 
Figure 3.28: SBEDS Arching Variable Brick Type .......................................................... 82 
Figure 3.29: Moradi Arching Variable Brick Type .......................................................... 83 
Figure 3.30: UFC Arching Variable Gap Width ............................................................... 84 
Figure 3.31: SBEDS Arching Variable Gap Width .......................................................... 85 
ix 
Figure 3.32: Arching Ultimate Resistance Comparison ................................................... 87 
Figure 3.33: Energy Dissipation for Arching Methods .................................................... 88 
Figure 3.34: Loading used in SDOF ................................................................................. 89 
Figure 3.35: Non-Arching Dynamic Response ................................................................. 90 
Figure 3.36Arching Dynamic Response ........................................................................... 91 
Figure 3.37: Schematic of Boundary Conditions .............................................................. 93 
Figure 3.38: Pressure on Wall 2 ........................................................................................ 94 
Figure 3.39: Support Cracking of Wall 1.......................................................................... 95 
Figure 3.40: Wall 1 Dynamic Response  .......................................................................... 96 
Figure 3.41: Wall 2 Dynamic Response ........................................................................... 97 
Figure 3.42: Wall 3 Dynamic Response ........................................................................... 98 
Figure 3.43 Boundry Condition of Test Wall 4 ................................................................ 99 
Figure 3.44: Wall 4 Dynamic Response ........................................................................... 99 
Figure 3.45: Wall 4 Pullout of the Frame ....................................................................... 100 
Figure 4.1: Cable Resistance Function Development ..................................................... 105 
Figure 4.2: Sample Calculation for Cable Method ......................................................... 106 
Figure 4.3: Flowchart for Seide Resistance Curve ......................................................... 108 
Figure 4.4: Free-Body Diagram of Deformed Membrane .............................................. 111 
Figure 4.5: Flow Chart of Lane?s Resistance Curve ....................................................... 112 
Figure 4.6: Stress Conversion Comparison .................................................................... 115 
Figure 4.7: True Stress-Strain Curve for Polypropylene ................................................ 116 
Figure 4.8: Comparison of Membrane Resistance Curves ............................................. 117 
Figure 4.9: Steel Membrane Resistance Curve Comparison .......................................... 118 
x 
Figure 4.10: Membrane Resistance Curve for 6-ft Wall ................................................. 119 
Figure 4.11: Complete Comparison of Resistance Functions with Changing Variables 120 
Figure 4.12: Membrane Mesh ......................................................................................... 122 
Figure 4.13: Linear Material Resistance Curves ............................................................. 123 
Figure 4.14: In-Plane Stresses of Membrane at 60 psi ................................................... 124 
Figure 4.15: Stress Strain for Nonlinear Material Comparison ...................................... 125 
Figure 4.16: Nonlinear Material Comparison ................................................................. 126 
Figure 4.17: Principal In-Plane Stresses for Nonlinear Model ....................................... 127 
Figure 5.1: Plate Response .............................................................................................. 129 
Figure 5.2: Rotation of the Plate for Elastic and Plastic Phases ..................................... 130 
Figure 5.3: Angle Detail of Plate and Membrane ........................................................... 132 
Figure 5.4: Routine to Determine Deflection with Slack in Membrane ......................... 135 
Figure 5.5: True Stress True Strain of Membrane .......................................................... 136 
Figure 5.6: Effect of Plate on membrane Resistance ...................................................... 137 
Figure 5.7: Mesh and Geometry of FE Model for Full Retrofit System......................... 139 
Figure 5.8: Retrofit Configuration 1 for FEM ................................................................ 141 
Figure 5.9: Retrofit Configuration 2 for FEM ................................................................ 142 
Figure 5.10: Retrofit Configuration 3 for FEM .............................................................. 143 
Figure 5.11Bearing Failure of Membrane ...................................................................... 144 
Figure 5.12: Retrofit Configuration 4 for FEM .............................................................. 144 
Figure 5.13: Retrofit Configuration 5 for FEM .............................................................. 145 
Figure 5.14: Retrofit Configuration 6 for FEM .............................................................. 146 
Figure 6.1: SDOF Theory (a) Real System (b) Simplified Model .................................. 149 
xi 
Figure 6.2: Deflected Shapes .......................................................................................... 150 
Figure 6.3: Loading Diagram for Free Air Blast ............................................................ 155 
Figure 6.5: (a) CMU Test Block (Johnson 2009) (b) CMU Dimensions ....................... 156 
Figure 6.6: (a) Interior of Test Wall (b) Exterior of Test Wall (Johnson 2009) ............. 156 
Figure 6.7: Wall Pressure from Test ............................................................................... 157 
Figure 6.8: Post Test Wall (Johnson 2009) ..................................................................... 158 
Figure 6.9: Dynamic Response Comparison .................................................................. 158 
Figure 6.10: True Stress-Strain Plot for PVC Liner ....................................................... 159 
Figure 6.11: Plate Connection for Test 2 ........................................................................ 160 
Figure 6.12: Pretest Constructed Wall (a) Interior (b) Exterior ...................................... 160 
Figure 6.13: Normalized Loading for Test 2 .................................................................. 161 
Figure 6.14: Test 2 Deflection Comparison .................................................................... 161 
Figure 6.15: Post Test Exterior View ............................................................................. 162 
Figure 6.16: Post Test Fragment Pressing Against Membrane ....................................... 163 
Figure 7.1: Isometric View of Building Identifying Walls ............................................. 165 
Figure 7.2: Elevation View of Building .......................................................................... 165 
Figure 7.3: Design Loading ............................................................................................ 166 
Figure 7.4: Membrane True Stress/Strain ....................................................................... 166 
Figure 7.5: Dynamic Results for Masonry Alone ........................................................... 167 
Figure 7.6: Total Resistance of System for Retrofit Trial 1 ........................................... 168 
Figure 7.7: Response of Wall #2 without Retrofit .......................................................... 170 
Figure 7.8: Response Comparison of Wall #2 ................................................................ 171 
xii 
List of Tables 
Table 2.1 Modulus of Rupture for Masonry, psi (kPa) (ACI 530-08) .............................. 10 
Table 2.2: Structural Damage Associated with Building Levels of Protection (U.S Army 
Corps of Engineers 2008) ................................................................................................. 32 
Table 2.3: Component Descriptions (U.S. Army Corps of Engineers 2008) ................... 33 
Table 2.4: Response Limits for Unreinforced Masonry ................................................... 35 
Table 3.1: Masonry Properties of Three CMU Blocks ..................................................... 48 
Table 3.2: Material Properties of Varying Blocks ............................................................ 52 
Table 3.3: Wiehle Method Summary of Deflections at Peak Resitance for Non-Arching 
Masonry with Varying Brick Types ................................................................................. 54 
Table 3.4: SBEDS Method Summary of Deflections at Peak Resistance for Non-Arching 
Masonry with Varying Brick Types ................................................................................. 55 
Table 3.5: Input Parameters for Wall Thickness Comparison of Arching Methods ........ 77 
Table 3.6: Input Parameters for Brick Comparison for Arching Methods ....................... 80 
Table 3.7: Response Criteria for Masonry Blast Design (Oswald and Zehrt 2010) ......... 89 
Table 3.8: Wall Response Comparisons ......................................................................... 100 
Table 5.1: Retrofit Configuration ................................................................................... 140 
Table 6.1: Equivalent Transformation Factors for One-Way Beams and Slabs ............. 152 
Table 7.1: Wall Properties .............................................................................................. 167 
  
xiii 
 
Notation 
a = length over which arching force acts 
A = Cross sectional area 
b = Unsupported length of membrane 
B = distance from support to yield location divided by span 
length = 0.5 
c = Distance from centroid to most extreme fiber 
C1 = Compression force per unit width in shell 
C2 = Compression force per unit width in web 
CMU = Concrete masonry unit 
E = Modulus of elasticity of retrofit 
Ec = Concrete masonry modulus of elasticity  
FBD = Free body diagram 
fcr = Cracking stress of masonry 
f'm = Crushing stress of masonry 
fr = Modulus of rupture 
gmax = Maximum allowable gap between wall and support 
H = Height of wall 
H = Horizontal reaction 
h' = Distance between supports 
Hbottom = Horizontal reaction at the bottom of the wall 
HT = Horizontal reaction at the top of the wall 
I = Moment of inertia for membrane 
Ig = Gross (uncracked) moment of inertia 
j = Constant 
k = Solid ratio of masonry through webs (k = 0 assumed for 
horizontal span arching if not fully grouted) 
Ke = Elastic Stiffness 
L = Length of membrane 
L' = Arch length of membrane 
Ld = Original length of diagonal for half wall 
Lp = Length of plate from hinge to edge 
M = Moment caused by lateral pressure 
P = Lateral pressure 
xiv 
P = Axial Load 
Q1 = Maximum allowable flexural resistance 
Q2 = Pressure of decaying wall 
Qy = Yield load 
R = Vertical reaction  
r1 = Resistance due to flexure 
r2 = Decaying resistance 
Rm = Moment arm of arching masonry wall 
SBEDS = Single-degree-of-freedom Blast Effects Design Spreadsheet 
SFR = Solid fraction ratio 
T = Thickness of wall 
T = Tension force at ends of retrofit 
tp = Thickness of plate 
Ts = Thickness of membrane 
tms = Masonry shell thickness 
Tv = Vertical component of membrane axial force 
UFC = Unified facilities criteria 
URM = Unreinforced masonry 
Vtop = Arching force at top of the wall 
Wi = Self-weight of wall 
? = Support fixity 
? = Curvature ratio 
? = Incremental displacement for retrofit 
? = Shortening of wall 
? = Lateral deflection at mid-span 
?1 = Deflection at Q1 
?2 = Deflection at Q2 
c = Distance midspan moves while closing gap 
?cr = Deflection at mid-span at cracking of CMU wall 
?crG = Deflection at mid-span at crack growth 
?g = Gap distance between wall and support 
?max elastic = Maximum elastic mid-span deflection of CMU wall 
?st = Deflection at the point of snap though  
? = Strain 
?' = Compression strain due to arching forces and support movement 
strains 
?d = Strain of diagonal in CMU at failure 
? = Angle of membrane at ends 
? = Angle of masonry wall 
? = Change in rotation of plate 
xv 
? = Poisson's ratio 
? = Stress in extreme fiber 
?t = Ultimate dynamic tensile stress of masonry 
?y = Stress in extreme fiber at yielding 
? = Angle of rotation of plate 
? = Additional displacement added by rotation of plate 
1 
Chapter 1: Introduction 
1.1 Background 
Terrorist attacks have increased in frequency over the years. In a study conducted 
by the National Research Council (NRC), explosions account for 46 percent of all 
terrorist attacks worldwide. This number is substantially higher within the United States, 
77 percent (NRC 1995). Explosives are cheap and easy to make but can cause significant 
damage to both the structure and the occupants.  
 In 1993, the New York World Trade Center was one of the first major U.S. 
buildings to come under attack. A Terrorist drove a 1500 lb explosive into the basement 
(Terrorism 2004). When detonated, a 100-ft crater was created killing six people and 
injuring over a 1000 (FBI 2008). The explosive was constructed using readily available 
materials at a cost of less than $400 (NRC 1995).  At that time, this attack was the largest 
terrorist attack using explosives on American soil. 
 On April 19th 1995, the Alfred P. Murrah Federal Building came under attack 
from a homegrown terrorist. A rented trunk placed next to the building contained the 
explosive. Upon detonation, 168 people were killed and over five hundred were wounded 
(Terrorism 2004). Damage from the attack is shown in Figure 1.1. 
 
2 
 
Figure 1.1 Structural Damage of the 1995 Oklahoma Building Bombing (AP Photo, 
Newman 2013) 
 In 1998, two US embassies were attacked simultaneously; one in Kenya, the other 
in Tanzania. Again the explosives were thought to be stored inside of a truck parked near 
the buildings. The explosion in Kenya killed 11 while injuring 86. The Tanzania 
explosion killed 213 people, 12 of which were Americans, and injured over 5000 
(Terrorism 2004). 
 The use of explosives can be attributed to several factors: to obtain publicity, 
political or symbolic statements, destroy assets, religious imperative, injure and kill 
occupants, and lastly for revenge. For these reasons, commercial and military structures 
are the two main targets (NRC 1995). Due to these attacks the military has implemented 
strict security measures in attempts to prevent explosives from entering the building. As a 
result, exterior elements are more prone to attacks and increase the need to be able to 
resist blast loads. Government facilities may require blast loading as a part of design 
3 
loads whereas most of the commercial market does not require a building be designed to 
resist an explosion.  
 Blast loading is dynamic in nature. It is commonly compared to earthquake 
loading, which is incorrect (NRC 1995). An earthquake has a low intensity and a long 
duration, whereas a blast typically has a high intensity over a very short duration (NRC 
1995). For this reason new techniques need to be developed to mitigate damage.  
 Unreinforced, ungrouted masonry (URM) walls are commonly found throughout 
the world. Masonry buildings have been around for thousands of years and comprise over 
70% of the buildings worldwide (The Mason Contractors Association of America, n.d.). 
Today buildings with a high risk of being subjected to a blast loading have been designed 
to withstand external detonation. However, many existing buildings were not designed to 
resist blast loading. Davidson et al. (2005) state that most injuries sustained during a blast 
are not directly caused by the detonation: pressure, heat or container fragmentation. 
Rather most injuries result from secondary fragmentation resulting in blunt trauma or 
penetration resulting from the disintegration of the CMU and the windows being 
propelled at high velocity. Because of secondary fragmentation it is important to ensure 
that these elements are able to resist the loading without producing hazardous debris.  
Since Wiehle and co-researchers at The Stanford Research Institute started 
researching how to develop a static resistance function for URM walls subjected to a 
lateral load in 1969, many different theories and methodologies have been developed. 
Due to the brittle nature of URM walls, laboratory testing is very difficult and unreliable, 
especially during the decay phase of the resistance function. For these reasons, many 
4 
have developed their own resistance functions based on differing assumptions. These 
assumptions can change the resistance curve significantly.  
 The three types of systems to be considered in this report are: 1) non-arching 
walls 2) arching walls 3) unbonded retrofit. The first two systems involve URM walls 
while the third system describes how an unbonded membrane will resist an out-of-plane 
load. The first URM wall system will describe how a simply supported wall, where the 
supports do not resist rotation, will respond to the lateral load, while the second details 
how a wall with an arching force will react to the lateral load.  
Since 1995, a great deal of research and testing has been conducted by the Air 
Force Research Laboratory (AFRL) at Tyndall Air Force Base, Florida to develop 
inexpensive and practical methods to strengthen URM walls against blast (Davidson et al. 
2004). An important factor that is considered is the ability for the retrofit system to be 
implemented worldwide where limited equipment is available during the installation. For 
this reason AFRL, along with other agencies, have explored the use of fiber reinforced 
composites, aramid composite fabrics, etc.  
 For unbonded membrane many different methods have been developed to 
generate a resistance definition. These methods are examined in the same manner as the 
masonry methods. Several laboratory tests have been conducted to examine how the 
retrofit connections will affect the resistance curve, but no analytical method has been 
developed to describe this effect. 
5 
1.2 Objective 
 The overall objective of the research represented by this report was to develop a 
resistance function to be used in SDOF-based blast analyses of membrane retrofitted 
unreinforced masonry walls.   
1.3 Methodology and Scope 
 The following tasks were required to achieve that broader goal stated above: 1) 
assess resistance definitions that are currently used for blast analyses of non-arching and 
arching unreinforced masonry walls, compare them to experimental data, and determine 
which is the most accurate; 2) evaluate the different membrane resistance definitions and 
validate them using finite element software; and 3) develop a method to incorporate the 
connection flexibility into the membrane resistance function used for the analyses of 
unbonded retrofit masonry walls subjected to blast loading and validate the method using 
full-scale explosion test results.  
 Over the years several different methods have been developed to form a resistance 
curve for both the wall and retrofit system. It is important to know how each of these 
methods was developed and what assumptions were used to develop these methods. 
Finally with each of these components, which method should be used for design 
purposes?  
 Research was conducted to find valid resistance functions for various types of 
URM walls. For each method the original source was reviewed to validate the reasoning 
and ensure it should be used. This process was also used for the membrane resistance 
methods.  
6 
 The connection algorithm was developed to mimic the response of the retrofit 
against previous lab tests and finite element models. The algorithms were developed 
using equilibrium, compatibility, and the stress-strain curve. The assumptions made to 
generate this algorithm produce a slightly conservative resistance curve by generating a 
slightly lower resistance. For design purposes this is acceptable.  
 The resistance of the wall system is a critical input component of single-degree-
of-freedom models used for blast design. For design of retrofit systems, it is important to 
determine the peak deflection of the system. This deflection corresponds to a wall 
rotation that is compared to response limits. These response limits determine the level of 
protection that is provided.  
 
  
7 
 
Chapter 2:  Literature Review 
2.1 Overview 
Over the past several years much research has been done in order to evaluate 
methods of retrofitting unreinforced concrete masonry (URM) walls. Many older 
buildings are not reinforced against lateral loads. Due to the brittle nature of concrete 
masonry there is little resistance when an out-of-plane load is applied. Buildings being 
constructed for a high blast resistance typically use grouted CMU with steel 
reinforcement. It is both expensive and impractical to demolish and rebuild the wall. A 
practical solution is to retrofit the existing wall with a membrane.  Many different 
materials can be applied as a backing to the CMU wall to add resistance to a blast load.  
As a blast wave propagates through a wall it encounters several forms of resistance that 
will help provide strength, ductility and dissipate the energy to reduce the lethality of a 
blast. Every part of the wall and retrofit system should be considered to ensure an 
accurate and proper design.  
According to Knox et al. (2000) the overall objective of a blast analysis is to 
determine the maximum deflection that would be acceptable to ensure minimal injury or 
death to the occupants. The use of a single-degree-of-freedom system simplifies the 
analysis as a mass on a spring with a defined resistance. Analytical formulas have been 
developed to formulate the resistance of the concrete masonry unit (CMU) wall and the 
unbonded retrofit. With some assumptions, Knox et al. (2000) stated that the overall 
8 
resistance of the wall is the sum of the resistance of the individual components at the 
corresponding displacement.  
Many analytical models have been developed to describe the resistance of the 
CMU and the unbonded retrofit. The way the retrofit is installed has been studied 
extensively and determined that the connection can increase the ductility of the system 
(Hoemann et al. 2010; Johnson et al. 2009).  
2.2 Masonry Wall  
 CMU walls can resist lateral loading in one of two ways. If the wall supports 
cannot resist rotations, the only way the wall will resist the load is through flexural 
resistance (Drysdale et al. 1994). If the supports are rigid the wall will have a 
compressive force to resist rotation of the wall. When the wall is bent due to the lateral 
loads the tension face will elongate. If the supports are rigid this elongation is met with a 
compression force at the supports. This type of interaction is referred to as an arching 
boundary condition. The two types of walls are separated based on the support conditions 
and will be classified as either arching walls or non-arching walls.   
 Buchan and Chen (2007) looked at 40 CMU walls subjected to explosions 
reported in 6 different papers. The tests included both URM walls and partially grouted, 
lightly reinforced walls. Of those categories some were retrofit and others were not. The 
retrofit under consideration was a fiber reinforced polymer membrane. It was concluded 
that the retrofitted walls resisted a peak pressure 50% greater than the non-retrofit walls. 
Along with an increase in peak pressure, the retrofit walls had a reduction in debris 
scatter compared to the non-retrofit walls.  
9 
 For all masonry walls, ACI 530-08 outlines the design criteria that must be 
satisfied for strength design of ungrouted unreinforced masonry walls subjected to both 
moments and an axial load. The four design assumptions used in ACI 530-08 are: 
a) Strength design of masonry members shall be in accordance with principles of 
engineering mechanics 
b) Strain in masonry shall be directly proportional to the distance from the neutral 
axis 
c) Flexural tension in masonry shall be assumed to be directly proportional to the 
strain 
d) Flexural compressive stress in combination with compressive stress shall be 
assumed to be directly proportional to strain 
Using these assumptions the strength of the masonry can be determined so that the 
compressive stress in the masonry does not exceed 0.80 'mf  and the tensile stress in the 
masonry does not exceed the modulus of rupture for masonry. Design values for the 
modulus of rupture for masonry is given in Table 2.1.  
10 
Table 2.1 Modulus of Rupture for Masonry, psi (kPa) (ACI 530-08) 
 
2.2.1 Non-Arching walls 
 Wiehle et al. (1969) was one of the first to outline the process of developing a 
static resistance function for masonry walls. Wiehle et al (1969) assumed the resistance 
can be idealized in five regions: elastic, elastic-plastic, strain hardening, plastic, and 
decaying. An idealized representation of the resistance functions is shown in Figure 2.1. 
11 
 
Figure 2.1 Idealized Resistance Curves 
The resistance definition used by Wiehle et al. (1969) considered the wall as only 
having two phases: an elastic and decaying phase. During the elastic phase the wall 
behaves in the same way as an elastic beam. Wiehle et al. (1969) does consider the tensile 
bond of the mortar. Once cracked, the wall is idealized as two rigid beams connected at 
the crack. After cracking, the wall will have a discontinuity due to the sudden loss of the 
flexural strength if no axial load is applied. The geometry of the wall along with the 
weight and axial load are the only forces considered resisting the lateral load. A 
preexisting axial load increases the resistance by keeping the crack closed. Wiehle et al. 
(1969) place the axial load at the wall centerline and assumes the eccentricity negligible. 
 Slawson (1995), Johnson et al. (2008), and Jones (1989) use this resistance 
function as the analytical method for URM walls.  
12 
   Paulay and Priestley (1992) describe the mechanics of masonry walls under 
lateral loads and what determines failure of a CMU wall. ?The formation of cracks does 
not constitute wall failure? (Paulay and Priestley 1992). The failure of the wall is 
dependent on the resultant of the compressive force moving past the line of action for the 
axial load and self-weight of the wall.  
Paulay and Priestley (1992) make the assumption that there will be no tensile 
resistance of the masonry wall. Figure 2.2 illustrates how the resultant force moves as the 
crack propagates.  
 
Figure 2.2 Compressive Resultant Force (Paulay and Priestley 1992) 
It is noted that the intensity of the compressive stress increases; however, the area 
decreases as the crack spreads. Because of this, the resultant force, R, remains the same 
throughout if there is no arching force. Paulay and Priestley (1992) state that as a 
13 
conservative estimate, the deflection of the wall increases proportionally with the 
curvature of the centerline.  
 The bond of the mortar and CMU can define the maximum tensile stress and 
therefore the ultimate strength of the wall (Drysdale et al. 1994). However, the strength 
of the bond can vary by as much as 20% in the laboratory testing. With a highly variable 
mortar tensile bond the crack can easily be initiated before the ultimate strength is 
reached. It only takes a small portion of the mortar in a bed to have lower strength to 
compromise the entire wall. ?The crack initiates along two or three units [?] and then 
immediately propagates for the full length of the wall? (Drysdale et al. 1994). The mortar 
tensile bond itself is very low and reduced because of its variability making it appear 
negligible. However, the addition of tensile stresses to the walls resistance can more than 
double the strength of the wall (Drysdale et al 1994). 
 Based on the ideas from Drysdale et al. (1994) and Paulay and Priestley (1992), 
Moradi and Davidson (2008) developed a resistance definition for non-arching URM 
walls. However, one major difference is that the Moradi resistance definition assumes 
that tensile stresses are negligible. The ACI 530-08 code indicates that some tensile stress 
can be used; however Moradi and Davidson use a value of zero mortar tensile bond. One 
major advantage of Moradi and Davidson?s method is that it considers partial fixity at the 
supports. If the fixity is below 75% the wall will fail at the mid-span. The conservative 
assumption is to assume no fixity at the top and bottom; however, if some restraint is 
known it will help increase the resistance of the wall. The axial load applied will remain 
along the centerline of the wall all the way through failure. With this resistance function, 
tensile stresses cannot be resisted and result in cracking of the wall. The wall will fail 
14 
when the resultant of the compressive stresses moves outside the line of action of the 
gravity loads. The compressive stresses developed in the wall cross-section can reach the 
linear elastic ultimate compressive stress without constituting failure of the wall.  
 Moradi and Davidson?s (2008) general resistance definition is not a closed form 
solution; it must be solved by incrementally changing the displacement in order to 
achieve the full resistance function. The overall process involves solving for the pressure 
and deflection just before cracking. After cracking, the crack length is increased 
incrementally allowing for the deflection and pressure to then be calculated. Since 
Moradi and Davidson?s resistance function does not consider tensile stresses it will fail 
well below the Wiehle et al. (1969) curve. 
 Single-Degree-of-Freedom Blast Effects Design Spreadsheet (SBEDS) 
Methodology Manual (PDC 2008) does not recommend using non-arching since static 
test shows little resistance of the masonry past the yield point. It is stated that the moment 
capacity of the wall is controlled primarily by the tensile bond strength between units. 
SBEDS Methodology Manual (PDC 2008) recommends using a dynamic mortar tensile 
bond of 200 psi if no test data is available; however for static resistance this should be 
reduced.  
 Because there is no restoring force the resistance of a simply supported CMU wall 
will rely solely on flexural resistance. After the onset of the crack the masonry resistance 
drops to an almost negligible value (Johnson 2008). It is noted that without arching 
supports, there is little resistance to blast loading (Johnson 2008).   
15 
2.2.2 Arching Wall Resistance Functions 
 Arching walls provide a significant increase in resistance due to the addition of 
compressive force acting on the wall after cracking. If masonry walls are placed inside a 
rigid frame substantial resistance can be developed. McDowell et al. (1956) states that the 
resistance of an arching wall can be up to six times the resistance of a plain CMU wall. 
Again it is assumed that the wall can be represented as two rigid beams rotating about the 
supports and hinged at the center. Once the wall cracks, coupled forces develop at the 
supports and the center crack. Theses coupled forces are what psrovides the additional 
resistance (McDowell 1956), the coupled forces are shown in Figure 2.3. 
  
Figure 2.3 Idealized Arching Wall Forces 
16 
Where a represents the arching force applied by the supports and R is the restoring force 
at the midspan. The forces are developed from the crushing of the masonry wall. Once 
the coupled forces vanish the wall is destroyed (McDowell 1956). 
 Several assumptions about the geometry and the material properties are needed. 
The material properties of the masonry wall play a significant role. The first major 
assumption is the inability of the masonry to withstand any tensile forces. It is assumed 
that the wall behaves as a classic elastic-plastic material. The plastic phase represents the 
crushing of the masonry. Once the masonry is crushed the masonry will not be able to 
develop any additional stress. Therefore, the masonry does not have any strength 
recovery (McDowell 1956).  
The first development of the resistance function for arching wall is presented in  
Wiehle et al. (1969). All the previous assumptions made by Wiehle et al. (1969) about 
non-arching walls are still viable. One major distinction between arching and non-arching 
walls is that the maximum elastic resistance does not occur at cracking as it does for non-
arching walls. For arching walls, the maximum elastic resistance is developed when the 
supports reach their maximum moment. The decay phase of the arching wall is assumed 
to behave in the same manner as the non-arching wall. This method was implemented by 
Jones (1989), Slawson (1995) and Johnson (2008). 
As the axial forces and moments at the supports increase, the wall will crush at 
the wall ends first followed by the crushing at the midspan (Drysdale et al. 1994). A three 
hinged arch will be formed by crushing at these three locations. The arching force is 
dictated by the mortar joint not the masonry properties (Drysdale 1994).  
17 
Drysdale et al. (1994) address the issue of gap arching. The presence of a gap will 
reduce the resistance of the wall compared to fully fixed supports and will have larger 
resistance than a simply supported wall. Gaps between the wall and support are common 
and have several causes, including poor construction, shrinkage and creep, or intentional. 
Even though a gap is present, arching forces can still develop. When a lateral load is 
applied, the wall must rotate about the base until the gap is closed. Once the gap has been 
closed the wall will begin to resist the load. Figure 2.4 illustrates how a gapped wall 
moves under lateral loading.  
 
Figure 2.4 Arching Mechanism for Gapped Wall 
The gap increases the deflection at cracking since the wall must rotate before cracking 
can happen. There is a maximum amount of gap allowed to ensure the wall will touch the 
18 
support and develop the compressive reaction. The maximum allowable gap is given by 
Drysdale et al. (1994) and presented in Equation 2.1.  
 22m axg h t h? ? ?  2.1 
Where gmax is the maximum allowable gap, h is the height of the wall and t is the 
thickness of the wall.  
Hoemann et al. (2010) looked at the connection?s ability to increase the arching 
resistance of a gapped wall. If an angle is placed at the top of a wall it will eliminate the 
need for the wall to rotate about the base before arching resistance can be exerted. The 
angle will allow the wall to form the hinged condition needed to exert the downward 
thrust if the roof support can be considered rigid. 
 Moradi and Davidson (2008) examined the use of mechanical properties of the 
masonry and the geometric properties to develop a resistance function for arching walls. 
Using the ideas developed by Paulay and Priestly (1992) and Drysdale et al. (1994), 
Moradi and Davidson developed a resistance function for non-gapped CMU walls. 
Assuming the arching force is developed as the wall deflects and increases in intensity as 
the deflection grows until failure. The arching force is assumed to act only on the face 
shell thickness or if the face thickness is unknown using the simplified formula below 
(Drysdale et al. 1994).  
 0.1at?   2.2 
Where a is the length over which the arching force acts and t is the thickness of the 
masonry block. The axial load applied to the wall is assumed to act at the top center of 
19 
the wall for the duration of the loading. This assumption is for simplicity since the axial 
load moving to the point of contact once arching exists will complicate the calculations. 
 SBEDS Methodology Manual (2008) developed the resistance functions based on 
Park and Gamble (2000). Park and Gamble (2000) developed resistance functions for 
reinforced concrete slabs based on mechanical and geometric properties as well as 
experimental data. The formulas developed by SBEDS neglect reinforcement and 
incorporate voids to simulate ungrouted CMU. SBEDS incorporates not only tensile 
strength of the mortar but the dynamic mortar tensile bond which should be reduced for 
use in static resistance curves. Most masonry walls will have a gap of 0.25 in. to 0.5 in. to 
allow for thermal expansion. This gap is often filled with a flexible material and needs to 
be examined when developing a resistance function for existing walls (SBEDS 
Methodology 2008).  
2.3 Unbonded Retrofit System 
 Since URM is very brittle and lacks ductility there is a need to retrofit walls to 
provide ductility. The retrofit may also catch the fragmented parts of the wall which 
could be deadly if unrestrained. There are two methods of retrofitting a pre-existing CMU 
walls: bonded and unbonded. The unbonded method is commonly referred to as a catcher 
system since it will catch the wall debris. An unbonded retrofit system drawing and 
installation are shown in Figures 2.5 and 2.6 respectively. The installation picture comes 
from testing done at the Air Force Research Laboratory (AFRL). 
20 
 
Figure 2.5 Membrane Retrofit System Drawing 
 
Figure 2.6 Installation of Membrane Retrofit System 
 
21 
An unbonded system is commonly attached to the floor and ceiling behind the wall. 
Many different materials can be used for the retrofit. The materials range from stiff steel 
to a plastic membrane commonly found at hardware stores.  
 Three methods for determining the resistance of the membrane have been 
developed. Young et. al (2012) presents a formula for the resistance of thin elastic 
flexible membranes. The deflection at mid-span corresponds to a cubic root relationship. 
The formula was developed assuming the membrane acts as a slender perfectly flexible 
cable that is loaded transversely and in tension. Figure 2.7 is the free body diagram of the 
cable (Young et a. 2012). 
 
Figure 2.7 Free Body Diagram of Cable Analogy 
The use of Young?s formula was proposed by Salim et al. (2006). After yielding 
the use of a formula for plastic deformation proposed by Lane (2003) could be used. This 
set of equations will provide an approximate elastic-plastic resistance function for the 
retrofit (Salim 2006). The formula presented by Young (2012) can be used after yielding 
if the modulus of elasticity is updated at each step.  
  The second method was developed by Seide (1977). This formula was developed 
assuming all edges are straight with negligible in-plane shear stress and do not translate 
once the load is applied. The width to height ratio is assumed to be large, greater than 5. 
The formula proposed by Seide (1977) was considered for use by Sudame (2004).  
22 
 One of the more widely used methods for developing the resistance function of 
the membrane starts with theory developed by Lane (2003). Lane determined that steel 
studs subjected to lateral pressures deflect in a parabolic shape. Lane (2003) developed a 
method to solve for the arc length of the stud assuming a parabolic shape. This method 
was used and re-derived by Fitzmaurice et al. (2006), Salim et al. (2007), Johnson (2008 
and 2009) and referenced by Hoemann et al. (2010). This method involves incrementing 
the deflection and calculating the new arc length of the retrofit. The development and 
process will be detailed in Chapter 4. 
 Two Engineering Technical Letters (ETL) provide guidelines for use of geotextile 
fabrics for use in an unbonded system. ETL 00-09 defines geotextile as ?a planer, 
permeable, polymeric textile material, which may be nonwoven, knitted, or woven? 
(Department of the Air Force 2000). Both ETL 110-3-494 and ETL 00-09 developed 
guidance based on a 8-in thick masonry and a 12-ft wall. The selection of these 
parameters will allow for a design for shorter and thicker walls that will have lower 
deflections (Protective Design Center 1999). It should be noted that this method only 
works for walls without windows (Department of the Air Force 2000). Figure 2.8 is a 
schematic illustrating the suggested design.  
23 
 
Figure 2.8 Suggested Catcher System Design (Department of the Air Force 2000) 
A 4-in anchor is suggested as a minimum for 4000 psi concrete restraints. Longer 
anchorage will allow for more ductile failure of the connection (Department of the Air 
Force 2000). The average cost of a geotextile with installation is approximately $80 per 
linear foot along the width of the wall with labor and material consuming most of the 
expense (Department of the Air Force 2000).  
2.4 Unbonded Retrofit System Connection 
 Several tests have been performed to demonstrate the importance and need for an 
analytical expression for the connections role in the overall resistance and ductility of the 
system. The first of these laboratory tests were performed by Salim et al. (2006). Two 
different setups were used to determine both the ductility and strength of the connection 
and how it will react with the membrane?s lateral loading. To determine the ductility of 
24 
the connections, an axial tensile test was performed on three different thickness 
membranes bent at 90? at the top and bottom. Two different thickness plates were used to 
connect the sheets to the loading frame with a 12 and 16 inch bolt spacing. Ten tensile 
tests were reported. The second setup was a 16 point loading tree that would simulate a 
lateral load applied to the retrofit system. Only one test was performed to assess the 
viability of the resistance definition developed.  
 More in-depth tensile testing was performed by Fitzmaurice et al. (2006). Using 
the same setup as Salim et al. (2006); however, Fitzmaurice et al. used polymers instead 
of steel sheeting. Sixteen different tests were conducted to determine which connection 
scheme would allow for adequate strength without cumbersome construction. Three bolt 
spacing?s were used: 8, 12, and 16 inches. Two different thicknesses of plate were used to 
clamp the membrane to the loading frame: 0.125 in and 0.25 in. The goal of the plate is to 
allow the membrane to develop significant stresses without contributing to premature 
failure. It was noted that most of the test conducted with the 0.125 in. plate, the plate was 
bent significantly, which allowed for the membrane to tear out at the bolts. For this 
reason, Fitzmaurice et al. (2006) recommend using a 0.25 in. plate. The polymer allows 
for significant ductility of the system (Fitzmaurice et al. 2006).  
Using energy dissipation as the criteria to determine which bolt spacing would be 
adequate, it was determined that the 5/8-in bolts spaced at 16 in. allowed for sufficient 
energy dissipation (Fitzmaurice et al. 2006). 12 in. spacing was considered adequate as 
well. The plate?s preparation before attaching the retrofit can decrease the strength of the 
membrane significantly. If the edges are allowed to remain square at the surface in 
contact with the membrane, the edge might cut the membrane preventing it from 
25 
achieving its full strength before failure. These sharp edges may reduce the capacity of 
the retrofit by as much as 50% (Fitzmaurice et al 2005). 
 Salim et al. (2007) performed a 16 point loading tree test on several different 
materials with plate configurations in order to determine if the plate edge affects the 
ultimate strength of the system. Three tests were conducted, two of the tests used toed 
plates. The toed plates were bent to eliminate a sharp edge from the system. The toed 
edges allowed for more ductility of the system. However, the toed plate bent more than a 
flat plate, allowing a bearing failure of the membrane at the bolts and decreasing the 
ultimate pressure (Salim et al. 2007).  
 Johnson et al. (2009) conducted two series of test on retrofit systems to determine 
which connection configuration would allow the membrane to achieve its ultimate 
strength. Series A used an axial tension approach to determine if sandwiching the 
membrane between an angle and a plate would eliminate the problem of a sharp edge. 
Figure 2.9 details the tensile set up used by Johnson et al. (2009). 
26 
 
Figure 2.9: Tensile Test using Angles for Connection 
This test revealed that the angles and plates did not bend, however they still 
allowed for bearing stresses around the bolts to form. These bearing stresses resulted in a 
59% reduction in theoretical strength predicted by the analytical model. The bolt spacing 
used by Johnson et al. (2009) were 2 in., 4 in., and 8 in. The bolt spacing that produced 
the largest resistance reduced of strength by 60% used a 2 in bolt spacing. This 
configuration is very labor intensive, cumbersome, and ineffective. 
 Johnson et al. (2009) performed a second series of test using the same 16 point 
loading tree as Salim et al. (2006). This test was conducted with four different 
configurations to determine adequate bolt spacing and plate thickness. Using a bolt 
spacing of 8 in. and 12 in., the test again only achieved 41% of the optimum strength 
27 
before the test was terminated. However, instead of the test being truncated because of 
failure, the loading frame ran out of travel room making additional loading impossible. 
The average deflection of the four tests was a little over 20 in. All of the tests produced 
data lower than the analytical model because the ductility of the wall is heavily affected 
by the connections. With a thick membrane, two forms of failure were exhibited by the 
system. The first was a bearing failure of the bolts on the plate. The bolts were ripped out 
of the 0.125 in. thickness plate. The second failure was a bearing failure of the membrane 
on the bolts experienced by previous research. The membrane was allowed to bear on the 
bolts after the plate was bent approximately 45? (Johnson et al. 2009). 
In an attempt to simplify the design and detailing of anchorage, Hoemann et al. 
(2010) simplified the anchorage into three cases. Case 1 involves the tension of the 
membrane being transferred directly into the plate with no moment arm. Without a 
moment arm the anchorage needed to resist the tension is simply the tensile force. In 
Case 2, the anchorage is placed in the center of the plate creating a pivot at the end 
opposite the applied load. The pivot point will amplify the tensile force causing the 
anchorage to be double the tension force. Case 3 is a transition case that accounts for the 
plate yielding and a plastic hinge forming in the plate. The force applied will bend the 
plate decreasing the moment arm until it is negligible. In Case 3 the anchorage force 
transitions from twice the applied load to the applied load. Figure 2.10 is an illustration of 
the three cases.  
28 
 
Case 2 is ideal for very soft low moduli membrane materials such as PVC 
membranes due to the inability to significantly bend or yield the connection (Hoemann et 
al. 2010). For stiffer materials that can bend the plate Case 3 should be considered. The 
capabilities of most drills will create a moment arm since the bolt will be a distance from 
the wall. Most drills will not be able to tighten the bolt if it is less than 3 in. from the wall 
(Hoemann et al. 2010).  
2.5 Combination of Systems 
 A composite resistance curve, including both the wall and retrofit resistance, is 
needed in order to implement a SDOF model for blast analyses. In order to create a 
complete model, two methods may be used to determine the total resistance function. The 
first is a laboratory test of the entire wall system. This method is both expensive and 
cumbersome to do for every wall. The second is to determine a way of taking individual 
resistance functions and combining them into one system.  
Figure 2.10: Idealized FBD of Mechanical Connections (Hoemann et al. 2010) 
29 
 Knox et al. (2000) proposed a theory to combine different components of the wall 
system. In order to develop this model, Knox et al. (2000) made several assumptions 
about the system. The first assumption is that all the components will have the same 
deflection at midspan at every interval. The second is that no shear stresses will be 
transmitted between components. This assumption implies that all surfaces are 
frictionless and therefore unbonded. And lastly all static pressures will be applied normal 
to the surface of the wall. If all assumptions are satisfied the individual resistances can be 
added to determine the total resistance of the wall. Figure 2.11 is an idealized resistance 
curve combination if only the URM wall and retrofit are considered.  
 
Figure 2.11: Idealized Resistance Curve Combination 
The URM wall will have significant resistance at the onset of the load application 
and then dissipate once the wall has failed. The retrofit requires tension to develop before 
significant resistance can be observed. Individually these two components may not have 
adequate resistance to a blast loading whereas their combination may provide the needed 
resistance.  
30 
2.6 Material Selection  
 The effectiveness of the retrofit is primarily based in the selection of the material 
used. Many different materials have been tested for use in blast retrofitting. No one 
material will be ideal for every situation therefore it is necessary to explore different 
materials. Some materials have been designed for use as blast retrofits and provide 
adequate resistance. However, many materials used in other industries can be used for 
retrofitting URM walls. Examples of these materials are PVC membrane liners 
(Hoemann 2010), geotextiles (ETL 2000), steel membranes (Salim 2006), and 
polypropylene (Johnson 2009).  
 The selection of a material is heavily dependent on the material properties, such 
as modulus of toughness and percent elongation at failure, but construction may also 
contribute to the selection process. Stiff materials, like steel and chain link fence, will 
have low deflections with very large ultimate pressures; however they come with large 
construction cost (Hoemann 2010). The thickness of the material may also play a 
contributing role in selection. Thicker materials have shown to resist larger pressures 
however bending a 3-mm membrane of polypropylene requires significantly more effort 
and time, up to 12 hours more than a thinner membrane (Johnson 2009).  
 Thornburg (2004) reported ASTM standard uniaxial tension tests on 3 samples of 
53 different materials.  The three main characteristics observed were: ultimate stress, 
percent elongation at failure, and modulus of toughness. The ultimate stress is used to 
determine the ultimate load a membrane can withstand. The percent elongation correlates 
to the amount of deflection that might be exhibited during loading. Lastly, the modulus of 
toughness is a representation of the amount of energy that will be absorbed by the system 
31 
before failure (Thornburg 2004). The three categories are what designers should consider 
when selecting a material but materials tend to perform well in two categories and poorly 
in a third (Thornburg 2004). Materials ranged from stiff to very soft. Stiff materials tend 
to have low elongations with large ultimate pressures but low modulus of toughness, 
whereas soft materials have large modulus of toughness and adequate pressures but very 
large deflections.  
 The material properties needed for analysis depend significantly on the method 
used to perform the analysis. SBEDS Methodology (PDC 2008) requires a lot of material 
data about masonry including the dynamic properties, where Wiehle et al. (1969) requires 
only basic properties about the masonry. The same is true about the membrane material. 
The method proposed by Lane (2003) requires a true-stress true-strain curve and the 
method used by Young et al. (2012) only use area and Young?s modulus of elasticity. 
2.7 Response Limits 
 New building design has the ability to estimate what the threat level might be and 
design the structure to resist the loads to a certain level of protection (LOP). Retrofit 
systems do not have the ability to always bring a building within a high level of 
protection if the threat is high. The idea of the retrofit is to bring the structure within an 
acceptable and feasible LOP. For retrofitting a building the process goes as follows:  
1) Define loading 
2)  Analyze the masonry without retrofit 
3) Determine what the provided LOP is 
4) Analyze the structure with the retrofit system 
5) Determine the change in the LOP 
32 
6) Iterate if the LOP with the retrofit is not adequate 
The LOP of a building is directly related to the expected damage to the system. The 
LOPs of building structures from PDC-TR 06-06 ?Single-Degree-of-Freedom Structural 
Response Limits for Antiterrorism Design? are outlined in Table 2.2. 
Table 2.2: Structural Damage Associated with Building Levels of Protection (U.S Army 
Corps of Engineers 2008) 
Building 
Level of 
Protection 
Descriptions of Potential Overall Structural Damage 
Below AT 
standards1 
Severe Damage - Progressive collapse likely. Space in and around 
damaged area is unusable. 
Very Low Heavy Damage - Onset of structural collapse. Progressive collapse is unlikely. Space in and around damaged area is unusable. 
Low Unrepairable Damage - Progressive collapse will not occur. Space in and around damaged area is unusable. 
Medium Repairable Damage - Space in and around damaged area can be used and is fully functional after cleanup and repairs. 
High Superficial Damage - No permanent deformations. The facility is immediately operable. 
1 - This is not a level of protection, and should never be a design goal. It only defines a realm of 
more severe structural response, and may provide useful information in some cases. 
 
For masonry structures, the masonry is commonly cracked and failed as a result of 
loading. Different structural elements are allowed to have different response levels 
depending on the type of element. This means that for a given building not all the 
elements must conform to the same criteria. The types of elements are outlined in Table 
2.3. 
 
 
 
33 
Table 2.3: Component Descriptions (U.S. Army Corps of Engineers 2008) 
Component Description 
Primary 
Structural 
Members whose loss would affect a number of other components 
supported by that member and whose loss could potentially affect the 
overall structural stability of the building in the area of loss. Examples of 
primary structural components include: columns, girders, and other 
primary framing components directly or in-directly supporting other 
structural or non-structural members, and any load-bearing structural 
components such as walls. 
Secondary 
Structural 
Structural component supported by a primary framing component. 
Examples of secondary structural components include non-load bearing 
infill masonry walls, metal panels, and standing seam roofs. 
Non- 
Structural 
Components whose loss would have little effect on the overall structural 
stability of the building in the area of loss. Examples of non-structural 
components include interior non-load bearing walls, and architectural 
items attached to building structural components. 
 
For masonry walls, depending on whether or not they are load bearing will fall 
into either the primary structural or secondary structural category. The damage associated 
with a structural component is related to the weight and standoff of the explosive and 
shown in Figure 2.12.  
34 
 
Figure 2.12: Component Damage Levels (PDC-TR 06-08 2008) 
 The damage to a component is specified as visible effects after loading happens. 
However, for design purposes response limits were developed to determine if damage 
would be declared superficial or hazardous. These response limits are described using 
two criteria. The first is a ductility ratio, ?, which is the ratio between the maximum 
deflection and the deflection at first yield. The second is the rotation of the wall at the 
support. Table 2.4 are the response limits for masonry walls from PDC-TR 06-08 
modified to only show unreinforced masonry.  
35 
Table 2.4: Response Limits for Unreinforced Masonry 
Member B1 B2 B3 B4 ? ? ? ? ? ? ? ? 
Flexure 1 - - 1.5? - 4.0? - 8? 
Combined 
Flexure & 
Compression 
1 - - 1.5? - 1.5? - 1.5? 
 
The design of a retrofit should bring the masonry wall within acceptable standards 
for a specific level of protection. As stated previously, due to the flexural response of 
unreinforced masonry superficial damage may not be applicable regardless of the amount 
of retrofit applied.  Chapter 8 presents a design example that illustrates how the entire 
system masonry wall, membrane and membrane connection will work and how design of 
a retrofit could be done.  
 
  
36 
 
Chapter 3: Review of Design Resistance Functions for Unreinforced Masonry  
3.1 Overview 
The brittle nature of unreinforced non-grouted masonry walls makes a resistance 
definition difficult to derive and test. The resistance of a masonry wall depends heavily 
on the support conditions. Two support conditions have been considered. The first is 
when the supports provide no rotational resistance, which is represented as a simple 
support. With simple supports the wall resist lateral loads using only its flexural strength 
and any axial loads such as self-weight. Since unreinforced masonry does not have a 
large flexural resistance, it is generally considered negligible (PDC 2008).  
 The second support condition is a fully rigid support. If the wall supports are 
rigid, the support will resist rotation as the wall attempts to rotate. As the wall attempts to 
rotate into the support, the support reacts and pushes back at the corner of the wall. Since 
the support is assumed to be at the longitudinal edge of the wall, it will create a moment 
arm. The moment arm and force create a restoring moment. This restoring moment   
increases the resistance of the wall significantly (Drysdale et al. 1994).  
3.2 Non-Arching Masonry Resistance Functions 
 If the wall supports cannot resist the rotation of the wall then the wall is assumed 
to be simple supported. The wall must be attached to the supports in such a way that no 
translation is permitted at the supports. If the wall is simply supported it is considered to 
be a beam with some flexural resistance up to cracking. After the wall cracks at midspan, 
37 
the resistance of the wall decreases significantly with very little additional displacement 
if no axial load is applied. 
3.2.1 Wiehle Resistance Function 
 One of the first methods derived for the development of resistance functions for 
simply supported masonry walls against lateral loads was presented by Wiehle et al 
(1969). Wiehle et al. describe the resistance function in two phases. First is an elastic 
phase where the masonry wall is uncracked and resists the load through its flexural 
strength. The second phase is a decaying phase. The decaying phase depends heavily on 
the axial loads applied to the wall as the source of resistance.  
 To develop a resistance curve, the elastic peak pressure, p1, and deflection at the 
maximum elastic pressure, ?1, is the first point calculated 
 ? ?
1 243 rt f t Php ??
  3.1 
 41
1 5? 384 gphEI?
  3.2 
Where t is the thickness of the wall, h the height of the wall, and P is the axial load 
applied through the center of the wall. fr corresponds to either the modulus of rupture or 
the tensile bond strength depending on the design assumptions. Johnson (2009) used 
bond strength of 65 psi. This value is widely variable: 30 psi to 250 psi (Drysdale et al. 
1994). The elastic resistance of the wall in Equation 3.1 uses a moment of inertia for a 
solid rectangular block, which does not incorporate the hollow sections of the CMU.  
38 
 Once the elastic portion of the curve is known, the decay portion is determined 
using the formula for p2 and incrementing the deflection.  
 
2224 ( ? )(2 )itPp h W? ? ?
  3.3 
Depending on the axial load applied to the wall, three different decay phases are 
possible, Figure 3.1. The axial load can increase the resistance above the elastic phase. 
The axial load may allow the decay phase to start where the elastic portion ends or if the 
load is low enough the decay phase will start below the elastic portion.  
 
Figure 3.1: Decaying Phases Assumed by Wiehle et al. (1969) 
 There is an assumed linear relationship from the origin to the elastic limit. If the 
decay phase is below the elastic limit a vertical drop is assumed. When the ultimate 
pressure is above the elastic limit the transition has the same slope as the elastic portion 
until it reaches the decay phase.  
39 
3.2.2 Moradi Method 
 Moradi and Davidson (2008) developed a resistance function that would allow for 
different degrees of fixity of the supports. Using theories developed by Paulay and 
Priestly (1992) and those presented in Drysdale et al. (1994), Moradi and Davidson 
(2008) developed a resistance function. The major distinction of the Moradi method is 
that the tensile strength of the mortar is neglected, which reduces the resistance of the 
URM wall and allows for a very conservative design. Figure 3.2 is a free body diagram of 
the wall used by Moradi (2008). 
40 
 
Figure 3.2: Free Body Diagram of Non-Arching Wall (Moradi et al. 2010) 
 
 To start, the pressure at the elastic limit is determined. The elastic pressure is 
calculated by setting Equations 3.4 and 3.5 equal to one another.  
 ? ? 232?
6 3 1 2 2 4i phty WP??? ? ? ?
  3.4 
 ? ? 4
m a x
54? 384
e la s tic cg
phEI???   3.5 
41 
Where Ec is the concrete masonry modulus of elasticity and Ig is the gross (uncracked) 
moment of inertia, ? is the support fixity, with 0 being a pinned condition and 1 being a 
fully fixed condition, and y is the crack length of the URM wall. For the elastic phase 
there is no crack in the wall, therefore y=0. Once the pressure at cracking is known, the 
maximum elastic deflection can be calculated using Equation 3.5.  
 For the decay phase of the resistance function, the crack length is increased 
incrementally. Moradi and Davidson (2008) assumed that the displacement at midspan is 
proportional to the curvature of the wall. The deflection after cracking is calculated as 
follows: 
 ??crG cr??   3.6 
 2
2W h e re : ()c r Gcr ttRR y? ? ?
  3.7 
Where cr denotes cracking and crG denotes crack growth. Since R does not change as the 
crack grows, Rcr and RcrG cancel each other out leaving: 
 2
2()tty?? ?
  3.8 
Using the deflection after cracking and plugging it into Equation 3.4, the pressure can be 
determined. This process is continued until failure. The process for developing the 
resistance function is outlined in Figure 3.3.  
42 
 
Figure 3.3: Moradi Method for a Non-Arching Resistance Curve Flow Chart 
3.2.3 SBEDS Methodology 
SBEDS User Manual (PDC 2008) considers two forms of URM wall response. 
The first is an elastic-plastic flexural response. This response is not recommended since 
static testing shows that masonry walls do not exhibit a plastic response (PDC 2008). 
SBEDS User Manual (PDC 2008) explains that the only reason it is included is due to the 
limited plastic response that has been used in the past. The second response type 
considered and is recommended is brittle flexural response with axial load. 
From the bending stress formula the maximum moment capacity can be 
calculated. 
 
t P McAI? ??
  3.9 
   Where A t SFR??           3.10 
Where ?t is the ultimate tensile stress of the masonry and SFR is the solid fractional ratio 
of the masonry. If the exact cross sectional area is known, that should be used. Once the 
43 
maximum moment is known, beam mechanics are used to determine the lateral load that 
causes that moment as well as the deflection due to elastic bending.  
 
1 28 w h e r e :  tMPr M S fht??? ? ?????
  3.11 
 1
m ax elastic
45?
384 cgrhEI?
  3.12 
Where r1 is the resistance due to flexure and S is the elastic section modulus. A second 
point is calculated to determine if the axial load will increase or decrease the resistance of 
the wall and to what extent. The SBEDS User Manual (PDC 2008) suggests using 10% of 
the compressive strength or 200 psi, whichever is greater. First the resistance after 
cracking is calculated.  
 
2 28 ( ?) 2im a x Whr t Ph ??? ? ?????
  3.13  3.15 
Where ?max refers to the maximum elastic deflection found in Equation 3.12. The 
deflection corresponding to r2, ?2, is then calculated assuming a linear relationship. If the 
resistance after the elastic region is greater than the elastic resistance then the sign of the 
elastic stiffness, Ke, is positive; otherwise the sign is negative, Figure 3.4.   
44 
 
Figure 3.4: Idealized Resistance-Deflection Curve for SBEDS 
 21
2??max errK???
  3.14 
 1
?e maxrK ?
  3.15 
From this point the resistance curve is assumed linear until the wall has no resistance. 
The deflection corresponding to failure is assumed to be the thickness of the wall (PDC 
2008). This method was presented by Oswald and Zehrt (2010) for the update to UFC 3-
340-02. 
3.2.4 Unified Facilities Criteria (UFC) 3-340-02 Methodology  
 TM 5-1300 is superseded by UFC 3-340-02 so it was not considered. UFC (2008) 
does not derive or present any formulas for a resistance curve for a simply supported 
wall. If a wall is simply supported the modulus of rupture will determine the maximum 
resistance of the wall (UFC 2008). Additional axial loads may be considered if they exist. 
An example presented by UFC (2008) for a simply supported wall is shown in Figure 3.5.  
45 
 
Figure 3.5: UFC Simply Supported Wall 
The wall will have supports that resist lateral movements but cannot resist 
rotation. An update to the masonry chapter of UFC 3-340-02 was released by Oswald and 
Zerht (2010) and for non-arching walls the method presented is the same as the SBEDS 
method.  
46 
3.2.5 Comparison of Non-Arching Resistance Functions  
 A resistance function should be able to accurately predict the pressures required to 
deflect any size wall. Three different criteria are compared using each of the three 
methods. The three criteria examined are the wall thickness, brick type, and axial load. 
The baseline case will be an 8-in CMU brick wall with a height of 12 ft with no axial 
load. When exploring how these models change with varying wall properties, only one of 
the properties of the baseline case will be changed.  
 First how the models compare to one another is important to understand. Figure 
3.6 shows how the models compare to one another. All the models have similar shape 
and all follow the same linear elastic path. The main differences are the peak pressure and 
how much the wall decays after cracking. The modulus of rupture proposed by Johnson 
(2008) was used for the Wiehle Method. SBEDS has a minimum of 200 psi for mortar 
tensile bond and this number was used for the SBEDS method. 
47 
 
Figure 3.6: Comparison of Non-Arching Methods 
SBEDS gives the largest prediction for the peak pressure followed by the Wiehle 
method. The Moradi method produces a resistance curve much lower than the other two. 
This is due to the assumption that the wall cannot resist any tensile stresses. This 
assumption allows for the lowest peak resistance. The Moradi method determines a peak 
pressure of 0.095 psi while the SBEDS and Wiehle methods predict 0.55 psi and 0.27 psi, 
respectively. The second important consideration is that for the Moradi method the wall 
deflection is less than half that of the other two models at failure.  
48 
 To investigate how the thickness of the block changes the resistance function, 
three CMU block sizes were used, 6-in, 8-in, and 12-in. All of the properties were taken 
from SBEDS methodology (PDC 2008) and shown in Table 3.1.  
Table 3.1: Masonry Properties of Three CMU Blocks 
 
The other properties such as modulus of rupture was taken from Johnson (2009) (65 psi), 
mortar tensile bond from SBEDS (PDC 2008) (200 psi), and support fixity from Moradi 
(2007) (0%) all remain the same. 
 How the thickness of the wall affects the models is examined by looking at how 
each model changes with respect to varying thicknesses rather than comparing every 
curve. The first method examined is the Wiehle method presented in Figure 3.7.  
B r i c k
D e n s i t y  
( pc f )
W e i g h t  
( ps f )
F a c e  S h e l l  
T h i c kn e s s  
( i n )
S ol i d 
T h i c kn e s s  
( % )
M om e n t  of  
i n e r t i a  
( i n ? / i n )
S e c t i on  
M odu l u s  
( i n ? / i n )
6- i n 33.0 1.25 55 12.8 4.5
8- i n 39.2 1.25 49 28.4 7.4
12- i n 48.0 1.25 40 82.4 14.2
120
49 
 
Figure 3.7: Wiehle Non-Arching Variable Wall Thickness 
When the thickness of the wall is increased by 30% the resistance for the wall 
increases by a factor of approximately 1.75 and when the wall thickness is doubled the 
resistance is increased by a factor of four. Because of the way the formula is set up the 
thickness of the wall has a squared effect on the resistance of the wall. The deflection of 
the wall at the maximum pressure does not change significantly; however, it is important 
to note that the thinner walls have a larger midspan deflection at the ultimate pressure.   
50 
 
Figure 3.8: SBEDS Non-Arching Variable Wall Thickness 
Figure 3.8 shows how the SBEDS method varies with block thickness. Once 
again the larger bricks produce a larger resistance. This method produces much larger 
deflection compared to the Wiehle method. The 6-in wall produced a pressure of 0.33 psi, 
the 8-in wall produced a pressure of 0.55 psi, and the 12-in wall generated a pressure of 
1.1 psi at the peak pressure.  
51 
 
Figure 3.9: Moradi Non-Arching Variable Wall Thickness 
The effect of block thickness on the Moradi method is shown in Figure 3.9. The 
Moradi method again produces larger pressures for thicker walls. For the 6-in wall, the 
peak pressure is 0.032 psi, the 8-in wall pressure is 0.094 psi, and the 12-in wall peak 
pressure is 0.41 psi. As the thickness of the wall increases the deflection at the peak 
pressure also increases, which is contrary to the other two methods.  
 The second characteristic to examine is the type of brick used. Three styles of 
bricks are used to examine how the models react to varying brick materials. The three 
type of bricks selected are: an adobe brick, CMU, and an European block. All bricks will 
have a thickness of 8 inches the other parameters are shown in Table 3.2. 
52 
Table 3.2: Material Properties of Varying Blocks 
 
The tensile strength of the adobe is calculated using the equation presented by ACI 318. 
 '7.5tcff?   3.16 
The modulus of rupture for adobe is 50 psi (IBC 2006). Since the Wiehle method use the 
modulus of rupture as the primary means to determine the peak pressure, the resistance 
can change significantly depending on the testing of the adobe. The effect that block type 
has on the Wiehle method is shown in Figure 3.10. 
B r i c k D e n s i t y  ( p c f ) W e i g h t  ( p s f ) C o m p r e s i v e  s t r e n g t h  ( p s i ) T e n s i l e  S t r e n g t h  ( p s i )
S o l i d  
T h i c k n e s s  
( % )
M o m e n t  o f  
i n e r t i a  ( i n ? / i n )
S e c t i o n  
M o d u l u s  
( i n ? / i n )
Y o u n g ' s  
M o d u l u s  
( p s i )
A d o b e 115 7 6 .7 4 3 5 .1 156 100 4 2 .7 1 0 .7 435100
C M U 120 3 9 .2 1350 200 49 2 8 .4 7 .4 2000000
E u r o p e a n 90 3 0 .0 1350 200 50 1 2 .8 3 .2 2000000
53 
 
Figure 3.10: Wiehle Non-Arching Variable Brick Type 
The Wiehle method uses the modulus of rupture as the main criteria to determine 
when the wall cracks and loses its flexural rigidity. Since the modulus of rupture for 
concrete is the same for the CMU blocks and the European blocks the peak pressures are 
the same. Table 3.3 summarizes the resistance and deflections at the peak resistance.  
54 
Table 3.3: Wiehle Method Summary of Deflections at Peak Resitance for Non-Arching 
Masonry with Varying Brick Types 
 
Since the moment of inertia is lower for the European brick the deflection at the peak 
resistance is greater. The difference between modulus of rupture of concrete and adobe is 
only 10 psi. This causes the peak resistance to be very close. The effect of brick type on 
the SBEDS method is shown in Figure 3.11. 
 
Figure 3.11: SBEDS Non-Arching Variable Brick Type 
D e f le c t io n  ( in ) Pr e s s u r e  ( p s i)
C M U 0 . 0 2 6 0 . 2 6 7
A d o b e  B r ic k 0 . 0 6 2 0 . 2 0 6
E u r o p e a n  B r ic k 0 . 0 5 9 0 . 2 6 7
55 
The SBEDS method does not follow the same pattern as the Wiehle method. The 
adobe again produces a large deflection at peak resistance while the European brick has 
the lowest resistance at the same deflection as the CMU brick. Table 3.4 summarizes the 
deflections at the peak resistance.  
Table 3.4: SBEDS Method Summary of Deflections at Peak Resistance for Non-Arching 
Masonry with Varying Brick Types 
 
The moment of inertia of the bricks is important in determining the maximum 
pressure; however, it appears that the moment of inertia does not have a significant effect 
on deflection. The moment of inertia of the CMU block is twice that of the European 
block yet they have the same deflection at peak resistance. For this reason, the deflection 
of the European block and the CMU are the same. The effect of brick type on the Moradi 
method is shown in Figure 3.12. 
D e f le c t io n  ( in ) Pr e s s u r e  ( p s i)
C M U 0 . 0 5 4 0 . 5 4 8
A d o b e  B r ic k 0 . 1 9 4 0 . 6 4 4
E u r o p e a n  B r ic k 0 . 0 5 4 0 . 2 4 7
56 
 
Figure 3.12: Moradi Non-Arching Variable Brick Type 
The modulus of elasticity of the adobe is much less than that of the concrete 
blocks and the Moradi method uses this characteristic to determine the peak resistance. 
For this reason the peak resistance of the adobe is less than that of the other two blocks. 
In this graph it is easy to see how the cracking of the masonry affects the Moradi method. 
At the onset of the crack, there is not a significant amount of stresses developed to resist 
the lateral pressures. The crack must propagate to approximately 83% of the thickness 
until the second peak resistance is observed. The second peak is caused by the self-
weight of the wall moving and helping to restore the system. 
57 
 The last important factor that defines the shape of the resistance curve for non-
arching masonry walls is the axial load. An axial load may be present if the roof or upper 
floor is supported by the wall. If the presence of an axial load is unknown or the designer 
is unsure as to what amount should be assumed, then the axial load is taken as zero when 
determining the resistance curve. The axial load acts as a restoring force and increases the 
lateral load resistance. Three axial loads will be examined to determine how they affect 
the resistance curves: 0 lb/in, 220 lb/in, and 500 lb/in. How axial load effects the Wiehle 
method is shown in Figure 3.13. 
 
Figure 3.13: Wiehle Non-Arching Variable Axial Loads 
The large axial load generates the largest resistance after the flexural resistance of 
the wall vanishes allowing the larger axial load to produce a larger peak resistance and 
deflection. The two walls with axial loads do not have a drop in resistance after the 
flexural resistance is gone. This is because the middle axial load was selected specifically 
58 
so the decay function would start approximately at the same point where the elastic 
function ended. How axial load effects the SBEDS method is shown in Figure 3.14. 
 
Figure 3.14: SBEDS Non-Arching Variable Axial Loads 
The SBEDS method produces resistance curves that are similar to the Wiehle 
method. There is a similar pattern where the larger axial load generates the largest peak 
pressure followed by the middle axial load. One notable distinction between SBEDS and 
the Wiehle method is that the SBEDS pressures and deflection are about double that of 
the Wiehle method. Axial load effects on the Moradi method are shown in Figure 3.15. 
59 
 
Figure 3.15: Moradi Non-Arching Variable Axial Loads 
The Moradi method produces smoother curves due to the incremental changes 
made at every step. The resistances generated for the walls with an axial load are 
comparable to those generated by the Wiehle method. These pressures are still the lowest 
of the three methods but still follow the general order of which wall produces the largest 
pressures.  
 A comparison of the peak pressures for each variable and method is depicted in 
Figure 3.16. Three of the variables shown come from the same wall. One standard 
example was chosen to compare how changing variables affects the resistance curves. 
This is why the 8-in, CMU and P=0 bars all have the same pressure for a specific method.  
 
 
 
Figure 3.16: Comparison of All Non-Arching Walls 
55
 
61 
For all three characteristics, the Moradi method produces the lowest pressures. 
For most of the cases the differences between the Moradi method and the other two are 
significant. However, when an axial load is applied to the wall there is a substantial 
decrease in the difference between the methods. The SBEDS method produced the largest 
pressures for all but the European brick.   
 The deflections at the peak pressure do not follow a unique trend. For most of the 
cases the SBEDS method tends to generate the largest deflections. However, the Moradi 
and Wiehle methods both generate large deflections for different characteristics. Most of 
the deflections are approximately 0.05 in. with the maximum being approximately 0.25 
in. for walls with large axial loads applied. Figure 3.17 compares the deflections at 
midspan. 
 The amount of energy dissipation, proportional to the area under the curve, a wall 
will absorb before failure is a key characteristic for comparison. The comparison of how 
each wall did in terms of energy dissipation is shown in Figure 3.18. The ideal wall can 
dissipate more energy than the load can impart. When an axial load is applied to the wall 
the amount of energy dissipated before failure is drastically increased. The SBEDS 
method produces a larger energy dissipation for every case. The Wiehle method used a 
moment of inertia for a solid rectangle. This assumption is inaccurate for ungrouted 
CMU. Due to the wide range of bond strengths of mortar to masonry, the Moradi method 
does not consider tensile stress.  
 
 
Figure 3.17: Deflection at Peak Pressure for Non-Arching Walls 
57
 
 
 
Figure 3.18: Energy Dissipation for Non-Arching Walls
58
 
64 
3.3 Arching Masonry Resistance Functions 
 Depending on the support conditions, the wall may be modeled as a beam 
between two fixed supports. These supports must be able to resist the rotation of the wall. 
If the supports are stiff enough the wall can be modeled as an arching wall, which will 
give significant resistance to lateral loads (Pauley and Priestley 1992). When the wall is 
bent due to the lateral loads the tension face will elongate. If the supports are rigid this 
elongation is met with a compression force at the supports. This compressive force will 
act as a restoring force and help increase the resistance of the wall by almost 2.5 times 
(Drysdale et al. 1994).  
 There are four methods considered when developing the arching resistance 
function. The first one developed was by Wiehle et al. (1968) followed by Moradi and 
Davidson (2008), SBEDS Methodology (PDC 2008), and lastly UFC (DOD 2008). 
3.3.1 Wiehle Arching Resistance Curve 
 The Wiehle method is used by Johnson (2008), Slawson (1995), and Jones 
(1989).  Several assumptions were needed to generate the resistance curve. The first is 
that the lateral loads are registered entirely by the compressive forces generated and any 
flexural strength from tension is neglected. The second is that the wall acts as two rigid 
halves rotating about the supports. Any additional axial load applied is neglected.  
 The resistance function starts by calculating the deflection at the point when the 
wall starts to crack. Once the deflection is known the pressure at yielding can be 
calculated.  
 
65 
 
'
1
 ?
2
md
c d
t f L h
E L? ?
  3.17 
 Where: 2 2
2d hLt????????
  3.18 
 ? ?' 2
122 ?my f thp ??
  3.19 
These values identify the yield load for the masonry wall. This is the only point 
that will describe the elastic region of the wall. When the wall deflection is greater than 
the yield deflection, the midspan deflection is incremented and the resulting load is 
calculated.  
 ? ?' 2
2122 ? ??????????? ?mf tp h? ? ?
  3.20 
Once the deflection at midspan is the same as the thickness of the wall, the resistance is 
assumed to go to zero.  
The UFC presents a set of equations for arching walls with a gap. This method 
works if the wall is braced at the top to resist translation but allows rotation of the wall 
until the wall is in contact with the rigid support. Figure 3.19 describes how the wall 
deflects during the loading.  
66 
 
Figure 3.19: UFC (DoD 2008) Gapped Wall Schematic 
In the above illustration, the distance the midspan, point m, travels before 
touching the support is described as ?c. While the wall is rotating, only inertial affects are 
resisting the movement and are considered negligible. 
 1 / 22?
2cd htL
????? ? ?????
????
  3.21 
 1/ 22 2'
2d hLt
??????????
????
  3.22 
Shortening of the diagonal at failure along with the elastic modulus will be used to 
determine the stress in the concrete.  
67 
 2d
d
d
hL
L?
??   3.23 
The deflection corresponding to the ultimate resistance is shown in Equation 3.24.  
 
1 ( ?)??c ccmtE ????
  3.24 
The peak resistance is equal to: 
 ? ?2'
1122 ?mp fth??
  3.25 
After the peak resistance, the deflection is incremented until the deflection equals the 
thickness of the wall.   
 ? ?2'
2 22 ?mfp th??
      ?>?1 3.26 
When there is no gap present, these equations produce the same curve as the Wiehle 
method.  Many URM walls have a small gap, making the use of the UFC formulas more 
viable and representative of how the wall will react. Gaps are present to allow for 
shrinkage and creep. They may also be the result of construction error. The effects of a 
gap are shown in Figure 3.20. 
 
68 
 
Figure 3.20: UFC Gapped Wall Comparison 
The stiffness of the wall is not affected by the addition of the gap. The 
displacement at the peak resistance is shifted to the right. This is because of the initial 
deflection required before arching can start. The peak resistance is adversely affected 
with the addition of a gap. For this reason it is important to incorporate a gap if 
applicable. The UFC method also does not incorporate an axial load that will produce a 
conservative curve.  
3.3.2 Moradi Arching Resistance Curve 
 The resistance function created by Moradi et al. (2010) uses mechanics to derive a 
process to generate a resistance curve. One main difference between the Moradi method 
69 
and the Wiehle method is that Moradi included an additional axial load for load bearing 
walls. The FBD used by Moradi et al. (2010) is shown in Figure 3.21. 
 
Figure 3.21: FBD of Moradi Arching Wall (Moradi et al. 2010) 
70 
 An equation was developed for the pressure as a function of displacement, the 
arching force, and the crack length.  
 
2 2 2488 ??2 6 3 2 2 2 2iiT WWt y t a P t a t ap P Vh h h? ? ??? ? ? ? ? ? ?? ? ? ? ? ? ? ? ?? ? ? ? ? ??? ? ? ? ? ? ???
  3.27 
Where VT represents the arching force. When the wall is not cracked, the arching force is 
assumed to be zero along with the crack length. If these values are substituted into 
Equation 3.27 and solving for displacement results in: 
 
2
1
2 6 8?
4
i
i
W t p hP
WP
??????????
?????
?
 3.28 
Before cracking, the wall resists the lateral load due to pure flexural resistance assuming 
no mortar tensile bond. Using the assumption that the wall ends are fixed, the deflection 
due to flexure comes from Euler?Bernoulli beam theory.  
 4
1? 384 gphEI?
  3.29 
Setting Equations 3.28 and 3.29 equal, the pressure that will result in maximum flexural 
resistance can be determined.  
 4226
38 4 4 8
i
i
cg
W tP
p
WhhP
EI
?? ??? ????
?????
?? ????
?? ????
??
  3.30 
Once the resistance is known the displacement can be calculated. The arching force on 
top of the wall is derived using theory developed by McDowell et al. (1956). The strain 
71 
created in the wall due to the shortening of the corner and the support is shown in 
Equation 3.31. 
 2
2
h???
  3.31 
Where ? is the shortening of the wall. The definition for the shortening is shown in Figure 
3.22 and using the geometry of deformation the definition was developed by Moradi and 
Davidson (2008).   
 
Figure 3.22: Shortening of URM Wall 
 ? ? ? 2 ?ta n   
2
th
th
???? ? ? ?  3.32 
Assuming the masonry is elastic until crushing, Hooke?s Law is applicable, ?=E?. 
 
2
2 ?448 ?
ccc
a r c h i n g
tEE E th
h h h?? ? ? ?  3.33 
72 
 
28 ?cT EtV h?
  3.34 
To determine the deflection after cracking, Equations 3.6 and 3.7 should be used 
while incrementing the crack length, y. The resultant force is the summation of the axial 
load, the self-weight, and the arching force. 
 2i
T WR P V? ? ?
  3.35 
 Since the stress relies on the arching force, which is a function of displacement, 
small incremental steps allow for the use of the stress from the previous iteration to 
calculate the curvature ratio. Once the deflection is known, the pressure can be 
determined using Equation 3.27. This process is continued until the resistance of the wall 
goes to zero which constitutes failure. 
 The process to develop a resistance curve using the Moradi method is detailed in 
the flow chart in Figure 3.23 below. 
 
 
 
 
68
 
Figure 3.23: Flow Chart of Moradi Arching Resistance Curve Development 
74 
  3.3.3 SBEDS Arching Resistance Curve 
 The SBEDS method is similar in the Wiehle method in that it assumes the flexural 
resistance is significantly less than the arching action. For this reason the flexural 
resistance is neglected. The methodology used to generate the resistance curve comes 
from Parks and Gamble (2000) ?Reinforced Concrete Slabs?. The equations developed 
neglect the presence of steel, allows for ungrouted voids, and a support movement for 
gapped walls.  
 The SBEDS arching resistance curve is constructed using only three points. The 
first is the displacement of the wall needed to close the gap between the wall and the 
supports. At this point there is no resistance from the wall and only deflection.  
 22 2
22 gc
hhth ??????? ? ? ? ? ????
?? ??
  3.36 
This is the starting point of the resistance curve. The second point needed is the 
maximum resistance of the wall and its corresponding deflection.  
 ? ?
m a x 1 2 1 2 12 0 . 8 58 222 2 2 2m s m smst a tttp C C t C Ch ? ? ?? ? ? ?? ? ? ? ? ? ? ?? ? ? ???? ? ? ???
  3.37 
 21 '
22 gtah h??????? ? ? ?????
 3.38 
 
1
1
2
1
1
0 .8 5 ' 2412
' ' 12
1 0 .2 1 2 5
dc
c dc
c
tf
tE h S fh
tE h S
?
? ??
? ? ??????
???? ????
? ? ??? ??
?? ?? ??
? ??
  3.38(a) 
75 
 '1 0.85 m msC f t?  3.39 
 '2 0 .8 5 (0 .8 5 )m m sC f k c t?? 3.40 
 
1 m in ,3 0 2 cht??? ? ? ?????
 3.41 
Where: ?1 ? Midspan deflection at maximum resistance 
 ?' - Compression strain due to arching forces and support movement strain 
 ?g - Gap between edge of wall and rigid supports 
 tms - Masonry shell thickness 
k - Solid ratio of masonry through webs (k = 0 assumed for horizontal span 
arching if not fully grouted) 
 C1 - Compression force per unit width in shell 
 C2 - Compression force per unit width in web 
 ? - Distance from support to yield location divided by span length = 0.5   
 S ? Lateral stiffness of the supports =0.2Ec 
 f'dc ? Dynamic ultimate compressive strength of concrete 
?1 ? Ratio of concrete compression block depth to the distance from the 
compression face to the neutral axis =0.85 for f?c ? 4000 psi and reduced 
by 0.05 for every 1000 psi increase in strength but not less than 0.65   
 Ec ? Modulus of elasticity of the concrete   
The deflection at ultimate resistance is assumed to be approximate to half the wall 
thickness based on experimental data presented in Parks and Gamble (2000). Based on 
the results of the experiments it was observed that the deflection is affected by the 
stiffness of the supports. For stiffer supports ?it would seem that a reasonably 
conservative estimate of the ultimate load would be obtained at a central deflection of 0.5 
the slab thickness? (Parks and Gamble 2000).  Once the maximum point has been defined 
76 
the last point needed is the point of failure. When the wall exhibits snap through the wall 
will not be able to resist any loading. The deflection at the point of snap through is 
defined as:  
 ? ?m i n 0 .0 8 , 0 .8s t ch t t? ? ? ? ?  3.42 
These three points are the only points used to define a resistance curve in SBEDS. Since 
there are only three points, a triangular resistance curve will always result.   
3.3.4 Comparison of Arching Resistance Functions 
 Three characteristics will be examined to see how these resistance functions will 
be affected. The first is the thickness of the wall. Three different sized bricks will be 
selected, 6-in, 8-in and 12-in. The second characteristic of the wall to be examined is the 
type of brick. The bricks selected are an 8-in CMU, European brick, and a solid adobe 
brick. The last characteristic is the gap between the walls. A zero gap, 0.1 in. gap, and a 
0.25 in. gap will be used. Only two of the resistance functions are designed to incorporate 
a gap so only these two will be used for this portion. The axial load can influence the 
resistance; however, since only the Moradi method is capable of incorporating axial loads 
this characteristic will be neglected.  
 The first characteristic that was examined was the thickness of the wall. The three 
wall thicknesses selected were 6-in, 8-in and 12-in CMU. When the thickness of the wall 
increases the moment of inertia as well as the self-weight of the wall increase. These two 
values were updated as well. All values correspond to SBEDS Methodology (2008) 
recommended properties for medium weight CMU. An increase in the wall thickness 
77 
should increase the ultimate pressure of the system. The input parameters are shown in 
Table 3.5 and Figure 3.24 shows the effect of wall thickness on the Wiehle method.  
Table 3.5: Input Parameters for Wall Thickness Comparison of Arching Methods 
 
 
Figure 3.24: Wiehle Arching Variable Wall Thickness 
W a l l  T h i c kn e s s  ( i n ) 6 8 12
W a l l  H e i g h t  ( i n )
A x i a l  L oa d ( l b/ i n )
M odu l u s  of  E l a s t i c i t y  of  M or t a r  ( ps i )
G r os s  M om e n t  of  I n e r t i a  ( i n ? / i n ) 12.8 28.4 82.4
W e i g h t  of  W a l l  ( l b/ i n ) 22.9 27.2 33.3
S ol i d Fr a c t i on a l  R a t i o ( % ) 55 49 40
C om pr e s s i on  B l oc k ( i n )
f ' m  ( ps i )
1350000
144
1350
1.25
0
78 
 As expected, the larger the wall the greater the peak resistance. The 12-in wall 
predicts a resistance almost eight times greater than what would be expected for a 6-in 
thick wall. Because the wall weight increases with increase in thickness, this helps resist 
the pressures as well as increasing the moment arm of the arching forces. The thicker 
walls produce a smaller deflection at the ultimate pressure. The effect of wall thickness 
on the SBEDS method is shown in Figure 3.25. 
 
Figure 3.25: SBEDS Arching Variable Wall Thickness 
 Similar to the Wiehle curves, the peak pressure increases with an increase in the 
wall thickness. These ultimate pressures are significantly lower than those from the 
Wiehle model. The deflections are also much greater for the SBEDS method. The snap 
79 
through deflection predicts a collapse of the wall at much lower deflections than the 
Wiehle method. The weight of the wall is not considered for the SBEDS method and 
could account for the lower pressures. How the Moradi method is effected by the block 
thickness is shown in Figure 3.26. 
 
Figure 3.26: Moradi Arching Variable Wall Thickness 
The Moradi method produces resistance curves that fall between the other two 
methods. The stiffness of the walls is not greatly affected by the wall thickness compared 
to the other two methods. The other two methods show a significant increase in stiffness 
with a larger wall thickness. This is partly due to the increased moment of inertia of the 
thicker walls.  
80 
 The second factor that was looked at is the type of brick. Three common bricks 
were selected: CMU, European brick, and solid adobe bricks. The CMU block is a 
common building material in the U.S. and the European brick is used widely overseas. In 
the Middle East, adobe bricks are still widely used due to the ease of construction and 
availability. The input parameters for variable brick type are shown in Table 3.6. 
Table 3.6: Input Parameters for Brick Comparison for Arching Methods 
 
The compression block of the adobe was not changed to reflect that suggested by 
Drysdale et al. (1994), a=0.1t. This was done in order to compare how the different 
bricks compare as a material. The effect of brick type on the Wiehle method is shown in 
Figure 3.27. 
C M U E u r ope a n A dobe
W a l l  T h i c kn e s s  ( i n )
W a l l  H e i g h t  ( i n )
A x i a l  L oa d ( l b/ i n )
M odu l u s  of  E l a s t i c i t y  of  M or t a r  ( ps i ) 1350000 1800000 465100
G r os s  M om e n t  of  I n e r t i a  ( i n ? / i n ) 28.4 12.8 82.4
W e i g h t  of  W a l l  ( l b/ i n ) 27.2 27.8 55.6
S ol i d F r a c t i on a l  R a t i o ( % ) 49 50 100
C om pr e s s i on  B l oc k ( i n )
f ' m  ( ps i ) 1350 1800 465.1
8
144
1.25
0
81 
 
Figure 3.27: Wiehle Arching Variable Brick Type 
 The European brick produces the largest ultimate pressure according to the 
Wiehle method. The European brick has a large amount of material in the core of the 
brick compared to the CMU brick as well as a larger compressive stress. The 
compressive stress of the adobe greatly affects the peak pressure of the adobe wall. The 
deflections at the ultimate resistance are directly related to the peak resistance. The 
European brick produced both the largest ultimate pressure and the largest deflection at 
peak resistance.  The ultimate pressure produced by the European brick is approximately 
8 psi. How the brick type effects the SBEDS method is shown in figure 3.28. 
82 
 
Figure 3.28: SBEDS Arching Variable Brick Type 
 The SBEDS curves follow the same trend as the Wiehle method, with the 
European brick producing the largest peak resistance. The deflection corresponding to the 
ultimate strength is independent of the brick properties. The geometry of the wall dictates 
this deflection point. The deflection corresponding to the peak resistance is half the width 
of the wall. These resistances are substantially lower than those produced by the Wiehle 
method. The maximum resistance produced by the European brick is 2.03 psi. The effect 
of brick type on the Moradi method is shown in Figure 3.29. 
83 
 
Figure 3.29: Moradi Arching Variable Brick Type 
The resistance curves produced by Moradi closely resembles the Wiehle curves. 
The peak resistances are below the Wiehle resistance but not as significantly as the 
SBEDS numbers. The resistance of the European brick at the ultimate strength is 5.30 psi 
and has a deflection of 0.70 in. at the ultimate strength. All of the walls collapse at the 
same deflection of approximately 7.30 inch.  
 The final feature to be examined is the gap spacing. The gap will lower the 
resistance of the wall by approximately 50% (Drysdale et al. 1994) and increase the 
deflections at the ultimate strength and collapse. Only UFC and SBEDS consider a gap in 
the development of the resistance curves so the Moradi method will not be considered in 
84 
this section. The three gap sizes to consider are zero, 0.1-in and 0.25-in. gaps. The wall 
will be a standard 8-in. CMU wall with a height of 144 in. between supports. The gap?s 
effect on the UFC method is shown in Figure 3.30. 
 
Figure 3.30: UFC Arching Variable Gap Width 
 The deflection at the start of the resistance curve is heavily influenced by the gap 
width. The wall must rotate in order to close the gap between the supports and the wall. 
The deflection at the onset of the resistance curves for the 0.1-in gap and the 0.25 inch 
gap are 0.43 in. and 1.05 in. respectively. The stiffness of the wall is not affected by the 
presence of a gap. The peak pressures however are adversely affected as well as the 
deflection corresponding to the peak pressure. The deflection of the wall at collapse 
85 
remains the same for all cases since the method assumes when the wall deflection at 
collapse equals the wall thickness. The effect of the gap length on the SBEDS method is 
shown in Figure 3.31. 
 
Figure 3.31: SBEDS Arching Variable Gap Width 
 The deflection at the beginning of the resistance curve is similar to the deflections 
of the UFC method. The collapse deflection is shifted over by the same amount as the 
initial deflection. The stiffness of the wall is adversely affected by the presence of a gap 
unlike the UFC model. The larger gaps have a greater effect on the peak resistance 
compared to the UFC method.  
86 
 How the models are affected by an individual characteristic is important. Another 
critical comparison is how the models compare to one another. Figure 3.32 compares the 
peak resistance of each of the models. In almost every instance the Wiehle method 
produced the largest resistance. These resistances were significantly larger in some cases 
such as the 12-in CMU wall. The Moradi method fell between the other two models in 
most cases. The SBEDS method always produced the smallest resistance. This is due to 
the over simplification of the SBEDS method. The low peak resistance can be seen as 
conservative for design. The European bricks produced a larger ultimate pressure for all 
methods. As the thickness of the wall increased, the peak resistances increased as well for 
all methods. The gap width does affect the peak resistance of the wall adversely as 
expected.  
The energy dissipation is an approximate method to determine the effectiveness 
of a wall?s resistance to blast loading. A comparison of the energy dissipation for the 
different methods is shown in Figure 3.33. For this reason it is used to compare various 
resistance functions for the same wall. Each class of wall has the same properties the only 
difference is the method used to calculate the pressure and deflection. The ideal system is 
one that dissipates more energy. However, the method used to calculate it should not 
overestimate the resistance of the wall. 
 
 
Figure 3.32: Arching Ultimate Resistance Comparison 
82
 
88 
 
Figure 3.33: Energy Dissipation for Arching Methods 
3.4 SDOF Implications 
 The static resistance function is an integral part of the SDOF model. The concern 
is not how much energy a wall can dissipate but how much it will deflect under a certain 
load. The amount a wall deflects must be controlled for blast applications. The response 
limits proposed by Oswald and Zehrt (2010) are shown in Table 3.7.  
89 
Table 3.7: Response Criteria for Masonry Blast Design (Oswald and Zehrt 2010) 
 
Based on this response criteria a load was selected that would push the limits of a 
simply supported wall. The loading used is shown in Figure 3.33. 
 
Figure 3.34: Loading used in SDOF  
An SDOF model was used to calculate the midspan deflection. Due to the 
fracturing of CMU during a dynamic load, the properties of the CMU change from the 
first cycle to the second. The resistance functions do not account for property changes. 
For this reason the SDOF model is only accurate in determining the first peak. A detailed 
look at the SDOF process is outlined in Chapter 7. With all of the wall properties 
90 
remaining the same as the 8-in CMU wall detailed in Table 3.5 the resistance for each 
method was plugged into the SDOF model. The results of the dynamic analysis for non-
arching walls is shown in Figure 3.35. 
 
Figure 3.35: Non-Arching Dynamic Response 
The Moradi method produces the largest deflection of 5.23 in corresponding to a 
rotation of 4.2? at the support. The second largest rotation was the Wiehle method with a 
rotation of 4.0? and lastly was the SBEDS method with a rotation of 3.8?. All three 
deflections were under the deflection to thickness ratio of 0.8. The percent difference is 
only 10 percent from the maximum rotation to the minimum. A 10% difference is a 
reasonable difference given the wide range of assumptions that went into the 
development of each method.  
 For arching wall the same loading and wall was used to generate the resistance of 
the three arching methods. It is expected that the rotation of the arching walls will be less 
than the non-arching walls since the loading will remain the same. The results of the 
dynamic analysis for arching walls are shown in Figure 3.36. 
91 
 
Figure 3.36Arching Dynamic Response 
 As expected, the deflection of all three arching methods is lower than the 
deflections observed in the non-arching analysis. There is a large difference in the 
response of these methods. The method with the lowest deflection is the SBEDS method 
with a deflection of 3.21 in. and a rotation of 2.6?. The Wiehle method produced the 
second largest deflection of 1.18 in. and a rotation of 0.94?. The Moradi method produces 
a deflection of 0.73 in. and a rotation of 0.58?. The difference between the SBEDS 
method and the Moradi method is 126% and the difference between the Wiehle method 
and the SBEDS method is 92%. The difference between the Moradi method and the 
Wiehle method is 47%. These differences are very different and greatly influence the 
response category the wall will fall into. The Moradi and Wiehle methods fall into 
Category 1 where the SBEDS method falls into Category 2. 
92 
3.5 Dynamic Testing  
3.5.1 Testing Overview 
 A test series was performed by AFRL to investigate the effects that boundary 
conditions have on plane masonry subjected to blast loading. In this series four walls 
were constructed all with the same height and wall thickness. Of the four walls 
constructed, two had non-arching supports and the other two had arching supports. For 
each support condition, all walls were constructed with 32 lb lightweight blocks. Walls 1 
and 3 were constructed as arching walls. Walls 2 and 4 were constructed to simulate non-
arching walls. For these non-arching walls, different methods were used to connect the 
wall to the support that would allow for a horizontal reaction to develop but would not 
have a moment when the wall rotated. Three support configurations were used. The first 
used a dowel that would prevent the wall form translating inward but allow rotation at the 
supports. The second placed the CMU blocks against the support and grouted the gap to 
prevent any rotation of the wall. The last configuration placed an angle behind the wall. 
The angle catches the wall and acts similarly to the first configuration. Hoemann et al. 
(2010) detailed the suggested support conditions shown in Figure 3.37.  
 
93 
 a) Walls 1 and 2 b) Wall 3 c) Wall 4 
Figure 3.37: Schematic of Boundary Conditions 
The height of all the walls is 120 in. with a wall thickness of 10 in. All the walls 
were and tested at the same time. The compressive strength of the CMU blocks was 
reported as 2890 psi and a mortar tensile bond of 100 psi.  The support configuration for 
each wall is shown in Figure 3.37. Pressure gages recorded the pressure for each wall. All 
four walls had approximately the same pressure.  
The loading was the same for each wall. The weight and stand off (distance 
between explosive charge and wall) was selected to ensure that collapse would not occur. 
Figure 3.38 presents the normalized measured pressure on Wall 2. The pressure applied 
to each wall was used in the SDOF model however they are all very similar. 
94 
  
Figure 3.38: Pressure on Wall 2 
3.5.2 Dynamic Test Results 
 Wall 1 exhibited an unexpected response. During the post test inspection it was 
noticed that the corners of the support framing around the wall were cracked, which 
likely decreased the stiffness of the wall and allowed for larger deflections. Cracking of 
the frame is shown in Figure 3.39. 
95 
 
Figure 3.39: Support Cracking of Wall 1 
The wall response compaired to the analytical methods is presented in Figure 
3.40. Because the support cracked, the largest deflections happened after the first cycle. 
During the loading significant damage occured in the wall rendering the static resistance 
function inaccurate after the first peak. This is true for all the resistance curves since 
cracking of the wall is assumed for all the walls. 
 
96 
 
Figure 3.40: Wall 1 Dynamic Response   
 The SBEDS method produces a much higher deflection while the other two 
methods generate lower deflections. The maximum deflection of the first phase was 0.54 
in. and 1.63 in. for the second phase. The second phase was three times greater than the 
first phase. The deflection of the negative phase was -1.21 in., 2.24 times larger than the 
first phase deflection. The maximum deflections for this wall and the other three are 
reported in Table 3.8.  
 Wall 2 was constructed with steel dowels grouted into the top and bottom of the 
wall. It is assumed that the grout in the top and bottom row of the CMU will not add 
significant resitance. The deflections from the test and the SDOF analysis are shown in 
Figure 3.41.  
 
97 
 
 
Figure 3.41: Wall 2 Dynamic Response 
 The deflections generated from the SDOF analysis have a smaller spread and 
more closely resemble the test results for Wall 2. The methods do not generate 
significantly different static resistance curves allowing for all three curves to closely 
resemble one another. The equipment used to measure the deflection stopped at 11.2 
msec and continued again at 96.6 msec. Because of this when plotted the response 
appears to be shifted when it is not. A similar problem occurred with Wall 4.The 
response of Wall 3 is shown is Figure 3.42. Wall 3 was another arching wall with only 
the weight of the block and compressive strength changing from Wall 1.  
98 
 
Figure 3.42: Wall 3 Dynamic Response 
 Wall 3 falls between the SBEDS method and the other two in the same way Wall 
1 does. The peak deflection occurs within the first cycle unlike Wall 1 since the support 
frame did not crack. This could be influenced by the wall being in the center of the 
support frame where Wall 1 is located at the edge.  
 Wall 4 was the last wall in this test series and was constructed to represent a non-
arching wall. Figure 3.43 shows the boundary conditions of Wall 4.The dynamic 
response is shown in Figure 3.44.  
99 
 
Figure 3.43 Boundry Condition of Test Wall 4 
 
 
Figure 3.44: Wall 4 Dynamic Response 
 Wall 4 had a similar response at Wall 2. One important factor is the walls ability 
to move out of the frame since it is only restrained from moving inward. During the post-
100 
test inspection Figure 3.45 was taken to show how Wall 4 pulled out from the support 
frame.  
 
Figure 3.45: Wall 4 Pullout of the Frame 
This skews the later cycles but does not affect the peak deflection. After the test the wall 
was pulled 2 in. out of the support frame. All three methods produced deflections greater 
than the test results.  
Table 3.8: Wall Response Comparisons 
Wall/   
Method 
Wiehle SBEDS Moradi Test 
in. in. in. in. 
1 0.140 1.23 0.096 0.578* 
2 1.446 1.558 1.599 1.636 
3 0.161 1.083 0.111 0.782 
4 1.746 1.954 1.803 1.478 
* Maximum deflection for first cycle only  
 
101 
Walls 1 and 3, arching walls, have a significant spread with the test data falling in 
the middle. The first wall has a strange anomaly where the peak deflection occurs on the 
second cycle. Table 3.8 only lists the deflection of the first cycle; the true maximum 
deflection is 1.62 in. The SBEDS method was developed with assumptions that 
overestimating the deflection of the wall. The other two methods produce similar results 
however these results under predict the maximum deflection making the design unsafe. 
The non-arching walls, Wall 2 and 4, all had very similar results. This is because the 
methods used to generate the resistance curves use very similar assumptions and only 
have slight differences.  
3.6 Conclusion 
Based on the results presented, conclusions can be made as to which method 
should be used for design. The non-arching resistance functions produced reasonably 
close results to the test data whereas the arching methods that over predicted or under 
predicted the deflection. For design purposes, the method that over predicts deflection 
will generate a safer design. 
The Wiehle method accurately predicted the response of the wall during the 
dynamic test. For this reason the Wiehle method was chosen as the method to generate 
the non-arching resistance function in Chapter 7.  
None of the arching methods did an accurate job of predicting the response of the 
wall for the dynamic test. For this reason the method used should generate the safest 
design without over designing the retrofit. The SBEDS method is the only method that 
over predicted the deflection. Over predicting the deflection will cause the retrofit to be 
overdesigned in order to be able to meet the response criteria.   
102 
 
Chapter 4: Comparison of Unbonded Membrane System Analytical Models 
4.1 Overview 
 As previously stated in Chapter 2, three different approaches to the resistance 
function for the unbonded system have been used and presented in the literature. 
However, no current used method incorporates the connection ductility in the resistance 
definition. The development of an analytical method that includes the connections will be 
discussed in Chapter 5.  
 The major challenges of developing a resistance function for the membrane are 
the large displacements and the nonlinear material behavior. The pressure will always act 
perpendicular to the surface of the membrane. At large displacements, the shape of the 
surface needs to be known in order to know the force within the membrane that has 
developed to resist the loading. Since most of the material will exhibit softening the 
material definition is no longer a linear definition complicating the problem.  
It will be shown that these three methods do not coincide with one another but it 
will be shown that the parabolic method should be used since it coincides with the FE 
model the most. Because of the large displacements and nonlinear behavior of the 
material a program should be used to develop a resistance curve since they all require 
incrementing the deflection or pressure in small steps.  
103 
4.2 Cable Analogy Method 
 The first method developed has the easiest computations. The function was 
developed for a cable with no flexural resistance subjected to simultaneous lateral and 
axial loads. This method was presented by Young et al. (2012). The method only uses the 
cross-sectional area (A), length of the membrane (L), and the modulus of elasticity (E) as 
a function of stress. The axial tension is assumed to be the only force contributing to the 
stress. The stress in the membrane is calculated using Equation 4.1, the deflection in 
terms of pressure can be derived, Equation 4.2. To be able to calculate the stress, the axial 
force will be needed, Equation 4.3. 
  rPA??   4.1 
 1/33
64 tpLL EA???? ????
  4.2 
 28
r plP? ?
  4.3 
Where ? is the stress in the extreme fiber, Pr is the axial load in the retrofit, ? is the 
deflection at mid-span, Et is the tangent modulus of elasticity from the previous step, and 
p is the uniform lateral load applied. If a unit width is considered, the area of the retrofit 
reduces to the thickness of the material, A=ts.  
 This method was originally developed for a linear elastic material. Because of the 
nonlinear aspect of the material definition an incremental approach was taken. The 
pressure is incremented and the deflection of that change in pressure is then calculated. 
To incorporate this additional deflection Equation 4.4 is used.  
104 
 333 1,i i p inc?? ? ? ? ?  4.4 
Where ?i  is the total deflection for that step, ?i-1 is the total deflection from the previous 
step, and ?p,inc is the deflection due to the change in pressure. 
 The process of developing the resistance curve is detailed in the flow chart in 
Figure 4.1 and a sample of the calculation is shown in Figure 4.2. The process is 
continued until the membrane fails. The membrane can fail in one of two ways. The first 
is if the stress developed within the membrane exceeds the tensile limits of the material. 
The second failure method is when the membrane exceeds a predetermined displacement. 
A recommended value for the maximum deflection is between 15 and 20 in. At this 
displacement the FE model shows that the membrane starts to transition from a parabolic 
shape to a circular shape. The second reason this deflection is chosen is because the 
response limits set by UFC 3-340-02 are exceeded at these deflections.
 
 
 
Figure 4.1: Cable Resistance Function Development 
 
 
   
105
 
 
 
 
 
Figure 4.2: Sample Calculation for Cable Method 
 
106
 
107 
4.3 Seide Resistance Function 
 Paul Seide (1977) generated a formula for large deflections of rectangular 
membrane. The membranes are assumed to have edges that are fixed against rotation but 
with no in-plane shear stress and fixed from translation. All edges have these support 
conditions, which is not how the membrane is attached in reality. The formula developed 
works reasonably well for walls with larger width to length ratios (Seide 1977). For this 
reason, Sudame (2004) chose to use this formula to compare to finite element models of a 
retrofit system. The formula created by Seide (1977) is the simplest method for 
generating a resistance curve. The two formulas used to generate the resistance curve are 
Equation 4.5 and Equation 4.6. Equation 4.5 is a constant used to simplify the 
calculations. 
 23 3*(1 )j ???  4.5 
The formula for center deflection of the plate in terms of pressure is: 
 4
3 4? 4 s
ts
pbt Etj?   4.6 
In these equations, ? is Poisson?s ratio, ? is the center deflection, ts is the thickness of the 
membrane, p is the lateral pressure, Et is the tangent modulus of elasticity and b is the 
unsupported length of the membrane (Sudame 2004). The strain should be calculated to 
determine if the membrane has exceeded its ultimate strength and to determine the 
tangent modulus of elasticity. A formula proposed by Sudame (2004) for the strain at 
mid-span is presented as Equation 4.7.  
108 
 23
2s st pbj Et?
??? ??
??
  4.7 
This strain definition does not reasonably represent the strain compared to the 
other methods. For this reason the strain is calculated the same way as the parabolic 
method presented in section 4.3. The research done by Sudame (2004) concluded that the 
formula closely resembled the resistance of the finite element model. The analytical 
function overestimated the pressure by about 5% to 10% around 15 in of deflection 
(Sudame 2004). The process for developing a resistance curve is simple with the process 
shown in Figure 4.3. The failure limits are the same as those specified in Section 4.2. 
 
Figure 4.3: Flowchart for Seide Resistance Curve 
  
4.4 Parabolic Deflection Method 
 The method developed by Lane (2003) assumes that the membrane with a unit 
width deflects in a parabolic shape. This method starts by using geometric properties to 
109 
assess how the membrane reacts and then correlates the change in length to strain and 
stresses to determine the pressures in the membrane. This method is one of the most 
widely used methods by researchers to compare experimental works to an analytical 
model. This model is used by Lane (2003), Fitzmaurice et al. (2006), Salim et al. (2007), 
Johnson et al. (2009) and Hoemann et al.  (2010). The development of this method uses 
the midspan of the membrane as the origin, refer to Figure 4.4. Assuming a parabolic 
shape, the general form of the equation is: 
 ? ? 2y x A x B x C? ? ?  4.8 
Using the boundary conations below, A, B, and C can be determined. 
 ? ?? ?
? ?
' 0  at  0  
? ?at? 0
0  at  / 2
y x x
y x x
y x x L
??
??
??
  4.9 
The slope of the membrane is need and found by taking the first derivative of the general 
form giving: 
 ? ?'2y x Ax B??  4.10 
Using Equations 4.8 and 4.10 and substituting the boundary conditions into them, the 
coefficients can be determined.  
 
2
4?
0
?
A L
B
C
??
?
?
  4.11 
Substituting the coefficients back into Equations 4.8 and 4.9 results in: 
110 
 ? ? 2? 1 4 xyx
L
??????????
????
  4.12 
 ? ?
28?'y x xL??
  4.13 
The next step in deriving the resistance function is to solve for the arc length as a 
function of mid-span deflection. The equation for arc length is: 
 ? ?/2 2
0
' 2 1 ( ' )LL g x???   4.14 
Substituting 4.11 in for g?(x), the arc length of the membrane can be solved.  
 2 4 3 2 4 3 2
2 2 2 2
4 ?' l n l n4 6 4 ? 3 2 ? 2 4 6 4 ? 3 2 ? 8 ?L L L L L L L LL L ????? ? ? ? ? ???? ? ? ? ? ???? ? ? ? ? ?
? ? ? ? ? ?????
  4.15 
The arc length is used to determine the strain in the membrane at each iteration using 
Equation 4.16.  
 'LLL? ??  4.16 
Using a true stress-strain curve the stress can be determined. Using the FBD in Figure 
4.4, the next step is to sum the forces in the y-direction in order to relate pressure to a 
known variable.  
111 
 
Figure 4.4: Free-Body Diagram of Deformed Membrane 
 0 2 s in ( )yF p L T ?? ? ??   4.17 
Substituting equation 4.11 in to 4.17 and solving for ? results in: 
 
28?2 sinT x p LL????????
  4.18 
It can be shown that T=?A where A is the area of the membrane. If only a unit width is 
considered the area of the membrane can be reduced to A=ts where ts is the thickness of 
the membrane. Since the angle of interest is at the ends, L/2 can be substituted in for x. 
Implementing these substitutions and solving for pressure gives: 
 2 4?sinstp
LL? ??? ????
  4.19 
This process is complex and requires a true stress true strain curve. The process for 
creating a resistance curve is detailed in the flow chart in Figure 4.5. 
 
112 
 
Figure 4.5: Flow Chart of Lane?s Resistance Curve 
4.5 Stress Strain Conversion 
 Johnson et al. (2009) discuss the importance of using the true stress conversion 
from the coupon test. The engineering stress/strain relationship does not consider the 
change in cross-sectional area as the specimen elongates. The true stress of a material 
will be larger because of the decrease in area. This larger stress will generate a greater 
pressure for the same deflection according to Johnson et al. (2009). However, the strain 
conversion was not considered. Engineering strain is calculated from the arc length and 
used to find the true stress in the material. The proper way to use the stress strain curve is 
to use the engineering stresses if the area is assumed constant or to update the area of the 
membrane and convert the strain values as well. 
 The engineering strain is only valid for small displacements. As the strain 
increases the cross-sectional area decreases. Some of the materials considered can have 
rupture strains of several hundred percent. There is one method to achieve true strain and 
is derived in Equation 4.20. 
113 
  
 
00 0 0l n ( ) l n ( ) l n ( ) l n
fL fLft L f
L
LdL L L LLL? ??? ? ? ? ? ??
???
  4.20 
 Two methods have been developed to convert engineering stress into true stress. 
The difference between these methods depends on the use Poisson?s ratio. The first 
method does not use Poisson?s ratio. Constant volume is assumed and there are no 
localized deformations for the first method. This method is derived in Equations 4.21 
through Equation 4.24.  
 00AL AL?   4.21 
 00
0 0 0 0w h e r e  1 1 e
A L LL L LA L L L L ??? ?? ? ? ? ? ?  4.22 
 0
1 eAA ????
  4.23 
 
0 (1 )t
PPAA??? ? ?  4.24 
Where L0 and A0 are the initial length and cross-sectional area, P is the axial load, ?t is the 
true stress, ?e is the engineering strain. The second method looks at the change in cross-
sectional area and incorporates Poisson?s ratio. The second method for converting stress 
is derived below. 
 
 
e Pab??
  4.25 
114 
 ''
t Pab? ?
  4.26 
 if x y z? ? ???  4.27 
 ,,
,
t y z
tx
?? ???   4.28 
 ,,' (1 )t x t xa a a a?? ??? ? ? ?  4.29 
 ,,' (1 )t x t xb b b b v?? ?? ? ? ?  4.30 
 2,' ' ' (1 )txA a b a b v?? ? ?  4.31 
 0ePA??   4.32 
 0
22,,' (1 ) (1 )e e et t x t xA a bA a b v v? ? ?? ??? ? ???
  4.33 
The effects of Poisson?s ratio can change the outcome substantially. Figure 4.6 
shows how these two methods vary. The first method is referred to as the logarithmic 
stress/strain method and the second is the Poisson?s ratio method. 
115 
 
Figure 4.6: Stress Conversion Comparison 
While the selection of the proper conversion method is important for high strains 
there is a limit to the amount of deflection that the wall can exhibit. At a rotation of 16?, 
double the allowable rotation, for a 10 foot wall the deflection is 17.2 in. and corresponds 
to a strain of 0.051 in/in. This strain is low enough that the various methods do not have a 
significant impact.  
4.6 Comparison of Resistance Functions  
 These three methods produce very similar resistance curves. Figure 4.7 is the true-
stress true-strain definition used to compare these methods. Figure 4.8 shows how these 
methods compare to one another. Using a retrofit system that has the properties: 
116 
 L=120 in 
Thickness=0.079 inches 
 Modulus of Elasticity = 610,000 psi 
 Yield stress = 2500 psi 
 Poisons Ratio (?) = 0.2 
And the stress-strain curve in Figure 4.7 
 
Figure 4.7: True Stress-Strain Curve for Polypropylene 
  
117 
 
Figure 4.8: Comparison of Membrane Resistance Curves 
 These methods all produce similar resistance curves despite the different 
assumptions. 
 The material used changes how the resistance curves function. How these 
methods are effected by the use of a different material is shown in Figure 4.9. Using A36 
steel greatly increases the elastic phase of the area under the curve.  
118 
 
Figure 4.9: Steel Membrane Resistance Curve Comparison 
The steel has a higher modulus of elasticity and will allow for much larger 
pressures to be developed for the same thickness. At 15 in. of deflection all the methods 
produce a pressure of about 25 psi compared to 5 psi for the polypropylene.   
 A second important comparison between the models is the height of the wall. The 
example wall was a 10 ft membrane. A smaller wall will allow for a larger development 
of pressures and can affect the way the resistance functions develop. A 6-ft wall was 
chosen to compare how these curves react. This dimension was not selected as a real 
world example but merely to demonstrate the difference. The effect of wall height on the 
resistance methods is shown in Figure 4.10.     
119 
 
Figure 4.10: Membrane Resistance Curve for 6-ft Wall 
The shorter membrane increases the pressures allowed on the wall. The general 
shape and slopes do not appear to have changed.  
 The ability to change variables and have a complete resistance curve is 
imperative. Each system is unique and should be designed as such. For design purposes 
using the model that produces the strongest wall may reduce the cost of the retrofit but 
may overestimate the strength of the material while underestimating the deflection. 
Figure 4.11 illustrates how changing a variable affects the resistance curve for all three 
methods.  
 
 
Figure 4.11: Complete Comparison of Resistance Functions with Changing Variables 
120
12
 
121 
 
 It is shown that using a shorter membrane with a higher modulus of elasticity and 
yield point will decrease the deflection of the membrane. These methods tend to follow 
one another closely. Any could be used for design purposes if only the membrane is 
considered. However, for the purpose of incorporating the connections a more rigorous 
method is needed.   
 The method proposed by Lane (2003) has been studied extensively.  Kennedy 
(2005), Salim et al. (2006 and 2007) Fitzmaurice et al. (2006) and Johnson et al. (2009) 
have all performed test to evaluate this analytical model. Many of the tests concluded that 
the analytical model represents the membrane sufficiently. There are some deviations due 
to the connection ductility and testing errors. At large deflections the 16 point loading 
frame slips and skews the results (Fitzmaurice 2006). 
4.7 Finite Element Validation of Membrane Methods 
Abaqus/Standard was used to conduct all finite element modeling. Finite element 
modeling was done in three stages. The first stage was to compare all the methods 
described in Chapter 4 assuming a linear stress strain relationship. The second stage 
compared all the membrane resistance functions for nonlinear stress strain relationships. 
The final stage was to validate the algorithm detailed in Chapter 5.  
4.7.1 Linear Material Definition Validation 
For the linear material definition, the membrane was defined using shell elements. 
A 120 in x 120 in membrane was created. The top and bottom edges were confined using 
pinned boundary conditions. All nodes were constrained to prevent transverse movement. 
122 
This boundary condition was used to simulate an infinite wall. 1 in. x 1 in. elements were 
created generating 900 elements and 961 nodes, shown in Figure 4.12. Geometric 
nonlinear effects were taken into account in the FE model. 
 
 
Figure 4.12: Membrane Mesh 
The membrane thickness was defined as 0.039 in. and the modulus of elasticity used was 
1270000 psi. A lateral pressure was applied and the pressure vs. displacement of the 
midspan node was plotted against the different resistance definitions, Figure 4.13. 
123 
 
Figure 4.13: Linear Material Resistance Curves 
 The FE model corresponds well with all the methods. The parabolic method 
however was closer to the FE results. At larger deflections, the parabolic method deviates 
slightly due primarily to the change in shape of the membrane. At large deflections and 
pressures the membrane assumes a cylindrical shape rather than a parabolic shape that is 
assumed, Figure 4.14. 
124 
 
Figure 4.14: In-Plane Stresses of Membrane at 60 psi 
The strain is assumed to be constant throughout the height of the membrane in all 
the algorithms.  The FE results show this may not be true; however the percent difference 
is only 0.004%.  
4.7.2 Nonlinear Material Definition Validation 
 For the nonlinear material modeling, the elastic model was used where only the 
material properties were changed to allow for a plastic definition to be defined. The stress 
strain points used for both the FE model and the algorithms is presented in Figure 4.15. 
125 
  
Figure 4.15: Stress Strain for Nonlinear Material Comparison 
A more robust definition of the material could be used in the algorithms however 
the FE model has difficulty with a large amount of data points in the stress strain 
definition. For comparison purposes however a simple definition will suffice. The results 
were plotted against the same methods, Figure 4.16, and the same stress-strain points 
were used in the numerical method.  
126 
 
Figure 4.16: Nonlinear Material Comparison 
 Similar to the linear analysis, the parabolic method better correlates to the FE 
model. The cable method as well as the Seide method have some difficulties when the 
stress strain diagram is not perfectly linear. These methods were adapted from their 
specified use to be able to use a tangent modulus instead of a constant modulus of 
elasticity. It is assumed that the stress in the membrane is constant throughout the length. 
Figure 4.17 shows the FEM results of the principal stress throughout the membrane. 
127 
 
Figure 4.17: Principal In-Plane Stresses for Nonlinear Model 
The stresses in the membrane are completely uniform for the nonlinear material 
definition. The deflected shape at 20 psi shows the non-parabolic shape that result in very 
large pressures.  
  
128 
 
Chapter 5:  Resistance Function Incorporating Connections 
5.1 Overview 
 Many laboratory tests have been performed to understand how the connection will 
affect the ductility of the retrofit system (Fitzmaurice et al. 2006, Salim et al. 2007, 
Johnson et al. 2009, and Hoemann et al. 2010). Johnson et al. (2009) conducted a loading 
tree test on several samples of a polypropylene composite with several connection 
configurations. It was observed that the test produced a much more ductile system than 
the analytical models when the connections are not considered. 
 Considering an unbonded system with a simple plate that does not allow for local 
stresses concentrations at the corner of the plates, an analytical model was developed to 
incorporate the membrane and the connections as one system. The analytical model will 
be compared to a finite element analysis. 
5.2 Unbonded Retrofit Connection Resistance Function Development 
 To develop a resistance curve for an unbonded retrofit system, the first step was 
to determine what method should be used for the resistance of the membrane. The 
parabolic method was chosen because of its accuracy. It is assumed that the membrane 
will react the same with and without the connections and that the plates will add 
flexibility to the system as it is bent upwards.  
 Two significant assumptions were needed to define how the plates will react to 
the loading. The first is that the plate will be made of a material that behaves elastic-
129 
perfectly plastic and forms a plastic hinge. The second assumption is that the plate can be 
modeled as a cantilever beam with the fixed support being at the location of the center of 
the bolt. This assumption is illustrated in Figure 5.1 along with the anticipated curvature 
of the plate.  
 
 
 
Theoretical Behavior 
Static Loading 
(Johnson 2009) 
Blast Testing by AFRL 
Figure 5.1: Plate Response 
 Depending on the state of the plate, elastic or plastic, one of two methods is used. 
If the plate is elastic a small displacement is added to the membrane and the rotation of 
the plate is calculated and the additional displacement from the plate is added to the total 
displacement. If the plate has reached its plastic moment capacity, a small rotation of the 
plate is added. With this new rotation, the tension in the membrane is solved, which will 
generate the plastic moment in the plate. How the plate moves with respect to the elastic 
and plastic phase is illustrated in Figure 5.2. 
130 
 
Figure 5.2: Rotation of the Plate for Elastic and Plastic Phases 
 To start the development of the resistance curve of the retrofit system is similar to 
the membrane resistance curve. A small displacement, ?, of the membrane is placed on 
the system and the arc length, L?, of the membrane is calculated using the distance 
between the tips of the plates as Li. If there is no slack in the system, the initial length and 
the distance between the supports are the same. At the first iteration ?=?. 
 2 4 3 2 4 3 2
2 2 2 2
4 ?' l n l n4 6 4 ? 3 2 ? 2 4 6 4 ? 3 2 ? 8 ?i i i i i i i i
i
L L L L L L L LL L ?? ??? ? ? ? ? ???? ? ? ? ? ???? ? ? ? ? ?
? ? ? ? ? ?????
  5.1 
Using the arc length and the original length of the membrane the true strain in the 
membrane can be calculated.  
 'ln 1 iLL
L? ?????????
  5.2 
131 
With this new strain the thickness of the membrane can be updated to account for 
Poisson?s effect.  
 it t t???? ?  5.3 
Using a true stress-strain curve the stress in the membrane can be determined. This 
process can be adapted if engineering stress strain definition is used. The reason this is 
allowable is for a 12 ft membrane with 12 in. of deflection the engineering strain is only 
2.6%. Once the stress is known the pressure required to cause that stress can be 
calculated.  
 2 4sini
i
tp LL? ???? ????  5.3 
In order to determine the force on the plate, the axial force within the membrane is 
required.  
 
42sin
i
pLT
L
? ???
????
  5.4 
The forces applied to the plate are shown in Figure 5.3. 
132 
 
Figure 5.3: Angle Detail of Plate and Membrane 
The angle of the retrofit is: 
 4
iL?
??   5.5 
The vertical and horizontal components of the tensile force in the membrane at the plate 
are: 
 cos( )vTT ??   5.6 
 sin( )hTT??   5.7 
Using beam mechanics the rotation at the tip of the plate and the vertical displacement 
can be calculated.  
 2
2vpTLEI??
  5.8 
133 
 3
3vpp TLEI? ?
  5.9 
The geometric effects from the bending of the plate are taken into account and used to 
find the moment at the bolt. For the elastic phase the shortening of the plate is assumed 
negligible.  
 bolt v p h pM T L T ???  5.10 
The moment at the bolt location is checked against the plastic bending moment to 
determine if the plate is still elastic.  
 pyM Z f?   5.11 
If the plate is still elastic the rotation of the plate tip increases the angle of rotation of the 
membrane and increases the deflection. This additional displacement is calculated by 
taking the change in the plate rotation and calculating how that effects the displacement.  
 4
plate L????
  5.12 
Where ?? is the change in rotation of the plate. The total displacement of the membrane 
is the summation of the displacement of the membrane plus the deflection added by the 
plate. 
 total plate? ????   5.13 
The total displacement of the membrane is used in the following step with an additional 
displacement increment. 
134 
 If the moment of the plate at the bolt reaches the plastic moment capacity a 
separate process must be used. Instead of incrementing the displacement of the 
membrane, the plate is rotated about the base. 
  1i i increment? ? ????   5.14 
For the first step of the plastic phase ?i-1 is equal to zero. With this new rotation the 
displacement at the tip of the plate can be calculated. The shortening of the plate is 
significant from this point forward and used in the calculations. 
  ' (1 sin( ))ppLL ???   5.15 
 , sin( )p p elastic? ? ???  5.16 
With the upwards deflection of the plate, additional membrane material will be added to 
the initial length of the membrane. A routine to determine what displacement corresponds 
to this additional material is needed. This routine finds what deflection will correspond to 
the zero strain state in the material. Figure 5.4 is the routine created to determine how the 
slack will affect the deflection. 
135 
  
S l a c k L L s u p p o r t??? ? ? 0?
?
1
0?
n 2?
?
n
?
n 1?
0 . 0 0 1??
A r c 4
?
n
L s u p p o r t
L s u p p o r t
2
4
L s u p p o r t
4
64 ?
n? ?
2
?
?
L s u p p o r t? ?
3
32 ?
n? ?
2
??
?
??
??
?
??
ln
L s u p p o r t
2
L s u p p o r t
2
4
L s u p p o r t
4
64 ?
n? ?
2
?
??
??
?
?
?
??
?
?
?
??
L s u p p o r t
3
32 ?
n? ?
2
??
?
??
??
?
??
ln
L s u p p o r t
2
8 ?
n
?
?
?
?
?
?
?
?
?
??
?
?
?
??
?
?
?
??
?
A r c L?
L
?
?
?
?
?
?
?
n n 1??
? 0 . 0 0 0 0 0 0 0 0 0 1?w h i l e
?
n 1?? ?
??
 
Figure 5.4: Routine to Determine Deflection with Slack in Membrane 
With the above routine the deflection of the slack alone can be found. The rotation at 
ends of the membrane is needed to determine what the tension in the membrane will 
maintain a plastic hinge in the plate.  
 
1 4 slacki L??? ???
  5.17 
With this as an initial trial value for the rotation, the tension in the membrane can be 
calculated.  
 
' c o s ( ) s in ( )pppMT L ? ? ?? ?
  5.18 
Using numeric iteration the deflection that corresponds to this tension can be 
found. Once the deflection that causes the tension in the membrane is known the 
deflection associated with the slack is added to this deflection. The actual rotation of the 
membrane at the support is calculated using Equation 5.5 and the pressure is calculated 
using Equation 5.3. The complete routine can be found in Appendix A. 
136 
 The addition of the plate into the routine will increase the deflection of the 
membrane for a given pressure. A sample resistance curve is shown in Figure 5.6 with 
retrofit properties listed below and the true-stress true-strain definition shown in Figure 
5.5.  
Length of Membrane 120 in. 
Thickness of Membrane 0.079 in. 
Thickness of Plate 0.125 in. and 0.25 in. 
Length of Plate (tip to bolt centerline) 3 in. 
Yield Stress of Plate 36 ksi 
`  
Figure 5.5: True Stress True Strain of Membrane 
137 
 
Figure 5.6: Effect of Plate on membrane Resistance 
The thickness of the plate contributes greatly to the resistance of the membrane. 
When a plastic hinge forms in the plate, the resistance of the system is controlled by the 
yield stress in the plate since the plate is assumed to exhibit elastic-plastic behavior. The 
ideal system will have a plate that will not yield and allow the membrane to soften.    
5.3 Finite Element Validation of the Plate Algorithm  
 The FE model for membrane with plate connections required a more complex 
model. Due to its size, only a quarter of the retrofit was modeled and symmetric 
boundary conditions were used. The bolt spacing and size was not explicitly modeled. 
The algorithm assumes the bolt spacing is small enough that local effects on the plate are 
not present.  
138 
 The reaction block was modeled with 8 node 3-D stress reduced integration solid 
elements and very stiff material properties. The size of the solid elements are 1 x 1 x ? 
and the elements are fully elastic with a modulus of 29 x 109 psi. The membrane was 
modeled using 4 node full integration shell elements. Unlike with the membrane only 
models, the membrane not only spanned vertically but also was bent over to be 
sandwiched between the steel plate and the reaction block. The vertical portion of the 
membrane used 1x1 elements and the horizontal portion used 1x1/4 elements. The steel 
plate was modeled using shell elements as well with a mesh size of 1x1/4. The steel plate 
had to have a small mesh in order to fully encapsulate the stress profile from the line of 
bolts the tip of the plate. The mesh and geometry of the model is shown in Figure 5.7. 
139 
 
Figure 5.7: Mesh and Geometry of FE Model for Full Retrofit System 
 The vertical span of the membrane was 60 in. and with a symmetric boundary 
condition this corresponds to a 120 in. clear span. The depth of the plate and support 
block were 6 in. and the nodes at the halfway point, 3 in., were tied together to simulate 
bolt connections. The entire model width is 18 in. with symmetric boundary conditions. 
In all there are 3040 node and 2376 elements 1944 of which are shell elements.  
 The purpose of the FE analysis was to determine the validity of the algorithm 
presented earlier in this chapter. Many different material and cross-section variations 
were studied. Only a few will be discussed in this chapter the rest can be found in 
Appendix B. 
140 
 Two main factors control the strength of a retrofit for a given wall, the membrane 
material and the membrane/plate thickness ratio. Two types of material were considered 
for the membrane material, A36 steel and a woven polymer. Three plate thicknesses were 
considered: 1/8 and 1/4 in. plates. All plates are A36 steel. The membrane thicknesses are 
1 and 3 mm for the polymer and 20 ga for the steel. Table 5.1 lists all the different 
configurations. The results of the analysis for configuration 1 is shown in Figure 5.8. 
Table 5.1: Retrofit Configuration 
Configuration Plate Thickness (in) Membrane Material Membrane 
Thickness (in) 
1 1/8 Polymer 0.039 
2 1/8 Polymer 0.118 
3 1/8 Steel 0.0359 
4 1/4 Polymer 0.039 
5 1/4 Polymer 0.118 
6 1/4 Steel 0.0359 
 
141 
 
Figure 5.8: Retrofit Configuration 1 for FEM 
 Configuration 1 produces relatively accurate results with a maximum error of 
30% at 6 in. of deflection. The point with the most discrepancy is the point where the 
plate transitions from elastic to plastic. The cause of the difference is the algorithms 
ability to make the transition from elastic plate to plastic plate gradually and with small 
steps. As a result, there is a large jump in deflection when the plate transitions to its 
plastic state. The last cause for the discrepancy is the membrane will push along the 
plate?s length reducing the moment arm. The results for Configuration 2 are shown in 
Figure 5.9. 
142 
 
Figure 5.9: Retrofit Configuration 2 for FEM 
 The same problem with the transition to the plastic plate state persists in 
configuration 2. However, a more important observation is the increase in membrane 
thickness does not increase the resistance of the retrofit. As the thickness of the plate 
decreases, the system starts to be controlled by the strength of the plate. For an 
economical system, there must be a balance between the thickness of the plate and 
membrane. The results for Configuration 3 are shown in Figure 5.10. 
143 
 
Figure 5.10: Retrofit Configuration 3 for FEM 
 Configuration 3 uses a steel membrane. One important consideration to notice 
about the difference between these results is what happens after 13 in. of deflection. The 
algorithm assumes that the membrane will always bear on the plate and bend it. The FE 
model shows that after a certain amount of rotation of the plate, the membrane will start 
to transmit the load directly to the bolts. When the membrane begins to bear directly on 
the bolts, large stresses are developed and cause the membrane to tear along the line of 
bolts. In laboratory test, when the membrane starts to bear on the bolts significant bearing 
stresses rapidly develop and the membrane fails along the bolt line (Johnson 2009). 
Figure 5.11 illustrates the tension failure observed in testing performed by Johnson 
(2009). Figure 5.12 is the results of the analyses for configuration 4. 
144 
 
Figure 5.11Bearing Failure of Membrane 
 
Figure 5.12: Retrofit Configuration 4 for FEM 
 Configuration 4 is similar to 1 except the plate thickness increases. This increase 
in plate thickness allowed for the membrane to develop larger tensile forces and greater 
pressures. From this observation it can be noted that the ability of the plate will control 
the overall resistance of the system as long as the membrane is strong enough. The results 
of the analysis for Configuration 5 is shown in Figure 5.13. 
145 
 
Figure 5.13: Retrofit Configuration 5 for FEM 
 The FE model for configuration 5 would not run after 1.12 psi of pressure was 
applied. This is the range in which the plate has reached the plastic moment capacity. The 
plate has completely yielded at this pressure. The FE model becomes unstable and will no 
longer run beyond this point unless post-yield hardening slope is defined for the plate. 
Figure 5.14 show the results of the analyses for Configuration 6. 
146 
 
Figure 5.14: Retrofit Configuration 6 for FEM 
 With a larger plate the membrane does not begin to bear on the bolts as they did 
in Configuration 3.  Even though the steel membrane has a higher modulus of elasticity 
and significantly higher yield point it did not produce substantially larger pressures. This 
again is contributed to the plate controlling the system. The tension in the membrane that 
causes a plastic hinge to form is approximately the same for both materials. Therefore, 
even though the steel is stronger it will not produce a higher resistance curve. 
 Overall the results produced by the algorithms at most underestimate the 
resistance by 30% which is reasonable considering the assumptions made. One 
consideration that was not made in the algorithm is the idea that the membrane is bearing 
along the length of the plate. The algorithm assumes the force of the membrane is 
transmitted to the plate only at the tip. The load is actually distributed along the plate 
which reduces the moment arm that will generate a moment at the hinge. This will 
147 
account for the FE model being larger since the plate will remain elastic longer. For 
design purposes, the deflection is what will control the design in the SDOF model. For a 
given deflection the resistance is typically smaller resulting in a conservative design.  
  
148 
 
Chapter 6: Dynamic Modeling and Comparison 
6.1 Overview 
 SDOF is the most commonly used method for design of wall components. The 
static resistance is a critical aspect of the SDOF methodology. The SDOF methodology is 
largely compiled by Biggs (1964), and is general for any linear or nonlinear resistance 
definition. This methodology does not depend on knowing the elastic and plastic modes.  
 This chapter will detail the SDOF methodology and the response criteria set by 
PDC (2008). SDOF will be used to compare the algorithm presented in Chapter 5 against 
two tests. The first test uses a polypropylene membrane and the second uses a PVC 
membrane. 
6.2 Single-Degree-of-Freedom Development 
 The system is idealized as a mass suspended from a spring, Figure 6.1 (b). In this 
system the mass is assumed to remain constant throughout the loading while the 
resistance and force are time dependent.  
149 
 
Figure 6.1: SDOF Theory (a) Real System (b) Simplified Model 
The equation of equilibrium for dynamic systems is: 
 ( ) ' ( ) ''F t C y k y y M y? ? ?  6.1 
Where k(y) is the resistance as a function of the displacement, M is the mass of the 
structure, C is the damping coefficient, y? is the velocity, and y? is the acceleration. For 
dynamic analysis of masonry walls the damping force is not typically considered. Three 
main reasons are stated by UFC: 
1) The first cycle and peak response are the only responses of interest which 
damping has little effect on. 
2) Plastic deformations dissipate significantly more energy the normal structural 
damping. 
3) Ignoring damping will generate larger deflections. 
A fourth reason is given by Kiger and Salim (1998) is that damping of masonry walls 
subjected to explosive loading cannot be experimentally evaluated to determine the 
validity of a damping coefficient. For these reason, damping will not be considered in the 
following SDOF formulation.  
150 
 As a simplified approach, equivalent mass and load factors will be used. These 
factors are generated based on the deformed shape of the structure during the elastic and 
plastic phases. With these equivalent factors Equation 7.1 reduces to: 
 ''( ) ( ) ( )e e eM y t k y t F t??  6.2 
The deflected shapes of the wall are given by Biggs (1963) and the deflected shape of the 
membrane is assumed to be parabolic as discussed in Chapter 4, these deformed shapes 
are shown in Figure 6.2. 
 
Figure 6.2: Deflected Shapes 
The deflected shape for the masonry wall is described by fourth order polynomial 
for the elastic phase and a triangle for the plastic phase described in Biggs (1963). The 
parabolic shape for the membrane is defined in Chapter 4. In order to achieve an 
equivalent factor the deflected shape must be divided by the maximum deflection. This 
results in the deflection being 1 at midspan for all mode shapes. The elastic phase of the 
masonry as defined by Biggs (1963) is:  
 ? ?3 3 4
416( ) 25x L x L x xL? ? ? ?
  6.3 
The equivalent mass is found by evaluating:  
 2 ()LeM m x dx?? ?   6.4 
151 
The equivalent mass factor, KM, is the ratio of equivalent mass to actual load mass.  
 2 ()o r  Le
MMt
m x d xMKK
M m L
??? ?  6.5 
Where the actual mass is Mt=mL, where m is the mass per unit length. The equivalent 
load, Fe, is found by evaluating: 
 ( ) ( )L
eF p x x d x?? ?
  6.6 
The equivalent load factor is the ratio of equivalent load to actual applied load, Ft. 
 ( ) ( ) o r  Le
LLt
p x x d xFKK
F p L
??? ?  6.7 
In order for there to be deformation compatibility, ?stiffness is numerically equal 
to the total load of the same distribution which would cause a unit deflection at the point 
where the deflection is equal to that of the equivalent system? (Biggs 1963). From this 
statement it can be said that: 
 e
RLkKKk??
  6.8 
Using these equivalent factors Equation 6.1 can be rewritten as: 
 ''( ) ( ) ( )m t L LK M y t K K y t K F t??  6.9 
152 
The resistance of the wall is equal to the stiffness of the wall times the displacement, 
()R z Kz? . Dividing both sides by the equivalent load factor and defining L
M
KK  as the 
load-mass factor, KLM, the final equation of motion can be written as: 
 ''( ) ( ( ) ) ( )L M tK M y t R y t F t??  6.10 
 For a given deflected shape and boundary conditions the load-mass is constant. A 
wall can have different deflected shapes depending on what stage the wall is in: elastic, 
plastic, or transition. Biggs (1963) lists these factors for various deflected shapes and 
loadings. The ones that are most relevant to this type of system and loading are the 
simply-supported beam and the fixed-fixed beam both with distributed loads. The 
parabolic load-mass factor was derived using the methods and equations listed above. 
Table 6.1 list all the factors for both the elastic and plastic phase.  
Table 6.1: Equivalent Transformation Factors for One-Way Beams and Slabs 
 Equivalent Load Factor  
KL 
Equivalent Mass Factor  
KM 
Equivalent Load-Mass 
Factor  KLM 
Elastic Plastic Elastic Plastic Elastic Plastic 
Simple 
Supports 0.64 0.50 0.50 0.33 0.78 0.66 
Fixed-
Fixed 0.64 0.50 0.41 0.33 0.77 0.66 
Parabolic 
Shape 0.66 0.53 0.80 
 
For masonry walls retrofit with an unbonded membrane, the initial resistance is 
dictated by the masonry wall. Once the wall cracks, there is a moment when the 
membrane may not have the ability to assume the parabolic shape. The resistance of the 
membrane is much greater than the plastic masonry wall at large deflections. The 
153 
membrane will then force the masonry wall to assume the parabolic shape. This transition 
is very small in terms of both displacement and time. For these reasons the plastic load-
mass factor of the masonry will be neglected and only the parabolic load-mass factor will 
be used after the masonry wall goes plastic.  
 Many different approximate methods have been developed to solve the problem. 
Biggs details how to use the constant velocity procedure, linear acceleration method, the 
Newmark ?beta? method, and several finite difference methods. The Newmark ?beta? 
method is one of the most versatile methods. When beta =0 and ? = ? the Newmark 
?beta? method is equivalent to the central difference method.  The central difference 
method is a direct approach for solving for the displacement. At any given time the 
acceleration of the system is: 
 
? ?1122'' i i iy y yz t????? ?
  6.11 
This formula is presented by Chopra (2007) is an explicit solution and does not require 
iteration. This formula allows for the deflection at time ?ti+1 to be calculated. Substituting 
Equation 6.11 into Equation 6.10 and solving for yi+1 results in Formula 6.12. 
 
2
11'' 2
( ) ( ( ) )W he r e : ''
i i i
i
LM t
y z t y y
F t R y tz
KM
??? ? ? ?
??
  6.12 
This formula only works if the time step, ?t, is constant and small enough. If the time 
step is too large the results will be meaningless (Chopra 2007). Chopra?s time step 
requirement is: 
154 
 1
n
tT ???   6.13 
Where Tn, the natural period of the wall system presented by Biggs as: 
 122 LM t
n KMT K? ??? ????
  6.14 
K is defined by SBEDS Methodology (PDC 2008) as the initial elastic stiffness of the 
system. For simply supported walls this is 
33845EIL
 and 
3384EIL
 for fixed walls. If the time 
step over the natural period is less than 0.1 the response is adequate. For this reason 
Chopra (2007) suggest using a time step of 0.01 to 0.02.   
 Knox et al. (2000) describes the method used to combine the different resistance 
functions. At a given displacement, the resistance is the sum of all the parts of the wall 
and the retrofit at that displacement. 
6.3 Blast Loading 
 For design purposes, the airblast loading to the exterior of a building due to an 
external explosion is typically idealized as a uniform pressure over the surface. The 
loading is represented as the pressure at a point vs. time and consists of two phases: 
positive phase and negative phase, shown in Figure 6.3. The positive phase is triangularly 
shaped and will start at the peak pressure and decay. The negative phase results as 
negative pressure forms. The negative phase is less important for design of buildings 
since the peak negative pressure is much less than that of the positive phase. Secondly, 
ignoring the negative phase generates higher deflections.  
155 
 
Figure 6.3: Loading Diagram for Free Air Blast 
 Loading effects from explosives are typically reported as the weight of TNT at a 
standoff distance. Using these two values the pressure-time data was extracted from 
SBEDS for use in the design example presented in Chapter 8.  
6.5 Dynamic Testing 
The Wiehle method is used to generate the resistance of the non-arching walls. 
The SBEDS method is used to generate a resistance definition for arching walls. The 
membrane and plate resistance function used is the one described in Chapter 5. 
6.5.1 Test 1 Using Polypropylene  
 Experimental validation was performed using the test data presented in Johnson 
(2009). The test was performed by AFRL and involves a full scale explosive field test. 
The height of the wall was 110.2 in. and simulated poor construction of the mortar joints. 
The CMU block used is shown in Figure 6.5 along with dimensions.  
156 
  
Figure 6.4: (a) CMU Test Block (Johnson 2009) (b) CMU Dimensions 
The tensile bond between the mortar and blocks was estimated to be 50 psi. The wall was 
retrofit with a 0.039 in. thick membrane of polypropylene, whose stress strain curve is 
shown in Figure 4.6. A steel plate with thickness 0.25 in. and a width of 6 in. was used to 
connect the membrane to the supports. The plate edge was placed 1.5 in. from the wall. 
Bolts were placed along the centerline of the plate spaced at 8 in. on center. Washers 
were used to help connect the bolts to the plate. The washers had a 1.5 in diameter and a 
thickness of 0.25 in. Figure 6.6 shows both the interior and exterior face of the wall.  
  
Figure 6.5: (a) Interior of Test Wall (b) Exterior of Test Wall (Johnson 2009) 
 A detailed pressure history was taken for this experiment. The wall was open in 
the back allowing pressures to develop. This was considered and pressure gages were 
placed both inside the structure and outside. The net pressure on the wall was calculated 
and shown in Figure 6.7 below. 
157 
 
Figure 6.6: Wall Pressure from Test 
 The algorithm does not incorporate holes in the plate and the additional thickness 
the washers add. Lastly the algorithm assumes that the plate and membrane are flush 
against the wall. After the test only the bottom course of CMU remained. Figure 6.8 is 
the exterior of the wall after the test. The results of the test and the analysis are shown in 
Figure 6.9. 
0 100 200 300
T i m e  ( m s e c )
P
r
e
s
s
u
r
e
 (
p
s
i
)
158 
 
Figure 6.7: Post Test Wall (Johnson 2009) 
 
Figure 6.8: Dynamic Response Comparison 
The dynamic analysis resulted in a very close approximation to the actual test. 
The algorithm under-predicted the deflection by 2.3% compared to the measured results. 
This is reasonable considering the factors that did not go into the analysis, such as holes 
in the plate, washer stiffness, gaps between the retrofit and the wall, and effects of high 
strain rate. 
159 
6.5.2 Test 2 using PVC Membrane  
 Test two was conducted by AFRL at Tyndall Air Force Base, Florida, using a 
PVC liner. The wall was constructed to be 120 in. tall using 8 in. CMU blocks with 49% 
of the cross-sectional area being solid and a moment of inertia of 28.4 in? (PDC 2008). A 
1.25 in. gap was present at the top of the wall and steel dowels were grouted into the top 
cells; refer to Figure 3.35 (a) for support schematic. A 40 mil PVC liner was used as the 
membrane material; the stress-strain relationship is shown in Figure 6.10. A 0.25 in. 
aluminum plate, E=10,000 ksi and fy=37 ksi was used for the connections. A 5 in. plate 
was fixed to the floor using concrete nails 1 in. from both edges, as shown in Figure 3.11. 
Figure 3.12 shows the exterior and interior view before testing.  
 
Figure 6.9: True Stress-Strain Plot for PVC Liner 
160 
 
Figure 6.10: Plate Connection for Test 2 
  
(a) Interior  (b) Exterior 
Figure 6.11: Pretest Constructed Wall (a) Interior (b) Exterior 
 The loading was selected to generate significant damage to the CMU wall without 
causing a blowout of the entire wall/retrofit system. Figure 6.13 is a normalized pressure-
time plot of the measured pressure.  
161 
 
Figure 6.12: Normalized Loading for Test 2 
 The wall was damaged and exhibited significant deflections during the test. The 
algorithm over predicted the peak deflection by 8.5%. The peak deflection recorded was 
8.03 in. while the algorithm predicted a deflection of 10.3 in. The assumptions that went 
into the development of the algorithm play a major role in this increased deflection. 
Figure 6.14 compares the measured deflection and the predicted deflection. 
 
Figure 6.13: Test 2 Deflection Comparison 
162 
For this test assumptions that played a role that were not relevant for Test 1 is that 
the connection plate is assumed to be elastic perfectly plastic. Unlike steel, aluminum 
does not have a well-defined yield point. A second factor is the exact dimensions and 
moment of inertia of the CMU blocks were estimated using the SBEDS Methodology 
(PDC 2008). With deflection being a controlling criteria, the algorithm over predicted 
peak deflection by 28% making the algorithm conservative. Figure 6.15 is an exterior 
view of the wall after the test.  
 
Figure 6.14: Post Test Exterior View 
 The top course of blocks was damaged during the testing. The wall was left with a 
permanent deflection of 3.6 in. The membrane remained intact and was capable of 
preventing fragments from entering the building. Figure 6.16 shows a fragment pressing 
on the membrane after the test did not penetrate the envelope of the building. 
163 
 
Figure 6.15: Post Test Fragment Pressing Against Membrane 
 
  
164 
 
Chapter 7: Design Examples 
7.1 Overview 
 The general procedure for the design of a retrofit will be as follows: 
For a given loading: 
1) Determine the resistance of the masonry alone 
2) Determine what level of protection is available 
3) Design an initial retrofit system and the complete resistance definition of the 
wall and retrofit system 
4) Determine what level of protection will be provided by the retrofit 
5) Modify retrofit system if adequate protection is not provided and iterate 
The loading of the structures will be presented as pressure-time data rather than weight 
and standoff.  
7.2 Problem Statement 
 The owner of an existing two story building, shown in Figure 7.1, wishes to 
determine the appropriate retrofit to prevent catastrophic failure. There is a high threat to 
this particular building. The building consists of concrete columns and beams and infill 
masonry walls; Figure 7.1 and 7.2 detail the geometry of the building. The loading on the 
building is shown in Figure 7.3 and the material properties for the membrane are shown 
in Figure 7.4 and Table 7.1 list the wall properties. There are two different walls to 
165 
consider for this building. The first wall is ground floor wall. This wall has boundary 
conditions that allow for it to be considered an arching wall. The second wall is on the 
second floor and the roof load is transmitted to this wall. The second story wall is 
assumed to be simply supported at both ends.  
 
Figure 7.1: Isometric View of Building Identifying Walls 
 
Figure 7.2: Elevation View of Building 
166 
 
Figure 7.3: Design Loading 
The loading on the walls should vary slightly since the second story wall will have a 
larger standoff distance. However, the difference between these two loadings is small 
enough to neglect.  
 
Figure 7.4: Membrane True Stress/Strain 
167 
Table 7.1: Wall Properties 
Wall Thickness 
(in) 
Support Axial Load 
(lb/in) 
Height     
(in) 
#1 8 Arching NA 144 
#2 8 Pin-Pin 100 120 
 
 The design process will look at each wall individually starting with Wall #1. The 
complete design process and calculations are provided in Appendix D 
7.3 Wall #1 
 The first step in the design process is to determine how the wall will react alone to 
determine if any retrofit is necessary. Assuming arching boundary conditions, the 
resistance of the wall was determined using the UFC method assuming no gap was 
present. Figure 7.5 shows the deflection of the wall to its peak deflection.  
 
Figure 7.5: Dynamic Results for Masonry Alone 
 The maximum deflection of the wall alone is 32.1 in. this deflection should be 
compared to the deflection limits from Table 6.4. To prevent blowout, the rotation must 
168 
be less than 8?. With 25.5 in. of deflection the rotation at the support is 24.0? which is 
larger than what is acceptable. For an initial retrofit design the plate will be 6 in. wide 
with bolts along the center line at 8 in. spacing on center. The thickness of the plate will 
be 0.25 in. and the membrane thickness will be 0.79 in.  
 The maximum deflection with this configuration is 21.7 in. and a rotation of 
16.7?. This again is larger than the allowable response limit of 8?. Figure 7.6 is the total 
resistance of the wall and retrofit system.  
 
Figure 7.6: Total Resistance of System for Retrofit Trial 1 
 By inspecting the resistance curve it can be noticed that the steel plate is yielding 
before the membrane can develop significant resistance. To prevent this, a thicker plate 
should be used.  
For the second trial, a 0.75 in. plate and 0.51-in thick polymer will be used. The 
results of this trial are that the deflection is 11.9 in. and a rotation of 9.4?. This system is 
169 
still not adequate and an increase in membrane thickness should be tried. However, this 
system is not very cost effective for both material and labor. Because of these reasons the 
membrane material is changed to A36 steel membrane.   
For the third trial, the steel plate will be 0.057 in. and the plate thickness is 7/8 in. 
The maximum deflection of the system is 10.1 in with a support rotation of 7.97?. based 
on the response limits outlined in Table 2.4, this system brings the wall out of the 
blowout range. The steel plate is more difficult to transport than a polymer and both will 
have to be bent at the support to attach to the supports. 
With this configuration, the maximum deflection will be used to determine the 
forces on the bolts. The axial and shear force on the bolts is 1170 lb and 341 lb, 
respectively. Using a A325 5/8 in. diameter bolt spaced at 8 in. on center the normal 
stress in the bolts is 56,700 psi and the shear stress is 16,000 psi. For Grade A325 bolts 
the normal tensile strength is 90 ksi and the shear strength is 54 ksi. For this 
configuration the bolt design is adequate.  
170 
7.4 Wall #2 
 The design process for Wall #2 will be the same as Wall #1. The membrane used 
will also be the steel membrane. The response of the wall is a complete blowout. 
Mathematically the maximum deflection is 48.6 in. but this simply corresponds to 
complete catastrophic collapse of the wall. 
 
Figure 7.7: Response of Wall #2 without Retrofit 
The first trial retrofit will consists of a 0.102 in. steel membrane with ? in. 
connection plate. This configuration brings the deflection to 12.3 in. with a rotation of 
11.6?. This rotation is still too large.  
The second trial will use a 0.229 in. steel membrane and a 13/16 in. connection 
plate. The response of the system is now 6.80 in. with a rotation of 7.94?. This rotation is 
within the limit to prevent blow out if the wall is flexural only. There is an axial force 
applied to the masonry; however, PDC-TR 06-08 states if load is below 20% of the 
ultimate compressive capacity of the masonry then a flexural response should be used. 
Figure 7.8 shows how the retrofit affected the response.   
171 
 
Figure 7.8: Response Comparison of Wall #2 
With this configuration, the maximum deflection will be used to determine the 
forces on the bolts. The axial and shear force on the bolts is 2,336 lb and 670 lb, 
respectively. Using a A325 5/8 in. diameter bolt spaced at 8 in. on center the normal 
stress in the bolts is 60,900 psi and the shear stress is 17,500 psi. For Grade A325 bolts 
the normal tensile strength is 90 ksi and the shear strength is 54 ksi. For this 
configuration the bolt design is adequate.  
 
 
  
172 
 
Chapter 8: Conclusion 
8.1 Masonry Walls  
 For unreinforced ungrouted masonry, there are many different ways to formulate 
a resistance curve. The first major distinction that can drastically change the resistance 
depends on the support conditions. If the supports do not resist rotation the wall can only 
resist the load using its flexural strength. If rotation is resisted then the wall will have a 
significant compressive force to restore the wall to its original location.  
 For non-arching walls, there are three methods to generate a resistance curve. The 
Wiehle method is the most common and generates a reasonable response. This method 
accounts for most of the geometric and material properties of the wall. The second 
method was generated by PDC and detailed in SBEDS Methodology. This method is 
reasonable and while it generates a graph only using four points, these points correspond 
closely with the Wiehle method. The last method was developed by Moradi and predicts 
the lowest resistance. This method has a lower resistance because it does not account for 
the tensile strength of the mortar bond.  
 For design purposes the best method is one that is reliable without being overly 
complicated to formulate. The Wiehle method was selected as the method that should be 
used. While this method requires more computation than the SBEDS method it can be 
assumed that a program can be used for this incremental method. While the Moradi 
173 
method has the lowest resistance, which may be desirable for design purposes, the 
computation is rigorous and will result in an over designed retrofit system.  
 Arching walls generate a significantly larger peak pressure and therefore a larger 
resistance. However, the use of the arching method can only be used if the supports are 
significantly stiffer than the wall. The support must have significant strength to resist the 
rotation of the wall. For arching walls four methods have been developed, the three 
sources above plus UFC. The Wiehle method is again a simple incremental method. This 
method is widely used and referenced. The method presented by UFC appears to be more 
complicated. However, the UFC method accounts for a gap between the wall and the 
supports. If no gap is present the UFC method reduces to the Wiehle method. The Moradi 
method is again more complicated to use however this method better aligns with the 
Wiehle method. The SBEDS method is drastically different than the other two methods. 
SBEDS should use the method presented in UFC; however, it uses a different method. 
This new method grossly over estimates the area under the curve compared to the other 
two methods. Secondly, the deflection at the peak pressure is much greater.  
 The resistance method that was chosen as the most appropriate for arching walls 
was the UFC method. This method generates results that are logical without being overly 
complicated. This method accounts for the gap of the wall which in reality often exists. 
 8.2 Membrane Resistance Functions 
 The resistance of the retrofit is what typically provides the strength of the wall 
system at larger pressures and deflections. Because it is an unbonded membrane, there is 
a large amount of deflection that is required to generate significant stresses in the 
174 
membrane that will resist the lateral loading. For the membrane alone three models have 
been suggested.  
 The first method was presented by Seide and was developed for plates subjected 
to large displacements. This method can generate relatively accurate results even though 
the boundary conditions that were used to generate this method are not accurate for this 
system. The second method is to treat the membrane as a cable and use formulas 
presented in Young (2012). This method more closely follows the anticipated resistance. 
Both of the previous methods were developed for a linear material definition. These 
methods were modified to use a tangent modulus so they could account for the softening 
of the material. The last method was developed using an assumed deflected shape. This 
last method is the one that has been validated the most with laboratory testing. The 
deflected shape of the membrane does change with very large displacements; however, 
because of the response limits, large deflections of the wall are not allowed so this does 
not have to be changed to a more complex system.  
 For the development of the algorithm to include the plate, the last method, the 
parabolic shape function method, was chosen. This method was the one most used and 
has been validated with testing. This method also allows for forces and stresses in the 
membrane to be calculated which are needed to incorporate the plate?s response. 
 Finite element modeling was conducted to validate these resistance methods. All 
of the methods were relatively close to one another especially during the linear phase. 
The parabolic method most closely followed the FEM and only had significant deviation 
at large deflections for the reasons stated above.  
175 
 One notable observation is that the parabolic method proposes using a true stress 
vs. engineering strain plot to determine what stresses are in the membrane. This was done 
to account for the larger stresses that will develop in the membrane. However, the 
stresses are only larger because the thickness of the membrane is decreasing and this is 
not accounted for in the parabolic method. The proper way to use the stress strain curve is 
to use the engineering stresses if the area is assumed constant or to update the area of the 
membrane and convert the strain values as well. 
8.3 Full Resistance Curve Including Plate Connections Algorithm 
  The algorithm developed that would incorporate the plate generates a lower 
resistance curve than the FE model. The plate is assumed to deflect as a cantilever beam 
with the fixed base being at the center line of the bolts. While the membrane reacted 
against the length of the plate it was assumed that the membrane will transmit the load 
only at the tip of the plate.  
 This last assumption has a large impact on lowering the resistance. The load will 
have a larger moment arm causing plastification of the plate to happen at lower pressures. 
This assumption was made to eliminate the geometric changes of the plate and the 
frictional resistance that can result from the membrane sliding across the surface of the 
plate.  
 This algorithm was validated using finite element modeling. It was shown that the 
algorithm produces lower resistance. Many deferent geometries and materials were used 
in the FEMs to ensure that the algorithm was not specific to a single case. It was shown 
that the plate can drastically decrease the resistance of the retrofit. If the plate is too thin 
176 
it will yield early and diminish a large amount of the strength that may be assumed if the 
membrane is sufficiently attached to the supports. Larger plates however do not increase 
the resistance beyond what the membrane alone can do. The most a plate can do is allow 
the membrane to generate sufficient stresses to resist the lateral load. Another notable 
point that was observed is that for a given system the use of a stiffer membrane material 
will not generate larger resistance. The plate will control what the maximum pressure will 
be as long as the membrane does not fail before the plate. The advantage of stiffer 
materials is that a thinner membrane can be used which will help with both material cost 
and labor cost. 
8.4 Single-Degree of Freedom Analysis 
 Biggs (1964) is the standard that has been used to determine the maximum 
deflection of the system. The deflected shapes of the masonry wall are well defined and 
using the simplified equivalent load mass factors for the masonry alone can be selected 
from Biggs. However, the membrane has a different deflected shape than the plastic 
phase of the masonry. The load-mass factor for a parabolic membrane was created to 
accurately model the system.  
  
177 
 
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Appendix A 
182 
 
Appendix A 
Routine for Resistance Definition of Retrofit with Plate Connections
Appendix A 
183 
  
Appendix A 
184 
 
Appendix A 
185 
  
Appendix A 
186 
 
Appendix A 
187 
 
Appendix A 
188 
 
Appendix B 
189 
Appendix B 
Finite Element Results Compared to Algorithm 
 
 
  
Appendix B 
190 
 
Table B-1: Additional Test Configurations 
# Height (in) Material 
Plate 
Thickness 
(in) 
Membrane 
Thickness 
(in) 
Length of 
Plate (in) 
1 120 Polymer 0.5 0.039 3 
2 120 Polymer 0.75 0.039 3 
3 120 Steel 0.5 0.039 3 
4 120 Steel 0.75 0.039 3 
5 120 Polymer 0.25 0.079 3 
6 120 Polymer 0.5 0.079 3 
7 120 Steel 0.25 0.079 3 
8 120 Steel 0.5 0.079 3 
9 72 Polymer 0.25 0.039 3 
10 120 Polymer 0.25 0.039 5 
 
Appendix B 
191 
 
Figure B-1: Configuration 1 
 
Figure B-2: Configuration 2 
Appendix B 
192 
 
Figure B-3: Configuration 3 
 
Figure B-4: Configuration 4 
Appendix B 
193 
 
Figure B-5: Configuration 5 
 
Figure B-6: Configuration 6 
Appendix B 
194 
 
Figure B-7: Configuration 7 
 
Figure B-8: Configuration 8 
Appendix B 
195 
 
Figure B-9: Configuration 9 
 
Figure B-10: Configuration 10
Appendix C 
196 
Appendix C 
Other Routines 
  
Appendix C 
197 
 
  
C o m b i n e R e s i s t a n c e W a l l M e m b r a n e??( ) N r o w s M e m b r a n e( )?
R w a l l l i n t e r p W a l l 1
? ?
W a l l 2
? ?
?? M e m b r a n e
i 1??
???
?
?
?
?
?
i
M e m b r a n e
i 1??
?
p
i
R w a l l M e m b r a n e
i 2??
??
i 1 N???f o r
a u g m e n t ? p??( )
??
W i e h l e N o n A r c h i n g E t?? L?? P?? F?? I?? w??( ) ?
1
0?
p
1
0?
p
2
4 t?
3 L
2
?
F t? P?( )??
?
2
5 p
2
? L
4
?
3 8 4 E? I?
?
i 2?
i i 1??
?
i
?
i 1?
0 . 0 1??
p
i
4
L
2
t ?
i
?? ?? P
w
2
?
?
?
?
?
?
?
??
?
i
t?w h i l e
?
i
t?
p
i
0?
R e s u l t s a u g m e n t ? p??( )?
R e s u l t s
??
SBEDS method adapted for one way spanning CMU wall with arching along  
vertical face. Therefor c, C2 and ? are not used. Assumes face shell, ts, is 1.25 in.  
Appendix C 
198 
 
S B E D S A r c h i n g t L?? f
m
?? F
r
?? E?? G a p??? ? ?
g
t
L
2
?
?
?
?
?
?
2
t
2
?
L G a p?
2
?
?
?
?
?
?
2
???
?
1
0?
p
1
0?
?
m
m i n
L
30
t
2
??
?
?
?
?
?
?
?
g
??
s 0 . 2 E??
B
1
0 . 8 5?
C
1
0 . 8 5 f
m
? 1 . 2 5??
?
2
G a p?
p
2
0?
?
3
?
m
?
p
3
8
L
2
2 C
1
?
t
2
1 . 2 5
2
?
?
?
?
?
?
?
? C
1
?
m
??
?
?
?
?
?
?
??
?
4
m i n m i n 0 . 0 8 L? 0 . 8 t??( ) ?
g
? t??? ??
p
4
0?
p
5
0?
?
5
1 0 0?
G a p 0?if
?
2
?
m
?
p
2
8
L
2
2 C
1
?
t
2
1 . 2 5
2
?
?
?
?
?
?
?
? C
1
?
m
??
?
?
?
?
?
?
??
?
3
m i n m i n 0 . 0 8 L? 0 . 8 t??( ) ?
g
? t??? ??
p
3
0?
p
4
0?
?
4
1 0 0?
G a p 0if
a u g m e n t ? p??( )
??
Appendix C 
199 
 
 
T r a n s i t i o n W a l l R e s u l t s( ) N r o w s W a l l R e s u l t s( )?
M a x m a x W a l l R e s u l t s 2
? ?? ?
?
? W a l l R e s u l t s i 1??? W a l l R e s u l t s i 2?? M a xif
i 1 N???f o r
?
??
Appendix C 
200 
S D O F p i n p i n m L?? Lo a d?? R e s i s t a n c e?? ? t?? K
LM
?? T r a n s??? ? n r o w s Lo a d( )?
N
Lo a d
n 1??
? t
?
M
t
m L??
?
1
0?
?
2
0?
t
1
0?
P h a s e " El a s t i c "?
t
i
? t t
i 1?
??
w l i n t e r p Lo a d
1
? ?
Lo a d
2
? ?
?? t
i
??
?
?
?
?
?
R l i n t e r p R e s i s t a n c e
1
? ?
R e s i s t a n c e
2
? ?
?? ?
i
??
?
?
?
?
?
z ' '
w L?
K
LM
1
M
t
?
R L?
K
LM
1
M
t
?
??
?
i 1?
z ' ' ? t
2
? ?
i 1?
? 2 ?
i
???
j i?
P h a s e " P l a s t i c "? ?
i 1?
T r a n s?if
t r a c e P h a s e i??( )
b r e a k ?
i 1?
?
i 1?
?if
i 2 N???f o r P h a s e " El a s t i c "if
t
i
? t t
i 1?
??
w l i n t e r p Lo a d
1
? ?
Lo a d
2
? ?
?? t
i
??
?
?
?
?
?
R l i n t e r p R e s i s t a n c e
1
? ?
R e s i s t a n c e
2
? ?
?? ?
i
??
?
?
?
?
?
z ' '
w L?
K
LM
2
M
t
?
R L?
K
LM
2
M
t
?
??
?
i 1?
z ' ' ? t
2
? ?
i 1?
? 2 ?
i
???
b r e a k ?
i 1?
?
i 1?
?if
i j N???f o r P h a s e " P l a s t i c "if
t
i 1?
t
i
? t??
R e s u l t a u g m e n t t ???( )?
R e s u l t
??
Appendix D 
201 
Appendix D 
Design Example Computations 
 
  
Appendix D 
202 
  
 
 
WALL #1 
Wall Properties: 
        Non-Arching  
 
     
    
 
 
Resistance Function 
 
SDOF  
 
   
 
L 144??twall 7.625?? P 0?? Fr 65?? Ec fm 1000???
Gap 0?? wwall 29.30?? fm 1350??
m w wa l l144 3 . 8 6 1 10 4??? ?? 5 2 6 . 9 9 4???
W a l l S B E D S A r c h i n g t w a l l L?? f m?? F r?? Ec?? G a p??? ???
0 50 100 150
10?
10
20
30
40
50
T i m e  ( m s e c )
P
r
e
s
s
ur
e
 (
ps
i)
?t 0.01?? T r a n s T r a n s it io n W a ll( ) 3 . 8 1 3??? K L M W all 0 .7 70 .6 6??? ?????
S D O F w a l l S D O F p i n p i n m L?? L o a d?? W a l l?? ? t?? K L M W a l l?? T r a n s??? ???
Appendix D 
203 
  Response Limits 
 
 
 
 
  
B2 t a n 1 .5 d e g( ) L2? 1 .8 8 5???
B3 t a n 4 d e g( ) L2? 5 .0 3 5???
B4 t a n 8 d e g( ) L2? 1 0 .1 1 9???
0 50 100 150
10
20
30
W a l l  A l o n e
T i m e  ( m s e c )
D
e
f
l
e
c
t
i
o
n
 (
i
n
)
M a x? w a l l m a x S D O F w a l l 2? ?? ? 3 2 . 1 2 1??? a n g l e a t a n M a x? w a l lL
2
??
??
??
?? 2 4 . 0 4 3 d e g????
Appendix D 
204 
  
Trial 1  
Membrane Properties: 
  
  
 
  
 
Resistance Definition: 
 
 
 
SDOF: 
 
 
  
0 0. 0 5 0. 1 0. 1 5 0. 2
10 000
20 000
30 000
40 000
50 000
S t r a i n ( i n/ i n)
S
tr
es
s
 (
ps
i)
L 144?? ? 5 10 2????
t 0.0359 2??? tp .25??
fy 36000??
? 0.2??Lp 3??
E 29000000??
M e m b r a n e S t a t i c R e s i s t a n c e L t?? L p?? E?? t p?? ??? f y?? P o l y m e r?? ???? ???
T o t a l R e s i s t a n c e C o m b i n e R e s i s t a n c e W a l l M e m b r a n e??( )( )??
0 10 20 30
0 . 5
1
1 . 5
2
R e s i s t a nc e
To ta l W al l R es is t an ce
D e f l e c t i on ( i n)
P
r
e
s
s
ur
e
 (
ps
i)
KLM 0.770.8??? ?????
S D O F S D O F p i n p i n m L?? L o a d?? T o t a l R e s i s t a n c e?? ? t?? K LM?? T r a n s??? ???
M a x D e f l e c t i o n m a x S D O F 2? ?? ? 2 1 . 6 5 9??? a n g l e a t a n M a x D e f l e c t i o n
L
2
??
??
??
?? 16.742 d e g????
Appendix D 
205 
  
 
 0 50 100 150
10
20
30
W i t h R e t r of i t
W a l l  A l one
T i m e  ( m s e c )
D
e
f
l
e
c
t
i
on (
i
n)
Appendix D 
206 
  
Trial 2  
Membrane Properties: 
  
  
 
  
 
Resistance Definition: 
 
 
 
SDOF: 
 
 
  
0 0. 0 5 0. 1 0. 1 5 0. 2
10 000
20 000
30 000
40 000
50 000
S t r a i n ( i n/ i n)
S
tr
es
s
 (
ps
i)
L 144?? ? 5 10 2????
t 0.0359 13??? tp 0.75??
fy 36000??
? 0.2??Lp 3.00??
E 29000000??
M e m b r a n e S t a t i c R e s i s t a n c e L t?? L p?? E?? t p?? ??? f y?? P o l y m e r?? ???? ???
T o t a l R e s i s t a n c e C o m b i n e R e s i s t a n c e W a l l M e m b r a n e??( )( )??
0 10 20 30
10
20
30
R e s i s t a nc e
To ta l W al l R es is t an ce
D e f l e c t i on ( i n)
P
r
e
s
s
ur
e
 (
ps
i)
KLM 0.770.8??? ?????
S D O F S D O F p i n p i n m L?? L o a d?? T o t a l R e s i s t a n c e?? ? t?? K LM?? T r a n s??? ???
M a x D e f l e c t i o n m a x S D O F 2? ?? ? 1 1 . 9 4 3??? a n g l e a t a n M a x D e f l e c t i o n
L
2
??
??
??
?? 9.418 d e g????
Appendix D 
207 
  
 
 0 50 100 150
10
20
30
W i t h R e t r of i t
W a l l  A l one
T i m e  ( m s e c )
D
e
f
l
e
c
t
i
on (
i
n)
Appendix D 
208 
  Trial 3
  
Membrane Properties: 
  
  
 
  
 
Resistance Definition: 
 
 
 
SDOF: 
 
 
  
 
0 0. 0 5 0. 1 0. 1 5 0. 2
10 000
20 000
30 000
40 000
50 000
S t r a i n ( i n/ i n)
S
tr
es
s
 (
ps
i)
L 144?? ? 5 10 2????
t 0.0673?? t
p 78??
fy 36000??
? 0.2??Lp 3.00??
E 29000000??
M e m b r a n e S t a t i c R e s i s t a n c e L t?? L p?? E?? t p?? ??? f y?? P l a t e?? ???? ???
T o t a l R e s i s t a n c e C o m b i n e R e s i s t a n c e W a l l M e m b r a n e??( )( )??
0 10 20 30 40
10
20
30
40
R e s i s t a nc e
To ta l W al l R es is t an ce
D e f l e c t i on ( i n)
P
r
e
s
s
ur
e
 (
ps
i)
KLM 0.770.8??? ?????
S D O F S D O F p i n p i n m L?? L o a d?? T o t a l R e s i s t a n c e?? ? t?? K LM?? T r a n s??? ???
M a x D e f l e c t i o n m a x S D O F 2? ?? ? 1 0 . 0 8 2??? a n g l e a t a n M a x D e f l e c t i o n
L
2
??
??
??
?? 7.971 d e g???? m a x S D O F 3? ?? ? 6 9 4 .7?
Appendix D 
209 
  
 
 
Bolt Loads: 
 
 
 
    
 
 
 
 
   
0 50 100 150
1 3 . 3 3 3
2 6 . 6 6 7
40
3 3 3 . 3 3 3
6 6 6 . 6 6 7
1 10
3
?
W i t h R e t r of i t
W a l l  A l one
t r a c e  3
T i m e  ( m s e c )
D
e
f
l
e
c
t
i
on (
i
n)
0 10 20 30 40
5
10
T r i a l  3
T i m e  ( m s e c )
D
e
f
l
e
c
t
i
o
n
 (
i
n
)
T A x i a l l i n t e r p M e m b r a n e 1? ? A x i a l M e m b r a n e( )?? M a x D e f l e c t i o n??? ? 2 1 7 5 . 1???
T S h e a r l i n t e r p M e m b r a n e 1? ? S h e a r M e m b r a n e( )?? M a x D e f l e c t i o n??? ? 6 1 0 . 8???
A r e a B o lt
5
8
??
?
??
?
2
??
4 0 .3 0 7???
? n o r m a l T Ax ia l 8?Ar e a Bo lt 5 6 7 1 7 . 1??? psi ? s h e a r T S h e a r 8?
A r e a Bo lt 1 5 9 2 7 . 7??? psi
Appendix D 
210 
WALL #2 
  
  
 
Wall Properties: 
  
    
    
 
Resistance Function 
 
SDOF  
 
   
 
fm 1350?? twall 7.625?? P 100?? Fr 65?? L 120??
Gap 0?? wwall 29.30?? I 28.4?? Ec fm 1000???
m w wa l l144 3 . 8 6 1 10 4??? ?? 5 2 6 . 9 9 4???
W a l l W i e h l e N o n A r c h i n g Ec t w a l l?? L?? P?? F r?? I?? w w a l l??? ???
0 50 100 150
10?
10
20
30
40
50
T i m e  ( m s e c )
P
r
e
s
s
ur
e
 (
ps
i)
?t 0.01?? T r a n s T r a n s it io n W a ll( ) 0 .0 3??? K LM W all 0 .7 70 .6 6??? ?????
S D O F w a l l S D O F p i n p i n m L?? L o a d?? W a l l?? ? t?? K L M W a l l?? T r a n s??? ???
Appendix D 
211 
  Response Limits 
 
 
 
 
  
B2 t a n 1 .5 d e g( ) L2? 1 .5 7 1???
B3 t a n 4 d e g( ) L2? 4 .1 9 6???
B4 t a n 8 d e g( ) L2? 8 .4 3 2???
0 50 100 150
10
20
30
40
W i t h R e t r of i t
W a l l  A l one
T i m e  ( m s e c )
D
e
f
l
e
c
t
i
on (
i
n)
M a x? w a l l m a x S D O F w a l l 2? ?? ? 4 8 . 5 5 2??? a n g l e a t a n M a x? w a l lL
2
??
??
??
?? 3 8 . 9 8 d e g???
Appendix D 
212 
  
Trial 1  
Membrane Properties: 
  
  
 
  
 
Resistance Definition: 
 
 
 
 
SDOF: 
 
 
  
0 0. 0 5 0. 1 0. 1 5 0. 2
10 000
20 000
30 000
40 000
50 000
S t r a i n ( i n/ i n)
S
tr
es
s
 (
ps
i)
L 120?? ? 5 10 2????
t 0.1345?? tp 0.50??
fy 36000??
? 0.2??Lp 3??
E 29000000??
M e m b r a n e S t a t i c R e s i s t a n c e L t?? L p?? E?? t p?? ??? f y?? P l a t e?? ???? ???
T o t a l R e s i s t a n c e C o m b i n e R e s i s t a n c e W a l l M e m b r a n e??( )( )??
0 5 10 15 20
2
4
6
8
R e s i s t a nc e
To ta l W al l R es is t an ce
D e f l e c t i on ( i n)
P
r
e
s
s
ur
e
 (
ps
i)
KLM 0.770.8??? ?????
S D O F S D O F p i n p i n m L?? L o a d?? T o t a l R e s i s t a n c e?? ? t?? K LM?? T r a n s??? ???
M a x D e f l e c t i o n m a x S D O F 2? ?? ? 1 2 . 2 7 7??? a n g l e a t a n M a x D e f l e c t i o n
L
2
??
??
??
?? 1 1 .5 6 4 d e g???
Appendix D 
213 
  
 
0 50 100 150
20
40
60
80
100
W i t h R e t r of i t
W a l l  A l one
T i m e  ( m s e c )
D
e
f
l
e
c
t
i
on (
i
n)
Appendix D 
214 
  
Trial 2  
Membrane Properties: 
  
  
 
  
 
Resistance Definition: 
 
 
 
SDOF: 
 
 
  
0 0. 0 5 0. 1 0. 1 5 0. 2
10 000
20 000
30 000
40 000
50 000
S t r a i n ( i n/ i n)
S
tr
es
s
 (
ps
i)
L 120?? ? 5 10 2????
t 0.239?? t
p 1316??
fy 36000??
? 0.2??Lp 3??
E 29000000??
M e m b r a n e S t a t i c R e s i s t a n c e L t?? L p?? E?? t p?? ??? f y?? P l a t e?? ???? ???
T o t a l R e s i s t a n c e C o m b i n e R e s i s t a n c e W a l l M e m b r a n e??( )( )??
0 5 10 15 20
10
20
30
R e s i s t a nc e
To ta l W al l R es is t an ce
D e f l e c t i on ( i n)
P
r
e
s
s
ur
e
 (
ps
i)
KLM 0.770.8??? ?????
S D O F S D O F p i n p i n m L?? L o a d?? T o t a l R e s i s t a n c e?? ? t?? K LM?? T r a n s??? ???
M a x D e f l e c t i o n m a x S D O F 2? ?? ? 8 . 3 7 1??? a n g l e a t a n M a x D e f l e c t i o n
L
2
??
??
??
?? 7 .9 4 2 d e g???
Appendix D 
215 
 
Bolt Loads: 
 
 
 
psi   psi 
 
 
 
 
   
0 50 100 150
20
40
60
80
100
W i t h R e t r of i t
W a l l  A l one
T i m e  ( m s e c )
D
e
f
l
e
c
t
i
on (
i
n)
T S h e a r l i n t e r p M e m b r a n e 1? ? S h e a r M e m b r a n e( )?? M a x D e f l e c t i o n??? ? 6 7 0 . 4???
T A x i a l l i n t e r p M e m b r a n e 1? ? A x i a l M e m b r a n e( )?? M a x D e f l e c t i o n??? ? 2 3 3 6 . 6???
A r e a B o lt
5
8
??
?
??
?
2
??
4 0 .3 0 7???
? n o r m a l T Ax ia l 8?Ar e a Bo lt 6 0 9 2 8 . 9??? ? s h e a r T S h e a r 8?A r e a Bo lt 1 7 4 8 0 . 1???
Appendix D 
216