Breaching of Concrete Masonry Unit Walls due to
Direct Shear when Subjected to Blast Loading
by
Daniel Glenn Brannon
A master?s thesis submitted to the Graduate Faculty of
Auburn University
in partial fulfillment of the
requirements for the Degree of
Master?s of Science
Auburn, Alabama
May 7, 2011
Keywords: concrete masonry units, direct shear, partially grouted
finite element modeling, breaching shear, quasi-static, LS-DYNA
Copyright 2012 by Daniel Brannon
Approved by
James S. Davidson, Chair, Associate Professor of Civil Engineering
Robert W. Barnes, James J. Mallett Associate Professor of Civil Engineering
Justin D. Marshall, Assistant Professor of Civil Engineering
ii
Abstract
This report details the steps needed to formulate an engineering design equation to
address the potential for breaching of concrete masonry walls in between grout cells when
subjected to blast loading. Prior to the beginning of this research effort, large-scale dynamic
tests were performed by Air Force Research Laboratory. These tests were used to verify finite
element models created using LS-DYNA. A comparisons of finite element models and testing
were performed to help verify the finite element model analysis. A short suitability study was
carried out to understand the cause of the breaching shear in concrete masonry walls. Two
analytical models were created to provide an understanding of the mechanics of the breaching.
A comparison of the FEM and analytical models were carried out. Finally, a design shear
equation was formulated, and a maximum pressure for partially grouted construction was found.
iii
Acknowledgments
I would like to thank Dr. James Davidson at Auburn University for taking me on as a
research assistant and for providing funding for furthering my education; he also provided
technical and professional advising, as well as, focusing my research efforts. I would like to
acknowledge the efforts of the researchers in the Air Force Research Laboratory, Mechanics and
Explosives Effects Research Group based at Tyndall Air Force Base in Florida for performing
dynamic testing of several concrete masonry walls. I also would like to thank Dr. Jun Suk Kang
at Georgia Southern University for providing technical advising that was fundamental to creating
finite element models for the project. I am appreciative to all the men and women who have kept
me on the straight and narrow, who have taught me what how to be a true man and a true man of
God. I thank my parents for providing an infinite amount of support, financially and
emotionally, over the last twenty-four years and who have always been loving and accepting of
everything I wanted to do. You are my rock which without I would have long been pulled under.
Finally, I express gratitude to the Father, the Son, and the Holy Spirit for providing me new life,
new strength, and infinite joy; you have provided me the ability to accept any situation thrust
upon me and are the only reason I am here.
iv
Table of Contents
Abstract.........................................................................................................................................ii
Acknowledgments........................................................................................................................iii
List of Tables ..............................................................................................................................vii
List of Figures..............................................................................................................................ix
Chapter 1 Introduction ................................................................................................................. 1
1.1 Overview ................................................................................................................... 1
1.2 Objective ................................................................................................................... 3
1.3 Scope and Methodology ........................................................................................... 3
1.4 Organization of Report ............................................................................................. 3
Chapter 2 Literature Review ........................................................................................................ 5
2.1 Overview ................................................................................................................... 5
2.2 Concrete Masonry Units ........................................................................................... 6
2.2.1 Flexural Behavior......................................................................................... 6
2.2.2 Shear Behavior............................................................................................. 8
2.3 Mortar Properties ...................................................................................................... 10
2.4 Blast Loading .......................................................................................................... 12
2.5 Finite Element Modeling .......................................................................................... 14
2.5.1 Constitutive Models for CMU ................................................................... 14
2.5.2 CMU Models ............................................................................................. 15
v
Chapter 3 Finite Element Model Development and Methodology ............................................ 18
3.1 Overview................................................................................................................... 18
3.2 Dynamic Testing Overview ...................................................................................... 19
3.2.1 Test Set-up ................................................................................................. 19
3.2.2 Test Results................................................................................................ 24
3.3 Unit System............................................................................................................... 27
3.4 Geometry and Meshing............................................................................................. 27
3.4.1 Concrete Masonry Units ............................................................................ 28
3.4.2 Mortar and Grout ....................................................................................... 32
3.4.3 Steel Reinforcing ....................................................................................... 35
3.5 Material Modeling .................................................................................................... 36
3.5.1 Cementitious Material Model .................................................................... 36
3.5.2 Reinforcement Material Model.................................................................. 39
3.5.3 Boundary Material Model.......................................................................... 40
3.6 Element Modeling..................................................................................................... 40
3.7 Load Modeling.......................................................................................................... 42
3.7.1 Gravity Preloading..................................................................................... 42
3.7.2 Blast Loading............................................................................................. 44
3.8 Boundary Modeling .............................................................................................................. 44
3.9 Contact Modeling.................................................................................................................. 45
3.9.1 Mortar-Block Interface .............................................................................. 46
3.9.2 Block-Boundary Interface.......................................................................... 48
3.10 FEM Validation ...................................................................................................... 49
vi
3.11 FEM Results and Suitability Study of Breaching................................................... 66
Chapter 4 Design Shear Resistance ............................................................................................ 95
4.1 Introduction............................................................................................................... 95
4.2 Structural Dynamics.................................................................................................. 95
4.2.1 Single Degree-of-Freedom Model ............................................................. 96
4.2.2 Pressure-Impulse Simplifications ............................................................ 100
4.3 Modeling of Breaching Response........................................................................... 104
4.3.1 Dynamics of Face Shell Model................................................................ 105
4.3.2 Dynamics of Between Grout Cells Beam ................................................ 108
4.3.3 Direct Shear Modeling............................................................................. 114
4.4 Resistance Equation Derivation.............................................................................. 114
4.4.1 Development of Resistance Equation ...................................................... 115
4.5 Comparison between FEM and Analytical Stress .................................................. 116
4.5.1 Face Shell Beam Comparison.................................................................. 116
4.5.2 Between Grout Cells Comparison ........................................................... 118
4.5.3 Differences between FEM and Analytical Shear Stress .......................... 120
4.6 Breaching Shear Design Equation .......................................................................... 121
Chapter 5 Summary and Recommendations ............................................................................ 124
5.1 Conclusions............................................................................................................. 124
5.2 Recommendations .................................................................................................. 125
References ............................................................................................................................... 126
Appendix................................................................................................................................... 128
vii
List of Tables
Table 2.1. Modulus of Rupture Stresses from ACI 530-11...................................................... 11
Table 3.1. Scaled Standoffs of the Dynamic Testing ............................................................... 19
Table 3.2. Details of Wall Construction ................................................................................... 20
Table 3.3. Material Properties................................................................................................... 24
Table 3.4. Units System............................................................................................................ 27
Table 3.5. CMU Material Model Selection .............................................................................. 37
Table 3.6. Maximum Deflection for Test 2 and FEM .............................................................. 66
Table 3.7. Maximum Out-of-Plane Shear Stresses of Various Geometries of Walls............... 79
Table 3.8. Maximum Effective Stresses of Various Geometries of Walls ............................... 79
Table 3.9. Stresses for Compressive Strength Suitability Study .............................................. 82
Table 3.10. Stresses for Compressive Strength Suitability Study .............................................. 84
Table 3.11. Stresses from Loading Shape Suitability Study....................................................... 88
Table 3.12. Statistical Data on Loading Shape Suitability Study............................................... 88
Table 3.13. Values of Stresses for the 5x5 Wall from the Peak Pressure Suitability Study....... 89
Table 3.14. Values of Stresses for the 10x3 Wall from the Peak Pressure Suitability Study..... 89
Table 3.15. Values of Stresses for the 3x10 Wall from the Peak Pressure Suitability Study..... 90
Table 3.16. Grouted vs. Non-Grouted Maximum Stresses......................................................... 92
Table 4.1. Representative Numbers for 8-in. CMU................................................................ 107
Table 4.2. Loading Regime Ranges......................................................................................... 107
viii
Table 4.3. Quasi-static details for between grouted cells beam............................................... 110
Table 4.4. Shear Stress Comparison for Single Block Beam................................................... 117
Table 4.5. Shear Stress Comparison for Between Grout Cells Beam...................................... 118
Table 4.6. Maximum Pressure for Single Block Beam............................................................ 122
Table 4.7. Maximum Pressure for Between Grout Cells Beams ............................................. 122
ix
List of Figures
Figure 2.1. Pressure Loading Profile ........................................................................................ 13
Figure 3.1. Front view of 8 in. CMU panel .............................................................................. 21
Figure 3.2. Details of construction and reinforcement of the 8-in. CMU wall......................... 22
Figure 3.3. Panels in reaction structure prior to testing ............................................................ 23
Figure 3.4. Breaching of 8 in. CMU wall after test 2 ............................................................... 25
Figure 3.5. Breaching of 8 in. CMU wall after test 2 ............................................................... 26
Figure 3.6. Comparison of real CMU and FEM CMU............................................................. 29
Figure 3.7. 3-D view of normal block....................................................................................... 30
Figure 3.8. U-block used in FEM ............................................................................................. 31
Figure 3.9. Half-high block used in FEM ................................................................................. 32
Figure 3.10. Mortar and grout meshing ...................................................................................... 33
Figure 3.11. Grout columns ........................................................................................................ 34
Figure 3.12. Bond beam and blocks............................................................................................ 34
Figure 3.13. Beam embedded in grout........................................................................................ 35
Figure 3.14. Base force of wall under gravity loading ............................................................... 43
Figure 3.15. Boundary modeling ................................................................................................ 45
Figure 3.16. Mortar-block interface............................................................................................ 48
Figure 3.17. Boundary-block interface ....................................................................................... 49
Figure 3.18. Test set-up and instrumentation position................................................................ 50
x
Figure 3.19. Normalized reflected pressure from dynamic testing............................................. 51
Figure 3.20. Normalized impulse form dynamic testing ............................................................ 51
Figure 3.21. Video captures of panel 2 during test 1 .................................................................. 53
Figure 3.22. Screen captures of FEM of Panel 2 during test 1 ................................................... 55
Figure 3.23. Deflection of panel 2 and of FEM from Test 1 ...................................................... 57
Figure 3.24. Video capture of panel 2 during test 2.................................................................... 58
Figure 3.25. Screen captures of FEM of Panel 2 during test 2 ................................................... 60
Figure 3.26. Cross-section of FEM............................................................................................. 61
Figure 3.27. Stress contours for FEM 1 ms after loading........................................................... 62
Figure 3.28. Deflection of panel 2 and of FEM from test 2........................................................ 64
Figure 3.29. Deflection of panel 2 and of FEM from test 3........................................................ 65
Figure 3.30. Contour plots of out-of-plane shear stress at various times ..............................68-69
Figure 3.31. Contour plots of effective shear stress at various times ....................................70-71
Figure 3.32. Contour plots of out-of-plane shear stress at various times ..............................72-73
Figure 3.33. Contour plots of effective stress at various times..............................................74-76
Figure 3.34. Plot of out-of-plane shear stress vs. time................................................................ 77
Figure 3.35. Plot of effective stress vs. time............................................................................... 77
Figure 3.36. Plot of geometry vs. out-of-plane shear stress........................................................ 80
Figure 3.37. Plot of geometry vs. effective stress....................................................................... 80
Figure 3.38. Plot of out-of-plane shear stress vs. compressive strength..................................... 82
Figure 3.39. Plot of effective stress vs. compressive strength .................................................... 83
Figure 3.40. Plot of out-of-plane shear stress vs. unit weight..................................................... 84
Figure 3.41. Plot of effective stress vs. unit weight.................................................................... 85
xi
Figure 3.42. Plot of normalized loading ..................................................................................... 86
Figure 3.43. Plot o out-of-plane shear stress vs. peak pressure .................................................. 90
Figure 3.44. Plot of effective stress vs. peak pressure ................................................................ 91
Figure 3.45. Out-of-plane shear stresses for grouted and non-grouted walls ............................. 93
Figure 3.46. Effective stresses for grouted and non-grouted walls............................................. 93
Figure 4.1. Single degree-of-freedom model............................................................................ 96
Figure 4.2. Typical response spectrum and P-I diagram ........................................................ 102
Figure 4.3. Loading regimes ................................................................................................... 103
Figure 4.4. Face shell beam and cross section ........................................................................ 106
Figure 4.5. Single block beam representation......................................................................... 104
Figure 4.6. Representative beam and cross section for between grouted cells beam ............. 109
Figure 4.7. Minimum load duration vs. modulus of elasticity................................................ 111
Figure 4.8. Minimum load duration vs. thickness of face shell.............................................. 111
Figure 4.9. Minimum load duration vs. width of block.......................................................... 112
Figure 4.10. Minimum load duration vs. unit weight ............................................................... 112
Figure 4.11. . Minimum load duration vs. length between grout cells ..................................... 113
Figure 4.12. Correction factor vs. peak pressure for single block beam .................................. 117
Figure 4.13. Correction factor vs. peak pressure for between grout cells beam....................... 119
Figure 4.14. Correction factor vs. length for between grout cells beam................................... 120
1
CHAPTER 1
INTRODUCTION
1.1 Overview
Starting in World War II when bombs and explosives began to be used as
conventional weapons, researchers began to look into ways to mitigate the forces caused
by blast. In the Cold War, the threat of large scale nuclear bombs lead to research in
whole system structural response to blasts. However, the Oklahoma City Bombing and
World Trade Center Bombing in the 1990s showed the damaging effects of more
localized blasts. The usage of improvised explosive devices following 9/11 has given an
importance to research on localized response and local phenomenon.
Blast loading is a dynamic load. A dynamic load is loadings that changes over
time and can cause increased deflection and higher accelerations relative to a static load
of the same intensity. However, unlike forces resulting from wind and earthquakes, blast
loading cannot be readily transformed into equivalent static forces. This requires that the
structure to be modeled as a dynamic system. Blast loading, like earthquake, is not
typically expected to be endured without damage. Blast could cause varying levels of
damage, from a few cracked windows to complete collapse; therefore, the primary
objective of blast design criteria focuses on the preservation of life, instead of a
preventing damage. One of the main concerns is that the loading can produce breaching
2
of the cladding leading to high velocity fragmentation or allowing the blast wave to enter
into the structure. Both can cause loss of life.
A common type of building material is masonry. Masonry has been used for
millennia as a building material. The first buildings were crude stacks of natural stone;
this eventually transitioned into manufactured stone with mortar and into brick and
mortar. Starting in the 1800s, concrete masonry units (CMU) began to be used for a wide
range of building applications. In modern society, CMU is being used as shear walls to
resist lateral loads or as cladding on the exterior of structures. This is because this wall
type is relatively inexpensive, is easily and quickly constructed, and provides insulation
for the structure. However, unreinforced CMU walls are weak in flexure and must be
grouted and reinforced to handle flexural loading. Since grouting every cell of the CMU
can be costly, owners and contractors often only grout cells that have reinforcement
running in them.
Unified Field Criteria (UFC) 3-340-02 Structures to Resist the Effects of
Accidental Explosions (UFC, 2008) allows the use of partially grouted CMU walls along
as the wall is still designed to meet the flexural demand. However, a recent study
(Davidson et al., 2011) on partially grouted walls has shown that blast loading can cause
localized stress and cause fracturing leading to loss of life. Therefore, partially grouted
walls can behave similar to unreinforced masonry walls. Therefore, the overall objective
of the work represented by this thesis was to define the situations in which it is safe to use
partially grouted CMU walls.
3
1.2 Objective
The overall objectives of the research were (1) to develop an understanding of the
causes of breaching of partially grouted CMU walls subjected to blast loading by using
finite element modeling and (2) to develop an engineering-level design equation to
predict direct shear and breaching in partially grouted CMU walls.
1.3 Scope and Methodology
In order to complete the objectives, several different tasks had to be performed
which included a literature review, development of finite element models, a suitability
study, and development of a resistance equation. The finite element models were created
and visualized in LS-PrePost; and analyzed using the LS-DYNA finite element solver.
Full-scale static and dynamic testing results were used to verify the modeling approach.
The testing was performed as part of a previous study by the Air Force Research
Laboratory (Davidson et al., 2011) for evaluating minimally reinforced partially grouted
walls. The resistance equation was derived by examining the behavior of the finite
element models, and structural dynamic and quasi-static models of CMU were created to
approximate the breaching behavior.
1.4 Organization of Thesis
The thesis is broken into five chapters. Chapter 1 consists of an introduction,
objectives, scope and methodology, and organization of the thesis. Chapter 2 provides a
literature review including a brief look at the literature on blast loading, concrete masonry
units, mortar, and finite element modeling. Chapter 3 provides an overview of the full-
4
scale dynamic testing, a summary of the finite element methodology, a verification of the
finite element modeling, and a suitability study of the breaching phenomenon. Chapter 4
discusses the breaching and shearing behavior of CMU due to blast loading and details
the development of the shear design equation. Chapter 5 summarizes the thesis and gives
recommendations for designers and researchers.
5
CHAPTER 2
LITERATURE REVIEW
2.1 Overview
With the increase in terrorist activity across the world, there has arisen a focus on
making structures more resistant to blast loading. Several researchers have looked at a
vast array of materials including steel, reinforced concrete, and masonry. Since masonry,
especially CMU, is a common type of building material for exteriors walls, many
researchers have looked into improving the performance of masonry subjected to blast
loading.
Since masonry has very low tensile strength, the wall performs poorly in flexure
unless a ductile reinforcement is added into the assembly. In order to do this, reinforcing
steel is added into hollow sections of the CMU. To provide composite action between
the steel and CMU, grout, a flowable concrete mixture, is placed in the cell. If grout is
placed into every cell (with or without reinforcing steel), the wall is said to be fully
grouted. To minimize costs, many contractors and owners prefer to only add grout to
cells that have steel in them. If this is done, the wall is said to be partially grouted.
Over the years, a great deal of work has gone into modeling masonry walls that are
unreinforced, reinforced, or fully grouted; several researches have also looked at catcher
systems and energy absorption systems. However, there is a general lack of research into
partially grouted, reinforced walls and the difference in their failure mechanics. This
6
thesis focuses on the phenomenon where there is a direct shear failure that occurs
between the grout columns; this type of failure is called breaching. The direct shear or
breaching shear causes shear cracks to form in the block.
2.2 Concrete Masonry Units
2.2.1 Flexural Behavior
The flexural behavior of CMU has been researched thoroughly. The masonry
section of Unified Field Criteria (UFC) 3-340-02 (UFC, 2008) states that ?the method of
calculating ultimate moment of [combined joint and cell reinforced masonry] is the same
as that presented in [the chapter on concrete].? UFC then gives the ultimate moment
capacity, Mn, for a concrete beam or non-load bearing wall as
( - )n s dsM A f d a/2= (2-1)
where As = area of the reinforcement, fds = dynamic yield strength of the steel
reinforcement, d = distance between the centroid of the tension reinforcement and
extreme compression fiber, and a = the depth of the equivalent rectangular compressive
block. UFC 3-340-02 makes no differentiation between fully grouted and partially
grouted walls. This is because research has shown both types of grouting works well if
the wall is designed properly. Other sections of UFC 3-340-02 go into detail establishing
rules and guidelines for damage levels from ?lightly damaged? to ?collapse? and the
design of slabs for one-way and two-way actions.
Davidson et al. (2011) tested several partially grouted CMU walls under uniform
static pressure in vacuum chambers. These walls were made of 8-in and 6-in CMU with
minimum reinforcing and only the reinforced cells being grouted. The walls were loaded
7
only by pressure and self-weight; the pressure was the same along and across the wall
and increased to failure. These walls first cracked along the bed joint at the course
nearest to mid-height of the wall. These walls were able to carry additional load with
increased cracking and deflection. Eventually, the walls failed in flexure due to self-
weight and did not show any signs of shear failure, conventional or breaching. Plots of
midheight deflection versus applied pressure were created; these showed that the
resistance of the wall can be described by three behavioral regions. The first is linear-
elastic resistance until cracking. This is followed by a nonlinear resistance caused by
changing progression of cracking and straining of steel, and the last section is a ductile
displacement under a constant load until the wall?s failure. In a linked study, Davidson et
al. (2011) tested three of the identically constructed panels using blast loading. In several
of the walls, the failure mechanism changed from a ductile failure in flexure to a brittle
failure in breaching. The breaching occurred between the grouted cells, typically
occurring at the interface between grouted and ungrouted blocks. Davidson et al. (2011)
was used as the primary driver for this work and data from this report was used to
validate this work.
Burrett et al. (2007) tested many large scale CMU walls when subjected to
impact. In their research, they developed finite element models to model the impact and
resulting damage and found analytical resistance models of unreinforced, ungrouted
walls. In their analytical modeling, they performed a parametric study to find the key
components of the resistance functions.
Gilbert et al. (2002) developed a rigid-body mechanism analysis of unreinforced
masonry walls. In this analysis, rocking, sliding, and a combination of rocking and
8
sliding were analyzed for impact. Five wall-failure mechanisms were found as analysis
tools for the prediction of failure and displacement. The analysis allows for the
prediction of the peak displacement to within a ten percent upper and lower bound.
Analysis correlated well with the displacement for all failure modes, except one.
Sudame (2004) developed a finite element modeling approach that predicts the
overall resistance of ungrouted walls subjected to blast loading. Moradi (2008) further
developed on Sudame?s work by providing modeling of retrofits. The paper develops
resistance functions for three different retrofits. These were compared to data of walls
subjected to real blast loading for verification.
2.2.2 Shear Behavior
In Masonry Structures Behavior and Design (Drysdale and Hamid, 2008), the
nominal shear strength, Vn, is generalized by
n m sV V V= + (2-2)
where Vm = strength provided by the masonry, and Vs = strength provided by the steel
reinforcement. The strength provided by the masonry also takes into account the effects
of frictional forces caused by the axial load on the wall. In American Concrete Institute
(ACI) 530-11 (ACI, 2011), the nominal shear capacity provided by the masonry, Vm, of a
reinforced masonry wall using strength design provisions is given by
4.0 1.75 0.25um nv m u
u v
MV A f' P
V d
? ?? ?= ? +
? ?? ?? ?
? ?? ?
(2-3)
where Mu = the ultimate factored bending moment on the section, Vu = the ultimate
factored shear on the section, dv = the depth of the member in the shear direction, and Pu
9
= the ultimate factored axial load on the section. The nominal shear capacity provided by
the transverse steel, Vs, for a reinforced masonry wall using strength design provisions is
0.5 vs y vAV f ds? ?= ? ?
? ?
(2-4)
where Av = the area of the transverse steel, fy = the yield stress of the steel, and s = the
spacing between layers of transverse steel. Since most walls do not provide transverse
steel for out-of-plane bending, the shear contribution attributed to the steel is zero.
UFC 3-340-02 states, ?Cell reinforced masonry walls essentially consist of solid
concrete elements?.Shear reinforcement for cell reinforced walls may only be added to
the horizontal joint similar to joint reinforced masonry walls.? The shear capacity for
joint reinforced masonry, Vu, is
y v n
u
f A AV
bs
?= (2-5)
where ? = strength reduction factor equal to 0.85, An = the net area of the section, and b =
the width of the wall. For walls without joint reinforcement, the shear capacity is zero
according to UFC 3-340-02.
UCF 3-340-02 provides no additional details in the masonry section on direct or
breaching shear. However, in the concrete section, it gives a minimum area of steel to be
provided at supports, Ad, for a beam as
dA sin( )s d
ds
V b V
f ?
?= (2-6)
where Vs = the shear at the support of width b, Vd = the direct shear capacity of the
concrete, b = the width of the member, fds = the dynamic design stress of the steel, and
? = the angle formed by the plane of the diagonal reinforcement and the longitudinal
10
reinforcement. The section applies to masonry walls only when they are fully grouted.
In addition, a proposed update to UFC 3-340-02, Oswald et al. (2010), states that no
direct shear be designed for in wall with expected blast exceeding a standoff of 3 ft/lb1/3
and that only fully grouted walls be used for standoffs less than 3 ft/lb1/3. There are no
comments on how to design for between grouted section breaching.
Psilla and Tassios (2009) evaluated several design shear strengths and several
other research shear equations. The authors then developed shear strength equations using
?tensile strength and compression strength of masonry, masonry to masonry friction, and
pullout force.? The equations predicted three failure modes, 1.) diagonal cracking, 2.)
disintegration of web, and 3.) diagonal compression failure and were calibrated using
experimental data on the ultimate shear load. The shear equations were compared to
several design equations and found to better match experimental data than design
equations given in American Concrete Institute, New Zealand Standard, and Canadian
Standard Association.
2.3 Mortar Properties
Since mortar is inherently weak in tension, it is a major contributor to the failure
mechanism of CMU walls in bending and shear. Therefore, the mortar?s properties must
be defined. One of the major contributors to the strength of the mortar is bond strength.
Drysdale states ?bond is perhaps the most critical factor because it influences both the
long term strength and the serviceability of the finished masonry.? He later says ?mortar
should have sufficient bond for water tightness and to resist tensile stress due to external
loads? (Drysdale and Hamid, 2008). The bond contributes to both the tensile and shear
11
strength of the mortar. The bond strength is affected by properties of the masonry block,
mortar type, workmanship, water-cement ratio, and curing conditions. Most of these
parameters are not fixed and are determined in the field by the mason. Also, bond
usually has no fixed limits. Therefore, bond strength of the mortar can have a wide range
with few known values.
In order to quantify the mortar?s bond properties, researchers have looked into the
tensile bond strength. Hamid and Drysdale (1988) looked into the tensile block
arrangements. The modulus of rupture values are 30.5 psi (0.21 MPa) to 242 psi (1.67
MPa) for failures normal to bed joints; the values are 120 psi (0.83 MPa) to 290 psi (2.00
MPa) for failures parallel to bed joints. To limit the variables, only values for non-
grouted sections were used. The range of ungrouted arrangements is 30.5-128 psi (0.21-
0.88 MPa) for normal to bed joints; the range of ungrouted arrangements for parallel to
bed joints is 120-216 psi (0.83-1.49 MPa). The median values are 49.3 psi (0.34 MPa)
for ungrouted normal failures and 168 psi (1.16 MPa) for ungrouted parallel failures.
ACI 530-11 (ACI, 2011) provide values for modulus of rupture for different mortars (M,
S, and N); these are provided in Table 2.1. These values are not in the ranges found by
research; this would be because ACI 530-11 gives conservative values for modulus
tension failures.
Table 2.1. Modulus of rupture values from ACI 530-11
Portland or Mortar
Cement (psi) Masonry cement (psi) Direction of Flexural Masonry type
M or S N M or S N
Ungrouted 33 25 20 12 Normal to
bed joint Fully Grouted 86 84 81 77
Ungrouted 66 50 40 25 Parallel to
bed joint Fully Grouted 106 80 64 40
12
Several researchers have looked into the shear bond strength as well. Atkinson et
al. (1989) performed direct shear tests on clay bricks and compared this to reported
values for other clay bricks and concrete blocks. The Atkinson et al. also gave shear
strength for concrete blocks ranges from to studies. These values range from 0 to 399 psi
(0 to 2.75 MPa) for one study and 0 to 232 psi (0 to 1.6 MPa) for another.
2.4 Blast Loading
Blast loading occurs when a pressure wave caused by an explosion strikes an
object. Since explosions are very destructive and very brief, little data can be obtained
from each explosive test; however, with the proper equipment and some care, data, such
as maximum deflection, pressure histories, and high-speed video, can be obtained. The
data can provide a way to analyze a system?s response to blast loading.
An explosion can be described as a sudden release of potential energy from a
source; this release of energy develops into a thermal energy difference creating an air
burst with high kinetic energy. The air burst causes an increase to pressure traveling as a
spherical wave to the surrounding area. At the edge of the wave, there is a sudden
increase of pressure from the atmospheric pressure; the difference in these two pressures
is called the peak pressure. After the shock front passes, the pressure decreases in an
almost linear fashion to atmospheric pressure. After atmospheric pressure is reached, the
pressure continues to decrease until is reaches the peak negative pressure; at this point the
pressure increases to atmospheric pressure. This portion of the loading is curved. It is
called the negative phase and characterized by suction or negative pressure. In Figure
2.1, a idealized pressure loading profile is shown; this is characterized by a sudden
increase to pressure with a linear decrease to zero and a linear negative phase with the
13
maximum negative pressure occurring at a quarter of the negative time. The idealized
curve is created using table in UFC 3-340-02 and using a scaled standoff specified as the
distance the object is from the blast source compared to the weight raised to the 1/3
power.
-25
0
25
50
75
100
125
150
175
200
225
0 10 20 30 40 50 60 70 80 90 100
Time (ms)
Pr
ess
ur
e (
ps
i)
Figure 2.1. Pressure loading profile
As the pressure wave travels outwards, it strikes objects creating a reflected
pressure wave. The reflected pressure causes an increase of the pressure above the air
burst pressure and is usually the pressure that all structures are designed to resist. The
present work uses UFC 3-340-02 because it is the design standard for most building in
the United States when an explosion can occur at a facility.
There are many sources of explosions. Since different explosions have very
different characteristics, a generalized way of describing different explosives has been
developed. Any explosive can be compared to trinitrotoluene (TNT); TNT is used to
14
describe explosive effects because its properties and resultant pressure wave are well-
defined. The above discussion and figure are based on the properties of a TNT
explosion. UFC 3-340-02 gives the following equation to find the equivalent weight of
TNT, WEQV, as compared to a given weight of an explosive.
dEXP
EQV EXPd
TNT
HW W
H
= (2-7)
where HdEXP = heat generated by the explosive, HdTNT = heat generated by TNT, and
WEXP = weight of the explosive. This equation allows for the scaling of an explosive
where design guides can be used.
2.5 Finite Element Modeling
2.5.1 Constitutive Models for CMU
Since concrete components are essential in almost all buildings, finite element
models have been developed to model different structures in different environments.
These FEMs are primarily concerned with the performance of individual components of a
structure, especially the failure of components.
In order to better simulate the failure and cracking in concrete structures, several
models have been developed. Listed in the LS-DYNA User?s Manual, Volume I (2009),
there are 26 different material models that are described as being suited for soil, concrete,
or rock. The wide selection can be attributed to the many tests and properties needed to
clarify the behavior of concrete. Also, several models have been developed to help
modelers by providing simple inputs and parameter generation algorithms.
15
Davidson and Moradi (2008) looked at five material models (Soil and Foam, Soil
and Foam with Failure, Brittle Damage, Pseudo Tensor, and Winfrith Concrete) in order
to find the best model for CMU subjected to blast. In order to test these models, a blast
test was set up with single blocks at various standoff distances. The results were
compared to finite element simulations ran in LS-DYNA. It was concluded that the Soil
and Foam model best matched the test.
Magallanes et al. (2010) mentioned that CMU acts like lightweight concrete and
stated that the LS-DYNA material model Concrete Damage Release 3 ?can provide
excellent results if properly calibrated for these materials.? Very few others have looked
directly at modeling and determining the best model for CMU.
2.5.2 CMU Models
While there is little information on constitutive models for CMU, there is much
more information on performing finite element modeling of CMU. Most of the work
focuses exclusively on modeling walls.
Martini (1997) developed a one-way masonry wall model to help in the
investigation of rebuilding of Pompeii following an earthquake in 62 A.D. In the model
he proposed a block-interface model where the mortar is not modeled, but the interface
between blocks retains the failure condition of the mortar. This model matched well with
the work published on static tests of one-way walls. Martini (1998) used the same block-
interface model to model two-way bending of masonry walls. The model showed that as
deflection of a wall increases the reaction changes from the base carrying almost all the
reactions to the base only carrying vertical reactions and the side supports carrying lateral
16
reactions. He also noticed the blocks created moment couples along the edges. Finally,
he noticed the failure pattern matched well with yield line pattern of the reinforced
concrete slabs and created a method to apply the yield line analysis for masonry walls.
Burnett et al. (2006) performed finite element modeling on CMU walls subjected
to low-velocity impacts. The authors detailed the creation of a discrete-crack model that
employed tied interface contact definition with normal and shear interface failure
stresses, dilatant friction, gravity loads, and viscous hourglass control. The modelers also
developed a Mohr-Coulomb failure surface in the compression zone and a hemispherical
cap in the tension zone. This model looked at impacts of a steel plate on CMU and brick
walls and was used to determine wall failure modes, maximum displacement, and the
influence of bonding pattern.
Dennis et al. (2002) developed a CMU model of a single strip of blocks in the
vertical direction. The modeling approach used most of the same concepts as Burnett?s
model. In his model, he used both quasi-static pressure test and dynamic blast test to
verify the model taking into account maximum out-of-plane deflection and failure
analysis; the model was conservative by a slight amount but was not able to accurately
predict all failures.
Eamon et al. (2004) developed a model similar to the one in Dennis et al.;
however, his was able to accurately predict the failure modes. Their work showed that
there were three different failure modes for out-of-plane bending, (1) two-segment
arching with the block remaining intact at low pressures, (2) two-segment arching with
increased deflection and boundary block rotation leading to failure at medium pressures;
and (3) multiple segments being expelled from the wall at various velocities at high
17
pressure. The model also showed sensitivity to the material parameters; however, a
change in failure type and a change in expolsion velocity were relatively insensitive to
material parameters.
Browning (2008) modeled multi-wythe walls that were fully grouted and had a
brick veneer filled with a foam cavity. In his model, he replaced the grout and CMU with
a composite material with composite properties based on area. The brick veneer was not
modeled directly, but its mass was added into the foam. The foam was not modeled
directly because it was assumed to only provide energy damping. Using the model, he
developed engineering-level equations for out-of-plane bending using single degree-of-
freedom and multiple degree-of-freedom methods.
In addition to conventional CMU modeling, there has been some research into
modeling CMU retrofits. Sudame (2004) built a model for CMU wall with a spray-on
polymer retrofit attached to the interior side of the wall. In modeling the polymer, he
used a rubber material model, a tied contact definition with tension and shear failures
stresses, and a rupture failure definition. He also used a tied interface for the mortar joint
with failure stresses. His research included a parametric study. Moradi?s (2008) work is
an extension of Sudame; he developed a CMU model based on the work of Sudame?s
model; Moradi?s main focus was the development of a resistance equation for flexure
which takes into account the effects of the retrofit.
18
CHAPTER 3
FINITE ELEMENT MODEL DEVELOPMENT AND METHODOLOGY
3.1 Overview
Since concrete masonry walls are nonlinear both in their geometry and their
material properties, finite element modeling is a valuable tool for understanding the
behavior of these walls. The drawback of finite element is that it can be complicated in
developing a valid model for analysis; however, after the model has been validated, it can
be used to perform a large number of virtual testing of complicated systems at a high
efficiency. LS-DYNA produced by Livermore Software Technology Corporation is the
finite element solver chosen because it is an advanced, general-purpose solver with the
ability to run nonlinear, dynamic analysis. In order to do the preprocessing and post-
processing, LS-PrePost, also produced by Livermore Software Technology Corporation,
was used because of its compatibility with LS-DYNA. Any specifics in the following
sections are given for input into LS-DYNA; these can be modified to model CMU for
any appropriate finite element program, but these details should be viewed only as input
for this one phenomenon and not as instructions to model CMU. In addition to the input
detailed below, a typical keyword input file can be seen in the appendix.
19
3.2 Dynamic Testing Overview
Prior to the beginning of this analytical research, a series of full-scale dynamic
testing on partially grouted CMU walls were carried out by Air Force Research
Laboratory researchers at Tyndall Air Force Base in Florida. The full details of the
testing can be seen in Davidson et al. (2011). Some of the methodology and the dynamic
testing are shared here to help develop a finite element model and to show the suitability
of the model; the details and model should not be seen as a way to understand anything
other than the breaching phenomenon addressed in this paper; this includes using the
details to understand the flexural response or non-localized shear response.
3.2.1 Test Set-up
In the testing, three individual blast tests were carried out at three different scaled
standoffs. These are shown in Table 3.1.
Table 3.1. Scaled standoffs of the dynamic testing
Test
Scaled Distance
1/3ft lb
? ?? ?
? ?
1 8.0
2 6.5
3 5.2
In each test, three panels were tested, giving a total of nine panels; each test had one of
the following panels: (1) a 6-in. CMU wall, (2) an 8-in. CMU wall, and (3) a multi-wythe
cavity wall made up of an 8-in. CMU wall with a 4-in. clay brick veneer. Each panel was
112 in. wide by 136 in. high. The 6-in. walls and 8-in. walls were a single wythe thick;
the multi-wythe cavity walls were two wythe thick with a 2-in. polystyrene rigid board
20
insulation and 1-in. air gap between the veneer and wall. Table 3.2 describes some of the
panels? construction details. The 8-in. wall was the focus on this thesis and the finite
element modeling. Figure 3.1 shows the general dimensions of the 8-in. wall; the
grouting is shown as being shaded and the reinforcing bars runs through the center of
grout. Figure 3.2 shows the construction detailing of the walls. Figure 3.3 shows one of
the tests with all three panels in the reaction structures prior to testing.
Table 3.2. Details of wall construction
Panel Block Reinforcement Veneer
1 6 inch CMU # 3 bars ? 36 in. avg., 40 in. max None
2 8 inch CMU # 4 bars ? 52 in. avg., 56 in. max None
3 8 inch CMU # 4 bars ? 52 in. avg., 56 in. max 4 in. clay bricks
21
Figure 3.1. Front view of 8 in. CMU panel
22
Figure 3.2. Details of construction and reinforcement of the 8-in. CMU wall
23
Figure 3.3. Panels in reaction structure prior to testing
Material testing was carried out according to ASTM Standards. The material properties
can be seen in Table 3.3.
24
Table 3.3. Material properties
Material/Test Value
Masonry Prism/
Compression Strength
Grouted: 6-in. CMU = 4870 psi
8-in. CMU = 4270 psi
Hollow: 6-in. CMU = 2080 psi
8-in. CMU = 1290 psi
Clay Brick = 4460 psi
Masonry Block/
Density
Density: 6-in. CMU = 112 lb/ft3
8-in. CMU = 101 lb/ft3
Clay Brick = 138 lb/ft3
Mortar/
Compression Strength
3190 psi
Grout/
Compression Strength
7520 psi
Rebar/
Tensile Strength
Yield: #3 bars = 73900 ksi
#4 bars = 66800 ksi
Ultimate: #3 bars = 113000 ksi
#4 bars = 106000 ksi
Max. Strain: #3 bars = 0.141
#4 bars = 0.143
Additional details of the walls? construction or detailing can be seen in Davidson et al.
(2011).
3.2.2 Test Results
The tests? purpose was to investigate the flexural response of minimally
reinforced, partially grouted walls subjected to blast loading. In order to understand the
flexural response, data on deflection and pressure were taken; high-speed cameras were
also used on the outside and inside of the structure to capture the response. This data was
analyzed and compared to existing single degree-of-freedom analysis tools used for blast
design by industry. It was found that the wall?s flexural response was considered
conservative compared to the analysis tools. However, in testing the walls, it was found
that large sections of the walls were breached between the grouted columns. Figure 3.4
25
and Figure 3.5 show typical breaching for the walls during testing. Figure 3.4 is a time
progression of the breaching during the testing.
Figure 3.4. Breaching of 8-in. CMU wall during to test 2
26
Figure 3.5. Breaching of 8-in. CMU wall after test 2
In a linked study, these walls were also tested in a static pressure chamber and failed in
flexure around midheight (Salim et al., submitted 2011). Since the walls were designed
to fail in flexure, the shear breaching failure mode of the walls under dynamic loading
was unexpected. In the recommendations of the report by Davidson et al. (2001), the
researchers stated ?additional testing and analysis of the between-column breaching
phenomenon is needed.? They also suggested that all walls that might be subjected to
blast loading be fully grouted until a better diagnostic tool for breaching can be
developed.
A better understanding of the testing methodology, resulting analysis, and
conclusions and recommendations can be seen in Davidson et al. (2011). Also, the report
goes into more detail on the flexural response of the wall than will be discussed in this
thesis. Finally, this thesis will only use data and figures from Davidson et al. (2011) to
demonstrate the suitability of the finite element models.
27
3.3 Unit System
LS-DYNA does not use units explicitly. It instead makes the modeler
consciously express every input in consistent units. U.S. Customary units for force,
length, and time were used in the model. All other units that are used are a derivation
based on the units for force, length, and time. These are shown in Table 3.4.
Table 3.4. Unit system
Metric Unit
Force pound (lbf)
Length inch (in.)
Time seconds (s)
Mass lbf-s2/in.
Density lbf-s2/in.4
Stress lbf/in.2 (psi)
As a note, the unit millisecond (ms) will be used in this thesis because it is convenient to
discuss and display data in that unit for blast loading instead of thousandth of a second.
3.4 Geometry and Meshing
Most masonry finite element models found in literature only used a single column
of block in either running or stack pattern; this is a simplification that can be used for
ungrouted or fully grouted walls because the walls are assumed to be well represented by
a single column of blocks and to be homogeneous and the response is assumed to be
dominated by one-way flexure. Since the phenomenon of breaching occurs between the
grouted columns in partially grouted walls, the entire wall section had to be modeled; this
is further complicated by using running bond pattern which means each successive course
is offset by a half-block?s length. Therefore, geometric discrepancies had to be employed
to facilitate the modeling. These are discussed in the following sections.
28
3.4.1 Concrete Masonry Units
The typical 8-in. CMU is nominally 16 in. long by 8 in. wide by 8 in. high; in
reality the block is 15.625 in. by 7.625 in. by 7.625 in. The 0.375 in. difference is to
allow for mortar joints. With the addition of running bond, this forced the model to use a
slightly different geometry than the one of typical CMU blocks. Figure 3.6 shows a
comparison of a typical 8-in. CMU against a model 8-in. CMU used in modeling. As can
be seen, the width and height of the section is the same in both models; however, the
overall length of the members is reduced to 15.5 in. instead of 15.625 in. The outer webs
are 1 in. wide in both blocks, but the center web for the FEM is 1.5 in. wide instead of 1
in. This causes the overall volume of the FEM block to be 432.2 in3, instead of 415.1
in3; the overall volume increased of area is 1.041%. This additional material is
compensated for by reducing the mass density in the material modeling and will be
discussed later. In addition, the CMU blocks have corner fillets while the FEM block has
squared corners. The fillets have two effects. They change the mass of the block, and
they cause less of a stress concentration at these corners.
29
1 in
1 in 1 in
1.25 in
Figure 3.6. Comparison of real CMU and FEM CMU
Real
FEM
30
Figure 3.7. 3-D view of normal block
Figure 3.7 shows the three dimensional element size for the block is 0.5 in. by 0.5
in. by 1.525 in. This element size was used for all block. The size was chosen to provide
two elements through the thickness of the faceshells and because analyses demonstrated
that five elements through the height was efficient and accurate.
The model also employed U blocks and half-high blocks. U blocks are the same
as the normal blocks except that the webs are removed to allow for continuity of grout in
the bond beams. U blocks are modeled the same as normal CMU except the webs only
consist of the lowest line of elements for the webs. The half-high blocks are only half the
height of a normal block or 3.625 in. The half-high blocks only use 3 elements through
the height at 1.21 in. A U-block can be seen in Figure 3.8; a half-high block can be seen
in Figure 3.9.
1.525 in by
0.5 in mesh
size
1.525 in by
0.5 in mesh
size
0.5 in by 0.5
in mesh size
31
Figure 3.8. U-block used in FEM
Same as the regular
CMU except webs are
only meshed with the
bottom row of blocks
32
Figure 3.9. Half-high blocks used in FEM
3.4.2 Mortar and Grout
The running bond also caused challenges in meshing the mortar joints. Each head
joint was modeled as 0.5 in. thick instead of the customary 0.375 in. This was done to
facilitate the running bond pattern. The bed joints were 0.375 in. thick. Mortar was only
simulated on the face shells as common in construction.
The mortar was modeled with the same element size as the CMU it is attached to
(either 0.5 in. by 0.5 in. for bed joints or 1.52. in by 0.5 in. for head joints). This was
done to allow for appropriate tying of nodes together. The mortar was modeled with one
element through its thickness.
The grout was slightly affected by the running bond as the column was not
necessarily straight. In some of the columns, the grout zigzagged following the slight
1.21 in. by
0.5 in. mesh
density
0.5 in. by 0.5
in. mesh
density
1.21 in. by
0.5 in.
mesh
density
33
offset of each block from the one below it. The bond beams were not affected by the
running bond. The element size of the grout for columns and bond beam matched the
concrete block it is attached to (0.5 in. by 0.5 in. by 1.52 in. for normal blocks). This was
done to allow the exterior nodes of the grout to share nodes with the CMU it was adjacent
to and allowed a connection of the two. Figure 3.10 shows an 8 in. CMU with mortar for
the bed and head joints and with one cell grouted. Figure 3.11 shows a grouted column
without blocks encasing it; Figure 3.12 shows a bond beam with the bottom layer of
blocks shown.
Figure 3.10. Mortar and grout meshing
0.5 in. by 0.5 in.
mesh to match
CMU mesh
1.525 in. by
0.5 in. mesh
to match
CMU mesh
34
Figure 3.11. Grout columns
Figure 3.12. Bond beam and blocks
Column meshed
with offset to
match the offset of
the surrounding
CMU blocks.
Bond beams created
with holes to allow for
the webs and mortar of
the wall.
35
3.4.3 Steel Reinforcing
The steel reinforcing was meshed using one-dimensional beam elements. These
were placed in the center of the cell for columns and the center of the bond beams. The
main focus for modeling the beams was to make sure the beam shared a node with the
surrounding grout elements; in order to do this, beam elements were generated for every
grout section, and each beam was divided into sub-beams to attach at every grout node in
the same location. The nodes of the beam shared the nodes with the grout elements. This
was done to ensure compatibility between the steel and grout and thus to cause the beam
elements to be properly stressed. Figure 3.13 shows reinforcing coming out of the
grouting.
The steel reinforcing
modeled as beam with
nodes shared with grout
nodes.
Figure 3.13. Beam embedded in grout
36
3.5 Material Modeling
LS-DYNA has a library of over 200 different material models that can be used for
many different applications. The material models used for this work were chosen
because they produced favorable results for similar research efforts or because the
literature review highlighted these models as being a good approximation for the actual
materials used. The following sections summarize the material models used in the
model.
3.5.1 Cementitious Material Model
As mentioned earlier, there are 26 material models that have been developed for
geological and cementitious materials; six of these models were determined to be
appropriate for the present work. Table 3.5 shows the six models that were considered
and the advantages and disadvantages of each model. All of the models would have been
appropriate to model CMU with proper validation and material testing. At present, there
is little literature on the properties of a single CMU block except in uniaxial compression.
In addition to uniaxial, unconfined compression testing, Schwer (2001) list hydrostatic
compression, triaxial compression/extension, and uniaxial strain as being necessary to
properly characterize geomaterials. Since this data is not readily available and is needed
in calibrating Mat 5 and Mat 14, these models were not used. The input of Mat 84 and
Mat 85 is complicated and is built around reinforced concrete; therefore, these were
eliminated. Mat 72 R3 requires minimum input with model generation; this model was
developed to provide generic material and volumetric parameters around a 6610 psi (45.6
MPa) normalweight concrete. This material model was initially used because the input
37
was simple under the circumstances; however, upon running a few models it was
determined that model?s parameter generation did not fit CMU modeling because the
model assume homogeneity and normalweight concrete. The model with the full-scale
testing material inputs was too conservative giving deflections that were too low. Mat 96
is built around reinforced concrete section with several inputs just for reinforcing.
However, Mat 96 allows for tensile and shear damage and was used in previous CMU
wall modeling. The LS-DYNA Keyword User?s Manual Version 971 Release 4 says that
the model is ?an anisotropic damage model?[admitting] progressive degradation of
tensile and shear strengths across smeared cracks?under tensile loadings.?
Table 3.5. CMU material model selection
Mat.
Model Pros Cons
5 Soil &
Foam Many inputs for accurate modeling First geomaterial model; primitive
14 Soil &
Foam w/
Fail.
Same as Mat 5 except has tension
cutoff
72 Con.
Dam R3
Has the ability to generate material
parameters; has many inputs
Parameter input is based on
reinforced concrete and has not been
validated for CMU
84 Win.
Con w/
RE
Is smeared crack model
Built primarily for reinforced
concrete section with many inputs
for rebar
85 Win.
Con. Same as Mat 85
96 Brittle
Dam.
Simple input model with both shear
and tension damage modeled; has
been used recently to model CMU.
Built primarily for reinforced
concrete section with many inputs
for rebar
38
The CMU properties used in the suitability analyses were a unit weight of 96.8
lb/ft3 (or 1.450x10-4 lb-s2/in.4), Poisson?s ratio of 0.20, and ultimate compressive
strength (f?m) of 1290 psi. The density was reduced from 101 lb/ft3 from material testing
in the full-scale testing to account for the added volume of the model CMU block where
the overall mass would not be affected. The reduction factor was the ratio of the volume
of an actual CMU compared to a model CMU. The modulus of elasticity was 1,163,000
psi based on 900 f?m given by American Concrete Institute (ACI) 530 (ACI, 2011), the
tensile strength was 181 psi based on 6.7?(f?m)0.5 given by ACI 318 (ACI, 2008), and the
shear strength was 2? ( f?m)0.5 based also on ACI 318 (ACI, 2008). The following shows
an input for Mat 96 Brittle Damage.
*MAT_BRITTLE_DAMAGE
mid ro e pr tlimit slimit ftough sreten
1 1.4450E-04 1.163E+06 0.2 181 53.9 0.8 0.03
visc fra_rf e_rf ys_rf eh_rf fs_rf sigy
104 0 0 0 0 0 0
where mid is the material ID number, ro is mass density, e is Young?s modulus, pr is the
Poisson ratio, tlimit is tensile limit, slimit is the shear limit, ftough is the fracture
toughness, sreten is the shear retention, and visc is the viscosity of the concrete, and all
other parameter are not used or default.
Grout and mortar were modeled using the Mat 96 with the ultimate compressive
strength and mass density changed to reflect their material properties. The mortar
properties were a unit weight of 125 lb/ft3 and an ultimate compressive strength of 3190
psi; the mortar?s modulus of elasticity was the same as the CMU since 900 f?m was based
on the prism strength, and the tensile and shear limits were left the same since these will
be modeled more explicitly in the bond modeling. The grout properties were a unit
39
weight of 125 lb/ft3 and an ultimate compressive strength of 7000 psi (f?g); the grout
modulus of elasticity was given by 500 f?g based on ACI 530 (ACI, 2011), and the tensile
and shear limits were based on normal concrete limits for lighter weight concrete from
ACI. All properties were based on the material tests from Davidson et al. (2011) or on
design standards.
3.5.2 Reinforcement Material Model
The steel material properties were assumed to be elastic-perfectly plastic without
strain hardening. This was done for ease of modeling. Mat 3 Plastic Kinematic was
selected because it allows elastic-perfectly plastic stress-strain modeling and because it
works with beam elements. The steel was simulated as standard Grade 60 reinforcement.
The properties of the reinforcement are a unit weight of 490 lb/ft3, a yield strength of 60
ksi, a Young?s modulus of 29000 ksi, and a Poisson ratio of 0.30. The material properties
were based on the industry standards not on the material testing from the dynamic testing.
The following is a sample of the input for the reinforcing steel.
*MAT_PLASTIC_KINEMATIC_TITLE
mid ro e pr sigy etan beta
7 7.34E-04 2.90E+07 0.30 60000 0 0
src srp fs vp
0 0 0 0
where mid is the material ID number, e is Young?s modulus, ro is mass density, pr is the
Poisson ratio, sigy is the yield stress, etan is the tangent modulus, and all other values are
not used in the model.
40
3.5.3 Boundary Material Model
The boundary was assumed to be infinitely rigid. This was also done where the
boundary will not have any effect on the results. Mat 20 Rigid was specifically
formulated to keep the material rigid. The boundary was assumed to be made of steel
with a unit weight of 490 lb/ft3, a modulus of elasticity of 29000 ksi, and a Poisson ratio
of 0.30. The following shows a sample input for Mat 20 Rigid.
*MAT_RIGID_TITLE
mid ro e pr n couple m alias
6 7.34E-04 2.90E+07 0.30 0 0 0
cmo con1 con2
0 0 0
lco or a1 a2 a3 v1 v2 v3
0 0 0 0 0 0 0 0
where mid is the material ID number, e is Young?s modulus, ro is mass density, pr is the
Poisson ratio, and all other inputs are not used.
3.6 Element Modeling
The model used two distinctive element types, solid and beam. The solid
elements were used to model CMU blocks, mortar joints, grout, and boundary supports.
The constant stress element formulation was used to model all solids for most runs. This
formulation is an eight-node, hexagonal brick element with single point integration. This
was done because it vastly reduces the computational time and costs; the drawback was
the model was less accurate than the fully integrated solid elements. The fully integrated
S/R solid formulation was also used in some smaller models to accurately capture the
stress and strain gradient over the CMU. The CMU were the only elements with the fully
integrated formulation. The following show solid element inputs. The first is for
constant stress solid elements, and the second is fully integrated solid elements.
41
*SECTION_SOLID_TITLE
secid elform aet
1 1 0
*SECTION_SOLID_TITLE
secid elform aet
1 2 0
where secid is the section ID, elform is the element formulation specification, and aet is
the ambient element type.
Beam elements were used to model the steel reinforcement. The Hughes-Liu
beam element formulation was used. This formulation takes into account both bending
and axial actions. Even though steel reinforcement is not necessarily used in design with
its individual moment-resistance and moment of inertia, this formulation takes into
account the full-effect of the steel internal forces. In addition, the steel then can respond
in dowel action which is carried through axial straining of the beam as the grout bends.
The following shows the Hughes-Liu beam input for the model.
*SECTION_BEAM_TITLE
secid elform shrf qr/irid cst scoor nsm
2 1 1 2 1 0 0
TS1 TS2 TT1 TT2 NSLOC NTLOC
0.2 0 0 0 0 0
where secid is the section ID, elform is the element formulation specification, shrf is the
shear factor, cst is the cross section type (1 is tubular), nsm is the nonstructural mass per
unit length, TS1 is the outer diameter, TS2 is the inner diameter, and all other inputs are
either not used or are defaults.
42
3.7 Load Modeling
In loading of the walls, there are two major loadings, gravity loading and blast
wave. The effects are modeled through various ways using load keyword cards in LS-
DYNA.
3.7.1 Gravity Preloading
The gravity preload was used to generate the initial conditions due to self-weight.
This was done easily by adding a body load in the downward direction using Load Body
with the direction being in the vertical direction. The following shows the gravity
preload input for the model.
*LOAD_BODY_Z
lcid sf lciddr xc yc zc cid
1 1 0 0 0 0 0
where lcid is the load curve ID, sf is the load curve scale factor, and all other are not used
in the model. In order to use Load Body card, a load curve had to be defined; this was
done by using the Define Curve card. The following is a sample input for the gravity
curve.
*DEFINE_CURVE_TITLE
lcid sidr sfa sfo offa offo dattyp
1 0 1 384.6 0 0 0
a1 o1
0 0
0.02 1
1.0 1
where lcid is the load curve ID, sfa is the scale factor for the abscissa value, sfo if the
scale factor for the ordinate value, offa is the offset for the abscissa value, offo is the
offset for the ordinate value, a1 are the abscissa values, o1 are the ordinate values, and all
other are not used in the model.
43
The curve provided a gradual increase in the gravity effect to allow for smaller
stress gradients in the initial loading. The dynamic relaxation algorithms were not
explicitly used in the model; these algorithms would generate damping forces to remove
any movement at the beginning of simulation; however, the algorithms would cause the
model to take longer to run and be more costly. Another way to allow dynamic
relaxation was to not start the blast wave until the base reaction under gravity loading
reached a stable oscillation. In order to accomplish this, a ramp function was used to
decrease the initial oscillations. Figure 3.14 shows interface force of the verification wall
with gravity preloading on it. It can be determined that the base reaction met a normal
oscillation at 20 ms; this was confirmed by research which recommended 20 ms as well.
In addition, the overall axial stress at the base was less than 30 psi which was about 2%
of the masonry prism?s strength.
0
1000
2000
3000
4000
5000
6000
0.000 0.010 0.020 0.030 0.040 0.050
Time (ms)
Re
ac
tio
n F
or
ce (
lb)
Figure 3.14. Base force of wall under gravity loading
44
3.7.2 Blast Loading
Blast loading can be applied in a few ways. The easiest is to use the Load Blast
Enhanced card. This load card allows for simple inputs to generate a pressure-time curve
that is applied on the wall. However, this method does not allow for direct control of the
loading. The other way in which the blast loading can be carried out is by directly
inputting the pressure values in Load Segment Set. This can be done to specifically
control the pressure to match data from testing or to generate user specified loadings for
analysis. A sample input of Load Segment Set is shown.
*LOAD_SEGMENT_SET
ssid lcid sf at
1 3 1 0
where ssid is the segment set ID, lcid is the load curve ID, sf is the load curve scale
factor, and at is the birth time of pressure.
3.8 Boundary Modeling
The boundary was modeled to allow one-way bending behavior of the wall;
therefore, the boundary was modeled with rigid material model with all the degrees-of-
freedom fixed; the rigid material model does not allow the boundary parts to deform.
This prevents boundary?s deflection from interfering with deformation of the wall. The
wall rested on the boundary parts. Figure 3.15 shows the boundary members.
45
Figure 3.15. Boundary modeling
As can be seen in the figure, the boundary was simulated as a base plate and two braces,
one at the top and one at the bottom. This was done in order to match the full-scale
dynamic test as closely as possible. Two-way bending was not modeled as this would not
provide any better understanding of the breaching phenomenon, and the common
construction practice is to build the wall with only support at the top and bottom. The
boundary restrains movement of the wall by causing added fixity at the bottom of the
wall. This was done to keep the wall in the frame during the spring-back phase. The
wall could freely rotate both at the top and bottom. The braces were offset from the wall
by 0.05 in.; this kept the wall in the frame without causing problems with the
calculations.
3.9 Contact Modeling
In order to accurately capture the response of a CMU, different contact definitions
must be set-up. These include a mortar-block interface to properly represent the bond of
the mortar to the CMU and a boundary-wall interface to properly contain the wall?s
movements. The following sections describe the modeling methods that were taken to
properly model contact.
Front Brace
Back Brace
Base Plate
Front Brace
Back Brace
Base Plate ront Brace
Back Brace
Base Plate
Another back brace
positioned at top of the wall. All degrees of freedom
are fixed for all boundary
materials.
46
3.9.1 Mortar-Block Interface
The bond between mortar and blocks must remain intact until failure limits are
reached. In order to do this, there are two modeling approaches. The first is to force the
mortar and CMU to share nodes; in order to have bond failure, the mortar elements must
have erosive properties built-in it to allow elements to be deleted when they reach
limiting stress or strain values. The other way is to define a contact definition where
nodes are tied together. The contact definition has a built-in failure criterion that allows
the two surfaces to untie and slide independently of each other. Even though the former
was cheaper and faster, the latter approach is used. This is because element erosion can
change the mass of the system. Also, part of the shear resistance of the wall is provided
by friction; therefore, even using the erosion method, a contact definition would have to
be applied, and the contact surfaces would have to be adaptive to allow for erosion of
elements.
LS-DYNA has several contact cards to allow for tying nodes together. Initially,
Contact Tied Surface to Surface was used; however, this was changed to Contact
Tiebreak Surface to Surface because it allows for friction sliding after ties are broken.
Tiebreak Node to Surface was used because it allows for massively parallel processor
runs. This contact definition allows for a Mohr-Coulomb failure surface characterized by
1
NEN MESn s
NEN MES
f f
NFLF SFLF
+ > (3-1)
where fn = tension force in the model (if the stress is in compression, the value is zero.),
NFLF = tensile failure force, fs = shear force in the model, SFLF = shear failure force,
and NEN and MES = exponent for normal force and shear force, respectively (normally
47
2). Once the equation is greater than one, the node is released and can slide. The tensile
failure stress was the modulus of rupture for N Portland cement mortar from MJSC; the
shear failure stress was a median stress from Atkinson et al. (1989). These stresses were
multiplied by contact area of each element to transform the stress into a force. The
contact definition allowed for friction. The static coefficient of friction between mortar
and CMU was modeled as 0.8; the dynamic coefficient was 0.7. The coefficients of
frictions are based on recommended values, and as shown in Browning (2008) the energy
dissipation due to sliding was minimumal.
There were two mortar-block interfaces, contact at the head joints and one for
contact at the bed joints; Figure 3.16 shows these contact surfaces. Each contact
definition needed two segment sets, a master and a slave. The slave set for the head
joints was the outer elements of the mortar that would be attached to the block in real
construction; the master set was the heads of each block with the exception of the blocks
on the ends that do not have mortar attached. The slave set for the bed joint interface was
the bed joint mortar on one side; the master set was the top or bottom of the block that is
on the same side of the slave set. The other side of the block and the mortar shared
nodes; this removed the need for one more contact definition and made the models run
quicker. A sample input of Contact Tiebreak Node to Surface is shown.
*CONTACT_TIEBREAK_NODES_TO_SURFACE
ssid msid sstyp mstyp sboxid mboxid spr mpr
8 7 4 0 0 0 0 0
fs fd dc vc vdc penchk bt dt
0.7 0.8 1 0 0 0 0 1.E+20
sfs sfm sst mst sfst sfmt fsf vsf
1 1 0 0 1 1 1 1
nfls sfls nen mes
100 150 2 2
48
where ssid is the slave set id, msid is the master set id, sstyp is the slave set type, mstyp is
the master set type, fs is the static coefficient of friction, fd is the dynamic coefficient of
frciton, dc is the exponential decay coefficient, bt is the birth time, nflf is the tensile
failure stress, sflf is the shear failure stress, nen is the exponent for normal force, mes is
the exponent for shear force, and all others are either not used or are default.
Figure 3.16. Mortar-block interface
3.9.2 Block-Boundary Interface
The interface between the boundary and the wall was necessary to allow the
model to simulate the blast properly. This interface allowed frictional sliding and
prevented penetration. Contact Automatic Surface to Surface allows for sliding without
penetration. This contact definition also used segment sets, as well. The master surface
Nodes of block and mortar are shared on one
side; the other is master and slave
The slave sets are these
mortar nodes
Nodes at connection of head
joints and bed are tied together
49
sets were the top of the base, the back of the bottom brace, and the front of the top brace.
The slave surface set rested against the master sets; they were, in the same order as listed
above, bottom of the bottom row of blocks, the back of the bottom row of blocks, and the
back of the top row of blocks. A sample input for this contact definition is shown.
Figure 3.17 shows the boundary-block interface.
*CONTACT_AUTOMATIC_SURFACE_TO_SURFACE_ID
ssid msid sstyp mstyp sboxid mboxid spr mpr
8 7 0 0 0 0 0 0
fs fd dc vc vdc penchk bt dt
0.8 0.6 1 0 0 0 0 1.E+20
sfs sfm sst mst sfst sfmt fsf vsf
1 1 0 0 1 1 1 1
where ssid is the slave set id, msid is the master set id, sstyp is the slave set type, mstyp is
the master set type, fs is the static coefficient of friction, fd is the dynamic coefficient of
frciton, dc is the exponential decay coefficient, bt is the birth time, and all others are
either not used or are default.
Figure 3.17. Boundary-block interface
3.10 FEM Validation
In order to trust the results of any of the finite element model, a standard had to be
set. This standard is the response of Panel 2 from all three blast tests carried out by
AFRL. This wall was built using 8-in. CMU laid with 7 blocks horizontally and 17
The interface is just where parts
of the block and boundary come
into contact.
50
courses vertically; Figure 3.1 gives the wall layout including placement of reinforcing
and grouting; in addition, Figure 3.2 gives detailed side view of the wall giving
information on the support conditions, reinforcing, splicing, and other construction
details. Figure 3.18, as seen below, describes the instrumentation used in the AFRL
dynamic tests.
Figure 3.18. Test set-up and instrumentation position
During testing both free field (FF1 and FF2) and reflected pressures (RP1?RP4) were
taken; the reflected pressure were averaged across the four gauges, and reflected impulses
were calculated. Dynamic deflections gauges (D1-9) were used to find deflections at the
mid-height and quarter-heights of the walls. The normalized pressures for Test 1 (T1),
Test 2 (T2), and Test 3 (T3) can be seen in Figure 3.19, and the normalized impulse is
shown in Figure 3.20.
6? CMU Panel 8? CMU Panel Cavity Wall
51
-0.5
0
0.5
1
1.5
2
2.5
0 10 20 30 40 50 60 70 80 90
Time (ms)
Pr
ess
ur
e (
ps
i)
T1 T2 T3
Figure 3.19. Normalized reflected pressure from dynamic testing
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 10 20 30 40 50 60 70 80 90Time (ms)
Pr
ess
ur
e (
ps
i)
T1 T2 T3
Figure 3.20. Normalized impulse from dynamic testing
52
The validation of the finite element model involved the model having similar deflected
shape or breaching pattern within a desired error range, having a similar deflection-time
response, and having matching breaching stress patterns. Having similar deflected shape
is hard to compare since the testing occurs quickly without a true way to capture a good
representation of deflected shape. As a substitute, high-speed footage from inside a blast
chamber is used. Since the 8-in. CMU panel was used for comparison to finite element
model results; the deflection of instruments D7, D8, and D9 are the only ones of interest.
These points coincide with the quarter-points and halfway point on the wall.
Figure 3.21 shows several video captures of the dynamic testing from behind the
wall of Panel 2 from Test 1. Figure 3.22 shows several screen captures of the finite
element model run of Test 1.
53
a.) b.)
c.) d.)
Figure 3.21. Video captures of panel 2 during test 1 at a.) 10 ms, b.) 21 ms,
c.) 41 ms, and d.) 84 ms after loading starts
54
a.)
b.)
55
c.)
d.)
Figure 3.22. Screen captures of FEM of panel 2 test 1 at a.) 10 ms, b.) 21 ms,
c.) 41 ms, and d.) 84 ms after loading starts
56
As can be seen, the deflected shape of the finite element model and the dynamic testing
correlate well for Test 1. Both formed cracks around the mid-height of the wall and
around the quarter-point of the walls. The wall from the dynamic testing did show signs
of breaching; however, the breach was small and did not break all the way through the
wall. The finite element pictures shown have some plastic strain in a fringe contour
around the border of the grout columns, but this was minimal compared to other model
runs. (Plastic strain is any strain that is not described by a linear elastic, stress-strain
relationship such as yielding, rupture, or crushing.) Figure 3.23 shows the deflections-
time graphs of the dynamic testing and FEM for the quarter-points and mid-point of the
wall. The three deflections match up well between the FEM and the full-scale testing.
The FEM is less stiff at the beginning of the test but tends to have greater stiffness later in
the simulation. This difference in stiffness is accounted for by not correctly modeling the
boundary conditions. However, the difference of overall maximum deflections between
FEM and full-scale testing is low.
57
0
1
2
3
4
5
6
7
8
9
10
20 30 40 50 60 70 80 90
Time (ms)
De
fle
cti
on
(i
n)
D7 D8 D9
FEM D7 FEM D8 FEM D9
Figure 3.23. Deflection of panel 2 and of FEM from test 1
Figure 3.24 shows several video captures of the dynamic testing from behind the
wall of Panel 2 from Test 2. Figure 3.25 shows several screen captures of the finite
element model run of Test 2.
58
a.) b.)
c.) d.)
Figure 3.24. Video capture of panel 2 during test 2 at a.) 10 ms, b.) 19 ms,
c.) 43 ms, and d.) 76 ms after loading starts
59
a.)
b.)
60
c.)
d.)
Figure 3.25. Screen captures of FEM of panel 2 test 2 at a.) 10 ms, b.) 19 ms,
c.) 43 ms, and d.) 76 ms after loading starts
61
As can be seen, the FEM?s deflected shape does not match with the deflection of the
dynamic test. This is because the FEM does not breach, and the dynamic testing does.
The model uses continuum elements that will not allow deletion of the element or
cracking; the element does allow for plastic strain. This could have been forced by using
erosive elements to model the CMU, but this would not have helped in the understanding
of the breaching phenomenon. The breaching is captured in the FEM model by plastic
strain. Shear hinges form at the boundary of the grout columns which would match
cracking and breaching in a brittle material. (Shear hinges are an analytical tools used to
understand when ductile material fail in shear; they allow for a constant strength while
allow for progressive shear damage such as a plastic hinge in flexure.) Figure 3.26
shows a cross-section of the wall around mid-height 70 ms after the blast. Even though
the cross section seems to show a flexural action of the faceshell, by this time in testing
the block had sheared off. Also, the breaching pattern from the dynamic testing also is
matched by the stress resultant in the FEM. Three stress contours occurring 1 ms after
loading is shown in Figure 3.27. The element formulation did allow plastic strain. The
plastic strain gradient right after the blast is also shown in Figure 3.27.
Figure 3.26. Cross-section of FEM
Grouted Cells
62
a.) b.)
c.) d.)
Figure 3.27. Stress Contours for FEM 1 ms after Loading: a.) Effective Stress,
b.) XY-Shear Stress, c.) XZ-Shear Stress, and d.) Plastic Strain
Grouted
Cells
Grouted
Cells
63
Figure 3.27a shows the effective stress contour (Effective stress is stress combination
based on von Mises stress calculations.) on the wall immediately following the blast
wave hitting the wall. This contour plot shows there is a high concentration of stress
along the boundary of grout columns. This is affirmed by the contour plot of xy-stress
(b.), which also shows that there are high stresses along the web lines for blocks between
grout cells. The high stress locations do not appear in the grout columns. The contour
plot for xz-stress shows that there are also high stress localizations at the boundary of the
bond beams at the top and bottom of the wall. These shear stress localizations caused the
breaching in the full-scale test. Finally, the contour plot of plastic strain shows that there
is plastic strain at the boundary of the grout columns. This plastic strain shows up as
shear hinges in the FEM, as seen in Figure 3.26.
Finally, the deflection-time plot for the testing and FEM is shown in Figure 3.28.
64
0
2
4
6
8
10
12
0 10 20 30 40 50 60 70 80 90 100
Time (ms)
De
fle
cti
on
(in
)
D7 D8 D9
FEM D7 FEM D8 FEM D9
Figure 3.28. Deflection of panel 2 and of FEM from test 2
The FEM?s deflection and full-scale test?s deflection does and does not match. The
maximum deflections for the quarter points of the wall are similar; however, at mid-
height the maximum deflection of the FEM is higher. This was because the full-scale
testing breached venting some of the pressure. Breaching also allowed the wall to
dissipate the energy applied to the wall that the FEM cannot perform. The FEM model is
stiffer than the full-scale test as well.
Panel 2 from Test 3 had very heavy damage with large sections of the wall
blowing out breaching into the testing chamber. Since the FEM did not capture the
breaching any better than the Test 2 model did, the video captures and screen captures are
not shown. However, the FEM does show a better match of deflections. The deflections
of the full-scale testing and dynamic testing are shown in Figure 3.29.
65
0
2
4
6
8
10
12
0 10 20 30 40 50 60 70 80 90
Time (ms)
Di
sp
lac
em
en
t (
in)
D7 D8 D9
FEM D7 FEM D8 FEM D9
Figure 3.29. Deflection of panel 2 and of FEM from test 3
The deflection-time graph shows that the quarter points? deflection matched up well.
However, the midpoint deflection for the FEM is much higher. This can be attributed to
venting of the pressure caused by breaching as well as the energy dissipation caused by
breaching. For all three data points, the FEM shows the same stiffness problems as in the
other testing.
For all three comparisons, part of the difference between the deflections can be
attributed to not modeling the boundary exactly the same as the full-scale tests. In the
full-scale test, dowel rods inserted into the grout along the base of the wall provide a
semi-fixed boundary condition. The FEM did provide a limited amount of fixity by
having a front and back brace limit rotation of the bottom; however, this only occurred
after the wall rotated. Another explanation for the differences is that basic design
66
material parameters were used instead of parameters established by material testing.
With that in mind, Table 3.6 shows the maximum deflections for all three full-scale tests
with a comparison to the appropriate FEM.
Table 3.6. Maximum deflection for test 2 and FEM
Test 1 Test 2 Test 3
Full-
Scale FEM
Percent
Diff.
Full-
Scale FEM
Percent
Diff.
Full-
Scale FEM
Percent
Diff.
D7 4.414 4.426 0.3% 6.121 5.459 10.8% 5.678 4.951 12.8%
D8 8.093 8.777 8.4% 9.663 11.694 21.0% 9.848 10.106 2.6%
D9 5.044 4.777 5.3% 6.179 6.170 0.1% 5.127 4.681 8.7%
Since there was less than 10% error for seven out of nine points of the deflections and
less than 25% error for all, the deflections matches well between the FEM and full-scale
testing. Since the breaching pattern is similar, the deflections are similar, and the
deflected shape is similar, the FEM is valid for investigating the effects of shear
breaching of CMU wall subjected to blast loading.
3.11 FEM Results and Suitability Study of Breaching
Having validated the modeling approach, a short parameter variability study was
conducted to understand the variables in breaching of the CMU. In order to understand
this, several FEM were created ranging from a single block to a full wall. The analyses
were used to understand (1) time of response, (2) the change in the stress? magnitudes
with a change in wall geometry, (3) a change in the stress? magnitude with a change in
material properties, (4) a change in the stress? magnitude with a change with loading, and
(5) a change in the stress? magnitude with the addition of grouting.
67
Analysis showed that the time of response for high local shear occurred in the first
few milliseconds after the pressure is applied. This is shown in contour plots for out-of-
plane (defined earlier as XY-Shear Stress) shear stress and effective stress for the
verification wall (Figure 3.30 and Figure 3.31) and for a single block (Figure 3.32 and
Figure 3.33). This wall and block was loaded using the pressures from Test 2. Figure
3.34 and Figure 3.35 shows a plot of out-of-plane shear stresses for the single block
model over time. The stresses were found at the corners for both the backside and front-
side of the front face.
68
(0 ms after blast) (1 ms after blast)
(2 ms after blast) (3 ms after blast)
Figure 3.30. Contour plots of out-of-plane shear stress at various times
69
(5 ms after blast) (10 ms after blast)
(20 ms after blast) (50 ms after blast)
Figure 3.30. Contour plots of out-of-plane shear stress at various times
70
(0 ms after blast) (1 ms after blast)
(2 ms after blast) (3 ms after blast)
Figure 3.31. Contour plots of effective stress at various times
71
(5 ms after blast) (10 ms after blast)
Figure 3.31. Contour plots of effective stress at various times
72
(0 ms after blast)
(1 ms after blast)
Figure 3.32. Contour plots of out-of-plane shear stress at various times
73
(2 ms after blast)
(5 ms after blast)
Figure 3.32. Contour plots of out-of-plane shear stress at various times
74
(1 ms after blast)
(2 ms after blast)
Figure 3.33. Contour plots of effective stress at various times
75
(3 ms after blast)
(5 ms after blast)
Figure 3.33. Contour plots of effective stress at various times
76
(10 ms after blast)
(20 ms after blast)
Figure 3.33. Contour plots of effective stress at various times
77
Figure 3.34. Plot of out-of-plane shear stress vs. time
Figure 3.35. Plot of effective stress vs. time
The contour plots of a single block show that as the initial loading hits there was
high shear localization at the reentrant corners of the blocks, and as the loading was
continued the shear forces diminish as flexural response takes over. The flexural
78
response began to occur in 3 to 4 ms after the initial loading for the single block and fully
develop over the rest of the testing as seen in the effective stress plots at 10 and 20 ms.
The plots of the shear stress on the wall show the same response except the shear
response lasts a little longer. The graphs show that as the blast wave hits the shear stress
is high and almost instantly diminishes and never reaches the same magnitude.
Therefore, the shear loading time of response occurs very quickly before any flexural
response can occur; this is in the first 5 ms.
The next task was to find how the wall?s overall geometry affects the location and
maximum value for the shear stresses. In order to do this, several models were built
starting with only one block and continuing building block by block to 10 rows and 10
courses. These models all implored rigid roller supports at the top and bottom. The peak
pressure, loading duration, contact modeling, and material properties are the same for all
models where the only variable is the wall?s geometry. Table 3.7 and Table 3.8 display
the peak out-of-plane stress and the peak effective stress for the loading time displayed in
the contour plots. Figure 3.36 shows the out-of-plane stress the maximum stress of the
reentrant corners of blocks over the period of the analysis. Figure 3.37 shows same as
Figure 3.36 except it displays the effective stress for the models.
79
Table 3.7. Maximum out-of-plane shear stresses of various geometries of walls
XY Stress (psi)
Columns
1 2 3 4 5 7 10
2 244 252 247 250 257 255 249
3 252 276 251 252 252 252 252
4 255 260 259 260 259 260 260
5 255 280 259 258 254 254 255
7 252 258 254 257 259 253 253
10 255 266 259 259 259 259 259
12 257 273 265 260 260 264 264
Ro
ws
15 260 280 270 261 260 270 269
Table 3.8. Maximum effective stresses of various geometries of walls
Effective Stress (psi)
Columns
1 2 3 4 5 7 10
2 565 676 639 652 660 651 649
3 584 645 634 634 646 653 660
4 589 671 666 671 671 674 674
5 588 662 640 638 647 658 664
7 583 614 635 636 646 657 664
10 588 700 668 668 671 670 670
12 588 713 669 659 661 664 659
Ro
ws
15 588 706 645 636 64 645 645
80
225
250
275
300
325
350
0 1 2 3 4 5 6 7 8 9 10 11
Number of Columns
She
ar
Str
ess
(ps
i)
2 Rows 3 Rows
4 Rows 5 Rows
7 Rows 10 Rows
12 Rows 15 Rows
Figure 3.36. Plot of geometry vs. shear stress
550
600
650
700
0 1 2 3 4 5 6 7 8 9 10 11
Number of Columns
Ef
fec
tive
St
res
s (
psi
)
2 Rows 3 Rows
4 Rows 5 Rows
7 Rows 10 Rows
12 Rows 15 Rows
Figure 3.37. Plot of geometry vs. effective stress
81
The plot of the stresses versus number of blocks in a row demonstrates that there
is no significant relationship between geometry and maximum breaching shear stress.
These plots show that beyond a model of a single row or column, the geometry of the
wall is independent of the stress experienced. The single column data is not a direct
comparison because the wall section were modeled in stack bond pattern and the strength
and stress have been reported to vary greatly from running bond pattern. The single row
can be seen as extension of stack bond pattern since no blocks are above the bottom row.
Next, FEM were used to investigate the effects that material properties have on
the shear stresses. In order to do this, one simple wall model was used; the model had 5
rows by 5 blocks in a row. The investigation examined the effects of ultimate
compressive strength and unit weight.
Since most properties of cementitious products can be directly linked to ultimate
compressive strength f?m, this property was investigated first. The compressive strength
was varied while keeping the load duration, the peak pressure, wall geometry, and all
other properties constant. Since this property is connected with the modulus of elasticity,
this quality will be changed as well. Table 3.9 shows the maximum values for the shear
stress and effective stress obtained during the model-run. Figure 3.38 and Figure 3.39
show plots the data from Table 3.9.
82
Table 3.9. Stresses for compressive strength suitability study
Comp. Strength Shear Stress (psi) Effective Stress (psi)
1500 87 220
1750 97 221
2000 102 217
2250 99 226
2500 105 228
2750 103 225
3000 102 219
Average 99 222
Stand Dev 5.86 4.00
50
60
70
80
90
100
110
120
130
140
150
1250 1500 1750 2000 2250 2500 2750 3000 3250
Compressive Strength (psi)
Sh
ear
St
res
s (
ps
i)
Figure 3.38. Plot of out-of-plane shear stress vs. compressive strength
83
200
210
220
230
240
250
1250 1500 1750 2000 2250 2500 2750 3000 3250
Compressive Strength (psi)
Ef
fec
tiv
e S
tre
ss
(ps
i)
Figure 3.39. Plot of effective stress vs. compressive strength
Looking at the data in Table 3.9 and the following figures shows there is no direct
connection between the ultimate compressive strength and the stresses experienced in the
breaching shear response. If there is an effect caused by changing the ultimate strength
of the masonry, this effect can be compensated by working the compressive strength into
the resistance formulation.
The next property looked into was the unit weight of the CMU. The weight of the
CMU makes up most of the mass of the wall. Since the loading is dynamic, a change in
the overall mass or weight of the system should cause a change in the stress of the
response. In order to do this, the same wall as used for the compressive strength
suitability study was used and the unit weight is changed from lightweight concrete to
middleweight concrete. Even though the modulus of elasticity is connected with the unit
weight, the modulus of elasticity remained constant for these analyses. Table 3.10 shows
the maximum values for the shear stress and effective stress of the suitability study over
84
the run-time of the model. Figure 3.40 and Figure 3.41 show plots of the data in Table
3.10.
Table 3.10. Stresses for compressive strength suitability study
Unit Weight
(lb/ft3)
Shear Stress
(psi)
Effective Stress
(psi)
85 103 221
90 100 222
95 102 222
100 99.1 221
105 100 220
110 100 220
115 101 219
120 103 218
125 102 217
130 104 217
135 104 216
140 102 216
145 100 216
150 100 216
80
90
100
110
120
130
140
150
80 90 100 110 120 130 140 150
Unit Weight (lb/ft3)
Sh
ea
r S
tre
ss
(p
si)
Figure 3.40. Plot of out-of-plane shear stress vs. unit weight
85
200
210
220
230
240
250
80 90 100 110 120 130 140 150
Unit Weight (lb/ft3)
Ef
fec
tiv
e S
tre
ss
(p
si)
Figure 3.41. Plot of effective stress vs. unit weight
The effective stress plot demonstrates a downward trend of stress as the unit weight goes
up. This is to be expected as an increase in unit weight gives an increase in mass of the
system. However, the decrease is negligible over the range of unit weights. The other
figures show that there is a uniform stress value for the same loading for all unit weights
and that there is no direct relationship between breaching shear stress and unit weight.
Next, the effects of loading were investigated. The first investigation into loading
was whether a change in impulse affected the magnitude of the shear stress. The loading
shapes were the simplified triangular pulse load and rectangular load. The maximum
loading pressure and loading duration were held constant, and only the shape was
changed; therefore, the applied impulse for the rectangular loading was twice the impulse
as the triangular loading. These loading shapes are seen in Figure 3.42. Various FEM?s
86
from geometry testing were modified in order to analyze the loading shape?s influence on
the breaching phenomenon. Table 3.11 shows the out-of-plane shear stress and effective
stress for the triangular pulse shape. Table 3.12 shows the average and the standard
deviation.
Table 3.11 shows that there is a difference between the rectangular and triangular
load in all stresses. The difference is as high 40.8 psi for the shear stress. However, for
the most part, the difference in shear stress is less than 30 psi for 50 psi. This pattern is
carried through effective stress. All data shows to have an average percent difference of
less than 10% and a standard deviation of less than 10 psi. Therefore, this shows there is
a slight difference in stress values with different loading shapes, but the difference is
negligible. The data shows how the shear stress is not greatly affected by increasing the
impulse, such as doubling the impulse.
Figure 3.42. Plot of normalized loading
Po
Td Normalized Time
No
rm
ali
zed
Pres
su
re
87
Table 3.11. Stresses from loading shape suitability study
Shear Stress (psi) Effective Stress (psi) Wall
Geometry Tri. Rect. Tri. Rect.
3x3 267 290 646 695
3x5 284 307 668 715
3x7 267 290 647 694
3x10 252 290 660 694
5x3 265 288 646 693
5x5 282 306 671 718
5x7 265 306 646 718
5x10 282 306 671 719
10x3 265 288 657 713
10x5 265 288 659 717
10x7 265 288 665 715
10x10 265 288 659 716
Table 3.12. Statistical data on loading shape suitability study
Shear Stress (psi) Effective Stress (psi) Wall
Geometry Difference % Diff. Difference % Diff.
3x3 23 8.5% 49 7.6%
3x5 22 7.8% 47 7.1%
3x7 23 8.6% 47 7.3%
3x10 38 15% 34 5.2%
5x3 23 8.7% 48 7.4%
5x5 23 8.3% 48 7.1%
5x7 41 15.4% 72 11%
5x10 24 8.4% 47 7.1%
10x3 23 8.8% 56 8.6%
10x5 23 8.7% 58 8.8%
10x7 23 8.7% 51 7.6%
10x10 23 8.7% 58 8.7%
Aver. 26 9.7% 51 7.8%
Also, the differences in peak pressure were examined to see if it caused any difference in
magnitude of the shear stresses. Only three FEM?s were used, and only the peak pressure
in both triangular and rectangular loading shape was varied. Table 3.13, 3.14 and 3.15
show the data from the peak pressure suitability study for three different walls. Figure
3.43 and Figure 3.44 show plots of the data from Table 3.13, 3.14, and 3.15.
88
Table 3.13. Values of stresses for the 5x5 wall from the
peak pressure suitability study
Triangular Rectangular Peak
Pressure
(psi)
Shear
Stress
(psi)
Effect.
Stress
(psi)
Shear
Stress
(psi)
Effect.
Stress
(psi)
5 54 133 61 152
10 102 217 112 244
15 122 321 135 344
20 140 359 151 384
25 149 374 162 411
30 163 421 178 458
35 190 485 208 520
40 223 546 243 589
45 254 616 273 658
50 282 671 306 718
Table 3.14. Values of stresses for the 10x3 wall from the
peak pressure suitability study
Triangular Rectangular Peak
Pressure
(psi)
Shear
Stress (psi)
Effect.
Stress (psi)
Shear
Stress (psi)
Effect.
Stress (psi)
5 54.1 135.9 60.4 155
10 86.9 214 98.0 244
15 120 302.4 135 330
20 138 334.5 149 363
25 151 373.4 166 423
30 169 426.8 187 475
35 187 487.8 208 537
40 215 558.8 237 611
45 238 611.8 259 658
50 265 658.7 288 713
89
Table 3.15. Values of stresses for the 3x10 wall from the
peak pressure suitability study
Triangular Rectangular Peak
Pressure
(psi)
Shear
Stress
(psi)
Effect.
Stress
(psi)
Shear
Stress
(psi)
Effect.
Stress
(psi)
5 54.2 134 54.2 134
10 94.7 209 94.7 209
15 122 320 122 320
20 137 370 137 370
25 159 388 159 388
30 196 407 196 407
35 201 477 201 477
40 218 536 218 536
45 240 599 240 599
50 252 660 290 694
0
50
100
150
200
250
300
0 10 20 30 40 50 60
Load (psi)
Sh
ea
r S
tre
ss
(ps
i)
T 5x5 T 10x3 T 3x10
Figure 3.43. Plot o out-of-plane shear stress vs. peak pressure
90
0
200
400
600
800
1000
1200
0 10 20 30 40 50 60
Peak Pressure (psi)
Ef
fec
tiv
e S
tre
ss
(ps
i)
T 5x5 T 10x3 T 3x10
Figure 3.44. Plot of effective stress vs. peak pressure
The graphs and data show that there is a relationship between peak pressure and
breaching stress values. The relationship is approximately linear. The R-squared values
are all above 0.95 indicating a good fit for linear regression. Therefore, there is linear
relationship between pressure and stress values; this is expected in shear stress since a
simple structural analysis will result in the linear increase of shear forces with a given
pressure or line load.
The final investigation looks at the effects of grouting. In order to do this, several
wall models were created that include grout. All models using grout were partially
grouted. Table 3.16 shows values for the maximum out-of-plane shear stresses for both
grouted and non-grouted models occurring immediately after the blast wave hits the wall.
Figure 3.45 and Figure 3.46 show graphical version of Table 3.16.
91
Table 3.16. Grouted vs. non-grouted maximum stresses
Shear Stress (psi) Effective Stress (psi)
Non-Grouted Grouted Difference Non-Grouted Grouted Difference
5x3 265 266 0.9 646 780 135
5x5 282 280 2.2 671 870 199
5x7 265 289 24 646 833 187
5x10 282 294 12 671 839 168
7x3 254 280 27 635 799 164
7x5 259 284 25 646 717 71.2
7x7 253 289 36 657 717 59.5
7x10 253 282 29 664 726 62.1
10x3 265 268 3.8 657 897 240
10x5 265 277 13 659 827 168
10x7 265 281 17 665 751 86.6
10x10 265 277 13 659 783 124
12x3 259 266 7.0 669 847 177
12x5 265 276 11 661 823 162
12x7 265 276 12 664 788 124
12x10 265 277 12 659 774 115
15x3 265 267 2.2 645 702 56.8
15x5 265 293 28 636 720 84.5
15x7 265 277 12 645 714 68.2
15x10 265 299 34 645 726 80.5
92
250
255
260
265
270
275
280
285
290
295
300
305
2 4 6 8 10 12
Number Blocks along Length
Sh
ea
r S
tre
ss
(ps
i)
Non-Grouted Grouted
Figure 3.45. Out-of-plane shear stresses for grouted and non-grouted walls
600
650
700
750
800
850
900
950
2 4 6 8 10 12
Number of Blocks along Length
Ef
fec
tiv
e S
tre
ss
(ps
i)
Non-Grouted Grouted
Figure 3.46. Effective stresses for grouted and non-grouted walls
93
As can be seen from the above graph, the grouted sections resulted in higher shear
stresses; this was because the wall sections around the grouted columns are stiffer and
would attract a higher percentage of the load. The models were only grouted in the end
cells of the models, so an increase in the number of blocks would increase the distance
between grouted cells. There is however no correspondence between the distances
between grouted columns and the shear stress values. This indicates that shear stress
experience, while not completely independent of the grouting, is independent of where
the grout is placed for the breaching shear effect. The grout does cause additional stress
in the wall sections, and this was compensated for in Chapter 4.
94
CHAPTER 4
DESIGN SHEAR RESISTANCE
4.1 Introduction
The point of this chapter is to highlight the steps taken to develop an engineering-
level design equation for direct shear in CMU walls subjected to blast loading. This
design equation must provide an adequate prediction of the strength of the system
without being overly conservative.
Since this system is dynamic, an overview of structural dynamics is given with a
focus on approximate modeling towards understanding the breaching phenomenon of
CMU walls. The wall dynamics properties of the wall are determined, allowing a single
block or a group of blocks to be modeled as quasi-static. This allows the wall to be
analyzed in a static state and be conceptualized as a beam. Finally, the structural analysis
will allow for the development of a nominal shear force according to the maximum
pressure and to eventually develop a shear resistance equation.
4.2 Structural Dynamics
In order to better understand the breaching phenomenon, a structural dynamic
analysis of the wall was to be carried out. Most systems are too complicated to have a
full structural dynamics analysis performed; this is because the system is made of an
infinite number of parts that want to move in an infinite number of ways. This multiple
95
degree-of-freedom system is out of the scope of most practicing design engineers;
therefore, a complicated system needs to be turned to a simplified one-way system. The
following section gives an overview of single degree-of-freedom (SDOF) systems plus
added simplifications that can be taken for special cases. A full SDOF analysis was not
carried out, but parts of the analysis were used to determine dynamics properties of the
wall.
4.2.1 Single Degree-of-Freedom Model
A SDOF system is system where there is only one way for the system to move
hence a single degree-of-freedom. This is extended to a multiple degree-of-freedom
system where predominant motion can be described by a single motion and all other
motions can be described by this motion. This degree-of-freedom can be a displacement
or a rotation. Figure 4.1 shows free body diagram of a lumped-mass, SDOF system.
Figure 4.1. Single degree-of-freedom model
With the help of Bigg?s Introduction to Structural Dynamics, the system has four forces
acting on it. These forces are inertia-induced force, stiffness-induced force, damping-
induced force, and the applied force. Putting the above system in equilibrium produces
m k cF F F F(t)+ + = (4-1)
96
where Fm = mass-induced force, Fk = stiffness-induced force, Fc = damping-induced
force, and F(t) = forcing function or applied force according to time.
Using Newton?s law of motion, mass-induced force is equal to mass times the
acceleration given by
mF ma my= = dotnospdotnosp (4-2)
where m = mass of the system and a = ydotnospdotnosp = acceleration of the system. By definition, the
acceleration is defined as the second derivative of the displacement according to time.
The resistance-induced force is provided by a resistance or rigidity of the system.
This resistance can be conceptualized as a simple elastic spring; an elastic spring has a
constant stiffness per unit displacement. This then gives the resistance-induced force as
kF k? ky= = (4-3)
where k = resistance or stiffness of the system per unit displacement and ? = y =
displacement of the system.
The damping-induced force is provided by friction, cracking, or other energy-
absorption mechanism. All these damping effects can be approximated by a viscous
damper; this type of damper requires a constant force to move a body through a viscous
liquid at a certain speed. The damping-induced force is given by
cF cv cy= = dotnosp (4-4)
where c = damping coefficient of the system and v = ydotnosp = velocity of the system. The
velocity is defined as the first derivative of the displacement according to time.
Substituting Equation 4-2, 4-3, and 4-4 into Equation 4-1 gives
my cy ky F(t)+ + =dotnospdotnosp dotnosp (4-5)
97
which is the governing equation for a SDOF system. This equation can be solved to find
the exact solution for a lumped-mass system. With the full dynamic equation known,
other properties of the dynamic system can be obtained. The natural frequency of the
system ?n is given by
? =n km (4-6)
The natural period of the system Tn is given by
2pi
?=n nT (4-7)
The natural period is the amount of time that the system takes to complete a full cycle.
Unlike in the above lumped-mass system, the mass for most structures is
distributed over the entire volume. Equation 4-5 can still be used, but the analysis
becomes extremely difficult for systems, especially structures with continuous?mass
distribution. An approximate method then must be carried out to ease computation. The
approximate system is given by
e e em y k y F (t)+ =dotnospdotnosp (4-8)
where me, ke, and Fe(t) = effective mass, effective resistance, and effective forcing
function of the system, respectively. The damping forces are negligible in the blast
simulations since the time of interest is small, so this force is assumed to be zero.
Equation 4-6 can be expressed in terms of the real system by
m L LK my K ky K F(t)+ =dotnospdotnosp (4-9)
where Km and KL = mass and load factors, respectively. The damping-induced force is
assumed to be zero in the equivalent system. Since most of the system?s response occurs
98
in the first mode, the approximation idealizes that all response will occur in a mode close
to the first mode. The idealized system assumes the displacement of the system ?(x) to
be the same as that produced in static loading with the exception that the maximum
deflection of the system is set to one. This is done to give a better approximation of the
effective forces.
The equivalent mass, me, resistance, ke, and force, Fe, of a distributed mass
system is given by
2de
L
m m?(x) x= ? (4-10)
eF F(t)?(x)= ? (4-11)
e ek F= (4-12)
The mass factor Km and load factors KL are given by
2d
e Lm
m?(x) xm
K m m
?
= = (4-13)
d
e LL
F?(x) xF
K F F
?
= = (4-14)
Finally, if Equation 4-6 is divided through by KL, the equation is given by
LmK my ky F(t)+ =dotnospdotnosp (4-15)
where KLm = the load mass factor (ratio of Km to KL). The previous equation will give
the idealized SDOF response of a system; this system gives accurate displacement, and
this displacement can be related to the internal forces of the system. The natural
frequency of the equivalent system is given by
99
? = =ene
e Lm
k k
m K m (4-16)
More detail can be found Bigg?s Introduction to Structural Dynamics or other structural
dynamics books.
4.2.2 Pressure-Impulse Simplifications
A SDOF system can be a convenient simplification; another approach is to set up
pressure-impulse (P-I) diagrams to define whether the member may be damaged or not.
A normal response spectrum highlights the importance of some property of the structure
as compared to the response; this property can be duration of loading, natural period,
natural frequency, or other like properties. On the other hand, P-I diagrams uses the load
and impulse for a given response. Figure 4.2a shows the response of an undamped,
perfectly elastic SDOF system. Figure 4.2b shows an equivalent system in a P-I diagram
format where Po is the peak pressure, K is the stiffness, M is the mass, xmax is the
maximum displacement, I is the impulse, td is the loading duration, and T is the natural
frequency. Seen in both graphs, there exist three loading regimes: impulsive, dynamic,
and quasi-static. P-I diagrams clearly show all three loading regimes and transfer points
between the three regimes. The vertical asymptote is the impulsive regime, the
horizontal asymptote is the quasi-static regime, and the line that connects these two is the
dynamic regime. These are shown on the response spectrum, but the transfer points are
not clearly determined because the curve does not have asymptotic properties in the
impulsive regime.
100
Seen from the response spectrum diagram, the period is important. The period is
the amount of time it takes for system to complete a full-cycle of vibration. Very high
periods are characterized by large, flexible structures; while, very low periods can be
deemed to be inflexible or rigid.
101
a.) td/T
xm
ax
/(P
o/
K)
Quasi-Static RegimeImpulsive Regime Dynamic Regime
b.)
I/(KMxmax)0.5
Po
/(x
ma
xK
)
Quasi-Static Asymtope
Implusive Asymtope
Dynamic Region
I ul e sympt te
Quasi-Static Asymptote
Po
/(x
ma
xK
)
Figure 4.2. a) Typical response spectrum and b) P-I diagram
102
Some system can be even further simplified. Figure 4.3 displays the quasi-static
and impulsive regimes emphasizing the duration of loading, td, and time of response, tm.
a.)
Time
Fo
rce
tm td
Load Function
Resistance History
b.)
Time
Fo
rc
e
tmtd
Load Function
Resistance History
Figure 4.3. Loading regimes: a) quasi-static case and b) impulse case
103
The regime shown in Figure 4.3.a is a quasi-static loading case. Quasi-static regime is
characterized by the period T being much shorter than the duration of loading td. In
quasi-static loading regime, the peak resistance is reached before the loading value has
had time to dissipate. Therefore, the resistance of a quasi-static system depends on the
peak pressure and not on the loading duration.
The regime shown in Figure 4.3.b is an impulsive loading case. This case is
characterized by the period of the system being much longer than the duration of loading.
In the impulsive loading regime, the load is applied and dissipated before any significant
system response has occurred. Therefore, the loading duration has almost very little
impact on the system.
A third regime is the dynamic loading regime. The dynamic loading regime is
characterized by the loading duration being of the same order as the period of the system.
In this regime, the loading duration cannot be uncoupled from the maximum response of
the system, and the response is more complex to determine.
4.3 Modeling of Breaching Response
In analyzing the finite element models, there are two structural models that could
simulate the breaching response. The first is single-block beam analogy; the other is a
beam that runs between grout cells. Both models were investigated to determine the best
modeling approach to develop the resistance equation.
104
4.3.1 Dynamics of the Face Shell Beam Model
The finite element analysis showed that there are high shear stresses near the
webs of each block for 2 to 3 ms after the pressure reaches the wall. High stresses
existed near the webs even when there was grout present in the model. This indicated
that each individual block carries only the shear developed on its face with the mortar not
allowing transfer of shear. In addition, the geometry of the wall had no significant
influence on the maximum value of shear stress in the non-grouted geometry
investigation. Since breaching will not occur at the webs, the beam analogy was modeled
with only one cell of the block with the face shell connections to the webs acting as
perfectly fixed. Figure 4.4 shows a single cell of a block represented as a beam and a
cross section of the representative beam; Figure 4.5 shows a visualization of the beam as
fixed-fixed beam. From this model, the natural period can be obtained.
L
A
A
t
h
Figure 4.4. Face shell beam and cross section A-A
105
Figure 4.5. Single block beam representation
The beam has a span length L of one cell, a width h of the full height of the wall, and a
thickness t of one face shell. The representative beam has a moment of inertia of
3
12
htI = (4-17)
The mass and the stiffness of the beam are
g
?M Lht= (4-18)
3
384EIk
L
= (4-19)
where ? is the unit weight of the block and g is the acceleration due to gravity. For a
fixed-fixed beam under a uniformly distributed load, the assumed displacement equation
is
2 2 3 4
4
16 ( - 2 )?(x) L x Lx x
L
= + (4-20)
With this equation, the equivalent mass factor, equivalent load factor, and equivalent
load-mass factor are
0.4064MK =
0.5333LK =
0.7619LMK =
This gives the equivalent single degree-of-freedom response equation as
Po
106
3
3840.7619 EIMy y F(t)
L
+ =dotnospdotnosp (4-21)
Finally, the natural frequency is given by
3 23
3 4
384
384E g 16, 230
0.7619 12 0.7619n LM
EI
k ht EtL?
K M M L ?Lht ?L= = = =? ?
(4-22)
Table 4.1 shows representative numbers of an 8-in CMU. E is Young?s modulus, and f?m
is the ultimate compressive strength of the masonry assemblage.
Table 4.1. Representative numbers for 8-in. CMU
f?m 1500 psi
E 1400 ksi
t 1.25 in
? 125 lb/ft3
L 6.06 in
Substituting the values from Table 4.1, the natural frequency and natural period are
19000 / s 19kHzn? = =
-43.4 10 s 0.34msnT = ? =
Table 4.2 gives the following ranges for loading regimes from Blast Effects on Buildings
(Smith and Cormie, 2009) and from Modern Protective Structures (Krauthammer, 2008).
Table 4.2. Loading regime ranges
Regime Cormie Krauthammer
Impulsive td/Tn < 0.1 td/Tn < 0.0637
Dynamic 0.1 < td/Tn < 10 0.637 < td/Tn < 6.37
Quasi-Static td/Tn > 10 td/Tn > 6.37
The loading duration td is assumed to be 0.01 s or 10 ms at its shortest duration. (The
shortest duration of the dynamic testing was approximately 14 ms.) The natural period
107
gives loading duration to natural period ratio of approximately 30, which corresponds to
the quasi-static regimes from both books. The quasi-static loading can be assumed for
loading duration of 3.4 ms or more. Therefore, the breaching of the walls can be
assumed to behave as if quasi-static loading is applied and can be modeled by applying a
static pressure over the whole wall. This matches well with the FEM results found in the
suitability study presented in Chapter 3, as the shape of the loading did not influence the
shear stress but the peak pressure did.
4.3.2 Dynamics of Between Grout Cells Beam
The finite element analysis showed that there were high stresses at the interface
between grouted and non-grouted cells in the models with grout present; this showed that
there could be a beam analog that runs between grout cells with the columns idealized as
perfectly fixed supports. Figure 4.6 shows the representative beam and cross section of
the between grout cell beam (BGC beam). The same approach as used for the single
block beam was used with the exception that the beam had a length L of the distance
between grout cells, a thickness t of one face shell, width w of the block, and a height h of
the height of one block. This model used only the face shells because the breaching will
occur in the faceshell, not at the web. In addition, the webs have sufficient stiffness to
ensure that the face shells have composite action.
108
A
A
w
h
t
L
Figure 4.6. BGC beam and cross section A-A
The moment of inertia contributing to stiffness of the section was only the two
face shells of the sections assuming composite action; it is represented by
3 22(1/12 ( / 2 / 2) )I ht th w t= + ? (4-23)
All other properties are the same for BGC breaching; the effects of reinforcing in the
grout cells were assumed to be negligible because the shearing happens between
columns. Then, the natural frequency is
109
2 23
n 4
384
389,500 (1/12 / 4)
0.7619
EI
E t wL
M ?L?
+= = (4-24)
These values are given for CMU ranging from 6-in. to 16-in. in Table 4.3 with
corresponding natural frequencies, and natural periods; in addition, the table shows the
minimum loading duration in which quasi-static analysis can be used. The length is
based on maximum bar spacing of 6 times the block?s width subtracting the grouted parts
of the length.
Table 4.3. Quasi-static details for between grouted cells beam
Size of
CMU
w
(in)
L
(in)
?n
(103/s)
Tn
(ms)
Td,min
(ms)
6-in 5.625 27.69 11.5 0.547 5.47
8-in 7.625 39.69 83.2 0.832 8.32
10-in 9.625 51.69 5.62 1.12 11.2
12-in 11.625 63.69 4.47 1.41 14.1
14-in 13.625 75.69 3.71 1.70 17.0
16-in 15.625 87.69 3.17 1.99 19.9
In the table above, the minimum loading duration for which the quasi-static response can
be used. Since the blasts considered in this study are at least 15 ms long, 6-in to 12-in
CMU can be analyzed using quasi-static for almost all cases. When wider blocks are
being used, the loading duration must be found and checked against values determined
for natural period. Figure 4.7 through Figure 4.11 show the effect that different
parameters can have on the minimum load duration. The data was generated using inputs
for an 8-in CMU block with a unit weight of 125 lb/ft3 only varying the parameter of
interest. Theses graphs cannot be used to determine the minimum load duration; they are
only used to show the general effects the parameters have on the minimum load duration.
110
0
2
4
6
8
10
12
1350 1850 2350 2850 3350 3850 4350
Modulus of Elasticity (ksi)
Lo
ad
D
ur
ati
on
(m
s)
Figure 4.7. Minimum load duration vs. modulus of elasticity
0
2
4
6
8
10
12
0.5 1 1.5 2 2.5
Thickness of Faceshell (in)
Lo
ad
Dur
ati
on
(m
s)
Figure 4.8. Minimum load duration vs. thickness of face shell
111
2
4
6
8
10
12
14
16
5 10 15 20
Width of Block (in)
Lo
ad
Du
ra
tio
n (
ms
)
Figure 4.9. Minimum load duration vs. width of block
0
2
4
6
8
10
12
85 95 105 115 125 135 145 155 165
Unit Weight (lb/ft3)
Lo
ad
D
ura
tio
n (
ms
)
Figure 4.10. Minimum load duration vs. unit weight
112
0
10
20
30
40
50
60
70
25 35 45 55 65 75 85 95
Lenght between Shear Surfaces (in)
Load
D
ur
ati
on
Figure 4.11. Minimum load duration vs. length between grout cells
The charts above show that the minimum load duration will change less than 5 ms for the
variable ranges considered. This is especially true for the thickness of the face shells, t,
which caused less than a 1 ms difference over its range of values. For modulus of
elasticity, E, and the unit weight, ?, the variation was small as well. The width did
change more rapidly with a variation of 10 ms over the data; however, the width of block
would change along with the distance between grouted columns giving a lower variation.
This leaves only the distance between shear planes as a major variable; the graph shows
that the minimum duration load changes rapidly with length. Therefore, if a load
duration is too small to assume quasi-static response for a given wall geometry, the
length between grouted columns could be reduced until the quasi-static analysis can be
used.
113
4.3.3 Direct Shear Modeling
Since both beam approximations are quasi-static, the complex wall is simply
modeled as a one-way beam with fixed supports at both ends. Because of similarities
between the models, the beams can be modeled the same with slight variations. The
loading area of the beams AL are given by
LA hL= (4-25)
where h = the full height of a single block and L = the length of the member. The total
force of the pressure loading FP is given by
PF PhL= (4-26)
where P = the peak pressure resulting from the blast loading. When the pressure is
applied to the loading area, the shear force at the supports Vu become
2 2
Lu PAPhLV = = (4-27)
The reaction can quickly be determined to be the shear forces acting at the supports.
Therefore, the shear force on the each beam model depends on the height of the wall, the
length of the wall, and the peak pressure on the wall.
4.4 Resistance Equation Derivation
The next step is to develop each beam?s nominal resistance. These resistance
equations will be based on mechanics and design standards.
114
4.4.1 Development of Resistance Equation
In cementitious materials there are three ways that shear force is resisted. These
are the strength attributed to masonry, the strength attributed to gravity loading, and the
strength attributed to steel. Therefore, the nominal shear Vn resistance is given by
n m p sV V V V= + + (4-28)
where Vm = the shear resistance of the masonry, Vp = the shear resistance of the
overburden, and Vs = the shear resistance of the steel. Since breaching will occur at
sections of lowest resistance, the wall will breach at cells without grout. Therefore, the
breaching shear is not affected by the steel reinforcement and is negligible giving
0Vs = (4-29)
Also, the effects of axial overburden may contribute to the breaching shear. However,
the resistance mechanism does not occur where there is any significant frictional force
because the breaching occurs on a vertical axis, and the nominal shear resistance of the
axial loading is
0pV = (4-30)
This leaves only the shear resistance contributed by the masonry. In the masonry design
codes, the masonry shear resistance is given by a variation of
n m n vmV V CA f= = (4-31)
where C = coefficient to account for safety factors or dynamic increase factors or to fit
data, An = the nominal resisting area, and fvm = the ultimate shear stress of masonry. The
equation can be further simplified to
n vmV Chtf= (4-32)
115
This equation was derived for the face shell beam. The equation for the between grout
cells beam is
2n vmV Chtf= (4-33)
4.5 Comparison between FEM Stress and Analytical Stress
In order to account for the differences between the actual breaching shear stress and the
FEM shear stress, an equation for the shear stress on the face shell of the CMU needs to
be found. For rectangular sections, the stress caused by shear loading ? is
V?
A= (4-34)
where Q = first moment area and V = shear force at the section of interest.
4.5.1 Face Shell Beam Comparison
Even though the model assumes beam action, the breaching shear develops in a
punching shear pattern. Therefore, the shear stress is given by
2 2
PhL PL?
ht t= = (4-34)
The shear stress resulting from this equation is compared to shear stresses collected on a
single block subjected to varying peak pressure with 0.25 in. by 0.25 in. by 0.25 in.
element size with fully-integrated element formulation. Table 4.4 gives a comparison of
finite element results compared to calculated value only for a face shell beam. In this
table, there are also values for the finite element stresses divided by the calculated values.
Figure 4.12 shows the last three columns of Table 4.4 versus the peak pressure.
116
Table 4.4. Shear stress comparison for single block beam
Pressure
(psi)
FEM Stress
(psi)
Calc. Stress
(psi) FEM/Calc.
5 42 13 3.3
10 83 25 3.3
15 127 38 3.4
20 173 51 3.4
25 220 63 3.5
30 267 76 3.5
35 314 88 3.6
40 361 101 3.6
45 403 114 3.5
50 439 126 3.5
2.5
2.8
3.0
3.3
3.5
3.8
4.0
4.3
4.5
0 10 20 30 40 50 60
Peak Pressure (psi)
FE
M/
Ca
lc.
Figure 4.12. Comparison between FEM and calculated shear stress for face shell beam
In looking at the data, the comparison shows that ratio of FEM to calculated stress
has some variation; however, the variation is small with a standard deviation of 0.11 and
average of 3.5. There are several reasons the FEM and analytical model do not match
more closely; these are discussed in a later section.
117
4.5.2 Between Grout Cells Beam Comparison
The BGC beam has similar mechanics as the face shell beam. The difference is in the
cross section. The shear stress is given by
2 2
PhL PL?
ht t= = (4-34)
The comparison of the FEM and the calculated shear stress is seen in Table 4.5. Figure
4.13 show a graphical version of the data.
Table 4.5. Shear stress comparison for BGC beam
Shear Stresses (psi) FE Stresses /Calc. Stresses
Pres. 5x5 Calc. 10x3 Calc. 3x10 Calc. 5x5 10x3 3x10
5 86.4 63.0 72.2 143.0 73.9 31.0 1.37 0.50 2.38
10 124 126.0 106.8 286.0 108.8 62.0 0.98 0.37 1.75
15 134.5 189.0 129.7 429.0 122.1 93.0 0.71 0.30 1.31
20 143.7 252.0 146.5 572.0 141.6 124.0 0.57 0.26 1.14
25 167 315.0 155.5 715.0 161.5 155.0 0.53 0.22 1.04
30 194.4 378.0 201.4 858.0 206.1 186.0 0.51 0.23 1.11
35 235.5 441.0 236.9 1001.0 228.2 217.0 0.53 0.24 1.05
40 267.3 504.0 262.3 1144.0 248.4 248.0 0.53 0.23 1.00
45 277.9 567.0 280.1 1287.0 264.4 279.0 0.49 0.22 0.95
50 280.1 630.0 268.4 1430.0 261.8 310.0 0.44 0.19 0.84
118
0
0.5
1
1.5
2
2.5
3
0 10 20 30 40 50 60
Pressure (psi)
FE
M
/C
alc
.
5x5
10x3
3x10
Figure 4.13. Comparison of FEM vs. calculated shear
stress for between grouted cell beam
The figure shows that the comparison is length dependent. It shows that the
smaller the distance between the grout columns the greater the ratio of FEM to calculated
shear stress. Figure 4.14 shows a graph of correction factor versus length for the
geometry data acquired on grouted models.
119
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 50 100 150 200 250
Length (in)
FE
M
/C
alc
.
Figure 4.14. Ratio of FEM to calculated shear stress vs. length
The figure shows that the ratio is linear for length. This is evident looking at the
Equation 4-34 and the data on the geometry suitability study; since the values for the
suitability were almost constant, the correction factor differed only by the length.
4.5.3 Differences between FEM and Analytical Shear Stress
In both models, there are major differences between the shear stress calculated by
FEM and calculated using beam models. The differences exist for a few different
reasons. These reasons are (1) a lack of testing data on shear stresses in full-scale blast
testing, (2) a lack of understanding of the mechanics in the breaching, (3) a lack of
understanding of the concrete model used in FEM, (4) a lack of understanding in how to
properly model mortar-block bond in highly dynamic loading, and (5) a lack of static
testing to validate FEM modeling. More research is needed to understand most of the
120
reasons the FEM and analytical models are not the same. However, there is a lack of
testing data on shear stresses during blast loading which is almost impossible to obtain.
The rest can be found using static testing or dynamic testing.
4.6 Breaching Shear Design Equation
Finally, the shear strength required by both models is
2=u
PLV (4-43)
The shear resistance is given by
=n vmV Ctf (4-44)
In order to prevent breaching, the nominal strength needs to be equal to or greater that the
ultimate shear demand; this is given by
n?uV V (4-45)
or
2 ? vm
PL Ctf (4-46)
Solving for pressure results in
2? vmCtfP
L (4-47)
Using ACI 530.1 (ACI, 2011) and assuming that most building use type N mortar, the
minimum compressive strength of the concrete block f?b is 2150 psi for a masonry
assemblage compressive strength f?m of 1500 psi with Type N mortar. ACI 318 (ACI,
2011) gives the ultimate shear stress of concrete by
121
4 / 3 46psivm bf = ? f' = (4-49)
where ? = a correction factor for lightweight concrete (0.75). The shear stress strength is
for plain concrete instead of masonry; this is because the masonry shear stress is built
around masonry-mortar bond and frictional sliding, not on the shear strength of concrete.
Plugging Equation 4-38 into Equation 4-37 and solving for pressure gives
2 46psiCtP
L
? ?? (4-50)
With Equation 4-39, a maximum pressure can be found for each block or section. Table
4.6 shows the maximum pressure for a single block beam; Table 4.5 shows the maximum
for the BGC beams with length being based on maximum steel reinforcement spacing.
For now since the stress gradient in the FEM has not been calibrated with data from full-
scale testing stress data, the correction factor is taken as 1 for now.
Table 4.6. Maximum pressure for single block beam
Single Block Beam
L (in) t (in) P (psi)
6.0625 1.25 19
Table 4.7. Maximum pressure for BGC beams
Between Grout Cells Beam
Nominal
Block Size L (in) t (in) P (psi)
6-in 28 2.5 8.3
8-in 40 2.5 5.8
10-in 52 2.5 4.4
12-in 64 2.5 3.6
14-in 76 2.5 3.0
16-in 88 2.5 2.6
Only the single block beam provides a pressure that is independent of length between
grout cells, geometry of the wall, and material properties of the blocks which is seen in
122
the suitability study; the BGC beam is not independent of these. In addition, using the
full-scale tests the single block beam pressure matches closer to the results in the tests.
Test 1 did breach but not to fully indicating the pressure was around the cutoff between
breaching and flexure. Test 2 and Test 3 had a more severe breaching response, and the
peak pressure was much higher than the single block beam and BGC beam pressures.
Therefore, all masonry walls must be fully grouted if a pressure exceeding 28 psi is
expected. Using UFC 3-340-02?s Figure 2-7, a minimum scaled standoff for a free air
blast at sea level was obtained. This cutoff scaled standoff is 8.2 lb/ft1/3.
123
CHAPTER 5
CONCLUSION AND RECOMMENDATIONS
5.1 Conclusions
A full-scale dynamic test was conducted on several partially grouted walls. These
walls were made of either 6-in. or 8-in. CMU blocks and either had a veneer or not.
These walls failed in a breaching pattern. A finite element model was created to analyze
the failure of the breaching phenomenon.
The finite element methodology involved a number of steps: (1) creation of a
FEM representing a partially grouted, reinforced wall CMU wall, (2) validation of the
FEM against the full-scale dynamic test, and (3) a brief parametric study.
With the FEM analysis, the dynamics of CMU breaching was explored, and the
system was found to be quasi-static. The parametric study supported all the assumptions
made in finding the natural frequency of the system; it also showed that the shear stresses
were independent of wall geometry, material properties, grouting geometry, and loading
shape. It was found that the shear stresses were only dependent on the peak pressure and
were only based on a single cell of a single block.
Finally, two engineering-level design equations were created to find the shear
resistance provided a wall. It was determined that the shear resistance is only provided
by the masonry without any addition due to overburden or steel. A design resistance
124
versus design load required was developed. The design ratio was found to independent
of wall geometry and only dependent on the block?s clear span between webs, block?s
thickness of faceshell, peak pressure of the blast, and the block shear strength. A
coefficient was determined to fit the data obtained from finite element modeling.
5.2 Recommendations
It is recommended that all walls that are expected to have pressures greater than
19 psi or a scaled standoff less than 10 lb/ft1/3 need to be fully grouted. In addition, it is
recommended that more full-scale static and dynamic testing be used to better understand
CMU response; this testing should focus on how to establish a better understanding of the
breaching. This dynamic testing should look into both the flexural capacity and shear
capacity of the wall with stress monitoring.
125
References
American Concrete Institute (ACI) (2008). Building Code Requirements for Structural
Concrete and Commentary, ACI 318-08, Farmington Hills, MI.
ACI (2011). Building Code Requirements and Specification for Masonry Structures, ACI
530-11, Farmington Hills, MI.
Atkinson R. H., Amadei, B. P., Saeb, S., and Sture, S. (1989). ?Response of masonry bed
joints in direct shear.? Journal of Structural Engineering, vol. 115 no. 9, pp. 2276-
2296.
Biggs, J. M. (1964). ?Introduction to Structural Dynamics,? McGraw-Hill, New York.
Browning IV, R. S. (2008). Resistance of multi-wythe insulated masonry subjected to
impulse loads.? M.S. thesis, Auburn University, Auburn, AL.
Burnett, S., Gilbert, M., Molyneaus, T., Beattie, G., and Hobbs, B. (2006). ?The
performance of unreinforced masonry walls subjected to low velocity: finite
element analysis.? International Journal of Impact Engineering, vol. 34, pp. 1433-
1450.
Davidson, J. S., Moradi, L., and Dinan, R. J. (2006). ?Selection of a material model for
simulating concrete masonry walls subjected to blast.? AFRL-ML-TY-TR-2006-
4521.
Davidson, J. S., Hoemann, J. M., Salim, H. H., and Shull, J. S. ( 2010). ?Full-scale
experimental evaluation of partially grouted, minimally reinforced CMU walls
against blast demands.? AFRL-RX-TY-TR-2010-00.
Dennis, S. T., Baylot, J. T., and Woodson, S. C. (2002). ?Response of 1/4-scale concrete
masonry unit walls to blast.? Journal of Engineering Mechanics, vol. 128, no. 2,
pp. 134-142.
Department of Defense (2008). ?Structures to Resist the Effects of Accidental Explosions,?
UFC 3-340-02, Whole Building Design Guide, http://dod.wbdg.org/ (accessed Feb.
2011).
Drysdale, R. G., and Hamid, A. A. (2008). Masonry Structures Behavior and Design, 2nd
Ed., The Masonry Society, Boulder, CO.
Eamon, C. D., Baylot, J. T., and O?Daniel, J. L. (2004). ?Modeling concrete masonry walls
subjected to explosive load.? Journal of Engineering Mechanics, vol. 130, no. 9, pp.
1098-1106.
Gilbert, M., Hobbs, B., and Molyneaux, T. C. K. (2002). ?The performance of unreinforced
masonry walls subjected to low-velocity impacts: mechanism analysis.? International
Journal of Impact Engineering, vol. 27, pp. 253-275.
Hamid, A. A. and Drysdale, R. G. (1988). ?Flexural tensile strength of concrete block
masonry.? Journal of Structural Engineering. vol. 114, no. 1, pp. 50-66.
Krauthammer, T. (2008). ?Pressure-impulse diagrams and their applications.? Modern
Protective Structures. CRC Press, Boca Raton, FL, pp. 325-371.
126
Livermore Software Technology Corporation (LSTC) (2009). LS-DYNA Keyword User?s
Manual.
Magallanes, J. M., Wu, Y., Malvar, L. J., and Crawford, J. E. (2010). ?Recent Improvements
to release III of the K&C concrete model.? 11th International LS-DYNA Users
Conference. Livermore Software Technology Corporation, Livermore, CA, 3-37 - 3-
48.
Martini, K. (1996). ?Research in the out-of-plane behavior of unreinforced masonry.?
Ancient Reconstruction of the Pompeii Forum, School of Architecture, University
of Virginia.
Martini, K. (1998). ?Finite element studies in the two-way out-of-plane failure of
unreinforced masonry.? Ancient Reconstruction of the Pompeii Forum, School of
Architecture, University of Virginia.
Moradi, L. G., Davidson, J. S., and Dinan, R. J. ( 2008). ?Resistance of membrane retrofit
concrete masonry walls to lateral pressure.? Journal of Performance of
Constructed Facilities, vol. 22, no. 3, pp. 131-142.
Psilla, N. and Tassios, T. (2009). ?Design of reinforced masonry walls under monotonic
and cyclic loading.? Engineering Structures, vol. 31, pp. 935-945.
Salim H., Saucier A., Bell B, Hoemann J., Bewick B, Davidson J., Shull J.
?Experimental Evaluation of Full-Scale NCMA Walls Under Uniform Pressure
Using Vacuum,? draft report submitted to the Air Force Research Laboratory,
July 2011.
Schwer, L. (2001) Draft, ?Laboratory Tests for Characterizing Geomaterials? Livermore
Software Technology Corporation, Livermore, California.
Schwer, L. (2005). ?Simplified concrete modeling with Mat_Concrete_Damage_Rel3.?
JRI LS-DYNA User Week. LSTC, Livermore, CA.
Smith, P. and Cormie, D. (2009) ?Structural response to blast loading.? Blast Effects on
Buildings. D. Cormie, G. Mays, and P. Smith, eds., Thomas Telford Limited,
London, UK, pp. 80-102
Sudame, S. (2004). ?Development of computational models and input sensitive study of
polymer reinforced concrete masonry walls subjected to blast.? M.S. thesis,
University of Alabama at Birmingham, Birmingham, AL.
127
APPENDIX
LS-DYNA INPUT
*KEYWORD
*TITLE
*CONTROL_TERMINATION
$# endtim endcyc dtmin endeng endmas
0.01 0 0 0 0
*CONTROL_TIMESTEP
$# dtinit tssfac isdo tslimt dt2ms lctm erode ms1st
0 0.5 0 0 0 0 0 0
$# dt2msf dt2mslc imscl
0 0 0
*DATABASE_ELOUT
$# dt binary lcur ioopt
2.50E-05 3 0 1
*DATABASE_BINARY_D3PLOT
$# dt lcdt beam npltc psetid
5.00E-04 0 0 0 0
$# ioopt
0
*DATABASE_HISTORY_SOLID_SET
$# id1 id2 id3 id4 id5 id6 id7 id8
1 2 3 0 0 0 0 0
*BOUNDARY_SPC_SET
$# nsid cid dofx dofy dofz dofrx dofry dofrz
1 0 1 1 1 1 1 1
*SET_NODE_LIST_TITLE
NODESET(SPC) 1
$# sid da1 da2 da3 da4
1 0 0 0 0
$# nid1 nid2 nid3 nid4 nid5 nid6 nid7 nid8
82645 82646 82647 82648 82649 82650 82651 82652
82653 82654 82655 82656 82657 82658 82659 82660
82661 82662 82663 82664 82665 82666 82667 82668
.
.
.
82621 82622 82623 82624 82625 82626 82627 82628
82629 82630 82631 82632 82633 82634 82635 82636
128
82637 82638 82639 82640 82641 82642 82643 82644
.
.
.
*LOAD_BODY_Z
lcid sf lciddr xc yc zc cid
1 1 0 0 0 0 0
*LOAD_SEGMENT_SET_ID
$# id heading
1
$# ssid lcid sf at
101 2 1 0
*CONTACT_AUTOMATIC_SURFACE_TO_SURFACE_ID
$# cid title
1
$# ssid msid sstyp mstyp sboxid mboxid spr mpr
2 1 0 0 0 0 0 0
$# fs fd dc vc vdc penchk bt dt
0 0 0 0 0 0 0.0001.0000E+20
$# sfs sfm sst mst sfst sfmt fsf vsf
1 1 0 0 1 1 1 1
*SET_SEGMENT_TITLE
BCB
$# sid da1 da2 da3 da4
2 0 0 0 0
$# n1 n2 n3 n4 a1 a2 a3 a4
61777 61810 61811 61778 0 0 0 0
49784 49817 49818 49785 0 0 0 0
3831 3870 3871 3832 0 0 0 0
.
.
.
2623 2632 3551 3512 0 0 0 0
3830 3869 3870 3831 0 0 0 0
2631 2640 2641 2632 0 0 0 0
.
.
.
*PART
$# title
CMU
$# pid secid mid eosid hgid grav adpopt tmid
1 1 1 0 0 0 0 0
*SECTION_SOLID
$# secid elform aet
1 1 0
*MAT_BRITTLE_DAMAGE_TITLE
CMU
129
$# mid ro e pr tlimit slimit ftough sreten
1 1.79E-04 1.80E+06 0.2 224.7 67.08 0.8 0.03
$# visc fra_rf e_rf ys_rf eh_rf fs_rf sigy
104 0 0 0 0 0 0
.
.
.
*MAT_BRITTLE_DAMAGE_TITLE
Mortar
$# mid ro e pr tlimit slimit ftough sreten
2 1.87E-04 1.80E+06 0.15 224.7 67.08 0.8 0.03
$# visc fra_rf e_rf ys_rf eh_rf fs_rf sigy
104 0 0 0 0 0 0
*PART
$# title
HJ
$# pid secid mid eosid hgid grav adpopt tmid
202 1 2 0 0 0 0 0
.
.
.
*PART
$# title
Boundary
$# pid secid mid eosid hgid grav adpopt tmid
501 1 5 0 0 0 0 0
*MAT_RIGID_TITLE
Boundary
$# mid ro e pr n couple m alias
5 7.34E-04 2.90E+07 0.3 0 0 0
$# cmo con1 con2
0 0 0
$# lco or a1 a2 a3 v1 v2 v3
0 0 0 0 0 0
.
.
.
*MAT_BRITTLE_DAMAGE_TITLE
Grout
$# mid ro e pr tlimit slimit ftough sreten
3 1.87E-04 2.00E+06 0.2 317.8 94.87 0.8 0.03
$# visc fra_rf e_rf ys_rf eh_rf fs_rf sigy
104 0 0 0 0 0 0
*MAT_PLASTIC_KINEMATIC_TITLE
Rebar
$# mid ro e pr sigy etan beta
4 7.34E-04 2.90E+07 0.3 60000 0 0
$# src srp fs vp
130
0 0 0 0
.
.
.
*DEFINE_CURVE_TITLE
Triangular
$# lcid sidr sfa sfo offa offo dattyp
1 0 1 1 0 0 0
$# a1 o1
0 0
0.001 0
0.00101 50
0.0061 0
0.01 0
.
.
.
*SET_SEGMENT_TITLE
Blast
$# sid da1 da2 da3 da4
101 0 0 0 0
$# n1 n2 n3 n4 a1 a2 a3 a4
51733 51766 51765 51732 0 0 0 0
48510 48543 48542 48509 0 0 0 0
18064 18076 18242 18209 0 0 0 0
.
.
.
5097 5145 5144 5096 0 0 0 0
2050 2059 2058 2049 0 0 0 0
268 307 306 267 0 0 0 0
*ELEMENT_SOLID
$# eid pid n1 n2 n3 n4 n5 n6 n7 n8
1 1 1 2 5 4 10 11 14 13
2 1 2 3 6 5 11 12 15 14
3 1 4 5 8 7 13 14 17 16
.
.
.
10138 502 82743 82744 82846 82845 82947 82948 83050 83049
10139 502 82744 82745 82847 82846 82948 82949 83051 83050
10140 502 82745 82746 82848 82847 82949 82950 83052 83051
*NODE
$# nid x y z tc rc
1 0 0 0 0 0
2 0.5 0 0 0 0
3 1 0 0 0 0
.
131
.
.
83050 48 8.13 23.625 0 0
83051 48.5 8.13 23.625 0 0
83052 49 8.13 23.625 0 0