unconfined/unreinforced masonry wall/pier using Abaqus. Abaqus is a ...... Abaqus is a unitless software, therefore a consistent unit convention is described.
COMPUTATIONAL MODELING OF AN UNCONFINED/UNREINFORCED MASONRY WALL USING ABAQUS
2015
By: Engr. Adil Rafiq
Supervised by: Dr. Muhammad Fahad
IN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE AWARD OF THE DEGREE OF MASTER OF SCIENCE in CIVIL ENGINEERINGRING (STRUCTURAL EINGIENERING)
Department of Civil Engineering University of Engineering and Technology, Peshawar
DIRECTORATE OF POSTGRADUATE STUDIES UNIVERSITY OF ENGINEERING AND TECHNOLOGY, PESHAWAR, PAKSITAN
i
1 DEDICATION
Dedicated to My Parents, my wife and our beautiful child “SEENUM”
i
2
ACKNOWLEDGEMENT
All credit goes to the Almighty Allah who gave me the ability to accomplish this challenging job. I express deep gratitude to my research supervisor Dr. Muhammad Fahad. I always got support from him whether it was in the form of technical guidance or a moral one. There were a lot of uncertainties regarding this research project but he pushed me a lot and backed me on every difficult time. A lot of credit also goes to Dr. Khan Shahzada for he has been extraordinarily cooperative in providing research material.
ii
3
ABSTRACT
Masonry is an oldest and mostly used construction material in buildings. It is a heterogeneous material and bears a complex behavior under various load conditions. This study has been carried out to numerically model an unconfined/unreinforced masonry wall/pier using Abaqus. Abaqus is a Finite Element Analysis (FEA) software that is used to model and analyze any complex system. Various material properties and stress–strain relations for compression and tension were obtained from empirical relations which were developed for modeling of masonry in the literature. There are different material models available in Abaqus for masonry and other quasi brittle material. Concrete Damaged Plasticity model was selected for this study which is able to describe the behavior in tension and compression with non-linearity in plastic range. Masonry wall/pier selected for this study is taken from the Ph.D. work of Javed (2008). Masonry is modeled in three different ways i.e. Micro-scale, Meso-scale and Macro-scale, out of which macro-scale modeling was selected, where the constituents of masonry (i.e. unit, mortar and the interface) is homogenized as an isotropic or anisotropic continuum. The solution method used in Abaqus for the subject model was explicit. The loading is monotonically in-plane under a displacement controlled boundary condition, however in experiments the loading of wall/pier is cyclic. Yet a fair level comparison was done with the help of load displacement curves of numerical model and the test results of the pier only in positive direction. The model was also tested for different precompression ratios and its load displacement curves were plotted. The numerical results were compared with experimental results in terms of drift iii
ratios at different limit states or performance levels. Stress, Plastic Strain and Damage conditions at different performance levels were also noted visually and were found in good agreement with the experiments. The default history output in Abaqus consists of all the global energies for the whole model, out of which only the total energy versus time curve was presented. In the end it was concluded that the procedure was not found helpful in case of cyclic loading and therefore it was recommended for future study to probe into homogenous anisotropic material modeling for cyclic load using a FORTRAN compiler.
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4
TABLE OF CONTENTS
Dedication ....................................................................................................................................... i Acknowledgement ........................................................................................................................ ii Abstract ......................................................................................................................................... iii Table of contents ........................................................................................................................... v List of figures .............................................................................................................................. viii List of tables ................................................................................................................................... x 1.
2.
3.
INTRODUCTION ................................................................................................................. 1 1.1.
BACKGROUND ............................................................................................................ 1
1.2.
PROBLEM STATEMENT ............................................................................................. 2
1.3.
OBJECTIVES .................................................................................................................. 2
1.4.
SCOPE OF WORK......................................................................................................... 3
1.5.
OUTLINE OF THESIS .................................................................................................. 3
LITERATURE REVIEW........................................................................................................ 6 2.1.
OVERVIEW .................................................................................................................... 6
2.2.
MATERIAL PROPERTIES ........................................................................................... 6
2.2.1.
Uniaxial Behavior ...................................................................................................... 7
2.2.2.
Biaxial Behavior ....................................................................................................... 10
2.3.
CHOICE OF NUMERICAL MODELING ................................................................ 13
2.4.
MODELING TECHNIQUES ..................................................................................... 13
2.5.
EMPIRICAL ESTIMATION ....................................................................................... 14
2.5.1.
COMPRESSIVE STRENGTH ................................................................................. 15
2.5.2.
TENSILE STRENGTH ............................................................................................ 16
2.5.3.
ELASTIC MODULUS ............................................................................................. 17
2.6.
THE STRESS-STRAIN MODEL ................................................................................ 18
2.7.
STRESS-STRAIN MODEL ESTIMATION ............................................................... 20
ABAQUS – A SOFTWARE SYNOPSIS ............................................................................ 23 3.1.
OVERVIEW .................................................................................................................. 23
3.2.
PROCESSING STEPS.................................................................................................. 23 v
3.3.
4.
5.
THE PRODUCT SUITE .............................................................................................. 24
3.3.1.
Abaqus/CAE: ........................................................................................................... 24
3.3.2.
Abaqus/CFD: ........................................................................................................... 25
3.3.3.
Abaqus/Explicit: ...................................................................................................... 25
3.3.4.
Abaqus/Standard: ................................................................................................... 27
3.4.
ELEMENT TYPE FOR SOLIDS ................................................................................. 28
3.5.
MATERIAL MODELS ................................................................................................ 29
3.6.
CONCRETE DAMAGED PLASTICITY:.................................................................. 30
3.6.1.
Strain rate decomposition: ..................................................................................... 31
3.6.2.
Stress-Strain Decomposition:................................................................................. 31
3.6.3.
Hardening Variables: .............................................................................................. 32
3.6.4.
Uniaxial Condition .................................................................................................. 32
3.6.5.
Uniaxial Cyclic Condition ...................................................................................... 34
3.6.6.
Yield Function ......................................................................................................... 36
3.6.7.
Flow Rule ................................................................................................................. 38
METHODOLOGY ............................................................................................................... 40 4.1.
SUBJECT MODEL ....................................................................................................... 40
4.2.
CONSISTENT UNITS ................................................................................................. 44
4.3.
MODEL GEOMETRY ................................................................................................. 44
4.4.
MATERIALS ................................................................................................................ 46
4.4.1.
Elastic Properties ..................................................................................................... 46
4.4.2.
Inelastic Properties .................................................................................................. 47
4.4.2.1.
Plasticity ............................................................................................................... 47
4.4.2.2.
Compressive Behavior........................................................................................ 48
4.4.2.3.
Tensile Behavior .................................................................................................. 51
4.5.
STEP DEFINITION ..................................................................................................... 54
4.6.
ASSEMBLY .................................................................................................................. 54
4.7.
BOUNDARY CONDITIONS & LOADS .................................................................. 55
4.8.
OUTPUT REQUESTS ................................................................................................. 57
RESULTS AND DISCUSSION .......................................................................................... 59 5.1.
LOAD-DISPLACEMENT CURVE ............................................................................ 60
5.2.
COMPARISON WITH JAVED (2008) ...................................................................... 61 vi
6.
5.3.
STRESS-STRAIN VISUALIZATION ........................................................................ 64
5.4.
DAMAGE VISUALIZATION .................................................................................... 69
5.5.
VISUAL COMPARISON ............................................................................................ 73
5.6.
TOTAL ENERGY OF THE OUTPUT ....................................................................... 75
CONCLUSIONS AND RECOMMENDATIONS............................................................ 76 6.1
CONCLUSIONS & RECOMMENDATIONS .......................................................... 76
REFERENCES .............................................................................................................................. 79
vii
5
LIST OF FIGURES
Figure 2-1: Results of the Compression Test Perpendicular to the Bed Joint (Binda et al (1988)) . 7 Figure 2-2: Results of the Tensile Test Parallel to the Bed Joint (Backes (1985)) ........................... 8 Figure 2-3. Modes of Failure of Brick Masonry under Direct Tensile Test (Schubert (1988)) ....... 9 Figure 2-4: Results of the Tensile Test Perpendicular to the Bed Joint (Pluijm (1992)) ................. 9 Figure 2-5: Methods of Masonry Flexural Tensile Strength (ASTM C1072-2000) ..................... 10 Figure 2-6: Variations of the Experiments performed by Page (1981, 1983). ............................... 11 Figure 2-7: Yield Surface envelopes Found by Page (1981) .......................................................... 12 Figure 2-8: Yield Surfaces with Tension-Compression results found by Page (1981, 1983) ........ 12 Figure 2-9: Methods of Idealization ............................................................................................... 14 Figure 2-10: Constitutive Models for Compressive Behavior ........................................................ 19 Figure 2-11: Constitutive Models for Tensile Behavior ................................................................ 19 Figure 2-12: Analytical Model for Stress–Strain Curve of Masonry in Compression (Hemant et al. (2007))....................................................................................................................................... 20 Figure 2-13: Compression Model with Hardening and Subsequent ............................................. 21 Figure 2-14: Tension Model with Exponential Softening Criterion ............................................. 22 Figure 3-1: Flow Diagram of the Simulation Process ................................................................... 23 Figure 3-2: C3D8R for Solids ........................................................................................................ 29 Figure 3-3: Damaged Response to Uniaxial (a) Tension and ........................................................ 33 Figure 3-4: Tension-Compression Cyclic Model with Damage and.............................................. 36 Figure 3-5: Typical Yield Surface (Section 4.5.2 of Theory Guide)............................................... 38 Figure 4-1: Geometry of Wall/Pier-PI (Javed (2008)) (a) Isometric View (b) Front View (c) Side View ............................................................................................................................................... 41 Figure 4-2 : Test Set Up by Javed (2008) ...................................................................................... 43 Figure 4-3: (a) Concrete Top Beam (b) Masonry Wall .................................................................. 45 Figure 4-4: (a) Meshed Concrete Top Beam (b) Meshed Masonry Wall ....................................... 45 Figure 4-5: Results of Linear Elastic Behavior .............................................................................. 46 Figure 4-6: Compressive Stress-Strain Curve ............................................................................... 49 Figure 4-7: Definition of Compressive Inelastic Strains ............................................................... 50 Figure 4-8: Tensile Stress-Strain Curve ........................................................................................ 52 Figure 4-9: Definition of Tensile Inelastic Strains ........................................................................ 53 Figure 4-10: The Assembled Model ............................................................................................... 55 Figure 4-11: Applied Amplitude to the Displacement in Lateral Direction .................................. 56 Figure 4-12: Applied Amplitude to the Precompression Load....................................................... 56 Figure 4-13: Model showing Applied Loads and Boundary Conditions ....................................... 57 Figure 5-1: Typical Performance Levels ........................................................................................ 60 Figure 5-2: Load-Displacement Curve of the Numerical Model Loaded Monotonically............... 60 Figure 5-3: Results of the Numerical Model at different Precompression Ratios ......................... 61 Figure 5-4: Comparison of the Results .......................................................................................... 62 viii
Figure 5-5: Stiffness Degradation Curve....................................................................................... 62 Figure 5-6: Mises Stress at Flexural Crack Limit State ................................................................ 65 Figure 5-7: Mises Stress at Diagonal Crack Limit State ............................................................... 65 Figure 5-8: Mises Stress at Maximum Resistance ........................................................................ 66 Figure 5-9: Mises Stress at Ultimate Limit State .......................................................................... 66 Figure 5-10: Max. Principal Plastic Strains at Flexural Crack Limit State ................................. 67 Figure 5-11: Max. Principal Plastic Strains at Diagonal Crack Limit State ................................ 67 Figure 5-12: Max. Principal Plastic Strains at Maximum Resistance ......................................... 68 Figure 5-13: Max. Principal Plastic Strains at Ultimate Limit State ........................................... 68 Figure 5-14: Tension Damage at Flexural Crack Limit State ....................................................... 70 Figure 5-15: Tension Damage at Diagonal Crack Limit State ...................................................... 70 Figure 5-16: Tension Damage at Maximum Resistance ............................................................... 71 Figure 5-17: Tension Damage at Ultimate Limit State ................................................................. 71 Figure 5-18: Compression Damage at Ultimate Limit State ......................................................... 72 Figure 5-19: Pier PIc at the end of testing (Javed (2008)) ............................................................. 73 Figure 5-20: Typical crushing and spalling of masonry during final stages ................................ 74 Figure 5-21: Total Energy of the Output for Displacement Controlled Step Only ...................... 75
ix
6
LIST OF TABLES
Table 2.1: k Factor used by various researchers ................................................................... 18 Table 4.1: Characteristics Of The Specimen Pier (Javed (2008))........................................ 42 Table 4.2: Consistent Units....................................................................................................... 44 Table 4.3: Elastic Properties ..................................................................................................... 47 Table 4.4: Plasticity Parameters ............................................................................................... 48 Table 4.5: Compressive Behavior ............................................................................................ 51 Table 4.6: Tensile Behavior ...................................................................................................... 53 Table 5.1: Percentage Difference between Experimental and Numerical Results ........ 63 Table 5.2: Comparison between Experimental and Numerical Results at different Limit States.................................................................................................................................. 64
x
1. INTRODUCTION 1.1.
BACKGROUND Masonry is the most commonly used building material in Pakistan and all
over the world. In past masonry buildings were usually built on the basis of an impressive empirical wisdom without mathematical or predictive methods. It is still widely used due to its simplicity and ease of construction with low material cost and aesthetics. Laying blocks on top of each other with a cohesive material like mortar makes it a composite of two different materials i.e. mortar and a building unit of any specific material. Each material bears its own linear and nonlinear mechanical properties and behaves differently under various conditions. Therefore masonry has most frequently been used as part of load transfer mechanism where it has been good in carrying gravity loads. On the other hand, when it is subjected to in-plane lateral load without confinement or reinforcement, it performance is poor. The said weakness and the damage pattern has to be gauged both experimentally and numerically for a better usage of masonry as a load bearing system as well as a part of a frame system (in-filled walls). For this purpose an illustrative model is usually tested for various material and mechanical properties in physical to assess its behavior and then refined mathematical techniques are used to model it numerically in virtual. It is a significant challenge for a researcher in any computational framework to predict complex failure. The modern developments in mechanics have somehow enabled them to model masonry with many of its complexities using Finite Element Analysis. Complex 1
models can be created where every brick and joint is considered separately, but in practice the use of simplified models are of importance where low computational time is desired. For accurate and complex computations various computer applications are developed, such as LS-DYNA, ANSYS, ABAQUS, etc. where ABAQUS has been chosen as a tool for the subject study and is discussed in detail later.
1.2.
PROBLEM STATEMENT Now, having said that, Pakistan came across the most severe earthquake
in its history i.e. Kashmir Earthquake on October 8th 2005 and this devastation has resulted in fatalities around 73000 and injuries around 80000 and rendering many other homeless. Most of the damaged and/or collapsed structures observed were stone and brick masonry which were unconfined and unreinforced, bearing weakness as mentioned earlier. As a result of the event, the researchers were obligated to study masonry in all respect and come up with better engineering techniques for such hazardous situations. Afterwards an extensive amount of experimental work was carried out to assess the seismic behavior of masonry for all conditions (confined and unconfined, reinforced and unreinforced, burnt bricks as a unit and concrete block as a unit). However there is a lack of doing numerical study using a refined Finite Element Analysis (FEA) technique which is necessary to have an insight into the structural behavior that will support the derivation of rational design rules for masonry. Therefore a great deal of research is also required to be done on numerical part. The report will help in the enhanced understanding of the crack formation and its propagation in the masonry structure.
1.3.
OBJECTIVES The major goal of this study is to standardize and gauge a numerical
tool with experiments performed on brick masonry and to compare its behavior 2
both linearly and non-linearly. To serve our purpose, the application used is ABAQUS. This study would eventually come up with procedure in hand to model brick masonry using a finite element analysis tool i.e. ABAQUS. The report will ultimately help in the enhanced understanding of the crack formation and its propagation numerically in the masonry.
1.4.
SCOPE OF WORK The focus of this research is to model a 3D masonry wall/pier
homogenously in ABAQUS bearing isotropic material properties. The subject wall/pier is taken from a PhD work of Javed (2008) with all its geometric and basic material properties. The compressive and tensile stress-strain curves are empirically estimated through mathematical equations. The model is tied at the top to an RCC beam of very high elasticity and is monotonically loaded. Simulations are done using explicit solver and the results are compared both elastically and inelastically with the experimental load displacement curve at different performance levels or limit states.
1.5.
OUTLINE OF THESIS This
dissertation
is
entitled
“Computational
Modeling
of
an
Unconfined/Unreinforced Masonry Wall Using ABAQUS” and is organized in six chapters. Each chapter is outlined as follow, Chapter 1: It is an introductory chapter that illustrates a motivational background behind the research work along with problem statement, objectives and scope of this study. Chapter 2: This chapter discusses a review of literature related to this research work. A brief description of experimental work on masonry by previous researcher is presented which is necessary to understand the mechanical behavior of masonry for the numerical study. Furthermore a basis for the choice 3
of numerical model is discussed. Then there are three techniques described for numerical modeling which are used these days out many other modeling techniques for masonry. It is required for numerical study to estimate the mechanical behavior of brick masonry, therefore a thorough review on empirical equations used by various researchers for estimation is presented after the discussion on numerical techniques. Chapter 3: This chapter elaborates the software used for this study, i.e. Abaqus. Than a brief description on product suite and element type that is used for the numerical model is presented. In the end a thorough explanation is given on Concrete Damaged Plasticity Model which is used for material in this research. Chapter 4: The procedure adopted for numerical analysis of masonry in the study is described in this chapter. At first the model assumed is described that is taken from the Ph.D. research of Javed along with its useful material properties. Since Abaqus is a unitless software, therefore a consistent unit convention is described in this chapter. Afterwards a detailed procedure to develop the required model for analysis is elaborated, where the important section to note is the material model estimated through various empirical equations. Chapter 5: In this chapter the results are discussed with comparison to the experiments, both numerically and visually. Its starts with assessment of load displacement and stiffness degradation curve. Comparison of drift ratios at different limit states are also discussed which were found in good agreement with the experimental values. Further, stress, plastic strain damage conditions at different performance levels are elaborated visually and then were compared. In the end the numerical results of total energy of the system are briefly discussed. Chapter 6: This is the last chapter of the dissertation. It presents the conclusive remarks of the research work and gives recommendations for future study. 4
5
2. LITERATURE REVIEW 2.1.
OVERVIEW The purpose of this chapter is to present a sufficient amount of literature
that is available to cover the topic. Although masonry is an old building technique but the research in this area is quite young and has risen only in the past decade and a half. The traditional design of masonry was based on thumb rule and empirical formulae that required only compressive strength to calculate thickness of the walls using static dead and live loads. The research community took keen interest in its mechanics with the development of new computational techniques, trying to separate masonry from its building tradition of trial and error. But the only problem was the lack of information on the characteristic properties of masonry and its components. At first the researchers imported numerical tools available from more advanced fields, i.e. mechanics of concrete, but such methods were useful only under certain conditions. Therefore the particular features required appropriate tools to analyze structures built with brick masonry, both experimentally and numerically. It is also important to note here that due to the composition of brick masonry, it can be numerically model as homogenous isotropic/anisotropic or can be modeled by assembling brick units. This review will basically focus on homogenous isotropic modeling of brick masonry, which is the simplest and quickest technique.
2.2.
MATERIAL PROPERTIES To estimate constitutive models for material behavior of brick masonry for
numerical modeling, it is necessary to review the material experimentally. Since 6
brick masonry is a composite of two different material i.e. clay bricks and mortar, therefore it bears different directional properties. It has been tested under different load conditions and with different orientations of its bed joint. However, there is a satisfactory literature available on most of its important load condition and orientation. This section discusses some former experimental work done on compressive and tensile properties of masonry.
2.2.1. Uniaxial Behavior Binda et al (1988) performed experiments on masonry under compression in the direction perpendicular to the bed joint. His results can be shown in the Figure 2.1,
Figure 2-1: Results of the Compression Test Perpendicular to the Bed Joint (Binda et al (1988))
It can be clearly noted that masonry with high strength mortar acts stiffer and cracks more brittle as compared to the weaker mortar. The graph also shows that difference between the strengths of masonry and mortar increases with the use of high strength mortar. On the other hand the mortar with low stiffness and high Poisson’s ratio than unit brick, the mortar tends to get squeezed out between the bricks. This phenomenon is studied by Rotts (1991). There is however lack of literature on uniaxial compression of masonry in the direction parallel to the bed joint.
7
Backes (1985) studied the tensile behavior of brick masonry in the direction parallel to the bed joint. The resultant behavior of tests for two different phenomenon are shown in the Figure 2.2.
Figure 2-2: Results of the Tensile Test Parallel to the Bed Joint (Backes (1985))
The stress-displacement curves indicate two different crack patterns, i.e. a saw toothed crack pattern (a) and a straight crack pattern (b). The pattern depends on the relative tensile strengths of brick and the sandwiched mortar. Mostly bricks have a strength ten times the strength of mortar, hence bricks would never crack and a pattern (a) is found. It can also be observed in pattern (a) that the sample continues to have fracture energy after the cracks occur which is due to the friction at surface contact between broken parts. When bricks are relatively weak, pattern (b) would be observed where the curve suddenly changes its direction when the breakage of unit brick occurs. Schubert (1988) also determined tensile strengths of brick masonry by loading it in two different directions in direct tensile test, one perpendicular to the bed joint (Figure 2.3(a) and 2.3(b)) and other parallel to the bed joint (Figure 2.3(c) and 2.3(d)). The failure pattern is governed by relative strengths of the unit brick and mortar as stated earlier and also shown in the Figure 2.3. Figure 2.3(a) 8
and 2.3(d) shows the crack pattern through the brick units indicating that the mortar is relatively stronger than the bricks, while in Figure 2.3(b) and 2.3(c) the cracks occurs at the mortar indicating that the mortar is relatively weaker than the bricks.
Figure 2-3. Modes of Failure of Brick Masonry under Direct Tensile Test (Schubert (1988))
In case of tensile strength in perpendicular to the bed joint, we can refer to the experiments performed by Pluijm (1992). His results can be illustrated in Figure 2.3.
Figure 2-4: Results of the Tensile Test Perpendicular to the Bed Joint (Pluijm (1992))
This interfacial tensile behavior is also referred as Mode-I failure mechanism where exponential softening occurs. Pluijm (1992) found the tensile bond strengths ranging from 0.3N/mm2 to 0.9N/mm2 and Mode-I fracture energies (Gf) ranging from 0.005N/mm to 0.03N/mm. 9
The interfacial tensile strength of masonry is also evaluated by flexural test in two types of loading options, first as horizontal beam with the transverse loads applied vertically and second by Bond-Wrench Method (ASTM C10722000), as shown in the Figure 2.5.
Figure 2-5: Methods of Masonry Flexural Tensile Strength (ASTM C1072-2000)
2.2.2. Biaxial Behavior For masonry failure to be completely, a three-dimensional failure surface in terms of the principal stresses σ1 and σ2, and their respective orientations (θ) to the bed joint is required. Page (1981,1983) and then Dhanasekar et al (1985) experimented the biaxial behavior of masonry and various samples of masonry panels were loaded with different configurations and bed joint orientations i.e. in biaxial compression, biaxial tension compression, uniaxial compression and uniaxial tension up until failure (see Figure 2.6).
10
Figure 2-6: Variations of the Experiments performed by Page (1981, 1983).
The angle of bed joint was oriented with respect to the direction of loads. Masonry panels were loaded in compression with hydraulic jacks in orthogonal direction, each time with different ratio between horizontal and vertical load and the results were projected on σ1-σ2 plane. To minimize the effects of restraint in platen and thus to ensure a more uniform state of stress in the panel, steel brush platens were used. In practice, the most critical regions of masonry under inplane condition is the compression-compression and tension-compression state. Therefore yield surfaces obtained for masonry with 0o to 90o orientation is presented in Figure 2.7 and Figure 2.8 for the aforementioned conditions.
11
Figure 2-7: Yield Surface envelopes Found by Page (1981)
Figure 2-8: Yield Surfaces with Tension-Compression results found by Page (1981, 1983)
12
2.3.
CHOICE OF NUMERICAL MODELING There are a number of numerical models, based on different assumptions
and characterized by different level of details. The choice of model depends on the best suited case that takes into account the following,
2.4.
The information searched (damage, collapse, serviceability etc.)
The accuracy required (local or global behavior)
The input data needed (information about material)
The time needed to complete the analysis.
MODELING TECHNIQUES In Finite Element Analysis, there are several different ways to model
masonry, ranging from a very detailed micro level to a composite macro level, also discussed by Lorenco (1996) and van Noort (2012). The most common philosophies these days are: Micro-Scale Model: Also termed as Detailed Micro Model and is illustrated in Figure 2.9(b). Here a continuum element is used to describe a particular masonry unit and mortar while to represent a unit-mortar interface with a plane of a potential crack a discontinuous element is taken into account. This technique is the most accurate in describing the masonry behavior although it requires a high computational time and effort. Therefore a micro-scale model is generally used to elaborate the study of local response. Meso-Scale Model: This method is also termed as Simplified Micro Model where masonry units are expended up to half the mortar thickness and is shown in the Figure 2.9(c). Units are modelled with a continuum element while the joints and unit-mortar interfaces are considered discontinuous elements. Thus the units are directly bonded by a potential crack plane. Macro-Scale Model: In this type, all the constituents of masonry (i.e. unit, mortar and the interface) is homogenized as an isotropic or anisotropic continuum and 13
is illustrated in Figure 2.9(d). Due to low computational demand, this approach is considered to be most practice-oriented. Parameters, which describe the continuum, must be found out during the tests on specimens that are subjected to a homogenous state of stress.
Figure 2-9: Methods of Idealization
2.5.
EMPIRICAL ESTIMATION For linear elastic study brick masonry requires only the modulus of
elasticity, while for nonlinear inelastic study a complete stress-strain curve for brick masonry is required. These stress-strain curves including all its elastic and inelastic parameters are obtained through material testing, as discussed earlier. Since brick masonry is the composite of unit bricks and mortar, it is easy to predict the stress-strain curve through elastic properties of masonry and its constituents, rather than running an experimental program. Many researchers had worked on standardizing the material properties of brick masonry through various empirical/mathematical relations, some of which are presented in this review.
14
2.5.1. COMPRESSIVE STRENGTH It is not always viable to conduct compression testing of masonry prisms to get an actual prism strength. However it is relatively easy to evaluate the estimated mechanical properties of the masonry through mathematical expressions. There are some factors that affect the compressive strength of a masonry wall, such as workmanship, the properties of the masonry units, the thickness of mortar joints, the age of mortar and also the suction rate (Sahlin (1971)). Sahlin (1971) also reported that increasing mortar joint thickness lowers the compressive strength and the normal joint thickness of 10 mm is recommended. Paulay and Priestley (1992), as well as Drysdale et al. (1994) have concluded that the compressive strength of brick masonry prism (f’m) is usually less compared to unit compressive strength of a unit brick (f’cb). D’Ayala (1997) presented Equation 2.1 to compute the masonry compressive strength from those of masonry units and mortar. ( 2.1 ) Equation 2.1 can also be written as follow, to account for high receptivity of the masonry strength to the mortar height, ( 2.2 ) Where is the compressive strength of masonry is the compressive strength of a unit brick is compressive strength of the mortar is masonry wall height is masonry unit height is mortar height
15
According to Binda et al. (1994),
can be calculated by Equation 2.3. ( 2.3 )
Where is the tensile strength of the bricks under flexure ,
are the Poisson’s ratios for brick and mortar respectively
m is the ratio between elastic moduli of brick and mortar. r is the ratio between brick and mortar thickness According to EC6, the typical compressive strength for unreinforced masonry using normal mortar is calculated from Equation 2.4. ( 2.4 ) is the brick compressive strength in loading direction according NAD-EC6 (1997), clause 3.2.1.1 in N/mm2. k is a factor that takes values from 0.4 to 0.6. Hemant et al. (2007) presented a generalized relation to estimate the masonry compressive strength from Equation 2.5. ( 2.5 ) Where k, α and β are the constants. Base on the experimental results Hemant et al. (2007) obtained 0.63, 0.49 and 0.32 for k, α and β respectively. It is to note here that α must be higher than β, as
is found to be more dependent on the
strength of bricks used in masonry (Hemant et al. (2007)).
2.5.2. TENSILE STRENGTH In tension, the Australian Standard (SAA Masonry Code AS-3700) allows designers to estimate a characteristic flexural tensile strength for masonry of 0.20 MPa. Hendry et al. (1997), reported that in the stronger direction, the flexural tensile strength of clay brick ranges from about 0.2 to 0.8 MPa.
16
Tomaževic (1999) reported the correlation between the tensile compressive
and
strength for any type of masonry and is given by Equation 2.6. ( 2.6 )
As stated by Wijanto (2007), the tensile strength value from the flexural tests can be used for in-plane lateral forces and out-of-plane bending conditions in unreinforced masonry walls.
2.5.3. ELASTIC MODULUS The Young’s modulus of elasticity of brick units also shows a very wide variety and basically depends on the type of material and value of compressive strength. Typically, a secant modulus of elasticity, Em, is described by the slope of the stress-strain curve between 5% and 33% of the masonry ultimate compressive strength of each prism test or prismatic test (FEMA-274 (1997), ASTM E-111 and NEHRP 2000). Binda et al (1988) specified a relationship for the elasticity modulus of the masonry expressed as a function of its constituent’s moduli through Equation 2.7. ( 2.7 ) Where, r is the ratio between the brick height and mortar thickness. ,
are the elastic moduli for brick and mortar respectively. Sihna (1983) introduced Equation 2.8 to determine the elasticity modulus
as follow, ( 2.8 ) Empirical linear relationship between the compressive elastic modulus and the equivalent compressive strength from some researches are usually assumed by Equation 2.9.
17
( 2.9 ) Where k is a constant factor, (MPa) and
is elastic modulus of masonry in compression
is specified compressive strength of masonry (MPa). Some of the
correlations used by various researchers are shown in Table 2.1.
1
Table 2.1: k Factor used by various researchers Elastic Modulus of Masonry in Reference Compression NEHRP (2000) Clay Brick
2
Tomaževic (1999)
3
FEMA 273 (1997)
Clay Brick
4
Sahlin (1971), Crisafulli et al
Clay Brick
No.
Clay Brick
(1995) 5
Drysdale et al. (1994)
Concrete Clay Brick
6
Paulay and Priestley (1992)
Concrete Clay Brick
7
2.6.
Hemant et al. (2007)
THE STRESS-STRAIN MODEL This section details different ways of input for constitutive modeling of
compressive and tensile behavior. There are multiple stress-strain models that are available for compression and tension behavior (see Figure 2.9 and 2.10). For compression (Figure 2.9) most researchers tend to choose parabolic relation (g) while modeling inelastic behavior where there is parabolic hardening till maximum compressive strength of the material and then subsequent softening until complete rupture.
18
Figure 2-10: Constitutive Models for Compressive Behavior
For inelastic tension behavior (Figure 2.10) the mostly used model is exponential (e) relation. There is a linear increase in stress with respect to strain till maximum tensile strength and then exponential softening occurs.
Figure 2-11: Constitutive Models for Tensile Behavior
19
2.7.
STRESS-STRAIN MODEL ESTIMATION Here are discussed some analytical methods to get formerly mentioned
graphical behavior for compression and tension. For compression Hemant et al. (2007) presented Equation 2.10 to estimate the stress strain curve in his research article. ( 2.10 ) Where, and
are the compressive stress and its corresponding strain in masonry
respectively. and
are the compressive strength and peak compressive strain
corresponding to the compressive strength respectively. The said model is graphically shown in the Figure 2.11,
Figure 2-12: Analytical Model for Stress–Strain Curve of Masonry in Compression (Hemant et al. (2007))
The peak compressive strain can also be estimated using Equation 2.11 that is suggested by Hemant et al. (2007).
20
( 2.11 ) Where
is the peak strain,
mortar compressive strength and
is the compressive strength,
is the
is the young’s modulus of brick masonry.
The aforementioned analytical model is a very good representation of the compressive behavior but it is not very easy to control the curvature of the parabola and there is also no representation for the post peak softening. Therefore another model with hardening criterion with subsequent softening (see Figure 7), also mentioned by van Noort (2012), is presented as follow;
Figure 2-13: Compression Model with Hardening and Subsequent Softening Criterion (Van Noort (2012))
Mathematically, ( 2.12 )
( 2.13 )
( 2.14 ) Where: is the crushing strength is the peak strain at the crushing strength 21
(Initial stress) (Mid stress) (Rupture Stress) The masonry behaves in two different possible ways when subjected to tension parallel to the bed joint, as described in Figure 2.2 and more probable is the pattern (a). Keeping the said pattern in view, Belarbi et al. (1996) and Ghiassi et al. (2012) presented a mathematical expression for tensile behavior of masonry and other quasi brittle material as follow,
Figure 2-14: Tension Model with Exponential Softening Criterion (Belarbi et al. (1996) and Ghiassi et al. (2012))
Mathematically, ( 2.15 ) ( 2.16 ) Where; is the cracking strength in tension is the strain at fracture strength c is the stiffening parameter which defines the post cracking sharpness of the model curve. 22
3. ABAQUS – A SOFTWARE SYNOPSIS 3.1.
OVERVIEW Abaqus is a finite element program that helps solving problems ranging
from simple linear analysis to complex nonlinear simulations. This code can solve problems that involve different elements model by associating the geometry and the material of each element and then choosing their interactions. In nonlinear analyses, load increments and convergence tolerances are chosen by the program. In this way abaqus keeps on adjusting them to ensure that an accurate solution is obtained efficiently.
3.2.
PROCESSING STEPS A complete Abaqus analysis consists of three distinct steps, i.e. 1. Pre-Processing – Preparing the Model 2. Simulation – Running the Analysis 3. Post-Processing – Visualizing the Results
Figure 3.1 shows a brief flow diagram of the simulation process.
Figure 3-1: Flow Diagram of the Simulation Process
Preprocessing is the model preparation step where the model and all its interactions are put together using Abaqus internal and/or any other external tools (like AutoCAD, Solid Edge, CATIA etc.). Than simulation is an evaluation 23
and calculation step where the model problem is solved using an explicit or implicit procedure within the Abaqus Software. The results are than analyzed in the postprocessing step.
3.3.
THE PRODUCT SUITE The abaqus product suite is made up of four core software, 1. Abaqus/CAE 2. Abaqus/CFD 3. Abaqus/Explicit 4. Abaqus/Standard
3.3.1. Abaqus/CAE: Abaqus/CAE (Complete Abaqus Environment) is a computer aided engineering software application used in pre-processing to design and model the components and in post-processing to visualize the finite element results. The software is divided into modules that defines a logical aspect of the modeling process. The modules are,
Part: where the elements of the model is created.
Property: material and section of each part is defined.
Assembly: where the assembly is created and can be modified. Every model contains only one assembly, composed of instances of parts from the model.
Step: where it is possible to create analysis steps and specify output requests.
Interaction: where mechanical interactions (such as contacts) between regions are managed.
Load: where load and boundary conditions are defined.
Mesh: where the mesh is generated.
Job: where jobs are created and their progression is monitored. 24
Sketch: where a sketch is created.
Visualization: where the output database is analyzed. The model is build up by moving from module to module. Abaqus/CAE
generates an input file when a model is completed which is than submitted to analysis product of the abaqus suite. The input file is read by Abaqus/Explicit or Abaqus/Standard and the analysis starts. The resulting information is sent to Abaqus/CAE where the job progress can be monitored. At the end of job, an output database is generated and it can be visualized using the visualization module in Abaqus/CAE.
3.3.2. Abaqus/CFD: Abaqus/CFD is a Computational Fluid Dynamics software application which is new to Abaqus 6.10. This product is not used in this study, however it is briefly described. It is very similar to Abaqus/CAE, but it provides more sophisticated computational fluid dynamics capabilities with extensive support for pre-processing and post-processing. With Abaqus/CFD it is possible to solve nonlinear coupled fluid-thermal and fluid-structural problems.
3.3.3. Abaqus/Explicit: Abaqus/Explicit is able to solve problems that involve short or transient dynamic events (impact, blast, earthquake etc.) and is also very efficient for highly nonlinear problems involving contact conditions. This procedure is based on the implementation of an explicit integration rule using diagonal or “lumped” element mass matrices. If the applied load vector
, the internal force vector
known, the acceleration
and the diagonal lumped mass matrix
are
at the beginning of increment is shown by Equation
3.1.
25
( 3.1 ) Now the problem is to find the dynamic equilibrium of the rigid body at the time t , which is done by solving Equation 3.2 and Equation 3.3. ( 3.2 ) ( 3.3 ) where, is the degree of freedom, displacements and rotations of the system. is the velocity of the system. is the acceleration of the system. is the subscript that refers to the increment number in the dynamic analysis are the subscripts that refers to mid increment values Abaqus/Explicit is explicit because the kinematic analysis can process to that next increment knowing values of
and
from previous increment.
This method is computationally efficient if the mass matrix of the problem is diagonal because its transposition is easier. The explicit method does not require any iteration nor the calculus of the tangent stiffness matrix. In Abaqus/Explicit, the order of time increment is automatic and any user intervention is not required. The explicit procedure integrates through time by using many small time increments. Its stability is conditioned and the stability limit for the operator is given in terms of the highest eigenvalue in the system as:
neglecting damping;
Considering damping (
is the
fraction of critical damping in the highest mode). Abaqus/Explicit contains a global estimation algorithm, which regulates the maximum frequency of the entire model. This algorithm continuously updates the estimate for the maximum frequency. Abaqus/Explicit initially uses 26
an element by element estimates. Once the algorithm determines that accuracy of the global estimation is acceptable, the stability limit will be determined from global estimator. The conservation of stable time increment is given by a minimum taken over all the elements in Equation 3.4. ( 3.4 ) where, is the characteristic element dimension; is the dilatational wave speed of the material. For beam, shell or membrane elements, the thickness of the element is not considered as the smaller dimension, but stability is referred to mid-plane. Time increments must be minor than the stability limit. If this condition is not met, the solution of the problem would be instable.
3.3.4. Abaqus/Standard: Abaqus/Standard is able to solve a wide range of linear and nonlinear problems that involve either static or dynamic response of elements. Usually, models generated in abaqus are nonlinear and can involve many variables. Let u be the variable of the problem and
the force component, the
problem is to find out the solution , solving the equilibrium Equation 3.5: ( 3.5 ) This problem is history-dependent, therefore the solution should be determined for a series of “small” increments using Newton’s method. This numerical technique for solving nonlinear equilibrium equation was chosen due to its better convergence rate compared to modified Newton or quasi-Newton method.
27
Newton’s method says that after iteration , an approximation to the solution
is obtained. Let
be the difference between the numerical solution
and the exact solution. This can be expressed by Equation 3.6. ( 3.6 ) Expanding equation 3.6 in Taylor series: ( 3.7 ) Considering magnitude of
as close approximation to the exact solution, the
will be small at every increment. Therefore all except the first
two terms from Equation 3.7 can be neglected giving a linear system given by Equation 3.8. ( 3.8 ) where
is the stiffness matrix and
As the iteration continues, the next approximation to the solution is then given by Equation 3.9. ( 3.9 ) If all entries in
and in
are small, convergence of the method is granted.
This is checked by default in Abaqus/Standard solution. Newton’s method needs to calculate the Jacobian matrix and solve linear equations at every single iteration. Thus, this method is computationally expensive.
3.4. ELEMENT TYPE FOR SOLIDS There is a wide range of elements described in Section 27.1.1 of the Abaqus Analysis User’s Manual which provides a powerful set of tools for solving different numerical problems. There are five aspects of an element that characterize its behavior, i.e.
Family
Degree of freedom
Number of nodes 28
Formulation
Integration
The type of element for a problem depends, at what cost a researcher wants to achieve the results, whether it’s time or accuracy or a balance between both. For large models with simpler and quicker calculation time, a linear brick element of first order is mostly used in the literature. It has 8 nodes at corners and uses a linear interpolation in each direction. It is designated as C3D8R in the Abaqus Analysis User’s Manual as shown in Figure 3.2, in which C3D8 shows to which element family it belongs and how much nodes it has. The last letter illustrates whether the integration is reduced or full for a material response. It is a choice that can have a substantial effect on the accuracy of an element for a given problem. While an element’s formulation refers to the mathematical theory used to define an element’s behavior that is briefly described in Section 28.1.4 of the Abaqus Analysis User’s Manual.
Figure 3-2: C3D8R for Solids
3.5.
MATERIAL MODELS The core of this study is the material model for masonry and the study of
its behavior under in-plane shear tests. As already said in the introduction, an “equivalent” masonry (homogenized material) was chosen, trying to use several 29
material models already implemented in Abaqus. The choice was made after a rough classification of all possible material implementations available in Abaqus, some of which are named as follow: 1. Concrete Damaged Plasticity 2. Concrete Smeared Cracking 3. Elastic-Plastic Model 4. Damaged Elasticity 5. Extended Drucker-Prager 6. Cast Iron Plasticity etc. We look for the most suitable material model which is able to describe the behavior in tension and compression with non-linearity in plastic rang.
3.6.
CONCRETE DAMAGED PLASTICITY: Concrete
Damaged
Plasticity
model
is
implemented
for
both
Abaqus/Standard and Abaqus/Explicit which provides a general capability for analysis of concrete and also is suitable for masonry and other quasi-brittle materials, under monotonic, cyclic or any other type of dynamic loading. This constitutive theory captures the irreversible effects of damage that occur in concrete under low confining pressure. To describe this behavior, following features are considered: 1. Different yield strengths in tension and compression (with the initial yield stress in compression a factor of 10 or more higher than the initial yield stress in tension). 2. Softening behavior in tension while in compression initial hardening followed by softening. 3. Different degradation of the elastic stiffness in tension and compression. 4. Stiffness recovery effects during cyclic loading. 30
5. Rate sensitivity, especially an increase in the peak strength with strain rate. The theory references of this implementation are models propose by Lubliner et al. (1989) and by Lee and Fenves (1998) (also see Section 4.5.2 of the Theory Guide).
3.6.1. Strain rate decomposition: The rate independent model is governed by an additive strain rate decomposition as illustrated in Equation 3.10. ( 3.10 ) Where
is the total strain rate and the superscripts ‘ ’ and ‘
’ refer to the
elastic and plastic part of the strain rate respectively.
3.6.2. Stress-Strain Decomposition: Stress-Strain relations for this method is governed by Equation 3.11. ( 3.11 ) where
is the initial undamaged elastic stiffness of material,
degraded elastic stiffness and (undamaged material
is the
is the scalar stiffness degradation variable
, fully damaged material
). Hence, damage is
represented with an isotropic reduction of the elastic stiffness with the scalar factor . Thus, Equation 3.12 shows the Cauchy oStress. ( 3.12 ) Where the effective stress is defined as
. When impairment in
the model take place, the effective stress signifies the effective stress area that is
31
resisting the external loads. This is why it is better to express the plasticity problem in terms of .
3.6.3. Hardening Variables: There are two different hardening variables: the equivalent plastic strain in tension
and the equivalent plastic strain in compression
. Let
be the
vector of hardening variable, than their evolution is defined by Equation 3.13. ( 3.13 ) Besides micro-cracking and crushing, these hardening variables also controls the progression of yield surface and the degradation of elastic stiffness. Moreover, these variables are associated to the dissipated fracture energy that is required to produce micro cracks.
3.6.4. Uniaxial Condition Under uni-axial condition, it means that the stress curves have the form defined by Equation 3.14 for tension and Equation 3.15 for compression. ( 3.14 ) ( 3.15 ) Subscripts ‘ ’ and ‘ ’ refers to tension and compression respectively; and
are the corresponding plastic strain ratios, while and
are the corresponding plastic strain in
tension and compression; is the temperature of system and are other field variables in the model.
32
The effective plastic strain rates under uni-axial loading conditions are given by Equation 3.16 for uniaxial tension and Equation 3.17 for uniaxial compression. ( 3.16 ) ( 3.17 ) It is supposed in this model that,
are positive quantities signifying the
magnitude of uni-axial compressive stress, thus
.
Starting from any point of the strain softening branch of the stress-strain curves for both tension and compression, the response of concrete and alike material is enfeebled which is due to the damage or degradation of elastic stiffness of the material (see Figure 3.3).
Figure 3-3: Damaged Response to Uniaxial (a) Tension and (b) Compression (Section 4.5.2 of Theory Guide)
33
There is a significant difference in the degradation of the elastic stiffness in tension and compression tests, but either event shows a more distinct effect, as the plastic strain increases. The concrete degradation is described through two independent variables
and
as illustrated in Equation 3.18 and Equation
3.19, which are increasing function of the corresponding plastic strain temperature
and field variables
, the
: ( 3.18 ) ( 3.19 )
Thus, let
be the undamaged elastic stiffness, the stress-strain relations under
uni-axial loading are given by Equation 3.20 and Equation 3.21. ( 3.20 ) ( 3.21 ) The yield surface size is determined by the effective uni-axial cohesion: ( 3.22 ) ( 3.23 )
3.6.5. Uniaxial Cyclic Condition Under uni-axial cyclic loading conditions, the degradation involves the interaction of the micro-cracks that constantly open and close. The elastic stiffness recovers as the load changes sign and shifts from tension to compression. Under such condition, elastic modulus is given by Equation 3.24 as a function of undamaged modulus
and stiffness reduction variable d.
34
( 3.24 ) The stiffness reduction variable d for the model is a function of the uni-axial damaged variables
and
and is given by Equation 3.25. ( 3.25 )
where
and
signifies the stiffness recovery effects associated to stress
reversals:
and
The weight factors
and
describe material properties link to stiffness
recovery of the model. Figure 3.4 shows a default behavior of the material model used in abaqus where the compressive stiffness is recovered due to closure of cracks as the load changes from tension to compression
, while the
tensile stiffness is not retrieved as the load changes from compression to tension once crushing of micro cracks happens
.
35
Figure 3-4: Tension-Compression Cyclic Model with Damage and Recovery Effect (Section 4.5.2 of Theory Guide)
3.6.6. Yield Function The state of failure damage in the effective stress space for tension and compression is represented by a yield function given by Equation 3.26. ( 3.26 ) The final form of the plastic damage concrete model is given by Equation 3.27, which takes into account the evolution of strength under tension and compression. ( 3.27 ) where is the effective hydrostatic pressure in the model;
is the Mises equivalent effective stress; is the deviatoric part of the effective stress tensor ; is the maximum eigenvalue of . 36
Let
and
be the effective tensile and compressive stresses,
respectively, thus the function
is shown by Equation 3.28. ( 3.28 )
The coefficient
can be determined through Equation 3.29 from initial
equi-biaxial and uni-axial compressive yield stress,
and
. ( 3.29 )
Typical values of
are between 0.08 and 0.12.
The Macaulay brackets are used to describe the ramp function:
Thus in bi-axial compression, when
, Equation 3.27 moderates to
the Drucker-Prager yield condition. Coefficient
enters in Equation 3.27 only when the specimen is subjected
to tri-axial compression and
(typical value for concrete is
).
Figure 3.5, shows a typical yield surface in the deviatoric plane for planestress condition. It can be noted that besides all similarities that can be pointed out looking at the yield surface of Figure 3.4 and Lorenco’s yield criterion, the concrete damaged plasticity is not able to reproduce different behavior of the material when load changes direction.
37
Figure 3-5: Typical Yield Surface (Section 4.5.2 of Theory Guide)
3.6.7. Flow Rule The flow potential
governs the plastic flow with the flow rule, thus
given by Equation 3.30. ( 3.30 ) Where
is the non-negative plastic multiplier that obey, together with the yield
function , the Kuhn-Tucker conditions. is the Drucker-Prager hyperbolic function and is given by Equation 3.31. ( 3.31 ) Where pressure,
is the dilation angle, measured in the
plane at high confining
is the uni-axial tensile stress at failure and
is a parameter that
describes the rate at which the function approaches its asymptote. The flow potential tends to a straight line as the eccentricity tends to zero. This flow 38
potential, which is continuous and smooth, ensures that the flow direction is defined uniquely. The function asymptotically approaches the linear DruckerPrager flow potential at high confining pressure stress and intersects the hydrostatic pressure axis at 90o.
39
4. METHODOLOGY A detailed procedure adopted for modeling of a masonry wall is presented in this chapter. It was intended to be as simple as possible and thus an isotropic homogenous model was chosen for an already existing material model in ABAQUS i.e. Concrete Damaged Plasticity Model.
4.1.
SUBJECT MODEL The model considered for this study is taken from Javed (2008). The
important objective of his work was to study the seismic performance of the solid fired clay brick masonry shear piers constructed in Cement: Sand: Khaka (CSK) Mortar. The bricks used were solid fired clay bricks, having an average size of 9.29” (235.97mm) x 4.33” (109.98mm) x 2.09” (53.09mm) and an average compressive strength of 3200Psi, which were obtained from a demolished building in Harbin City. The mortar used was a composition of Cement: Sand: Khaka in a ratio of 1:4:4 by weight and the w/c ratio was 1.2. Walls/Piers with three different geometries were tested having aspect ratios (i.e. height to length ratio) of 0.66, 0.93 and 1.22. Pier with aspect ratio of 1.22 is selected for this study, which was marked as PIc by Javed (2008). The geometry of the tested pier can be drafted as shown in the Figure. 4.1.
40
(a)
(c)
(b)
Figure 4-1: Geometry of Wall/Pier-PI (Javed (2008)) (a) Isometric View (b) Front View (c) Side View
41
The characteristics of the specimen pier PI are tabulated in the Table 4.1. Table 4.1: Characteristics Of The Specimen Pier (Javed (2008)) Pier Series
Height, hp - ft.
Length, lp - ft.
Thickness, t - in.
Aspect Ratio = hp/lp
PreCompression, σo – psi
Relative PreCompression = σo/fm
PI
5.45
4.46
9.29
1.22
103
0.153
Following is the summary of material properties from experimental work of Javed (2008), Compressive Strength of Brick Units = 3200 Psi Flexural Tensile Strength of Brick Units = 387.5 Psi Compressive Strength of CSK Mortar = 719 Psi Compressive Strength of Brick Masonry Prism = 658.1 Psi Modulus of Elasticity of Brick Masonry Prism = 188358 Psi Principle Tensile Strength of Brick Masonry Prism = 24.5 Psi The wall/pier was subjected to shear-compression static cyclic test also known as quasi-static cyclic test. The test set up is shown in the Figure 4.2. The precompression (
) ratio used for the pier PI was 0.153 corresponding to
pressure.
42
Figure 4-2 : Test Set Up by Javed (2008)
43
4.2.
CONSISTENT UNITS Before defining any model it is necessary to decide which system of units
is to be used. Abaqus is basically a unit less application, therefore a great deal of care is required for the consistency of the units. Some common systems of consistent units are shown in Table 4.2.
Quantity
Table 4.2: Consistent Units Metric (m) Metric (mm) Imperial (ft)
Imperial (in)
Length
M
Mm
ft
In
Force
N
N
lbf
Lbf
Mass
kg
tonne (103 kg)
slug
lbf.s2/in
Time
s
s
s
S
Stress
Pa (N/m2)
MPa (N/mm2)
lbf/ft2
Psi (lbf/in2)
Energy
J
mJ (10-3 J)
ft.lbf
in.lbf
Density
kg/m3
tonne/mm3
slug/ft3
lbf/in3
The system of units used in this study is Metric (m). However the units are occasionally interconverted, as the experimental work was done in Imperial units.
4.3.
MODEL GEOMETRY There were two parts defined in the Part module, Concrete Top Beam and
the Masonry wall. The dimensions used were that from Figure 4.1, but in meters. The parts are shown in the Figure 4.3.
44
(a) (b)
Figure 4-3: (a) Concrete Top Beam (b) Masonry Wall
There were some datum points and partition cells defined for the beam, so that its mesh is consistent with the masonry wall. Now the parts were meshed in the Mesh module, but before that an approximate global seed size of 0.0762m (3in) was given in Seed>Part from the menu bar. Also from Mesh>Element Type, an Explicit 3D Stress element (C3D8R) is selected. Than the mesh was applied, as shown in the Figure 4.4.
(a) (b)
Figure 4-4: (a) Meshed Concrete Top Beam (b) Meshed Masonry Wall
45
4.4.
MATERIALS Material properties are defined in the Property module where the material
model required input for three parameters, i.e. Density and Elastic for both model parts and Concrete Damaged Plasticity for the Masonry Wall only. The procedure is described as under,
4.4.1. Elastic Properties In case of the top beam, the elastic modulus was assumed to a very high value to account for a stiff system at the top of the masonry wall. For masonry, the elastic modulus was estimated from the Equation 2.8. There were many trail tests performed for only linear elastic behavior of the system. Figure 4.5 shows test results of the model with different k values used in Equation 2.8 and k = 850 was chosen as a best fit for linear elastic behavior. Initially k=550, also used by Hemant et al. was also tested but was not found suitable.
35.000 30.000
LOAD (KIPS)
25.000 20.000 +ive DIRECTION (Javed (2006))
15.000
-ive DIRECTION (Javed (2006)) E=550xfm (Hemant et al. (2007))
10.000
E=800xfm E=850xfm
5.000
E=900xfm E=950xfm
0.000 0.000
0.050
0.100 DISPLACEMENT
0.150
Figure 4-5: Results of Linear Elastic Behavior
46
0.200
The concluded values are tabulated in Table 4.3 Table 4.3: Elastic Properties Concrete Top Beam
Masonry Wall
Density (kg/m3)
2400
1887
Modulus of Elasticity (MPa)
20000
4070.39
0.15
0.2
Poisson’s Ratio
4.4.2. Inelastic Properties Here the parameters required to define concrete damaged plasticity model are determined.
4.4.2.1.
Plasticity
For plasticity tab, following data need to be characterized, Dilation Angle: ψ (in Degrees) in the p-q plane. (default ψ = 370) Eccentricity:
a small positive number that defines the rate at which the
hyperbolic flow potential approaches its asymptote. (default = 0.1) fb0/fc0: that is σb0/σc0, the ratio of initial equi-biaxial compressive yield stress to initial uni-axial compressive yield stress. (default σb0/σc0 = 1.16) K: Kc must satisfy the yield condition, thus 0.5 < Kc < 1. (default Kc = 2/3) Viscosity Parameter: μ is used for the visco-plastic regularization on the constitutive equation in Abaqus/Standard analysis. (default μ = 0.0) After various trial tests, started with the default values, the resulting plasticity parameters are obtained in Table 4.4.
47
Table 4.4: Plasticity Parameters Dilation Angle
Eccentricity
fb0/fc0
K
Viscosity Parameter
45
0.1
1.3
0.5
0
4.4.2.2.
Compressive Behavior
The compressive strength is determined using Equation 2.5, where the resulting value (i.e. 4.79MPa) was found very close to the experimental value (i.e. 4.54MPa). In the thesis of Javed (2008), there was no information found regarding the strains for the model. Therefore an empirical method was adopted from Hemant et al. (2007), which is discussed in Section 2.7, to calculate the peak strain corresponding to the compressive strength. Since the exact value of the strain was unknown, therefore from the literature it was assumed to be around 0.004 (Hemant et al. (2007)). For this purpose, different decimal figures instead of 0.27 were used in Equation 2.11 and also trail tested, and finally 0.4 was used for the model. The rest of the stress strain values of the graph are calculated with the help of Equations 2.12 through Equation 2.14, which were modified for the subject model as presented from Equation 4.1 to Equation 4.3. ( 4.1 ) ( 4.2 )
( 4.3 ) Where, is the crushing strength 48
is the peak strain at the crushing strength (Initial stress) (Mid stress) (Rupture Stress) is the strain corresponding to mid stress, i.e. is assumed to be 2.25 times
.
The subsequent stress strain curve is shown in the Figure 4.6. The exponential softening part of the equations are not taken into account, since it was concluded not to damage the model up to that extent for our problem. 6
Compressive Stress (106 Pa)
5
4
3
2
1
0 0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
Compressive Strain Figure 4-6: Compressive Stress-Strain Curve
The compressive behavior outside the elastic range is defined with a tabular function of stress
over the inelastic deformation
(see Figure 4.7).
Abaqus automatically calculates plastic strains from already known inelastic strains. If the elastic strain in compression is defined by Equation 4.4,
49
( 4.4 ) Than through the relationship given in Equation 4.5, we can have inelastic deformation. ( 4.5 ) The plastic strains are calculated by Equation 4.6. ( 4.6 ) Equation 4.4 through Equation 4.6 are graphically illustrated in the Figure 4.7.
Figure 4-7: Definition of Compressive Inelastic Strains
Where
is the damage parameter, obtained from Equation 4.7. ( 4.7 )
In Equation 4.7,
is the compressive strength of masonry. The resulting
inelastic strains and damage parameters corresponding to yield stresses are tabulated in Table 4.5.
50
Table 4.5: Compressive Behavior Yield Stress (106 Pa)
Inelastic Strain
Damage Parameter
Yield Stress (106 Pa)
Inelastic Strain
Damage Parameter
2.088
0
0
4.762
0.00315
0.00549
2.552
0.00037
0
4.730
0.00341
0.01235
2.958
0.00052
0
4.684
0.00367
0.02196
3.315
0.00069
0
4.624
0.00393
0.03432
3.628
0.00086
0
4.552
0.00420
0.04942
3.899
0.00104
0
4.467
0.00447
0.06726
4.131
0.00124
0
4.368
0.00474
0.08786
4.327
0.00144
0
4.256
0.00502
0.11119
4.487
0.00165
0
4.131
0.00530
0.13727
4.612
0.00187
0
3.993
0.00559
0.16610
4.704
0.00209
0
3.842
0.00587
0.19768
4.762
0.00233
0
3.678
0.00616
0.23199
4.788
0.00257
0
3.500
0.00646
0.26906
4.789
0.00264
0
3.310
0.00675
0.30887
4.782
0.00289
0.00137
3.106
0.00705
0.35142
4.4.2.3.
Tensile Behavior
For masonry in tension, Equation 2.6 is used to compute the tensile strength. The factor used is 0.055, which was concluded as a result of numerous trails. Tension starts with a linear behavior from zero till the peak tensile strength and the stress strain values are calculated using Equation 2.14. Then the curve exponentially softens till the ultimate strain, i.e. 0.000389, which is also concluded from the trail runs. Here Equation 2.15 has been used to calculate the second part of the curve. It is important to note here that the masonry wall is subjected to a shearcompressive load and it is more likely to behave in a manner shown in the Figure 2.2 (a), which is already discussed in the Section 2.1.1. It is logically obvious that in saw tooth pattern, masonry doesn’t lose all its strength after the cracks opens because after cracking the friction between adjacent bricks comes into action, and 51
that is why there is a significant value of stress at the ultimate strain (see Figure 4.8). Subsequently, different values of c for Equation 2.15 were tested and c = 0.2 was found to match the experimental results. The resulting stress strain curve is shown in the Figure 4.8. 3
Tensile Stess (105 Pa)
2.5
2
1.5
1
0.5
0 0
0.0001
0.0002
0.0003
0.0004
Tensile Strain Figure 4-8: Tensile Stress-Strain Curve
Inelastic strains in tension are calculated in the same manner as that of compression. If the elastic strain in tension is defined by Equation 4.8, ( 4.8 ) Than through the relationship given in Equation 4.9, we can have inelastic deformation. ( 4.9 ) The plastic strains are calculated by Equation 4.10. ( 4.10 ) Equation 4.8 through Equation 4.11 are graphically illustrated in the Figure 4.9. 52
Figure 4-9: Definition of Tensile Inelastic Strains
Where
is the damage parameter, obtained from Equation 4.11: ( 4.11 )
In Equation 4.11,
is the tensile strength of masonry. The resulting inelastic
strains and damage parameters corresponding to the yield stresses are tabulated in Table 4.6. Table 4.6: Tensile Behavior Yield Stress Cracking Damage 6 (10 Pa) Strain Parameter 0.263 0 0 0.247
0.000029
0.06325
0.235
0.000057
0.10819
0.226
0.000084
0.14267
0.218
0.000111
0.17044
0.212
0.000138
0.19356
0.207
0.000164
0.21328
0.203
0.000190
0.23042
0.199
0.000216
0.24554
0.195
0.000242
0.25904
0.192
0.000268
0.27120
0.189
0.000293
0.28226
0.186
0.000319
0.29238
0.184
0.000345
0.30170
53
Values from Table 4.5 and Table 4.6 are assigned to the material model i.e. concrete damaged plasticity. To assign material properties it is necessary to create a section type, whether solid or shell. In this study a solid homogenous type section is created for each part in the property module, one is concrete and the other is masonry.
4.5.
STEP DEFINITION The type of numerical procedure is defined in the step module. As the
masonry wall has been subjected to a quasi-static in plane lateral load, the dynamic explicit option was found more suitable for this situation, which is also stated in the Manual Section 6.3.3. It uses an explicit central-difference time integration rule and efficiently performs analysis on large models with relatively short dynamic response times. It requires time period as a basic input, which is the total time for the loading procedure in that particular step. There is one step defined by default i.e. Initial. In this research, two more steps were defined i.e. PreComp for precompression load and DispCont for the displacement controlled lateral load. Input values used for the time period were 30sec for PreComp and 1200sec for DispCont. However, the analysis was found to be computationally costly. Therefore mass scaling was used to speed up the process with a scale factor of 1000 in the semi-automatic mass scaling option, which artificially increase the material density (see Section 6.3.3 of the Manual).
4.6.
ASSEMBLY Both parts created in Section 4.3 are assembled in the assembly module,
by placing the top concrete beam on the masonry wall. Masonry wall is tied to the beam by creating a Tie type constraint in the interaction module that uses a default discretization method. A reference point RP-1 is made and also is 54
constrained to the bottom surface of masonry wall through a Coupling type constraint. The assembled model is shown in the Figure 4.9 with and without mesh.
Figure 4-10: The Assembled Model
4.7.
BOUNDARY CONDITIONS & LOADS The model is now rigidly fixed at the bottom in the load module. The type
of boundary condition used is ENCASTRE and it is applied to the reference node RP-1. This will now sum up all the reaction forces on nodes of the bottom surface coupled to RP-1. Another boundary condition is applied to the top concrete beam which is a displacement type. The total lateral displacement in x-direction (i.e. U1) is 0.0085m (0.335in) and z-direction is restrained by putting U3=0. It is also necessary to select DispCont step while creation of the boundary condition. This displacement is applied with an amplitude shown in the Figure 4.11.
55
1.2
Amplitude
1 0.8 0.6 0.4 0.2 0 0
200
400
600
800
1000
1200
1400
Time (sec) Figure 4-11: Applied Amplitude to the Displacement in Lateral Direction
The model is kept under compression, initializing from PreComp step and propagating it through DispCont step till its end. The precompression value used is 0.71MPa (103Psi) but in Pa, that is applied with a Ramp amplitude, shown in the Figure 4.12.
1.2
AMPLITUDE
1 0.8 0.6 0.4 0.2 0 0
10
20
30
40
TIME (sec) Figure 4-12: Applied Amplitude to the Precompression Load
Figure 4.13 shows a loaded model, almost ready to be run with only one step left, i.e. to set output requests for post-processing.
56
Figure 4-13: Model showing Applied Loads and Boundary Conditions
4.8.
OUTPUT REQUESTS Abaqus analysis product computes values of many requested variables by
the end of the analysis process and usually only a small subset of that computed data is of interest. Here abaqus provides its users an option to specify the data that is to be written in output database through output request (see Section 14.4 of Abaqus/CAE User’s Guide for details). An output request consists of following information:
The variables or variable components of interest.
The region of the model and the integration points from which the values are written to the output database.
The rate at which the variable or component values are written to the output database. When a first step is created, Abaqus/CAE selects a default set of output
variables and its rate corresponding to the step’s analysis procedure. One is Field Output Request, where the output is generated from data that is spatially distributed over the entire model or over a portion of it. Another one is History 57
Output Request, where the output is generated from the data at specific points in a model. In our model, besides default variable settings, both output requests were set to a frequency of values at every 5 seconds of time and also compression and tension damage variables were added for all the analysis procedures. There were two more history outputs added, one for reactions in x-direction (i.e. RF1), where domain was a reference point RP-1 added as a Set. Another one was for the displacements in x-direction (i.e. U1), where domain was a node at the top of the masonry wall also added as a Set.
58
5. RESULTS AND DISCUSSION The model is all set for execution and is analyzed using Job module. After the analysis starts to run, model can be visualized using visualization module for every increment of the step time and also monitored by right clicking the job file created. The monitor window shows few important parameters, out of which energies are constantly under observation where Kinetic Energy should never exceed the total energy if it does than the solution is unstable. Also it is a good sign if Stable Time Increment is constant along the process. Finally after the job is completed X-Y Plots can be extracted from history output and also stress, strain and damage condition can be envisaged at every step time. Furthermore, results of the analysis are compared with reference to four performance levels, explained by Javed (2008) and described with the help of Figure 5.1. They are,
Operational Level (O) – related to drift ratio corresponding to flexural cracking of bed joint.
Immediate Occupancy Level (IO) – related to drift ratio when first diagonal crack is formed.
Life Safety Level (LS) – related to drift ratio when maximum resistance is gained by the pier.
Collapse Prevention Level (CP) – related to drift ratio at a phase when shear resistance either dropped to 80% of Vu or test is stopped due to extensive damage (which ever occurr first)
59
Figure 5-1: Typical Performance Levels
5.1.
LOAD-DISPLACEMENT CURVE The stress-strain behavior and damage is visualized in later sections of
this chapter. However the resulting load-displacement curve is shown in the Figure 5.2. 140.0
Lateral Force (kN)
120.0 100.0 80.0 60.0 40.0 20.0 0.0 0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Lateral Diaplacement (mm) Figure 5-2: Load-Displacement Curve of the Numerical Model Loaded Monotonically
Flexural tensile damage starts to appear at 0.479mm displacement where the lateral force is (16.43kips) 73.09kN and hence can be associated to Operational (O) Level performance. There also appears vertical damage due to 60
tension but is not momentous. At 0.949mm displacement, there appears a significant damage due to tension at the center of the model that expands diagonally towards corner and the corresponding lateral load is 100.27kN. However, substantial diagonal cracks appears at 1.473mm displacement and its corresponding load value is 101.56kN. This can be associated to Intermediate Occupancy (IO) Level performance. The peak lateral force for the model is 116.59kN at 3.785mm lateral displacement and is associated to Life Safety (LS) level. The analysis was stopped at 8.96mm lateral displacement where the corresponding lateral force was 113.25kN and is associated to Collapse Prevention Level. Moreover, the same model was tested for different precompression levels, i.e. 0.479MPa and 0.958MPa, where the precompression ratios are 0.1 and 0.2 respectively. The results are shown in the Figure 5.3. 140
Lateral Force (kN)
120 100 80 60 Precomp Ratio = 0.1
40
Precomp Ratio = 0.148 20
Precomp Ratio = 0.2
0 0
1
2
3
4
5
Lateral Displacement (mm)
6
7
8
Figure 5-3: Results of the Numerical Model at different Precompression Ratios
5.2.
COMPARISON WITH JAVED (2008) For comparison, hysteresis results in positive direction for the model from
Javed (2008) was digitized using GetData Graph Digitizer and was plotted as 61
shown in the Figure 5.4. Also its values are tabulated in Table 5.1. It is to note here that the units of the numerical results were converted from metric to imperial comparison. 30
Lateral Force (kips)
25 20 15 10 +ive DIRECTION (JAVED(2008))
5
this study
0 0.00
0.05
0.10
0.15
0.20
0.25
0.30
Lateral Displacement (inch) Figure 5-4: Comparison of the Results
Relation between stiffness degradation and drift ratio was also observed using the criteria mentioned in Javed (2008) and is shown in the Figure 5.5.
Stiffness Degradation K/Kp
2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.0
0.1
0.2
0.3
0.4
Drift Ratio % Figure 5-5: Stiffness Degradation Curve
62
0.5
0.6
A percentage difference of the numerical result with experimental result is presented in the Table 5.1. In this table the selected numerical values of displacements and their corresponding loads are those which are close to the experimental results. Table 5.1: Percentage Difference between Experimental and Numerical Results Javed (2008) This Study %age Difference Displ. (inch)
Load (kips)
Displ. (inch)
Load (kips)
Displ. (%)
Load (%)
0
0
0
0
0
0
0.010
11.925
0.010
9.564
0.544
19.793
0.030
20.650
0.031
21.593
2.609
4.568
0.058
23.473
0.058
22.837
0.725
2.712
0.066
23.490
0.066
23.259
0.641
0.984
0.071
23.114
0.071
22.275
0.359
3.628
0.082
23.097
0.082
23.328
0.045
1.001
0.092
24.209
0.092
24.122
0.784
0.358
0.111
24.380
0.111
23.683
0.294
2.858
0.128
24.859
0.129
25.714
0.446
3.439
0.149
24.482
0.149
26.213
0.124
7.068
0.178
25.988
0.180
25.887
1.064
0.387
0.195
26.176
0.197
25.934
0.826
0.925
0.218
24.602
0.217
24.085
0.134
2.103
0.238
23.867
0.238
23.908
0.199
0.173
0.271
23.849
0.273
22.707
0.492
4.789
0.304
22.156
0.305
22.716
0.094
2.530
0.348
21.044
0.353
25.460
3.133
20.987
63
Another comparisons between lateral load, displacement and drift ratios at different limit states or performance levels are shown in the Table 5.2. Table 5.2: Comparison between Experimental and Numerical Results at different Limit States Experimental Results (Javed (2008)) Numerical Results Limit Lateral Lateral Lateral Load Drift Ratio Lateral Load Drift Ratio States Displ. Displ. (kips) %δ (kips) %δ (inch) (inch) Flexural Vfcr 20.71 dfcr δfcr Vfcr 16.43 dfcr 0.019 δfcr 0.029 Crack Diagonal Vdcr 23.61 ddcr 0.058 δdcr 0.09 Vdcr 22.83 ddcr 0.058 δdcr 0.089 Crack Max. Vmax 26.17 dmax 0.196 δmax 0.3 Vmax 26.21 dmax 0.149 δmax 0.228 Resistance Ultimate
5.3.
Vu
21.05
du
0.348
δu
0.53
Vu
25.46
du
0.353
δu
0.539
STRESS-STRAIN VISUALIZATION In this section stress-strain contours of the output results are presented,
where the concentration of mises stresses and plastic strains are shown for various performance levels listed in Table 5.2. In Figure 5.6 through Figure 5.9, Mises stresses at each level is shown where compression can be observed at the two alternate top and bottom edges. Additionally it can be seen in Figure 5.7 that there is a reduction in stress along the diagonal which is due to tension damage and it progresses along each time step till Figure 5.9. It is note here that plastic strains occurs at the damaged areas of the wall/pier, primarily due to tension. Hence, in Figure 5.10 a maximum principal plastic strain can be observed at the corner due to flexure tension. Similarly, Figure 5.11 through Figure 5.13 shows the progress of maximum principal plastic strains at diagonal due the tension.
64
Observing mises stresses and plastic strains at each level, it can also be noted that the areas with zero plastic strain values are having the minimum level of compressive stress in the contour, since the system is under compression.
Figure 5-6: Mises Stress at Flexural Crack Limit State
Figure 5-7: Mises Stress at Diagonal Crack Limit State
65
Figure 5-8: Mises Stress at Maximum Resistance
Figure 5-9: Mises Stress at Ultimate Limit State
66
Figure 5-10: Max. Principal Plastic Strains at Flexural Crack Limit State
Figure 5-11: Max. Principal Plastic Strains at Diagonal Crack Limit State
67
Figure 5-12: Max. Principal Plastic Strains at Maximum Resistance
Figure 5-13: Max. Principal Plastic Strains at Ultimate Limit State
68
5.4.
DAMAGE VISUALIZATION This section shows the progress of damage at each level of performance
which can be observed in Figure 5.14 through Figure 5.18. One can also relate each level to the stress and strain condition discussed in the previous section. Since masonry is known to be weak in tension, a satisfactory visualization of damage in tension at ultimate level is observed for the model. It starts with damage at one corner of the wall where there is tension produced due to flexure and is shown in the Figure 5.14. As the analysis progress to its end, a pattern of flexural cracks can be observed at the alternate corners of the wall. Figure 5.15 to Figure 5.17 shows the progress of tensile damage at the diagonal. Masonry being strong in compression, a very fewer damage can be noticed at the ultimate level as shown in the Figure 5.18, where rushing at the toe can also be visualized up to some extent.
69
Figure 5-14: Tension Damage at Flexural Crack Limit State
Figure 5-15: Tension Damage at Diagonal Crack Limit State
70
Figure 5-16: Tension Damage at Maximum Resistance
Figure 5-17: Tension Damage at Ultimate Limit State
71
Figure 5-18: Compression Damage at Ultimate Limit State
72
5.5.
VISUAL COMPARISON Specimen PIc tested by Javed (2008) showed significant diagonal cracks as
well as vertical cracks in the middle during last stages of the cyclic test which can be seen in Figure 5.19. It was a core purpose of this study to numerically represent these damages for a laterally loaded brick masonry wall/pier, however cyclic tests couldn’t be performed on the model due to the limitations of the used material model in abaqus. But still there is a good visual representation of diagonal and vertical damages in the studied numerical model that is loaded unilaterally. The discussed damages occurs primarily due to tension, therefore this can be observed in Figure 5.17 for comparison, which shows tension damage of the numerical model at ultimate stage.
Figure 5-19: Pier PIc at the end of testing (Javed (2008))
73
Javed (2008) also noted typical crushing and spalling during final stages of the test which primarily occur due to compression and is shown in Figure 5.20. Interestingly, the studied numerical model for compression damage also showed the same behavior at ultimate limit state which can be observed in Figure 2.18.
Figure 5-20: Typical crushing and spalling of masonry during final stages of testing in the pier PIc
74
5.6.
TOTAL ENERGY OF THE OUTPUT The default history output consists of all the global energies for the whole
model. Here only the Total Energy versus time of the displacement controlled step for the system is presented in the Figure 5.20, for which the energy balance equation in Abaqus/Explicit is written as: ( 5.1 ) Where, = the internal energy (elastic, inelastic, “artificial” strain energy), = the energy absorbed by viscous dissipation, = the frictional dissipation energy of the model, = kinetic energy of the model, = the work done due to external forces on the model, = the total energy in the system. 250
Total Energy (N.m)
200
150
100
50
0 0
200
400
600
800
1000
1200
Displ. Controlled Step Time (sec) Figure 5-21: Total Energy of the Output for Displacement Controlled Step Only
75
6. CONCLUSIONS AND RECOMMENDATIONS 6.1.
CONCLUSIONS & RECOMMENDATIONS
Since past few the numerical study is becoming imperative alongside of the experimental work in this modern age of research and is almost considered incomplete without it. Therefore this numerical study was carried out on already experimented wall/pier by Javed (2008) and its load-displacement curve was compared at different limit states.
For this study, it was necessary to calibrate the numerical model elastically with the experimental results by using an empirical equation and hence the elastic modulus of 4070.39MPa was concluded.
The inelastic model used was Concrete Damaged Plasticity that can visualized the damage in both compression and tension. The plasticity parameters were determined from trial tests starting from the default values, whereas the compression and tension parameters were determined from the mathematical expressions.
The material in the FEA (Finite Element Analysis) model is assumed as isotropic one, while in the reality masonry wall is a composite of masonry units and mortar (anisotropic). However the results acquired were found to be in a good agreement with the experimental results.
The material assigned is considered isotropic, which is not the case in reality. Actually masonry is a composite of bricks and mortar and
76
therefore has to be modeled as anisotropic for more realistic numerical modeling.
For anisotropy, there can be three methods to adopt, o First one is the anisotropic homogenous method in which two different material properties are input in horizontal and vertical direction. Abaqus has anisotropy for elastic behavior only while for inelastic behavior it is recommended to use a user defined material compiled in FORTRAN. o
Second one is to build the model brick by brick and bind them using a cohesive material with negligible thickness or an interface. Here half the thickness of the mortar is added to every dimension of the brick. This method is highly sophisticated and mostly recommended by various researchers.
o Another method is to build the model brick by brick with a given thickness of mortar between the bricks. This method is too much time costly and is not recommended for larger model like walls/piers. However this method could be helpful to model masonry at micro level for homogenization.
The procedure was not found helpful in case of cyclic loading and therefore it is recommended for future study to probe into the material modeling for cyclic load using a FORTRAN compiler.
The study may be extended further to a more detailed level of experimentation, since many of the material parameters were assumed and concluded from empirical expressions. The empirical formulas used also needs to be standardized and thus require more experimental work.
77
78
7 REFERENCES 1. Binda, L., Fontana, A. & Frigerio, G. (1988), Mechanical Behavior of Brick Masonries derived from Unit and Mortar Characteristics, in `Proc. 8 Int. Brick and Block Masonry Conf.', eds. J.W. de Courcy, Elsevier Applied Science, London, England, pp. 205-216. 2. Rots, J. G. (1988), Computational Modeling of Concrete Fracture, PhD thesis, Delft University of Technology, Delft, The Netherlands. 3. Backes, H. P. (1985), Behavior of Masonry Under Tension in the Direction of the Bed Joints, PhD thesis, Aachen University of Technology, Aachen, Germany. 4. Schubert, P. (1988), Compressive and Tensile Strength of Masonry. In: procc. 8th International Brick and Block Masonry Conference, Ireland, 1988, pp. 406-419. 5. Pluijm, R.V. (1992), Material Properties of Masonry and its Components Under Tension and Shear, in 6th Proc. Canadian Masonry Symposium', eds. V.V. Neis, Saskatoon, Saskatchewan, Canada, pp. 675-686. 6. Page, A. W. (1981), `The Biaxial Compressive Strength of Brick Masonry', Proc. Intsn. Civ. Engrs. 2(71), 893-906. 7. Page, A. W. (1983), `The Strength of Brick Masonry Under Biaxial CompressionTension', Int. J. Masonry Constr. 3(1), 26-31. 8. Dhanasekar, M., Page, A. W. & Kleeman, P. W. (1985), `The Failure of Brick Masonry Under Biaxial Stresses', Proc. Instn. Civ. Engrs. 2(79), 295-313. 9. Lourenco, P.B. (1996), Computational Strategies for Masonry Structures, PhD thesis, Delft University of Technology, Delft, The Netherlands. 10. van Noort J.R. (2012), Computational Modeling of Masonry Structures, Master thesis, Delft University of Technology, Delft, The Netherlands. 11. Sahlin, S. (1971), Structural Masonry, Prentice-Hall Inc., Englewood Cliffs, New Jersey, 290 p. 79
12. Paulay, T. and Priestley, M.J.N. (1992), Seismic Design of Reinforced Concrete and Masonry Buildings, John Wiley & Sons, Inc., 744 p 13. Drysdale, R.G., Hamid, A.A. and Baker, L.R. (1994), Masonry Structures, Behavior and Design, Prentice Hall Inc., Englewood Cliffs, 784 p 14. D’Ayala, D.F. (1997), Numerical Modelling of Masonry Structures Reinforced or Repaired. In: Proc., Computer Methods in Structural Masonry-4, 3-5 September, Italy, 1997, pp.161-168. 15. Hemant B, Kaushik, Rai D.C., and Jain S.K. (2007), “Uniaxial Compressive Stress-Strain Model for Clay Brick Masonry”, Current Science, 92(4), Indian Academy of Sciences, Bangalore, India, 25 February 2007, pp. 497-501. 16. Hendry, A.W., Sinha, B.P and Davies, S.R. (1997), Design of Masonry Structures, Third Edition of Load Bearing Brickwork Design, E&FN Spoon, UK 17. Tomazevic, M. (1999), Earthquake-Resistant Design of Masonry Buildings, Series on Innovation in Structures and Construction, Volume I, Chapter 3, Masonry Materials and Construction Systems, Imperial College Press 18. Wijanto, L.S. (2007), Seismic Assessment of Unreinforced Masonry Walls, PhD thesis, University of Canterbury, Christchurch, New Zealand 19. Sinha, B.P. and Pedreschi, R. (1983), Compressive Strength and some Elastic Properties of Brickwork. In: The International Journal of Masonry Construction, 3(1983), No. 1. 20. Crisafulli, F.J., Carr, A.J. and Park, R. (1995), Shear Strength of Unreinforced Masonry Panels, Proceedings of The Pacific Conference on Earthquake Engineering, Volume 3, Melbourne, Australia, pp 77-86 21. Drysdale, R.G., Hamid, A.A., and Baker, L.R. (1994), Masonry Structures, Behavior and Design, Prentice Hall Inc., Englewood Cliffs, 784 p
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22. Belardi, A. Zhang, L. and Thomas, T.C. (1996), Constitutive Laws of Reinforced Concrete Membrane Elements. The 11th World Conference on Earthquake Engineering, paper No. 1208. 23. Ghiassi, B. Soltani, M. and Tasnimi, A.A. (2012), A Simplified Model For Analysis Of Unreinforced Masonry Shear Walls Under Combined Axial, Shear And Flexural Loading, The International Journal of Engineering Structures, Elsevier, Volume 42, September 2012, Pages 396–409 24. Lubliner, J. Oliver, J. Oller, S. and Onate, E. (1989), A Plastic-Damage Model for Concrete, International Journal of Solids and Structures, vol. 25, no. 3, pp. 229326. 25. Lee, J. and Fenves, G.L. (1998), A Plastic-Damage Model for Cyclic Loading of Concrete Structures. Journal of Engineering Mechanics, vol. 124, no.8, pp. 892900. 26. Javed, M. (2008), Siesmic Risk Assesment of Unreinforced Brick Masonry Buildings System of Northern Pakistan, Ph.D. Thesis, University of Engineering and Technology, Peshawar, Khyber Pakhtunkhwa, Pakistan 27. Nationales Anwendungs Dokument (NAD), Richtlinie zur Anwendung von DIN1996-1-1,
Eurocode
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Bemessung
und
Konstruktion
von
Mauerwerksbauten Teil 1-1: Allgemeine Regeln; Regeln für bewehrtes und unbewehrtes Mauerwerk, 1997. 28. ASTM, 2000, Standard Test Method for Measurement of Masonry Flexural Bond Strength, ASTM C1072-2000, West Conshohocken, Pa. 29. ASTM, 1997, Test Method for Young’s Modulus, Tangent Modulus, and Chord Modulus, ASTM E111-1997, West Conshohocken, Pa. 30. Federal Emergency Management Agency, 1997, NEHRP Commentary on the Guidelines for The Seismic Rehabilitation of Buildings, FEMA Publication 274, Washington D.C. 81
31. NEHRP 2000, Recommended Provisions for Seismic Regulations for New Buildings and Other Structures, 2000 edition 32. Federal Emergency Management Agency, 1997, NEHRP Guidelines for The Seismic Rehabilitation of Buildings, FEMA Publication 273, Washington D.C. 33. D.S. Simulia, Abaqus/CAE User’s Guide 34. D.S. Simulia, Abaqus Analysis User’s Guide 35. D.S. Simulia, Abaqus Theory Guide 36. D.S. Simulia, Abaqus Keywords Reference Guide
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