Computational Modeling of Irregular Masonry Failure

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Computational Modeling of Irregular Masonry Failure

Erasmus Mundus Programme

ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS i

DECLARATION

Name: Rony Betzer

Email: [email protected]

Title of the

Msc Dissertation:


Supervisor(s): Prof. Ing. Milan Jirsek, DrSc. & Doc. Ing. Jan Zeman, Ph.D.

Year: 2014

I hereby declare that all information in this document has been obtained and presented in accordance

with academic rules and ethical conduct. I also declare that, as required by these rules and conduct, I

have fully cited and referenced all material and results that are not original to this work.

I hereby declare that the MSc Consortium responsible for the Advanced Masters in Structural Analysis

of Monuments and Historical Constructions is allowed to store and make available electronically thepresent MSc Dissertation.

University: Czech Technical University

Date: 21.07.2014

Signature:

___________________________


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ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS iii

To my family


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ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS v

ACKNOWLEDGEMENTS

First and foremost I wish to express my deepest appreciation and gratitude to my supervisors, Prof.Milan Jirsek and Prof. Jan Zeman, who have supported, guided and encouraged me throughout my

thesis with their impeccable knowledge and unmatched patience. One could not wish for better or

friendlier supervisors.

I must also acknowledge my thanks and appreciation to Dr. Petr Havlsek for the technical support

and suggestions.

I would like to thank Prof. Paulo Loureno, Prof. Pere Roca, Prof. Claudio Modena, Prof. Petr Kabele,

Prof. Daniel Oliveira and Prof. Lus Ramos for their professional and personal commitment to the

SAHC program.

A very special thanks goes to Eng. Yaacov Schaffer and Eng. Meir Ronen, for their continued support

and mentoring.

My sincere gratitude goes to the Erasmus Mundus program and the MSc consortium for the generous

scholarship. I am truly fortunate to have had this opportunity.

Last but not least, I wish to thank my extended international family of SAHC students, for being the

highlight of this amazing year.


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ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS vii

ABSTRACT

The study discusses the potential influence of material and geometrical randomness on the behaviorof masonry walls. Accumulated knowledge and modeling approaches as well as documented

experimental studies from the literature are considered in the process of definition, validation and

calibration of the adopted finite element models. A typical stones and joints arrangement of an existing

masonry wall is defined as reference geometry. Statistical data are then collected by performing

numerical simulations of compression and shear tests, taking into account the natural randomness of

the various material properties on both meso and macro scale, as well as the 'semi-randomness' of

the geometrical parameters, which are mainly dominated by construction technology, available

building materials and somewhat by the masons' building technique. These parameters are randomly

generated in these simulations, while complying with the overall arrangement of the reference

geometry.

The objectives of the research are to characterize, depict and quantitatively investigate the structural

behavior of masonry walls, focusing on the possible sensitivity to random material and geometrical

factors on both local and global scales. For the in-plane loading conditions considered, the results

indicate that random distribution of material properties has fairly limited influence on global behavior,

whereas geometrical arrangement is dominant in the failure mechanism triggered and in post-failure

behavior.

Keywords: Masonry; random; stochastic; failure; simulation


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ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS ix

NUMERICK MODELOVN PORUEN NEPRAVIDELNHO ZDIVA

ABSTRAKT

Tato studie pojednv o monm vlivu nhodnho charakteru materilovch a geometrickch

vlastnost na chovn zd nch st n. P i tvorb , ov en a kalibraci pouitch kone nprvkovch

model se vyuv znalost, model a experimentlnch studi p evzatch z literatury. Typick

uspo dn kamen a spr ve skute n zd n st n je definovno jako referen n geometrie. Pot

jsou shromd na statistick data zaloen na numerickch simulacch tlakovch a smykovch

zkouek, kter berou v vahu p irozenou nhodnost r znch materilovch vlastnost v mezo- a

makrom tku, jako i ste nou nhodnost geometrickch parametr , kter jsou ovlivn ny zejmna

technologi vstavby, dostupnmi stavebnmi materily a do jist mry i specifickou stavebn technikou

konkrtnch zednk . Tyto parametry jsou v simulacch nhodn generovny, p i respektovncelkovho uspo dn referen n geometrie.

Clem provedenho vzkumu je charakterizovat, zobrazit a kvantitativn prozkoumat konstruk n

chovn zd nch st n, se zam enm na p padnou citlivost na nhodn materilov a geometrick

faktory na lokln i globln rovni. Vsledky ukazuj, e p i zat ovn ve st ednicov rovin m

nhodn rozloen materilovch vlastnost jen omezen vliv na globln chovn, zatmco

geometrick uspo dn je pro vznikl mechanismus poruen a pro chovn v postkritickm reimu

rozhodujc.

Klov slova: Zdivo; nhodnost; stochastick; poruen; simulace


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ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS xi

. ,

. .

,

, , '-' ,

, .

, .

, ,

. ,

, .

:


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ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS xiii

TABLE OF CONTENTS

1. INTRODUCTION ............................................................................................................................. 1

2. ADOPTED MESO MODELING ....................................................................................................... 5

2.1 Finite element mesh ................................................................................................................ 5

2.2 Constitutive material models ................................................................................................... 7

2.3 Modeling of uncertainties ...................................................................................................... 12

3. FEM SIMULATIONS AND RESULTS ........................................................................................... 15

3.1 Deterministic validation model ............................................................................................... 15

3.2 Random input models ........................................................................................................... 18 3.3 Random geometry models .................................................................................................... 28

3.4 Periodic geometry models ..................................................................................................... 36

4. CONCLUSIONS ............................................................................................................................ 47

REFERENCES ...................................................................................................................................... 49


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LIST OF TABLES

Table 1 Input parameter of SBETA material......................................................................................... 7

Table 2 Input parameter of Interface material .................................................................................... 11

Table 3 Input material parameters of the validation model ................................................................ 16

Table 4 - Elastic and inelastic properties of the material models .......................................................... 18

Table 5 Mean and standard deviation of random fields in material sample DG2RB1 ....................... 19

Table 6 Input variables of material properties in simulations 1-30 ..................................................... 20

Table 7 Input variables of material properties in simulations 31-41 ................................................... 20

Table 8 Input variables of material properties in simulations 42-47 ................................................... 23

Table 9 Results summary for simulations performed with a vertical load equal to -80 kN ................. 29

Table 10 Results summary for simulations 61-66 .............................................................................. 35

Table 11 Results summary for simulations 67 and 69 ....................................................................... 37

Table 12 Results summary for simulations 68 and 70 ....................................................................... 37


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ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS xv

LIST OF FIGURES

Figure 1 Modeling strategies for masonry structures: (a) detailed micro-modeling; (b) simplifiedmicro-modeling; and (c) macro-modeling, Loureno (2002). .......................................................... 1

Figure 2 Masonry wall defined as reference geometry, Cavalagli, Cluni and Gusella (2013). ............ 3

Figure 3 Blocks and interfaces representation of reference geometry. ............................................... 3

Figure 4 Representative 1mx1m specimens from reference geometry. .............................................. 3

Figure 5 Geometry of CCISOQUAD quadrilateral element, adapted from ! ervenka, Jendele and! ervenka (2013). ............................................................................................................................. 5

Figure 6 Geometry of CCISOGAP 2D interface element, ! ervenka, Jendele and ! ervenka (2013). . 6

Figure 7 Uniaxial stress-stain law, adapted from ! ervenka, Jendele and ! ervenka (2013) ............... 8

Figure 8 Exponential crack opening law, ! ervenka, Jendele and ! ervenka (2013) ........................... 8

Figure 9 Biaxial failure function, adapted from ! ervenka, Jendele and ! ervenka (2013) .................. 9

Figure 10 Interface element failure surface, adapted from ! ervenka, Jendele and ! ervenka (2013)

....................................................................................................................................................... 11

Figure 11 Division of the variable domain into intervals, Novk, Tepl, Kerner and Vo echovsk

(2002) ............................................................................................................................................ 12

Figure 14 Geometry and schematic loading arrangement, Loureno, Oliveira, Roca and Ordua

(2005) ............................................................................................................................................ 15

Figure 15 Geometry and finite element mesh of the validation model ............................................... 15

Figure 16 Load-displacement diagrams, Loureno, Oliveira, Roca and Ordua (2005) on the left,

validation model on the right ......................................................................................................... 16


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Figure 17 Principal compressive stresses (N/mm2) depicted on the incremental deformed mesh for a

horizontal displacement equal to (mm): (a) 1.0; (b) 2.0; (c) 3.0; and (d) 15.0, Loureno, Oliveira,

Roca and Ordua (2005). .............................................................................................................. 17

Figure 18 Principal compressive stresses (N/mm2) depicted on the deformed mesh of the validation

model for a horizontal displacement equal to (mm): (a) 1.0; (b) 2.0; (c) 3.0; and (d) 15.0 ............ 17

Figure 19 Random fields of material sample DG2RB1 ...................................................................... 19

Figure 20 Load-displacement diagram for simulations 1-30 .............................................................. 21

Figure 21 Load-displacement diagram for simulations 31-41 ............................................................ 21

Figure 22 Load-displacement diagram for simulations 1 and 31 ....................................................... 22


Figure 24 Load-displacement diagram for simulations 1 and 42 ....................................................... 24

Figure 25 Principal compressive stresses (N/mm2) depicted on the deformed mesh of simulation

model 1 for a horizontal displacement equal to 0.2 mm ................................................................ 25


model 42 for a horizontal displacement equal to 0.2 mm .............................................................. 25









Figure 31 Load-displacement diagram for simulations 1, 42 and 48-60 ............................................ 28

Figure 32 Load-displacement diagram for simulations 53 and 55 ..................................................... 30




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ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS xvii








model 53 for a horizontal displacement equal to 1.45 mm............................................................ 33


model 55 for a horizontal displacement equal to 1.45 mm............................................................ 33


Figure 40 Peak response versus applied vertical load for samples RG1-RG6 .................................. 35

Figure 41 Geometries of the stack bond and running bond models .................................................. 36

Figure 42 Load-displacement diagram for simulations 1, 42, 48, 67 and 69 ..................................... 37

Figure 43 Load-displacement diagram for simulations 61, 63, 68 and 70 ......................................... 38














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ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 1

1. INTRODUCTION

"I need to discuss science vs. engineering. Put glibly: In science if you know what you are doing youshould not be doing it. In engineering if you do not know what you are doing you should not be doingit. Of course, you seldom, if ever, see either pure state." (Richard W. Hamming, 1915-1998)

Numerical models of mechanical systems may be described as mathematical idealizations of theirphysical characteristics. They assume geometry, loads and material properties as well as thegoverning equations that link these variables with the response variables of the system(displacements, strains, stresses). The finite element method provides a powerful tool of numericalsolutions to such boundary value problems. Coupled with advanced formulations of constitutive laws,approximations can be achieved with high level of accuracy for engineering purposes.

The three main approaches to the modeling of masonry as an anisotropic material are detailed micro-modeling, simplified micro-modeling and macro-modeling. In the first approach, units and mortar jointsare represented by continuum elements whereas the unit-mortar interface is represented by adiscontinuum element. In the second approach, the mortar joints and the unit-mortar interface arelumped in discontinuum elements and the units are represented by continuum elements andexpanded to keep the geometry unchanged. In the third approach, units, mortar and unit-mortar

interface are smeared out in a homogenous continuum. The approaches differ in implementedassumptions and in complexity, which is linked to the cost of the numerical computation, the accuracyand the validity range, all of which should be adequately considered and prioritized when choosing amodeling strategy, depending on the desired performance scale. For most practice orientedapplications, macro-modeling is considered efficient whereas micro-modeling is considered moresuitable for smaller scale applications, where individual units and joints and their interaction are ofinterest. (Loureno 2002).

Figure 1 Modeling strategies for masonry structures: (a) detailed micro-modeling; (b) simplifiedmicro-modeling; and (c) macro-modeling, Loureno (2002).


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2 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS

One of the remaining challenges lies in the inherent randomness of material properties in thestructural components. The significance of both natural material randomness and manmadegeometrical arrangement has been subject to comprehensive research during the past few decades.

In the case of masonry structures in particular, many related studies focus on formulating descriptiveand optimal computational solutions to these 'built-in' yet statistically foreseeable imperfections, mostlywithin the framework of homogenization approaches to the definition of equivalent material properties(Zucchini & Loureno 2004) and equivalent periodic representations of irregular masonry (Spence,Gioffr & Grigoriu 2008).

In this study, these uncertainties are factored back into the models by means of stochastic orprobabilistic mechanics, with the objective of assessing the effects of randomness, as well as theaccuracy of the averaged deterministic approaches commonly used in civil engineering. Specifically,displacement controlled simulations of compression and shear tests are performed on masonry wallspecimens. An initial geometry is taken from an existing wall and a representative finite element modelis compiled as a 2D mesh, consisting of blocks and interfaces (simplified micro-modeling). Once themodel is validated by comparing simulation results with documented laboratory experiments, threetypes of assumptions are made in subsequent simulations: (i) deterministic material properties, (ii)stochastic material properties expressed as discretized random fields and (iii) probabilistic distributionof material properties expressed as random input variables per masonry block. A fourth type ofsimulation is performed assuming deterministic material properties and randomly generated geometry,

true to the reference model in terms of statistical distributions of the geometrical parameters.Simulations are repeated with a variety of isolated or coupled modified fields and variables in order toestimate the sensitivity of the structural behavior to each parameter.


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Figure 2 Masonry wall defined as reference geometry, Cavalagli, Cluni and Gusella (2013).

Figure 3 Blocks and interfaces representation of reference geometry.

Figure 4 Representative 1mX1m specimens from reference geometry.


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2. ADOPTED MESO MODELING

The FEM simulations are modeled inATENA nonlinear finite element software. The generalmathematical formulation of the problem follows a classical energetic approach. The principle of virtualwork is applied in its virtual displacement variation. Equilibrium and boundary condition equations areexpressed in an incremental form, where the structural response up to timet is assumed known and aload is applied att+ D t. Linearization is performed by neglecting 2nd order terms of the nonlinear strain

increment, arriving at the general governing equations balancing virtual internal work with work doneby external forces. The domain is decomposed by applying the Finite Element Method. Thediscretized displacement field is approximated for each element at each load increment and the nodal

displacements are determined from a system of nonlinear algebraic equations solved by an iterativeNewton-Raphson Method solver. Detailed numerical assumptions and formulations of the process andof the incorporated nonlinear material models and elements are found inATENA ProgramDocumentation.

2.1 Finite element mesh

The modeled finite element mesh consists ofCCIsoQuad plane stress quadrilateral isoparametricelements and CCIsoGap 2D interface elements, representing masonry blocks and joints, respectively.

Figure 5 Geometry ofCCIsoQuad quadrilateral element, adapted fromervenka, Jendele andervenka (2013).

The CCIsoQuad plane stress isoparametric element is bilinearly interpolated over its area and theintegrals arising in the finite element method are approximated using four Gauss integration points.The interpolation functions at each point are expressed in the local coordinate systemr,s :

(1)Node i 1 2 3 4

Function h i 1

4(1+r)(1+s)

1

4 (1-r)(1+s)

1

4 (1-r)(1-s)

1

4 (1+r)(1-s)


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The CCIsoGap 2D interface element is a linear approximation derived from the isoparametric element.It is defined by a pair of gapped lines, each located on opposing side of the interface. The degrees offreedom defined for the element are the relative displacementsD u, D v , expressed in the local

coordinate system, which is aligned with the gap direction.

Figure 6 Geometry ofCCIsoGap 2D interface element,ervenka, Jendele and ervenka (2013).

The initial state of the gap is closed, allowing full contact interaction. In addition, friction sliding ispossible within the gap. In open state, there is no contact between the lines. This behavior is modeledby employing a penalty method. A constitutive matrix of the interface is defined:

0

0tt

nn

F K uF Du

F K v

= = =

(2)

where D u, D v are the relative displacements of the interface sides (sliding and opening),K tt , K nn are

the shear and normal stiffness, respectively andF , F are the tractions at the interface. All values are

defined in the local coordinate systemr, s .The relative displacementsD u, D v are derived:

(3)

A numerical integration in two Gauss points is used to integrate the interface element stiffness matrix.

The stiffness coefficients depend on the gap state. The interface is considered open if the normal

1 2

2 1,4 1 2,3

2 1,4 1 2,3

1

1

2

1 2 2 1 2

1 2 2 1 3

3

4

4

1 1(1 ), (1 )

2 2

0 0 0 0

0 0 0 0

h r h r

h u h uuu

h v h vv

u

v

u

h h h h vu Bu

h h h h u

v

u

v

= + =

+ = = +

= =


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stress exceeds the interface tensile strength, in which case a stress free constitutive law is considered(F i =0) and the stiffness is set to a small, but nonzero value. The local stiffness matrix and internaltractions vector are transformed into the global coordinate system and then assembled in the general

problem governing equations.

2.2 Constitutive material models

The damage-based SBETA constitutive model, in which cracks are computed using a smearedapproach, is assigned to the block elements. The properties defined for a material point are validwithin a certain material volume, associated with a Gauss integration point. The mathematicalformulation is considered in the plane stress state. The principal directions of the computed stressesand strains may be identical or different, depending on the state of the material (un-cracked or

cracked). The input parameters used to construct the constitutive model are given in Table 1.

Table 1 Input parameter ofSBETA material

In the simulations described in Chapter 3, these input variables are defined as either deterministic orrandom and assigned values accordingly. The following effects of the material behavior are accountedfor by the listed input variables and relations formulations of the constitutive model:

nonlinear behavior in compression including softening tensile fracturing based on nonlinear fracture mechanics biaxial strength failure criterion reduction of compressive strength after cracking tensile softening effect rotated crack direction model

The material stiffness matrix is defined by the elastic constants derived from a stress-strain functionrepresenting an equivalent uniaxial law, where different laws are used for loading and unloading,allowing energy dissipation. The nonlinear behavior in the biaxial stress state is described by means ofa so-called effective stress ef and the equivalent uniaxial strain eq .

Parameter DefinitionE Elastic modulus Poisson's ratiof t Tensile strengthf c Compressive strengthG f Specific fracture energy c Compressive strain at uniaxial compressive strength

c Compressive strength reduction factor due to cracksw d Critical compressive displacement Specific material weight


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eq i

i E

= (4)

The effective stress is in most cases a principal stress. The equivalent uniaxial strain is introduced inorder to eliminate the Poissons effect in the plane stress state and may be considered as the strainthat would be produced by the stress i in a uniaxial test with modulusE i associated with the directioni . Peak compressive and tensile stresses reflect biaxial stress state.

Figure 7 Uniaxial stress-stain law, adapted fromervenka, Jendele and ervenka (2013)

The tangent modulusE t is used in the material stiffness matrixD to construct an element stiffnessmatrix for the iterative solution. For numerical reasons,E is set to a minimum positive value near thecompressive peak and tensile softening ranges of the stress-strain curve.

Tensile behavior prior to cracking is assumed linear elastic. Post cracking tensile behavior is based onan exponential crack opening law and fracture energy, derived experimentally by Hordijk (1991) anddefined per unit area of a crack.

Figure 8 Exponential crack opening law,ervenka, Jendele and ervenka (2013)

The particular form of the crack opening law reads:

3

'

'

1 3 exp 6.93 28 exp( 6.93),

5.14

ef t c c c

f c ef

t

w w w f w w w

Gw

f

= +

=

(5)


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where w is the crack opening,w c is the crack opening at the complete release of stress, is thenormal stress in the crack,f t 'ef is the effective tensile strength andG f is the fracture energy . The crackopening displacementw is calculated as total strains normal to the crack direction multiplied by band

size, which is associated with the finite element mesh size and skew.Compressive behavior prior to peak stress is expressed as:

2' 0, ,

1 ( 2)ef ef

cc c

E kx x f x k

k x E

= = =+ (6)

where ef is the compressive stress, f c 'ef is the effective compressive strength, and c are the strain

and strain at peak stress respectively,E 0 is the initial elastic modulus andE c is the secant elasticmodulus at peak stress.Post peak compressive behavior is defined by a linearly descending softening law. In an analogy tothe tensile cracking theory, the shape of the crack opening law is associated with a plasticdisplacement and fracture energy defined and considered material properties. Both tensile andcompression failure bands are introduced in the formulation with the purpose of eliminating finiteelement size and orientation effects and are defined as projections of the finite element dimensions onthe failure planes, assumed normal to the principal stresses. For skewed meshes, band size isincreased with respect to the element orientation angle.

A biaxial stress failure criterion according to Kupfer et al. (1969) is used.

Figure 9 Biaxial failure function, adapted fromervenka, Jendele andervenka (2013)

Cracks are formed in the material when the principal stress exceeds the tensile strength. They areassumed uniformly distributed within the material volume. A rotated crack model is defined, in whichthe directions of the principal stress and principal strain are identical. Therefore, no shear strain occurs

on the crack plane and only two normal stress components are defined.


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Decreased compressive strength due to cracks is expressed by normal strain and reduction factorc

parallel to the cracks:

2(128 )' ' , (1 )ef c c c c f r f r c c e = = + (7)

The material stiffness matrix prior to cracking is defined as elastic isotropic and written in the globalx,y coordinate system:

2

1 0

1 01

10 0

2

E D

=

(8)

where E is the elastic modulus derived from the equivalent uniaxial law and is the Poisson's ratioregarded constant. In the cracked material, the stiffness matrix takes the form of an elastic orthotropicsolid. It is defined in a coordinate system coinciding with crack direction. Local direction 1 is definednormal to the crack and local direction 2 parallel to it. The matrix is derived as an inverse of amanipulated plane stress state flexibility matrix, in which 21= is assumed and symmetry relation

12 E 2= 21 E 1 is used to obtain 1122

E E

= .

The stiffness matrixD L found accordingly in the local coordinate system:

1 1

2 2

21 11

2 2

0

(1 ) 1 0

0 0

L

E E E E

E E D E

E E

G

=

(9)

and is transformed to the global coordinate system and assembled into the global matrix.


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The joints are assigned with an Interface material model which simulates the contact between theblocks and is based on Mohr-Coulomb criterion with tension cut off. The constitutive relation is given interms of tractions on the interface plane and relative sliding and opening displacements (2).

The initial failure surface corresponds to Mohr-Coulomb condition (10). Once stresses violate thiscondition, the surface collapses to a residual dry friction surface.

2 2

0 02 2

2

| | , 0

( )1 , , ,0

( ) 21

( )

0,

c t c t

t c t c

t c

t

c

f c f

f c f

f

f

= = =

= =

(10)

The tensile failure criterion is represented by an ellipsoid intersecting the normal stress axis atf t with avertical tangent and the shear axis atc (i.e. cohesion) with a tangent equivalent to .

Figure 10 Interface element failure surface, adapted fromervenka, Jendele andervenka (2013)

K nn and K tt denote the initial elastic normal and shear stiffness respectively. Theoretically, post failure

stiffness is zero, however, for numerical reasons both are set to a positive value of ~10 00 of the initial

stiffness. The input parameters of the interface material model are listed in Table 2. These parametersare defined as either deterministic or random in the FEM simulations and assigned with values

accordingly.

Table 2 Input parameter ofInterface materialParameter Definition

K nn Normal stiffnessK tt Shear stiffnessf t Tensile strengthC Initial cohesionF Friction coefficient

K nn Minimal normal stiffness for numerical purposes

K tt Minimal shear stiffness for numerical purposes


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2.3 Modeling of uncertainties

The random fluctuations in the mechanical properties of the materials are captured by introducingrandomized input variables in the FEM models. An initial set of deterministic variables is defined,assigned with values representing average physical properties of natural building stones and mortar joints (reduced to 2D interfaces in the models). These values and their probability density functions(PDF) are estimated based on experimental and numerical studies by Oliveira (2003), ejnoha,ejnoha, Zeman, Skora and Vorel (2008) and Naghoj, Youssef and Maaitah (2010). Randomizedinput values are generated using the Latin Hypercube Sampling (LHS) technique. Utilizing thissampling strategy, the need for a large number of samples is avoided, as the range of the probabilitydensity functions of the random variables is divided intoN equivalent intervals, whereN is the numberof simulations. The centroids of the intervals are then used in the simulation process; therefore, the

range of the PDF for each variable is divided intoN intervals of equal probability1/N. The cumulativeprobability density function (CPDF) is used directly for the random sampling:

1,

0.5i k i

k x

N = !

(11)

where x i,k is the k -th sample of thei -th variableX i and F i -1 is the inverse of the CPDF ofX i .

Figure 11 Division of the variable domain into intervals, Novk, Tepl, Kerner and Vo echovsk(2002)

Every interval of each variable is used once in the sampling, resulting in anN by n table, wheren isthe number of variables. Two types of samples are collected implementing this method: (i) randomizedinput variables associated with material properties of the blocks, generated per block and (ii) randomfields of both block and interface material properties, generated per integration point associated with afinite element. Each generated sample of random material properties replaces deterministic input in a

FEM simulation.


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The geometrical characteristics of the reference wall are statistically analyzed and artificiallyreproduced using a set of random parameters, derived from their respective PDF, accounting for theheight and width distribution of the blocks. The length distribution of the bed joints is used as control

criteria to prevent vertical overlap of head joints. If a sample fails the control criteria it is rejected and anew sample is generated. The samples are converted into coordinate input for FEM models, as wellas the two samples taken directly from the actual wall. Compression and shear test simulations areperformed, considering deterministic material properties, thus isolating the effects of the geometricalarrangement. The performed simulations are reported and discussed in Chapter 3.

(a) (b) (c)Figure 12 Estimation of probability density function for the reference wall shown in Figure 2:(a) stones width, (b) stones height, (c) bed joints' length, Cavalagli, Cluni and Gusella (2013).

Figure 13 Randomly generated geometries


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3. FEM SIMULATIONS AND RESULTS

3.1 Deterministic validation model

For the purpose of procedure validation, an initial FEM model is developed, representing a 1m X 1mspecimen from the reference geometry. The model is numerically analyzed with deterministic inputvariables calculated using the same considerations and basic configuration implementation to matchexperimental and simulation studies by Loureno and Ramos (2004) and Loureno, Oliveira, Rocaand Ordua (2005). The compared specimens differ in geometrical arrangement and parameters,which can be characterized as periodic in the experimental study case and quasi-periodic in thereference geometry.

Figure 14 Geometry and schematic loading arrangement, Loureno, Oliveira, Roca and Ordua(2005)

Figure 15 Geometry and finite element mesh of the validation model


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The input variables for the material models are shown in Table 3.

Table 3 Input material parameters of the validation model

While both present similar global structural responses in terms of triggered failure mechanisms andload-displacement behaviors, each specimen exhibits a different stress path and a different post peakbehavior, most likely linked with their respective geometries, which dominate the location of thediagonal cracks propagating through the head and bed joints and the formation of struts.

Figure 16 Load-displacement diagrams, Loureno, Oliveira, Roca and Ordua (2005) on the left,validation model on the right

Material Property Value Unit Definition

Block

E 15500 MPa Young's modulus 0.2 - Poisson's ratiof t 3.7 MPa Tensile strengthf c - 80.3 MPa Compressive strengthG f 110 N/m Specific fracture energy c -3.277E-03 - Compressive strain at uniaxial compressive strengthc 0.8 - Compressive strength reduction factor due to cracks

w d -0.5 mm Critical compressive displacement

2.500E-02 MN/m3 Specific material weight

Interface

K nn 155000 MN/m3 Normal stiffness

K tt 64580 MN/m3 Shear stiffness

f t 0.015 MPa Tensile strengthC 0.019 MPa Initial cohesionF 0.62 - Friction coefficient

K nn MIN 100 MN/m3 Minimal normal stiffness for numerical purposes

K tt MIN 60 MN/m3 Minimal shear stiffness for numerical purposes


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Figure 17 Principal compressive stresses (N/mm2) depicted on the incremental deformed mesh for ahorizontal displacement equal to (mm): (a) 1.0; (b) 2.0; (c) 3.0; and (d) 15.0, Loureno, Oliveira, Roca

and Ordua (2005).

Figure 18 Principal compressive stresses (N/mm2) depicted on the deformed mesh of the validationmodel for a horizontal displacement equal to (mm): (a) 1.0; (b) 2.0; (c) 3.0; and (d) 15.0


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The comparison between the results leads to the development of two random modeling strategies.The first is achieved by performing repetitive simulations, altering material properties input to capturetheir natural physical randomness, whereas the second is realized by performing repetitive simulations

altering the geometrical arrangement which captures manmade irregularity of the masonry.

3.2 Random input models

The model described in the previous article is used in subsequent random input modeling. For thepurpose of this study, the material properties are redefined according to average values for limestonemasonry. The input variables of the block and interface material models are shown in Table 4. Thestatistical distribution around some of these mean values is assumed normal, with a coefficient ofvariation equal to 0.25, while the remaining values are assumed deterministic in the simulations, for anumber of reasons. For example, a numerical input parameter such asK nn MIN has no physical meaningand therefore is not randomized. On the other hand, the specific material weight which certainly is aphysical characteristic is not randomized to avoid changing the self weight load, since randomizedloads are not investigated within the scope of this study.

Table 4 - Elastic and inelastic properties of the material models

In the initial simulation, deterministic values are defined according to Table 4. The self weight load isapplied to the blocks and a compressive distributed vertical load of -80 kN/m is applied on the top ofthe wall. A master slave condition is defined on the top nodes in the vertical direction and a prescribedhorizontal displacement is incrementaly applied. Simulations 2 through 30 are performed with identicalgeometrical and loading configurations, while the input variables are randomized in isolated andcoupled variations as reported in Table 6. An illustration of randomized fields is shown in Figure 19.

Material Property Value Unit Definition

Block

E 35610 MPa Young's modulus 0.178 - Poisson's ratiof t 4 MPa Tensile strengthf c - 40 MPa Compressive strengthG f 80 N/m Specific fracture energy c -2.257E-03 - Compressive strain at uniaxial compressive strengthc 0.8 - Compressive strength reduction factor due to cracks

w d -0.5 mm Critical compressive displacement 2.300E-02 MN/m Specific material weight

Interface

K nn 350000 MN/m Normal stiffnessK tt 150000 MN/m Shear stiffnessf t 0.225 MPa Tensile strengthC 0.35 MPa Initial cohesionF 0.75 - Friction coefficient

K nn 350 MN/m Minimal normal stiffness for numerical purposes

K tt 150 MN/m Minimal shear stiffness for numerical purposes


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E f t f c

Figure 19 Random fields of material sample DG2RB1

Table 5 Mean and standard deviation of random fields in material sample DG2RB1

All samples are defined to mimic natural randomness of mechanical properties, with the exception ofthose generated for simulations 21 through 23, in which the variables are defined to represent aprobabilistic scatter of properties among the stones, but not within them. In each randomized sample,

the mean value of any particular input variable is slightly shifted from the mean of the population, whilethe coefficient of variation may differ significantly, as described in Chapter 2. Nevertheless, theresponses obtained in all simulations are remarkably similar, practically identical before failure. Thisphenomenon is due to the global nature of the mechanical behavior, in which localized materialfailures have lesser influence on the response than the overall geometrical arrangement thatdetermines the triggered mechanisms. However, it has been well established that the vertical loadapplied to a wall plays a significant role in its shear capacity and failure modes (rocking vs. crushing).As the investigated wall exhibits minimal cracks, simulations 31 through 41 are performed with anincreased vertical load equal to -100 kN. The samples used in these simulations are reported in Table7. As expected, the responses computed with the increased vertical load configuration exhibit anelevated horizontal force peak. The peak loads in the latter simulations appear at exactly the samehorizontal displacement as in the former. The responses suggest that the failure modes triggered areidentical for both the lower and the higher stress capacities, which complies with experimentalevidence of correlation between the equivalent stiffness and the applied vertical load.

Field E f t f c Mean 36800 MPa 3.8043 MPa -38.974 MPa

Standard deviation 8436.7 MPa 1.0241 MPa -10.744 MPa


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Table 6 Input variables of material properties in simulations 1-30


Simulation Input variables of material propertiesNo. Sample ID Blocks Joints

1 DG1D Deterministic Deterministic2 DG1RE1 RandomE field Deterministic3 DG1RE2 RandomE field Deterministic4 DG1RE3 RandomE field Deterministic5 DG1RE4 RandomE field Deterministic6 DG1RE5 RandomE field Deterministic7 DG1RK1 Deterministic Random K nn field8 DG1RK2 Deterministic Random K nn field9 DG1RK3 Deterministic Random K nn field10 DG1RK4 Deterministic Random K nn field11 DG1RK5 Deterministic Random K nn field12 DG1RB1 RandomE , f t , f c fields Deterministic13 DG1RB2 RandomE , f t , f c fields Deterministic14 DG1RB3 RandomE , f t , f c fields Deterministic15 DG1RB4 RandomE , f t , f c fields Deterministic16 DG1RB5 RandomE , f t , f c fields Deterministic17 DG1RJ1 Deterministic Random K nn , K tt , f t fields18 DG1RJ2 Deterministic Random K nn , K tt , f t fields19 DG1RJ3 Deterministic Random K nn , K tt , f t fields20 DG1RJ4 Deterministic Random K nn , K tt , f t fields21 DG1RBB1 RandomE , f t , f c input per unit Deterministic22 DG1RBB2 RandomE , f t , f c input per unit Deterministic23 DG1RBB3 RandomE , f t , f c input per unit Deterministic

24 DG1RFC1 Deterministic Random C ,F

, f t fields25 DG1RFC2 Deterministic Random C ,F , f t fields26 DG1RFC3 Deterministic Random C ,F , f t fields27 DG1RFC4 Deterministic Random C ,F , f t fields28 DG1RFC5 Deterministic Random C ,F , f t fields29 DG1RFC6 Deterministic Random C ,F , f t fields30 DG1RFC7 Deterministic Random C ,F , f t fields

Simulation Input variables of material propertiesNo. Sample ID Blocks Joints31 DG1D Deterministic Deterministic32 DG1RE1 RandomE field Deterministic33 DG1RE2 RandomE field Deterministic34 DG1RE3 RandomE field Deterministic35 DG1RE4 RandomE field Deterministic36 DG1RE5 RandomE field Deterministic37 DG1RK1 Deterministic RandomK nn field38 DG1RK2 Deterministic RandomK nn field39 DG1RK3 Deterministic RandomK nn field

40 DG1RK4 Deterministic RandomK nn field41 DG1RK5 Deterministic RandomK nn field


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Figure 20 Load-displacement diagram for simulations 1-30



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For the first loading configuration, a peak response with a median value of 46.88 kN is obtained whenthe prescribed displacement reaches 0.79 mm (median value). At this point, the head joints along thepropagated diagonal crack have all reached failure. Stresses redistribute, as the bed joints along the

diagonal crack are partially in contact. The second peak marks the triggering of the second failuremode. As the bed joints along the diagonal crack fail, the boundary conditions associated with thepartial confinement of the specimen allow the corner bed joints of the crack to establish partial contact,which results in high stress concentrations and eventual crushing. As the secondary failure mode is alocalized one, dominated by the stress concentrations, the scatter in the responses of the differentspecimens is directly linked to the randomness of their mechanical material properties. The weaker thespecimen is at the crucial remaining contact areas, the sooner a crushing mechanism appears, i.e. atlower values of displacement and force. Once the crushing mechanism is activated and the residualavailable stress paths are exhausted, the structure yields. The yielding mechanism may becharacterized as a combined failure mode of rocking and crushing, as the scattered responses appearto converge back into a global rather than local failure mode. A similar behavior is exhibited by theresponses computed for the second loading configuration. Once the primary failure mode is activated,the structure reaches a peak response with a median value of 56.56 kN, at a displacement of 0.8 mm.The secondary failure mode of the specimens is scattered on the load-displacement curve, whiletriggered at slightly lower displacement values with respect to the first configuration, due to the higherstresses. It is noted that for both loading configurations, the responses of the deterministic samplemarked DG1D are of intermediate values in comparison to those of the randomized samples.

Moreover, as the localized crushing mechanism is governed primarily by the material properties of thestone and since the random samples used in the second set of simulations are the same as thoseused in the first set, the order in which the crushing mechanism is triggered in these samples isidentical for both loading configurations.

Figure 22 Load-displacement diagram for simulations 1 and 31


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In the mechanical sense, these results suggest that the random fluctuations of material propertieshave negligible effects on the global response of the structure and are only significant when a weakestlink type of localized failure is dominant, in which case, the performance of a structure with a narrower

scatter of material properties is not necessarily superior, but is more predictable.The last set of samples with randomized input variables is generated and used in simulations 43through 47, as reported in Table 8. A FEM model is compiled using geometrical input derived from asecond specimen of the reference wall, in which the number of blocks (and their mean height) isidentical to the first specimen, while their width (and consequential area) has a higher variation. Thesimulations are performed with a loading configuration identical to the one used in simulations 1through 30. As the expected scatter of the responses is dominated by the material properties of theblocks, the interface material properties are not randomized in these simulations. All six specimensexhibit similar behavior, almost identical before the peak response, after which a low scatter isobserved. The responses of deterministic material model marked DG2D are of intermediate valueswithin said scatter. The peak response has a median value of 50.02 kN, at a median displacementvalue of 1.38 mm.



Simulation Input variables of material propertiesNo. Sample ID Blocks Joints42 DG2D Deterministic Deterministic43 DG2RB1 Random E, f t , f c fields Deterministic44 DG2RB2 Random E, f t , f c fields Deterministic45 DG2RB3 Random E, f t , f c fields Deterministic46 DG2RB4 Random E, f t , f c fields Deterministic47 DG2RB5 Random E, f t , f c fields Deterministic


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Comparing the structural behaviors of the two geometrical specimens, taken from the same referencewall and applied with the same loading, several differences are noted, suggesting that the geometricalarrangement within the wall plays a significant role in the response, while the fluctuations in material

properties play a minor one. In the initial linear phase, no noticeable differences are present, butthroughout the nonlinear phase, each specimen presents a different response, associated with thedifferent failure mechanisms. The first cracks appear in the specimen marked DG1D before the peakload is reached, resulting in a semi brittle behavior, local softening, stress redistribution and combinedfailure modes of rocking and crushing. In the specimen marked DG2D, the first cracks appear after thepeak load is reached, resulting in a more ductile behavior.



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-1.576E+00

-1.395E+00

-1.215E+00

-1.035E+00

-8.550E-01

-6.600E-01

-4.800E-01

-3.000E-01

-1.200E-01

4.751E-02

Figure 25 Principal compressive stresses (N/mm2) depicted on the deformed mesh of simulationmodel 1 for a horizontal displacement equal to 0.2 mm

-1.339E+00

-1.185E+00

-1.005E+00

-8.250E-01

-6.600E-01

-4.800E-01

-3.000E-01

-1.350E-01

4.500E-02

2.013E-01



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-3.528E+00

-3.150E+00

-2.730E+00

-2.310E+00

-1.890E+00

-1.470E+00

-1.050E+00

-6.300E-01

-2.100E-01

1.556E-01


-2.718E+00

-2.430E+00-2.100E+00

-1.770E+00

-1.440E+00-1.140E+00

-8.100E-01

-4.800E-01

-1.500E-011.523E-01



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-8.423E+00

-7.470E+00

-6.480E+00

-5.490E+00

-4.500E+00-3.510E+00

-2.520E+00

-1.530E+00

-5.400E-012.846E-01


-3.987E+00

-3.560E+00

-3.080E+00

-2.600E+00

-2.120E+00

-1.680E+00

-1.200E+00

-7.200E-01

-2.400E-01

1.715E-01



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3.3 Random geometry models

Considering the results obtained in simulations 1 through 47, simulations 48 through 60 are performedby generating random geometries as described in Chapter 2 and keeping the material properties fixedto the values listed in Table 4. As the former simulations demonstrate the effect of the random materialproperties on the structural behavior, the latter are meant to demonstrate the significance of thegeometrical arrangement of the masonry. It is understood that given a fixed set of building stones, afinite number of possible arrangements exists and determining between them is up to the mason. Thegeometries are randomly generated taking into account the statistical distribution of stone dimensionsfrom the reference geometry, as well as incorporating the mason's discretion by applying a controlcriteria derived from the statistical distribution of the bed joint dimensions. The generated 1m X 1mspecimens are applied with the same loading configuration described in the previous article. The

scatter found in the responses of the random geometries is significantly higher than the scatterassociated with the randomness of the material properties and is apparent throughout the load-displacement curve. It is noted that the responses of the samples marked DG1D and DG2D takenfrom the reference geometry are of intermediate values in comparison to those obtained in the randomgeometry simulations.

Figure 31 Load-displacement diagram for simulations 1, 42 and 48-60


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Table 9 Results summary for simulations performed with a vertical load equal to -80 kN

GeometrySimulation Responses for vertical load=-80 kN

No. Sample ID Horizontal disp. at peak load [mm] Peak horizontal load [kN]

Referencewall

specimen#1

1 DG1D 0.80 46.882 DG1RE1 0.80 46.853 DG1RE2 0.80 46.884 DG1RE3 0.80 46.825 DG1RE4 0.80 46.876 DG1RE5 0.75 46.697 DG1RK1 0.80 46.898 DG1RK2 0.80 46.889 DG1RK3 0.80 46.88

10 DG1RK4 0.80 46.8811 DG1RK5 0.80 46.8812 DG1RB1 0.80 46.8113 DG1RB2 0.75 46.8214 DG1RB3 0.80 46.8415 DG1RB4 0.70 47.0516 DG1RB5 0.70 46.7517 DG1RJ1 0.80 46.8818 DG1RJ2 0.80 46.8719 DG1RJ3 0.80 46.8720 DG1RJ4 0.80 46.8821 DG1RBB1 0.80 46.8722 DG1RBB2 0.80 46.8823 DG1RBB3 0.80 46.8724 DG1RFC1 0.80 46.91

25 DG1RFC2 0.80 46.9126 DG1RFC3 0.80 46.9127 DG1RFC4 0.80 46.9128 DG1RFC5 0.80 46.9129 DG1RFC6 0.80 46.9130 DG1RFC7 0.80 46.91

Referencewall

specimen#2

42 DG2D 1.35 50.0043 DG2RB1 1.35 49.8144 DG2RB2 1.35 49.8845 DG2RB3 1.40 50.0346 DG2RB4 1.40 50.1747 DG2RB5 1.40 50.44

Random 48 RG1 1.13 46.98Random 49 RG2 0.80 52.90Random 50 RG3 0.85 44.41Random 51 RG4 0.70 43.45Random 52 RG5 0.70 44.64Random 53 RG6 1.49 57.20Random 54 RG7 1.01 51.00Random 55 RG8 1.45 45.63Random 56 RG9 1.09 46.52Random 57 RG10 0.65 42.17Random 58 RG11 1.05 47.72Random 59 RG12 0.80 46.82Random 60 RG13 0.80 47.56


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Samples RG6 and RG8 (simulations 53 and 55, respectively) both exhibit high values in terms of thedisplacement corresponding to the peak response (1.49 mm and 1.45 mm respectively), but while thepeak response obtained by sample RG6 has the highest value (57.20 kN), the peak response value of

sample RG8 (45.63 kN) is lower than the median value (46.82 kN). In comparison with the othersamples, the geometrical assemblies of both these samples sufficiently allow diagonal struts to formthrough partial contact in the joints. However, in sample RG6 there are larger stones located in thediagonal path compared to those in sample RG8 and therefore these paths are exhausted at a muchlater stage in comparison, postponing the formation of cracks in the stones. These occurrences lead tothe high deformability of both samples as well as to the high equivalent stiffness and load capacity ofsample RG6 and the corresponding lower values of sample RG8.



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-5.046E+00-4.510E+00

-3.905E+00

-3.300E+00

-2.695E+00-2.090E+00

-1.485E+00

-8.800E-01

-2.750E-012.723E-01



-6.200E+00

-5.525E+00

-4.810E+00

-4.030E+00

-3.315E+00

-2.600E+00

-1.820E+00

-1.105E+00

-3.900E-01

2.867E-01


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-5.564E+00

-4.920E+00

-4.260E+00

-3.600E+00

-2.940E+00-2.280E+00

-1.620E+00

-9.600E-01

-3.000E-013.018E-01


model 53 for a horizontal displacement equal to 1.45 mm


-7.706E+00

-6.880E+00

-6.000E+00

-5.040E+00

-4.160E+00

-3.280E+00

-2.320E+00

-1.440E+00

-5.600E-01

2.559E-01


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The next set of simulations (61-66) is performed using samples RG1 through RG6 with an increasedcompressive load equal to -200 kN. As the influence of the geometrical arrangement is underinvestigation, it is important to establish whether the scatter in the responses is somehow linked to the

loads and to what extent. It is noted that the increase in the peak horizontal response and thecorresponding displacement was more or less predictable, meaning that each sample performed aswas expected, with the exception of sample RG6, which while obtained the expected peak response,had no significant increase in the displacement corresponding to it. Although the limited number ofsimulations renders these results inconclusive, it is decided to proceed to the final modeling approachfor further validation.



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Table 10 Results summary for simulations 61-66

Figure 40 Peak response versus applied vertical load for samples RG1-RG6

Simulation Responses for vertical load=-200 kNNo. Sample ID Horizontal disp. at peak load [mm] Peak horizontal load [kN]

61 RG1 2.60 103.7062 RG2 1.65 112.1063 RG3 1.30 94.9264 RG4 1.30 99.6165 RG5 1.93 102.0066 RG6 1.75 124.10


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3.4 Periodic geometry models

To demonstrate the significance of the geometrical arrangement, the final set of simulations isperformed by placing a set of averaged sized blocks in what is considered a poor arrangement,meaning that the head joints are aligned throughout the height of the masonry wall (stack bond). Thisoverlapping is expected to reduce the capacity of the wall and bring to an early onset of failure, evencompared to the lowest results obtained in former simulations. Then, the blocks are rearranged in theoptimal running bond manner, meaning that the bed joints are all the same length and no head jointsoverlap. It is important to emphasize that the reference wall has an irregular geometry that iscomparable to a running bond, in which each stone is in contact with six adjacent stones and not to astack bond, in which each stone is in contact with four adjacent stones. Taking that into account,

simulations 67-68, performed with the sample marked SBOND, are meant to represent an extremesituation that is not applicable to the reference wall, while simulations 69-70, performed with thesample marked RBOND, represent a regular periodic assembly with geometrical characteristics thatresemble median values of the reference wall and are therefore comparable. The material propertiesand model configurations are defined the same as in the previous simulations. The structuralbehaviors are compared for applied vertical loads of -80 kN and -200 kN. The response of the runningbond specimen RBOND is very much like the responses of the formerly analyzed samples. The failuremechanisms triggered and the formation of diagonal struts occurs within the parameters of theobserved scatter in the former simulations. As expected, the stack bonded specimen SBOND exhibitsa significantly different structural behavior.

Figure 41 Geometries of the stack bond and running bond models


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Table 11 Results summary for simulations 67 and 69

Figure 42 Load-displacement diagram for simulations 1, 42, 48, 67 and 69

Table 12 Results summary for simulations 68 and 70

Simulation Responses for vertical load=-80 kNNo. Sample ID Horizontal disp. at peak load [mm] Peak horizontal load [kN]67 SBOND 3.25 24.0269 RBOND 0.80 45.65

Simulation Responses for vertical load=-200 kNNo. Sample ID Horizontal disp. at peak load [mm] Peak horizontal load [kN]68 SBOND 7.45 57.2570 RBOND 1.60 110.90


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Figure 43 Load-displacement diagram for simulations 61, 63, 68 and 70

The stack bond has apparent diminished equivalent stiffness compared to the running bond and to thereference geometry. Under the compression and shear loads, the geometry of the stack bond, which

resembles an array of pillars, dictates a behavior that conforms to this structural assumption. Thisgeometrical arrangement of the wall allows for very limited lateral interaction which is necessary inorder to absorb lateral loading and form diagonal struts. Therefore, the failure mode is very differentthan those seen in the former simulations, the deformations are substantially higher and the loadbearing capacity is substantially lower. In the running bond sample, interaction is established viapartial contact in the joints, allowing diagonal struts to form in a uniform fashion. It is noted that thereference geometry samples and the randomly generated geometries support the same type ofinteraction but in a more scattered fashion, as no two contact joints are of the same length and thestress paths are formed accordingly, for better or worse. In the cases of the irregular geometry, if the

stones on the diagonal path happen to be large and happen to have a large contact surface with theadjacent stones, the formation of struts is accomplished with relative ease and a larger part of thestructure participates in the load bearing. As there is no guaranty of that happening in any of theirregular walls, there is a scatter in the structural responses which is associated with these geometricalparameters and the periodic sample RBOND obtains responses of intermediate values within thisscatter.


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-1.185E+00-1.050E+00-9.150E-01-7.650E-01-6.300E-01-4.950E-01-3.450E-01-2.100E-01-7.500E-025.715E-02


-1.990E+00-1.780E+00

-1.540E+00

-1.320E+00

-1.080E+00

-8.600E-01

-6.200E-01

-4.000E-01

-1.600E-01

5.141E-02



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-1.538E+00

-1.365E+00

-1.185E+00

-1.005E+00

-8.250E-01

-6.450E-01

-4.650E-01

-2.850E-01

-1.050E-01

6.907E-02


-3.500E+00

-3.115E+00

-2.695E+00

-2.275E+00

-1.855E+00

-1.470E+00

-1.050E+00

-6.300E-01

-2.100E-01

1.448E-01



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-3.736E+00

-3.320E+00

-2.880E+00

-2.440E+00

-2.000E+00

-1.560E+00

-1.120E+00

-6.800E-01

-2.400E-01

1.740E-01


-8.578E+00

-7.650E+00

-6.660E+00

-5.670E+00

-4.680E+00

-3.600E+00

-2.610E+00

-1.620E+00

-6.300E-01

2.886E-01




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-4.794E+00

-4.250E+00

-3.700E+00

-3.150E+00-2.600E+00

-2.000E+00

-1.450E+00

-9.000E-01

-3.500E-01

1.890E-01


-3.523E+01-3.150E+01

-2.730E+01

-2.310E+01

-1.925E+01

-1.505E+01

-1.085E+01

-7.000E+00

-2.800E+00

7.841E-01




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The running bond sample, which is considered comparable with the reference wall and indeed exhibitssimilar global behavior, has an important difference embedded in the periodicity, which is thesymmetry and uniformity of dimensions. The efficiency of these geometrical characteristics can be

regarded as both an advantage and a drawback. The periodicity of the arrangement is clearly anadvantage in terms of predictability. The symmetry in geometry translates to a higher uniformity in theresponses within the wall, clearly an important aspect since the loads themselves are rarely aspredictable. On the other hand, it seems that in the reference geometry, the larger stones docompensate for the lack of regularity and an efficient arrangement can be achieved that is of highercapacity in comparison to regular running bond. That being said, it is emphasized that thesesimulations only predict the compression and shear capacity and under different conditions or even adifferent location of the same load, the larger stones advantage may be lost or irrelevant.


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4. CONCLUSIONS

The simulations performed in this study provide a compelling insight to the sensitivity of the structuralbehavior to the randomness of the material properties as well as the geometrical arrangement. It isclear from the results that in masonry structures, the geometrical characteristics have a much greaterinfluence on the structural behavior than that of the random fluctuations in the material propertieswithin the components, which can be regarded as minor in the full range of the response andnegligible in the linear range.

The analyzed simulations suggest that predictions associated with the randomness of the material

properties can be achieved with high accuracy and that the safety factors that should be consideredcan be derived from their estimated scatter. On the other hand, it is obvious that the geometricalarrangement should be more carefully studied, as any assumptions regarding its characteristics thatare applied will profoundly alter the obtained results. Although this study is limited to just one form ofstructural response and neglects some aspects related to it, such as out of plain behavior and thePoisson effect of the mortar joints, the simulations which were carried out confirm the applicability ofthe deterministic approach regarding the material properties of the structural components. Moreover, itis stated that even given the high dependency on the geometrical arrangement, it can still bequantified to a sufficient extent, in the sense that such educated assessments when appropriatelyapplied enable simplified modeling strategies to provide precise predictions for engineering purposes,especially for estimations conducted in the linear range.

As far as computational effort is concerned, it is advisable to construct numerical models on a case bycase basis, according to the desired level of accuracy in the predictions. While the simplest solutionsare suffice for most practice oriented applications, an experienced user should always consider all theavailable tools and not allow the effort of computation to dictate the modeling strategy, especiallywhen dealing with a complex subject matter.

It is emphasized that the conclusions of the study can be inferred only in such cases where a globalresponse is the subject of interest and that any assumptions regarding the material properties shouldbe made with respect to the

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Computational Modeling of Irregular Masonry Failure

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