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# 2D Nonlinear Finite Element Analysis of Masonry Bedjoint

Abstract (Summary)
Mortars with crushed brick particles were used for construction of historic buildings. These ancient masonry structures proved to be more durable especially in seismic regions. The purpose of this work was to analyse the masonry bedjoint subjected to shear loading. In particular, the attention was paid to crack localization, when the joint is reinforced by crushed brick particles of different quality. A 2D nonlinear finite element analysis utilizing an anisotropic damage model was used for the analysis and the calculations were performed using the OOFEM software. The results indicate that the crushed brick particles of lower quality than the surrounding matrix can contribute to a bigger ductility of the joints and suppress the localization of damage into a single major crack.
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Advisor:doc.Ing. Jan Zeman, Ph.D.

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Source Type:Other

Keywords:masonry, mortar, crushed brick particles, FEM, shear load, damage model, damage localization

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Date of Publication:05/13/2013

Document Text (Pages 1-10)

CZECH TECHNICAL UNIVERSITY IN PRAGUE

FACULTY OF CIVIL ENGINEERING

Department of Mechanics

BACHELOR THESIS

2D Nonlinear Finite Element Analysis of
Masonry Bedjoint

Andrej Osvald

2013

Supervisor: Jan Zeman

Page 2

Honesty Declaration

I declare that this bachelor thesis has been carried out by me and only with the
use of materials that are stated in the literature sources.

May 10, 2013 Andrej Osvald

............................................

Page 3

Acknowledgement

I would like to thank my supervisor, Doc. Ing. Jan Zeman, Ph.D., for leading
and his willingness to me. I also want to thank Ing. Václav Nežerka for introducing
me to the issues of numerical modelling and explanation me all necessary
knowledge, his support, patience and kindness.
And not at least, special thanks belongs to my parents for their constant
support and trust in me.

Page 4

Abstract

Mortars with crushed brick particles were used for construction of historic
buildings. These ancient masonry structures proved to be more durable especially in
seismic regions. The purpose of this work was to analyse the masonry bedjoint
subjected to shear loading. In particular, the attention was paid to crack localization,
when the joint is reinforced by crushed brick particles of different quality. A 2D
nonlinear finite element analysis utilizing an anisotropic damage model was used for
the analysis and the calculations were performed using the OOFEM software. The
results indicate that the crushed brick particles of lower quality than the surrounding
matrix can contribute to a bigger ductility of the joints and suppress the localization
of damage into a single major crack.

Keywords: masonry, mortar, crushed brick particles, FEM, shear load, damage
model, damage localization

Abstrakt

Při konstrukci historických budov byla použita malta s příměsí drcených cihel.
Tyto historické konstrukce vykazovaly větší odolnost převážně v seismických
oblastech. Účelem této práce bylo analyzovat ložní spáru zdiva vystavenou
smykovému napětí. Zvláštní pozornost byla věnována lokalizaci trhlin ve spáře,
která byla vyztužená drcenými cihlami o různé kvalitě. Při analýze byla použita 2D
nelineární analýza metodou konečných prvků, uplatňující anisotropní model
poškození. Výpočty byly provedeny softwarem OOFEM. Výsledky poukazují na to,
že kousky drcených cihel s nižší kvalitou než obklopující matrice přispívají k vyšší
duktilitě spáry a zamezují soustředí poškození do jedné hlavní trhliny.

Klíčová slova: zdivo, malta, kousky drcených cihel, MKP, smykové namáhání,
model poškození, lokalizace poškození

Page 5

Contents

Introduction......................................................................................................................... 7

PART I: THEORETICAL BACKGROUND ...................................................................... 8

1 Finite Element Method.............................................................................................. 9
1.1 History of FEM .................................................................................................... 9
1.2 Basics of FEM ..................................................................................................... 10

1.2.1 Pre-processing .............................................................................................. 10
1.2.2 Solution ......................................................................................................... 11
1.2.1 Post-processing ............................................................................................ 12

1.3 Linearity.............................................................................................................. 12

1.3.1 Non-Linear Analysis ................................................................................... 13
1.4 Applications ....................................................................................................... 13
1.5 Limitations of FEM............................................................................................ 14
1.6 Convergence criteria ......................................................................................... 14
1.7 Nonlinear solution methods ............................................................................ 15

1.7.1 Newton-Raphson method .......................................................................... 15
1.7.2 Modified Newton-Raphson method......................................................... 17

2 Theory of elasticity................................................................................................... 19
2.1.1 Elasticity ........................................................................................................ 19
2.1.2 Stress .............................................................................................................. 19
2.1.1 Notation for forces and stresses................................................................. 20

2.2 Strain-displacement relations .......................................................................... 21
2.3 Stress-strain relations........................................................................................ 21
2.4 Equilibrium equations ...................................................................................... 22

3 Damage Models ........................................................................................................ 23
3.1 Isotropic damage model ................................................................................... 23
3.2 Anisotropic damage model.............................................................................. 23

4 Damage localization ................................................................................................ 25
4.1 Numerical solution of tensile test with softening material ......................... 25
4.2 Damage work..................................................................................................... 26
4.3 Fracture energy .................................................................................................. 27

PART II: CALCULATIONS ............................................................................................... 29

5 Calculation with Crushed Brick Particles ........................................................... 30
5.1 Pure Shear........................................................................................................... 32
5.2 Shear and Compression.................................................................................... 35

Conclusion ......................................................................................................................... 39

Page 6

REFERENCES ....................................................................................................................... 40

Page 7

INTRODUCTION 7

Introduction

Mansory is one of the oldest building materials and it still have a wide usage
in a nowadays construction. The main reason of its development is its simplicity.
Masonry is a composite material consisting of two elements units and joints. Many
alternatives were used through time. Units can be bricks, blocks, ashlars, adobes,
irregular stones and others. Mortar can be for example clay, bitumen , chalk, cement
and glue. There is a countless number of possible combinations generated by the
geometry, nature and arrangement of units.

Fig.: Several types of masonry: stone masonry, rubble masonry and brick masonry

Time passing and structure degradation showed that we lack a great deal of
knowledge about behaviour of masonry. Only recently the scientific community
started to explore this issue and showed interest in advanced testing (under
displacement control). The lack of experience is obvious in comparison with other
fields such as concrete, soil, rock or composite mechanics.
With the current capabilities, the finite element method is very common to
achieve sophisticated simulations of the structural behaviour. These numerical tools
help to predict the behaviour of the structure from linear elastic stage, through cracks
development and degradation until complete failure.
All masonry types have generally a common feature in mechanical behaviour
- a very low tensile strength. Due this fact, characteristic ancient structures have
redesigned their shape. A numerical modelling of masonry structures leads towards
the micro-modelling of the individual components (units and mortar) or the macromodelling
of masonry as a composite.
The failure mechanism of the masonry structure involves a tensile failure of
units and joints, shear failure of joints or compressive failure of the composite. A big
advantage of micro-modelling strategy is that all mentioned phenomena can be
covered with these methods and that is because joints and units are represented
separately.
The purpose of this work was to analyse the masonry bedjoint subjected to
shear loading. In particular, the attention was paid to crack localization, when the
joint is reinforced by crushed brick particles of different quality. In next chapters all
used methods, such as finite element method, anisotropic damage model and
damage localization, are explained.

Page 8

PART I:

THEORETICAL

BACKGROUND

Page 9

FINITE ELEMENT METHOD 9

1 Finite Element Method

The finite element method is a numerical method for solving problems
described by partial differential equations or as functional minimization. The method
is represented by the finite number of elements. This computational method is used
to obtain approximate solutions of boundary value problems. It is a mathematical
problem in which dependent variables must suit a differential equation and satisfy
boundary conditions. Depending on the type of physical problem being determined,
variables can carry for example displacement, temperature, heat flow and fluid
velocity.

1.1 History of FEM

The mathematical roots of the finite element method dates at least a half
century in past. The origins of approximate methods for differential equations are
even older. It was Lord Rayleigh and Ritz who used trial functions to approximate
differential equations solutions. Galerkin method used a similar solution concept and
it laid the strong foundations of the finite element method. In comparison with
modern finite element method, trial functions had to be applied over the entire area
of the problem, until 1940s, when Courant provided the concept of piecwisecontinous
functions in subarea, which is considered as a finite element method real
start.
In the late 1940s, aircraft industry needed more sophisticated method for
solving airframe structures. Engineers wanted to implement a jet engine in to a
commercial aircrafts, without the benefit of modern computers; they developed
matrix methods of force analysis, better known as the flexibility method with
prescribed displacements and unknown forces. The most often used form of the
finite element method, refers to the displacement method, in which the unknowns
are system displacements related to applied force systems. The term displacement is
quite general and does not necessary mean a position change; it can be represented
by physical displacement, temperature or fluid velocity, for instance. The collocation
finite elementwas first used by Clough in 1960 in combination of plane stress
analysis and has been used since present.
Decades 1960s and 1970s expanded the finite element method applications to
plate bending, shell bending, pressure vessels and general 3-dimensional problems in
elastic structural analysis as well as fluid flow and heat transfer.
The finite element method is computationally demanding and has got large
time requirements. In the past, mainframe computers were considered to be very
powerful, with high-speed tools for engineering usage. The United States space
exploration brought a first finite element software code NASTRAN. After NASTRAN
development, many commercial software capable of the finite element analysis were
invented, for example ANSYS, ALGOR and COSMOS/M. Nowadays, most of these
software are supported on desktop computers and engineering workstations to

Page 10

FINITE ELEMENT METHOD 10

obtain solutions to large problems in static and dynamic structural analysis, fluid
flow, heat transfer, electromagnetics and seismic simulations.

1.2 Basics of FEM

Finite element method is based on replacing original shape with an
approximately equivalent network of simple elements. This network is called a mesh
(Fig. 1.1) and is unique for every new project. It needs to be decided what kind of
element will be used. In 1-dimensional model it is a beam. Joining these simple
elements together we make a truss consists of rods and nodes. In 2-dimensional
models, a triangle can be used and for 3-dimensional
models often rely on pyramids. The results depend on
the size of elements. For linear models the accuracy is
increasing with the decreasing size of elements. On the
other hand, the computational cost is increasing with
the increasing number of elements. Therefore, a
reasonable number of elements should be selected, in
order to achieve sufficiently low errors for a low
computational cost. It also depends in case to case,
according to size, difficulty and outputs of project
demands.

Fig. 1.1: Example of finite element mesh

In 2-dimensional analysis we assume that each node is capable of moving in
two directions horizontally and vertically. It is necessary to define boundary
conditions of mesh to reach the desired solution of the model. The effect of boundary
conditions may be observed in the reduction of unknowns and corresponding
equations.
Finally we need to determine the material characteristics such as elastic
properties (Young’s modulus of elasticity, Poisson’s ratio, compressive, tensile and
sheer strength) and the applied external and body loads.

1.2.1 Pre-processing

This step, generally, describes the model and its sub-steps are summarized next:
define the domain of the problem
define the element type
define the material characteristics of the elements (length, area)
define the element connectivities (finite element mesh)
define the physical constraints (boundary conditions)
define the loadings
In most cases the pre-processing step is critical. The more accurate model
definition we have, the better result we get. A perfectly solved finite element model
is absolutely no value if input definitions are wrong.

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