# 2D Nonlinear Finite Element Analysis of Masonry Bedjoint

Advisor:doc.Ing. Jan Zeman, Ph.D.

School:

School Location:

Source Type:Other

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

ISBN:

Date of Publication:05/13/2013

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

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

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

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.

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

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

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

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.

PART I:

THEORETICAL

BACKGROUND

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 element” was 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

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.