Report

CC2013: Analysis, Modelling and Design of Masonry

Structures

Mesoscale Modelling of Masonry Structures

Accounting for Brick-Mortar Interaction

Francisco B. Xavier, Lorenzo Macorini, Bassam A. Izzuddin

Department of Civil & Environmental Engineering, Imperial College London

Project Funding

Outline

Introduction

- Standard Mesoscacle Modelling

- Importance of Brick-Mortar Interaction

Enhanced Meoscale Modelling

- Interface FE Formulation

Verification Examples under Uniaxial Compression

- Elastic Analysis of Single Prism

- Crack Initiation on Masonry Wall

Closure

- Ongoing Work

Numerical Analysis of Masonry Panels

Bed Joint

Brick Unit

Head Joint

Numerical Analysis of Masonry Panels

a) Micro-Model

b) Simplified Micro-Model –

Mesoscale Model

Increasing

Computational

Expense

c) Homogenised Macro-Model

Mesoscale Modelling

20-Noded Solid

Element

Elastic Material

•Brick Units

•Brick-Mortar Interfaces

•“Brick-Brick” Interfaces

16-Noded Interface

Element

Material Nonlinearity,

Mix-Mode Cohesive

Cracking, Crushing,

Damage

Mesoscale Modelling - Drawback

Brick Mortar Interaction Leading to Unit Cracking

e.g.: Masonry Prism – Uniform Compression

Tension

assuming Eb > Em

Compression

Mesoscale Modelling - Drawback

Brick Mortar Interaction Leading to Unit Cracking

e.g.: Masonry Prism – Uniform Compression

assuming Eb > Em

However, with standard interface modelling there is no

coupling between in-plane and normal deformations:

z k z

0

0 z

Approximate

Solution

0 x

x 0 k x

0

0 k y y

y

at Interface Material Level

Tension & Shear

x y

2

2

“Crushing” Failure Surface

No Lateral Tension Develops in the Units

z

Enhanced Mesoscale Modelling

Brick-Mortar Interaction

a) Micro-Model

- Typically Captured with Refined

Micro-Models

Modified Interface

Element Kinematics

b) Simplified Micro-Model –

Mesoscale Model

Enhanced Mesoscale Modelling

Considering interface finite elements representing an actual volume, in which one

of the dimensions is considerable smaller than the other two – in this case the

mortar joint thickness h

It is possible to introduce triaxial stresses and

deformations into a zero-thickness interface,

while maintaining its capabilities for cohesive

crack modelling

Enhanced Mesoscale Modelling

•

Assuming displacements inside the mortar layer as linear

function of top and bottom surfaces:

u ( x, y, z )

1

(u u )

2

•

(u u )

h

A representative average strain vector is obtained as:

h

av

1

h

h

2

dz

h

1

h

2

•

z

2

Lu ( x, y, z )dz

h

2

Introducing a further simplification with regards to shear

strain definition in the x-z and z-y planes:

xz

'

u x

z

; yz

'

u y

z

Enhanced Mesoscale Modelling

•

Assuming displacements inside the mortar layer as linear

function of top and bottom surfaces:

u ( x, y, z )

1

(u u )

2

•

(u u )

h

A representative average strain vector is obtained as:

h

av

1

h

h

2

dz

h

1

h

2

2

•

z

Lu ( x, y, z )dz

h

2

Assemble matrix L as:

x

L 0

0

0

y

0

0

0

z

z

0

0

0

z

0

y

x

0

T

Enhanced Mesoscale Modelling

The strain vector for the enhanced interface

element yields:

x

y

z

'

xz

'

yz

xy av

1 (u x u x )

2

x

1 (u y u y )

2

y

uz uz

h

ux ux

h

uy uy

h

1 (u u ) 1 (u u )

y

y

x

x

x

2

y

2

Considering the conjugate

stress vector:

Average of top and bottom surface

T

engineering

av x strain

y z xz yz xy

The local elastic constitutive

relationship is:

av D av

z

1with:

x

Av

0

0

0

A (1 v ) h A v

Av

A(1 y v)

Av

0

0

0

Av

Av

A (1 v )

0

0

0

Typical Interface displacement

D

0

0

0

Gx

0

0

discontinuities

uniformly smeared over the

0

0

0

0

Gy

0

height of the mortar layer

(1

2

v

)

0

0

0

0

0

A

2

Enhanced Mesoscale Modelling

3D Constitutive matrix:

A (1 v )

Av

Av

D

0

0

0

A

Av

Av

0

0

A (1 v )

Av

0

0

Av

A (1 v )

0

0

0

0

Gx

0

0

0

0

Gy

0

0

0

0

E

(1 v )(1 2 v )

Coupling between interface opening

and normal strains at mid-surface

Interface stiffness to sliding

In-plane shear stiffness

at mid-surface

0

0

0

0

(1 2 v )

A

2

0

Directly obtained with shear test

Enhanced Mesoscale Modelling

Co-rotational Framework

•Large Displacements

Out-of-Plane Response under

Extreme Loading

Enhanced Mesoscale Modelling

Comparison between full continuum and enhanced

interface elastic response at detailed level

Mortar joints

detailed with solid

FE

Masonry prism under uniform compression

Symmetry Boundary

Conditions

•10 mm thick mortar joints

•250x120x55 mm3 units

•Eb>Em

Mortar joints lumped into

zero-thickness enhanced

interfaces

Enhanced Mesoscale Modelling

•Lateral Tensile Stresses in Brick Units

Full Continuum

With Interfaces

Z

X

•Lateral Stresses

in Pattern

Mortar Joint

Similar

in Z-Y Plane

Importance of 3D Modelling

Continuum Mortar Joint

Good Match especially in

the region where tensile

cracks are expected to

develop

Interface Mortar Joint

Enhanced Mesoscale Modelling

Brick-Mortar

Interface

Enhanced Formulation

Full

Continuum

Detailed

with

Interfaces

Mesoscale a)

Symmetry Boundary

Conditions

Brick-Brick

Interface

Standard Formulation

Brick-Mortar

Interface

Brick-Brick

Interface

Enhanced Mesoscale Modelling

Full

Continuum

Detailed

with

Interfaces

Mesoscale a)

Mesoscale b)

Lateral tensile Stresses in the Brick Units

Mesoscale c)

Enhanced Mesoscale Modelling

Comparison in terms of global stiffness

Response obtained

with standard

interfaces

No lateral stresses

Computational Cost

DOFs

Full

Continuu

m

27951

Detailed Mesoscale Mesoscale Mesoscale

w/

a)

b)

c)

interfaces

23535

1440

2880

10560

Enhanced Mesoscale Modelling

Unreinforced Masonry Wall – Uniaxial Compression test

Mesoscale a)

•

Head and Bed mortar joints 10 mm thick

•

Mesocale Model a) – 1 solid element along the height

of brick units

•

Mesocale Model b) – 2 solid elements along the

•

height of brick units

Mesoscale b)

Symmetry Boundary Conditions

Head Mortar Joints Modelled with standard interfaces

Enhanced Mesoscale Modelling

Enhanced Mesoscacle

Elastic

12

Initiation of cohesive cracking in the

Mesoscale model

11

10

Compressive Stress (MPa)

9

Onset of cracking recorded experimentally

8

7

Experimental

6

5

4

3

2

Brick Cracking

Activated

1

0

0

2

4

6

8 10 12

Vertical Strain (x103)

14

Closure

Further Improvements on the enhanced interface element:

•

Adapt previous cohesive model (Macorini & Izzuddin, 2011) to accommodate

new stress components in the new interface, i.e., allow mix-mode fracture

(Tension & Shear) in brick-mortar interfaces (bed joints)

•

Introduce failure surface at interface level, accounting for triaxial stress state in

order to capture the actual failure of confined mortar material

•

Non-linear response of masonry prisms by the knowledge of individual

components properties, as opposed to composite properties dependent on the

prism characteristics

Closure

•

Despite mechanically sound, full potential of this enhanced mesoscale modelling

strategy is only achieved if realistic material properties for both mortar and brick

units are available

•

Current published research underlines mortar material properties when part of a

masonry assemblage or taken from single specimen to be markedly different

•

There is the need to establish procedures to assess the actual mortar material

properties, thus enabling the composite behaviour o masonry panels to be

characterized by its individual constituents properties

Thank You!

Questions?