### CC2013 (FBX_LM_BAI) - Workspace

Slide 1

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

Slide 2

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

Slide 3

Numerical Analysis of Masonry Panels
Bed Joint

Brick Unit

Slide 4

Numerical Analysis of Masonry Panels

a) Micro-Model

b) Simplified Micro-Model –
Mesoscale Model

Increasing
Computational
Expense
c) Homogenised Macro-Model

Slide 5

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

Slide 6

Mesoscale Modelling - Drawback
Brick Mortar Interaction Leading to Unit Cracking
e.g.: Masonry Prism – Uniform Compression
Tension

assuming Eb > Em

Compression

Slide 7

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

Slide 8

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

Slide 9

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

Slide 10

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

Slide 11

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

Slide 12

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

Slide 13

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

Slide 14

Enhanced Mesoscale Modelling

Co-rotational Framework
•Large Displacements

Out-of-Plane Response under

Slide 15

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

Slide 16

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

Slide 17

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

Slide 18

Enhanced Mesoscale Modelling

Full
Continuum

Detailed
with
Interfaces

Mesoscale a)

Mesoscale b)

Lateral tensile Stresses in the Brick Units

Mesoscale c)

Slide 19

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

Slide 20

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

Slide 21

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

Slide 22

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

Slide 23

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

Slide 24

Thank You!
Questions?