CC2013: Analysis, Modelling and Design of Masonry Structures

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

Mesoscale Modelling of Masonry Structures Accounting for Brick-Mortar Interaction

Department of Civil & Environmental Engineering, Imperial College London

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

Brick Unit

Bed Joint

Head Joint

Numerical Analysis of Masonry Panels

a) Micro-Model

b) Simplified Micro-Model – Mesoscale Model

c) Homogenised Macro-Model

Increasing

Computational Expense

Mesoscale Modelling

20-Noded Solid Element

Elastic Material

16-Noded Interface Element

Material Nonlinearity, Mix-Mode Cohesive Cracking, Crushing, Damage

• Brick Units

• Brick-Mortar Interfaces

• “Brick-Brick” Interfaces

Brick Mortar Interaction Leading to Unit Cracking

Mesoscale Modelling - Drawback

e.g.: Masonry Prism – Uniform CompressionTension

Compressionassuming Eb > Em

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:

0 0

0 0

0 0

z z z

x x x

y y y

k

k

k

z

2 2x y Tension & Shear

“Crushing” Failure Surface

No Lateral Tension Develops in the Units

Approximate Solution at Interface Material Level

Enhanced Mesoscale Modelling

a) Micro-Model

b) Simplified Micro-Model – Mesoscale Model

Brick-Mortar Interaction

- Typically Captured with Refined Micro-Models

Modified Interface Element Kinematics

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

1( , , ) ( ) ( )

2

zu x y z u u u u

h

• Assuming displacements inside the mortar layer as linear function of top and bottom surfaces:

• A representative average strain vector is obtained as:

2 2

2 2

1 1( , , )

h h

avh h

dz Lu x y z dzh h

• Introducing a further simplification with regards to shear strain definition in the x-z and z-y planes:

' '; yxxz yz

uu

z z

Enhanced Mesoscale Modelling

1( , , ) ( ) ( )

2

zu x y z u u u u

h

• Assuming displacements inside the mortar layer as linear function of top and bottom surfaces:

• A representative average strain vector is obtained as:

2 2

2 2

1 1( , , )

h h

avh h

dz Lu x y z dzh h

• Assemble matrix L as:

0 0 0

0 0 0

0 0 0 0 0

T

x z y

Ly z x

z

Enhanced Mesoscale Modelling

The strain vector for the enhanced interface element yields:

'

'

( )1

2

( )1

2

( ) ( )1 1

2 2

x x

y y

x

y z z

z

xz x x

yz

y yxy av

y y x x

u u

x

u u

y

u u

h

u u

h

u u

h

u u u u

x y

1z

x

y

h

Typical Interface displacement discontinuities uniformly smeared over the height of the mortar layer

Average of top and bottom surface engineering strain

Considering the conjugate stress vector:

T

av x y z xz yz xy

The local elastic constitutive relationship is:

av avD

with:

(1 ) 0 0 0

(1 ) 0 0 0

(1 ) 0 0 0

0 0 0 0 0

0 0 0 0 0

(1 2 )0 0 0 0 0

2

x

y

A v Av Av

Av A v Av

Av Av A vD G

G

vA

Enhanced Mesoscale Modelling

(1 ) 0 0 0

(1 ) 0 0 0

(1 ) 0 0 0

0 0 0 0 0

0 0 0 0 0

(1 2 )0 0 0 0 0

2

x

y

A v Av Av

Av A v Av

Av Av A vD G

G

vA

3D Constitutive matrix:

(1 )(1 2 )

EA

v v

Coupling between interface opening and normal strains at mid-surface

Interface stiffness to sliding

In-plane shear stiffness at mid-surface

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

Masonry prism under uniform compression

• 10 mm thick mortar joints• 250x120x55 mm3 units• Eb>Em

Mortar joints detailed with solid FE

Mortar joints lumped into zero-thickness enhanced interfaces

Symmetry Boundary Conditions

Enhanced Mesoscale Modelling

Full Continuum With Interfaces

• Lateral Tensile Stresses in Brick Units

• Lateral Stresses in Mortar Joint

Good Match especially in the region where tensile cracks are expected to develop

Continuum Mortar Joint Interface Mortar Joint

Z

X

Similar Pattern in Z-Y PlaneImportance of 3D Modelling

Enhanced Mesoscale Modelling

Full Continuum

Detailed with Interfaces

Symmetry Boundary Conditions

Brick-Brick Interface

Standard Formulation

Brick-Mortar Interface

Enhanced Formulation

Mesoscale a)

Brick-Brick InterfaceBrick-Mortar

Interface

Enhanced Mesoscale Modelling

Full Continuum

Detailed with Interfaces

Mesoscale b) Mesoscale c)

Lateral tensile Stresses in the Brick Units

Mesoscale a)

Enhanced Mesoscale Modelling

Comparison in terms of global stiffness

Response obtained with standard interfaces

No lateral stresses

Full Continuu

m

Detailed w/

interfaces

Mesoscale a)

Mesoscale b)

Mesoscale c)

DOFs 27951 23535 1440 2880 10560

Computational Cost

Enhanced Mesoscale Modelling

Unreinforced Masonry Wall – Uniaxial Compression test

• Head and Bed mortar joints 10 mm thick

Symmetry Boundary Conditions

Mesoscale a) Mesoscale b)

• Mesocale Model a) – 1 solid element along the height of brick units

• Mesocale Model b) – 2 solid elements along the height of brick units • Head Mortar Joints Modelled with standard

interfaces

Enhanced Mesoscale Modelling

Experimental

0 2 4 6 8 10 12 140

1

2

3

4

5

6

7

8

9

10

11

12

Vertical Strain (x103)

Com

pres

sive

Str

ess

(MP

a)

Enhanced Mesoscacle Elastic

Brick Cracking Activated

Onset of cracking recorded experimentally

Initiation of cohesive cracking in theMesoscale model

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?