COMPDYN 2019

7th ECCOMAS Thematic Conference on

Computational Methods in Structural Dynamics and Earthquake Engineering

M. Papadrakakis, M. Fragiadakis (eds.)

Crete, Greece, 24–26 June 2019

MACROSCALE MODEL CALIBRATION FOR SEISMIC ASSESSMENT

OF BRICK/BLOCK MASONRY STRUCTURES

Corrado Chisari1, Lorenzo Macorini1, and Bassam A. Izzuddin1

1 Department of Civil and Environmental Engineering, Imperial College London

South Kensington Campus, London SW7 2AZ, United Kingdom

e-mail: {c.chisari12, l.macorini, b.izzuddin}name@e-mail.address

Abstract

The accurate prediction of the response of masonry structures under seismic loading is one of

the most challenging problems in structural engineering. Detailed heterogeneous models at the

meso- or microscale, explicitly allow for the specific bond and, if equipped with accurate ma-

terial models for the individual constituents, generally provide realistic response predictions

even under extreme loading conditions, including earthquake loading. However, detailed meso-

or microscale models are very computationally demanding and not suitable for practical design

and assessment. In this respect, more general continuum representations utilising the finite

element approach with continuum elements and specific macroscale constitutive relationships

for masonry assumed as a homogeneous material represent more efficient but still accurate

alternatives. In this research, the latter macroscale strategy is used to model brick/block ma-

sonry components structures, where a standard damage-plasticity formulation for concrete-like

materials is employed to represent material nonlinearity in the masonry. The adopted material

model describes the softening behaviour in tension and compression as well as the strength and

stiffness degradation under cyclic loading. An effective procedure for the calibration of the

macroscale model parameters is presented and then used in a numerical example. The results

achieved using the calibrated macroscale model are compared against the results of simula-

tions where masonry is modelled by a more detailed mesoscale strategy. This enables a critical

appraisal of the ability of elasto-plastic macroscale nonlinear representations of masonry mod-

elled as an isotropic homogenised continuum to represent the response of masonry components

under in-plane and out-of-plane earthquake loading.

Keywords: Mesoscale modelling, Macroscale modelling, Inverse analysis, Material calibra-

tion, Cyclic loading.

Corrado Chisari, Lorenzo Macorini and Bassam A. Izzuddin

1 INTRODUCTION

Even though in the last century many building techniques have been developed and utilised,

most of the historical structures around the world are made of unreinforced masonry (URM).

Buildings, monuments and bridges, most of which are still in use, were built mainly following

rules-of-thumb and trial-and-error procedures without proper consideration of the effects in-

duced by earthquakes. This has led to significant damage and partial collapse of historical URM

structures caused by previous seismic events. Thus to prevent future failures, an urgent need

for realistic assessment of existing URM constructions under extreme loading has emerged in

recent years.

The seismic behaviour of URM buildings and monuments is very complex. It can be accu-

rately predicted only allowing for masonry material nonlinearity, which is due not only to the

independent behaviour of the masonry constituents (e.g. mortar and units) but it is also related

to the specific masonry bond. In the last two decades, several numerical strategies for nonlinear

analysis of URM structures have been developed. These include micro- or mesoscale models

[1, 2, 3], where the individual masonry constituents are modelled separately, and macroscale

models [4, 5, 6] which represent masonry as a homogeneous material. In particular, in

mesoscale masonry models the contribution of both mortar and brick-mortar interfaces is rep-

resented using zero-thickness nonlinear interface elements. This enables the analyst to account

also for damage-induced anisotropy achieving realistic predictions of crack propagation within

any masonry element. Another advantage of mesoscale modelling is the relatively easy calibra-

tion of the material properties, which, being related to the individual components of masonry,

needs less invasive and less expensive tests. However, the computational cost of representing

masonry at such level of detail can become unfeasible for ordinary resources and the use of

mesoscale modelling is at the moment restricted to the research community for relatively small

structures, while seismic analysis of entire buildings is to date an open issue.

On the contrary, macromodelling, which spans from the use of homogenous continuum [7]

to specific macroelements for entire structural components [8, 6] requires much less computa-

tional resources and can be used for the nonlinear analysis of buildings and monuments. The

drawback is represented by the difficulty of defining an accurate phenomenological constitutive

law for masonry as material and calibrating model parameters for such representations, which

should be based on the mechanical behaviour of entire walls or spandrels. A compromise is

represented by multiscale approaches, also called FE2 method, based on homogenisation of

Representative Volume Elements [9, 10, 11], where a macroscale model is still used for the

analysis, but its constitutive behaviour is obtained by the solution of a Boundary Value Problem

(BVP) for the corresponding RVE. While this methodology in theory represents the compro-

mise between the two scales, it is still characterised by high computational cost, as at each

integration point a nonlinear BVP must be solved and the historical variables are to be stored.

Localisation of damage, typical of softening materials as masonry, must be captured maintain-

ing mesh objectivity and this implies careful consideration of regularisation approaches at both

scales [10, 11]. Recent advances [12] propose off-line homogenisation for in-plane loading, in

which the mechanical behaviour at the macroscale is defined once and for all at the beginning

of the analysis.

In this paper, a different approach based on multilevel calibration is proposed. The approach

utilises two scales of representation: a mesoscale model, through which a number of virtual

tests are simulated and the responses evaluated, and a widely used isotropic homogenous mac-

roscale model, which is calibrated against the pseudo-experimental outcomes. The calibration

is based on the application of inverse analysis, which the authors has already successfully pro-

Corrado Chisari, Lorenzo Macorini and Bassam A. Izzuddin

posed for the calibration of the mesoscale material parameters based on real experimental evi-

dence [13]. The proposed methodology is then applied to a case study where different virtual

tests are considered. The accuracy of the macroscale model and its capability of simulating

several masonry collapse modes is finally assessed.

2 MODELLING AND CALIBRATION

2.1 Calibration through virtual experiments

The calibration procedure developed in this paper starts from a realistic parameter identifi-

cation of a mesoscale model which can be achieved by means of simple tests on units and mortar,

or more advanced experimental-strategies involving inverse analysis [13]. Once such model

has been calibrated, it can serve as virtual test for the calibration of the macroscale model. After

the assessment of the modelling error involved in the change of scale, the calibrated macroscale

model can finally be used for analysis of masonry structures under earthquake loading.

The passage from the mesoscale to the macroscale model is carried out considering an en-

ergy equivalence at component level, i.e. at the scale of a wall. In this respect, such procedure

differs from homogenisation procedures as described in [10] or [14], where the equivalence is

imposed at integration point level of the macro-scale which corresponds to a Representative

Volume Element (RVE) in the meso- or micro-scale (Hill-Mandel principle, [15, 16].

Considering a quasi-static virtual test, the energy equivalence can be imposed minimising a

suitable discrepancy functional of the external work:

𝐹𝑖𝑛𝑑 𝒑 𝑠. 𝑡. min 𝜔(𝒑) (1a)

𝜔(𝒑) = ∑ ∫ (𝑑𝑊𝑒,𝑖𝑀(𝒑, 𝑡) − 𝑑𝑊𝑒,𝑖

𝑚(𝑡))2𝑡𝑓𝑖𝑛

𝑡0𝑖

(1b)

where dWe,i(t) is the incremental work done by external forces Fi for the displacement incre-

ments dui at time t varying from t0 and tfin and subscripts M and m refer to macroscale and

mesoscale model, respectively. In Eq. (1) it is underlined that, as the mesoscale model acts as

virtual experiment, the dependency of the functional on the macroscale material parameters p

is represented by the terms 𝑑𝑊𝑒,𝑖𝑀(𝒑).

In case of displacement-only boundary conditions applied to the same nodes at the two scales,

Eq. (1b) can be reduced to:

𝜔(𝒑) = ∑ ∫ (𝑑𝑊𝑒,𝑖𝑀(𝒑, 𝑡) − 𝑑𝑊𝑒,𝑖

𝑚(𝑡))2𝑡𝑓𝑖𝑛

𝑡0𝑖

= ∑ ∫ ((𝐹𝑖𝑀(𝒑, 𝑡) − 𝐹𝑖

𝑚(𝑡)) ⋅ 𝑑𝑢𝑖(𝑡))2

𝑢𝑖,𝑓𝑖𝑛

𝑢𝑖,0𝑖

(2)

and, if the displacement increments are constants, the minimisation of energy discrepancy fi-

nally reduces to the minimisation of force time history curves.

2.2 Mesoscale modelling

In the mesoscale approach employed in this work [3], a masonry element is modelled by

explicitly representing units and mortar joints. Mortar and unit–mortar interfaces are lumped

Corrado Chisari, Lorenzo Macorini and Bassam A. Izzuddin

into 2D 16-noded zero-thickness nonlinear interface elements. Masonry units are represented

by elastic 20-noded solid elements, and possible unit failure in tension and shear is accounted

for by means of zero-thickness interface elements placed at the vertical mid-plane of each block.

The typical discretisation for the structure, as proposed in [3], consists of two solid elements

per unit connected by a unit-unit interface (Figure 1).

Figure 1. Mesoscale representation of masonry (from [3]).

The interface local material model is formulated in terms of one normal and two tangential

tractions 𝝈 = {𝜏𝑥, 𝜏𝑦, 𝜎}𝑇 and relative displacements 𝒖 = {𝑢𝑥, 𝑢𝑦, 𝑢𝑧}

𝑇 evaluated at each inte-

gration point over the reference mid-plane. In the linear range, they are linked one another by

uncoupled elastic stiffnesses, which simulate the linear response of the mortar joints, 𝝈 = 𝒌 𝒖.

The nonlinear behaviour of the material model is based on the coupling of plasticity and

damage [17]. This approach is able to simulate all the principal mechanical features of a mortar

joint or a dry frictional interface - when mortar is absent – with an efficient formulation that

ensures numerical robustness. In particular, it can simulate i) the softening behaviour in tension

and shear, ii) the stiffness degradation depending on the level of damage, iii) the recovering of

normal stiffness in compression following crack closure and iv) the permanent (plastic) strains

at zero stresses when the interface is damaged.

The yield criterion is represented in the stress space by a conical surface which simulates the

behaviour in shear according to the Coulomb law, corresponding to mode II fracture. This sur-

face, governed by cohesion c and friction angle φ, is capped by two planar surfaces representing

failure in tension and compression respectively (Figure 2).

Figure 2. Multi-surface yield criterion for the interfaces in the mesoscale representation.

The evolution of the effective stresses is elastic perfectly-plastic, except for the case where

the plastic surface F1, representing failure in tension, is traversed. The damage of the interfaces

is defined by a diagonal damage tensor D which controls stiffness degradation and is governed

Corrado Chisari, Lorenzo Macorini and Bassam A. Izzuddin

by the plastic work corresponding to each fracture mode. By applying damage to the effective

stresses �̃�, corresponding to the physical stresses developed in the undamaged part of the inter-

face, it is possible to obtain the nominal stresses 𝝈, defined as:

𝝈 = (𝑰 − 𝑫)�̃� = (𝑰 − 𝑫)𝑲(𝜺 − 𝜺𝒑) (3)

This way the implicit solution of the plastic problem and the damage evolution are decoupled,

thus allowing for increased efficiency and robustness at the material level. Further details about

the material model may be found in [17].

2.3 Macroscale modelling

The isotropic macroscale model used in this paper is a slightly modified version of the plas-

tic-damage model proposed by Lee and Fenves [18]. In this model a standard decomposition of

strains 𝜺 in elastic 𝜺𝒆 and plastic 𝜺𝒑 components is considered:

𝜺 = 𝜺𝒆 + 𝜺𝒑 (4a)

𝜺𝒆 = 𝑲𝒆−𝟏𝝈 (4b)

where 𝑲𝒆 is the fourth-order isotropic elastic stiffness tensor and 𝝈 is the nominal Cauchy

stress tensor. According to continuum damage mechanics, the nominal stress tensor 𝝈 is

mapped into an effective stress tensor �̅�:

𝝈 = (1 − 𝑑)�̅� (5)

where 𝑑 = 𝑑(�̅�, 𝜿) is a scalar global damage variable depending on the stress state and two

historical variables 𝜿 = (𝜅𝑡 , 𝜅𝑐)𝑇 representing the evolution of damage in tension and in com-

pression. The effective stress �̅� is defined as the theoretical stress for given strain if the stiffness

was equal to the initial one 𝑲𝟎, and thus:

�̅� = 𝑲𝟎(𝜺 − 𝜺𝒑) (6)

From (4)-(6) the standard plastic-damage constitutive relationship is obtained:

𝝈 = 𝑲𝒆(𝜺 − 𝜺𝒑) = (1 − 𝑑)𝑲𝟎(𝜺 − 𝜺𝒑) (7)

where, following the approach proposed by [18], the evaluation of 𝜺𝒑 is performed working

in the effective stress space. Non-associated plasticity with Drucker-Prager-like flow potential

and the yielding function proposed by [19] is used in the incremental plastic problem, which is

governed by the evolution of two plastic strain-driven historical variables, respectively in ten-

sion kt and in compression kc.

In order to achieve improved robustness in the local problem, effective uniaxial strength has

been assumed always hardening, while the softening branch of the nominal stress is introduced

through the use of damage. Two damage variables, dt=dt(kt) and dc=dc(kc), are evaluated at each

increment depending on the solution of the plastic problem, thus following a similar approach

to that used for the mesoscale material model to decouple plasticity and damage.

The scalar variable d is then evaluated as:

𝑑(�̅�, 𝜿) = 1 − [1 − 𝑠𝑡(�̅�) 𝑑𝑐(𝜅𝑐)][1 − 𝑠𝑐(�̅�) 𝑑𝑡(𝜅𝑡)] (8)

where 𝑠𝑡(�̅�), 𝑠𝑐(�̅�), which depend on the stress state, govern stiffness recovery which sim-

ulates crack closing.

3 NUMERICAL APPLICATION

A numerical case study is considered here to illustrate the multilevel calibration procedure

described in the previous Section. The masonry studied here is a running bond composed of

250mm×250mm ×175mm units assembled with thin mortar bed joints and unfilled head joints.

Corrado Chisari, Lorenzo Macorini and Bassam A. Izzuddin

The material properties used for the units and the interfaces simulating bed joints, head joints

and in-brick cracks are displayed in Table 1.

Parameter Value Parameter Value Brick Young’s modulus 13620 MPa Brick Poisson’s ratio 0.253 Concrete Young modulus 30000 MPa Concrete Poisson’s ratio 0.15 Bed joint axial stiffness 34.0 N/mm3 Brick-brick axial stiffness 104

N/mm3

Bed joint shear stiffness 16.5 N/mm3 Brick-brick shear stiffness 104

N/mm3

Bed joint tensile strength 0.35 MPa Brick-brick tensile

strength

1.49 MPa

Bed joint cohesion 0.28 MPa Brick-brick cohesion 2.235

MPa

Bed joint friction angle atan(0.55) Brick-brick friction angle atan(1.0) Bed joint fracture energy

(mode I)

0.01 N/mm Brick-brick fracture en-

ergy (mode I)

0.1 N/mm

Bed joint fracture energy

(mode II)

0.2 N/mm Brick-brick fracture en-

ergy (mode II)

0.5 N/mm

Bed joint fracture energy

(compression)

0.5 N/mm Brick-brick fracture en-

ergy (compression)

5 N/mm

Bed joint damage parameter 0.1 Brick-brick damage pa-

rameter

0.1

Bed joint compressive

strength

23.6 MPa Brick-brick compressive

strength

23.6 MPa

Table 1. Material properties of the mesoscale model in the virtual tests.

To calibrate the macroscale model, three virtual tests, involving different failure modes in

the masonry, are considered (Figure 3). In tests (a) and (b) a stiff elastic element was applied

on the top of the specimen to transfer the vertical load p=0.5MPa uniformly. Then a horizontal

displacement history was applied to the top. Constraints were applied to the stiff element to

couple the vertical displacements and keep the top element horizontal. In test (c), which simu-

lates an out-of-plane test, a uniform stress was applied on one surface of the wall, while all

edges except the top one were constrained. A load spreader element was utilised to apply a load

history on the specimen by controlling the mean displacement of the load application nodes.

Load protocol for all tests was characterised by a parabolic curve with maximum equal to

2.5mm for tests (a) and (b) and 1.25mm for tests (c). This protocol was designed to evaluate

the cyclic behaviour of the specimens.

Corrado Chisari, Lorenzo Macorini and Bassam A. Izzuddin

(a) (b) (c)

Figure 3. Virtual tests for the calibration of the macromodel.

(a) (b) (c)

Figure 4. Deformed shapes (magnification factor 50) and force-displacement plots for the virtual tests.

Corrado Chisari, Lorenzo Macorini and Bassam A. Izzuddin

In Figure 4, the deformed shapes and the force-displacement plots of the three specimens are

shown. It is possible to appreciate the different failure modes of tests (a) (flexural failure) and

(b) (shear failure with crack opening at the toes). Diagonal cracks are observed in test (c), with

additional flexural crack opening at the base. In terms of force-displacement, the first two tests

show quite stable plastic behaviour and reduced dissipation, characteristic of rocking behaviour.

On the contrary, the out-of-plane behaviour is characterized by large loss of strength at maxi-

mum displacement and significant stiffness degradation at unloading.

The virtual tests were then modelled with the macroscale approach. Twenty-node solid ele-

ments with average dimensions equal to 250mm×250mm ×175mm were used, and the material

model described in Section 2.3 considered for their nonlinear mechanical behaviour. The ma-

terial parameters and their ranges of variation are displayed in Table 2

Param-

eter

Definition Minimum Maximum

E Young’s modulus 100 MPa 5000 MPa

ν Poisson’s ratio 0.001 0.499

𝑓𝑏𝑜 Ratio between biaxial and uniaxial compressive

strength

0.9 1.5

ψ Dilation angle 0° 90°

ϵ Flow potential eccentricity 0.05 0.15

wt Tension stiffness recovery factor 0.0 1.0

wc Compression stiffness recovery factor 0.0 1.0

ft0 Initial uniaxial tensile strength 0.01 MPa 1.0 MPa

Gt Fracture energy in uniaxial tension 1e-5 N/mm 0.1 N/mm

μ Parameter controlling stiffness degradation in ten-

sion

0.0 1.0

fc,max Maximum uniaxial compressive strength 5.0 MPa 30.0 MPa

𝑓𝑦 Ratio between uniaxial yielding stress and maxi-

mum strength in compression

0.01 1.0

𝑘𝑐,𝑓𝑚𝑎𝑥 Plastic strain in compression at fc,max 1e-5 1e-2

ρc Ratio of 𝑘𝑐,𝑓𝑚𝑎𝑥 where damage in compression

starts

0.0 1.0

Table 2. Material parameters for the micromodel.

The optimization problem (1) was solved by means of a Genetic Algorithm implemented in

software TOSCA-TS [20] for each of the virtual tests. Taking into account Eq. (2), the input

parameters for the macro-model were defined as to fit the mesoscale model force-displacement

plot. As more than one solution in terms of parameters gave similar levels of fitting, all these

models were considered solutions of the calibration problem. The material parameters so cali-

brated were consequently applied to the models simulating the other tests to investigate the

robustness of the solution in validation tests [21].

The results shown in Figure 5 allow to draw some conclusions on the calibration procedure

and on the macromodel employed. On the main diagonal of the matrix, the force-displacement

plots of the solutions are compared to the pseudo-experimental results coming from the virtual

test used in the calibration. It is possible to see that in all cases a good agreement is obtained,

meaning that: (i) there exists at least a set of material parameters fitting with satisfactory accu-

racy any of the three tests used in the procedure, and (ii) the optimisation procedure is able to

find such solution. Conclusion (i) however does not guarantee that such three sets are coincident,

Corrado Chisari, Lorenzo Macorini and Bassam A. Izzuddin

that is, there exists a unique set of parameters fitting with satisfactory accuracy all three virtual

tests at the same time. It is important thus to study how the solutions predict the response in

tests not used in the calibration. This is seen in the off-diagonal plots in Figure 5.

In case of a calibration performed by means of test (a) or (c), the calibrated parameters ap-

plied to the other tests show both large variability and low accuracy (first and third rows in

Figure 5). In particular, maximum load and global stiffness are largely overestimated in both

tests (b) and (c) with the solutions coming from test (a), while with the solutions coming from

test (c) stiffness degradation is overestimated in tests (a) and (b), and initial stiffness is under-

estimated. The curves in the second row in Figure 5, however, show that a calibration performed

with test (b) can predict reasonably well the behaviour of the specimen under test (a), while,

again, relatively large variability of the prediction is observed for test (c). Among this variabil-

ity, it is possible to find some solutions fitting with sufficient accuracy the response of the out-

of-plane virtual test.

Figure 5. Results of the calibration and validation.

Among all the solutions obtained by calibration with test (b), the one with minimum dis-

crepancy from the mesoscale test (c) results was then assumed as final solution. This is charac-

terised by the values displayed in Table 3.

Corrado Chisari, Lorenzo Macorini and Bassam A. Izzuddin

Parameter Value

E 2850 MPa

ν 0.31

𝑓𝑏𝑜 1.23

ψ 16°

ϵ 0.13

wt 0.87

wc 0.16

ft0 0.215 MPa

Gt 9.11e-3 N/mm

μ 0.216

fc,max 18.4

𝑓𝑦 1.0

𝑘𝑐,𝑓𝑚𝑎𝑥 7e-4

ρc 0.299

Table 3. Final solution of the macroscale calibration.

The deformed shape and tensile damage pattern, which is an indicator of crack in the con-

tinuum, are displayed in Figure 6. Comparing those results with Figure 4, it is possible to ap-

preciate that a correctly calibrated model seems to be able to capture the main damage patterns

as observed in the mesoscale representation in all cases.

(a) (b) (c)

Figure 6. Deformed shapes (same amplification as Figure 4) and tensile damage patterns for the calibrated mac-

romodels.

4 CONCLUSIONS

In this paper, a novel multilevel calibration strategy has been proposed. Two scale of repre-

sentation are utilised in a hierarchic approach, where a mesoscale model is firstly completely

calibrated from simple and low invasive experimental tests on units and mortar, and then serves

Corrado Chisari, Lorenzo Macorini and Bassam A. Izzuddin

as virtual experiment for the calibration of a more approximate but computationally advanta-

geous homogeneous continuum model. Three virtual tests are considered for the calibration,

characterised by different failure modes: (a) an in-plane shear test on a slender wall with flex-

ural cracks, (b) an in-plane shear test on a square wall with the formation of diagonal and hori-

zontal cracks, and (c) an out-of-plane loading test. The results seem to indicate that an

inconsiderate calibration may lead to large errors in the prediction of the response in different

loading cases, and maximising the information content of the calibration test is paramount for

the correct application of the methodology. In the studied case, the most informative test seems

to be test (b), where compression, tension and shear are all present.

Ongoing work will explore the use of multi-objective optimisation for calibration of masonry

structures, as well as the use of surrogate models to decrease the computational burden of the

procedure, which can increase when larger tests are to be considered. Further validations against

experimental tests and simulations of full-scale structures are also underway.

5 ACKNOWLEDGEMENTS

This research has been supported by the European Commission through the Marie Skłodow-

ska-Curie Individual Fellowship “MultiCAMS – Multi-level Model Calibration for the Assess-

ment of Historical Masonry Structures”, Project no. 744400.

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