A DIRECT METHOD TO COMPUTE THE INELASTIC DISPLACEMENT DEMAND OF MASONRY STRUCTURES … ·...

1

A DIRECT METHOD TO COMPUTE THE INELASTIC

DISPLACEMENT DEMAND OF MASONRY STRUCTURES

Francesco GRAZIOTTI1, Guido MAGENES

2 and Andrea PENNA

3

ABSTRACT

Estimating the maximum inelastic deformation demands for a given earthquake excitation is essential

for the evaluation of the seismic vulnerability of existing structures. There is not a general consensus

on a specific, easy and reliable method to compute the displacement demand for low-period

unreinforced masonry structures. In some of the available methods, the maximum inelastic

displacement is directly calculated starting from the spectral displacement at initial period, others

consider an equivalent linearization in which the maximum deformation is estimated by means of an

equivalent linear elastic system with lower lateral stiffness and higher damping coefficient. In order to

validate existing formulations and propose new ones, a technique for the definition of single degree of

freedom (SDOF) nonlinear systems representing the global behaviour of multi degree of freedom

(MDOF) structural models has been developed to perform fast dynamic analyses. Both SDOF and

MDOF systems are based on a nonlinear macro-element model able to reproduce the in-plane shear

and flexural cyclic behaviour of pier and spandrel elements. The comparison of the results in terms of

maximum displacement obtained for the SDOF system and for the MDOF system proves the

feasibility and reliability of the proposed approach. In this paper, parametric analyses were run on the

SDOF and used to develop and verify a non-iterative method based on secant stiffness specific for

masonry structures. The result of the study is a trustworthy formulation able to predict the ductility

demand for masonry structures with combined flexural and shear behaviour.

INTRODUCTION

The quantification of displacement demands for a structure is a crucial step in performance-based

assessment. Currently, a simple, yet reliable method to compute displacements in masonry structures

is lacking within the literature. The absence of such a procedure is rooted in two key limitations: first

of all, the complex cyclic behaviour of masonry structures does not allow for simple hysteretic rules to

be assigned to them and, secondly, the extremely short periods typical of masonry structures does not

allow the use of the “equal displacement rule” utilised in numerous simplified procedures for other

structural typologies. Nonlinear dynamic analysis can be used to simulate the dynamic response of a

masonry structure, provided that the model is capable of capturing the principal failure modes and

cyclic decay. In many cases, the global displacement response can be modelled though the

implementation of an equivalent frame technique; allowing for straightforward interpretation of the

expected damage pattern (Penna et al., 2014).

One of the objectives of this work is to develop a single degree of freedom (SDOF) model that

allows for the estimation of the seismic response of a masonry structure in a reliable, yet efficient

1 Postdoctoral Researcher, EUCENTRE, Pavia, [email protected]

2 Associate Professor, University of Pavia, EUCENTRE, [email protected]

3 Assistant Professor, University of Pavia, EUCENTRE, [email protected]

2

manner. The reduced computational complexity of simplified SDOF analysis methods greatly

enhances the feasibility of conducting parametric studies using nonlinear dynamic analyses. The

SDOF analyses can be used to study the correlations between the displacement demand and various

seismic intensity measures (e.g. Shome et al., 1998), perform simplified incremental dynamic analyses

(e.g. Azarbakht and Dolšek, 2011), calculate limit state-dependent fragility curves (e.g. Luco et al.,

2004; Mouyiannou et al. 2014) or to create simplified methods that can estimate the maximum

displacement demands on masonry structures (similar to what was done for general structures in Vidic

et al., 1994 and by Graziotti et al., 2014 for masonry ones). Due to the lack of studies in the latter

mentioned field of application, this work utilises the SDOF system to derive a corrected strength and

displacement reduction factors (ηin-R) relationship by means of a parametric study. The results

provided a simplified formulation for determining the inelastic displacement of a masonry structure

initiating from an idealized push-over and an elastic response spectrum, trying to bridge the evident

gap in this particular field of study. The proposed procedure is based on an equivalent period Teff that

was proposed by Lin and Miranda (2004) in their “optimal stiffness” method.

THE SINGLE DEGREE OF FREEDOM MODEL

Concept of the model

In order to perform simplified nonlinear dynamic analyses, a single degree of freedom model (SDOF)

was created. This model is able to synthetically interpret the seismic response of a multi degree of

freedom (MDOF) model representing a complete masonry building (if the MDOF is governed by a

single dominating mode of deformation, typical in regular buildings). It consists of two macro-

elements in parallel characterised by nonlinear behaviours typical of masonry panels. The two

elements are connected by a top rigid link. Such approach arises from the aim of completely

decoupling the shear and flexural/rocking mechanisms (or behaviours) consequently facilitating an

independent calibration of the parameters. The variables that govern the model are the geometry of the

elements, their axial compression, the mechanical characteristics (related to the flexural behaviour in

one and shear behaviour in the other), and the inertial mass. The single degree of freedom is the top

horizontal displacement of the two elements where the mass is concentrated; the rotation of the top

edge of the elements is restrained. Figure 1 reports a simplified illustration of the SDOF model. The

model was created in the TREMURI computer program (Lagomarsino et al., 2013; Penna et al.,

2014), a nonlinear analysis program capable of performing monotonic and cyclic pushover analyses as

well as time-history analyses of masonry buildings.

M

Axial rigid link

F1 F2

Fh

Figure 1. Scheme of the simplified SDOF model obtained assembling two pier elements, one governed by

flexural and one by shear behaviour.

Reference experimental campaign

An experimental campaign was used to calibrate the MDOF three-dimensional numerical model, it

consisted of a cyclic quasi-static test on an unreinforced masonry structure. The full-scale bricks

Δs

F. Graziotti, G. Magenes and A. Penna 3

structure was tested at the laboratory of Department of Structural Mechanics of University of Pavia in

1994 (Magenes et al., 1995).

Figure 2. Plan and views of the building specimen used to calibrate the MDOF three-dimensional numerical

model (Magenes et al., 1995).

It is important to notice that façade D is decoupled from the walls A, C and that the pushing

forces were equal at the two floor levels. Thanks to the above mentioned test it was possible to

determine the parameters of the MDOF model that allowed a fair simulation of the experimental

results. In particular the calibrated masonry mechanical properties are: elastic modulus E, shear

modulus G, compression strength fm, the cohesion c and friction coefficient μf. Values are reported in

Table 1.

Table 1. Calibrated masonry mechanical model.

E [MPa] G [MPa] fm [MPa] c [MPa] μf [-]

3000 500 2.8 0.14 0.15

The so calibrated macro-element model was used to create two larger symmetrical models that

represent realistic two story buildings with rigid floors. One is characterized by a flexural dominated

cyclic behaviour and the other by a shear one. The flexural model was created imaging a symmetrical

structure composed by two mirrored parts with geometrical characteristics equal to the ones of the lab-

tested structure. On the other hand, in order to create a model that exhibits a shear related failure it was

chosen to create a squatter pier by removing one of the two doors in wall D in one of the models.

Figure 3 shows the plan views and the equivalent frame idealization of the two models.

The parameters related to the flexural dominated structure (F) are indicated with superscript

“f” while those related to the shear dominated structure (S) with “s”. The total masses of the structures

are Mf=224.5 t and M

S=228.2 t, the first modal elastic periods are 0.21 s and 0.17 s and the

participating masses for the first vibration mode are 185 t (82.4%) and 188 t (82.3%) respectively.

Figure 4 plots the nonlinear pushover analyses used to define an equivalent bilinear capacity curve.

These analyses are run considering a limit shear drift equal to 5‰, i.e. the lateral strength and stiffness

of elements exceeding such drift value is set to zero.

4

E68 E69

E70 E71

72 73 74

75 76 77

n37

n38

n39

N13

N14

N15

N16

N17

N18

t112 t113

t114 t115

E68

E70

72 74

75 77

N13

N14

N15

N16

N17

N18

t112

t114

Figure 3. F Building (top) and S Building (bottom): plan view, three-dimensional models and equivalent frame

idealization (red, piers; green, spandrels; blue, rigid nodes)..

0

100

200

300

400

500

600

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8

Ba

se s

he

ar [k

N]

Top displacement [cm]

dMU

FMax

70%FMax

FMy

dMy0

100

200

300

400

500

600

700

0 0.2 0.4 0.6 0.8 1 1.2

Ba

se s

he

ar

[kN

]


dMU

FMax

70%FMax

FMy

dMy

Figure 4. F Building (left) and S Building (right) pushover curve and equivalent bilinearization.

The equivalent bilinear curve is defined according to the recommendations reported in the

Commentary to the Italian Building Code (MIT, 2008 and 2009). The modal participation factors is

equal to Γf= Γ

s =1.2 in both cases and the masses of the equivalent SDOF are Ms

f=154 t and Ms

s=157 t,

respectively. The periods of vibration of the equivalent bilinear system are calculated as:

2S My

My

M dT

F

(1)

where the meaning of the quantities are explained in Figure 4.

F building

S building


The periods of vibration are Tf=0.21 s and T

s=0.17 s, respectively.

Calibration of the model

The SDOF model needs to be calibrated by a two-step procedure described in (Graziotti, 2013). The

first calibration to be performed involves the use of nonlinear static analysis. The goal of this

calibration is to obtain a SDOF model with a cyclic behaviour similar to the global one of the

reference MDOF model.

It is useful to repeat more than one cycle for each displacement level, the macro-element

implemented in the TREMURI model is able to take in account mechanical property degradation. A

SDOF cyclic pushover is run at the same maximum displacements of the one run for the MDOF. In

particular the pushover run for the MDOF model was adaptive, with a force distribution related to the

current deformed shape (Galasco et al., 2006). The parameters that characterize the system were

calibrated comparing the static responses of the two analyses. In particular, the variables that have

been checked are: the maximum base shear, the initial stiffness, the unloading and reloading stiffness,

the shear deformation and stiffness and the area of the hysteresis loops. The properties to be calibrated

on the SDOF are: the geometry of the elements, their mechanical characteristics, the load axial forces

and model parameters controlling post-peak softening and nonlinear shear deformation.

Figure 5 shows the comparison between the MDOF and SDOF hysteresis cycles in terms of

base shear/displacement. It is possible to notice how the SDOF models are able to interpret the static

behaviour of the MDOF systems. The shear dominated MDOF model has a non-symmetrical

force/deformation relationship due to the geometry of the walls D. This obviously could not be

simulated by the simplified SDOF models.

In order to create a dynamically calibrated model there is the need to assign a mass MS and the

Rayleigh damping parameter αS (second step of the calibration). The mass MS is assigned using the

equation of the modal analysis:

T

SM φ M1 (2)

where the vector φ is the mode shape normalized for dMax=1 and M is the mass matrix of the MDOF

system. In this way, the mass of the SDOF dynamic system is chosen considering the first modal

elastic vibration mode.

Concerning the Rayleigh damping to assign to the model, Eq. 3 gives the damping matrix for a

general MDOF system according to Rayleigh hypothesis:

M M C M K (3)

where αM and βM are constants with units of s−1

and s, respectively, and K is the linear stiffness matrix

of the structure when the initial tangent stiffness is used. Thus, C consists of a mass-proportional term

and a stiffness-proportional term. The MDOF models have Rayleigh damping parameters αM=0.43 and

βM=7.8.10

-4, which gives to the models a nearly constant damping of 2% between the first elastic

period and the secant period at collapse (minimum value 1.8%). The damping coefficient of the

simplified system is not necessarily equal to the ones used in the MDOF analyses, so a calibration is

needed.

6

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-600

-400

-200

0

200

400

600


Bas

e s

hear

[kN

]

-1.5 -1 -0.5 0 0.5 1 1.5

-600

-400

-200

0

200

400

600


Bas

e sh

ear

[kN

]

Figure 5. Hysteretic response of MDOF (red) and SDOF (black) model of the flexural (top) and shear (bottom)

dominated building.

The Rayleigh parameters of the two SDOF models were calculated by mean of 10 nonlinear

dynamic analyses run on each MDOF model. In the SDOF system the viscous damping matrix C is a

1-by-1 (scalar). Note that the stiffness-proportional part of C does not contribute to the total damping

force in SDOF systems. The SDOF damping parameters that are able to best replicate the nonlinear

analyses output are αSf=1.87 and αS

s=4.19 (which means ξ

F=3% at elastic period for F structure and

ξS=5% at elastic period for S structure).

The slight increase of damping is likely due to the contribution of higher modes that are not

taken in account by the SDOF model and the choice to use the elastic modal participation factor Γ.

Validation of the method

In order to validate the method, a large database of natural accelerograms was used. The database

employed is SIMBAD (Smerzini et al., 2013). Nonlinear dynamic analyses were run using both

models (complete and simplified). Maximum displacements obtained by means of two models were

plotted as a function of different earthquake intensity measures. The equivalent frame modelling has

computational times lower than detailed finite element models, but the time histories analyses still take

more than one hour to be completed (standard structures run on a normal PC). Conversely, the

analyses for the SDOF equivalent models take only a few seconds to be completed.

The multiple nonlinear dynamic analyses allowed the comparison of displacement demands of

the two systems. Figure 6 shows the correlation between the top displacements of the SDOF and

MDOF systems. The graphs do not consider displacements higher than the collapse point dfMC=2.5 cm

and dsMC= 1.3 cm. Lines for mean and ± mean one standard deviation are annotated on the graphs. In

particular, all the Figures below refer to top displacements of the building. It is noticeable how the

SDOF model is able to interpret the response of the MDOF one. The correlation coefficients between

the vectors of maximum MDOF and SDOF displacements are 0.97 and 0.98 respectively for the

flexural and the shear dominated models.

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

Max displacement m-d.o.f. [cm]

Max

dis

pla

cem

ent

s-d

.o.f

. [c

m]

0 0.2 0.4 0.6 0.8 1 1.20

0.2

0.4

0.6

0.8

1

1.2

Max displacement m-d.o.f. [cm]

Max

dis

pla

cem

ent

s-d

.o.f

. [c

m]

Figure 6. Correlation between top displacements obtained with the flexural (first) and shear (second) dominated

SDOF and MDOF models.


The use of un-scaled natural accelerograms allows for the correlation of the structural

responses with different spectral intensity measures (Graziotti et al., 2013a; 2013b; Graziotti, 2013).

SIMPLIFIED PREDICTION OF THE INELASTIC DISPLACEMENT DEMAND: AN

OPTIMUM STIFFNESS METHOD

Models combination

Starting from the two calibrated structures (Section 2), five others with intermediate behaviours

(mixed shear and flexural response) were created. This allowed to have a bunch of seven structures, to

represent the nonlinear dynamic behaviour of a variety of masonry buildings. The idea was to combine

in different ways the two structures in order to create more than one system with an intermediate

behaviour. The easiest way to obtain these models is to parallelize the two calibrated systems

weighting them to obtain different configurations. In particular, 7 systems were created from the shear

dominated one (T=0.165 s) to the flexural dominated one (T=0.212 s). Figure 7 (left) shows a

schematic representation of one of the models created while Figure 7 (right) plots in one graph the

cyclic push overs of all the models in terms of equivalent acceleration and displacement. It could be

noticed that the more flexural the response is, the thinner the hysteretic loops are (solid black line).

The Jacobsen (1960) method was used in order to characterize the 7 SDOF structures by

comparing their cyclic dissipated energy at ultimate displacement. The hysteretic damping ξhyst varies

almost linearly from a value of 19.9 % for the shear governed structure to a value of 13.8% for the

flexure dominated one.

-4

-3

-2

-1

0

1

2

3

4

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

Acc

eler

atio

n [m

/s2

]

Displacement SDOF [cm]

0%

20%

33%

50%

66%

80%

100%

Figure 7. Scheme of one of the combined systems (20% shear) (left) and direct comparison of the cyclic

pushovers of the 7 SDOF systems (right).

Effective period

For existing structures, the value of strength reduction factor (R) is known while that of μ is unknown.

In an assessment process, calculating the ductility demand is the final goal. Lin & Miranda (2004)

proposed a method that defines an effective period Teff and an equivalent damping coefficient ξeq as a

function of R.

The aim of the proposed assessment method is to calculate the displacement demand by means

of a non-iterative procedure that defines the effective period Teff and the displacement reduction factor

ηin as a function of R.

In particular, concerning the equivalent period, Lin & Miranda (2004) proposed:

8

1 ( 1)eff el

h

RT T

R

(4)

where αh is the stain hardening ratio and Tel is the elastic period calculated by Eq. (1). In case of

elasto-plastic approximation the equation becomes:

eff elT T R (5)

ηin–R relations

The seven calibrated SDOF structures described in a previous section were used in order to run several

nonlinear analyses. Natural records of SIMBAD database were used as input. The procedure to

calculate ηin for each analysis is straight forward and it is here summarized:

1. Run the analysis in order to calculate Sd

2. compute the strength reduction factor as: R=m.Sa(Tel)/Fy

3. Calculate Teff=Tel.R

0.5

4. Compute Sde(Teff) as the spectral displacement

5. Compute the displacement reduction factor by mean of Eq. (6).

Eq. (6) was derived based on the results obtained from the nonlinear time history analyses

reported earlier, the following ηin-R expressions are proposed:

1.82(8.94 0.4) 1 1 1

R

in hyst

e

e R

(6)

The proposed equations are plotted together with the numerical results in Figure 8. In

particular, the graphs represent the displacement reduction factor ηin vs. strength reduction factor R for

structures with different Jacobsen damping ξhyst. The black line refers to the mean value and the red

line is the regression line with a shape given by Equation (8). The displacement reduction factor is

smaller than 1 in a range of R between 1 and 3. The minimum value of R is approximately 1.7. The

scatter is almost constant for all the ductility values and for all the structures and increases for higher

R.

Eq. (7) is calculated by regression using the results of the seven structures, it is plotted on

graph in Figure 8. The formulation could be simplified if a cyclic pushover analysis is not available,

considering an average value:

1.821.1 1 1 1

R

in

e

e R

(7)


0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

R [-]

in

[-]

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

R [-]

in

[-]

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

R [-]

in

[-]

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

R [-]

in

[-]

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

R [-]

in

[-]

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

R [-]

in

[-]

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

R [-]

in

[-]

Figure 8. Displacement vs. strength reduction factor for structures with different Jacobsen damping (mean, in

black; mean ± 1 standard deviation, in blue; proposed regression, in red).

Displacement prediction by optimal stiffness DBA

In the previous sections all the parameters to use the optimal stiffness displacement based assessment

(DBA) method were calculated in terms of capacity of the structure (dy, Fy) and in terms of spectral

characteristics. With these data it is possible to compute the predicted displacement demand.

ξhyst=19.9%

ξhyst=13.8%

ξhyst=18.5%

ξhyst=17.7% ξhyst=17.1%

ξhyst=15.4% ξhyst=14.9%

10

To perform the displacement-based assessment procedure the following steps must be

sequentially performed:

1. Computation of a force-displacement response by mean of a pushover analysis identifying a

yield displacement dy (bilinear approximation, e.g. Figure 4)

2. Calculation the effective mass m* and the elastic period T

*(=Tel)

6. Calculation of the spectral acceleration at T*

and computation of the strength reduction factor

as: R=m*.Sa(T

*)/Fy

3. Calculation of ηin as a function of R (Figure 8, Eq. (6) or, for simplicity Eq. (7))

4. Calculation of Teff= T.R

0.5

5. Calculation of the spectral displacement at Teff, Sde(Teff)

6. Sd(predicted)= ηin. Sde(Teff)

The predicted displacement (or ductility) was compared with the same quantity calculated by

the nonlinear 7 SDOF systems defined in previous sections. Figure 9 shows the predicted ductility

demand (by mean of optimal stiffness method) vs. the calculated ductility demand (computed by mean

of nonlinear SDOF systems) for the two outermost systems (F and S).

0 1 2 3 4 5 6 70

1

2

3

4

5

6

7

Ductility SDOF [-]

Duct

ility

op

tim

um

sti

ffn

ess

met

ho

d [

-]

0 1 2 3 4 5 6 70

1

2

3

4

5

6

7

Ductility SDOF [-]

Duct

ility

op

tim

um

sti

ffn

ess

met

ho

d [

-]

Figure 9. Predicted ductility demand vs. inelastic SDOF ductility (mean, in black; mean ± 1 standard deviation,

in blue; on left 100% shear, ξhyst=19.9%, and on right 0% shear, ξhyst=13.8%).

In particular, the predicted vs. calculated by inelastic SDOF ductility demand and the relative

variation of the prediction are plotted in the graphs, that are repeated for S and F structure (from shear

to flexural behaviour). Graziotti (2013) reported similar graphs for all the analyzed structures. On the

graphs lines for mean and for mean ± one standard deviation are reported. The red dashed line on the

diagonal of the graphs represent the best possible prediction: μ(opt.stiff.)=μSDOF. From the graphs it

is possible to derive some considerations:

- There is a dispersion in the results also for μ<1 (elastic range) because the model used for the

SDOF is elastic nonlinear in this range.

- The method is able to approximate with an acceptable accuracy the mean value of the

displacement demand for all the considered ductility values. A slight overestimation was

noticed in all the structures considered. (The correlation coefficient is 82% for the shear

dominated structure and 85% for the flexural dominated one.)

- The coefficient of variation of the predicted values was calculated. The CoV of the results is

similar remains quasi constant for all the ductility levels. This is very important facilitating the

introduction of a safety factor, as example, in a code formulation. The average CoV of the

predicted ductility is higher for the shear dominated structure (0.43), and lower for the flexural

one (0.36).


The optimal stiffness method provide a slight better prediction for flexural dominated

structures. An opposite trend was observed for the corrected N2 method (Graziotti et al., 2014a) which

better approximates the nonlinear behaviour of shear dominated structures. However, good results

were obtained also using the simplified formulation reported in Eq. (7).

CONCLUSIONS

A simplified system to simulate the nonlinear behaviour of masonry building was studied. In particular

an equivalent SDOF system was developed and calibrated statically and dynamically. It synthesized,

with sufficient accuracy, the seismic response of the case study MDOF system under investigation,

while minimizing the computational effort.

The SDOF model was used to evaluate the displacement demand of a masonry structure in a

reliable and fast way proposing a relation between strength reduction factor and displacement

reduction factor. The proposed method is based on secant period spectral displacement. In particular a

non-iterative optimum stiffness (Lin & Miranda, 2004) method was calibrated using the available

nonlinear analysis results. The method based on this simplified formulation is able to give a fair

average prediction, in particular for flexure dominated structures. The encouraging results suggest

continuing the study. For example, nonlinear time history analyses on larger statistical samples

(models) should be performed in order to validate the formulation with a wider structural database.

ACKNOWLEDGMENTS

This work was carried out with the partial financial support of the EUCENTRE Executive Project

2012-2014 e3 “Seismic vulnerability of masonry buildings”, funded by the Italian Department of Civil

Protection. The authors would also like to thank Dr. Alessandro Galasco and Mr. Emanuele Bossi for

their valuable support in the numerical analyses.

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A DIRECT METHOD TO COMPUTE THE INELASTIC DISPLACEMENT DEMAND OF MASONRY STRUCTURES … ·...

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