Seismic analysis of masonry buildings: equivalent frame …paginas.fe.up.pt/~eurodyn2014/CD/papers/031_MS01_ABS_1648.pdf · ABSTRACT: The seismic analysis of masonry buildings by

ABSTRACT: The seismic analysis of masonry buildings by means of the equivalent frame simplified methodology has

received considerable attention in the last decades; the walls are modeled with macroelements to represent the in-plane non-

linear behavior of the panels. The mechanical characterization of these macroelements is the crucial point that defines the

specific frame approach. In the present paper, the spandrels and the piers of the masonry wall are modeled through fiber section

force-based elements that accounts for both axial-flexural and shear deformations, while the connecting nodes are assumed as

rigid. The approach is investigated using the open source computational platform OpenSees that allows coupling between

flexural and shear responses through a section aggregator procedure. The interaction between axial load and bending behavior is

automatically accounted for by the fiber section model. The shear response is given by a phenomenological law and flexure -

shear coupling is enforced at the element level. The approach is very simple and quite promising for both research and practice

and the first numerical tests on sample cases show both numerical robustness in monotonic and cyclic analyses under vertical

and horizontal loads and a satisfactory agreement with available experimental test results.

KEY WORDS: masonry, macro-element, finite element, frame-equivalent model, nonlinear analysis, fiber section, N-M

interaction diagram.

1 INTRODUCTION

A significant part of the historic heritage in Italy and many

other European countries consists of old masonry

constructions and this system is also widespread all over the

world. Usually, older unreinforced masonry (URM) buildings

were conceived to carry only vertical loads and most of them

have experienced a continuous process of modification over

the years. Moreover, masonry is a composite material, whose

components (bricks or stones and mortar) can be very

different due to historical or technological reasons, and this

makes its behavior difficult to predict. This difficulty is also

due to different possible failure modes and to non-

uniformities in construction quality.

2 GENERALITIES ABOUT MASONRY BUILDINGS

2.1 Local and global response of masonry buildings

Damage observed in past earthquakes showed that masonry

buildings are vulnerable to local failures, mainly due to the

out-of plane response of walls. These failure mechanisms are

mainly caused by poor connections between the orthogonal

walls and between walls and floors.

Without a box behaviour, the seismic vulnerability mainly

depends on the out-of-plane collapse mechanisms of the

resisting macroelements – e.g. masonry external walls or

portions of them – rather than on the in-plane ultimate strain

state in the masonry. In buildings with well connected walls

the box behavior governs the seismic response; this buildings

are the focus of the present paper.

2.2 The Equivalent frame models

To deal with the global response of real buildings, many

researchers introduced a modelisation of the masonry walls as

one-dimensional macro-elements, in such a way to represent

the walls by means of framed structures, and to apply then

conventional methods of structural mechanics (e.g. [1] and

papers there quoted). This idea has been indeed investigated

and developed by many researchers, that led for example to

the POR method developed in the seventies [2], and that was

more recently used in programs such as 3Muri [3] or SAM

[4]. Some common assumptions are made in these type of

models: the wall deformation is assumed to be lumped in piers

and spandrels and the other parts of the wall are considered

rigid. A frame equivalent model is shown in Figure 1, where

the piers and the spandrels are represented by columns and

beams, respectively.

Figure 1. A frame equivalent representation of a wall

Seismic analysis of masonry buildings: equivalent frame approach with fiber beam

elements Vincenzo Sepe, Enrico Spacone, Eva Raka & Guido Camata

University “G. D’Annunzio” of Chieti-Pescara, Department of Engineering and Geology, Italy

email: [email protected], [email protected], [email protected] & [email protected]

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014Porto, Portugal, 30 June - 2 July 2014

A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)ISSN: 2311-9020; ISBN: 978-972-752-165-4

237

The main advantage of this numerical approach is its

computationally efficiency and for this reason the equivalent

frame method may also be suitable for studying very large

buildings or aggregates.

With an appropriate formulation, in fact, the macroelement

model can represent with a good approximation the cyclic

shear and the flexural response of masonry panels.

In particular, the failure mechanisms of masonry piers

subjected to horizontal (e.g. seismic) loads are the following

[5][6] (see Figure 2):

a) rocking failure: the failure is related to the crushing of

the pier in the compressed zone;

b) shear-diagonal failure: the failure of the pier is due to

excessive shear stresses and consequently formation of

inclined diagonal cracks;

c) shear-sliding failure: the failure is associated to

horizontal cracks in the bed-joints.

Figure 2. Failure mechanisms of a masonry pier: a) rocking

failure; b) shear-diagonal failure; c) shear-sliding failure

(adapted from [6]).

3 THE EQUIVALENT FRAME MODEL USING FIBER

FRAME ELEMENTS

3.1 Formulation of the fiber section beam element

In the equivalent-frame approach discussed here the flexural

behavior is computed considering a force-based frame

element with the fiber section model and performing a

moment-curvature analysis. The approach is investigated

using the open source computational platform OpenSees

[7][17] and the masonry walls are modeled as

"NonlinearBeamColumn" elements. This command is used to

construct a beam element object, which is based on the

iterative force-based formulation, that has been originally

developed for the simulation of reinforced concrete members

under seismic actions [8].

In the distributed inelasticity used in this model the element

response is determined by numerical integration of the

nonlinear response at several monitored sections along the

element.

The flexure and axial behavior is modeled using a nonlinear

fiber stress-strain relation (material nonlinearity) (see

Figure 3.

Figure 3. Fiber section discretization

3.2 Shear implementation: the section aggregator

Shear deformations, which are often neglected in more

simplified procedures, have an important role in the total

nonlinear displacements in masonry walls, and for this reason

they have been included in the proposed model by means of a

trilinear force displacement law with pinching hysteretic

behavior. A section aggregator in OpenSees framework is

then used to combine flexural and shear behavior. This

command aggregates into a single section force-deformation

model the flexural response - described here by means of a

fiber-section approach - and the shear response -described by

the trilinear V- law.

Each Material-Object represents the section force-

deformation response for a particular degree-of-freedom, and

at the section level there is no interaction between responses

in different dofs.

Figure 4. Section Aggregator

Using the force-based formulation [8], equilibrium is

imposed at the element level, thus enforcing shear-bending

interaction.

4 PARAMETRIC ANALYSIS OF MASONRY WALLS

4.1 Opensees parametric analysis implementation

In the present Section, the fiber model approach is used to

simulate the seismic response of two masonry panels. The

accuracy in describing the material properties, combined with

the low computational effort required to perform the analyses

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014

238

give the possibility to carry out several analyses. In particular,

pushover analyses were performed on two columns with

different slenderness ratios to determine the drifts at which

failure of the column occur. The panels geometry analyzed

are shown in Figure 5.

Figure 5. Panels geometry (N: total load; dimension in [m])

The Opensees input data of the two panels are summarized

in Table 1.

Table 1. Geometry of panels in Figure 5 (tCol : thickness).

LCol [m] HCol [m] tCol[m]

First panel 1.6 3 0.25

Second panel 3 3 0.25

4.2 Masonry constitutive law

In the proposed model the masonry behavior under

compression load is described by the modified Kent and Park

model [9][10]. Although originally introduced for concrete,

this law (see Figure 6) can reproduce many of the properties

of masonry nonlinear behaviour: linear elasticity, nonlinear

relationship before the peak stress and softening branch.

Figure 6. Concrete01 Kent and Park constitutive law

The following expressions [10][17] are used for the

modified Kent and Park model:

00

0 2cc

cpc f

(1)

cp

ccu

ccpcucucu fff

0

0

(2)

cucu f (3)

where fcp, fcu , c0 and cu denote the compressive strength and

the residual strength, respectively, and their corresponding

strains.

The second constitutive relation used in this study, that

includes tensile strength branch, is called in Opensees

Concrete06 (see Figure 7).

Figure 7. Concrete06 constitutive law

The compressive constitutive law of Concrete 06 is defined

as the Thorenfeldt - base curve, which is similar to the one

defined by Popovics [11][17]:

nk

c

c

cc

n

n

f

0

0'

1

(4)

where f 'c and ε0 are the compressive strength and the

corresponding strain, respectively, while n and k are

parameters.

According to Belarbi and Hsu [12], the tensile envelope is

described by

c

cr

crccrc

f

(5)

b

c

crcrccrc f

(6)

where fcr and εcr are the tensile strength and strain,

respectively, and b is a parameter.


239

4.3 Parametric analysis of the slender wall, panel one

The panel is modeled with a fiber section force-based element

described in section 3.1. The lateral force resistance of this

wall depends primarily on the boundary conditions, on the

effect of the axial loads and on the characteristics of the

material constitutive law.

The parameters considered in this work to investigate their

influence on the wall global behavior are: the material

constitutive law softening branch, the initial axial load and the

tensile strength. The geometry, boundary and load conditions

are shown in Figure 8.

Figure 8. Boundary conditions and geometry of first panel

In the first analysis the constitutive law chosen to represent

the behavior masonry is Concrete 01. Different configurations

of the softening branch of the constitutive law are varied as

shown in the Table 2 and Figure 9.

Table 2. Concrete 01 - Material properties (symbols in Fig. 6).

fcp[MPa] fcu[MPa] c0 cu

material 1 4 3.5 0.004 0.02

material 2 4 2.5 0.004 0.02

material 3 4 1.5 0.004 0.02

Figure 9. Effect of different materials constitutive law

parameters

The base shear-displacement curves for each material

constitutive law (see Figure 9) are presented in Figure 10. The

maximum axial load capacity for the slender panel is equal to

Nmax=1600kN. The pushover analyses shown in Figure 10

have all the same value of initial axial load 0.5Nmax =

800kN. It is evident from the graph that the behavior of

material 3 is more brittle than the other ones.

Figure 10. Shear-displacement curves for the different

material constitutive laws (see Figure 9).

The softening branch influences the behavior of the post

peak response of the panel.

The second parameter investigated is the initial vertical

load. The shear-displacement curves shown in Figure 11 are

obtained with the concrete law material 3 (see Figure 9) for

different value of axial load. The graph shows that for the

axial load equal to 0.1Nmax = 160kN the panel behavior is

more ductile, but shows a lower lateral force resistance.

Figure 11. Shear-displacement curves for different values of

axial load imposed on the column

It is clear that by increasing the initial axial load, the wall

shear strength increases but the wall behavior becomes more

0

1

2

3

4

5

0 0,005 0,01 0,015 0,02 0,025

[M

Pa]

"Material 2"

"Material 3"

"Material 1"

0

50

100

150

200

250

0 5 10 15 20 25

Vb

ase[

kN]

Displacement d[mm]

Material 3

Material 1

Material 2

0

50

100

150

200

250

0 5 10 15 20

Vb

ase[

kN]

Displacement d[mm]

Material 3 N=0.5 Nmax





240

brittle The maximum axial load used in this analysis is equal

to 0.5Nmax = 800kN. This value correspond to the maximum

moment capacity of the wall section as shown in the axial

load-moment interaction diagram shown in Figure 13.

Figure 12. Slender wall fiber section

Figure 13. M-N Interaction diagram for the section described

in Figure 12

Another important parameter investigated in this study is

the masonry tensile strength. The analyses are performed

using the material Concrete 06 and Concrete 01 described in

section 4.2. Concrete 01 is defined as material 3 in Table 2

and Concrete 06 is described in Table 3.

It should be noted (Figure 14) that the softening curve of

Concrete 06 is calibrated to match as close as possible

material 3, therefore the only parameters varied is the material

tensile strength.

Table 3. Concrete 06 Material properties (symbols in Fig. 7)

f 'c[MPa] fcr[MPa] 0[-] cr[-] n k b

M6 4 0.3 0.004 4. E-05 2 1 2

The two stress-strain constitute laws are shown in Figure 14.

Figure 14. Two materials constitutive law with tensile strength

(concrete06) and without tensile strength (concrete 01)

Figure 15 shows the horizontal load-displacement curves

obtained with Concrete 01 (material 3) and Concrete 06. The

curves show that after reaching the maximum force, the load

decreases abruptly.

The post peak behavior is controlled by the softening

branch of the curve. The results show that the tensile strength

influences slightly the wall behavior.

Figure 15. Pushover shear-displacement curves using the

materials of Figure 14 (N = 0.5 Nmax).

Figure 16 shows the stress distribution in the section during

the analyses. At the beginning of the analyses, point A, the

stress distribution is constant. At point B the section starts

cracking, at point C the section is cracked and the neutral axis

shifted, the stress on the last fiber reached the maximum

compressive strength of 4MPa. Point D shows that increasing

the displacement the stresses redistribute along the section,

and part of the section is on the post peak branch.

Figure 16. Section stress distribution [MPa] for the steps

indicated in Figure 15

0

100

200

300

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

M [

kN

m]

N/Nmax

-1

0

1

2

3

4

5

-0,005 1E-17 0,005 0,01 0,015 0,02 0,025

[M

Pa]

"Concrete 01"

"Concrete 06"

0

50

100

150

200

250

0 2 4 6 8 10

Vb

ase[

kN]

Displacement d[mm]

B

C

A

DB

C

D

A

Concrete 01

Concrete 06


241

Additional analyses were performed with a lower axial load,

0.1Nmax = 160kN. The results are shown in Figure 17. The

shear - displacement curves are almost identical indicating

that the tensile strength does not play a major role on the load-

displacement behavior.

Figure 17. Pushover shear-displacement curves using the

materials of Figure 14 (N = 0.1 Nmax).

4.4 Parametric analysis of the squat wall, panel two.

The deformation capacity strongly depends on the type of

failure mechanism (shear or flexure). The panel described in

the previous paragraph is slender and the failure is governed

by flexure. For this reason a second panel is investigated in

order to analyse the shear failure. The lowest deformation

capacities, in terms of horizontal drift θ = δ/H (horizontal

deflection/ height of the panel) were found in correspondence

of diagonal cracking failures, involving cracking of the units.

The second panel is modelled with the material 3 constitutive

law described in Figure 9, with other properties shown in

Table 4, where denotes the friction coefficient.

Figure 18. Boundary conditions and geometry of panel two

Table 4. Panel two data

E [MPa] G [MPa] ftu [MPa]

Panel two 2000 1000 0.15 0.4

The second panel is modelled with fixed-fixed boundary

conditions (see Figure 18). The shear strength is reached when

the stress reaches the tensile strength of the masonry ftu =0.15

MPa (see Table 4).

In this examples the resulting shear strength is calculated

using the following expression [5][13]:

tu

tud

fb

ltfV 01

(7)

where ftu is the masonry tensile strength, 0 the compression

stress of the section (0=N/lt), l the width, t the thickness of

the wall section and b is a parameter that depends on the

aspect ratio h/l.

The shear behaviour of the wall is described by means of a

trilinear lateral force-displacement law with pinching

hysteretic behavior. The constitutive law V- with initial

axial load equal to N=300kN (i.e. 0.1 Nmax) is described in

Figure 19.

Figure 19. V-shear spring

Figure 20 shows the flexure (Opensees analysis) and shear

failure curves (eq.7) of the panel for different axial load ratio.

The figure shows that for an axial load lower than 0.074

Nmax the panel fails in flexure and for a higher axial load the

panel fails in shear.

0

20

40

60

80

100

0 2 4 6 8 10

Vb

ase[

kN]

Displacement d[mm]

Concrete 01

Concrete 06

0

50

100

150

200

250

0 0,001 0,002 0,003 0,004 0,005

Vb

ase[

kN]


242

Figure 20. The V-N Interaction diagram (shear spring 1)

Figure 21. V-d base shear displacement curve

The last analyses are performed to investigate the influence of

the constitutive law V-(shear spring behavior). Figure 22

shows the two laws used, one with maximum shear strength

equal to Vd1 =199.2 kN and the other with Vd2 =240 kN.

The two base shear curves for N=300kN (i.e. 0.1 Nmax)

without and with shear spring are shown in Figure 21.

The base shear curves for N=240kN (i.e. 0.08 Nmax) without

shear springs and with the two shear springs are shown in

Figure 23. The behavior of the curve without shear springs is

stiffer than the other two, due to the fact that the shear spring

gives the possibility to consider in addition to the flexure

deformation also the shear deformation. The curves show

also that for spring 2 the failure is in flexure because the shear

strength is higher than the flexural strength. In the case of

spring 2 the failure is in shear because the shear strength is

lower than the flexural strength.

Figure 22. V-shear springs

Figure 23. V-d base shear displacement curve

CONCLUSIONS

Masonry wall in-plane failure modes can be represented

efficiently with the nonlinear fiber-beam equivalent-frame

model presented in this study.

Both shear and flexural deformations, which play an

important role in the global response of the masonry walls, are

taken into account through constitutive laws widely accepted

in the scientific literature, combined by means of a section

aggregator in OpenSees framework.

The parametric analyses on two different panels show the

influence of various parameters on the panel behavior, and in

particular slenderness ratio, axial load (that plays a major role

on the strength and ductility of the panel), material

constitutive law.

The force-based fiber section model appears to be a very

promising approach to model the masonry behavior. Both

flexural behavior and shear behavior can be represented with a

good approximation, the numerical model is very stable and

0

100

200

300

400

500

600

700

800

0 0,1 0,2 0,3 0,4 0,5

V(k

N)

N/Nmax

Flexure failure (Opensees)

Diagonal shear failure

0

50

100

150

200

250

300

0 5 10 15 20

Vb

ase[

kN]

Displacement d[mm]

Shear and flexure

Flexure

0

50

100

150

200

250

300

0 0,002 0,004 0,006 0,008 0,01

Vb

ase[

kN]

Shear spring1

Shear spring 2

0

50

100

150

200

250

0 5 10 15 20 25

Vb

ase[

kN]

Displacement d[mm]

Flexure

Shear spring1 and flexure

Shear spring 2 and flexure


243

efficient and can be easily extended to model complex

masonry buildings.

Other tests on sample cases show also both numerical

robustness in monotonic and cyclic analyses [14] under

vertical and horizontal loads and a satisfactory agreement with

available experimental test results for masonry panels [15]

and 3D buildings [16].

REFERENCES

[1] R. Marques and P.B. Lourenco, Possibilities and comparison of

structural component models for the seismic assessment of modern

unreinforced masonry buildings, Computers and Structures, 89, pp.

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[2] M.Tomaževič, The computer program POR. Report ZRMK (in

Slovene), 1978.

[3] A. Brencich and S. Lagomarsino, A Macro-Element dynamic model for

masonry shear walls. In G.N. Pande & J. Middleton (ed.s), Computer

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[4] G. Magenes and A. Della Fontana, Simplified non-linear seismic

analysis of masonry buildings, 5th International Masonry Conference,

Poc. of the British Masonry Society, 1998.

[5] G. Magenes and G.M. Calvi, In-plane seismic response of brick

masonry walls, Earthq. Engin. and Struct. Dyn., Vol. 26, pp. 1091-

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[6] Caliò, I, Marletta M. and Pantò B., A simplified model for the

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Civil-Comp Press, 2005

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[16] G. Magenes G. M. Calvi and R. Kingsley, (1995), Seismic testing of a

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Seismic analysis of masonry buildings: equivalent frame …paginas.fe.up.pt/~eurodyn2014/CD/papers/031_MS01_ABS_1648.pdf · ABSTRACT: The seismic analysis of masonry buildings by