1. No category

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
Porto, Portugal, 30 June - 2 July 2014
A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)
ISSN: 2311-9020; ISBN: 978-972-752-165-4
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]
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 nonlinear 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 nonuniformities in construction quality.
2
2.1
GENERALITIES ABOUT MASONRY BUILDINGS
Local and global response of masonry buildings
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.
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.
Figure 1. A frame equivalent representation of a wall
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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
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
3.1
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 forcedeformation response for a particular degree-of-freedom, and
at the section level there is no interaction between responses
in different dofs.
THE EQUIVALENT FRAME MODEL USING FIBER
FRAME ELEMENTS
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 4. Section Aggregator
Using the force-based formulation [8], equilibrium is
imposed at the element level, thus enforcing shear-bending
interaction.
4
4.1
PARAMETRIC ANALYSIS OF MASONRY WALLS
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
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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
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.
   c 0    f cp
 
 
 2 

 c 0   c 0 
(1)
    c0 
  f cp
 cu     cu     f cu  f cp 
  cu   c 0 
(2)
   cu    f cu
(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 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).
First panel
Second panel
4.2
LCol [m]
1.6
3
HCol [m]
3
3
tCol[m]
0.25
0.25
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 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]:
 
n c 
 0 
 c  f c'
nk
 c 
n  1   
 0 
(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
f 
 c   cr   c   cr  c
  cr 
Figure 6. Concrete01 Kent and Park constitutive law
The following expressions [10][17] are used for the
modified Kent and Park model:
 
 c   cr   c  f cr  cr 
 c 
(5)
b
(6)
where fcr and εcr are the tensile strength and strain,
respectively, and b is a parameter.
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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
4.3
Parametric analysis of the slender wall, panel one
250
Material 1
200
Vbase[kN]
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.
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.
150
Material 2
100
50
Material 3
0
0
5
10
15
20
25
Displacement d[mm]
Figure 10. Shear-displacement curves for the different
material constitutive laws (see Figure 9).
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).
material 1
material 2
material 3
fcp[MPa]
4
4
4
fcu[MPa]
3.5
2.5
1.5
c0
0.004
0.004
0.004
cu
0.02
0.02
0.02
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.
250
5
Material 3 N=0.5 Nmax
200
Material 3 N=0.35 Nmax
4
"Material 1"
"Material 2"
Vbase[kN]
[MPa]
3
Material 3 N=0.25 Nmax
150
100
Material 3 N=0.1 Nmax
2
"Material 3"
50
1
0
0
0
0
0,005
0,01

0,015
0,02
Figure 9. Effect of different materials constitutive law
parameters
0,025
5
10
15
20
Displacement d[mm]
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
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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
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 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.
250
Concrete 06
Figure 12. Slender wall fiber section
C
200
C
300
Vbase[kN]
150
200
M [kNm]
Concrete 01
B
D
D
B
100
50
100
A
0
0
0
0,2
0,4
0,6
0,8
1
N/Nmax
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.
0
A
2
4
6
8
10
Displacement d[mm]
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.
Table 3. Concrete 06 Material properties (symbols in Fig. 7)
f 'c[MPa]
M6
4
0[-]
cr[-]
n k b
0.004
4. E-05
2 1 2
fcr[MPa]
0.3
The two stress-strain constitute laws are shown in Figure 14.
5
4
"Concrete 01"
[MPa]
3
"Concrete 06"
2
1
-0,005
0
1E-17
-1
0,005
0,01
0,015
0,02
0,025

Figure 14. Two materials constitutive law with tensile strength
(concrete06) and without tensile strength (concrete 01)
Figure 16. Section stress distribution [MPa] for the steps
indicated in Figure 15
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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
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 loaddisplacement behavior.
100
Concrete 06
80
Vbase[kN]
Concrete 01
60
Table 4. Panel two data
Panel two
E [MPa]
G [MPa]
ftu [MPa]

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]:
Vd 
40
f tu lt
b
1
0
(7)
f tu
20
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
0
0
2
4
6
8
10 aspect ratio h/l.
The shear behaviour of the wall is described by means of a
Displacement d[mm]
trilinear lateral force-displacement law with pinching
hysteretic behavior. The constitutive law V- with initial
Figure 17. Pushover shear-displacement curves using the
axial load equal to N=300kN (i.e. 0.1 Nmax) is described in
materials of Figure 14 (N = 0.1 Nmax).
Figure 19.
4.4
Parametric analysis of the squat wall, panel two.
250
200
Vbase[kN]
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.
150
100
50
0
0
0,001
0,002

0,003
0,004
0,005
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.
Figure 18. Boundary conditions and geometry of panel two
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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
800
300
Flexure failure (Opensees)
700
Shear spring 2
250
600
Diagonal shear failure
400
300
Vbase[kN]
V(kN)
Shear spring1
200
500
150
100
200
100
50
0
0
0,1
0,2
0,3
0,4
0
0,5
0
N/Nmax
0,002
0,004
0,006
0,008
0,01

Figure 20. The V-N Interaction diagram (shear spring 1)
Figure 22. V-shear springs
300
250
Flexure
Flexure
250
Shear spring 2 and flexure
Shear spring1 and flexure
200
Vbase[kN]
Vbase[kN]
200
150
100
150
100
Shear and flexure
50
50
0
0
5
10
15
20
Displacement d[mm]
0
0
5
10
15
Displacement d[mm]
20
25
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 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
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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
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].
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