-
Material Model For URM With A Application On The Seismic
Assessment Of A Hospital In Tyrol, Austria
Suikai Lu1, Rudolf Heuer2, Marian Ralbovsk3, Rainer Flesch4
1Dipl.-Ing., Scientist, Business Unit Transport Routes
Engineering, arsenal research, 1030 Vienna, Austria,
[email protected]
2Professor, CMSD Center of Mechanics and Structural Dynamics,
Vienna University of Technology, 1040 Vienna, Austria,
[email protected]
3Dipl.-Ing., Scientist, Business Unit Transport Routes
Engineering, arsenal research, 1030 Vienna, Austria,
[email protected]
4Professor, Head of Business Unit, Business Unit Transport
Routes Engineering, arsenal research, 1030 Vienna, Austria,
[email protected]
1) Abstract The content of this paper presents a new model for
unreinforced masonry (URM), based on the plastic material model by
Ganz [1]. Here, the idea of Ganz [1] was followed, to use a
combination of yielding surfaces in the stress space, where each
surface describes one failure mode. Compared to the material model
of Ganz [1], the new model was extended to cover even the tension
strength aspects of masonry constructions in both directions (in
plane, orthogonal and parallel to the horizontal joints) for each
failure mode. The model still consists of only 5 convex yielding
surfaces which describe the following failures, respectively: -)
tension failure -) compression failure -) shear failure -) sliding
along the horizontal joints -) tension failure in the horizontal
joints Additionally to the theoretical background an application is
shown, where laboratory experiments are used to test and calibrate
the material model and its parameters. Therefore, this model was
implemented into the Finite Element Software ANSYS. The
implementation involves all the failure modes and an automatic
searching for the positions of the masonry structures in the global
FE-model. The result of this numerical implementation is the
display of the cracked and yielded areas of the wall, respectively.
The accuracy depends on the size of the finite elements chosen in
the model. Furthermore, this model was applied to seismic
assessment of a hospital located in Innsbruck, Tyrol, Austria. In a
first step, in-situ measurements have been carried out to detect
the dynamic parameters (natural frequencies and mode shapes) for
updating the finite element model with these properties.
Afterwards, the FE-model was analyzed by response spectra method
using SRSS combination rule to simulate a code earthquake, as
required in the Austrian national code for designing of buildings
with seismic actions, NORM B 4015 [9]. Finally, the new material
model was introduced to display cracks in the structure occurring
due to the earthquake excitation. 2) Introduction Based on the
classical theory of plasticity, Ganz [1] formulated two material
models in 1985, where he described yielding surfaces for each
failure mode in masonry. First he formulated a model for URM, which
covers only 5 yielding surfaces. Then he developed a model to
include tension strength, where 12 surfaces were needed. The new
model, presented in this paper extends the basic Ganz-model
(without tension strength) by including tension in a new effective
approach capturing the main failure modes of unreinforced masonry
structures. Both the analytical derivation as well as the software
implementation in form of a macro for the FE Software ANSYS, are
presented. To verify the model, experimental laboratory tests were
analyzed numerically by using this macro, in order to confirm and
prove the analytical work. In the last part of this paper, a
practical application on a lifeline structure, a hospital in Tyrol,
Austria was analyzed by using this new approach.
-
3) Original Model by Ganz In 1985, Ganz [1] formulated a
material model for URM, where the two components brick and joints
were split. For the component brick, he focused on the most
generally form, perforated bricks (see Figure 1). Within a limiting
approach this theory can be also applied to a solid brick.
Figure 1 Brick Element
The considered forces and cross sections are defined as
follows:
2,1, xxx FFF += (1)
xF stands for the normal force, and 1,xF , 2,xF act upon xA ,
xyA , respectively.
2,1, yyy FFF += (2)
yF stands for the horizontal force, and 1,yF , 2,yF act upon xA
, xyA , respectively.
2,1, xyxyxy FFF += (3)
xyF stands for the shear force, and 1,xyF , 2,xyF act upon xA ,
xyA respectively. The total cross sectional area reads
0AAAA xyx ++= (4) By combining the uni- and biaxial parts of the
forces using principal forces, the following three equations for
failure in brick result:
021 = yxxyf tension failure in brick (5)
0))((22 ++= cyycxxxy fff compression failure in brick (6)
0)(23 ++= cyyyxy ff shear failure in brick (7) For the component
mortar, supposing that the vertical joints are not filled, it is
only necessary to focus on the horizontal joints. With this
assumption the model equations are on the conservative side.
Sliding in the joints is modeled by means of the Mohr-Coulombs
law,
0))tan(( 224 = xxy cf sliding along the horizontal joints
(8)
-
Finally, a tension cut-off for the Mohr Coulombs friction law is
formulated,
0)24
tan(225
+++=
cf xxxy tension failure in the horizontal joints (9)
(Eq 5 to Eq 9) describe the law for URM according to Ganz
considering the components of compressive strength cxf , cyf ,
respectively. 4) Modified Material Model The new model, developed
within this research work, was expanded by the former model to
consider also tension stresses. Taking the uniaxially exposed parts
of the brick section, the governing equations can be written
as:
xcxxt AFA 1, (10)
01,1, xyy FF (11) where c , t denote compression strength and
tension strength of brick, respectively. For the biaxially exposed
parts, the inequation can be written by using the principal forces
in terms of
xycxyyxyx
xyt AFFFFF
FA +
+= 2 2,
22,2,2,2,
2,1 22)(
(12)
Combining (Eq 10 to Eq 12), and by substitution of
txxxy
t fAAA
=+
)( , tyxy
t fAA
= (13)
the former derived material laws for brick (Eq 5 and Eq 7) can
be replaced by
0))((21 = tyytxxxy fff (14)
( ) 023 ++= tycyytycyyxy fffff (15) The surface function 2f (Eq
6) remains unchanged. To enclose tension in joints, criteria 4f (Eq
8) can be remained unchanged too, but the equation for tension
cut-off 5f (Eq 9) has to be modified (Figure 2).
-
Figure 2 Tension cut-off
Finally, 5f can be written in terms of
02)(sin1sincos
2 2225 +
++= txtxxtxxtx
xxy ffffc
f
(16)
The new material model (Eq 14, 6, 15, 8, 16) can be displayed
graphically as a combined yielding surface (Figure 3):
Figure 3 Combined Yielding Surface of the extended model for
URM
The failures of the corresponding five yielding surfaces can be
interpreted in the same way as in the original model by Ganz
(compare to the previous section). 5) Input URM parameters for
analytical analysis For numerical implementation of the URM model,
some main input parameters are necessary. The following section
presents an overview of the evaluation of the data used for the
analysis.
-
-) Compression strength orthogonal to the horizontal joints [2]:
cxf 25.075.0
mcccx fKf = (17) with the module 5.10.1 =K , and the compression
strength of the considered mortar mcf . -) Compression strength
parallel to the horizontal joints [3]: cyf URM consisting of solid
brick:
cxcy ff = 75.0 (18) URM consisting of perforated brick:
cxcy ff = 5.0 (19) -) Tension strength orthogonal to the
horizontal joints [4]: txf The tension strength of URM depends only
on the tension strength of the used mortar mtf and can be written
as:
mttx ff = 32
(20)
-) Tension strength parallel to the horizontal joints [3]: tyf
In case of tension strength parallel to the horizontal joints, two
different crack types should be treated separately. Crack Type A
(Figure 4) occurs, if bricks are made of low quality materials and
if large portions of normal stresses x are exposed to masonry
members.
Figure 4 Crack Type A, taken from [5]
( )mbhorizbtb
ABTty hhfh
f+
=
2,
_, (21)
where horizbtf , denotes the horizontal tension strength of used
brick Crack Type B (Figure 5) is typical for high strength bricks
in combination with low quality mortar and/or if the exposing
normal stress x is very small.
Figure 5 Crack Type B, taken from [5]
-
( )mbub
BBTty hhlf
+
=2_,
(22)
where u stands for shear strength. -) Young's modulus orthogonal
to the horizontal joints [2]: Ex
cxx fE = 1000 (23) -) Young's modulus parallel to the horizontal
joints [6], [7]: EY
fE
E XY +=
21 (24)
where b
b
lh
f
4
.
-) Friction angle: ; and shear strength under no compressive
stress: c The Friction angle varies normally between 20 and 40, and
the shear strength under no compressive strength is between 0.2 2.5
MPa where some numbers for c are listed in [2] in table 3.4. -)
Shear strength
( )xu c += )tan( (25) The shear strength is in accordance to a
conventional Mohr-Coulomb friction law, where x denotes the normal
stress. 6.) Implementation of the new model into the FE-Software
ANSYS: To implement the model into a FE-software, it should be
taken into account, that
tancftx . (26)
If this condition is not maintained, txf has to be set equal to
tanc
, before continuing the analysis.
The stress state of the analyzed object must be verified. Four
different positions can be distinguished (Figure 6). Therefore, an
extra condition (Eq 27), has to be considered.
Figure 6 Positions of the analyzed stress state
-
0010
sincos2/5 isresultthethenisresultthethen
ffcf xtxtxT
+
++=
,
(28) the four positions P1 - P4 can be separated numerically in
the following way: P1: 0102/55 == Tff ; P2: 0002/55 == Tff ; P3:
0012/55 == Tff ; P4 1112/55 == Tff (29) Only in case of P4, the
analyzed stress point is outside the combined yielding surface,
which means that cracks will occur. The combined yielding surface
was implemented into ANSYS (Figure 7) including the conditions
described previously.
Figure 7 Screenshot of the implemented Program for URM
The implementation also involves an automatic detecting of the
positions of masonry structures in the global coordinate system of
the FE-model. 7) Verification of the material model by numerical
analysis of laboratory tests: In 1982, Ganz et al [8] made
experimental tests on URM. They exposed the test specimen stepwise
until collapse. Those walls were exposed either uniaxially and/or
biaxially. Also the angle of the horizontal joints of each specimen
varied between 0; 22.5; 45; 67.5 and 90 degrees. To verify the
material model and to demonstrate the implemented macro, these
experimental tests have been recalculated numerically and were
compared to the test results. Input data for the computer
simulation are given in Table 1.
-
Table 1 Input data for computer simulation
density [kg/m] 905 strengths
cxf [N/mm] 7.6
cyf [N/mm] 2.7
txf [N/mm] 0.03
tyf [N/mm] 0.00
Cohesion c [N/mm] 0.06 Friction angle [] 39
In Figure 8, K3 of the test series is displayed graphically, to
show the effectiveness of the implementation.
Figure 8 left: numerical implementation; right: laboratory
experiment on real test specimen
A summary of additional results is given in Table 2, where
stresses were measured in the middle of the wall and were taken
from the middle element of the FE-Model.
Table 2 Comparison between test results and numerically analyzed
results
Test Angle of horiz. joints
Ratio FH / FV
Measured crack at laboratory (taken from [8])
Numerically analyzed crack
X Y XY X Y XY
[] [ ] [N/m] [N/m] [N/m] [N/m] [N/m] [N/m] K1 22.5 1 / -10.9
-8.00 e4 -9.20 e5 4.20 e5 -7,89e4 -9,10e5 4,14e5 K3 0.0 0 / -1 0
-7.63 e6 0 0 7.61e6 0 K4 90.0 0 / -1 -1.83 e6 0 0 -2.70e6 0 0 K6
45.0 0 / -1 -3.20 e5 -3.20 e5 3.20 e5 -3.19e5 -3.19e5 3.19e5 K7
22.5 0 / -1 -3.90 e5 -2.25 e6 9.30 e5 -3.99e5 -2.33e6 9.64e5 K8
67.5 0 / -1 -2.20 e5 -4.00 e4 9.00 e4 -2.28e5 -3.91e4 9.43e4
K10 0.0 -1 / -3.2 -2.11 e6 -6.44 e6 0 -2.40e6 -7.3e6 0 K11 22.5
-1 / -3.1 -2.04 e6 -4.49 e6 1.23 e6 -2.07e6 -4.36e6 -1.13e6 K12
45.0 -1 / -3.2 -2.03 e6 -2.03 e6 1.08 e6 -2.05e6 -2.05e6
-1.05e6
-
8) Seismic application of the material model to the Hospital LKH
Innsbruck, Tyrol, Austria This hospital (Figure 9) was built
approximately in 1945 where the structure consists of unreinforced
masonry (URM) with Ast-Molin (reinforced ribbed arch concrete)
slabs and flat reinforced concrete slabs at the aisles. It exhibits
a basement, a ground floor and five upper floors including one
attic floor. Innsbrucks geographically position is: 11.390
longitudes and 47.263 latitudes. The soil under this building is
composed of middle dense to dense sandy and stony flint. The
maximum PGA in accordance to Austrian national code for designing
of buildings with seismic actions, NORM B 4015 [9], is 1.54 m/s.
However, the strongest earthquake was in 1572, and its PGA has been
assessed to 1.48m/s with a main duration of 3.4 sec and a dominant
frequency of 2.7 Hz according to the Austrian earthquake catalogue
of ZAMG (Zentralanstalt fr Meteorologie und Geodynamik, the
Austrian Central Institute of Meteorology and Geodynamics). I.e.,
structural designing according to the current Austrian code NORM B
4015 leads to mechanically conservative results.
Figure 9 Photograph of LKH-Innsbruck
In a first step, the natural frequencies of the structure were
measured by ambient vibration excitations as wind, humans in the
structure, traffic and earth micro tremors. The advantage of this
method is that no extra artificial exciter is needed. Therefore,
the structure was instrumented by 25 sensor positions (partly
triaxial) very sensitive seismic accelerometers (Wilcoxon Research
731A-P31), positioned as shown in Figure 10 and Figure 11.
Figure 10 Sensor position at the fourth upper floor
-
Figure 10 Sensor position in the staircases
The results of the measured data were analyzed by the software
MACEC, to get the natural frequencies and corresponding mode
shapes. For comparisons sake a finite element model (Figure 11) was
built and modal analysis was performed by means of the software
ANSYS, where mainly shell (shell 63) and line (line 188) elements
were used. In this model, the total number of elements was
approximately 80,000. In Figure 12, the numerical results are
compared to the experimentally evaluated parameters.
Figure 11 Finite Element model of LKH Innsbruck
-
Dynamic behavior: (mode shapes and natural frequencies)
Experimentally evaluated by In Situ Measurements Numerically
analyzed by Finite Element
Software, ANSYS f1=2.423 Hz
f1=2.3443 Hz
=-3%
f2=2.580 Hz
f2=2.5874 Hz
=0%
f3=2.945 Hz
f3=3.1026 Hz
=+5%
f4=3.265 Hz
f4=4.1136 Hz
=+20%
-
f5=3.855 Hz
f5=4.1236 Hz
=+7%
Figure 12 Comparison of Mode shapes and natural frequencies
The material parameters of the existing masonry were derived by
a simple compression test on a test specimen taken from the
original structure (see Figure 13 to Figure 14). The bricks are of
the dimensions l/w/h = 250/100/60 [mm].
Figure 13 Test Specimen of the real structure
Figure 14 Axial compression test The evaluated axial compression
strength oriented orthogonal to the horizontal joints is cxf =1.68
N/mm. The parameters according to Eq (17) - Eq (25) are summarized
in Table 3.
-
Table 3 URM Parameters
Parameter Floor MinValue MaxValue
[N/mm] [N/mm] [N/mm] Compression
strength
Orthogonal to the horizontal joints
1.68
Parallel to the horizontal joints
0.92 1.18
Tension strength
*
Orthogonal to the horizontal joints *
4th floor 0.08 3rd floor 0.19 2nd floor 0.25 1st floor 0.35
Ground floor 0.38 Parallel to the
horizontal joints
4th floor 0.25 0.30 3rd floor 0.35 0.45 2nd floor 0.40 0.54 1st
floor 0.49 0.68 Ground floor 0.52 0.72
Shear strength
4th floor 0.14 0.17 3rd floor 0.20 0.25 2nd floor 0.23 0.30 1st
floor 0.28 0.38 Ground floor 0.29 0.41
Youngs Modulus
Orthogonal to the horizontal joints *
2000 2500
Parallel to the horizontal joints
1500 1850
Shear Modulus
850 1000
-
After determination of the mode shapes and natural frequencies,
a response spectrum analysis was performed considering the code
spectra of NORM B4015 [9], where the assumed earthquake was acting
in the weaker direction (east-west), see Figure 10. Finally, the
material model described above was used to analyze the cracks
occurred due to the earthquake. Figure 15 - Figure 21 show the
effected cracks, and how many criteria are violated: number 5 (red)
stands for no crack and the number 0 (blue) indicates that all
criteria are violated. Basement:
Figure 15 Crack pattern of basement
Ground Floor
Figure 16 Crack pattern of ground floor
-
1st upper floor
Figure 17 Crack pattern of 1st upper floor
2nd upper floor
Figure 18 Crack pattern of 2nd upper floor
3rd upper floor
Figure 19 Crack pattern of 3rd upper floor
-
4th upper floor
Figure 20 Crack pattern of 4th upper floor
Attic floor
Figure 21 Crack pattern of attic floor
9) Conclusion A powerful and precise tool for analyzing the
bearing capacity of URM has been developed. The result of this
research work is a macro implementation into the Finite Element
software ANSYS. The user, e.g. a practical engineer, has only to
enter the material parameters into the input mask, which are in
most cases available (or, at least, can be estimated from the
literature or specific building codes), and so a precise analysis
of URM becomes applicable. Although this procedure is a nonlinear
approach, the implementation was elastic, which fulfils most of the
practical demands of a practical engineer. As result of this
application the computer shows the occurrence, localisation and
even the failure mode of the cracks in specific URM members. The
accuracy depends on the chosen element size of the finite elements.
10) Acknowledgements The research work of this paper was funded by
arsenal research (Fund No: 2.05.00187.4.0). The authors are
grateful for this support and want to thank them. The research work
on LKH-Innsbruck hospital was supported in part by TILAK, Tiroler
Landeskrankenanstalten Ges.m.b.H. The authors want to also thank
them.
-
References [1] Ganz, H. R., Mauerwerksscheiben unter Normalkraft
und Schub. Institut fr Baustatik und Konstruktion, ETH Zurich,
report no. 148, 1985. [2] Code, EN 1996-1-1, Design of Masonry
Structures, part 1-1 common rules for reinforced and unreinforced
masonry structures. [3] Glitzka, H., Druckbeanspruchung parallel
zur Lagerfuge. Mauerwerkskalender 1988, pp 489 496, Ernst &
Sohn, 1988. [4] Tassios , , , . , 1986. [5] Vratsanou, V., Das
nichtlineare Verhalten unbewehrter Mauerwerksscheiben unter
Erdbebenbeanspruchung. Ph. D. Thesis, Institut fr Massivbau und
Baustofftechnologie, Universitt Fridericiana zu Karlsruhe TH, 1992.
[6] Graubner, C. A., Glock, C., Meyer, G., Abschtzung der Knicklnge
mehrseitig gehaltener Wnde aus groformatigen Mauersteinen.
Bauingenieur, Juni 2004, pp 300-305, Springer, 2004. [7] Gross, D.,
Seelig, T., Bruchmechanik mit einer Einfhrung in die Mikromechanik.
3. edt, Springer, 2001. [8] Ganz, H. R., Thrlimann, B., Versuche
ber die Festigkeit von zweiachsig beanspruchtem Mauerwerk. Institut
fr Baustatik und Konstruktion, ETH Zurich, report no. 7502-3, 1982.
[9] Code, NORM B4015 (2002), Belastungsannahmen im
BauwesenAuergewhnliche Einwirkungen-Erdbebeneinwirkungen, Grundlage
und Berechnungsverfahren
-
Paper should be fully refereed, please
