Numerical Investigations on masonry elements with different angles of inclination of the bed joints under biaxial deformation states

Abstract

The present paper describes the numerical investigations performed on masonry
elements with different angles of inclination of the bed joints under biaxial deformation states by using the nonlinear finite element program MASA. First, the numerical model was calibrated on uniaxial compression tests. A micro-modelling approach was used, in which threedimensional volume (solid) elements represented the stones and the joints and the constitutive law was described using the microplane model with relaxed kinematic constraint. The bonding properties were considered directly in the material behavior of the mortar. To reduce the computational effort, the geometry of the whole specimen was modelled as a quarter of the real dimensions using symmetry in two directions. With this model, an extensive numerical parametric study was carried out. Thereby the influence of the angle of bed joints, α was changed from 0 to 75 degrees with the horizontal in the interval of 15 degrees. For each angle of bed joints, different displacement ω1 and ω3 were applied in the principal direction 1 and 3.
The displacements ω1 and ω3 were chosen in a manner that a tension-tension, a tensioncompression, a compression-tension and a compression-compression state results. For each of this case, the ratio ω1/ω3 was changed between specified ranges. As the results of the numerical study, the characteristic of the constitutive law, the elastic properties and the softening behavior will be investigated. In addition to the model calibration, first results of the numerical study were shown in this paper. The investigation serves as a basis for the development of a consistent mechanical model to describe the load-deformation behavior of masonry structures.

Full text

10th IMC
10th International Masonry Conference
G. Milani, A. Taliercio and S. Garrity (eds.)
Milan, Italy, July 9-11, 2018
NUMERICAL INVESTIGATIONS ON MASONRY ELEMENTS WITH
DIFFERENT ANGLES OF INCLINATION OF THE BED JOINTS
UNDER BIAXIAL DEFORMATION STATES
M. Weber1, P. Hahn2 and A. Sharma3
1 PhD Student at University of Stuttgart
Pfaffenwaldring 4, 70569 Stuttgart, Germany
marius.weber@iwb.uni-stuttgart.de
2 Student at University of Stuttgart
Pfaffenwaldring 4, 70569 Stuttgart, Germany
e-mail: Philipphahn@web.de
3 Jun.-Prof. at University of Stuttgart
Pfaffenwaldring 4, 70569 Stuttgart, Germany
akanshu.sharma@iwb.uni-stuttgart.de
Keywords: numerical modelling, angle of bed joint, biaxial stress, masonry elements
Abstract. The present paper describes the numerical investigations performed on masonry
elements with different angles of inclination of the bed joints under biaxial deformation states
by using the nonlinear finite element program MASA. First, the numerical model was cali-
brated on uniaxial compression tests. A micro-modelling approach was used, in which three-
dimensional volume (solid) elements represented the stones and the joints and the constitutive
law was described using the microplane model with relaxed kinematic constraint. The bond-
ing properties were considered directly in the material behavior of the mortar. To reduce the
computational effort, the geometry of the whole specimen was modelled as a quarter of the
real dimensions using symmetry in two directions. With this model, an extensive numerical
parametric study was carried out. Thereby the influence of the angle of bed joints, α was
changed from 0 to 75 degrees with the horizontal in the interval of 15 degrees. For each angle
of bed joints, different displacement ω1 and ω3 were applied in the principal direction 1 and 3.
The displacements ω1 and ω3 were chosen in a manner that a tension-tension, a tension-
compression, a compression-tension and a compression-compression state results. For each of
this case, the ratio ω1/ω3 was changed between specified ranges. As the results of the numeri-
cal study, the characteristic of the constitutive law, the elastic properties and the softening be-
havior will be investigated. In addition to the model calibration, first results of the numerical
study were shown in this paper. The investigation serves as a basis for the development of a
consistent mechanical model to describe the load-deformation behavior of masonry structures.
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P. Hahn, M. Weber and A. Sharma
1 INTRODUCTION
Neglecting the restraining effect of loading plates, masonry walls are mainly subjected to
normal forces Nx, shear forces Vy and moments Mz, see figure 1 (a). Because of the material
composition and the resulting inherent weakness along the head and bed joints, the masonry is
highly anisotropic. An elegant experimental way to study the anisotropic material behavior of
masonry is to perform element tests. The basic idea is to study the behavior of a single wall
element under different combinations of the stresses σx, σy and τyx (=τxy), see figure 1(a). Due
to the difficulty of introducing the shear stresses τyx the masonry element is rotated by the
principal stress angle θσ. This leads to the element tests with the angle of inclination to the bed
joints α relative to horizontal. By varying the principal stresses σ1 and σ3 and the angle α, dif-
ferent possible stress states according to figure 1 (b) to (e) can be examined. Extensive tests
on two axial stress states were conducted in [1], [2], [3] and [4]. In particular the introduction
of tensile forces and the capture of the softening behavior are extremely difficult. For simplic-
ity, most studies have been performed under uniaxial compressive loading (σ1=0, σ3<0). Un-
der this background a fairly large number of experiments with different stones, mortars and
joint designs had been carried out at the ETH Zurich. These tests are summarized in [7].
However, the constitutive behavior under biaxial stress states cannot be completely derived
from its constitutive behavior under uniaxial stress states. In all the experiments (biaxial and
uniaxial) mentioned above, emphasis only laid to the evaluation of the ultimate load. Based
on this, numerous empirical and mechanical failure criterions for masonry were developed
([1], [2], [3], [6], [7]). Consequently, the knowledge of the full constitutive behavior of ma-
sonry is still insufficiently characterized.
For closing this gap and due to the limited possibilities in the experiments, nonlinear finite
element (FE) method is used in this paper to contribute the understanding of the anisotropy of
masonry. As in the experimental investigations mentioned above, masonry elements with dif-
ferent angles of inclination of the bed joints α will be modelled under different biaxial states.
To capture the softening behavior, the applied displacement ω1 and ω3 in principal direction
θσ is varied instead of the state of stresses σ1 and σ3 (see figure 1 (b) to (e)). With a compre-
hensive numerical study, the characteristics of the stress-strain curves, the elastic properties
and the softening behavior are investigated depending on the parameters α and ω1/ω3. The
main focus of this paper is the calibration of the numerical model on uniaxial compression
element tests, as well as the discussion of the first results of the numerical study.
Figure 1: masonry elements: (a) masonry shear wall under in-plane loads; masonry elements under (b) tension-
tension; (c) tension-compression; (d) compression-tension; (e) compression-compression.
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P. Hahn, M. Weber and A. Sharma
2 FE-MODELLING
2.1 Modelling approach
A micromodelling approach is used to perform the FE analysis. For this the masonry ele-
ments were modelling with the FE-program MASA, developed at the Institute of Construction
Materials (IWB), University of Stuttgart [8]. Three dimensional volume (solid) elements rep-
resented the stones and the joints and the constitutive law was described using the microplane
model with relaxed kinematic constraint [10]. The bonding properties were considered direct-
ly in the material behavior of the mortar.
2.2 Modelling the masonry elements
Figure 2 (a) shows the numerical model for a masonry element under the biaxial defor-
mations ω1 and ω3. The model is constrained in the vertical direction on the lower and in hori-
zontal direction on the left side. To prevent local failure, the constraints were placed on linear-
elastic elements. A small cap between the linear-elastic elements allowed the lateral exten-
sions. On the right and upper side, the displacements ω1 and ω3 were introduced, respectively.
The deformation ω1 acts in 1-direction (horizontal) and the deformation ω3 in 3-direction (ver-
tical). The applied displacement ω1 and ω3 were increased proportional and monotonically.
To reduce the computational effort the geometric symmetry of the model is used. As it
seen in figure 2 (b) only half of the wall is modeled and on the symmetric line the FE-nodes
are constrained in z-direction. The webs of the bricks were modelled in detail. The choice of
the geometric dimensions l=h=990mm and t/2=57.5 mm, as well as the arrangement of the
bricks and joints is based on uniaxial element tests, on which the FE-Model is calibrated (see
chapter 2.4).
Figure 2: FE-modelling: (a) original FE-model; (b) top view; (c) reduced FE-model.
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P. Hahn, M. Weber and A. Sharma
2.3 Numerical study
The parameters α and ω1/ω3 were varied in the numerical study. For the applied displacements
ω1>0 and ω3>0 a tension-tension (t-t), for ω1>0 and ω3<0 a tension-compression (t-c), for
ω1<0 and ω3>0 a compression-tension (c-t) and for ω1<0 and ω3<0 a compression-
compression (c-c) state is induced. In a pure compression state, the applied displacement ω1 is
zero. For each state the angles of inclination of the bed joints α was changed from 0 to 75 de-
grees in the interval of 15 degrees. In addition the ratio ω1/ω3 was changed within specified
ranges (see table 1). For each state every possible combination of the angle α and the ratio
ω1/ω3 will be calculated. Therefore, a total of 156 different combinations are possible. The
parameters of the numerical study are summarized in table 1.
ω3 ω1 α ω1 / ω3
Tension Tension 0°, 15°, 30°, 45°, 60°, 75° 0.0, 0.1, 0.2, 0.6, 1.0, 1.4
Tension Compression 0°, 15°, 30°, 45°, 60°, 75° 0.0, 0.1, 0.2, 0.6, 1.0, 1.4
Compression
Tension 0°, 15°, 30°, 45°, 60°, 75° 0.0, 0.2, 0.6, 1.0, 1.4, 1.8, 2.2
Compression
Compression 0°, 15°, 30°, 45°, 60°, 75° 0.0, 0.2, 0.6, 1.0, 1.4, 1.8, 2.2
Table 1: Parameters of the numerical study.
2.4 Materialparameters
To calibrate the FE-Model, element tests were used which were performed at the IWB. In
figure 3 the test set up is shown. The test specimens were tested under uniaxial compression
(σ1=0) with the angle of inclination of the bed joints α varied between 0 and 90 degrees in the
interval of 15 degrees. The tests were performed in displacement control in order to capture
the post-peak response as well. A detailed description of the tests can be found in [9].
Figure 3: test set up from [9].
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P. Hahn, M. Weber and A. Sharma
Due to the shrinkage process during manufacturing progress of the bricks, the inner webs
are cracked before the test began. Therefore, different tension strength ft were used in the FE-
analyses to the webs and the bowls of the bricks. As a result of the not fully filled head joints,
the chosen compression and tension strength fc and ft , respectively, as well as the elasticity
modulus E, differ from the one of the bed joints. So totally four different materials were used
in the FE-analyses, see table 2. It should be noted that the same calibrated model and thus the
same material properties were used in all recalculations.
material brick-bowls brick-webs bed joints head joints
fc [N/mm
2
] 20 20 5.9 4
ft [N/mm
2
] 0.3 0 0.03 0.023
E [N/mm
2
] 8000 8000 3360 2000
Table 2: material parameters.
With the FE-model in chapter 2.2 and the material parameters mentioned above, the stress-
strain curves (σ3-ε3) and the failure mechanisms of the element tests were recalculated. The
stresses σ3 were calculated by dividing the reaction force with the cross sectional area. The
strain ε3 is determined by dividing the vertical displacement ω3 with the height h of the spec-
imen. Figure 4 (a) shows the stress-strain curves of the numerical calculations (FE-orig.) and
the test for α=0, 15, 30, 45, 60 and 75 degrees. In figure 3 (b) and (c) the crack patterns were
compared at the maximum load level. In general, a good agreement between experimental and
numerical results is found. The largest deviations between the calculated stress strain (σ3-ε3)
curve and the one from the tests is found for the angles α=15 and α=30 degrees (see figure 4
(b) and (c)). This is due to the fact that the failure occurred in the joints and bricks simultane-
ously and as a result, the calibration of the FE-model becomes very difficult. The initial devia-
tions in the simulation for α=75 degrees is due to an increase in stiffness observed at the
beginning of the test, as a result of slipping in the test set up. In the softening zone, instability
was observed in the test, which led to the sudden reduction of the stress σ3, instead of a slow
degradation as seen in the calculated curves. This slippage behavior was not explicitly mod-
eled in the analysis.
3 PARAMETRIC STUDIES
3.1 Reduced model
To reduce the computational effort of the comprehensive numerical study, the geometry of
the whole (original) specimen was modelled as a quarter of the real dimensions, see figure 2
(c). The dimensions of elements stay unchanged. To assess the influence of the size effect, the
stress-strain curves from the reduced model are compared with the stress-strain curves ob-
tained for the original model under uniaxial compression. The corresponding results (FE-red.)
are shown in figure 3 (a) to (f). Although the strength obtained by the analysis of reduced
model is slightly higher than at the original model, the overall behavior and the failure mech-
anism obtained from the reduced model is very similar to that obtained by the original model.
Therefore, to study the influence of various parameters relatively, the reduced model was used
in the analyses.
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P. Hahn, M. Weber and A. Sharma
Figure 4: model calibration: (a) to (f) stress-strain curves; (g) to (l) calculated crack patterns; (m) to (r) crack
patterns from test [9].
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P. Hahn, M. Weber and A. Sharma
3.2 Results of the numerical analyses
Due to the large number of calculations, only selected results are presented in this paper.
The focus will be on the elastic modulus E3(α,ω1/ω3) as a function of the parameters α and
ω1/ω3. The elastic modulus E3(α,ω1/ω3) correspond to the initial tangent modulus as show in
figure 4 (c). As a consequence of the orthogonality of the principal direction 1 and 3, the be-
havior of the elasticity modulus E1(π/2-α,ω1/ω3) is equal to E3(α,ω1/ω3).
3.3 First results of the numerical study
Figure 5 (a) to (c) show the normalized elasticity modulus E3(α,ω1/ω3)/E3(α=0,ω1/ω3=0) for
the tension-tension, tension-compression, compression-tension and the compression-
compression state.
For both tension-tension (figure 5 (a)) and tension-compression (figure 5 (b)), the angle α and
the ratio ω1/ω3 has no significant influence on the elasticity modulus E3(α,ω1/ω3). The varia-
tion depending on the parameters α and ω1/ω3 is about max. 7% for tension-tension and max
15% for tension-compression. The reason for this is that the behavior in direction 3 always led
to a tensile failure in the mortar. Since the mortar was assumed to be isotropic in the model,
the loading direction does not matter.
The described behaviour changes under a compression-tension state which can be observed in
figure 5 (c). The elasticity modulus E3(α,ω1/ω3) decreases with increasing ratio ω1/ω3. Due to
an increasing deformation ω1, the crack opening in the horizontal direction occurs earlier,
which results in a reduction of the stiffness in 3-direction. With increasing the angle α, the
elasticity modulus E3(α,ω1/ω3) decreases to α=60 degrees and then it remains constant for ra-
tios ω1/ω3<=0.2 or increase again for ratios ω1/ω3>0.2. As seen by the uniaxial element tests
in figure 4 (c), depending on the angle α, the failure occurs either in the bricks or in the mortar
or in both. In case of failure in the bricks, a stiffer and in case of a failure in the joints a softer
behavior is obtained. Similar failure mechanism was found in the numerical analysis as well.
The curves for compression-compression state are shown in figure 5 (d). From an angle α=30
degrees the elasticity modulus E3(α,ω1/ω3) shows a decreasing trend and the influence of the
ratio ω1/ω3 gains in influence. It can be seen that the elasticity modulus increases with in-
creasing ratio ω1/ω3. The reason for this is that a larger horizontal deformation ω1 lead to
higher hindrance of the transverse strain from the vertical direction and the misalignment of
the masonry element. As a result a stiffer behavior in 3-direction is observed. In case of small
angles (α=0 or 15 degrees), there is no significant misalignment of the masonry element and
the expansion in horizontal direction is based only on the transverse strain from the vertical
direction. Since the horizontal expansion only increases with cracking and due to the fact that
the elasticity modulus E3 (α,ω1/ω3) was determined at small strains, the elasticity modulus E3
(α,ω1/ω3) remains almost constant and the parameters α and ω1/ω3 has no significant influence.
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P. Hahn, M. Weber and A. Sharma
Figure 5: normalized modulus of elasticity (a) tension-tension; (b) tension-compression; (c) compression-tension;
(d) compression-compression.
4 CONCLUSION
In the present paper, 3D numerical investigations are performed on masonry elements using
the microplane model for bricks and mortar utilizing the FE program MASA. The aim was to
numerically investigate the anisotropic behaviour on masonry element by variation of angles
of inclination of the bed joints α and ratios ω1/ω3. The displacements ω1 and ω3 were applied
as an external action in a manner that would result in a compression-compression, a tension-
tension, a compression-tension and a tension-compression state.
First, it could be shown that with the modeling approach used, the stress-strain curves and the
failure mechanism of uniaxial element tests could be simulated. Even by reducing the geome-
try to a quarter of the original size, good overall results could be achieved. Using the validated
model, the evaluation was conducted with respect to the influence of the angle α and the ratio
ω1/ω3 on the modulus of elasticity E3(α,ω1/ω3) . It could be shown that especially in compres-
sion-tension both the angle α and the ratio ω1/ω3 had a significant influence on the E3(α,ω1/ω3)
modulus.
Further results of the numerical analysis, such as the characteristics of the stress-strain curves
and the softening behavior depending on the parameter α and ω1/ω3 will be presented in future.
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P. Hahn, M. Weber and A. Sharma
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