Content uploaded by Amirhosein Shabani
Author content
All content in this area was uploaded by Amirhosein Shabani on Jan 07, 2022
Content may be subject to copyright.
17th World Conference on Earthquake Engineering, 17WCEE
Sendai, Japan - September 27th to October 2nd, 2021
A COMPARATIVE STUDY ON THE INITIAL IN-PLANE STIFFNESS OF
MASONRY WALLS WITH OPENINGS
A. Shabani(1), V. Plevris(2), M. Kioumarsi(3)
(1) PhD Candidate, Department of Civil Engineering and Energy Technology, Oslo Metropolitan University, Oslo, Norway,
amirhose@oslomet.no
(2) Associate Professor, Department of Civil and Architectural Engineering, Qatar University, Doha, Qatar, vplevris@qu.edu.qa
Professor, Dept. of Civil Engineering and Energy Technology, Oslo Metropolitan University, Oslo, Norway, vageli@oslomet.no
(3) Associate Professor, Department of Civil Engineering and Energy Technology, Oslo Metropolitan University, Oslo, Norway,
mahdik@oslomet.no
Abstract
Masonry buildings have been used for centuries in various locations around the world, including areas with high
seismicity. Studies about the behavior of masonry structural components subjected to lateral loadings and retrofitting
techniques for improving their performance have gained much attraction lately. Various simplified methods have been
presented in the literature for the seismic vulnerability assessment of masonry buildings. The initial in-plane stiffness of
masonry walls is a key parameter which significantly affects the nonlinear backbone curve of the masonry walls as well
as their ultimate in-plane strength.
Different simplified analytical methods have been proposed for deriving the initial in-plane stiffness of masonry
buildings with regular or irregular openings by considering the flexible spandrels that can translate and rotate under
lateral load and flexible piers’ endings. In the analytical methods, the initial in-plane stiffness of each pier will be
computed from the equations by considering the geometry of each component as input. Each structural component is
considered as a spring and the stiffness of the whole system is computed based on equations of springs in series or in
parallel.
The finite element method is considered as a reliable tool for verifying the analytical methods. For this purpose, a
homogenization method has been employed for modeling the masonry walls and lateral loads have been applied on the
walls with the assumption of linear material to derive the initial in-plane stiffness of the walls. For this purpose, three
categories of masonry walls have been considered with one, two, and three openings where the openings’ geometries
also vary to investigate the effect of opening placements and irregularities on the initial in-plane stiffness of the walls.
Afterwards, the stiffnesses computed from the analytical methods are compared with the stiffnesses that have been
derived from the finite element analysis to investigate the accuracy of the analytical methods. It is shown that the
analytical methods can be utilized for deriving the initial in-plane stiffness of masonry walls with openings, providing
fast and accurate solutions in comparison to more detailed and time-consuming finite element implementations.
Keywords: Initial stiffness; masonry walls; in-plane stiffness; analytical methods; finite element analysis
17th World Conference on Earthquake Engineering, 17WCEE
Sendai, Japan - September 27th to October 2nd, 2021
2
1. Introduction
Unreinforced masonry (URM) buildings can be considered as the oldest construction technique in the world
[1] that consists of URM shear walls as a load-bearing system [2, 3]. Moreover, nowadays, URM walls have
been utilized in moment-resisting frames as an infill wall, effective on the building responses to the different
types of loadings [4, 5]. The initial in-plane stiffness (IIPS) of each structural component is considered as a
key parameter for design purposes and deriving the nonlinear analysis’s backbone curve [6, 7], which is
significantly effective on the nonlinear analysis results. Therefore, calculating an accurate enough value for
the IIPS of URM walls could be critical for seismic performance evaluation of URM buildings [8-10] and
designing the modern buildings with URM infill walls [11]. Instead of performing finite element (FE)
analysis, different analytical hand methods have been developed for the estimation of IIPS of URM walls
with less computational effort. For the URM walls without openings, the estimation of the IIPS by assuming
the wall as a deep beam is easy and accurate enough since rigid boundary conditions are considered in both
the theory and equations. Nevertheless, in terms of perforated URM walls, the estimation of this parameter is
not accurate enough due to the possible flexibility of pier ends [9].
As the easiest method for the estimation of the IIPS of URM walls with openings, the wall is
discretized to piers, and the IIPS of each pier can be derived based on the deep beam theory neglecting the
flexible boundary conditions. It was investigated that the perforated wall’s IIPS is overestimated using this
method [9]. Another well-known analytical hand method is called the interior strip method [12]. By
comparing the results with the results of FE analysis, it was investigated that the interior strip method is not
accurate enough and overestimate the IIPS of the perforated URM wall in some cases [12]. Moreover, an
analytical method was proposed in [13] considering flexible endings for piers by modifying the boundary
conditions stiffnesses, and design tables were provided to facilitate the estimation process of the IIPS. The
method’s accuracy was then verified by comparing the results with the FE analysis results [13, 14].
Furthermore, the effective height method is an analytical method proposed in [9]. Modification of the pier
stiffness due to the flexible boundary conditions has been performed using regression analysis based on the
FE analysis of cantilever piers with different boundary conditions. The method has been validated by
comparing the results with the FE analysis results of four perforated walls [9].
The last two mentioned analytical methods are chosen in the current study to investigate their
performance against the FE analyses. Due to the low number of case studies investigating the performance of
the methods in previous studies, a broader level of URM walls with different configurations of openings is
needed to be developed. Firstly, a FE model has been developed and validated based on an experimental test
performed by [15]. Afterward, URM wall case studies with openings in different configurations have been
modeled and analyzed. Then, the IIPS of the walls is derived based on the modified boundary conditions
stiffness method and the effective height method. Finally, the results from the two analytical methods have
been compared with the FE analysis results to determine each analytical method’s accuracy, and
modifications have been proposed to improve the accuracy of the analytical methods.
2. Method
In this section, details about all the analysis types for estimating IIPS of URM walls are presented and
investigated. Firstly, the experimental test is presented as the most robust method. Then the FE modeling
procedure and the procedure of the two analytical methods utilized in this study are presented.
2.1 Experimental test
Quasi-static and monotonic tests on a single-leaf tuff masonry URM wall with an opening were performed
by [15], where the geometrical data of the tested wall is shown in Figure 1. Vertical forces of 200 kN were
applied to the piers by hydraulic jacks to simulate gravity loads [16]. A prescribed monotonic displacement
was applied on one side of the wall through the test procedure, and the horizontal resistant force of the wall
and the deformation were recorded.
17th World Conference on Earthquake Engineering, 17WCEE
Sendai, Japan - September 27th to October 2nd, 2021
3
2.2 FE modeling
Different methods have been presented for the numerical modeling of URM walls. Among them, the
continuum-based method is utilized in this study. In this method, the masonry unit will be considered as a
homogenous texture, and the masonry blocks and mortar joint have not been modeled in detail [2].
Based on a database from the test to derive the shear modulus (G) of masonry, see [17], it is concluded
that G=0.15E, where E is the modulus of elasticity. This is a reasonable estimation equation for calculating
the accurate enough G parameter. Using G=0.4E by assuming the masonry as an isotropic material
overestimates the G parameter and the URM wall’s stiffness [17]. The FE model of the test wall has been
developed in DIANA FEA software [18] considering the mentioned assumptions with the material properties
summarized in Table 1.
Table 1 – Material properties of masonry for the FE model validation.
E(GPa)
G(GPa)
ρ (kg/m3)
Tuff masonry (compression parallel to bed joints)
2.07
0.31
1600
Tuff masonry (compression perpendicular to bed joints)
2.22
Furthermore, two blocks on top of each pier have been modeled to simulate the test set up with a
specific density to simulate the constant vertical applied load of 200 kN as illustrated in Fig.1 [16]. However,
it was investigated that the effect of vertical loads in FE analysis is negligible on the IIPS of URM walls, see
[13].
(a)
(b)
Fig. 1 – (a) Geometry and (b) FE model of the test wall (Dimensions in cm).
2.3 Analytical methods
The deep beam is considered a suitable structural model for solid, prismatic, and unperforated shear walls. In
deep beam theory, the cross-sections are assumed to remain plane, and unlike in Bernoulli beam theory,
cross-sections do not remain perpendicular to the beam axis after deformation [12]. The elastic in-plane
shear stiffness of the wall can be obtained from Eq. (1) that combines the flexibility of the wall due to shear
and flexure:
(1)
where flexural stiffnesses for a cantilevered and two fixed end walls (Kflex) are calculated based on Eq. (2)
and Eq. (3), respectively:
(2)
(3)
17th World Conference on Earthquake Engineering, 17WCEE
Sendai, Japan - September 27th to October 2nd, 2021
4
Moreover, the shear stiffness for a rectangular cross-section wall (Kshear) is calculated from Eq. (4):
(4)
where E is elastic modulus, is the moment of inertia for the gross section, is the height of the pier, G is
the shear modulus, and is the cross-section area. Two ends of a pier are not stiff enough in perforated
walls to satisfy the predefined stiffness boundary conditions of Eq. (2) and Eq. (3). For the estimation of the
IIPS of perforated URM walls, in this paper, two analytical methods, (a) the effective height method (EHM)
and (b) the modified boundary conditions stiffness method (MBCSM), have been studied in detail.
2.3.1 Effective Height Method (EHM)
In EHM, the pier is divided into equally two cantilever piers, and the stiffness of each segment can be
calculated based on Eq. (2). The shear stiffness of the cantilever segment is calculated based on Eq. (4), but
for the flexural stiffness, Eq. (5) is utilized.
(5)
Three parameters are defined based on the geometry of the pier segment to calculate the r factor: the
aspect ratio of the pier , the ratio of the depth of the spandrel component to the pier , and the
symmetry factor of the pier end . The first two parameters can be calculated based on the geometry of the
pier and the spandrel. The third parameter defines the asymmetry of the end region, which is described in [9].
After calculating the three mentioned parameters from the geometry of the pier and the spandrel the stiffness
of the pier segments, the r factor can be derived using Eq. (6):
(6)
After deriving the in-plane shear stiffness of two cantilever pier segments, the IIPS of the whole pier
can be calculated based on the stiffness of the top (Ktop) and bottom (Kbot) cantilever pier segments using Eq.
(7):
(7)
For estimating the stiffness of a perforated wall, the wall can be discretized to horizontal (spandrels)
and vertical (piers) elements, as illustrated in Fig.2b. Then the stiffness of the whole wall is defined by using
the series or parallel spring rules for the elements, as is shown in Fig.2c.
(a)
(b)
(c)
Fig. 2 – (a) A perforated URM wall, (b) dividing the wall to the spandrel and piers, and (c)
composite spring model of the wall.
17th World Conference on Earthquake Engineering, 17WCEE
Sendai, Japan - September 27th to October 2nd, 2021
5
The IIPS of each spandrel can be roughly estimated based on Eq. (2), which is derived from Eq.
(3), assuming a deep cantilever beam in a conservative way for all configurations of the spandrel. Then the
effective stiffness of a perforated wall is calculated as described in [9].
is derived based on the shear stiffness of the components; however, the in-plane bending action
of the wall needs to be taken into account. This effect will become larger when the wall aspect ratio increases
[9]. For this purpose, should be modified based on Eq. (8):
(8)
where is the bending stiffness of a perforated wall and calculated based on Eq. (9):
(9)
In Eq. (9), the term corresponds to the perforated wall’s moment of inertia and is the total height
of the perforated wall. The term 𝜌 is a correction factor to consider the opening effects calculated based on
Eq. (10):
(10)
where is the ratio of the area of the openings to the area of the wall in percentage [9].
2.3.2 Modified Boundary Conditions Stiffness Method (MBCSM)
In MBCSM, the rotational deformations of the top and bottom spandrel of a pier are considered, but the
shear stiffness term of Eq. (1) is not changed, and the flexural stiffness has been modified and calculated
based on Eq. (11) [14]:
(11)
where and are equal to and respectively. Making the calculation procedure easier, a
simplified nondimensional relationship for estimating the IIPS of a pier is introduced [14]. Firstly, three
nondimensional parameters should be defined as follows:
(12)
(13)
(14)
Furthermore, the stiffness nondimensional parameter will be calculated from Eq. (15):
(15)
17th World Conference on Earthquake Engineering, 17WCEE
Sendai, Japan - September 27th to October 2nd, 2021
6
where the term p is calculated based on where:
(16)
and
(17)
After deriving the flexural stiffness part of the pier from Eq. (15), the pier’s IIPS can be estimated
based on Eq. (1) [14]. Note that the effect of asymmetry of pier ends, stiffness of spandrels and bending
stiffness of the whole wall have been neglected in the MBCSM method.
2.4 Developed case studies
Totally 15 walls with an equal height, including the experimental test wall (model Ex) with one, two, and
three openings in different configurations, have been developed for performing the comparative study.
Geometry, opening configurations, and allocated name of each case study are presented in Fig.3.
1a
1b
1c
1d
1e
1f
1g
2a
2b
2c
2d
3a
17th World Conference on Earthquake Engineering, 17WCEE
Sendai, Japan - September 27th to October 2nd, 2021
7
3b
3c
Fig. 3 – Geometry and opening configurations of the URM wall case studies (Dimensions in m).
2.5 Performance of the analytical methods
The IIPS of the case studies will be estimated using the two mentioned analytical methods. However, for the
MBCSM, three scenarios have been considered. Firstly, the stiffness has been calculated just by summing
the piers’ stiffnesses. In the second scenario, the effect of spandrel stiffness has been considered
(MBCSM+SE), and in the third scenario, the bending effect of the whole perforated wall is taken into
account in the calculations (MBCSM+SE+BE).
2.5.1 Quantitative approach
The values of coefficient of determination (R2), root mean square error (RMSE), and mean absolute error
(MAE) are calculated based on Eqs. (18), (19), and (20), respectively, to evaluate the performance of the
analytical methods.
(18)
(19)
(20)
where N is the number of the values in both datasets, and are the values from two datasets and and
are the corresponding mean values. It is noted that a larger value of the R2 and lower values of RMSE and
MAE show a better correlation between the two datasets.
2.5.2 Qualitative approach
In the qualitative approach, the scatter plot of the results has been provided. The deviation of the equality
line (Y=X) from the best fitted polynomial line (i.e., Y=aX+b) shows the correlation of the result of each
method to the obtained results from the FE analysis; and therefore, the robustness of each analytical method.
3. Results and discussion
3.1 FE model validation and mesh sensitivity analysis
The effect of mesh element size has been investigated to achieve the most efficient and accurate enough
meshing size. Table 3 shows the four maximum mesh element sizes assigned to the FE model of the test wall
and the corresponding number of the elements.
Table 2: Mesh sizes and the number of elements for performing the mesh sensitivity analysis.
Mesh size (m)
0.02
0.05
0.1
0.2
Number of elements
36764
5859
1466
403
17th World Conference on Earthquake Engineering, 17WCEE
Sendai, Japan - September 27th to October 2nd, 2021
8
A displacement with the values of 1mm has been applied on the loading position, and the IIPS is
calculated as the ratio of the base shear and the prescribed displacement. Figure 12 shows the ratio of the
IIPS derived from the FE model to the experimental test and the mesh sensitivity analysis results. Based on
Fig.4, the maximum mesh size of 0.1 m is considered the most efficient mesh size, and the FE model is
validated with adequate accuracy.
Fig. 4 – Results of the mesh sensitivity analysis.
3.2 FE analyses results
The material properties and the thickness of the developed case study walls are considered equal to the
experimental tests. However, the elastic moduli in both X and Y directions are the same with a value of 2.07
GPa. After developing the FE models, the analysis has been done by applying a load on the top left of the
wall and recording the displacement at the top right side of the wall. Based on the test procedure, a
displacement-based analysis has been done for the validation of the FE model of the test wall. Nevertheless,
for the analysis of the case studies, a load-based method has been utilized by applying a force and recording
lateral displacement. Note that based on the previous studies on the perforated URM walls, the results from
the displacement-based procedure are more conservative than the load-based procedure, see [19]. Moreover,
the load-based method better reflects the loading that would be applied during a seismic event compared to
the displacement-based procedure [19]. Fig.5 shows the displacement contour of the case study walls in the
X direction from the FE analysis.
Ex
1a
1b
1c
1d
1e
17th World Conference on Earthquake Engineering, 17WCEE
Sendai, Japan - September 27th to October 2nd, 2021
9
1f
1g
2a
2b
2c
2d
3a
3b
3c
Fig. 5 – Displacement contour of the URM wall case studies in X direction obtained from FE analyses.
3.3 Comparative study of the perforated URM wall case studies
All the results from the FE analyses and four analytical methods have been derived, and the IIPS of the
perforated URM walls are shown in Table 5. For the results from the FE analyses, the IIPS values of the case
studies are calculated by dividing the applied force by the recorded displacement. The results in Table 3
show that for models 1f, 1e, and 1g, the IIPS values calculated from the analytical methods are the same, but
the FE analysis results are different. Therefore, the location of opening that affects the IIPS is not effective
on the results derived from the analytical methods that can be the weakness of the analytical methods. For
this purpose, analytical methods for walls with symmetric configurations of openings give more accurate
results.
Table 3 –IIPS of the case studies from FE analysis and the analytical methods in (kN/mm)
Model name
FE
EHM
MBCSM
MBCSM+SE
MBCSM+SE+BE
Ex
57.1817
57.0973
87.7436
72.0029
64.3741
1a
37.7489
39.4851
106.95
55.178
40.1496
1b
6.9093
6.8225
8.9125
8.5899
8.1169
1c
27.5064
26.4202
44.5625
37.276
28.755
1d
36.3148
32.7925
68.448
46.2486
34.5878
1e
44.9309
39.1383
114.4097
56.0822
40.6262
1f
32.3233
39.1383
114.4097
56.0822
40.6262
1g
32.1328
39.1383
114.4097
56.0822
40.6262
2a
104.5415
99.0443
182.9935
116.2014
105.6221
2b
103.246
101.7857
188.5906
118.4334
107.463
2c
114.7652
111.0057
318.3407
126.3638
114.1944
2d
49.1843
64.1954
157.2289
71.5514
65.5155
3a
52.7841
48.4768
53.6991
49.5441
42.5073
3b
8.9398
10.2374
10.9736
10.7159
10.0469
3c
31.6113
36.7344
48.9335
34.4814
30.8833
17th World Conference on Earthquake Engineering, 17WCEE
Sendai, Japan - September 27th to October 2nd, 2021
10
The values of R2, RSME, and MAE are illustrated in Fig.6 for investigating the accuracy of each
analytical method. As illustrated in Fig. 6, the value of R2 for the EHM is the largest, and the values of
RSME and MAE are the lowest compared to other analytical methods. This method can be considered the
most robust method compared to other analytical methods. Moreover, it is illustrated that the accuracy of the
MBCSM is not enough to be employed for estimating the IIPS of URM walls. By considering the spandrel
stiffness effects, the results improve, and by taking to account the bending effect stiffness, the results become
more accurate. The values of R2 for EHM and modified MBCSM are 0.97 and 0.96, respectively, which
confirm them as the accurate methods for estimating the IIPS of URM walls.
(a)
(b)
(c)
Fig. 6 – (a) R2, (b) RMSE, and (c) MAE values for the four mentioned analytical methods.
Based on the scatter plot of EHM in Fig.7 (a), the results of EHM are accurate enough, but the best-
fitted polynomial line of the MBCSM is not close enough to the equality line as illustrated in Fig.7 (b). The
modifications by considering the spandrel stiffness effects and bending stiffness effects are taken into
account to increase the accuracy of MBCSM that can be seen in Fig.7 (c) and (d).
(a)
(b)
17th World Conference on Earthquake Engineering, 17WCEE
Sendai, Japan - September 27th to October 2nd, 2021
11
(c)
(d)
Fig. 7 – Scatter plot and the equality line for the results of FE analysis and (a) EHM, (b)
MBCSM, (c) MBCSM + SE, and (d) MBCSM + SE + BE (in kN/mm).
4. Conclusion
The IIPS of URM walls is considered an effective parameter on the structural vulnerability assessment of
URM buildings and designing modern structural systems with URM infill walls. FE modeling is considered
as a more robust method for deriving the IIPS of the URM walls with openings compared to the analytical
methods. Nevertheless, expertise and high computational efforts are two main barriers that have limited the
application of the FE method. Therefore, different analytical methods have been proposed for calculating the
IIPS of URM walls with openings. The MBCSM and EHM are chosen as the analytical methods to
investigate their performance against the FE analyses’ results. For this purpose, URM wall case studies with
different openings configurations have been developed, and the IIPS of the walls have been derived using the
FE analyses and the mentioned analytical methods. The accuracy of each analytical method is evaluated
quantitatively by calculating the RSME and MAE, and R2 parameters and qualitatively by providing the
scatter plots. Performance evaluations show that results using EHM have enough accuracy but results from
MBCSM show a high deviation from the FE results. Two modifications have been applied to MBCSM.
Firstly, the effect of spandrel stiffness has been considered, and through the second modification, the effect
of bending stiffness of the wall is added to the previous one. The comparative studies show that the modified
MBCSM is accurate enough to estimate the IIPS of URM walls with openings.
5. Acknowledgements
This work is a part of the HYPERION project. HYPERION has received funding from the European Union’s
Framework Programme for Research and Innovation (Horizon 2020) under grant agreement No 821054. The
contents of this publication are the sole responsibility of Oslo Metropolitan University (Work Package 5,
Task 2) and do not necessarily reflect the opinion of the European Union.
6. References
[1] Shabani, A., M. Kioumarsi, V. Plevris, and H. Stamatopoulos (2020): Structural Vulnerability Assessment of
Heritage Timber Buildings: A Methodological Proposal. Forests. 11(8): p. 881 https://doi.org/10.3390/f11080881.
[2] D’Altri, A.M., V. Sarhosis, G. Milani, J. Rots, S. Cattari, S. Lagomarsino, E. Sacco, A. Tralli, G. Castellazzi, and
S. de Miranda (2020): Modeling strategies for the computational analysis of unreinforced masonry structures:
review and classification. Archives of Computational Methods in Engineering. https://doi.org/10.1007/s11831-
019-09351-x.
[3] Shabani, A., M. Kioumarsi, and M. Zucconi (2021): State of the art of simplified analytical methods for seismic
vulnerability assessment of unreinforced masonry buildings. Engineering Structures. 239: p. 112280
https://doi.org/10.1016/j.engstruct.2021.112280.
17th World Conference on Earthquake Engineering, 17WCEE
Sendai, Japan - September 27th to October 2nd, 2021
12
[4] Ozturkoglu, O., T. Ucar, and Y. Yesilce (2017): Effect of masonry infill walls with openings on nonlinear
response of reinforced concrete frames. Earthquakes and Structures. 12(3): p. 333-347
https://doi.org/10.12989/eas.2017.12.3.333.
[5] Shabani, A. and S. Erfani (2020): Seismic performance evaluation of SSMF with simple beam–column
connections under the base level. International Journal of Steel Structures. 20(1): p. 89-100
https://doi.org/10.1007/s13296-019-00273-9.
[6] Moradi, M.J., M.M. Roshani, A. Shabani, and M. Kioumarsi (2020): Prediction of the load-bearing behavior of
spsw with rectangular opening by RBF network. Applied Sciences. 10(3): p. 1185
https://doi.org/10.3390/app10031185.
[7] Bypour, M., M. Kioumarsi, and M. Yekrangnia (2021): Shear capacity prediction of stiffened steel plate shear
walls (SSPSW) with openings using response surface method. Engineering Structures. 226: p. 111340
https://doi.org/10.1016/j.engstruct.2020.111340.
[8] Xu, H., C. Gentilini, Z. Yu, H. Wu, and S. Zhao (2018): A unified model for the seismic analysis of brick masonry
structures. Construction and Building Materials. 184: p. 733-751
https://doi.org/10.1016/j.conbuildmat.2018.06.208.
[9] Craig, J.I., B.J. Goodno, Towashiraporn, and J. Park (2002): Response Modification Applications for Essential
Facilities. Georgia Institute of Technology, School of Civil and Environmental Engineering.
[10] Park, J., P. Towashiraporn, J.I. Craig, and B.J. Goodno (2009): Seismic fragility analysis of low-rise unreinforced
masonry structures. Engineering Structures. 31(1): p. 125-137 https://doi.org/10.1016/j.engstruct.2008.07.021.
[11] Nicola, T., C. Leandro, C. Guido, and S. Enrico (2015): Masonry infilled frame structures: state-of-the-art review
of numerical modelling. Earthquakes and structures. 8(3): p. 733-759 https://doi.org/10.12989/eas.2015.8.1.225.
[12] Neuenhofer, A. (2006): Lateral stiffness of shear walls with openings. Journal of Structural Engineering. 132(11):
p. 1846-1851 https://doi.org/10.1061/(ASCE)0733-9445(2006)132:11(1846).
[13] Qamaruddin, M., A.W. Hag, S. Al-Oraimi, and S. Hamoud (1997): Investigation on the Lateral Stiffness of Shear
Walls with Openings. in 11th International Brick Block Masonry Conference, China.
[14] Qamaruddin, M. (1998): In-plane stiffness of shear walls with openings. Building and Environment. 34(2): p. 109-
127 https://doi.org/10.1016/S0360-1323(98)00006-7.
[15] Parisi, F., Non-linear seismic analysis of masonry buildings. 2010, PhD Thesis, Naples: University of Naples
Federico II (URL: http://wpage ….
[16] Augenti, N., F. Parisi, A. Prota, and G. Manfredi (2011): In-plane lateral response of a full-scale masonry
subassemblage with and without an inorganic matrix-grid strengthening system. Journal of Composites for
Construction. 15(4): p. 578-590 https://doi.org/10.1061/(ASCE)CC.1943-5614.0000193.
[17] Croce, P., M.L. Beconcini, P. Formichi, P. Cioni, F. Landi, C. Mochi, F. De Lellis, E. Mariotti, and I. Serra
(2018): Shear modulus of masonry walls: a critical review. Procedia Structural Integrity. 11: p. 339-346
https://doi.org/10.1016/j.prostr.2018.11.044.
[18] DIANA, DIANA- Finite Element Analysis user manual version 10.4. 2020: Delft, Netherlands.
[19] Liu, Z. and A. Crewe (2020): Effects of size and position of openings on in-plane capacity of unreinforced
masonry walls. Bulletin of Earthquake Engineering. 18(10): p. 4783-4812 https://doi.org/10.1007/s10518-020-
00894-0.
