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Title
Abstract
Figures
Introduction
Experimental Tests
Theoretical and Experimental Parameters
Conclusions
References

Modeling the Dynamic Behavior of Masonry Walls as Rigid Blocks

Profile image of Fernando PeñaFernando PeñaProfile image of Jose V LemosJose V Lemos

2006, III European Conference on Computational Mechanics

https://doi.org/10.1007/1-4020-5370-3_282
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17 pages

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Abstract

This paper addresses the numerical modeling of masonry walls as rigid blocks. Models based on rigid block assemblies provide a suitable frame work for understanding their dynamic behavior under seismic actions. In this context, the problem is primarily concerned with Rocking Motion dynamics. The numerical tool is based on the Discrete Element Method (DEM) specially effective for the numerical modeling of rigid blocks. Some authors have been used successfully the DEM in the study of block structures. However, they have used experimental test to calibrate their models and to obtain the parameters used in the DEM; because these parameters cannot be obtained directly form the characteristics of the stones. In this context, a new methodology is proposed to find the parameters of the DEM by using the parameters of the classical theory. Special attention regards about the damping factor, since in the DEM a viscous damping is considered, but in reality the damping is due to impulsive forces that they can be considered as a type of Dirac-δ forces. An extensive comparison between numerical and experimental data has been carried out to verified the proposed methodology. Very good agreement between the numerical model and experimental data is achieved.

Figures (20)
Figure 1: Modeling URM walls as single rigid block; a) real behavior, b) collapse mechanism, c) rigid block model. VHGLYe PIUUULGS d 1lUoo UL CHELYY UUS W Gil LIVULOIVeE LULUD. In this context, the study of the out-of-plane behavior of unreinforced masonry (URM) walls can be studied as an assemblage of rigid blocks. Figures|1) and /2|show two simply models of a URM wall. The first one presents the case in which the URM wall is detached from the rest of the structure. The collapse mechanism is due to overturning of the wall. This behavior can be study by a single rigid block under RM (Figure|1). The second model is made by an ensemble of two rigid blocks. Particularly, this model is useful if the wall has different width along its height or if the upper part of the wall is restrained (Figure|2).
Figure 1: Modeling URM walls as single rigid block; a) real behavior, b) collapse mechanism, c) rigid block model. VHGLYe PIUUULGS d 1lUoo UL CHELYY UUS W Gil LIVULOIVeE LULUD. In this context, the study of the out-of-plane behavior of unreinforced masonry (URM) walls can be studied as an assemblage of rigid blocks. Figures|1) and /2|show two simply models of a URM wall. The first one presents the case in which the URM wall is detached from the rest of the structure. The collapse mechanism is due to overturning of the wall. This behavior can be study by a single rigid block under RM (Figure|1). The second model is made by an ensemble of two rigid blocks. Particularly, this model is useful if the wall has different width along its height or if the upper part of the wall is restrained (Figure|2).
Figure 2: Modeling URM walls as ensemble of two rigid blocks; a) real behavior, b) collapse mechanism, c) rigid blocks model. between numerical and experimental data [10] has been carried out to verified the proposed model. Very good agreement between the numerical model and experiment is achieved. 2 PARAMETERS OF THE CLASSICAL FORMULATION
Figure 2: Modeling URM walls as ensemble of two rigid blocks; a) real behavior, b) collapse mechanism, c) rigid blocks model. between numerical and experimental data [10] has been carried out to verified the proposed model. Very good agreement between the numerical model and experiment is achieved. 2 PARAMETERS OF THE CLASSICAL FORMULATION
Figure 3: Test specimens, a) single blocks, b) bi-block structure. The experimental tests were carried out on four single blue granite stones (Figure 3a) and an ensemble of two blue granite stones (Figure 3b). Each stone has different geometrical dimen- sion (Tables 1/and 2). The dimensions of the single specimens 1, 2 and 3 were fixed to achieve a Height- Width ratio (h/b) of 4, 6 and 8, respectively. In addition, single specimen number 4 was specifically designed with a different geometry than the others specimens, in order to compare its performance with the rest of the stones. It has a large 45 degree cut (40 mm) at the base. Moreover, a foundation of the same material was used as the base where the blocks are free to rock. This foundation was fixed to the shaking table by means of four steel bolts.
Figure 3: Test specimens, a) single blocks, b) bi-block structure. The experimental tests were carried out on four single blue granite stones (Figure 3a) and an ensemble of two blue granite stones (Figure 3b). Each stone has different geometrical dimen- sion (Tables 1/and 2). The dimensions of the single specimens 1, 2 and 3 were fixed to achieve a Height- Width ratio (h/b) of 4, 6 and 8, respectively. In addition, single specimen number 4 was specifically designed with a different geometry than the others specimens, in order to compare its performance with the rest of the stones. It has a large 45 degree cut (40 mm) at the base. Moreover, a foundation of the same material was used as the base where the blocks are free to rock. This foundation was fixed to the shaking table by means of four steel bolts.
Table 1: Test specimens dimensions.
Table 1: Test specimens dimensions.
Table 2: Dimensions of the blocks used in the bi-block structure.
Table 2: Dimensions of the blocks used in the bi-block structure.
Figure 4: Reference system of the data acquisition system and typical location of LEDs in the RB. errors and enables accurate position measurement. The main data obtained were: rotations around Y and Z axes, and linear displacements and Y (see Figure 4 for the system coordinate). Rotations Y and Z were directly measured by means of a mirror linked to the blocks surface on the West face of the specimens. Two accelerometers were placed at the top of each block. One triaxial accelerometer was located in the North face and one biaxial accelerometer was located in the South face. The displacements and accelerations of the shaking table were also measured.
Figure 4: Reference system of the data acquisition system and typical location of LEDs in the RB. errors and enables accurate position measurement. The main data obtained were: rotations around Y and Z axes, and linear displacements and Y (see Figure 4 for the system coordinate). Rotations Y and Z were directly measured by means of a mirror linked to the blocks surface on the West face of the specimens. Two accelerometers were placed at the top of each block. One triaxial accelerometer was located in the North face and one biaxial accelerometer was located in the South face. The displacements and accelerations of the shaking table were also measured.
Figure 5: Generation of synthetic earthquakes; a) typical accelerogram, b) response spectra of four different records. OEE OIE IA) ODI EE SLI IIE eee eee eee NN Ne Ee Nt Ne —=—o~ Thirty synthetic eartha vakes compatibles with the design spectrum proposed by the Eu- rocode 8 [4] were generated. In order to identify them, they were named consecutively with the number of generating. Figure |5|shows a typical synthetic record and the response spectrum of four different records. The constant branch of the spectrum is located between 0.1 and 0.3 seconds, with a spectral acceleration of 7 m/s? while the maximum ground acceleration is 2.8 m/s. The main aim of the study is to address stability of RM under random motion. 3.4 Theoretical and experimental parameters
Figure 5: Generation of synthetic earthquakes; a) typical accelerogram, b) response spectra of four different records. OEE OIE IA) ODI EE SLI IIE eee eee eee NN Ne Ee Nt Ne —=—o~ Thirty synthetic eartha vakes compatibles with the design spectrum proposed by the Eu- rocode 8 [4] were generated. In order to identify them, they were named consecutively with the number of generating. Figure |5|shows a typical synthetic record and the response spectrum of four different records. The constant branch of the spectrum is located between 0.1 and 0.3 seconds, with a spectral acceleration of 7 m/s? while the maximum ground acceleration is 2.8 m/s. The main aim of the study is to address stability of RM under random motion. 3.4 Theoretical and experimental parameters
Table 3: Theoretical and experimental classic parameters of single blocks. Table 4: Theoretical and experimental classic parameters of bi-block structure. T =Theoretical, E = Experimental, % = Error (in percentage) and was taken from [7, 13]. The commercial UDEC code, based on the DEM, has been used to perform the numerical analyses.
Table 3: Theoretical and experimental classic parameters of single blocks. Table 4: Theoretical and experimental classic parameters of bi-block structure. T =Theoretical, E = Experimental, % = Error (in percentage) and was taken from [7, 13]. The commercial UDEC code, based on the DEM, has been used to perform the numerical analyses.
Figure 8: Typical DE models in free rocking motion; a) specimen 2, b) specimen 4. The free while Figure rocking motion tests were used to calibrate the DE model. Table geometrical ch 8 1 shows the aracteristics of the four specimens used in the experimental testing program; depicts typical DE models during free rocking motion. ) CALIBRATION OF THE MODELS .1 Single block model
Figure 8: Typical DE models in free rocking motion; a) specimen 2, b) specimen 4. The free while Figure rocking motion tests were used to calibrate the DE model. Table geometrical ch 8 1 shows the aracteristics of the four specimens used in the experimental testing program; depicts typical DE models during free rocking motion. ) CALIBRATION OF THE MODELS .1 Single block model
Figure 9: Typical free rocking motion response of DE model using theoretical and fitting parameters (Specimen 1). Table 5: Parameters used in the definition of the DE models.
Figure 9: Typical free rocking motion response of DE model using theoretical and fitting parameters (Specimen 1). Table 5: Parameters used in the definition of the DE models.
Figure 10: Typical free rocking motion response of DE model using fitting parameters (Specimen 1, with initial conditions from Figure{9); a) Horizontal acceleration, b) Vertical acceleration.
Figure 10: Typical free rocking motion response of DE model using fitting parameters (Specimen 1, with initial conditions from Figure{9); a) Horizontal acceleration, b) Vertical acceleration.
Figure 11: Fitting curve for empirical formulation of equivalent damping.
Figure 11: Fitting curve for empirical formulation of equivalent damping.
Figure 13: Typical free rocking motion response of DE model using fitting parameters (bi-block structure); a) Upper block, b) Lower block. Figure 12: Typical bi-block model in free rocking motion.
Figure 13: Typical free rocking motion response of DE model using fitting parameters (bi-block structure); a) Upper block, b) Lower block. Figure 12: Typical bi-block model in free rocking motion.
Table 6: Parameters used in the definition of the bi- block models.
Table 6: Parameters used in the definition of the bi- block models.
Figure 14: Typical results of harmonic motion simulation with a constant sine excitation; a) specimen 1, constant sine with frequency of 3.3 Hz and amplitude of 6 mm, b) specimen 3, constant sine with frequency of 3.0 Hz and amplitude of 4 mm. Figure |14| shows a typical response of RB under constant sine base motion. For clarity of the figure, there is a gap in the time axis that corresponds to the first part of the stationary state. The response of the model is almost the same as the experimental test even during the last cycles. The three states of the harmonic motions (transient, stationary and free rocking) are well reproduced by the model.
Figure 14: Typical results of harmonic motion simulation with a constant sine excitation; a) specimen 1, constant sine with frequency of 3.3 Hz and amplitude of 6 mm, b) specimen 3, constant sine with frequency of 3.0 Hz and amplitude of 4 mm. Figure |14| shows a typical response of RB under constant sine base motion. For clarity of the figure, there is a gap in the time axis that corresponds to the first part of the stationary state. The response of the model is almost the same as the experimental test even during the last cycles. The three states of the harmonic motions (transient, stationary and free rocking) are well reproduced by the model.
Figure 15: Typical results of random motion test; a) specimen 2, earthquake record 21 and load factor of 0.6, b) specimen 3, earthquake record 18 and load factor of 0.5.
Figure 15: Typical results of random motion test; a) specimen 2, earthquake record 21 and load factor of 0.6, b) specimen 3, earthquake record 18 and load factor of 0.5.
From the experimental test fi associated with the amplitude ai stant sine base motion with freq rst and corrobo uencies betwee with frequency of 6 Hz and amplitudes greater greater than 1 mm. The pattern 4 was recorded with constant si are 2 and 4 n4 and 5 Hzai rated by the numerical model, the principal pat- ferns that the bi-block structure tested exhibit nd frequency of (Figure (17). These patterns are the loads. The pattern 2 was record with con- nd amplitudes greater than 4 mm, than 3 mm and frequency 7 Hz with amplitudes between 1 and 5 Hz with amplitudes lesser than 4 mm. ce Pe ce ice ct Be EN Be st tp we cee se ec Fre = ES = ne base motions with frequencies
From the experimental test fi associated with the amplitude ai stant sine base motion with freq rst and corrobo uencies betwee with frequency of 6 Hz and amplitudes greater greater than 1 mm. The pattern 4 was recorded with constant si are 2 and 4 n4 and 5 Hzai rated by the numerical model, the principal pat- ferns that the bi-block structure tested exhibit nd frequency of (Figure (17). These patterns are the loads. The pattern 2 was record with con- nd amplitudes greater than 4 mm, than 3 mm and frequency 7 Hz with amplitudes between 1 and 5 Hz with amplitudes lesser than 4 mm. ce Pe ce ice ct Be EN Be st tp we cee se ec Fre = ES = ne base motions with frequencies
Figure 17: Typical rocking patterns fora bi-block structure found experimentally and numerically; a) Pattern 2, b) Pattern 4.
Figure 17: Typical rocking patterns fora bi-block structure found experimentally and numerically; a) Pattern 2, b) Pattern 4.
Figure 19: Typical result of harmonic motion simulation with a constant sine excitation (Upper block); a) frequency of 3.0 Hz and amplitude of 3 mm, b) frequency of 4.0 Hz and amplitude of 1 mm.
Figure 19: Typical result of harmonic motion simulation with a constant sine excitation (Upper block); a) frequency of 3.0 Hz and amplitude of 3 mm, b) frequency of 4.0 Hz and amplitude of 1 mm.

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