study on the correlation between shear & rupture

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KEY WORDS: Digitization, Chandigarh fault, rupture velocity, applied element ... a correlation between shear wave and rupture velocity of Chandigarh fault.
STUDY ON THE CORRELATION BETWEEN SHEAR & RUPTURE WAVE VELOCITY OF CHANDIGARH FAULT Mohammed Ahmed Hussain a*, Chenna Rajaram b and Ramancharla Pradeep Kumar c a*

Associate Professor, Department of Civil Engineering, Nizam College of Engineering and Technology, Hyderabad, India – [email protected] b Ph.D Scholar, Earthquake Engineering Research Centre, International Institute of Information Technology, Hyderabad, India – [email protected] c Professor, Earthquake Engineering Research Centre, International Institute of Information Technology, Hyderabad, India – [email protected]

KEY WORDS: Digitization, Chandigarh fault, rupture velocity, applied element method

ABSTRACT: Recent large earthquakes show that different fault segments may rupture at different velocities and rupture may propagate at comparable speed to that of shear wave velocity. To investigate the seismic rupture process, kinematic source models can be used. Generally the characteristics of rupture process such as rupture velocity, rise time, or the stress drop may depend on the geometry of the fault. In the absence of the exact value, the rupture velocity is usually taken as 0.8 times of shear wave velocity, however, it should be noted that the velocity is different at different fault segments during earthquake. An attempt has been made to calculate the rupture velocity using Applied Element Method. The present work illustrates a correlation between shear wave and rupture velocity of Chandigarh fault. Using digitization, raster data is converted into vector data from Open Jump tool around 100 kms radius of Chandigarh. A1:15 scaled numerical model of Chandigarh fault is created. Parameters used are as follows; length 3 km, width 0.5 km and depth 0.2 km. The shear wave velocity is considered as 500 m/s. The boundary condition at the left is fixed and at the right of the model is assumed to be free so as to move in vertical and horizontal directions. Bedrock displacement of 2 m is given to the hanging wall with a dip angle of 45°. To study the effect of slip rate, three models of slip rise times i.e., 10, 5 and 2.5 sec are considered. The rupture velocity is calculated at every time step during rupture process. From the analysis, it is observed that for rise time of 2.5 sec, most of the fault region has cracked due to high frequency waves. Also, from numerical model, it is understood that the rupture velocity is approximately equal to 0.683 times of shear wave velocity, which re-confirms the theoretical approximations.

1. INTRODUCTION Locations of the surface rupture and its propagation through the soil deposit, reaching the ground surface are important in city planning, especially in the design of critical structures located near the zone of faulting. However, the damage caused by the recent earthquakes has posed many new challenges to focus the study on the fault motion and its effects on the surface behavior. Earthquakes in different geological regions show drastic variations in their effect. Carrying out the experimental studies to understand such kind of behavior is difficult, because of severe complications involved in replicating the exact site conditions. Moreover, many experiments of high precision are to be conducted to establish relationships between various parameters, such as width, soil properties, and thickness of deposit and dip angle. Numerous researchers have attempted to study this phenomenon through experiments and tried to establish the relationship between various parameters of soil and resulting surface characteristics. Numerical modelling allow us to investigate a number of aspects of the fault rupture propagation, which are difficult to study from the examination of case histories or the conduct of physical model tests. Numerical simulations of earthquake fault rupture have the much advantage of being more flexible to

investigate a number of aspects of the fault rupture propagation phenomenon than analytical solutions. One of the first numerical solutions of earthquake rupture (Andrews, 1976), used a two-dimensional finite difference method (FDM) to model rupture propagation under a slip-weakening friction law; the post-failure stress on a point was found to decrease linearly as a function of slip at the point. It was developed further by numerous authors (Day,1982a,b; Mikumo and Miyatake, 1995; Harris and Day, 1993) and can be used to study rupture propagation in heterogeneous elastic media. A simulation of the 1992 Landers earthquake (Olsen et al., 1997) demonstrated the ability of FDM and a slip-weakening friction model to produce reasonable rupture behaviour. The 2D Lattice numerical model was to study dynamic simulation of thrusting rupture behaviour and the associated near-fault strong ground motions (Shi et al., 1998). It has been seen that when rupture reaches the toe of the fault outcrop, the hanging wall breaks away from the foot wall and creates a large opening vibration at the hanging wall toe. Applied element method (AEM) in 2D which is based on discrete modelling was used to investigate the response on the ground surface due to seismic base fault movement (Pradeep et al., 2000). The main advantage of this method of modelling is it has the ability of crack initiation based on the material failure and propagation of crack till the collapse. It was found that the ground acceleration very near to the fault is not maximum

* Corresponding author. This is useful to know for communication with the appropriate person in cases with more than one author.

instead the peak value is little away from the fault trace due the high nonlinearity of the soil. Both the vertical and normal peak ground motion has been seen to occur on the hanging wall side. The dynamic rupture process of a normal fault was also studied using 2D lattice particle model (Shi et al., 2003). Using the same 2D lattice particle model (Shi et al., 2005) the characteristics of near-fault ground motions was studied by dynamic thrust faulting with a sedimentary layer on the surface. With the sedimentary layer (lower shear wave velocity) present only on footwall the largest peak ground motions, both in terms of velocity and acceleration, occur on the footwall side. The foam rubber an experiment was used to study (Day et al., 2008) the rupture directivity along strike slip. The behavior of fault rupture (Ahmed, 2012) was studied for various dip angles using AEM. In this study an attempt has been made to calculate the rupture velocity using AEM.

2. MODELING PARAMETERS OF FAULT Chandigarh city (30.7500N, 76.7800E) is located and over a thick sequence of clay, silt sand and pebble of the Older Alluvium. The geotechnical tests classify the strata of Chandigarh into Class D, (stiff soil) of the NEHRP Code Provisions. The shallow surface shear-wave velocity, empirically derived from N-values, varies from 216 to 305 m/s. Predominant frequency and ground amplification vary from 0.61.66 Hz and 1.31-5.37 Hz respectively. A 14 m long, 2 to 4 m deep and 4.5 m wide E-W trench was dug at the base of a 16-20 m high fault scarp across the Chandigarh fault. The raster image of Chandigarh fault scrap is shown in figure 1. The raster image of Chandigarh fault is extracted through Geological Survey of India (GSI). The fault image is converted into vector through digitization process using OPEN JUMP and QGIS packages. Figure 2 represents vectorization of Chandigarh fault.

3. NUMERICAL METHOD: 3D APPLIED ELEMENT METHOD 3.1 Numerical Modeling Applied Element Method is an efficient numerical tool based on discrete modeling (Hatem, 1998). The two elements shown in figure 3 are assumed to be connected by the set of one normal and two shear springs. Each set is representing the volume of elements connected. These springs totally represents stress and deformation of that volume of the studied elements. Six degrees of freedom are assumed for each element. These degrees of freedom represent the rigid body motion of the element. Although the element motion is as a rigid body, its internal deformations are represented by spring deformation around each element. This means that the element shape doesn’t change during analysis, which means that the element is rigid, but the behaviour of element collections is deformable. To have a general stiffness matrix, the element and contact spring’s locations are assumed in a general position. The stiffness matrix components corresponding to each degree of freedom are determined by assuming a unit displacement in the studied degree of freedom direction and by determining forces at the centroid of each element. The element stiffness matrix size is (12 x 12). However, the global stiffness matrix is determined by summing up the stiffness matrices of individual pair of springs around each element.

Figure 3. Element formulation in 3DAEM

Figure 1. Northwall view of E-W trench at the base of Chandigarh fault scarp near Mansa Devi

We compute the displacement time histories by the threedimensional dynamic elasticity equation given by eq. (1),

{}

{}

&& + [C] U & + [K ]{U} = {P( t )} [ M] U

(1)

Where [M], [C] and [K] are the mass, damping and global stiffness, respectively; U the displacement vector and [P(t)] the applied load vector. Here mass proportional damping matrix is used with 10% damping coefficient. The above differential equation is solved numerically by Newmark’s method. The material model adopted in AEM is the two parameter model called hyperbolic model. It is logical to assume that any stressstrain curve of soils is bounded by two straight lines that are tangential to it at small strains and at large strains as shown in figure 4. The tangent at small strains denoted by Go, represents the elastic modulus at small strains and the horizontal asymptotic at large strain indicates the upper limit of the stress τf, namely the strength of soils. The stress-strain curve for the hyperbolic model can be obtained directly from eq. (2) Figure 2. Location of fault map

τ=

G oγ 1+

γ γΓ

(2)

The above equation has been extensively used for representing the stress-strain relations of a variety of soils. Since, the target of this study is to show the new application of AEM, so we adopted the material model which is based on only two parameters, namely, initial modulus, Go and reference strain, γ Γ = τ f / G o , where τ f is the upper limit of the stress. However, any type of material model can be adopted in AEM. For further details on material modelling please refer O. Hardin 1972.

compression (yc). If the F value is greater or equal to zero the spring is said to be failed. The normal and shear forces in the failed springs are redistributed in the next increment by applying the forces in the reverse direction. These redistributed forces are transferred to the element centre as a force and moment, and then these redistributed forces are applied to the structure in the next increment. The redistribution of spring forces at the crack location is very important for following the proper crack propagation. For the normal spring, the whole force value is redistributed to have zero tension stress at the crack faces. Although shear springs at the location of tension cracking might have some resistance after cracking due to the effect of friction and interlocking between the crack faces, the shear stiffness is assumed zero after crack occurrence. Having zero value of shear stress indicates that the crack direction is coincident with the element edge direction. In shear dominant zones, the crack direction is mainly dominant by shear stress value. This technique is simple and has the advantage that no special treatment is required for representing the cracking.

Figure 4. Non-linear behavior of soil - Skeleton curve 3.2 Failure Criteria To define the failure criteria we need to find the threedimensional state of stress at each point where the spring is defined. The three-dimensional state of stress is defined at each spring location point. After obtaining all the components of stress tensor we shall define the failure criteria. A Mohr Coulomb failure criterion has been adopted here. Mohr Coulomb invariants I1, J2 and θ (smith et. al., 2004) has been calculated using three dimensional stress components. After defining the Mohr Coulomb invariants soil's internal friction angle ‘Φ’ and cohesion ‘c’ is calculated using uni-axial tension capacity yt and uni-axial compression capacity yc and from eq. 3 & eq. 4 (Boresi et. al., 2002).

 yt   − 2 tan −1   yc  2   y  y  c = t − c  2  y t 

φ=

π

(3)

(4)

Using the above invariants the Mohr-Coulomb failure envelops is defined by eq. 5. (Smith et. al. 2004) In principal stress space, this criterion takes the form of an irregular hexagonal cone, as shown in figure 5.

 1 sin θ sin φ   − c cosφ F = I1 sin φ + J 2  cos θ − 3 3   Failure if F ≥ 0

(5)

The failure envelop F depends on the invariants discussed above and the cohesion c and the friction angle ‘Φ’ which depends on the soil uniaxial tension (yt) and uniaxial

Figure 5. Mohr Coulomb failure envelope in 3D

4. NUMERICAL MODELING OF CHADIGARH FAULT The study is mainly focusing on the near fault ground motion due to seismic bedrock movement (slip). In dynamic conditions we also need the slip rate in addition to the slip value at the bedrock level. Since we do not have the slip or slip rate for the Indian faults at the bedrock level, we try to approximate it from the paleo-seismic studies for the Chandigarh fault. From the paleo-seismic studies, a vertical slip of 1.47 m and horizontal shortening of 3.17 meter with a dip angle of 25° has been seen at a depth of 2-4 meters below the ground surface. The slip value which is seen is not the actual bedrock displacement and for the purpose of analysis, the slip value and the slip rate at the bedrock level are needed. In order to develop the correlation of the slip just below the surface with the actual bedrock displacement, a 1:15 scaled numerical model of Chandigarh fault is created with length 3km, width 0.5km and depth 0.2km as shown in figure 6. Bedrock displacement of 2 m is given to the hanging wall with a dip angle of 45°. The average shear wave velocity of the sediments down to 30 m depth at different sites around Chandigarh Urban Centre varies from less than 216 to 305 m/s. Here the shear wave velocity has been considered as 500 m/s. The base of the fault lies exactly at the centre of the model. The boundary condition at the left is fixed and at the right of the model is assumed to be free in vertical and horizontal direction.

(c)

Figure 6. 3D Numerical modelling of fault For applying the bedrock displacement value in the form of Pulse-like displacement time history that represents the base motion is considered (Mladen, 2000 and Pradeep, 2001). As an approximation, the corresponding displacement pulse can be assumed as Gaussian-type function shown in eq. 6. Where Vsp is the amplitude of static velocity pulse, Tp is velocity pulse duration, tc time instant, at which the pulse is centered, ‘n’ constant equal to 6 and t is the time. The term ‘Tp/n’ has the meaning of standard deviation and controls the actual spread of the pulse with respect to the given pulse duration and Φ is the normal probability function. d sp ( t ) =

 t − tc  2π  Vsp Tpφ   Tp /n  n  

Figure 8. Cross section of the numerical model and zone of tension and shear cracks for the slip rate with rise time of (a) 10 s, (b) 5 s, and (c) 2.5 s

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(6)

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Figure 9. Peak ground acceleration on the surface for different slip velocity (a) horizontal, (b) vertical

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To study the effect of slip rate three models of slip has been taken as shown in figure 7. The three slip time histories are with rise time 10, 5 and 2.5 sec respectively. By doing dynamic nonlinear analysis on the numerical model shows the zone of tension and shear cracks for the three different slip rate applied respectively (see figure 8(a) – (c)). From the crack pattern, it can be said that as the slip rate increases the model becomes unstable and gives unrealistic results. From figure 8(c), it can be seen that for the rise time of 2.5 sec most of the fault region has been failed due to high frequency waves and as it reaches the surface, ground acceleration values are unrealistic. Figure 9 (a) and (b) shows the comparison of the variation of horizontal and vertical acceleration on the surface for the different slip rates applied at the bedrock level. Increase in the response on the surface due to decrease in rise time values can be seen from the figure 9. From the above description it is decided that the rise time of 10 sec used for the rest of the analysis.

To study the ratio of rupture to shear wave velocity, the same model has been taken. A rise time of 10 sec displacement time history is applied at the bedrock level (see figure 7) and study the rupture propagation. Chandigarh is one of the places, where moderate earthquakes occur. For moderate earthquakes, the rupture velocities are commonly estimated to be 0.5-0.85 of shear wave velocity (Source: USGS). A study has been done to obtain the rupture velocity of Chandigarh fault. The rupture starts at 3.94 sec and its initial velocity is 16 m/s. As the rupture length increases, its velocity also increases. Finally it reaches to a maximum value of 342 m/s at 9.02 sec. It is assumed that the shear wave velocity is 500 m/s. The maximum ratio of rupture to shear is 0.68. It is concluded that Vr = 0.68 Vs for Chandigarh fault. Figure 10 represents the propagation of rupture length of Chandigarh fault. Figure 11 represents the rupture velocity and Vr / Vs ratio. Height (m)

Figure 7. Input vertical displacement applied at the base of the fault plane for different rise time

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Andrews, D. J., 1976b. “Rupture velocity of plane strain shear cracks”, J. Geophys. Res., 81(32), pp.5679-5687.

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Boresi Arthur and Richard J. Schmidt, 2002. “Advanced Mechanics of Materials”, John Wiley & Sons publications.

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Day, S. M., 1982a. “Three-dimensional finite difference simulation of fault dynamics: rectangular faults with fixed rupture velocity”, Bull. Seismol. Soc. Am., 72, pp.705-727.

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Day, S. M., 1982b. “Three-dimensional simulation of spontaneous rupture: the effect of non uniform pre-stress”, Bull. Seismol. Soc. Am., 72, pp.1881-1902.

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Day, S. M., Sarah H. Gonzalez, Rasool Anooshehpoor and James N. Brune, 2008. “Scale-Model and Numerical Simulations of Near-Fault Seismic Directivity". Bull. Seismol. Soc. Am., 98(3), pp. 1186–1206. Harris, R. A., and Day, S. M., 1999. “Dynamic 3D simulations of earthquakes on en echelon faults”, Geophys. Res. Lett. 26, pp.2089–2092. Hatem Tagel-Din, 1998. “A new efficient method for nonlinear, large deformation and collapse analysis of Structures”, Ph.D. thesis, Civil Eng. Dept., The University of Tokyo. Mikumo, T., and T. Miyatake, 1995. “Hererogeneous distribution of dynamic stress drop and relative fault strength recovered from the results of waveform inversion the 1984 Morgan Hill, California, earthquake”, Bull. Seismol. Soc. Am. 85, pp.178-193.

Figure 11. (a) Rupture velocity of fault w.r.t time and (b) ratio of rupture to shear wave velocity

Mladen, V. K., 2000. “Utilization of strong motion parameters for earthquake damage assessment of grounds and structures”, Ph.D. thesis, Civil Eng. Dept., The University of Tokyo.

5. CONCLUSIONS

Mohammed Ahmed Hussain., 2012. “Numerical approach to model near field fault normal ground motions in reverse fault scenario”, Ph.D. thesis, Civil Eng. Dept., International Institute of Information Technology.

The present work illustrates a correlation between shear wave and rupture velocity of Chandigarh fault. A1:15 scaled numerical model of Chandigarh fault is created. The shear wave velocity is considered as 500 m/s. The boundary condition at the left is fixed and at the right of the model is assumed to be free so as to move in vertical and horizontal directions. Bedrock displacement of 2 m is given to the hanging wall with a dip angle of 45°. 1.

2.

From the crack pattern, it can be said that as the slip rate increases the model becomes unstable and gives unrealistic results. For the rise time of 2.5 sec most of the fault region has been failed due to high frequency waves and as it reaches the surface, ground acceleration values are unrealistic. The maximum ratio of rupture to shear is 0.68. It is concluded that Vr = 0.68 Vs for Chandigarh fault. Also, from numerical model, it is understood that the rupture velocity is approximately equal to 0.683 times of shear wave velocity, which re-confirms the theoretical approximations.

Olsen, K. B., Madariaga, R., and Archuleta, R. J., 1997. “Threedimensional dynamic simulation of the 1992 Landers earthquake”,Science, 278, pp.834-838. Pradeep, R. K., and Kimiro Meguro, 2000. “Non-linear static Modelling of Dip-slip faults for studying ground surface deformation using Applied Element Method”, seisankenkyu, 52(12), pp. 602 -605 Pradeep, R.K., 2001. “Numerical analysis of the effects on the ground surface due to seismic base fault movement”, Ph.D. thesis, Civil Eng. Dept., The University of Tokyo. Shi, B., Anooshehpoor, A., Brune, J. N., and Zeng, Y., 1998. “Dynamics of thrust faulting: 2-D lattice model”, Bull. Seismol. Soc. Am. 88, pp.1484 -1494. Shi, B., Brune, J. N., Zeng, Y., and Anooshehpoor, A., 2003. “Dynamics of earthquake normal faulting: two-dimensional lattice particle model”, Bull. Seismol. Soc. Am. 93, pp.11791197.

6. REFERENCES Andrews, D. J., 1976a. “Rupture propagation with finite stress in antiplane strain”, J. Geophys. Res., 81(20), pp. 3575 -3582.

Shi, B., and James N. Brune, 2005. "Characteristics of NearFault Ground Motions by Dynamic Thrust Faulting: TwoDimensional Lattice Particle Approaches” Bull. Seismol. Soc. Am., Vol. 95(6), pp.2525–2533.

Smith, I. M., Griffiths, D. V., 2004. “Programmming the finite element method.” John Wiley & Sons publications.