Experimental validation of a numerical model for the interaction of dip-slip normal fault ruptures, sand deposits, and raft foundations M. Rokonuzzaman*1, A. El Nahas2 and T. Sakai3 An incomplete understanding of the failure mechanisms in fault rupture propagation has led to inconsistent and insufficient regulations in building codes. In the present study, a sophisticated numerical model is calibrated and validated in order to clarify a complex problem involving the interaction of fault ruptures, medium dense Fontainebleau sand deposits, and existing structures across the fault plane. Calibration is performed using direct shear test data. Repeatable centrifuge models of dip-slip normal faults with a dip angle of 60u in the free field condition and light and heavy rigid strong raft foundations are used for the validation. The present numerical model satisfactorily simulates the centrifuge models. Rigid rafts divert the shear bands so as to bypass the rafts, rather than rip them apart. The rafts tilt on the foundation soil during fault rupture. The raft-tilting increases as the raft bearing pressure on the soil decreases. Keywords: Sand deposits, Normal fault, Raft foundation, Centrifuge, Shear band, Interaction, Elastoplastic, FEM
*Corresponding author, email
[email protected]
active regions requires a better understanding of the complex problem of the interaction of fault ruptures, soil deposits, and existing structures. However, the main goals of previously published studies were to identify the failure pattern in the alluvium (Bray et al., 1994a; Cole and Lade, 1984; Roth et al., 1981; Lade et al., 1984) and the height of fault rupture in the model ground (Bray et al., 1994b), as well as the general criterion for surface faulting (Scott and Schoustra, 1974) in the free field condition. However, a number of field case studies (Bray et al., 1994a; Taylor et al., 1985; Anastasopoulos, 2005) were conducted to investigate the failure mechanisms in the soil during a fault rupture event and the effects of the resulting ground movements and deformations on existing structures across or near the fault rupture plane. Soil behavior after failure has been shown to play a major role in problems related to shear band formation and propagation, and this mechanism can be investigated using an accurate numerical model. Applying the finite element method (FEM) in combination with the elastic-perfectly plastic constitutive soil model with Mohr–Coulomb failure criterion, Scott and Schoustra (1974) obtained results that contradict both reality and experiments. In contrast, Bray (1990) and Bray et al. (1994a, 1994b), also using the FEM with a hyperbolic non-linear elastic constitutive law, achieved satisfactory agreement with small scale model test results. Walters and Thomas (1982) performed a sandbox experiment and FEM simulations and found that the nonassociated flow rule and proper strain softening were essential in the localization of rupture. Other researchers
ß 2015 W. S. Maney & Son Ltd Received 4 February 2014; accepted 12 March 2014 DOI 10.1179/1939787914Y.0000000057
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Introduction The co-seismic movements on the fault surface in the dip direction cause permanent soil deformations in the overlaying soil strata. In a relatively dense soil, the strains may be localized in narrow shear bands that propagate upward in the soil. If the soil is less dense, soil deformations during fault rupture may be associated with extensive shear bands in the overlaying soil. Terzaghi (1943) classified these failure patterns as general and local failures, and Bray et al. (1994a) reported that these failure patterns occurred in brittle and ductile ground materials, respectively. As a secondary effect, the resulting permanent soil displacements in the overlaying soil cause additional stresses in the geotechnical structures in and above the soil layer. The resulting strains or displacements of the structure may violate the serviceability requirements of the structure, or worse, cause structural collapse. The published guidelines by US governmental agencies (Cole and Lade, 1984) or other building design codes include inconsistent setback criteria to restrict the distances between the facilities and existing active or potentially active faults. Modifying the current design codes for these facilities in the seismically 1
Department of Civil Engineering, Khulna University of Engineering & Technology, Khulna-920300, Bangladesh Haskoning UK Ltd., 17, Hornbeam Drive, Tile Hill, Coventry CV4 9UJ, UK 3 Department of Environmental Science and Technology, Graduate School of Bioresources, Mie University, 1577 Kurimamachiya, Tsu, Mie, Japan 2
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1 Dimensions of the experimental apparatus installed in the Dundee University centrifuge (all dimensions are in millimeters and figure is not to scale)
(Roth et al., 1981, 1982; Nakai et al., 1995; White et al., 1994) applied the finite difference method with the elastoplastic constitutive model, Mohr–Coulomb failure criterion, and strain softening. In their simulations, rupture propagated through the sand and broke the ground surface with only a fraction of the displacement observed in experiments. However, the previous numerical models used for the analysis of fault rupture propagation through overlaying soil beds were not accurate and robust in the modeling of material hardening/softening or in handling the mesh size effect (due to softening) and the confining pressure effect, which indicates the necessity for an accurate numerical model and its validation. In the present study, a numerical model using an elastoplastic framework with a non-associated flow rule, a strain-hardening/softening law, and the shear band effect is calibrated and validated for the study of a complex target problem involving the interaction of normal fault ruptures, sand deposits, and rigid raft foundations. The calibration of the numerical model is performed using direct shear model test data. Centrifuge models, which can produce prototype stress level in the model ground of dip-slip normal fault events with a dip angle of 60u in the free field condition, and light and heavy rigid strong raft foundations are used to validate the numerical model, by comparing the failure mechanisms from experimental and numerical results. Such an integrated study will help to ensure that the proposed numerical model is applicable in the real-world design and analysis of such complex problems.
Testing procedures The beam centrifuge of the University of Dundee was used in the experiment on fault rupture propagation through a Fontainebleau sand deposit (D5050?24 mm, Cu51?33, Gs5 2?59, emax50?833, emin50?55, and fines content50%). The internal dimensions of the strongbox are 80065006500 mm (Fig. 1). The strongbox has front and back transparent Perspex plates through which the models were monitored during the tests. Two hydraulic cylinders were used to push the right part up or down to simulate reverse and normal faulting. A central guide (G) and three wedges (A1–A3) were used to guide the imposed displacement at the desired dip angle (60u). Sand was
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pluviated in the strongbox in 20–30-mm thick layers up to 217 mm. A line of dyed sand was laid on top of each layer in order to clearly visualize the shear bands. The density measuring cans were placed to verify the sand unit weights near the edges, at the bottom, and in the middle of the model ground. A series of digital images were taken for the displaced model ground after each stepwise fault dip slip of approximately 0?5–1?5 mm until the total machine allowable maximum vertical component of base dislocation hmax was reached, as shown in Table 1. The displacement vectors and shear strains in the model ground were analyzed using the deformation measurement system (Geo-PIV program of White et al., 2003). In addition, linearly variable differential transformers (LVDTs) were used to monitor the vertical settlements of the model ground surface and the vertical component of the base dislocation h. The parameters of the physical model used in the present study are defined in Fig. 2. After preparing the model, the strongbox was mounted on the centrifuge, and the centrifuge was spun to 115g. The dimensions and parameters of the prototype used in the experiments are listed in Table 1. Three tests were conducted in the repeatable centrifuge test setup in order to investigate the failure mechanism in the sand layer during the movement on an underlying normal fault. One test (Test12_R2) was conducted in the free field condition in order to confirm the outcropping location of the fault rupture on the ground surface, and other two tests (Tests 14_R and 15) were conducted in order to study the interaction between the fault rupture and existing rigid rafts for light and heavy structures, centered above the scarp position confirmed in the free field test. The nature and dimensions of the superstructures were not considered. Two model raft footings were used (shown by R in Fig. 1). Each had dimensions of 876500610 mm. Sheets of sand paper (No. 100) were glued to the bottom surfaces of the models in order to create a rough-base condition. In Tests 14_R and 15, the footings were made of steel and aluminum, respectively, in order to obtain foundation bearing pressures (q, Fig. 2) of 91 and 37 kPa, respectively. Details on the experiments of the present study can be found elsewhere (El Nahas et al., 2006).
Numerical modeling The finite element (FE) model of the present study uses an elastoplastic framework with a non-associated flow rule and a strain-hardening/softening law. An explicit dynamic relaxation method devised with a return mapping stress updating algorithm (Tanaka and Kawamoto, 1988; Tanaka and Sakai, 1993) is used for the fast solution of non-linear equations. The standard FE solutions of the strain-softening material are strongly dependent on mesh Table 1 Prototype dimensions centrifuge models
and
parameters
for
Test name
Dr/%
H*/m
L*/m
W*/m
hmax/m
q*/kPa
Test12_R2 Test15 Test14_R
60.2 59.2 62.5
24.6 24.9 24.6
75.9 75.9 75.9
24.15 24.15 24.15
3.15 3.15 2.89
0** 37*** 91***
*
the
These terms are defined in Fig. 2. Fault rupture in free field and foundation centered at the scarp position confirmed in the free field test, respectively. **, ***
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and sm is the mean stress (positive in compression), J2 is the second invariant of deviatoric stresses, h is the Lode angle, k is an internal variable, and dexp, deyp, dezp, and dexyp are the incremental deviatoric plastic strains along the x, y, and z axes. In the case of the Mohr–Coulomb model, the Lode angle function, g(h), is given as 3{sin Qmob gðhÞ~ pffiffiffi 2 3 cos h{2 sin h sin Qmob
(7a)
where " pffiffiffi # 1 {1 3 3 J3 h~ cos 3 2 J 3=2
(7b)
2
2 Definitions of the physical model with raft (figure is not to scale)
size. Several techniques have been proposed to resolve the mesh-dependent pathology of FE solutions. For example, Pietruszczak and Mroz (1981) proposed the concept of using a softening modulus scaled by the element size. Following this method, shear banding is introduced in the numerical model by a strain localization parameter S in the additive decomposition of total strain increments (de) into elastic (dee) and plastic (dep) parts, as follows de~dee zSdep
(1)
where S5Fb/Fe (Fb is the area of the shear band in each element, and Fe is the total area of an element). Here, Fb is decided based on the shear band thickness (SB). By ignoring the effects of the orientation of the shear band in each finite element, an approximated form of S is used in the present study SB S~ pffiffiffiffiffi Fe
(2)
The shear band thickness is known to be approximately 16–30 times the mean particle diameter (D50) (Vardulakis et al., 1981; Yoshida et al., 1993). In the present study, it is assumed that the deformation of a given soil element under uniform boundary stress conditions is homogeneous in the pre-peak regime and that the strain localization in a shear band starts suddenly at the peak stress state. The yield function (f ) corresponding to the Mohr– Coulomb model and the plastic potential function (Y), which is represented geometrically by the Drucker–Prager model, are given as follows pffiffiffiffiffi J2 f ~{3aðkÞsm z ~0 (3) gðhÞ pffiffiffiffiffi Y~{3a’ðkÞsm z J2 ~0
(4)
where ð k~ de- p 2 n 2 2 2 o 2 de- p ~2 dexp z deyp z dezp z dcxyp
and J3 is the third invariant of the deviatoric stresses. The mobilized friction angle wmob is given as ( pffiffiffi ) 3 3aðkÞ {1 pffiffiffi (8a) Qmob ~sin 2z 3aðkÞ where a(k) are the frictional hardening and softening functions and expressed as pffiffiffiffiffiffiffim 2 kef aðkÞ~ ap (hardening-regime : kƒef ) (8b) kzef aðkÞ~ar z ap {ar exp (
) k{ef 2 (softening regime : kwef ) { er
(8c)
where ef, er, and m are hardening/softening material parameters. The parameters of ap and ar are estimated using the following equations 2 sin Qp ap ~ pffiffiffi 3 3{sin Qp
(9a)
2 sin Qr ar ~ pffiffiffi 3ð3{sin Qr Þ
(9b)
where wp and wr are the peak and residual friction angles, respectively, which can be obtained from appropriate model tests in the laboratory. However, in order to consider the stress level effect, the peak friction angle is estimated from the empirical equation of Bolton (1986) Ir ~Dr f10{lnðsm Þg{1
(10a)
Qp ~3Ir zQr
(10b)
This equation agrees well with the direct shear test results at different confining pressures for medium dense Fontainebleau sand (Fig. 3). The results of this test are discussed in detail in the following section. The plastic potential function a9(k) is defined for the plane strain conditions as
(5)
(6)
tan y a’ðkÞ~ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 9z12 tan2 y
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3 Effect of confining pressure on peak frictional angle
The dilatancy angle y is estimated from the modified Rowe’s stress–dilatancy relationship sin y~
sin Qmob {sin Q’r 1{sin Qmob sin Q’r
(11b)
where "
(
2 )# k Q’r ~Qr 1{b exp { ed
(11c)
and ed and b are the stress–dilatancy material parameters. The elastic moduli are estimated using the modified equation proposed by Hardin and Black (1968) and, in the case of clean sand, are given by the following equations : ð2:17{e0 Þ2 sm 0 5 Pa ðPa ~98 kPaÞ (12a) Gmax ~G0 1ze0 Pa K~
2(1zn) Gmax 3(1{2n)
(12b)
where G0 is the initial-shear-modulus constant, e0 is the initial void ratio, Pa is atmospheric pressure, and v is Poisson’s ratio.
Calibration In order to use the numerical model to analyze fault problems, it is necessary to calibrate the material parameters for hardening/softening (ef, er, and m) and
the stress–dilatancy relationship (ed and b) of the incorporated sophisticated constitutive model. For this purpose, the model experimental data obtained from the direct shear tests are used, because this data closely mimics the shearing mechanism in the faults (El Nahas et al., 2006). The Fontainebleau sand (Dr563¡4%) was pluviated in the shear box. Direct shear tests were then conducted at a quasi-static displacement rate of 0?01 mm s21. The length and thickness of the specimen were 60 and 30?4 mm, respectively. The test procedures followed the test standards BS1377-7 (British Standards Institution, 1990). The FE mesh used to simulate the direct shear box test is shown in Fig. 4. The analysis was performed for the plane strain case with the following boundary conditions. The nodes along the bottom of ‘‘A’’ were fixed. Nodes along the upper box sides were given a prescribed displacement in the horizontal direction. Element row ‘‘C’’ represents the gap between the upper and lower boxes. The upper box was free to move relative to the lower box. The solid top element row ‘‘B’’ was the loading plate for transmitting the vertical load. The loading plate was free to rotate and to move in the vertical direction. The material of the elements for the loading plate and the side walls was linear elastic (Young’s modulus: 2?16104 MPa, Poisson’s ratio: 0?3). Interfaces were assumed to be a soil sample attached to all walls with a maximum frictional angle of 6u (Tatsuoka and Haibara, 1985). In order to clarify the effect of the material parameters of such a sophisticated constitutive model on the relationship between the average shear stress of the elements along the prescribed horizontal shear plane (Fig. 4) and the average vertical displacements of the loading plate with respect to the relative prescribed horizontal displacements in the direct shear tests, a detailed parametric study was conducted and some of the obtained results are shown in Fig. 5a. The hardening/ softening material parameters ef, er, and m influence the pre- and post-peak regimes of the relationship. The parameter ef is related to er. The parameter m influences the hardening-regime, and er influences the softeningregime. In addition, the stress–dilatancy material parameters (b and ed) control the mobilization of dilatancy and shear stress in the direct shear test box. As shown in Fig. 5b, there is close agreement between the numerical and experimental results for some of the direct shear tests with effective normal stresses (200 and 385 kPa) using the calibrated material parameters given in Table 2. Thus, the calibrated numerical model is valid for investigating the
4 Finite element mesh and constituents for direct shear test box analysis (‘‘A’’5soil, ‘‘B’’5loading plate, ‘‘C’’5gap, figure is not to scale)
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5 a Parametric study on direct shear test box with normal effective stress of 100 kPa, and b comparison of experimental and numerical results (using calibrated material parameters) with normal effective stress of 200 and 385 kPa
stress-level effect, the strain hardening–softening nature, and dilation in direct shear tests. Table 2 Material parameters of the numerical model Unit weight/kN m23 Initial void ratio (e0) Pressure coefficient at rest (K0) Initial-shear-modulus constant (G0/kPa) Residual friction angle (wr/u) Poisson’s ratio (v) Shear band thickness (SB/mm, model scale) ef er ed m b
15.57 0.64 0.5 50 30.2 0.3 3.84 0.2 0.6 0.3 0.8 0.2
FE techniques and mesh size effect Following the recommendation of Bray (1990), the FE discretizations of the sandbox tests, as shown in Fig. 6, were performed such that the width of the FE model was equal to 4H (H is the uniform soil deposit thickness, Fig. 2) in order to minimize undesired boundary effects. The discretization was finer in the central part of the model than at the two edges, where limited deformation was expected. The differential quasi-elastic displacement was applied in small consecutive increments to the right side of the model (hanging wall). The elements were four-node quadrilateral Lagrange elements
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6 Finite element mesh and boundary conditions (R5raft and figure is not to scale)
with reduced integration. Linear elastic elements with a Young’s modulus of 2?16104 MPa and a Poisson’s ratio of 0?3 were used for the footing. However, following Kotake et al. (1999), no special interface elements were used along the interface between the rough footing base and the ground. Since the conventional ‘‘zero thickness’’ interface elements are relevant to the case of displacement discontinuity in the interface and the parameters of these elements are not so clear, particularly for modeling of the dilatancy characteristics of the thin elements beside the contact. It was assumed that a shear band similar to bands that can develop inside the ground can also develop along the interface elements. Before selecting a particular size of mesh for the analyses of the fault problems, it is necessary to investigate how much the proposed numerical model is susceptible to mesh size. In this regard, for the analyses of mesh size sensitivity, finite elements of 161 m (width6height),
7 Analysis of sensitivity of mesh density (compared with Test 12_R2) for vertical displacements of ground surface
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1?561?5 m, and 262 m in size were used in the central part of the mesh, and coarser elements of 261 m, 361?5 m, and 462 m in size were used at the two edges. The sand (D5050?24 mm) modeled in the centrifuge sand box corresponds to a prototype material with a mean particle size diameter equal to nD50, where n is the scale
8 Comparison of experimental ground images (Test 12_R2) with deformed mesh and shear strain contour for FE sizes: 1, 1?5, and 2 m (figures are not to scale)
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9 Comparison of experimental and numerical vertical displacements of ground surface for Test 12_R2
factor. Therefore, the prototype shear band thicknesses, used for 115g (n5115) model test analyses, was 115616D50 (
