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Progressive failure and scale effect of anchor foundations in sand
Ocean Engineering xxx (xxxx) xxx
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Progressive failure and scale effect of anchor foundations in sand A.S.M. Riyad a, *, Md. Rokonuzzaman a, Toshinori Sakai b a b
Department of Civil Engineering, Khulna University of Engineering and Technology, Khulna, Bangladesh Department of Environmental Science and Technology, Graduate School of Bioresources, Mie University, Mie, Japan
A R T I C L E I N F O
A B S T R A C T
Keywords: Circular anchor foundations Dynamic relaxation Progressive failure Scale effect Shear band
Progressive failure is considered one of the least understood classical problems in geomechanics, gaining considerable attention in foundation related problems. However, the transformation of model test results to prototype is so much complex (scale effect). This paper highlights a FE ideology for evaluation of the response of shallow circular anchor foundations in sands with sophisticated strain hardening-softening behaviour. In these soils, the progressive failure can occur because of the non-mobilization of pinnacle strength of soil on potential failure surface and softening behaviour of soils. The presented paradigm allows this failure phenomenon to be properly evaluated by using a non-associated elastoplastic constitutive model coupled with explicit dynamic relaxation method considering yield surface as the MC material model and the potential surface is the smooth DP model having shear band effect. The FE results are compared with experimental outcomes to assess the reli ability. Numerical model well-predicted the experimental stress-strain data points from element tests very closely. The scale effect is deliberated due to the progressive failure with shear banding phenomenon, which is remarkable with the increase of embedment and predominant in the sand of higher density. The peak resistance factor and settlement are presenting a decreasing trend with the increase of foundation width.
1. Introduction Anchor problems are related to different types of geotechnical and civil engineering constructions, such as retaining walls, television and transmission towers, offshore platforms, submerged pipelines, and so on with different embedment, sizes, and shapes considered worldwide by enthusiastic engineers showing creativity in anchor usage technique, their behaviour, and design (Ilamparuthi and Muthukrishnaiah, 1999; Kim et al., 2015; Riyad and Rokonuzzaman, 2018; Sahoo and Ganesh, 2018). Numerous researchers (Majer, 1955; Mors, 1959; Balla, 1961; Baker and Konder, 1966; Matsuo, 1967; Meyerhof and Adams, 1968; Vesic, 1969; Hanna et al., 1972; Clemence and Veesaert, 1977; Davie and Sutherland, 1977; Rowe and Booker, 1979; Ovesen, 1981; Rowe and Davis, 1982; Vermeer and Sutjiadi, 1985; Murray and Geddes, 1987; Dickin, 1988; Merifield and Sloan, 2006; Deshmukh et al., 2010) have performed several studies on various types of anchors and proposed several design equations. Most of the equations are based on the limit state concepts, followed by rigid elastic-perfectly plastic hypothesis, which cannot correctly predict the real load displacement behaviour, because, the failure in real soil is highly progressive. The progressive failure phenomenon is supposed to elucidate the propagation of shear
stress and deformations on shear zone exhibiting the stress-strain behaviour of soil, together with post-peak strain-softening, dislocates the soil mass, and therefore, active failure and lack of support from the remoulded shear zone (Terzaghi and Peck, 1948; Skempton, 1964; Bishop, 1967, 1971; Bjerrum, 1967; Christian and Whitman, 1969; Urciuoli et al., 2007; Bernander, 2008, 2011). Nevertheless, traditional studies are basically focused on the principle of associated flow (AF) rule, whereas peak dilation angle (ψp) is almost equal to the peak fric tional angle (φp). It is also assumed that the influence of the dilation angle (ψ) can be uniquely captured by the frictional angle (φ). However, several researchers (Davis, 1968; Drescher and Detournay, 1993; Lou kidis et al., 2008; Sloan, 2013) stated that the AF rules over predict the foundation capacity of soils. It is also known that practically soils fric tional angle is significantly greater than the dilation angle (Schanz and Vermeer, 1996), which proved that the analytical solutions are invalid and leads to a non-associated flow rule. In 1985, non-associated elas to-perfectly plastic finite element model (FEM) was used by Vermeer and Sutjiadi to authenticate their introduced design equations (proposed by Borst and Vermeer, 1984). Moreover, the importance of the non-associated strain hardening-softening and mobilized dilatancy was presented by Walters and Thomas (1982) for proper modelling of the
* Corresponding author. E-mail addresses: [email protected] (A.S.M. Riyad), [email protected] (Md. Rokonuzzaman), [email protected] (T. Sakai). https://doi.org/10.1016/j.oceaneng.2019.106496 Received 24 April 2019; Received in revised form 24 September 2019; Accepted 25 September 2019 0029-8018/© 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: A.S.M. Riyad, Ocean Engineering, https://doi.org/10.1016/j.oceaneng.2019.106496
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Fig. 4. FE Mesh for Circular Anchor (H/D ¼ 2, D ¼ 10 cm and number of elements ¼ 2400).
Fig. 1. Testing apparatus.
Fig. 2. Test conditions; (a) Dense bed; (b) Medium bed; (c) Loose bed.
Fig. 5. FE mesh used for the analysis: (a) 2D FEM (elements ¼ 56); (b) 3D FEM (elements ¼ 1470).
behaviour of the anchor foundations, as none model can predict real soil behaviour accurately. In addition to that, many previous model tests were performed in different scales of the anchor (Ovesen, 1981; Hutchinson, 1982; Dickin, 1988; Tanaka and Sakai, 1993) revealing the importance of scale effect, which is linked to the degree of different progressive failure as the anchor scale changes. The scale effect is pictured because of the distinction between the strength of small size specimen within the laboratory which of a bigger specimen in real size (Omine et al., 2005). It has been stated that the progressive failure is linked to bifurcation or shear banding phenomena (Papanastasiou and Vardoulakis, 1989). Furthermore, the importance of the consideration of grain size effect has been detected by numerous researchers (Tatsuoka et al., 1997; Sakai et al., 1998; Cerato and Lutenegger, 2007; Athani et al., 2017) during the study of scale effect. If the grain size effect is ignored, it might presumably considerably overestimate the pull-out capacity of the anchor in full-scale (Athani et al., 2017). So, to use the
Fig. 3. Formation of shear band after peak: all plastic strain localized in shear bands.
proliferation of shear zone above trapdoors. The shear band can be defined as a thin material layer that’s delimited by the surface of two material discontinuities of the velocity gradient (Hill, 1962). Presently evolved, almost all the prevailing sophisticated models are very complex and incomplete in the sense that they do not define several important factors like inherent and induced anisotropies, strain localization, strain softening, and so on. Above important factors are overlooked in most of the current geotechnical models, which has a strong link between the FE model and a reliable set of experimental data. According to the author’s knowledge, no researchers have given attention to predict the settlement 2
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Fig. 6. Parametric Study by 2D finite element analysis in dense sand: (a) Deviator stress; (b) Volumetric strain.
framework of elastoplasticity in FEM to deal with the scale effect, an in-depth study is required in relation to calibration for the sophisticated hardening-softening constitutive models and verification as well as validation of rigorous FE models for some important problems. There fore, this research is designed to simulate the ultimate uplift capacity of a shallow circular anchor in different types of sand, using the widely used numerical technique, FEM, and a constitutive relation of sand, developed in such a way that it defines explicitly all the affecting factors in a relatively simple but sophisticated way. This research also pays special attention to the “scale effect” in the traditional small-scale models in relation to the proper modelling of progressive failure in the sand and tries to identify the different components of above-mentioned effects.
pulling out speeds of 0.03 mm/min and a displacement transducer is equipped on the top of the anchor bar to measure the upward displacement. The pull-out mechanism of anchors in the sand mass is observed by keeping contact in the side of a glass wall. A computerized data acquisition system is used to record the data. The movement of the anchor is controlled until the residual conditions are obtained with the help of a direct current (DC) motor. Air-dried Toyoura sand having specific gravity (Gs) ¼ 2.64, average particle size (D50) ¼ 0.016 cm, maximum void ratio (emax) ¼ 0.98, minimum void ratio (emin) ¼ 0.61, and no fine content less than 0.00075 cm, is used in this test. For the construction of the ground above the anchor foundation, air-pluviation method is used with the help of two sieves in the mould. The test is conducted on Toyoura sand of different densities (obtained by differing sieve opening size and sand falling height). However, the maintenance of unvarying density is necessary for the formation of the depth of the sand mass. Depending on the sand falling height and the deposition intensity during the air-pluviation process, various uniform densities can be obtained. To estimate the average unit weight in the homogenous sand mass, the sand is weighed after the anchor pull-out test. Owing to the difference of sieve opening size and sand falling height, the density of sand varies from dense to loose (relative density, Dr ¼ 95%–5%; and the density, γd ¼ 1630 kg/m3 to 1350 kg/m3, allowing an error of �10 kg/m3). The density of the ground is measured by using 6-hundred cc cylinder-shaped samplers (D ¼ 50 mm), which are placed on the soil
2. Physical modelling The experimental setup is presented in Fig. 1. A 590 mm diameter cylindrical container is used as a soil bin so that it does not give any boundary effects. Flat and circular steel anchor foundation with three different diameters (i.e., 5, 10, and 15 cm) with a 5 mm thick steel plate is used to conduct the anchor pull-out test. This test is directed at an embedment ratio (H/D, where H ¼ depth of sand mass and D ¼ diameter of the anchor) of 2 in a homogeneous bed as illustrated in Fig. 2. A load cell is linked with an anchor bar to measure the vertical pull-out load at 3
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Fig. 7. Parametric Study by 2D finite element analysis in medium sand: (a) Deviator stress; (b) Volumetric strain.
the bifurcation problems so that any specific measures need not be taken; easy to be coded and very small data storage are needed owing to its explicit nature; use of approximate critical damping for linear iso parametric elements with reduced integration can implicitly suppress the hourglass modes at failure. This method is incorporated in the personally developed FORTRAN program in conjunction with the generalized return mapping algorithm by considering an elasto-plastic framework incorporating isotropic hardening-softening law (Tanaka and Sakai, 1993) following the non-associated flow rule including shear band (Sakai and Tanaka, 2007). A shear band (SB) is a narrow band where all the plastic strains are localized after peak as illustrated in Fig. 3. The inclusion of a shear band is essential to make the constitutive model and its calibration independent on the size of FE. As the pre vailing solutions of strain-softening material are firmly reliant on the size of mesh, several approaches such as Pietruszczak and Mroz (1981) suggested an uncomplicated form of mesh-size reliant on hardening modulus technique for resolution of the mesh-dependent pathology of FE solutions. Following this technique, shear banding can be incorpo rated through a strain localization parameter, S, according to one assumption of elastoplasticity theory. The assumption is that total strain increments (dε) can be decomposed additively into elastic (dεe ) and plastic (dεp ) parts, as follows (Tanaka and Sakai, 1993; Sakai and
Table 1 Parameters for hardening and softening material models. Parameters
Dense
Medium
Loose
Density (kg/m3) Void ratio, e Relative density, Dr Coefficient of shear modulus, G0 Residual frictional angle, φr (degrees) Poisson’s ratio
1630 0.62 0.95 500 33 0.3 0.1 0.1 0.3 0.1 0.1
1480 0.78 0.53 500 33 0.3 0.1 0.4 0.3 0.2 0.1
1350 0.95 0.05 500 33 0.3 0.1 0.8 0.3 0.4 0.1
εf εr εd m β
bin’s bottom. 3. Numerical modelling In numerical modelling an explicit dynamic relaxation method (Tanaka and Kawamoto, 1987; Sakai and Tanaka, 2007) is used due to its robust ability to handle extremely nonlinear material; to deal with 4
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Fig. 9. Effect of shear band in dense sand using strain hardening-softening model: (a) Deviator stress; (b) Volumetric Strain.
Fig. 8. Comparison of experimental result with simulated stress-strain-volume with shear band by 2D and 3D FEM using sophisticated strain hardeningsoftening law: (a) Dense sand; (b) Medium sand.
deviatoric stress) and gðθÞ is the function of Lode angle, θ. In case of MC material model, it can be defined as:
Tanaka, 2007): dε ¼ dεe þ Sdεp
(1)
3 sinφmob gðθÞ ¼ pffiffi 2 3cos θ 2 sin θ sinφmob
where, S is the ratio of the area of a single shear band in each element (Fb) to the total area of the finite element (Fe). However, the decision of Fb is taken based on the SB thickness. Moreover, the effects of the SB orientation in an apiece element is ignored, and thus the approximate form of SB considered in this study can be defined as SB S ¼ pffiffiffiffiffi Fe
(4)
where 1 θ ¼ cos 3
(2)
1
" pffiffi # 3 3 J3D 2 J 3=2 2D
(5)
and J3D is the third deviatoric stress invariants. The plastic strain rates can be calculated from non-associated flow rules, expressed as,
The SB thickness for sands is about 10–30 times of mean grain size diameter of a particle (Mühlhaus and Vardoulakis, 1987; Yoshida et al., 1993). MC yield criterion is used as its material parameters are well known in soil mechanics model tests, and Drucker–Prager (DP) potential function is used due to its smoothness and easy differentiability. The yield function can be introduced as: pffiffiffiffiffiffi J2D F ¼ αðκÞI1 þ κ1 ¼ 0 (3) gðθÞ
dεp ¼ dλ
dG dσ
(6)
where, G and dλ are the plastic potential and a positive constant known as plastic rate multiplier, respectively. The plastic potential function takes a similar form as the yield function Equation (3) with gðθÞ ¼ 1:0 in Equation (6), as pffiffiffiffiffiffi G ¼ α’ðκÞI1 þ J2D κ2 ¼ 0 (7) The internal variable, κ, can be found by considering the incremental deviatoric plastic strains, expressed as:
where, I1 is the first invariant of stress (i.e., hydrostatic stress compo nent, positive in compression), J2D is the second stress invariant (i.e., the 5
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For plastic potential functions, the constant, α’ can be expressed as, 2 sin ψ 3ð3 sin ψ Þ
(13)
α’ ¼ pffiffi
From the modified Rowe’s stress dilatancy relationship, the dilat ancy angle ψ can be expressed as, � � sinφmob sin φ’r ψ ¼ sin 1 (14) ’ 1 sinφmob sin φr where, sinφmob ¼
pffiffi 3 3αðκÞ pffiffi 2 þ 3αðκÞ
8 > > < ’ φr ðκÞ ¼ φr 1 > > :
βe
(15)
� �2 9 > > κ = εd > > ;
(16)
where, β and εd are the stress dilatancy material parameters controlling the rate of dilatancy. So, therefore, these 5 numbers of indeterminate constitutive pa rameters (m, εf , εr , β, and εd ) have to be optimized in the calibration process by back prediction of the anchor pull-out test. The problems of circular anchor are analysed in this study, using the four-node quadrilateral isoparametric finite elements (Fig. 4). To over come the boundary effects, the mesh extends at a distance of a minimum of 4D from the anchor edges. Moreover, finer discretization is used in the centre with the quadrilateral elements than edges, where limited deformation is expected. Furthermore, along the anchor boundary, dif ferential quasi-elastic displacements are introduced until failure at minor successive increments and the corresponding nodal forces are averaged with respect to the displacement control nodes to ascertain the ultimate load. The normalized displacement factor, δ/D (where, δ presents the displacement of the ground), and the pull-out resistance factor, Np (¼P/ γdHA, γd is the effective unit weight of dry sand and A is the surface area of the anchor) are obtained by considering the proposals of Lindsay (1980), whereas, δ is the displacement, and P is the uplift load. 4. Calibration of hardening and softening material models
Fig. 10. Effect of shear band in medium sand using strain hardening-softening model: (a) Deviator stress; (b) Volumetric Strain.
Z � � �2 � κ ¼ 2 depii þ 1
�� � �12 2 δjk depjk
The accurate modelling of the stress-strain relationship obtained from the experiment depends on the accurate constitutive models which is so much challenging task for the practical engineers. In this context, the calibration of constitutive models of linear elastic material is very simple. On the other hand, most of the engineering materials found in nature, such as soils, display the nonlinear hardening-softening behav iour, which can be modelled by sophisticated strain hardening-softening material model having an elasto-plastic framework with a nonassociated flow rule, costing of numerous parameters. And so, calibra tion is necessary to fit those parameter values by laboratory test results, before used in the numerical analysis. Moreover, if the model is highly nonlinear, the mathematical complexities amalgamated with the accu racy and sophistication of the model can be overcome by costing more parameters (Rokonuzzaman and Sakai, 2010). For this reason, extensive numerical tests on the constitutive relations of Toyoura sand of different relative densities and the effects of different parameters of the consti tutive relations on the results are examined. Further, a parametric study is done to find out the appropriate material parameter values to use in the sophisticated strain hardening-softening model. A series of drained triaxial compression tests are performed on cy lindrical saturated Toyoura sand (Gs ¼ 2.64, emax ¼ 0.977, emin ¼ 0.605, Uc ¼ 1.46, no fines) samples, prepared by air-pluviation method, of diameter and height is 70 mm and 150 mm, respectively, for different densities and confining pressures (Fukushima and Tatsuoka, 1984). FE
(8)
Due to the use of dry sand in simulation and test in this study, the cohesion can be ignored, and so the function κ can be ignored in Equations (3) and (6). The frictional function α(κ) can be expressed as, � pffiffiffiffiffiffi �m � 2 κεf hardening regime : κ � εf (9) αðκÞ ¼ αp κ þ εf �
αðκÞ ¼ αr þ αp
�
αr e
κ εf εr
�2 softening regime : κ > εf
�
(10)
where, m, εf and εr are the constants, which considers the hardeningsoftening material condition, and 2 sinφp
�
(11)
2 sinφr 3ð3 sinφr Þ
(12)
αp ¼ pffiffi
3 3
αr ¼ pffiffi
sinφp
where, φp is the peak and φr is the residual frictional angle, respectively. 6
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Fig. 11. Validation and Effect of Constitutive Models in sand of different densities: (a) Dense Sand; (b) Medium Sand; (c) Loose Sand.
Fig. 12. Mesh-size effect (H/D ¼ 2, D ¼ 10 cm): (a) fine mesh (number of elements ¼ 2400); (b) coarse mesh (number of elements ¼ 1536); (c) very coarse mesh (number of elements ¼ 600).
meshes used for the analysis are presented in Fig. 5. The results of the parametric studies on the hardening-softening material parameters are presented in Figs. 6 and 7. By parametric study, it is detected that when the hardening-softening material parameters: εf, εr, and m are 0.1; the calculated result coincides with the experimental result in dense sand. On the other hand, when the hardening-softening material parameters: εf, εr, and m are 0.1, 0.4, and 0.2, respectively, the calculated outcome is agreed with the experimental outcome in medium sand. The parameters for loose sand are adopted from Sakai and Tanaka (2007). After the
parametric study, the selected parameters are used for the analysis is presented in Table 1. Here, 0.3 cm wide shear band is adopted, which equals to twenty times of mean particle diameter (D50) of the Toyoura sand (Yoshida et al., 1993). Fig. 8 presents the effect of two-dimensional (2D) and threedimensional (3D) FEM model on stress-strain-volume change relation ship. Small difference is observed on stress-strain-volume change rela tionship using the 2D and the 3D FEM models due to the difference in their kinematic boundary conditions. The 2D model constraints are 7
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Fig. 13. Uplift resistance-displacement factor relationship by different mesh-size: (a) Dense sand with shear band; (b) Medium sand with shear band; (c) Loose sand with shear band; (d) Dense sand without shear band; (e) Medium sand without shear band; (f) Loose sand without shear band.
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Fig. 14. Mesh dependency in shear strain and deformation pattern (δ/D ¼ 0.024; H/D ¼ 2, D ¼ 10 cm).
outcomes with respect to the other models with the experiment which is validated for the further study.
greater than the 3D models. As the 3D numerical simulation needs a larger amount of elements, and therefore a larger amount of integration points, nodes, time and resources are needed in comparison with the 2D model. However, there is no effect of aspect ratio in the circular anchor as in the rectangular anchor. Volumetric strain is more in 3D model. The differences between the results from those models are negligible and so 2D model is used for the analysis. Figs. 9 and 10 presents the stress-strain-volume change relationship achieved from the research recommended by Fukushima and Tatsuoka (1984) and the corresponding FE analysis. The effect of the shear band on the material parameters are also analysed in this study. For the analysis, a single element and multi-element of 56 is used. The experi mental result indicates close agreement with the analysis considering maximum deviator stress. Without the shear band, the softening part is affected due to the element size, especially in dense sand. And so, due to the mesh size effect after the peak, the shear band is introduced in this study.
6. Effect of mesh size, calculating step size and shear band Also, the effect of mesh size and calculating step size, constants of the constitutive models and shear band option on the anchor foundation are studied. Due to the sensitiveness of mesh size in the FE solutions, the numerical model integrating hardening-softening rule must be authen ticated before used for the analysis of the anchor foundation. For the analysis of the anchor foundation (H/D ¼ 2, D ¼ 10 cm), 2400 quadri lateral finite elements 5 mm in size, 1536 quadrilateral finite elements 6.25 mm in size and 600 quadrilateral finite elements 10 mm in size in the central zone are used as presented in Fig. 12. The curves in Fig. 13 point out the relationship between the displacement factor and the pullout resistance factor. Due to having the shear band thickness in the constitutive model, the load-displacement behaviour of anchor foun dation is independent on the mesh size. Thus, re-meshing is avoided by scaling up a smaller proportional model in the parametric study. In the pre-peak regime, deformation of a given sand element under a uniform boundary stress condition is analogous. Strain localization starts sud denly at the peak stress state. It is assumed that shear bands start to develop at the peak. For the post-peak parts, strains are the ones defined locally within a band including a shear band. It is assumed that the rate of strain softening is independent of bedding plane orientation angle and φmob becomes a constant value, φr at the same relative displacement across a shear band. The stress-strain relationship inside a shear band is independent of the size, shape and boundary conditions of a given element containing the shear band. This relation and shear band thickness is unique for a given mass of sand, and independent of the density and pressure level. In the absence of a shear band, the effect of mesh size is predominant after peak especially for softening dense sand
5. Validation and Effect of Constitutive Models Besides, the effects of different constitutive models on the anchor foundation are examined. The curves shown in Fig. 11, depict the experimental and numerical relationships between the pullout resis tance and the displacement factor of the circular anchor foundation (H/ D ¼ 2 and D ¼ 10 cm) at different density condition by considering different constitutive models. From all the curves, three distinct phases are evident: a sharp increase of pullout resistance with anchor displacement in the initial phase, followed by a shallow decrease, and, finally, in the residual phase, the pullout resistance remains unaffected with the further uplifting of the anchor. From the overall shape of the load-displacement curve, it can be assumed that progressive failure is attributed and strain hardening softening model is presented better 9
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Fig. 15. Influence of calculating step in different meshes in loose sand: (a) Number of elements ¼ 2400; (b) Number of elements ¼ 1536; (c) Number of elements ¼ 600.
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Fig. 16. Influence of calculating step in different meshes in medium sand: (a) Number of elements ¼ 2400; (b) Number of elements ¼ 1536; (c) Number of elements ¼ 600.
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Fig. 17. Influence of calculating step in different meshes in dense sand: (a) Number of elements ¼ 2400; (b) Number of elements ¼ 1536; (c) Number of elements ¼ 600.
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Fig. 18. Mesh dependency in different calculating steps in loose sand: (a) Step ¼ 1000; (b) Step ¼ 2000; (c) Step ¼ 5000; (d) Step ¼ 15000; (e) Step ¼ 20000.
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Fig. 19. Mesh dependency in different calculating steps in medium sand: (a) Step ¼ 1000; (b) Step ¼ 2000; (c) Step ¼ 5000; (d) Step ¼ 15000; (e) Step ¼ 20000.
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Fig. 20. Mesh dependency in different calculating steps in dense sand: (a) Step ¼ 1000; (b) Step ¼ 2000; (c) Step ¼ 5000; (d) Step ¼ 15000; (e) Step ¼ 20000.
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Fig. 21. Pullout resistance–displacement factor curves using sophisticated strain hardening-softening model: (a) Dense Sand; (b) Medium Sand; (c) Loose Sand.
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7. Effect of material parameters It is assumed that the progression of localized deformation is ob tained from the triaxial compression test with the pull-out test. The hardening-softening material parameters values (εf, εr, and m) are evaluated on the back prediction of the anchor pullout test by comparing the experimental and the finite-element analysis results. Calculated and experimental results for D ¼ 10 cm at H/D ¼ 2 are presented in Fig. 21, which presents the pullout resistance and displacement factor curves. After the parametric study, the same values are obtained as observed in calibration results using the sand of different densities. Table 1 already summarized the parameters used in the finite-element analysis. 8. Study of progressive failure According to Terzaghi and Peck (1948), it can be expressed as “The term progressive failure indicates the spreading of the failure over the potential surface of sliding from a point or line towards the boundaries of the surface. While the stresses in the clay near the periphery of this surface approach the peak value, the shearing resistance of the clay at the area where the failure started is already approaching the much smaller ultimate’ value. As a consequence, the total shearing force that acts on a surface of sliding at the instant of complete failure is consid erably smaller than the shearing resistance computed on the basis of the peak values.” Thus, progressive failure can be referred to as the state of affairs which indicates the consecutive failure of individual elements of soil in a soil mass. This method spreads in an area and needs time to occur, whereas time aspects might not continuously be necessary beneath some circumstances, an understanding of spatial progressive failure is most vital (Lo, 1972). Lo also stated that the prime necessary condition for the occurrence of progressive failure is that the material should follow the stress-strain softening relationship. A study of Taylor (1937) on shearing resistance on sands using effective stress concept stated the associated progression of failure with the redistribution of shear stress on a potential failure surface and neglected the propagation of progression of failure direction. But, the importance of the determi nation of propagation of progressive failure is so much crucial for esti mating the available shearing resistance along the sliding surface. The failure in the ground when loaded with an anchor is highly progressive in a view that the soil’s peak strength is not mobilized along potential failure surfaces; while the perfectly-plastic assumption leads to that the peak soil strength is mobilized concurrently along the potential failure surface. Therefore, the soil shear strength is not concurrently mobilized at all points of a slip surface. It is also supposed that the strength properties of soil are unaltered even if large strains are introduced by loading. However, this hypothesis is insignificant for the soils catego rized by noticeable strain-softening behaviour, like dense sands. Actu ally, in these materials, it happens that a few segments of the sand initially fail due to loading, with the shear strain that is situated in a zone of restricted thickness (shear band). With the increase of strain inside this zone, soil quality decreases from the peak towards the critical state. Because of the consequential stress redistribution, the shear band spreads in the sand and a slip surface dynamically creates up to causing the collapse of the soil-foundation framework. In the case of the failure condition, the average strength mobilized along the slip surface is not as much as the pinnacle quality and more prominent than the quality at the critical condition of the sand under consideration. For the documenta tion of this failure phenomenon, the amassed plastic deformation zones above the anchor calculated for any loading stage (D ¼ 10 cm at H/D ¼ 2) at different densities of sand are illustrated in Fig. 22 for the same δ/D ratio (δ is larger for larger D). As the shear bandwidth is in dependent of pressure-level, its ratio with D decreases proportionally as D increases. Then, once shear bands have appeared, the relative displacement across a shear band at corresponding points becomes larger for larger D, resulting in a larger ratio of shear band length relative to D and more non-uniform distribution of φmob, thus a larger
Fig. 22. Accumulated plastic deformation zones above the anchor (figures are not to scale): (a) Loading stage considered in analysis; (b) Average shear strain distribution calculated at any loading stage with respect to denseness of sand.
and so the result is influenced. And so, a shear band is used in the further analysis of anchor problem. Fig. 14 shows the mesh size dependency in shear strain and the deformation pattern of the specimen at different density conditions. Though the overall distributions of the shear strain are almost similar in all meshes, it is much sharper in finer mesh than in the coarser mesh. On the other hand, there is not much difference in the deformation pattern. Figs. 15–17 shows the influence of the calculating steps in different meshes. From all the curves, it can be recapitulated that to obtain a stable result, more than 5000 calculating steps are needed. After that, the relationship between the pull-out resistance and the displacement factor will be the same in all the calculating steps. Figs. 18–20 shows the mesh size dependency in different calculating steps. It is observed that more than the calculating steps 5000, the peak strength is not affected at all in different meshes while the residual strength changed in such a tendency that the finer the mesh is, less the residual strength will be. When the number of elements is 1536 and 2400, the difference between the residual strength is almost zero. And so, for further study, calculating step 15000 is selected. 17
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Fig. 23. Scale effect: relationship between experimental and numerical results at H/D ¼ 2: (a) Dense Sand; (b) Medium Sand.
Fig. 24. Scale effect: uplift resistance-displacement factor relationships at H/D ¼ 1: (a) Dense Sand; (b) Medium Sand; (c) Loose Sand.
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Fig. 25. Scale effect: uplift resistance-displacement factor relationships at H/D ¼ 2: (a) Dense Sand; (b) Medium Sand; (c) Loose Sand.
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Fig. 26. Scale effect: uplift resistance-displacement factor relationships at H/D ¼ 3: (a) Dense Sand; (b) Medium Sand; (c) Loose Sand.
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Fig. 27. Scale effect: uplift resistance-displacement factor relationships at H/D ¼ 4: (a) Dense Sand; (b) Medium Sand; (c) Loose Sand.
degree of progressive failure. In Fig. 22a, some loading stages (i.e., A–C) are presented to support the demonstration of the outcomes. Fig. 22b shows that at Stage C, the slip surface is inclined in case of dense sand and vertical in case of loose sand. It is also evident that in dense sand failure is not fully happened and on the other hand, for loose sand, the failure is outcropped on the surface. And so, it can be said that the failure of dense sand is more progressive than the others, as less unloading is required to initiate the failure that spreads over a larger space within the soil mass.
may be attributed to the combined effect of particle size and stress level according to Tatsuoka et al. (1991) and increases with the increase of the embedment. This trend of scale effect with varying embedment is attributed to the various degree of progressive failure with shear band development. Due to high confining pressure and particle size effect or more progressive nature of the failure of the material, the scale effect is predominant in dense sand. On the other hand, less softening in the material behaviour so less scale effect in medium and/or loose sand. In 1 g test result, the scale effect consists of the effect of the pressure level and particle size. The latter effect is observed in centrifuge tests with varying the g level using the same anchor size and the same type of sand. As a small-scale physical model has high dimensionless power, care must be taken while extrapolating the laboratory model test results (small- scale) to the full-scale structures.
9. Width (scale) effect The problems of the transformation of the model test results to a prototype is mainly due to width (scale) effect. The scale effects are examined by varying H and D, proportionally (e.g. the FE mesh for H/ D ¼ 2, D ¼ 5 cm is scaled up by a factor of 10 to have anchor with H/ D ¼ 2, D ¼ 50 cm, and this procedure is supported by such kind of nu merical model, insensitive to mesh size effect). Fig. 23 presents the relationship between the experimental results and the numerical results at H/D ¼ 2 (D ¼ 5, 10 and 15 cm), which is presented that numerical results well predict the experimental results. Figs. 24–27 presents the uplift resistance-displacement factor relationships at different embed ment. Figs. 28 and 29 display that Npu and δu/D decreases with D, which
10. Conclusions Progressive failure behaviour mainly occurs due to the nonmobilization of peak strength of soil in potential failure regions, whereas, scale effect is predominant in softening soil. It is observed that, based on calibration test results, the numerical models can well-predict the experimental stress-strain data points from element tests very closely using the appropriate set of material constants of the constitutive models 21
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Fig. 28. Scale effect: peak resistance factor as a function of D and H/D: (a) H/D ¼ 1; (b) H/D ¼ 2; (c) H/D ¼ 3; (d) H/D ¼ 4.
Fig. 29. Scale effect: peak displacement factor as a function of D and H/D: (a) H/D ¼ 1; (b) H/D ¼ 2; (c) H/D ¼ 3; (d) H/D ¼ 4.
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and yield surface as the MC model and potential surface as the smooth DP model with shear band effect using the explicit dynamic relaxation method framework. The sophisticated hardening-softening material model shows better performance as it is capable to closely predict the pull-out resistance and settlement patterns costing five numbers of material parameters. Due to the consideration of the shear band thick ness in the constitutive model, the load-displacement behaviour of an chor foundation is independent on the mesh size. The peak resistance factor presents a decreasing tendency with the increase of the width of the foundation, which is scale effect. The failure pattern of dense sand is more progressive than loose sand, and so, scale effect is predominant in dense sand. The settlement at the ultimate bearing capacity of the an chor foundation always increases with the decrease of width.
Ilamparuthi, K., Muthukrishnaiah, K., 1999. Anchors in sand bed: delineation of rupture surface. Ocean. Eng. 26 (12), 1249–1273. https://doi.org/10.1016/S0029-8018(98) 00034-1. Kim, Y.H., Hossain, M.S., Wang, D., 2015. Effect of strain rate and strain softening on embedment depth of a torpedo anchor in clay. Ocean. Eng. 108, 704–715. https://do i.org/10.1016/j.oceaneng.2015.07.067. Lindsay, J.A., 1980. The use of physical models in design. Des. Parameters Geotech. Eng. 4, 324–326. Lo, K.Y., 1972. An approach to the problem of progressive failure. Can. Geotech. J. 9 (4), 407–429. https://doi.org/10.1139/t72-042. Loukidis, D., Chakraborty, T., Salgado, R., 2008. Bearing capacity of strip footings on purely frictional soil under eccentric and inclined loads. Can. Geotech. J. 45 (6), 768–787. https://doi.org/10.1139/T08-015. Majer, J., 1955. Zur berechnung von zugfundamenten [To calculate train foundations]. Osterreichische Bauzeitschrift 10 (5), 85–90 (in German). Matsuo, M., 1967. Study on the uplift resistance of footing (I). Soils Found. 7 (4), 1–37. https://doi.org/10.3208/sandf1960.7.4_1. Merifield, R.S., Sloan, S.W., 2006. The ultimate pullout capacity of anchors in frictional soils. Can. Geotech. J. 43 (8), 852–868. https://doi.org/10.1139/t06-052. Meyerhof, G.G., Adams, J.I., 1968. The ultimate uplift capacity of foundations. Can. Geotech. J. 5 (4), 225–244. https://doi.org/10.1139/t68-024. Mors, H., 1959. The behaviour of mast foundations subjected to tensile forces. Bautechnik 36 (10), 367–378. Mühlhaus, H.B., Vardoulakis, I., 1987. The thickness of shear bands in granular materials. Geotechnique 37 (3), 271–283. https://doi.org/10.1680/geot.1987.37 .3.271. Murray, E.J., Geddes, J.D., 1987. Uplift of anchor plates in sand. J. Geotech. Eng. 113 (3), 202–215. https://doi.org/10.1061/(ASCE)0733-9410(1987)113:3(202). Omine, K., Ochiai, H., Yasufuku, N., 2005. Evaluation of scale effect on strength of cement-treated soils based on a probabilistic failure model. Soils Found. 45 (3), 125–134. https://doi.org/10.3208/sandf.45.3_125. Ovesen, N., 1981. Centrifuge tests of the uplift capacity of anchors. In: Proc., 10th Conf. Soil Mech. Found. Eng., Balkema, Netherlands, pp. 717–722. Papanastasiou, P.C., Vardoulakis, I.G., 1989. Bifurcation analysis of deep boreholes: II. Scale effect. Int. J. Numer. Anal. Methods Geomech. 13 (2), 183–198. https://doi. org/10.1002/nag.1610130206. Pietruszczak, St, Mroz, Z., 1981. Finite element analysis of deformation of strainsoftening materials. Int. J. Numer. Methods Eng. 17 (3), 327–334. https://doi.org/1 0.1002/nme.1620170303. Riyad, A.S.M., Rokonuzzaman, M., 2018. Numerical modeling of anchor foundation in dense sand. In: Proc., 4th Int. Conf. Civil Engineering for Sustainable Development, Khulna, Bangladesh, vol. 5088, pp. 1–9. Rokonuzzaman, M., Sakai, T., 2010. Calibration of the parameters for a hardeningsoftening constitutive model using genetic algorithms. Comput. Geotech. 37 (4), 573–579. https://doi.org/10.1016/j.compgeo.2010.02.007. Rowe, R.K., Booker, J.R., 1979. A method of analysis for horizontally embedded anchors in an elastic soil. Int. J. Numer. Anal. Methods Geomech. 3 (2), 187–203. https://doi. org/10.1002/nag.1610030207. Rowe, R.K., Davis, E.H., 1982. The behaviour of anchor plates in sand. Geotechnique 32 (1), 25–41. https://doi.org/10.1680/geot.1982.32.1.25. Sakai, T., Erizal, V., Tanaka, T., 1998. Particle size effect of anchor problem with granular materials. Appl. Numer. Methods Geotech. Probl. 397, 191–200. https ://doi.org/10.1007/978-3-7091-2512-0_18. Sakai, T., Tanaka, T., 2007. Experimental and numerical study of uplift behaviour of shallow circular anchor in two-layered sand. J. Geotech. Geoenviron. Eng. 133 (4), 469–477. https://doi.org/10.1061/(ASCE)1090-0241(2007)133:4(469). Sahoo, J.P., Ganesh, R., 2018. Vertical uplift resistance of rectangular plate anchors in two layered sand. Ocean. Eng. 150, 167–175. https://doi.org/10.1016/j. oceaneng.2017.12.056. Schanz, T., Vermeer, P.A., 1996. Angles of friction and dilatancy of sand. Geotechnique 46 (1), 145–151. https://doi.org/10.1680/geot.1996.46.1.145. Skempton, A.W., 1964. Long-term stability of clay slopes. Geotechnique 14 (2), 77–102. https://doi.org/10.1680/geot.1964.14.2.77. Sloan, S.W., 2013. Geotechnical stability analysis. Geotechnique 63 (7), 531–571. https://doi.org/10.1680/geot.12.RL.001. Tanaka, T., Kawamoto, O., 1987. Three-dimensional finite element analysis for collapse loads of foundations and slopes using dynamic relaxation. Trans. Jpn. Soc. Irrig. Drain. Reclam. Eng. 1987 (131), 79–86. https://doi.org/10.11408/jsidre1965.19 87.131_79. Tanaka, T., Sakai, T., 1993. Progressive failure and scale effect of trap-door problems with granular materials. Soils Found. 33 (1), 11–22. https://doi.org/10.3208/sandf1 972.33.11. Tatsuoka, F., Okahara, M., Tanaka, T., Tani, K., Moromoto, T., Siddiquee, M.S.A., 1991. Progressive failure and particle size effect in bearing capacity of footing on sand. In: Proc., Geotechnical Engineering Congress, New york, pp. 788–802. Tatsuoka, F., Goto, S., Tanaka, T., Tani, K., Kimura, Y., 1997. Particle size effects on bearing capacity of footing on granular material. In: Proc. Int. Symp. On
References Athani, S., Kharel, P., Airey, D., Rognon, P., 2017. Grain-size effect on uplift capacity of plate anchors in coarse granular soils. G�eotech. Lett. 7 (2), 167–173. https://doi. org/10.1680/jgele.17.00002. Baker, W.H., Konder, R.L., 1966. Pullout load capacity of a circular earth anchor buried in sand. Natl. Acad. Sci. Highw. Res. Rec. 108, 1–10. http://onlinepubs.trb.org/On linepubs/hrr/1966/108/108-001.pdf. Balla, A., 1961. The resistance to breaking-out of mushroom foundations for pylons. In: Proc., 5th. Int. Conf. Soil Mech. Found. Eng., Paris, France, pp. 569–576. Bernander, S., 2008. Down-hill Progressive Landslides in Soft Clays, Triggering Disturbance Agents, Slide Propagation over Horizontal or Gently Sloping Ground, Sensitivity Related to Geometry. Research Report. Department of Civil and Mining Engineering, Luleå University of Technology, Luleå, Sweden. Bernander, S., 2011. Progressive Landslides in Long Natural Slopes, Formation, Potential Extention and Configuration of Finished Slides in Strain-Softening Soils. PhD dissertation. Department of Civil and Mining Engineering, Luleå University of Technology, Luleå, Sweden. Bishop, A.W., 1967. Progressive failure - with special reference to the mechanism causing it. In: Proc., Geotech. Conf., Norwegian Geotechnical Institute, Oslo, Norway, vol. 2, pp. 142–150. Bishop, A.W., 1971. The influence of progressive failure on the choice of the method of stability analysis. Geotechnique 21 (2), 168–172. https://doi.org/10.1680/geot .1971.21.2.168. Bjerrum, L., 1967. Progressive failure in slopes in overconsolidated plastic clay and clay shales. Terzaghi Lecture J. Soil Mech. Found. Div. 93 (5), 3–49. Borst, R.D., Vermeer, P.A., 1984. Possibilities and limitations of finite elements for limit analysis. Geotechnique 34 (2), 199–210. https://doi.org/10.1680/geot.1984.34 .2.199. Cerato, A.B., Lutenegger, A.J., 2007. Scale effects of shallow foundation bearing capacity on granular material. J. Geotech. Geoenviron. Eng. 133 (10), 1192–1202. https://doi.org/10.1061/(ASCE)1090-0241(2007)133:10(1192). Clemence, S., Veesaert, C., 1977. Dynamic pullout resistance of anchors in sand. In: Proc., Int. Symp. On Soil-Structure Interaction, Roorkee, India, pp. 389–397. Christian, J.T., Whitman, R.V., 1969. A one-dimensional model for progressive failure. In: Proc., 7th Conf. Soil Mech. Found. Eng., Mexico City, Mexico, vol. 2, pp. 541–545. Davie, J.R., Sutherland, H.B., 1977. Uplift resistance of cohesive soils. J. Geotech. Eng. Div. 103 (9), 935–952. Davis, E.H., 1968. Theories of Plasticity and the Failure of Soil Masses. Soil mechanicsSelected topics. American Elsevier, Newyork, pp. 341–380. Deshmukh, V., Dewaikar, D., Choudhury, D., 2010. Analysis of rectangular and square anchors in cohesionless soil. Int. J. Geotech. Eng. 4 (1), 79–87. https://doi.org/ 10.3328/IJGE.2010.04.01.79-87. Dickin, E.A., 1988. Uplift behaviour of horizontal anchor plates in sand. J. Geotech. Eng. 114 (11), 1300–1317. https://doi.org/10.1061/(ASCE)0733-9410(1988)114:11 (1300). Drescher, A., Detournay, E., 1993. Limit load in translational failure mechanisms for associative and non-associative materials. Geotechnique 43 (3), 443–456. https://do i.org/10.1680/geot.1993.43.3.443. Fukushima, S., Tatsuoka, F., 1984. Strength and deformation characteristics of saturated sand at extremely low pressures. Soils Found. 24 (4), 30–48. https://doi.org/10.320 8/sandf1972.24.4_30. Hanna, T.H., Sparks, R., Yilmaz, M., 1972. Anchor behaviour in sand. J. Soil Mech. Found. Div. 98 (11), 1187–1208. Hill, R., 1962. Acceleration waves in solids. J. Mech. Phys. Solids 10 (1), 1–16. https://doi.org/10.1016/0022-5096(62)90024-8. Hutchinson, N.J., 1982. Groups of Plate Anchors in Sand. International Research Report CEI/1982. Department of Civil Engineering, University of Glasgow.
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Vesic, A.S., 1969. Breakout Resistance of Objects Embedded in Ocean Bottom. Report Number CR.69.031. U. S. Naval Civil Engineering Laboratory, Port Hueneme, California. Walters, J.V., Thomas, J.N., 1982. Shear zone development in granular materials. In: Proc., 4th Int. Conf. Numerical Methods in Geomechanics, Edmonton, Canada, pp. 263–274. Yoshida, T., Tatsuoka, F., Siddiquee, M.S.A., Kamegai, T., 1993. Shear banding in sands observed in plane strain compression. In: Proc., Localisation and Bifurcation Theory for Soils and Rocks, Balkema, pp. 165–179.
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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Progressive failure and scale effect of anchor foundations in sand A.S.M. Riyad a, *, Md. Rokonuzzaman a, Toshinori Sakai b a b
Department of Civil Engineering, Khulna University of Engineering and Technology, Khulna, Bangladesh Department of Environmental Science and Technology, Graduate School of Bioresources, Mie University, Mie, Japan
A R T I C L E I N F O
A B S T R A C T
Keywords: Circular anchor foundations Dynamic relaxation Progressive failure Scale effect Shear band
Progressive failure is considered one of the least understood classical problems in geomechanics, gaining considerable attention in foundation related problems. However, the transformation of model test results to prototype is so much complex (scale effect). This paper highlights a FE ideology for evaluation of the response of shallow circular anchor foundations in sands with sophisticated strain hardening-softening behaviour. In these soils, the progressive failure can occur because of the non-mobilization of pinnacle strength of soil on potential failure surface and softening behaviour of soils. The presented paradigm allows this failure phenomenon to be properly evaluated by using a non-associated elastoplastic constitutive model coupled with explicit dynamic relaxation method considering yield surface as the MC material model and the potential surface is the smooth DP model having shear band effect. The FE results are compared with experimental outcomes to assess the reli ability. Numerical model well-predicted the experimental stress-strain data points from element tests very closely. The scale effect is deliberated due to the progressive failure with shear banding phenomenon, which is remarkable with the increase of embedment and predominant in the sand of higher density. The peak resistance factor and settlement are presenting a decreasing trend with the increase of foundation width.
1. Introduction Anchor problems are related to different types of geotechnical and civil engineering constructions, such as retaining walls, television and transmission towers, offshore platforms, submerged pipelines, and so on with different embedment, sizes, and shapes considered worldwide by enthusiastic engineers showing creativity in anchor usage technique, their behaviour, and design (Ilamparuthi and Muthukrishnaiah, 1999; Kim et al., 2015; Riyad and Rokonuzzaman, 2018; Sahoo and Ganesh, 2018). Numerous researchers (Majer, 1955; Mors, 1959; Balla, 1961; Baker and Konder, 1966; Matsuo, 1967; Meyerhof and Adams, 1968; Vesic, 1969; Hanna et al., 1972; Clemence and Veesaert, 1977; Davie and Sutherland, 1977; Rowe and Booker, 1979; Ovesen, 1981; Rowe and Davis, 1982; Vermeer and Sutjiadi, 1985; Murray and Geddes, 1987; Dickin, 1988; Merifield and Sloan, 2006; Deshmukh et al., 2010) have performed several studies on various types of anchors and proposed several design equations. Most of the equations are based on the limit state concepts, followed by rigid elastic-perfectly plastic hypothesis, which cannot correctly predict the real load displacement behaviour, because, the failure in real soil is highly progressive. The progressive failure phenomenon is supposed to elucidate the propagation of shear
stress and deformations on shear zone exhibiting the stress-strain behaviour of soil, together with post-peak strain-softening, dislocates the soil mass, and therefore, active failure and lack of support from the remoulded shear zone (Terzaghi and Peck, 1948; Skempton, 1964; Bishop, 1967, 1971; Bjerrum, 1967; Christian and Whitman, 1969; Urciuoli et al., 2007; Bernander, 2008, 2011). Nevertheless, traditional studies are basically focused on the principle of associated flow (AF) rule, whereas peak dilation angle (ψp) is almost equal to the peak fric tional angle (φp). It is also assumed that the influence of the dilation angle (ψ) can be uniquely captured by the frictional angle (φ). However, several researchers (Davis, 1968; Drescher and Detournay, 1993; Lou kidis et al., 2008; Sloan, 2013) stated that the AF rules over predict the foundation capacity of soils. It is also known that practically soils fric tional angle is significantly greater than the dilation angle (Schanz and Vermeer, 1996), which proved that the analytical solutions are invalid and leads to a non-associated flow rule. In 1985, non-associated elas to-perfectly plastic finite element model (FEM) was used by Vermeer and Sutjiadi to authenticate their introduced design equations (proposed by Borst and Vermeer, 1984). Moreover, the importance of the non-associated strain hardening-softening and mobilized dilatancy was presented by Walters and Thomas (1982) for proper modelling of the
* Corresponding author. E-mail addresses: [email protected] (A.S.M. Riyad), [email protected] (Md. Rokonuzzaman), [email protected] (T. Sakai). https://doi.org/10.1016/j.oceaneng.2019.106496 Received 24 April 2019; Received in revised form 24 September 2019; Accepted 25 September 2019 0029-8018/© 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: A.S.M. Riyad, Ocean Engineering, https://doi.org/10.1016/j.oceaneng.2019.106496
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Fig. 4. FE Mesh for Circular Anchor (H/D ¼ 2, D ¼ 10 cm and number of elements ¼ 2400).
Fig. 1. Testing apparatus.
Fig. 2. Test conditions; (a) Dense bed; (b) Medium bed; (c) Loose bed.
Fig. 5. FE mesh used for the analysis: (a) 2D FEM (elements ¼ 56); (b) 3D FEM (elements ¼ 1470).
behaviour of the anchor foundations, as none model can predict real soil behaviour accurately. In addition to that, many previous model tests were performed in different scales of the anchor (Ovesen, 1981; Hutchinson, 1982; Dickin, 1988; Tanaka and Sakai, 1993) revealing the importance of scale effect, which is linked to the degree of different progressive failure as the anchor scale changes. The scale effect is pictured because of the distinction between the strength of small size specimen within the laboratory which of a bigger specimen in real size (Omine et al., 2005). It has been stated that the progressive failure is linked to bifurcation or shear banding phenomena (Papanastasiou and Vardoulakis, 1989). Furthermore, the importance of the consideration of grain size effect has been detected by numerous researchers (Tatsuoka et al., 1997; Sakai et al., 1998; Cerato and Lutenegger, 2007; Athani et al., 2017) during the study of scale effect. If the grain size effect is ignored, it might presumably considerably overestimate the pull-out capacity of the anchor in full-scale (Athani et al., 2017). So, to use the
Fig. 3. Formation of shear band after peak: all plastic strain localized in shear bands.
proliferation of shear zone above trapdoors. The shear band can be defined as a thin material layer that’s delimited by the surface of two material discontinuities of the velocity gradient (Hill, 1962). Presently evolved, almost all the prevailing sophisticated models are very complex and incomplete in the sense that they do not define several important factors like inherent and induced anisotropies, strain localization, strain softening, and so on. Above important factors are overlooked in most of the current geotechnical models, which has a strong link between the FE model and a reliable set of experimental data. According to the author’s knowledge, no researchers have given attention to predict the settlement 2
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Fig. 6. Parametric Study by 2D finite element analysis in dense sand: (a) Deviator stress; (b) Volumetric strain.
framework of elastoplasticity in FEM to deal with the scale effect, an in-depth study is required in relation to calibration for the sophisticated hardening-softening constitutive models and verification as well as validation of rigorous FE models for some important problems. There fore, this research is designed to simulate the ultimate uplift capacity of a shallow circular anchor in different types of sand, using the widely used numerical technique, FEM, and a constitutive relation of sand, developed in such a way that it defines explicitly all the affecting factors in a relatively simple but sophisticated way. This research also pays special attention to the “scale effect” in the traditional small-scale models in relation to the proper modelling of progressive failure in the sand and tries to identify the different components of above-mentioned effects.
pulling out speeds of 0.03 mm/min and a displacement transducer is equipped on the top of the anchor bar to measure the upward displacement. The pull-out mechanism of anchors in the sand mass is observed by keeping contact in the side of a glass wall. A computerized data acquisition system is used to record the data. The movement of the anchor is controlled until the residual conditions are obtained with the help of a direct current (DC) motor. Air-dried Toyoura sand having specific gravity (Gs) ¼ 2.64, average particle size (D50) ¼ 0.016 cm, maximum void ratio (emax) ¼ 0.98, minimum void ratio (emin) ¼ 0.61, and no fine content less than 0.00075 cm, is used in this test. For the construction of the ground above the anchor foundation, air-pluviation method is used with the help of two sieves in the mould. The test is conducted on Toyoura sand of different densities (obtained by differing sieve opening size and sand falling height). However, the maintenance of unvarying density is necessary for the formation of the depth of the sand mass. Depending on the sand falling height and the deposition intensity during the air-pluviation process, various uniform densities can be obtained. To estimate the average unit weight in the homogenous sand mass, the sand is weighed after the anchor pull-out test. Owing to the difference of sieve opening size and sand falling height, the density of sand varies from dense to loose (relative density, Dr ¼ 95%–5%; and the density, γd ¼ 1630 kg/m3 to 1350 kg/m3, allowing an error of �10 kg/m3). The density of the ground is measured by using 6-hundred cc cylinder-shaped samplers (D ¼ 50 mm), which are placed on the soil
2. Physical modelling The experimental setup is presented in Fig. 1. A 590 mm diameter cylindrical container is used as a soil bin so that it does not give any boundary effects. Flat and circular steel anchor foundation with three different diameters (i.e., 5, 10, and 15 cm) with a 5 mm thick steel plate is used to conduct the anchor pull-out test. This test is directed at an embedment ratio (H/D, where H ¼ depth of sand mass and D ¼ diameter of the anchor) of 2 in a homogeneous bed as illustrated in Fig. 2. A load cell is linked with an anchor bar to measure the vertical pull-out load at 3
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Fig. 7. Parametric Study by 2D finite element analysis in medium sand: (a) Deviator stress; (b) Volumetric strain.
the bifurcation problems so that any specific measures need not be taken; easy to be coded and very small data storage are needed owing to its explicit nature; use of approximate critical damping for linear iso parametric elements with reduced integration can implicitly suppress the hourglass modes at failure. This method is incorporated in the personally developed FORTRAN program in conjunction with the generalized return mapping algorithm by considering an elasto-plastic framework incorporating isotropic hardening-softening law (Tanaka and Sakai, 1993) following the non-associated flow rule including shear band (Sakai and Tanaka, 2007). A shear band (SB) is a narrow band where all the plastic strains are localized after peak as illustrated in Fig. 3. The inclusion of a shear band is essential to make the constitutive model and its calibration independent on the size of FE. As the pre vailing solutions of strain-softening material are firmly reliant on the size of mesh, several approaches such as Pietruszczak and Mroz (1981) suggested an uncomplicated form of mesh-size reliant on hardening modulus technique for resolution of the mesh-dependent pathology of FE solutions. Following this technique, shear banding can be incorpo rated through a strain localization parameter, S, according to one assumption of elastoplasticity theory. The assumption is that total strain increments (dε) can be decomposed additively into elastic (dεe ) and plastic (dεp ) parts, as follows (Tanaka and Sakai, 1993; Sakai and
Table 1 Parameters for hardening and softening material models. Parameters
Dense
Medium
Loose
Density (kg/m3) Void ratio, e Relative density, Dr Coefficient of shear modulus, G0 Residual frictional angle, φr (degrees) Poisson’s ratio
1630 0.62 0.95 500 33 0.3 0.1 0.1 0.3 0.1 0.1
1480 0.78 0.53 500 33 0.3 0.1 0.4 0.3 0.2 0.1
1350 0.95 0.05 500 33 0.3 0.1 0.8 0.3 0.4 0.1
εf εr εd m β
bin’s bottom. 3. Numerical modelling In numerical modelling an explicit dynamic relaxation method (Tanaka and Kawamoto, 1987; Sakai and Tanaka, 2007) is used due to its robust ability to handle extremely nonlinear material; to deal with 4
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Fig. 9. Effect of shear band in dense sand using strain hardening-softening model: (a) Deviator stress; (b) Volumetric Strain.
Fig. 8. Comparison of experimental result with simulated stress-strain-volume with shear band by 2D and 3D FEM using sophisticated strain hardeningsoftening law: (a) Dense sand; (b) Medium sand.
deviatoric stress) and gðθÞ is the function of Lode angle, θ. In case of MC material model, it can be defined as:
Tanaka, 2007): dε ¼ dεe þ Sdεp
(1)
3 sinφmob gðθÞ ¼ pffiffi 2 3cos θ 2 sin θ sinφmob
where, S is the ratio of the area of a single shear band in each element (Fb) to the total area of the finite element (Fe). However, the decision of Fb is taken based on the SB thickness. Moreover, the effects of the SB orientation in an apiece element is ignored, and thus the approximate form of SB considered in this study can be defined as SB S ¼ pffiffiffiffiffi Fe
(4)
where 1 θ ¼ cos 3
(2)
1
" pffiffi # 3 3 J3D 2 J 3=2 2D
(5)
and J3D is the third deviatoric stress invariants. The plastic strain rates can be calculated from non-associated flow rules, expressed as,
The SB thickness for sands is about 10–30 times of mean grain size diameter of a particle (Mühlhaus and Vardoulakis, 1987; Yoshida et al., 1993). MC yield criterion is used as its material parameters are well known in soil mechanics model tests, and Drucker–Prager (DP) potential function is used due to its smoothness and easy differentiability. The yield function can be introduced as: pffiffiffiffiffiffi J2D F ¼ αðκÞI1 þ κ1 ¼ 0 (3) gðθÞ
dεp ¼ dλ
dG dσ
(6)
where, G and dλ are the plastic potential and a positive constant known as plastic rate multiplier, respectively. The plastic potential function takes a similar form as the yield function Equation (3) with gðθÞ ¼ 1:0 in Equation (6), as pffiffiffiffiffiffi G ¼ α’ðκÞI1 þ J2D κ2 ¼ 0 (7) The internal variable, κ, can be found by considering the incremental deviatoric plastic strains, expressed as:
where, I1 is the first invariant of stress (i.e., hydrostatic stress compo nent, positive in compression), J2D is the second stress invariant (i.e., the 5
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For plastic potential functions, the constant, α’ can be expressed as, 2 sin ψ 3ð3 sin ψ Þ
(13)
α’ ¼ pffiffi
From the modified Rowe’s stress dilatancy relationship, the dilat ancy angle ψ can be expressed as, � � sinφmob sin φ’r ψ ¼ sin 1 (14) ’ 1 sinφmob sin φr where, sinφmob ¼
pffiffi 3 3αðκÞ pffiffi 2 þ 3αðκÞ
8 > > < ’ φr ðκÞ ¼ φr 1 > > :
βe
(15)
� �2 9 > > κ = εd > > ;
(16)
where, β and εd are the stress dilatancy material parameters controlling the rate of dilatancy. So, therefore, these 5 numbers of indeterminate constitutive pa rameters (m, εf , εr , β, and εd ) have to be optimized in the calibration process by back prediction of the anchor pull-out test. The problems of circular anchor are analysed in this study, using the four-node quadrilateral isoparametric finite elements (Fig. 4). To over come the boundary effects, the mesh extends at a distance of a minimum of 4D from the anchor edges. Moreover, finer discretization is used in the centre with the quadrilateral elements than edges, where limited deformation is expected. Furthermore, along the anchor boundary, dif ferential quasi-elastic displacements are introduced until failure at minor successive increments and the corresponding nodal forces are averaged with respect to the displacement control nodes to ascertain the ultimate load. The normalized displacement factor, δ/D (where, δ presents the displacement of the ground), and the pull-out resistance factor, Np (¼P/ γdHA, γd is the effective unit weight of dry sand and A is the surface area of the anchor) are obtained by considering the proposals of Lindsay (1980), whereas, δ is the displacement, and P is the uplift load. 4. Calibration of hardening and softening material models
Fig. 10. Effect of shear band in medium sand using strain hardening-softening model: (a) Deviator stress; (b) Volumetric Strain.
Z � � �2 � κ ¼ 2 depii þ 1
�� � �12 2 δjk depjk
The accurate modelling of the stress-strain relationship obtained from the experiment depends on the accurate constitutive models which is so much challenging task for the practical engineers. In this context, the calibration of constitutive models of linear elastic material is very simple. On the other hand, most of the engineering materials found in nature, such as soils, display the nonlinear hardening-softening behav iour, which can be modelled by sophisticated strain hardening-softening material model having an elasto-plastic framework with a nonassociated flow rule, costing of numerous parameters. And so, calibra tion is necessary to fit those parameter values by laboratory test results, before used in the numerical analysis. Moreover, if the model is highly nonlinear, the mathematical complexities amalgamated with the accu racy and sophistication of the model can be overcome by costing more parameters (Rokonuzzaman and Sakai, 2010). For this reason, extensive numerical tests on the constitutive relations of Toyoura sand of different relative densities and the effects of different parameters of the consti tutive relations on the results are examined. Further, a parametric study is done to find out the appropriate material parameter values to use in the sophisticated strain hardening-softening model. A series of drained triaxial compression tests are performed on cy lindrical saturated Toyoura sand (Gs ¼ 2.64, emax ¼ 0.977, emin ¼ 0.605, Uc ¼ 1.46, no fines) samples, prepared by air-pluviation method, of diameter and height is 70 mm and 150 mm, respectively, for different densities and confining pressures (Fukushima and Tatsuoka, 1984). FE
(8)
Due to the use of dry sand in simulation and test in this study, the cohesion can be ignored, and so the function κ can be ignored in Equations (3) and (6). The frictional function α(κ) can be expressed as, � pffiffiffiffiffiffi �m � 2 κεf hardening regime : κ � εf (9) αðκÞ ¼ αp κ þ εf �
αðκÞ ¼ αr þ αp
�
αr e
κ εf εr
�2 softening regime : κ > εf
�
(10)
where, m, εf and εr are the constants, which considers the hardeningsoftening material condition, and 2 sinφp
�
(11)
2 sinφr 3ð3 sinφr Þ
(12)
αp ¼ pffiffi
3 3
αr ¼ pffiffi
sinφp
where, φp is the peak and φr is the residual frictional angle, respectively. 6
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Fig. 11. Validation and Effect of Constitutive Models in sand of different densities: (a) Dense Sand; (b) Medium Sand; (c) Loose Sand.
Fig. 12. Mesh-size effect (H/D ¼ 2, D ¼ 10 cm): (a) fine mesh (number of elements ¼ 2400); (b) coarse mesh (number of elements ¼ 1536); (c) very coarse mesh (number of elements ¼ 600).
meshes used for the analysis are presented in Fig. 5. The results of the parametric studies on the hardening-softening material parameters are presented in Figs. 6 and 7. By parametric study, it is detected that when the hardening-softening material parameters: εf, εr, and m are 0.1; the calculated result coincides with the experimental result in dense sand. On the other hand, when the hardening-softening material parameters: εf, εr, and m are 0.1, 0.4, and 0.2, respectively, the calculated outcome is agreed with the experimental outcome in medium sand. The parameters for loose sand are adopted from Sakai and Tanaka (2007). After the
parametric study, the selected parameters are used for the analysis is presented in Table 1. Here, 0.3 cm wide shear band is adopted, which equals to twenty times of mean particle diameter (D50) of the Toyoura sand (Yoshida et al., 1993). Fig. 8 presents the effect of two-dimensional (2D) and threedimensional (3D) FEM model on stress-strain-volume change relation ship. Small difference is observed on stress-strain-volume change rela tionship using the 2D and the 3D FEM models due to the difference in their kinematic boundary conditions. The 2D model constraints are 7
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Fig. 13. Uplift resistance-displacement factor relationship by different mesh-size: (a) Dense sand with shear band; (b) Medium sand with shear band; (c) Loose sand with shear band; (d) Dense sand without shear band; (e) Medium sand without shear band; (f) Loose sand without shear band.
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Fig. 14. Mesh dependency in shear strain and deformation pattern (δ/D ¼ 0.024; H/D ¼ 2, D ¼ 10 cm).
outcomes with respect to the other models with the experiment which is validated for the further study.
greater than the 3D models. As the 3D numerical simulation needs a larger amount of elements, and therefore a larger amount of integration points, nodes, time and resources are needed in comparison with the 2D model. However, there is no effect of aspect ratio in the circular anchor as in the rectangular anchor. Volumetric strain is more in 3D model. The differences between the results from those models are negligible and so 2D model is used for the analysis. Figs. 9 and 10 presents the stress-strain-volume change relationship achieved from the research recommended by Fukushima and Tatsuoka (1984) and the corresponding FE analysis. The effect of the shear band on the material parameters are also analysed in this study. For the analysis, a single element and multi-element of 56 is used. The experi mental result indicates close agreement with the analysis considering maximum deviator stress. Without the shear band, the softening part is affected due to the element size, especially in dense sand. And so, due to the mesh size effect after the peak, the shear band is introduced in this study.
6. Effect of mesh size, calculating step size and shear band Also, the effect of mesh size and calculating step size, constants of the constitutive models and shear band option on the anchor foundation are studied. Due to the sensitiveness of mesh size in the FE solutions, the numerical model integrating hardening-softening rule must be authen ticated before used for the analysis of the anchor foundation. For the analysis of the anchor foundation (H/D ¼ 2, D ¼ 10 cm), 2400 quadri lateral finite elements 5 mm in size, 1536 quadrilateral finite elements 6.25 mm in size and 600 quadrilateral finite elements 10 mm in size in the central zone are used as presented in Fig. 12. The curves in Fig. 13 point out the relationship between the displacement factor and the pullout resistance factor. Due to having the shear band thickness in the constitutive model, the load-displacement behaviour of anchor foun dation is independent on the mesh size. Thus, re-meshing is avoided by scaling up a smaller proportional model in the parametric study. In the pre-peak regime, deformation of a given sand element under a uniform boundary stress condition is analogous. Strain localization starts sud denly at the peak stress state. It is assumed that shear bands start to develop at the peak. For the post-peak parts, strains are the ones defined locally within a band including a shear band. It is assumed that the rate of strain softening is independent of bedding plane orientation angle and φmob becomes a constant value, φr at the same relative displacement across a shear band. The stress-strain relationship inside a shear band is independent of the size, shape and boundary conditions of a given element containing the shear band. This relation and shear band thickness is unique for a given mass of sand, and independent of the density and pressure level. In the absence of a shear band, the effect of mesh size is predominant after peak especially for softening dense sand
5. Validation and Effect of Constitutive Models Besides, the effects of different constitutive models on the anchor foundation are examined. The curves shown in Fig. 11, depict the experimental and numerical relationships between the pullout resis tance and the displacement factor of the circular anchor foundation (H/ D ¼ 2 and D ¼ 10 cm) at different density condition by considering different constitutive models. From all the curves, three distinct phases are evident: a sharp increase of pullout resistance with anchor displacement in the initial phase, followed by a shallow decrease, and, finally, in the residual phase, the pullout resistance remains unaffected with the further uplifting of the anchor. From the overall shape of the load-displacement curve, it can be assumed that progressive failure is attributed and strain hardening softening model is presented better 9
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Fig. 15. Influence of calculating step in different meshes in loose sand: (a) Number of elements ¼ 2400; (b) Number of elements ¼ 1536; (c) Number of elements ¼ 600.
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Fig. 16. Influence of calculating step in different meshes in medium sand: (a) Number of elements ¼ 2400; (b) Number of elements ¼ 1536; (c) Number of elements ¼ 600.
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Fig. 17. Influence of calculating step in different meshes in dense sand: (a) Number of elements ¼ 2400; (b) Number of elements ¼ 1536; (c) Number of elements ¼ 600.
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Fig. 18. Mesh dependency in different calculating steps in loose sand: (a) Step ¼ 1000; (b) Step ¼ 2000; (c) Step ¼ 5000; (d) Step ¼ 15000; (e) Step ¼ 20000.
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Fig. 19. Mesh dependency in different calculating steps in medium sand: (a) Step ¼ 1000; (b) Step ¼ 2000; (c) Step ¼ 5000; (d) Step ¼ 15000; (e) Step ¼ 20000.
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Fig. 20. Mesh dependency in different calculating steps in dense sand: (a) Step ¼ 1000; (b) Step ¼ 2000; (c) Step ¼ 5000; (d) Step ¼ 15000; (e) Step ¼ 20000.
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Fig. 21. Pullout resistance–displacement factor curves using sophisticated strain hardening-softening model: (a) Dense Sand; (b) Medium Sand; (c) Loose Sand.
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7. Effect of material parameters It is assumed that the progression of localized deformation is ob tained from the triaxial compression test with the pull-out test. The hardening-softening material parameters values (εf, εr, and m) are evaluated on the back prediction of the anchor pullout test by comparing the experimental and the finite-element analysis results. Calculated and experimental results for D ¼ 10 cm at H/D ¼ 2 are presented in Fig. 21, which presents the pullout resistance and displacement factor curves. After the parametric study, the same values are obtained as observed in calibration results using the sand of different densities. Table 1 already summarized the parameters used in the finite-element analysis. 8. Study of progressive failure According to Terzaghi and Peck (1948), it can be expressed as “The term progressive failure indicates the spreading of the failure over the potential surface of sliding from a point or line towards the boundaries of the surface. While the stresses in the clay near the periphery of this surface approach the peak value, the shearing resistance of the clay at the area where the failure started is already approaching the much smaller ultimate’ value. As a consequence, the total shearing force that acts on a surface of sliding at the instant of complete failure is consid erably smaller than the shearing resistance computed on the basis of the peak values.” Thus, progressive failure can be referred to as the state of affairs which indicates the consecutive failure of individual elements of soil in a soil mass. This method spreads in an area and needs time to occur, whereas time aspects might not continuously be necessary beneath some circumstances, an understanding of spatial progressive failure is most vital (Lo, 1972). Lo also stated that the prime necessary condition for the occurrence of progressive failure is that the material should follow the stress-strain softening relationship. A study of Taylor (1937) on shearing resistance on sands using effective stress concept stated the associated progression of failure with the redistribution of shear stress on a potential failure surface and neglected the propagation of progression of failure direction. But, the importance of the determi nation of propagation of progressive failure is so much crucial for esti mating the available shearing resistance along the sliding surface. The failure in the ground when loaded with an anchor is highly progressive in a view that the soil’s peak strength is not mobilized along potential failure surfaces; while the perfectly-plastic assumption leads to that the peak soil strength is mobilized concurrently along the potential failure surface. Therefore, the soil shear strength is not concurrently mobilized at all points of a slip surface. It is also supposed that the strength properties of soil are unaltered even if large strains are introduced by loading. However, this hypothesis is insignificant for the soils catego rized by noticeable strain-softening behaviour, like dense sands. Actu ally, in these materials, it happens that a few segments of the sand initially fail due to loading, with the shear strain that is situated in a zone of restricted thickness (shear band). With the increase of strain inside this zone, soil quality decreases from the peak towards the critical state. Because of the consequential stress redistribution, the shear band spreads in the sand and a slip surface dynamically creates up to causing the collapse of the soil-foundation framework. In the case of the failure condition, the average strength mobilized along the slip surface is not as much as the pinnacle quality and more prominent than the quality at the critical condition of the sand under consideration. For the documenta tion of this failure phenomenon, the amassed plastic deformation zones above the anchor calculated for any loading stage (D ¼ 10 cm at H/D ¼ 2) at different densities of sand are illustrated in Fig. 22 for the same δ/D ratio (δ is larger for larger D). As the shear bandwidth is in dependent of pressure-level, its ratio with D decreases proportionally as D increases. Then, once shear bands have appeared, the relative displacement across a shear band at corresponding points becomes larger for larger D, resulting in a larger ratio of shear band length relative to D and more non-uniform distribution of φmob, thus a larger
Fig. 22. Accumulated plastic deformation zones above the anchor (figures are not to scale): (a) Loading stage considered in analysis; (b) Average shear strain distribution calculated at any loading stage with respect to denseness of sand.
and so the result is influenced. And so, a shear band is used in the further analysis of anchor problem. Fig. 14 shows the mesh size dependency in shear strain and the deformation pattern of the specimen at different density conditions. Though the overall distributions of the shear strain are almost similar in all meshes, it is much sharper in finer mesh than in the coarser mesh. On the other hand, there is not much difference in the deformation pattern. Figs. 15–17 shows the influence of the calculating steps in different meshes. From all the curves, it can be recapitulated that to obtain a stable result, more than 5000 calculating steps are needed. After that, the relationship between the pull-out resistance and the displacement factor will be the same in all the calculating steps. Figs. 18–20 shows the mesh size dependency in different calculating steps. It is observed that more than the calculating steps 5000, the peak strength is not affected at all in different meshes while the residual strength changed in such a tendency that the finer the mesh is, less the residual strength will be. When the number of elements is 1536 and 2400, the difference between the residual strength is almost zero. And so, for further study, calculating step 15000 is selected. 17
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Fig. 23. Scale effect: relationship between experimental and numerical results at H/D ¼ 2: (a) Dense Sand; (b) Medium Sand.
Fig. 24. Scale effect: uplift resistance-displacement factor relationships at H/D ¼ 1: (a) Dense Sand; (b) Medium Sand; (c) Loose Sand.
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Fig. 25. Scale effect: uplift resistance-displacement factor relationships at H/D ¼ 2: (a) Dense Sand; (b) Medium Sand; (c) Loose Sand.
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Fig. 26. Scale effect: uplift resistance-displacement factor relationships at H/D ¼ 3: (a) Dense Sand; (b) Medium Sand; (c) Loose Sand.
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Fig. 27. Scale effect: uplift resistance-displacement factor relationships at H/D ¼ 4: (a) Dense Sand; (b) Medium Sand; (c) Loose Sand.
degree of progressive failure. In Fig. 22a, some loading stages (i.e., A–C) are presented to support the demonstration of the outcomes. Fig. 22b shows that at Stage C, the slip surface is inclined in case of dense sand and vertical in case of loose sand. It is also evident that in dense sand failure is not fully happened and on the other hand, for loose sand, the failure is outcropped on the surface. And so, it can be said that the failure of dense sand is more progressive than the others, as less unloading is required to initiate the failure that spreads over a larger space within the soil mass.
may be attributed to the combined effect of particle size and stress level according to Tatsuoka et al. (1991) and increases with the increase of the embedment. This trend of scale effect with varying embedment is attributed to the various degree of progressive failure with shear band development. Due to high confining pressure and particle size effect or more progressive nature of the failure of the material, the scale effect is predominant in dense sand. On the other hand, less softening in the material behaviour so less scale effect in medium and/or loose sand. In 1 g test result, the scale effect consists of the effect of the pressure level and particle size. The latter effect is observed in centrifuge tests with varying the g level using the same anchor size and the same type of sand. As a small-scale physical model has high dimensionless power, care must be taken while extrapolating the laboratory model test results (small- scale) to the full-scale structures.
9. Width (scale) effect The problems of the transformation of the model test results to a prototype is mainly due to width (scale) effect. The scale effects are examined by varying H and D, proportionally (e.g. the FE mesh for H/ D ¼ 2, D ¼ 5 cm is scaled up by a factor of 10 to have anchor with H/ D ¼ 2, D ¼ 50 cm, and this procedure is supported by such kind of nu merical model, insensitive to mesh size effect). Fig. 23 presents the relationship between the experimental results and the numerical results at H/D ¼ 2 (D ¼ 5, 10 and 15 cm), which is presented that numerical results well predict the experimental results. Figs. 24–27 presents the uplift resistance-displacement factor relationships at different embed ment. Figs. 28 and 29 display that Npu and δu/D decreases with D, which
10. Conclusions Progressive failure behaviour mainly occurs due to the nonmobilization of peak strength of soil in potential failure regions, whereas, scale effect is predominant in softening soil. It is observed that, based on calibration test results, the numerical models can well-predict the experimental stress-strain data points from element tests very closely using the appropriate set of material constants of the constitutive models 21
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Fig. 28. Scale effect: peak resistance factor as a function of D and H/D: (a) H/D ¼ 1; (b) H/D ¼ 2; (c) H/D ¼ 3; (d) H/D ¼ 4.
Fig. 29. Scale effect: peak displacement factor as a function of D and H/D: (a) H/D ¼ 1; (b) H/D ¼ 2; (c) H/D ¼ 3; (d) H/D ¼ 4.
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and yield surface as the MC model and potential surface as the smooth DP model with shear band effect using the explicit dynamic relaxation method framework. The sophisticated hardening-softening material model shows better performance as it is capable to closely predict the pull-out resistance and settlement patterns costing five numbers of material parameters. Due to the consideration of the shear band thick ness in the constitutive model, the load-displacement behaviour of an chor foundation is independent on the mesh size. The peak resistance factor presents a decreasing tendency with the increase of the width of the foundation, which is scale effect. The failure pattern of dense sand is more progressive than loose sand, and so, scale effect is predominant in dense sand. The settlement at the ultimate bearing capacity of the an chor foundation always increases with the decrease of width.
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