canadian geotechnical journal - university of toronto t-space · 2018. 12. 7. · rowe and davis...

Draft

Vertical uplift resistance of horizontal plate anchors for

eccentric and inclined loads

Journal: Canadian Geotechnical Journal

Manuscript ID cgj-2017-0515.R1

Manuscript Type: Note

Date Submitted by the Author: 13-Mar-2018

Complete List of Authors: Kumar, Jyant; Indian Institute of Science, Civil Engineering Rahaman, Obaidur; Indian Institute of Science bangalore, civil engineering

Keyword: anchors, limit analysis, lower bound analysis, conic optimization, failure load

Is the invited manuscript for consideration in a Special

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Not applicable (regular submission)

https://mc06.manuscriptcentral.com/cgj-pubs

Canadian Geotechnical Journal

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Technical Note

Vertical uplift resistance of horizontal plate anchors for eccentric and inclined

loads

Jyant Kumar1

and Obaidur Rahaman2

Abstract: The vertical uplift resistance of horizontal plate anchors embedded in sand has been

computed for inclined and eccentric pullout loads. The analysis has been performed by using the

lower bound theorem of the limit analysis in combination with finite element and second order

cone programming (SOCP). The methodology is based on the Mohr-Coulomb yield criterion and

the associated flow rule. Several combinations of the eccentricity (e) and the vertical inclinations

(α) of the resultant pullout loads have been considered. The computations have revealed that the

magnitude of the vertical uplift resistance decreases with an increase in the values of both e and

α. The reduction of the vertical pullout resistance with eccentricity and inclination becomes

more prominent for smaller values of embedment ratio. The anchor-soil roughness angle (δ)

hardly affects the uplift capacity factor as long as the value of α remains smaller than δ.

Keywords: anchors; limit analysis; lower bound analysis, conic optimization, failure load

1,2Civil Engineering Department, Indian Institute of Science, Bangalore-560012, India.

Corresponding author: Jyant Kumar (e-mail: [email protected]).

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Introduction

Anchors are often employed to resist pullout forces for the foundations of transmission

towers, buried pipelines under water, dry docks, bulkheads and sheet piles walls. A number of

investigations have been reported to predict the pullout capacity of plate anchors. Among the

earlier studies, Mors (1959) introduced a simplified cone method to determine the uplift

resistance for shallow circular plate anchors. Balla (1961) employed the Kötter’s equation on the

curvilinear slip surface to compute the vertical ultimate uplift capacity of horizontal circular

plate anchors in sand. Meyerhof and Adams (1968) presented a semi-theoretical approach based

on the limit equilibrium method to estimate the vertical uplift capacity of horizontal strip,

circular, and rectangular plate anchors. From the expansion of cylindrical and spherical cavities

below the surface of a semi-infinite, homogeneous and isotropic half space, Vesic (1971) derived

a theoretical expression for computing the vertical uplift resistance of strip and circular plate

anchors. Ovesen (1981) carried out centrifuge model tests to predict the uplift capacity of

anchors. Rowe and Davis (1982) performed an elasto-plastic finite element analysis to examine

the effect of anchor plate roughness and soil dilatancy on the pullout resistance of horizontal and

vertical anchor plates. Simplified upper and lower bound limit analyses were carried out by

Murray and Geddes (1987) and Basudhar and Singh (1994) to estimate the vertical uplift

resistance of horizontal strip plate anchors. Uplift capacity factors due to the components of

cohesion, surcharge and unit weight were established by Rao and Kumar (1994) using the

method of stress characteristics. Kumar (2001) examined the effect of an inclusion of seismic

forces on the uplift capacity of anchors. A rigorous numerical investigation, on the basis of the

lower and upper bound finite elements limit analysis for both horizontal and vertical plate anchor

was carried out by Merifield and Sloan (2006). This methodology was also later extended by

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Kumar and Kouzer (2008) to compute the vertical uplift resistance of plate anchors in sand.

Jesmani et al. (2013) carried out small scale model tests as well as the elasto-plastic finite

elements analysis to predict the uplift capacity of anchors in clays with different inclinations of

the plate. Recently, Nouri et al. (2017) performed three-dimensional finite element analysis for

assessing the undrained response of plate anchors considering translation and torsional loads. It

is noticed that most of the existing studies deal with the determination of the pullout resistance

considering the case when the pullout load acts in a direction normal to the anchor plate without

any eccentricity. However, in many practical cases anchors are frequently subjected to eccentric

and inclined pullout loads due to the presence of wind, water waves and earthquake body forces.

Although some model and field tests were performed by Meyerhof (1973) and Das and Seeley

(1975) to examine the response of anchor piles and square plate anchors under oblique pullout

loads. However, hardly any extensive research seems to have been carried out for determining

the pullout resistance of strip anchors in the presence of eccentricity as well as the inclination of

the pullout load. This is the primary objective of the current research. In the present note, an

attempt has been made to examine the effects of eccentric and inclined pullout loads on the

magnitude of the vertical uplift resistance of horizontal plate strip anchors buried in a sandy

medium. The analysis has been carried out by using the rigorous lower bound finite element

technique in combination with the second order cone programming (SOCP). The effects of

embedment ratio, soil internal fiction angle and anchor-soil interface friction angle on the uplift

resistance for different combinations of load eccentricity and inclination of loading have been

studied. Failure patterns have also been explored for different cases. The results obtained from

the analysis have been compared with that available from literature.

Problem definition

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A strip plate anchor of width � with negligible thickness is embedded horizontally at a depth � from horizontal ground surface in a cohessionless soil medium. The anchor is subjected to pullout load (i) with an eccentricity � from center of the plate, and (ii) having an inclination � with the vertical as shown in Fig.1(a); the positive values of e and α have been marked in this

figure. The soil mass is assumed to follow the Mohr-Coulomb failure criterion and an associative

flow rule. The objective is to determine the vertical pullout resistance of the plate anchor in the

presence of (i) eccentricity (�), (ii) inclination (�) and (iii) combination of both eccentricity and inclination. The ultimate vertical pullout resistance (��) is expressed in terms of a non-dimensional pullout capacity factor (�) as defined herein: (1) � = �� where � is unit weight of soil medium; all the results have been presented in a non-dimensional fashion and the chosen value of γ does not affect the value of �. Depending on signs of e and α, the anchor may be subjected to either Type I or Type II loading condition as shown in Fig 1(b);

for the Type I loading condition, the signs of both e and α remain either positive or negative

simultaneously, on the other hand for the Type II loading, the signs of e and α need to be

opposite in nature.

It should be mentioned that in the lower bound analysis, the strains rates (velocities) do

not enter into the formulation. Ideally, the analysis is applicable for small strains when the

ultimate shear failure has just initiated.

Mesh details and boundary conditions

A rectangular domain PQRS as shown in Fig. 1(a) has been chosen. The horizontal

distance (�) between the anchor edge and the chosen vertical boundary, and the depth (D) of the

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bottom boundary (QR) from the level of anchor plate were chosen sufficiently large such that (i)

the elements near the chosen boundaries do not yield and (ii) a further increment in the size of

the domain does not alter any change in the magnitude of the collapse load. These two criteria

were found to be generally quite acceptable in fixing the locations of the distance boundaries as

were noted from the sensitivity analysis by varying the positions of these chosen boundaries. For

the different embedment ratios and soil friction angles, (i) the value of L was varied from 3B to

15B, and (ii) the value of D was varied between 1.5B and 6B. The rectangular domain PQRS was

discretized into a number of three noded triangular elements and the sizes of the elements were

reduced approaching towards the edges of the anchor plate. The stress variation was assumed to

vary linearly over the area of the element by using the linear shape functions. Typical finite

element meshes for � = 2 and 4 are shown in Fig. 2. For the particular case � = 2, the total number of (i) elements (��) are 9936, (ii) the stress discontinuities (��) are 14794, (iii) the nodes along soil-anchor interface (��) are 64, and (iv) the nodes on top boundary on which stress boundary conditions are imposed (��) are 124. The total number of nodes are 3 × ��. The values of vertical normal stresses (��) and shear stresses (� �) along the ground surface (PS) are kept equal to zero. For the remaining three chosen boundaries, no stress

boundary condition needs to be imposed. Along the soil-anchor interface, the following

inequality constraint needs to be imposed:

(2) !� �! ≤ #$ cot ( − ��* tan - here c is the cohesion of soil medium which is kept zero for this study. The different boundary

conditions have been shown in Fig.1(a). It should be mentioned no exclusive elements have been

chosen to model the anchor plate and the rigidly of the plate does not enter in the lower bound

formulation.

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Methodology

In most of the earlier research on the limit analysis involving the Mohr-Coulomb yield

criterion (Lysmer 1970; Pastor 1978), the problem was solved by using the linear programming

technique. Later on, new algorithms were developed for solving the problem on the basis of

nonlinear optimization techniques (Zouain et al. 1993; Lyamin and Sloan 2002). However, the

nonlinear optimization approach requires the smoothening of the yield surface in pi as well as

meridional planes, and moreover, the associated computational task becomes quite cumbersome.

These issues can be successfully resolved in the second order cone programming (SOCP)

technique (Makrodimopoulos and Martin 2006). The present analysis involves the application of

the lower bound theorem of the limit analysis in conjunction with finite elements and SOCP

(Makrodimopoulos and Martin 2006; Tang et al. 2014).

Horizontal normal stress (� ), vertical normal stress (��) and shear stress (� �) at the nodes are taken as the basic unknown variables. Following conditions need to be satisfied to determine

the statically admissible stress field (Sloan 1988):

(i) Equilibrium condition within each element:

(3a) ./0.1 +.304.5 = 0 (3b)

./4.5 +.304.1 = �

(ii) The continuity of normal and shear stresses along the interface of two adjacent elements i and

j with the nodal pairs (1,2) and (3,4); the nodes 1 and 3 are associated with the element i and the

nodes 2 and 4 belong to the element j. For an interface with n and t representing normal and

tangential directions, respectively:

(4) �7,9 = �7,:; �7

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(iv) The Mohr-Coulomb yield criterion with tensile normal stress taken as positive:

(5) ?#� − ��*: + #2� �*: ≤ 2$ cos( − #� + ��* sin( Defining the new variables �B, C andC � as given by the following expressions: (6) �B = 0.5#� + ��*;C = 0.5#� − ��*; C � = � � The yield criterion as defined by Eq. (5) then takes the following form:

(7) ?C : + C �: ≤ $ cos( − �B sin( Introducing an auxiliary variable, �G� = $ cos( − �B sin(. The Mohr-Coulomb yield criterion can be expressed in terms of a second order conic constraint:

(8) ?C : + C �: ≤ �G� The relationship between the conic constraints’ variables H C C ��G� I and the basic stress variables

H � �� I can be written in the matrix form as follows:

(9) J −0.5 0.5 00 0 −10.5 sin∅ 0.5 sin∅ 0 M H� �� I + H

C C ��G� I = H00$ cos ∅I

Effect of eccentricity and inclination

In order to consider the effect of eccentricity (�) and the inclination (�) of the applied pullout load, the following two equality constraints associated with the stresses along the anchor-

soil interface have been formulated. These two equality constraints can be derived by

considering the force and moment equilibrium along the soil-anchor interface in a manner as

follows:

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The ultimate pullout capacity (��) can be expressed as a combination of vertical force (��), horizontal force (��N) and a moment (��B) as shown in Fig.1(b). The total vertical load (��) and horizontal load (��N) acting on the anchor plate area (�)are (10) �� = O#−��P + ��Q*RST and��N = O#−� �P + � �Q *RS

T

where ��P and � �P are the vertical normal stresses and shear stresses acting over the nodes above the anchor plate, and likewise, ��Q and � �Q refer to the vertical normal stresses and shear stresses acting over the nodes below the anchor plate. Since the direction of the resultant pullout load

makes an angle � with the vertical, the following equality constraint relating ��N and �� can be written as:

(11) ��N$UC� − ��CVW� = 0

(12) cos � O#−� �P + � �Q *RST − sin�O#−��P + ��Q*RST = 0

Further, in order to certify that the resultant pullout load acts at an eccentricity (�) from the centre of the plate, the following expression can be written:

(13) �O#−��P + ��Q*RST −O#−S��P + S��Q*RST = 0

Finally, the magnitude of the vertical component (��) of the collapse load is maximized subjected to a set of linear and conic constraints. The problem was solved by writing the code in

MATLAB (2016). For dealing with SOCP, the optimization tool MOSEK (2016) was employed.

Results

The values of the uplift capacity factor (�), as defined earlier by Equation (1), for different combinations of eccentricity (�) and inclination (�) have been computed. The value of � has been varied from 1 to 6, and the soil internal friction angle (() has been kept between 250 – 450

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with an interval of 50. Four different values of e/B, namely 0, 0.1, 0.2, 0.3 and four different

values of � (⁄ , which were, 0, 1/3, 1/2, and 2/3 have been chosen. The interface friction angle (δ) between soil mass and anchor plate was varied from 0 to (. Since the magnitude of Quv is hardly affected by δ, as will be shown later, the uplift capacity factors for different cases have

been computed for a rough anchor with - = (. In all the cases, the results were obtained for both Types I and II loading conditions. However, the final charts for obtaining the values of the uplift

factors were established for the most critical case, that is, for the one which provides the least

magnitude of the vertical uplift resistance.

The variation of YZ The values of the pullout capacity factor (�) for different combinations of e/B and � (⁄

have been presented in Fig. 3 for both the loading types with � = 2 , ( = 30[ and - = (. Note that as compared to the Type II loading, the magnitude of the � becomes always smaller for the Type I loading condition; when the value of either e or α becomes equal to zero, both the types

of the loading conditions provide exactly the same value of �. Accordingly, all the results were finalized for the Type I loading condition. The variation of Fγ with e/B for different combinations

of α/φ, λ and φ have been shown in Figs. 4-6 with - = (. It can be observed that the value of � decreases continuously with an increase in the

eccentricity. The rate of reduction of Fγ with e/B tends to become greater for larger values of e/B

and for smaller values of λ. For instance, for λ = 1, φ = 35o and α/φ = 0, with an increase in e/B

(i) from 0.1 to 0.2, the factor Fγ decreases by 5.68%, and (ii) from 0.2 to 0.3, the factor

Fγ decreases by 11.40%. However, for λ = 3 again with φ = 35o and α/φ = 0, the decrease in Fγ is

(i) only 0.94% with an increase in e/B from 0.1 to 0.2, and (ii) 2.20% with an increase in e/B

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from 0.2 to 0.3. Furthermore, for λ = 2 with α/φ = 0, with an increase in e/B from 0.2 to 0.3, the

factor Fγ decreases by (i) 12.82% for φ = 25o and (ii) 5.07% for φ = 35o. Note that the reduction

of � with e/B becomes smaller for greater values of φ. It can also be noted that the magnitude of Fγ decreases continuously with an increase in

α/φ. The reduction of Fγ with an increase in α/φ becomes more prominent for greater values of φ

and for lower values of λ. For λ = 1 and e/B = 0, with an increase in α/φ from 0 to 1/3, the factor

Fγ decreases by (i) 1.92% for φ = 35o, and (ii) 2.18% for φ = 45o. Likewise, for λ = 3 and e/B =

0, with an increase in α/φ from 0 to 1/3, the factor Fγ (i) reduces by 1.38% for φ = 35o, and (ii)

1.67% for φ = 45o.

Effect of anchor roughness

It is known from literature that for α = e = 0, the variation of the interface friction angle

(-) between soil mass and anchor plate hardly affects the vertical uplift resistance (Rowe and Davis 1982). To examine the effect of δ in the presence of α and e, the variation of Fγ with -/( for different α/φ and e/B has been illustrated in (i) Fig. 7(a) for different values of α/φ, and (ii)

Fig. 7(b) for different values of e/B. As long as the value of α remains smaller than δ, the

magnitude of Fγ is hardly affected with the changes in δ. For α ≥ δ, the slippage will simply

occur along the interface of the anchor plate and cohesionless soil mass and it will lead to failure.

Failure patterns

The proximity of state of stress to shear failure at any point has been defined based on the

value of a ratio ]/R; where ] = #� − ��*: + #2� �*: and R = #� + ��*:CVW:(. The value of ]/R =1 implies the yield state and the value of this ratio lesser than unity indicates a non-plastic

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state (before yield). For � = 2 and ( = 30[,the failure patterns for different combinations of e/B and α/φ have been illustrated in Fig. 8. Failure patterns for both the Type I and II loading

conditions have been drawn. Fig. 8 (a) is associated with the case of a pure vertical uplift load

with � � = 0⁄ for which case the failure pattern becomes symmetrical about the axis of the anchor, and a non-plastic zone forms at the middle and just above the anchor plate. As the

resultant load becomes inclined and eccentric, the failure patterns become asymmetrical as

illustrated in Figs. 8(b)-8(e). The anchor roughness hardly affects the failure patterns as can be

noted from the comparisons of Figs. 8(a) and 8(f). For an inclined load, the plastic zones tend to

bend in a direction opposite to the direction of the inclination angle α. For an eccentric load, the

plastic zones tend to bend towards the eccentricity of the loading; the size of the plastic zone

towards e also becomes larger in size.

Comparisons

With e =α =0, the results obtained from the present analysis were compared with that

reported by (i) Murray and Geddes (1987) based on the simplified upper bound analysis using

the planar rupture surface, (ii) Merifield and Sloan (2006) on the basis of lower and upper bound

finite elements limit analysis using the nonlinear programming, and (iii) Khatri and Kumar

(2011) by using the lower bound finite elements limit analysis with the linear optimization

technique. For ( = 30^ and 40^ , the comparison of all these results, in terms of the variation of Fγ with λ, is indicated in Fig. 9. It can be seen that the present values of � are found to be a little smaller than the corresponding upper bound solution of Merifield and Sloan (2006) but remain

very close to their lower bound solution as well as the upper bound results of Murray and Geddes

(1987). On the contrary, the lower bound values of Fγ by Khatri and Kumar (2011) are found to

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be marginally smaller than the present results: this is on account of the fact that the SOCP

provides more accurate solution as compared to the linear optimization approach.

Conclusions

The vertical pullout resistance of horizontal strip plate anchors embedded in sand and

subjected to various combinations of eccentric and inclined loading has been computed by using

the lower bound finite elements limit analysis in conjunction with the SOCP. The magnitude of

the vertical pullout resistance decreases with an increase in the magnitudes of eccentricity (e) and

the vertical inclination (α) of the resultant pullout load. The reduction in � due the presence of e and α becomes more prominent for lower values of λ. The reduction of � with an increase in e/B becomes greater for larger values of �/� and with smaller values of φ. The reduction of Fγ with an increase in α/φ becomes, however, more extensive for greater values of φ. In the

presence of the eccentricity and inclination of the pullout load, the failure mechanism becomes

asymmetrical. The anchor roughness hardly affects the uplift resistance as long as α remains

smaller than δ.

References

Balla, A. 1961. The resistance of breaking-out of mushroom foundations for pylons.

Proceedings, 5th International Conference on Soil Mechanics and Foundation

Engineering, Paris, 1:569-576.

Basudhar, P.K., and Singh, D.N. 1994. A generalized procedure for predicting optimal lower

bound break-out factors of strip anchors. Geotechnique, 44 (2):307-318.

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Das, B.M., and Seeley, G.R. 1975. Inclined load resistance of anchors in sand. Proceedings of

the American Society of Civil Engineers 2 (GT 9): 995–1003.

Jesmani, M., Kamalzare, M., and Nazari, M. 2013. Numerical study of behavior of anchor plates

in clayey soils. International Journal of Geomechanics, 13(5):502-513.

Khatri, V.N., and Kumar, J. 2011. Effect of anchor width on pullout capacity of strip anchors in

sand. Canadian Geotechnical Journal, 48(3): 511-517.

Kumar, J. 2001. Seismic vertical uplift capacity of strip anchors. Geotechnique, 51(3): 275- 279.

Kumar, J., and Kouzer, K.M. 2008. Vertical uplift capacity of horizontal anchors using upper

bound limit analysis and finite elements. Canadian Geotechnical Journal, 45(5): 698-704.

Lyamin, A.V., and Sloan, S.W. 2002. Lower bound limit analysis using non-linear programming.

International Journal for Numerical Methods in Engineering, 55(5):573–611.

Lysmer, J. 1970. Limit analysis of plane problems in soil mechanics. Journal of the Soil

Mechanics and Foundations Division (ASCE), 96:1311–1334.

Makrodimopoulos, A., and Martin, C.M. 2006. Lower bound limit analysis of cohesive‐

frictional materials using second‐order cone programming. International Journal for Numerical Methods in Engineering, 66(4): 604-634.

MATLAB R2016a [Computer software]. MathWorks, Natick, MA.

Merifield, R.S., and Sloan, S.W., 2006. The ultimate pullout capacity of anchors in frictional

soils. Canadian Geotechnical Journal, 43(8): 852-868.

Meyerhof, G.G., and Adams, J.I. 1968. The ultimate uplift capacity of foundations. Canadian

Geotechnical Journal, 5(4), 225–244.

Meyerhof, G.G. 1973. The uplift capacity of foundation under oblique load. Canadian

Geotechnical Journal, 10(64): 64–70.

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Mors, H. 1959. The behaviour of most foundations subjected to tensile forces. Bautechnik

36(10):367-378.

MOSEK ApS. 2016. The MOSEK optimization tools manual, Version 7.0. Available from

http://www.mosek.com.

Murray, E.J., and Geddes, J.D. 1987. Uplift of anchor plates in sand. Journal of Geotechnical

Engineering, ASCE, 113(3): 202-215.

Nouri, H., Biscontin, G., and Aubeny, C.P. 2017. Numerical prediction of undrained response of

plate anchors under combined translation and torsion. Computers and Geotechnics, 81:

39-48.

Ovesen, N. K. 1981. Centrifuge tests on the uplift capacity of anchors. Proc., 10th Int. Conf. on

Soil Mechanics and Foundation Eng., A. A. Balkema, Rotterdam, Netherlands, 1: 717–

722.

Pastor, J. 1978. Analyse limite: détermination numérique de solutions statiques complètes.

Application au talus vertical. Journal de Mécanique appliquée, 2(2):167–196.

Rao, K.S.S., and Kumar, J. 1994. Vertical uplift capacity of horizontal anchors. Journal of

Geotechnical Engineering, ASCE, 120: 1134-1147.

Rowe, R.K., and Davis, E.H. 1982. Behaviour of anchor plates in sand. Geotechnique, 32(1): 25-

41.

Sloan, SW. 1988. A steepest edge active set algorithm for solving sparse linear programming

problems. International Journal for Numerical Methods in Engineering, 26(12):2671–

2685.

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Tang, C., Toh, K.C., and Phoon, K.K. 2014. Axisymmetric lower-bound limit analysis using

finite elements and second-order cone programming. Journal of Engineering

Mechanics, 140(2): 268-278.

Vesic, A.S. 1971. Breakout resistance of objects embedded in ocean bottom. J. Mech. Found.

Div., ASCE, 97(9): 1183-1205.

Zouain, N., Herskovits, J., Borges, L.A. and Feijóo, R.A. 1993. An iterative algorithm for limit

analysis with nonlinear yield functions. International Journal of Solids and Structures,

30(10):1397–1417.

Figure captions

Fig.1. (a) Problem definition, boundary conditions, and sign conventions for e and α; (b) two

different loading types.

Fig. 2. Typical meshes for (a) � = 2 ; (b) � = 4. Fig. 3. The variation of � with e/B for different values of α/φ with two different loading types for � = 2, ( = 30 and - = (. Fig. 4. The variation of � with e/B for different values of φ with δ = φ for (a) � = 1, � (⁄ = 0 & 1/3; (b)� = 1, � (⁄ = 1/2 & 2/3; (c) � = 2, � (⁄ = 0 & 1/3 ; (d) � = 2, � (⁄ = 1/2 & 2/3. Fig. 5. The variation of � with e/B for different values of φ with δ = φ for (a) � = 3, � (⁄ = 0 & 1/3; (b)� = 3, � (⁄ = 1/2 & 2/3; (c) � = 4, � (⁄ = 0 & 1/3 ; (d) � = 4, � (⁄ = 1/2 & 2/3. Fig. 6. The variation of � with e/B for different values of φ with δ = φ for (a) � = 5, � (⁄ = 0 & 1/3; (b)� = 5, � (⁄ = 1/2 & 2/3; (c) � = 6, � (⁄ = 0 & 1/3 ; (d) � = 6, � (⁄ = 1/2 & 2/3. Fig. 7. The variation of � with δ for � = 1, 2 and 3 with ( = 30o for (a) e/B =0 and α/φ =0,1/3, 1/2 & 2/3; (b) α/φ =0 and e/B = 0, 0.1, 0.2 & 0.3.

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Fig. 8. Failure patterns for � = 2 , ( = 30o with (a)-/( = 1, � � =⁄ 0, � ( =⁄ 0 ; (b) -/( = 1, � � =⁄ 0, � ( =⁄ 0.5; (c)-/( = 1, � � =⁄ 0.2, � ( =⁄ 0; (d)-/( = 1, � � =⁄ 0.2, � ( =⁄ 0.5; (e) -/( = 1, � � =⁄ 0.2, � ( =⁄ -0.5 ; (f)-/( = 0, � � =⁄ 0, � ( =⁄ 0. Fig. 9. A comparison of the variation of Fγ with λ from different approaches for e/B =α/φ = 0

with Type I loading condition.

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Fig.1. (a) Problem definition, boundary conditions, and sign conventions for e and ; (b) two

different loading types.

(a)

(b)

P

Q

S

R

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Fig. 2. Typical meshes for (a) 𝜆 = 2 ; (b) 𝜆 = 4.

(a)

(b)

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Fig. 3. The variation of 𝐹𝛾 with e/B for different values of /with two

different loading types for 𝜆 = 2, 𝜙 = 30 and 𝛿 = 𝜙.

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Fig. 4. The variation of 𝐹𝛾 with e/B for different values of with = for (a) 𝜆 = 1, 𝛼 𝜙⁄ = 0 &

1/3; (b) 𝜆 = 1, 𝛼 𝜙⁄ = 1/2 & 2/3; (c) 𝜆 = 2, 𝛼 𝜙⁄ = 0 & 1/3 ; (d) 𝜆 = 2, 𝛼 𝜙⁄ = 1/2 & 2/3.

(a) (b)

(c) (d)

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Fig. 5. The variation of ( with e/B for different values of I with G = I for (a) ã L u, Ù ö¤ L 0 &

1/3; (b)�ã L u, Ù ö¤ L 1/2 & 2/3; (c) ã L 4, Ù ö¤ L 0 & 1/3 ; (d) ã L 4, Ù ö¤ L 1/2 & 2/3.

(a) (b)

(c) (d)

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Fig. 6. The variation of 𝐹𝛾 with e/B for different values of with = for (a) 𝜆 = 5, 𝛼 𝜙⁄ = 0 &

1/3; (b) 𝜆 = 5, 𝛼 𝜙⁄ = 1/2 & 2/3; (c) 𝜆 = 6, 𝛼 𝜙⁄ = 0 & 1/3 ; (d) 𝜆 = 6, 𝛼 𝜙⁄ = 1/2 & 2/3.

(a) (b)

(c) (d)

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Fig. 7. The variation of 𝐹𝛾 with for 𝜆 = 1, 2 and 3 with 𝜙 = 30o for (a) e/B =0 and

/=0,1/3, 1/2 & 2/3; (b) /=0 and e/B = 0, 0.1, 0.2 & 0.3.

(a)

(b)

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Fig. 8. Failure patterns for 𝜆 = 2 , 𝜙 = 30o with (a) 𝛿/𝜙 = 1, 𝑒 𝐵 =⁄ 0, 𝛼 𝜙 =⁄ 0 ; (b) 𝛿/𝜙 = 1,

𝑒 𝐵 =⁄ 0, 𝛼 𝜙 =⁄ 0.5; (c) 𝛿/𝜙 = 1, 𝑒 𝐵 =⁄ 0.2, 𝛼 𝜙 =⁄ 0; (d) 𝛿/𝜙 = 1, 𝑒 𝐵 =⁄ 0.2, 𝛼 𝜙 =⁄

0.5; (e) 𝛿/𝜙 = 1, 𝑒 𝐵 =⁄ 0.2, 𝛼 𝜙 =⁄ -0.5 ; (f) 𝛿/𝜙 = 0, 𝑒 𝐵 =⁄ 0, 𝛼 𝜙 =⁄ 0.

(a) (b)

(c) (d)

(e) (f)

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Fig. 9. A comparison of the variation of F with from different approaches for

e/B =/= 0 with Type I loading condition.

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canadian geotechnical journal - university of toronto t-space · 2018. 12. 7. · rowe and davis...

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