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Proceedings of the XVI ECSMGEGeotechnical Engineering for Infrastructure and DevelopmentISBN 978-0-7277-6067-8

© The authors and ICE Publishing: All rights reserved, 2015doi:10.1680/ecsmge.60678

Figure 4 shows that the bearing capacity significantly increases due to suction developed within the soil foundation. The numerical predictions of the bearing capacity are compared with that obtained by the equation proposed by Vanapalli & Mohamed (2007). The difference between the two solutions becomes larger for large suctions. It may be the result of the different stress state frameworks thus different model parameters adopted in the two methods.

4 CONCLUSION

A finite element model for fully coupled flow-deformation analysis of unsaturated soil is presented. The parameters required in this model are significantly reduced for the application of the effective stress principle, compared to the concept of two independent state variables. It is found that the results predicted by this model are in a reasonable agreement with the experimental tests data on samples of unsaturated soil. The finite element model is also applied to the bearing capacity problems for both saturated and unsaturated soils. Good agreement is obtained between the numerical predictions and theoretical solutions. Overall, this study provides a useful tool to predict the bearing capacity of footings on unsaturated soil.

REFERENCES

Bishop, A.W. 1959. The principle of effective stress, Teknisk Ukeblad 106(39), 859-863. Bishop, A.W. & Donald, I.B. 1961. The experimental study of party saturated soil in the triaxial apparatus. Proceedings, 5th Int. Conf. On Soil Mechanics and Found Eng, 13-21. Carter J.P. & Balaam N.P. 1995. AFENA users’ manual. University of Sydney, Australia. Centre for Geotechnical Research. Fredlund, D.G. & Rahardjo, H. 1986 Unsaturated Soil Consolidation Theory and Laboratory Experimental Data. ASTM: Fort Lauderdale, FL, USA. Gatmiri, B. & Delage, P. 1995. A new void ratio state surface formulation for the non-linear elastic constitutive modelling of unsaturated soil: Code U-Dam. Proceedings, 1st International Conference on Unsaturated Soils, 745-750. Paris, France. Griffiths, D.V. 1982. Computation of bearing capacity factors using finite elements, Geotechnique 32,195-202.

Hansen, B. & Christiensen, N.H. 1969. Discussion of theoretical bearing capacity of very shallow footings, J. Soil Mech. Fdns Div., ASCE 95, 1568-1572. Hill, R. 1950. The mathematical theory of plasticity. Oxford: Clarendon. Khabbaz, M.H. 1997. An effective stress based numerical model for flow and deformation in unsaturated soils. University of New South Wales, Sydney, Australia. Khalili, N. Habte, M. & Zargargashi, S. 2008. A fully coupled flow deformation model for cyclic analysis of unsaturated soils including hydraulic and mechanical hysteresis, Computers and Geotechnics 35(6), 872-889. Khalili, N. & Khabbaz, M.H. 1998. A unique relationship for the determination of the shear strength of unsaturated soils, Geotechnique 48(5), 681-687. Khalili, N. Khabbaz, M.H.& Valliappan, S. 2000. An effective stress based numerical model for hydro-mechanical analysis in unsaturated porous media, Computational Mechanics 26, 174-184. Khalili, N. & Zargarbashi, S. 2010. Influence of hydraulic hysteresis on effective stress in unsaturated soils, Geotechnique 60, 729–734. Khoshghalb, A. & Khalili, N. 2013. A meshfree method for fully coupled analysis of flow and deformation in unsaturated porous media, Int J Numer Anal Methods Geomech 37(7), 716-743. Lloret, A. & Alonso, E.E. 1980. Consolidation of unsaturated soils including swelling and collapse behaviour, Geotechnique 30(4), 449-477. Meyerhof, G.G. 1953. The bearing capacity of foundations under eccentric and inclined loads. Proceedings, 3rd International Conference on Soil Mechanics and Foundation Engineering, 440-445. Zurich. Oh, W.T. & Vanapalli, S.K. 2008. Modelling the stress versus settlement behaviour of model footings in saturated and unsaturated sandy soils. Proceedings, 12th International Conference of IACMAG, 2126-2137. Goa, India. Oh, W.T. & Vanapalli, S.K. 2011. Modeling the applied vertical stress and settlement relationship of shallow foundations in saturated and unsaturated sands, Canadian Geotechnical Journal 48, 425-438. Oloo, S.Y. Fredlund, D.G. & Gan, J.K-M. 1997. Bearing capacity of unpaved roads, Canadian Geotechnical Journal 34, 398-407. Schrefler, B.A. & Xiaoyong Z. 1993. A fully coupled model for water flow and air flow in deformable porous media, Water Resource Research 29(1), 155-167. Sheng, D. Sloan, S.W. Gens, A. & Smith, D.W. 2003. Finite element formulation and algorithms for unsaturated soils. Part I: theory, Int J Numer Anal Methods Geomech 27(9), 745-765. Terzaghi, K. 1943. Theoretical soil mechanics, John Wiley, New York. Vanapalli, S.K. & Mohamed, F.M.O. 2007. Bearing capacity of model footings in unsaturated soils. Experimental Unsaturated Soil mechanics, Springer, New York, 483-493. Vesic, A.S. 1973. Analysis of ultimate loads of shallow foundations, Journal of Soil Mechanics and Foundations Division 99(1), 45-73. Wheeler, S.J. & Sivakumar, V. 1995. An elasto-plastic critical state framework for unsaturated soil, Geotechnique 45, 35-54.

Impact of anchor failure for deep excavation in soft clay based on 3D FEM analyses

Modélisation 3D par éléments finis de l'effet de la rupture d'ancrages dans des excavations profondes en argiles molles

Ø. Torgersrud*1, H.P Jostad1, K. Karlsrud1 and S. Kalstad1

1 Norwegian Geotechnical Institute, Oslo, Norway* Corresponding author

ABSTRACT The paper investigates 3D effects related to a deep excavation in soft clay supported by sheet pile walls and three levels of tieback anchors using a 3D finite-element program. The focus is on the impact of failure of an anchor and shows how earth pressures and structural forces are redistributed as a result of that. In Norwegian design practice anchor failure is commonly considered as an accidental limit state (ALS) case, but accounts for some redistribution of loads as a result of the failure. The numerical 3D analyses were performed with the Plaxis 3D finite element program. The NGI-ADP constitutive model, which is an elasto-plastic total stress based material model with anisotropic shear strengths and hardening functions, is used to describe the clay behaviour. The results of the analyses are presented in terms of soil and wall deformations, lateral earth pressure and loads in the supporting structure. The results shows that for the considered example, the load redistribution effects assumed in current design are considerably on the conservative side.

RÉSUMÉ Cet article présente l'analyse des effets tridimensionnels dans une excavation profonde en argile molle retenue par des pal-planches et trois niveaux d'ancrage au moyen d'un code 3D par éléments finis (MEF). L'étude se concentre en particulier sur l'effet de la rupture d'un ancrage et montre comment les pressions des terres ainsi que les forces sur les structures se redistribuent en conséquence. En Norvège, dans la pratique, la rupture d'un ancrage est généralement considérée comme un état limite accidentel (ou ALS, Accidental Limit State), mais prend en compte une possible redistribution des charges. Les analyses numériques ont été réalisées au moyen du code par élé-ments finis Plaxis 3D. Le comportement des argiles est décrit au moyen du modèle NGI-ADP, une loi de comportement basée sur les con-traintes totales élastoplastiques, prenant en compte la résistance au cisaillement et le durcissement. Les résultats de l'étude sont présentés en termes de déformations du sol et du mur de soutènement, des pressions des terres latérales ainsi que des charges sur les structures de soutè-nement. Les résultats indiquent que dans le cas considéré, la redistribution des charges prise en compte dans les calculs de design est lar-gement conservatrice.

1 INTRODUCTION

3D finite element analyses (FEA) of an excavation in soft clay supported by sheet piles and tie-back an-chors have been studied to evaluate the redistribution of forces due to failure of one of the anchors. A rec-tangular excavation is analysed and anchor failure at the centre of the excavation or near the corner is ex-amined to evaluate potential corner stiffening effects (3D effects). Some past studies on 3D effects in con-nection with supported excavations were presented by e.g.: Ou et al., 1996; Zdrakovic et al., 2005; Finno

et al. 2007; Karlsrud and Andresen (2008), but none of these have addressed failure of an anchor or strut.

1.1 Accidental limit state

Possible anchor failure is an issue often considered as an accidental limit state (ALS) loading case in com-mon Norwegian practice. The sheet pile wall should be stable for this condition without exceeding the de-sign capacity of the adjacent anchors or the plastic moment resistance of the wale beams. When one an-chor fails, the sheet pile wall will deflect locally around this anchor location and the force will partly

Geotechnical Engineering for Infrastructure and Development

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be redistributed through the wale beams to adjacent anchors. Due to the local wall deflection there will be an arching effect in both the vertical and horizontaldirection, which locally reduces the lateral earth pressure (Stille, 1976). The magnitude of the redis-tributed anchor forces and moments in the walls and wale beams depend on the change in lateral earth pressure against the wall.

In current Norwegian design practice possible an-chor failure is normally dealt with as follows (Nor-wegian Public Roads Administration, 2010):

1) Anchor forces: The ALS design load on the an-chor adjacent to the failing anchor is calculated by applying a form-factor on the SLS design value, Kformsuch that Rd,ALS = Rd,SLS · Kform. Kform is suggested to be in the range of 1.2 to 1.5, depending on the loca-tion of the failing anchor and the geometry of the ex-cavation, leaving the design-engineer to evaluate the appropriate value within the suggested range.

2) Wale beam moments: The ALS design moment is calculated as Md,ALS = qd,ALS · l2/16, where qd,ALS is a design strip load on the beam and l is two times the original anchor spacing. In order to account for the reduction of lateral earth pressure due to arching ef-fects a reduction factor is applied on the characteris-tic load: qd,ALS = λred · qd,SLS. λred is suggested to lie in the range of 0,7 to 1,0 depending on the mobilised shear stresses in the soil for the SLS-case and the ge-ometry of the retaining wall.

2 CASE STUDY

An excavation in a typical normally consolidated Os-lo clay was chosen for a case study. The rectangular excavation measures 112 m x 56 m in plan, and 9.7 m deep. The support system consists of an AZ 37-700 sheet pile wall (SPW) going down to bedrock at 18.5 m depth, and three levels of wale beams and tie-back anchors A-C down to bedrock. The anchors are set at 3.5 m horizontal spacing. The excavation was performed sequentially with successive installationand pre-stressing of the anchors as described in Table 1.

Figure 1 shows a cross section of the geometry and material parameters considered in the study. The soil profile comprises a layer of fill and dry crust down to 2.8 m depth, and soft clay below. The ground water head is initially located at 2.8 m depth

and is gradually lowered inside the excavation for each excavation step. A surface load of 13 kPa was acting on all sides around the excavation.

Figure 1. Illustration of the example problem.

Table 1. Construction phases.

Stage number

Description

1 Install sheet pile wall (wished-in-place in FE-model)2 Excavate to -1.0 m *3 Install Level A wale beams and anchors at -1.0 m,

pre-stress anchors to 764 kN / anchor4 Excavate to -2.8 m *5 Install Level B wale beams and anchors at -2.8m,

pre-stress anchors to 1104 kN / anchor6 Excavate to -6.3 m *7 Install Level C wale beams and anchors at -6.3 m,

pre-stress anchors to 1444 kN / anchor8 Excavate to final depth -9.7 m *

*lowering ground water inside the excavation to the exc. depth

3 FINITE ELEMENT MODELS

Both 2D plane strain and 3D finite element analyses (FEA) are performed in this study, using the com-mercial finite element programs Plaxis 2D AE and Plaxis 3D 2012.01 (www.plaxis.nl).

The 3D model of the rectangular excavation is shown in Figure 2. Due to symmetry a quarter of the excavation was modelled. The model consists of

about 85 000 10-noded tetrahedral soil elements. A mesh sensitivity study was performed to ensure that any error due to element discretisation was negligi-ble.

The 2D plane strain model consists of 685 15-noded triangular soil elements. This model was used to compare the results from the 3D analyses with plane strain conditions.

Figure 2. Perspective view of 3D FE model at final excavation depth

3.1 Modelling of soil and structure (base case)

The dry crust was modelled as a drained material us-ing the Hardening Soil model (Schanz et al. 1999). A cohesion c = 0 kPa, friction angle φ = 30°, reference shear modulus G50

ref =12 500 kPa and reference un-loading-reloading modulus Gur

ref = 3·G50ref was used.

The shear modulus is stress dependent with the refer-ence value at 50% shear mobilisation and a reference stress p' = 100 kPa. The coefficient of the lateral earth pressure at rest (Ko') is estimated from Jaky's equation Ko' = 1 – sin φ.

The clay layer was modelled as an undrained ma-terial using the material model called NGI-ADP(Grimstad et al., 2011). This is a strain hardening elasto-plastic model with total stress path dependent anisotropic undrained shear stress – strain curves.The input to the model is the spatial distribution of the undrained active shear strength su

A, the anisotro-py strength ratios su

DSS/suA (direct simple shear, DSS)

and suP/su

A (passive), the corresponding shear strains at failure, γf

A, γfDSS and γf

P, and initial elastic shear modulus ratio, Go/su

A. These parameters are obtained from anisotropically consolidated undrained triaxial compression and extension tests and direct simple shear (DSS) tests. Table 2 shows the input parame-ters used in the analyses.

The SPW was modelled with 6-node plate ele-ments and an anisotropic linear elastic material. An equivalent thickness d and Young's modulus E1 was used to give the correct bending stiffness about the horizontal axis. To simulate the low bending stiffness of the wall about the vertical axis, the stiffness was specified as E2/E1 = 1/20 in this direction. The stiff-ness parameters are summarised in Table 3. Full fric-tion in the idealised vertical interface plane between the soil and the SPW was assumed, and a moment free (hinged) connection was assumed in the corner and against bedrock.

The wale beams were modelled as beam elements with anisotropic linear elastic bending stiffness, ori-ented with the strong axis 45° to the horizontal plane. A moment free connection in the corner was mod-elled by using short dummy beams with low bending stiffness, EI, near the corner. The tie-back anchors were model as axial springs. Table 4 summarises the stiffness parameters for the beams and anchors.

Table 2. Soil parameters used in NGI-ADP model.

Parameter Value Unitsu

A 9 - 2.9z * kPasu

DSS/suA 0.7 -

suP/su

A 0.4 -τo/su

A 0.7 -Go/su

A 500 -γf

A 1.0 %γf

DSS 2.0 %γf

P 3.0 %γ (unit weight) 18.5 kN/m3

ν (Poisson ratio) 0.495 -Ko' 0.7*z defined negative downwards

Table 3. Sheet pile wall parameters.

Parameter ValueYoung's modulus, E1 (kPa) 1.87E7Young's modulus, E2 (kPa) 9.37E5Shear modulus, G12 (kPa) 9.37E5Shear modulus, G13 (kPa) 1.59E6Shear modulus, G23 (kPa) 4.76E5Poisson ratio, ν 0Equivalent thickness, d (m) 0.5

Table 4. Wale beam and anchor parameters.

Parameter Support levelA B C

Wale beamsAxial stiffness, EA (kN) 3.05E6 4.15E6 5.63E6Bending stiffness EI3 (kNm2) 6.83E4 1.42E5 2.82E5Bending stiffness EI2 (kNm2) 4.37E3 7.06E3 1.12E4

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be redistributed through the wale beams to adjacent anchors. Due to the local wall deflection there will be an arching effect in both the vertical and horizontaldirection, which locally reduces the lateral earth pressure (Stille, 1976). The magnitude of the redis-tributed anchor forces and moments in the walls and wale beams depend on the change in lateral earth pressure against the wall.

In current Norwegian design practice possible an-chor failure is normally dealt with as follows (Nor-wegian Public Roads Administration, 2010):

1) Anchor forces: The ALS design load on the an-chor adjacent to the failing anchor is calculated by applying a form-factor on the SLS design value, Kformsuch that Rd,ALS = Rd,SLS · Kform. Kform is suggested to be in the range of 1.2 to 1.5, depending on the loca-tion of the failing anchor and the geometry of the ex-cavation, leaving the design-engineer to evaluate the appropriate value within the suggested range.

2) Wale beam moments: The ALS design moment is calculated as Md,ALS = qd,ALS · l2/16, where qd,ALS is a design strip load on the beam and l is two times the original anchor spacing. In order to account for the reduction of lateral earth pressure due to arching ef-fects a reduction factor is applied on the characteris-tic load: qd,ALS = λred · qd,SLS. λred is suggested to lie in the range of 0,7 to 1,0 depending on the mobilised shear stresses in the soil for the SLS-case and the ge-ometry of the retaining wall.

2 CASE STUDY

An excavation in a typical normally consolidated Os-lo clay was chosen for a case study. The rectangular excavation measures 112 m x 56 m in plan, and 9.7 m deep. The support system consists of an AZ 37-700 sheet pile wall (SPW) going down to bedrock at 18.5 m depth, and three levels of wale beams and tie-back anchors A-C down to bedrock. The anchors are set at 3.5 m horizontal spacing. The excavation was performed sequentially with successive installationand pre-stressing of the anchors as described in Table 1.

Figure 1 shows a cross section of the geometry and material parameters considered in the study. The soil profile comprises a layer of fill and dry crust down to 2.8 m depth, and soft clay below. The ground water head is initially located at 2.8 m depth

and is gradually lowered inside the excavation for each excavation step. A surface load of 13 kPa was acting on all sides around the excavation.

Figure 1. Illustration of the example problem.

Table 1. Construction phases.

Stage number

Description

1 Install sheet pile wall (wished-in-place in FE-model)2 Excavate to -1.0 m *3 Install Level A wale beams and anchors at -1.0 m,

pre-stress anchors to 764 kN / anchor4 Excavate to -2.8 m *5 Install Level B wale beams and anchors at -2.8m,

pre-stress anchors to 1104 kN / anchor6 Excavate to -6.3 m *7 Install Level C wale beams and anchors at -6.3 m,

pre-stress anchors to 1444 kN / anchor8 Excavate to final depth -9.7 m *

*lowering ground water inside the excavation to the exc. depth

3 FINITE ELEMENT MODELS

Both 2D plane strain and 3D finite element analyses (FEA) are performed in this study, using the com-mercial finite element programs Plaxis 2D AE and Plaxis 3D 2012.01 (www.plaxis.nl).

The 3D model of the rectangular excavation is shown in Figure 2. Due to symmetry a quarter of the excavation was modelled. The model consists of

about 85 000 10-noded tetrahedral soil elements. A mesh sensitivity study was performed to ensure that any error due to element discretisation was negligi-ble.

The 2D plane strain model consists of 685 15-noded triangular soil elements. This model was used to compare the results from the 3D analyses with plane strain conditions.

Figure 2. Perspective view of 3D FE model at final excavation depth

3.1 Modelling of soil and structure (base case)

The dry crust was modelled as a drained material us-ing the Hardening Soil model (Schanz et al. 1999). A cohesion c = 0 kPa, friction angle φ = 30°, reference shear modulus G50

ref =12 500 kPa and reference un-loading-reloading modulus Gur

ref = 3·G50ref was used.

The shear modulus is stress dependent with the refer-ence value at 50% shear mobilisation and a reference stress p' = 100 kPa. The coefficient of the lateral earth pressure at rest (Ko') is estimated from Jaky's equation Ko' = 1 – sin φ.

The clay layer was modelled as an undrained ma-terial using the material model called NGI-ADP(Grimstad et al., 2011). This is a strain hardening elasto-plastic model with total stress path dependent anisotropic undrained shear stress – strain curves.The input to the model is the spatial distribution of the undrained active shear strength su

A, the anisotro-py strength ratios su

DSS/suA (direct simple shear, DSS)

and suP/su

A (passive), the corresponding shear strains at failure, γf

A, γfDSS and γf

P, and initial elastic shear modulus ratio, Go/su

A. These parameters are obtained from anisotropically consolidated undrained triaxial compression and extension tests and direct simple shear (DSS) tests. Table 2 shows the input parame-ters used in the analyses.

The SPW was modelled with 6-node plate ele-ments and an anisotropic linear elastic material. An equivalent thickness d and Young's modulus E1 was used to give the correct bending stiffness about the horizontal axis. To simulate the low bending stiffness of the wall about the vertical axis, the stiffness was specified as E2/E1 = 1/20 in this direction. The stiff-ness parameters are summarised in Table 3. Full fric-tion in the idealised vertical interface plane between the soil and the SPW was assumed, and a moment free (hinged) connection was assumed in the corner and against bedrock.

The wale beams were modelled as beam elements with anisotropic linear elastic bending stiffness, ori-ented with the strong axis 45° to the horizontal plane. A moment free connection in the corner was mod-elled by using short dummy beams with low bending stiffness, EI, near the corner. The tie-back anchors were model as axial springs. Table 4 summarises the stiffness parameters for the beams and anchors.

Table 2. Soil parameters used in NGI-ADP model.

Parameter Value Unitsu

A 9 - 2.9z * kPasu

DSS/suA 0.7 -

suP/su

A 0.4 -τo/su

A 0.7 -Go/su

A 500 -γf

A 1.0 %γf

DSS 2.0 %γf

P 3.0 %γ (unit weight) 18.5 kN/m3

ν (Poisson ratio) 0.495 -Ko' 0.7*z defined negative downwards

Table 3. Sheet pile wall parameters.

Parameter ValueYoung's modulus, E1 (kPa) 1.87E7Young's modulus, E2 (kPa) 9.37E5Shear modulus, G12 (kPa) 9.37E5Shear modulus, G13 (kPa) 1.59E6Shear modulus, G23 (kPa) 4.76E5Poisson ratio, ν 0Equivalent thickness, d (m) 0.5

Table 4. Wale beam and anchor parameters.

Parameter Support levelA B C

Wale beamsAxial stiffness, EA (kN) 3.05E6 4.15E6 5.63E6Bending stiffness EI3 (kNm2) 6.83E4 1.42E5 2.82E5Bending stiffness EI2 (kNm2) 4.37E3 7.06E3 1.12E4

Torgersrud, Jostad, Karlsrud and Kalstad

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AnchorsAxial stiffness, EA/L (kN/m) 11.4E3 16.6E3 27.8E3Pre-stress, tension (kN) 764 1104 1444

3.2 Solution procedure

The Ko' procedure was used to get the initial stress state. Then the analysis follows the sequence of staged construction shown in Table 1.

The ALS condition is simulated by deactivating one anchor in the model, either at the centre line orthe one being closest to the corner. This procedure was followed for each level of support, removing an-chors at level A after stage 4, level B after stage 6 and level C after stage 8.

3.3 Parametric studies

A set of 12 analyses were performed, where strength and stiffness properties of the soil and stiffness of the support system were varied. Table 5 summarises the input to the parametric studies, with the varied pa-rameter in each case in bold letters. Case A1 is the base case. For cases A2 to A7 the clay strength and stiffness were varied. The values are varied within typical ranges representative for Oslo clays. Analysis A8 is for interface roughness of 0.6, compared to 1.0 for the base case. Analysis B1 was made with a "soft" wall, by reducing the bending stiffness of the wale beams and the bending stiffness of the wall in the horizontal direction by a factor of 10. Similarly, B2 represents a "stiff" wall by increasing the same properties with a factor 10. In analysis B3, moment fixity was introduced in the corner, while in B4 the anchors were not pre-stressed, giving a behavioursimilar to an internally braced excavation.

4 RESULTS

Figure 3 presents horizontal wall displacements and bending moments after the last excavation step for the base case for two locations: one section at the centreline of the excavation and one near the corner,at the location of the inner anchors. It can be seen that the response at the centre of the wall gives close to perfect match with the plane strain model. This agrees with the findings in Finno et al. (2007), who concludes that a plane strain condition is relevantnear the centre when the length to height ratio of the excavation is greater than 6. Both displacements and moments are significantly lower close to the corner due to 3D effects, which also is found from field data reported in literature and in FE-studies (Zdrakovic et al., 2005; Ou et al., 1996).

Figure 3. Wall displacement and moment at final excavation depth for the base case (A1)

0.08 0.04 0 -0.04Wall displacement, δh (m)

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

Dep

th (m

)

800 600 400 200 0 -200Wall moment (kNm)

Table 5. Overview of parametric study.

suA

(kPa)Go/su

A γf Roughness SPW E2/E2,(A1)

Beam EI/EI(A1)

Corner fixity AnchorPre-stress

A1 9-2.9z 500 (1,2,3) 1 1 1 hinged yesA2 9-2.9z 250 (1,2,3) 1 1 1 hinged yesA3 9-2.9z 1000 (1,2,3) 1 1 1 hinged yesA4 7.6-2.4z 500 (1,2,3) 1 1 1 hinged yesA5 7.6-2.4z 250 (1,2,3) 1 1 1 hinged yesA6 7.6-2.4z 1000 (1,2,3) 1 1 1 hinged yesA7 9-2.9z 500 (2,4,6) 1 1 1 hinged yesA8 9-2.9z 500 (1,2,3) 0.6 1 1 hinged yesB1 9-2.9z 500 (1,2,3) 1 0.1 0.1 hinged yesB2 9-2.9z 500 (1,2,3) 1 10 10 hinged yesB3 9-2.9z 500 (1,2,3) 1 1 1 fixed yesB4 9-2.9z 500 (1,2,3) 1 1 1 hinged no

Figure 4(a) shows the lateral earth pressure at the centre of the wall for the case with all anchors intact compared with the situation where one anchor fails on level C. A significant reduction in the active earth pressure close to the location of the failed anchor is observed. Figure 4(b), which presents contours of the change in active earth pressure against the wall,shows also arching effects in the horizontal direction. This effect is rather local and almost limited to the adjacent anchor.

Figure 5 shows the increase in axial force in the remaining anchors when one anchor fails at one of the levels in the base case model. The vertical axis denotes the anchor force ratio, which is defined as the relative change in anchor force. The nearest an-chor on the same level, or the anchor directly above gets the largest relative increase, for the base case about 6%. No significant difference was seen be-tween a failure at the centre line or near the corner, however the absolute anchor forces near the corner are in general lower than at the centre of the excava-tion.

Figure 6 shows the bending moment in the wale beams before and after an anchor failure. The highest moment occurs at the anchor supports and is about twice the magnitude of the mid-span moments, ex-cept towards the corner where the mid-span moment increase due to the lack of corner fixity. When one anchor fails, the moment at the nearest support in-creases as expected, and for the base case the highest increase is about 40%. It is also seen that the mid-span moment increases significantly for an anchor failure near the corner, but is in this case not exceed-ing the support moment. By reducing the distance be-tween the first anchor and the corner, this effect will be reduced.

4.1 Parametric study

Figure 7 shows the results of the parametric study in terms of: (a) maximum anchor force ratio and (b) maximum beam moment ratio. The results are not very sensitive to the parametric variations, except for the case with no pre-stressing of the anchors where a higher relative increase in anchor force is observed. However, the absolute magnitude of the anchor forc-es are lower for this case than for the others.

The study shows that an anchor failure has a high-er relative impact on the wale beam bending mo-

ments than on anchor forces. As expected, the wale beam moments are highly sensitive to change in bending stiffness (Case B1 and B2) in addition to the case with no pre-stressing of the anchors (B4). These three cases are not shown in the Figure 7(b) due to large variations of the beam moment ratio.

Figure 4. (a) Lateral earth pressure before and after anchor failurefor the final excavation depth and base case model (A1). (b) Con-tours of the change in active earth pressure on the wall before and after anchor failure.

Figure 5. Anchor force ratio showing relative increase in anchor forces due to anchor failure (base case model, A1)

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

-300 -200 -100 0 100 200 300

σv - 2.3 suA

σv + 2.3 suP

Distance from centre line (m)

(a) (b)

0-10

1

1.02

1.04

1.06

1.08Anchor failure Level Aat x=0 m (Centre)

1

1.02

1.04

1.06

1.08

Anc

hor f

orce

ratio

Anchor failure Level Bat x=0 m (Centre)

0 3.5 7 10.5 14Position along wall, x (m)

1

1.02

1.04

1.06

1.08

Level A anchors Level B anchors Level C anchors

Anchor failure Level Cat x=0m (Centre)

1

1.02

1.04

1.06

1.08Anchor failure Level Aat x=52.5 m (Corner)

1

1.02

1.04

1.06

1.08Anchor failure Level Bat x=52.5 m (Corner)

38.5 42 45.5 49 52.5 56Position along wall, x (m)

1

1.02

1.04

1.06

1.08Anchor failure Level Cat x=52.5 m (Corner)

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AnchorsAxial stiffness, EA/L (kN/m) 11.4E3 16.6E3 27.8E3Pre-stress, tension (kN) 764 1104 1444

3.2 Solution procedure

The Ko' procedure was used to get the initial stress state. Then the analysis follows the sequence of staged construction shown in Table 1.

The ALS condition is simulated by deactivating one anchor in the model, either at the centre line orthe one being closest to the corner. This procedure was followed for each level of support, removing an-chors at level A after stage 4, level B after stage 6 and level C after stage 8.

3.3 Parametric studies

A set of 12 analyses were performed, where strength and stiffness properties of the soil and stiffness of the support system were varied. Table 5 summarises the input to the parametric studies, with the varied pa-rameter in each case in bold letters. Case A1 is the base case. For cases A2 to A7 the clay strength and stiffness were varied. The values are varied within typical ranges representative for Oslo clays. Analysis A8 is for interface roughness of 0.6, compared to 1.0 for the base case. Analysis B1 was made with a "soft" wall, by reducing the bending stiffness of the wale beams and the bending stiffness of the wall in the horizontal direction by a factor of 10. Similarly, B2 represents a "stiff" wall by increasing the same properties with a factor 10. In analysis B3, moment fixity was introduced in the corner, while in B4 the anchors were not pre-stressed, giving a behavioursimilar to an internally braced excavation.

4 RESULTS

Figure 3 presents horizontal wall displacements and bending moments after the last excavation step for the base case for two locations: one section at the centreline of the excavation and one near the corner,at the location of the inner anchors. It can be seen that the response at the centre of the wall gives close to perfect match with the plane strain model. This agrees with the findings in Finno et al. (2007), who concludes that a plane strain condition is relevantnear the centre when the length to height ratio of the excavation is greater than 6. Both displacements and moments are significantly lower close to the corner due to 3D effects, which also is found from field data reported in literature and in FE-studies (Zdrakovic et al., 2005; Ou et al., 1996).

Figure 3. Wall displacement and moment at final excavation depth for the base case (A1)

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suA

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Beam EI/EI(A1)

Corner fixity AnchorPre-stress

A1 9-2.9z 500 (1,2,3) 1 1 1 hinged yesA2 9-2.9z 250 (1,2,3) 1 1 1 hinged yesA3 9-2.9z 1000 (1,2,3) 1 1 1 hinged yesA4 7.6-2.4z 500 (1,2,3) 1 1 1 hinged yesA5 7.6-2.4z 250 (1,2,3) 1 1 1 hinged yesA6 7.6-2.4z 1000 (1,2,3) 1 1 1 hinged yesA7 9-2.9z 500 (2,4,6) 1 1 1 hinged yesA8 9-2.9z 500 (1,2,3) 0.6 1 1 hinged yesB1 9-2.9z 500 (1,2,3) 1 0.1 0.1 hinged yesB2 9-2.9z 500 (1,2,3) 1 10 10 hinged yesB3 9-2.9z 500 (1,2,3) 1 1 1 fixed yesB4 9-2.9z 500 (1,2,3) 1 1 1 hinged no

Figure 4(a) shows the lateral earth pressure at the centre of the wall for the case with all anchors intact compared with the situation where one anchor fails on level C. A significant reduction in the active earth pressure close to the location of the failed anchor is observed. Figure 4(b), which presents contours of the change in active earth pressure against the wall,shows also arching effects in the horizontal direction. This effect is rather local and almost limited to the adjacent anchor.

Figure 5 shows the increase in axial force in the remaining anchors when one anchor fails at one of the levels in the base case model. The vertical axis denotes the anchor force ratio, which is defined as the relative change in anchor force. The nearest an-chor on the same level, or the anchor directly above gets the largest relative increase, for the base case about 6%. No significant difference was seen be-tween a failure at the centre line or near the corner, however the absolute anchor forces near the corner are in general lower than at the centre of the excava-tion.

Figure 6 shows the bending moment in the wale beams before and after an anchor failure. The highest moment occurs at the anchor supports and is about twice the magnitude of the mid-span moments, ex-cept towards the corner where the mid-span moment increase due to the lack of corner fixity. When one anchor fails, the moment at the nearest support in-creases as expected, and for the base case the highest increase is about 40%. It is also seen that the mid-span moment increases significantly for an anchor failure near the corner, but is in this case not exceed-ing the support moment. By reducing the distance be-tween the first anchor and the corner, this effect will be reduced.

4.1 Parametric study

Figure 7 shows the results of the parametric study in terms of: (a) maximum anchor force ratio and (b) maximum beam moment ratio. The results are not very sensitive to the parametric variations, except for the case with no pre-stressing of the anchors where a higher relative increase in anchor force is observed. However, the absolute magnitude of the anchor forc-es are lower for this case than for the others.

The study shows that an anchor failure has a high-er relative impact on the wale beam bending mo-

ments than on anchor forces. As expected, the wale beam moments are highly sensitive to change in bending stiffness (Case B1 and B2) in addition to the case with no pre-stressing of the anchors (B4). These three cases are not shown in the Figure 7(b) due to large variations of the beam moment ratio.

Figure 4. (a) Lateral earth pressure before and after anchor failurefor the final excavation depth and base case model (A1). (b) Con-tours of the change in active earth pressure on the wall before and after anchor failure.

Figure 5. Anchor force ratio showing relative increase in anchor forces due to anchor failure (base case model, A1)

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Torgersrud, Jostad, Karlsrud and Kalstad

Geotechnical Engineering for Infrastructure and Development

4090

Figure 6. Wale beam moment distribution before and after anchor failure for the base case model (A1)

Figure 7. Anchor force and beam moment ratios for different par-ametric studies.

4.2 Comparison with current design practice

Current Norwegian design practice suggest that the acting force in an adjacent anchors to one that fails should be increased by 20 – 50 % compared to the SLS loading-case. Results from this study suggest anincrease by a maximum of 6 % only.

Based on the design guidelines in Norwegian Pub-lic Road Administration (2010), the design moment, Md,ALS for the wale beams due to anchor failure gave values of Md,ALS in the range 1100 to 1575 kNm, de-pending on the applied reduction-factor, λred. For the same load-case the FE base case model gave Mk,ALS ~515 kNm, which is 53 – 67 % lower than the hand calculated values.

5 CONCLUSION

For the considered case, a maximum increase of 6% in adjacent anchor force and 40% in wale beam mo-ment was observed when one of the pre-stressed tie-back anchors fails. By not pre-stressing the anchors, a higher increase in anchor force was obtained. No significant difference was seen between a failing an-chor at centre line or near the corner.

Comparison between the results from this studyand current design practice gives reasons to believe that the current design practice is quite conservative. This is especially critical for the wale beams where traditional design methods for the ALS-case often is the governing load-case.

ACKNOWLEDGEMENT

The work is part of a strategic research program at NGI called "SP5 – Advanced Soil-Structure Interac-tion" financed by the Norwegian Research Council.

REFERENCES

Finno, R.J. Blackburn, J.T. & Roboski, J.F. 2007. Three-dimensional effects for supported excavations in clay. J. Geotech. Geoenviron. Eng. 133(1), 30-36.Grimstad, G. Andresen, L. & Jostad, H.P. 2011. NGI ADP: An-isotropic Shear Strength Model for Clay, Int. Journal for Numeri-cal and Analytical Methods in Geomechanics 36, 483–497Karlsrud, K. & Andresen, L. 2008. Design and performance of deep excavations in soft clays. 6th Int. Conf. on Case Histories in Geotech. Engineering. Arlington, Virginia. Proceedings, SOAP lecture 12. 26pNorwegian Public Roads Administration, 2010. Geoteknikk i vegbygging. Vegdirektoratet. ISBN: 978-82-7207-672-5. Online available on http://www.vegvesen.no/Fag/Publikasjoner/HandbokerOu, C.Y. Chiou, D.C. & Wu, T.S. 1996. Three-dimensional finite element analyses of deep excavations. J. Geotech. Engrg. 122(5),337-345.Schanz, T. Vermeer, P.A. & Bonnier, P.G. 1999. Formulation and verification of the Hardening-Soil model. Beyond 2000 in Compu-tational Geomechanics, Balkema, Rotterdam, 281-290.Stille, H. 1976. Behaviour of anchored sheet pile walls, Royal In-stitute of Technology. Department of Soil and Rock Mechanics, Stockholm, Sweden.Zdrakovic, L. Potts, D.M. & St John, H.D. 2005. Modelling of a 3D excavation in finite element analysis. Géotechnique 55(7), 497-513.

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Numerical development of p-y curves for soft clays Développement numérique des courbes p-y pour les argiles molles

K.P. Tzivakos*1 and M.J. Kavvadas1 1 National Technical University, Athens, Greece

ABSTRACT The paper presents a 3D numerical study (using the finite element code ABAQUS) of the undrained lateral response of a sin-gle, free-head pile under monotonic loading. Soil conditions simulating saturated normally consolidated clays are examined (linearly in-creasing shear strength with depth). The study first compares the results of several p-y curve methods from the literature (e.g. diagrams of lateral load versus horizontal displacement of the pile head and distributions of the horizontal displacement and bending moment along the pile) and shows that the differences are very significant. Due to these differences, the present study revisits the subject by performing par-ametric 3D finite element analyses with the objective to develop a new set of p-y curves for saturated normally consolidated clays under undrained lateral loading. Soil behavior is modeled by the Modified Cam-Clay (MCC) constitutive model to investigate the effects of nor-mal compression line slope (λ) and critical state line slope (M) on the initial stiffness and the ultimate resistance of the p-y curves. Although the numerical model uses effective stresses, the loading was applied sufficiently rapidly to ensure practically undrained conditions. Inter-face elements are used along the periphery of the pile in order to simulate separation and slipping effects on the response. Based on the re-sults of the finite element analyses, a new set of p-y curves for soft clays is proposed and the key parameters of the curves are determined.

RÉSUMÉ Le document présente une étude numérique 3D (utilisant le code d'éléments finis ABAQUS) de la réponse latérale non drainé d'un seul pieu avec la tête libre sous chargement monotone. Les conditions du sol simulant des argiles saturée normalement consolidées sont examinées (résistance au cisaillement croissant linéairement avec le profond). L'étude compare d'abord les résultats des courbes p-y proposés dans la littérature (par exemple diagrammes de la charge latérale en fonction du déplacement horizontal de la tête du pieu et la distribution du déplacement horizontal et moment de flexion le long du pieu) et montre que les différences sont très significatives. En rai-son de ces différences, la présente étude revisite cette objet en effectuant les analyses paramétriques par 3D éléments finis avec l'objectif de développer un nouvel ensemble de courbes p-y pour les argiles normalement consolidées saturés sous charge latérale non drainée. Le com-portement du sol est modélisé par la Modified Cam-Clay (MCC) modèle constitutif pour étudier les effets de la pente de la ligne de com-pression normale (λ) et la ligne d'état critique (M) sur la rigidité initiale et la résistance ultime du sol des courbes p-y. Bien que le modèle numérique utilise contraintes effectives, le chargement a été appliqué suffisamment rapidement pour assurer des conditions pratiquement non drainées. Les éléments d'interface sont utilisés le long de la périphérie du pieu afin de simuler les effets de séparation et de glissement sur la réponse. Sur la base des résultats des analyses des éléments finis, une nouvelle forme de courbes p-y pour les argiles molles est pro-posée et les paramètres clés des courbes sont déterminés.

1 INTRODUCTION

P-y curves are nowadays a common practice for the calculation of bending moment and horizontal dis-placement along laterally loaded piles. They are ap-plicable as non-linear springs along a beam-on-nonlinear-Winkler-foundation (BNWF) simulation of the laterally loaded piles. P-y curves for static pile

head lateral loading are investigated in the current paper, due to their importance for the formulation of p-y curves for cyclic or dynamic loading.

P-y curve formulations consist of three parts: the initial small-strain stiffness, the yielding section of the curve and the ultimate lateral resistance. The scope of the present paper is the determination of such curves for soft, normally consolidated clays.