Displacement – based parametric study on the seismic response of gravity earth-retaining walls

Soil Dynamics and Earthquake Engineering 80 (2016) 210–224

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Displacement – based parametric study on the seismic response of gravity earth-retaining walls Manya Deyanova a,n, Carlo G. Lai b,c, Mario Martinelli d a Earthquake Engineering and Engineering Seismology (ROSE Programme), Institute of Advanced Study of Pavia (IUSS), UME School, c/o EUCENTRE, Via Ferrata 1, 27100 Pavia, Italy b Department of Civil Engineering and Architecture, University of Pavia, Italy c European Centre of Training and Research in Earthquake Engineering (EUCENTRE), c/o EUCENTRE, Via Ferrata 1, 27100 Pavia, Italy d Deltares, Boussinesqweg 1, 2629 HV Delft, P.O. Box 177, 2600 MH Delft, The Netherlands

art ic l e i nf o

a b s t r a c t

Article history: Received 26 December 2014 Received in revised form 3 August 2015 Accepted 15 October 2015

The essence of performance-based design of gravity earth-retaining structures lies in the estimation of the residual (i.e. permanent) displacements after a seismic event. The accomplishment of this task however can be very complicated due to two interacting phenomena: the coupled sliding and tilting rigid body motion of the wall on an inelastic base and the formation of failure surfaces in the soil backfill. In this study a large number of fully non-linear, time-history analyses of gravity retaining walls (GRW) were performed using advanced numerical modelling. Different types of soil parameters and varying wall geometry within a practical range were investigated. The influence of different ground motion parameters was discussed and the results were compared with some of the most common limit equilibrium Newmark's sliding block procedures, including the recommendations by Eurocode 8, Part 5 [20]. Lastly, some recommendations for fast preliminary assessment of the seismic permanent displacements of GRW were provided. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Performance-based design Gravity earth-retaining Sliding Non-linear Time-history analyses Arias intensity Newmark's methods EN 1998-5 FLAC 7.0

1. Introduction Gravity walls are the oldest type of earth-retaining structures and, although their design under gravity loads is considered simple from an engineering point of view, post-earthquake observations [1,2] and experimental test results [3-8] have shown that the prediction of their residual displacements and predominant failure modes under seismic loading is still a serious challenge for the present analytical and design methods. The issue becomes even more important in the light of the performance-based design concepts. By knowing the residual horizontal displacements and tilting of gravity walls induced by earthquakes, engineers would be able to base their design on prescribed performance levels and on desirable failure patterns like the ductile sliding failure mechanism. What is more, relationships between influential ground motion intensity measures (IM) and permanent displacements are valuable for the development of fragility functions, risk assessment and loss estimation.

n

Corresponding author. Tel.: þ 359 896 790 140. E-mail address: [email protected] (M. Deyanova).

http://dx.doi.org/10.1016/j.soildyn.2015.10.012 0267-7261/& 2015 Elsevier Ltd. All rights reserved.

The approaches for the evaluation of the residual displacements of gravity walls fall into two main categories: displacementbased simplified analytical techniques, which consider systems of rigid bodies displacing along predefined potential failure surfaces and numerical techniques, which account for the non-linear soil properties to predict the magnitude and pattern of stresses and deformations through finite element or finite difference numerical models (stress-deformation analysis methods) [9–12]. Due to their simplicity and ease of implementation the former approaches prevail in the design methods based on allowable displacements like the Newmark sliding-block procedure and its improved variants. Some of the most common ones are listed in Table 1, together with their main assumptions and limitations based on [13] and [14]. The yield acceleration ay of the potential failure mass, required by the listed analytical procedures, is usually estimated using the Mononobe–Okabe (M–O) active soil wedge coupled with the assumption of a constant seismic coefficient (typically 50–70% of the free-field ground acceleration) and an educated guess about the point of application of the soil thrust. Limit equilibrium slope stability computer methods can also be used to determine the yield acceleration. The limitations listed in the last column in Table 1 have been addressed by many researchers. For example, Matasovic et al. [15]

M. Deyanova et al. / Soil Dynamics and Earthquake Engineering 80 (2016) 210–224

Nomenclature CAV CoV DD FS IM M–O ODF PE PGA PGV UD agR ah amax ay c' d dperm. dr e Ed fmax G0 Gwall H Ia kh

α γI

cumulative absolute velocity coefficient of variation Dobry duration factor of safety intensity measure Mononobe–Okabe over-design factor (EN 1997-1) probability of exceedance peak ground acceleration peak ground velocity uniform duration reference peak ground acceleration on soil type A pseudo-static acceleration peak ground acceleration critical/yielding acceleration soil effective cohesion (in the Mohr–Coulomb sense) horizontal wall displacement residual horizontal displacement of a wall allowable horizontal displacement (EN 1998-5) void ratio of the soil design value of the effect of an action maximum frequency initial (tangent) shear modulus weight of the wall wall height Arias intensity horizontal seismic coefficient constant related to the geometry of the sliding block importance factor (EN 1997-1)

developed a trilinear model for degradation of the yielding acceleration as a function of displacement for geosynthetic surfaces, with which the residual displacements calculated with the Newmark procedure were significantly lower than those obtained with a constant yield acceleration based on residual strength parameters. NCHRP 12-70 [16] suggested that for sloping backfill and high accelerations, when the M–O equation leads to unrealistically large seismic active earth pressure, the limit equilibrium slope stability computer methods might be used instead. In fact, dynamic tests [3–8] have shown that the failure surface in the backfill is almost planar, which was also the conclusion reached by Chen and Liu [17], who used limit analysis theorems and obtained almost planar log-spiral slip surfaces. Another observation reported by the NCHRP 12-70 [16] and based on finite element wave scattering analyses was that the maximum average horizontal acceleration to be used for the evaluation of the earth pressure decreases with the wall height and is a function of the frequency content of the ground motion record. The noncompliant assumption in the conventional Newmark analysis, which treats the unstable mass as rigid, was addressed by Kramer and Smith [18]. The authors developed a two degree-of-freedom analytical model and the results showed that if the fundamental period of the unstable mass was close to the predominant period of the base motion, the conventional Newmark method overpredicts the displacements by up to 100%. This complied with the findings by Gazetas and Uddin [19]. The current design codes are based almost exclusively on the limit equilibrium pseud-static approach. For instance, Chapter 7 of Eurocode 8, Part 5 [20] begins by setting stringent requirements for the method of analysis of retaining structures. According to these requirements, phenomena like non-linear soil behaviour during

γR,h γR,v δd δf Δl ν μ ρ σ σ''0 φ' χ2 ψ kv K Ko KAE,M-O Mw PAE r Rd S tanθ T vmax Vs W w

211

partial resistance factor for sliding (EN 1997-1) partial factor bearing resistance (EN 1997-1) friction between wall and backfill friction between wall and foundation grid size Poisson ratio mean value bulk mass density standard deviation mean total stress soil effective friction angle (in the Mohr– Coulomb sense) goodness of fit dilation angle of the soil vertical seismic coefficient bulk modulus coefficient for lateral earth pressure earth pressure coefficient with M–O soil wedge moment magnitude resultant horizontal force reduction coefficient (EN 1998-5) design value of the resistance to an action soil factor (EN 1998-1) tilting of the wall predominant period at which occurs the maximum spectral acceleration at 5% structural damping peak ground velocity shear wave velocity of soil width of wall base settlement of the ground behind the wall

dynamic soil-structure interaction and the compatibility between soil deformations and the wall displacements should be taken into account. Nevertheless, the model proposed by the code uses simply the M–O active soil wedge with a constant horizontal and vertical acceleration. The horizontal pseudo-static seismic coefficient kh ¼(agR  γI  S)/(g  r) is calculated by means of a reduction coefficient r (Table 2), which correlates to a certain global factor of safety (FS) and a selected displacement limit (ductility). FS is a combination of partial factors for actions, soil parameters and resistance factors for different limit states, according to EN 1997-1 [21]. The admissible wall displacements dr are prescribed by EN 1998-5 [20] and “in the absence of specific studies” they should be calculated according to Table 2. For example, for European seismicity with agR  γI/g¼0.28 g and a soil coefficient S¼1.15 dr ¼64–97 mm. This is a rather narrow band of allowable displacements and no further clarifications are given whether these displacement limits refer to serviceability or ultimate limit state and what would be the reduction coefficient r for retaining structures with larger allowable displacements. The seismic coefficient for all wall types varies again within a narrow band kh ¼(0.5C0.67)agR  γI  S/g and as a result a few cases of gravity walls satisfy the stability requirement for sliding. For comparison, a Supplementary Guidance [22], released together with guidance for repair of the city of Christchurch (NZ) after the Canterbury sequence in 2010–2011, states that the seismic coefficient for pseudo-static design for ultimate limit state varies between 30% and 70% of the seismic design acceleration according to six different cases of application of the retaining structure, with an allowable displacement of 100–150 mm for FS¼ 1. The Italian building code NTC 2008 [23] sets the seismic coefficient at kh ¼(0.2C0.3)agR  γI  S/g for an allowable displacement of 200 mm. For lower displacement limits (dr ¼50 mm), kh increases to 0.47 [24]. Finally, the guidelines by

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M. Deyanova et al. / Soil Dynamics and Earthquake Engineering 80 (2016) 210–224

Table 1 Simplified Newmark's block-on-plane models for calculating earthquake induced displacements. Based on [13]. Reference Equation/Model

Assumptions

Limitations

[30]

– Soil response is idealised as rigid-plastic – Dynamic soil friction angle ¼ static soil friction angle – The internal friction of the backfill is negligible

– – – – – –

v2max amax 2ay ay ay amax 4 0:17

dperm: ¼

upper bound modified later to

  2 ay ay v2max dperm: ¼ 2a 1  amax amax y [31]

log




4dperm αamax T 2



a

y ¼ 0:85 3:91amax

for 50% PE
 4d ay log αa permT 2 ¼ 1  3:86amax max

for 5% PE 2

T 10 dperm;5%PE ¼ α∙amax 4

[32]

dperm ¼



a

y 1  3:86amax



dperm ¼ Rv Rz

37v2max amax exp




 9:4ay amax




a

 a

2

y y Rv ¼ 1:015  0:2amax þ 0:72 amax   Rz ¼ 0:7 þ 1:2ay 1  ay ay o 0:3

[34]



 ay v2 dperm ¼ amax exp 9:4 0:66  amax max 95% confidence that dperm. is not exceeded

[35]

dperm ¼

37v2max amax exp




 9:4ay amax



dperm ¼

35v2max amax exp




 6:91ay amax

mean upper bound

– Limited verification – Similar to [30] – Two-block-model: one rigid block for the – Tilting is not considered – Sensitivity of relative displacement to variation of base friction angle wall and one for the failing wedge behind is not considered – Accounts for time variation of dynamic earth pressure – Modification of [32] – Based on probabilistic assessment of uncertainties affecting dperm. – Take into account wall inertia – The failure occurs incrementally – Take into account vertical acceleration




ay amax

  0:38

– Similar to [32] – Similar to [32] – Less conservative than in [32] especially for ay amax o 0:3

Table 2 Values of factor r for the calculation of the horizontal seismic coefficient Based on [20]. Type of earth-retaining structure

– Similar to [32] – The method provides unrealistically heavy walls since the peak soil strength is not mobilised at displacements o0.01∙H – Constant wall acceleration – Sliding occurs only at the base

– Similar to [30] – Similar to [30] – Uses the results by [37] of sliding block – By using this approach it is necessary to apply a factor of safety to the calculated weight of the wall analyses of 14 ground motions

mean fit [36]

– Three type of pulses are used to approximate the actual acceleration: – Considers T of the input signal ay ay – Based on effective stress principle rectangular amax o 0:5, triangular amax 4 0:5 and half sine – The assumption for the pore pressure is – Pore pressure remains constant conservative – All models underestimate the displacements – Critical slip surface can be converted into a plane slip surface – Only horizontal acceleration – The critical acceleration is very large – No reversal of displacements occurs – Similar to [30] – Similar to [30] – The inertia force due to the mass of the wall – Conservative because it is an upper bound fit to upper bound displacement values computed for different earthquakes is included – The backfill failure wedge moves as a rigid body with the retaining wall – Displacements before cut-off acceleration are neglected

0:087v2max a3max a4y

upper bound

[33]

Tilting or rotation is not considered Vertical acceleration is negligible Seismic excitation is only from the base Failure occurs at a predefined plane with a constant resistance The earth pressure does not change with time Assumptions for ay are required

r

Free gravity walls that can accept a displacement up to dr ¼ 300aS [mm] 2 a¼ agR  γI/g Free gravity walls that can accept a displacement up to dr ¼ 200aS [mm] 1.5 a¼ agR  γI/g 1 Flexural reinforced concrete walls, anchored or braced walls, reinforced concrete walls founded on vertical piles, restrained basement walls and bridge abutments

NCHRP 12-70 [16] and FHWA [25] lead to a seismic coefficient kh ¼(0.4C0.5)PGA for smaller magnitudes, wall height up to 10 m and a ductility factor 0.5 (with the EN 1998-1 notations, this refers to a behaviour factor q¼2). Another important aspect of the procedures listed in Table 1 is the choice of IMs, on which the residual displacements depend. The formulae use peak ground acceleration (PGA), peak ground velocity (PGV) and predominant period (T) as the most influential IMs. Some other researchers consider parameters like Arias Intensity (Ia) and Dobry Duration (DD) [26] as critical. For example, Smith [27] analysed the Newmark displacements for six values of critical acceleration under 227 digitalised strong-motion records and showed a pattern between displacements and Ia. Similar research was performed by Jibson [28],

who proposed a relationship between Newmark displacement, Ia and ay. Garini et al. [29], on the contrary, came to the conclusion that Ia cannot alone be a reliable predictor for sliding, especially with motions containing acceleration pulses of long duration. However, all the aforementioned studies, together with others concerning the influence of the IMs on the response of gravity retaining walls, dealt with the ideal Newmark sliding model in which the only non-linearity is represented by the Coulomb sliding friction at the base. In the present study a large number of fully non-linear time-history analyses of gravity earth-retaining walls were performed using advanced numerical modelling, in which different soil conditions and several wall geometries were considered. The aim was to investigate the seismic response of monolithic GRWs from the point of view of the most influential IMs and to give recommendations for their displacement-based design by comparing the results with both the common limit equilibrium procedures based on the Newmark sliding block methods and the EN 1998-5 [20] pseudo-static provisions. This requires the definition of a very important parameter: the horizontal displacement at failure. The issue should be considered from two points of view. The first refers to the functionality of the wall and the expected level of damage in the supported structure. The second one addresses the soil strength mobilisation in the backfill with the subsequent soil strength degradation. For example, Huang et al. [38] investigated the failure displacements from the stand point of the mobilised friction angle along the shear band in a coarse-

M. Deyanova et al. / Soil Dynamics and Earthquake Engineering 80 (2016) 210–224

Backfill type B2 Backfill type B1

H

Foundation - 4 R

CM

W

Backfill - 3 Backfill - 2 Backfill - 2 Backfill - 1 Backfill - 1

6.4m 6.0m

0.50m

Foundation - 3

Base

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

W [m] Fig. 1. Wall geometries considered in the analyses corresponding to ODFsliding for static sliding greater than one according to EN1997-1 [21].

59.0m Fig. 2. Sketch of the soil layers and the soil types used in the models.

for H<6m =

28.0 m

Wall

R=5.5 R=5.8 R=6.1

1

6.3m

1

R=1.6 R=1.9 R=2.2 R=2.5 R=2.8 R=3.1 R=3.4 R=3.7 R=4.0 R=4.3 R=4.6 R=4.9 R=5.2

10 9.5 9 8.5 8 7.5 7 6.5 6 5.5 5 4.5 4 3.5 3 2.5 2 1.5 1

R=1.3

H [m]

The walls are trapeziums with fixed top width at 0.5 m and variable height and base width (Fig. 1). The soil type of the backfill is either loose sand B1 or dense sand B2. The foundation soil type is B2.

As shown in Table 3, two types of coarse-grained soils for the backfill were considered: loose sand (B1) and dense sand (B2). The foundation soil, which is the soil layer immediately underneath the wall base, is sand type B2. The backfill and the foundation soil were modelled as a non-linear media with shear degradation rule according to Darendeli [44] and with Mohr–Coulomb failure criterion. The base layer and the wall were modelled with linearelastic materials. The ratio K/G0 was calculated for the assumed Poisson ratio υ using the relationship K/G0 ¼2/3(1 þ υ)/(1 2υ). The soil layers were divided into sub-layers with constant geotechnical properties within their thickness as shown in Fig. 2. The soil was considered dry.

=

2.1. Wall geometry

2.2. Soil types and constitutive models

for H>=6m

Seventeen sets of ten non-linear time history analyses were performed with the explicit, finite difference, plane strain commercial software FLAC 7.0 [40]. Each model has its own unique name that gives the basic information of the case it represents, as it is shown on the scheme below:

= = =

2. Parametric study and description of the numerical model

The soil parameters are shown in Table 3 and the wall geometries are presented in Fig. 1 with circles and black dots for backfill B1 and B2 respectively. Two aspects were considered when selecting the wall geometry: on one hand, the wall geometry is within the range of common practical cases; on the other hand, only the cases that satisfy the requirements for static design according to EN 1997-1 [21] were selected. The code ensures that the design value of the resistance to an action is greater than the design value of the effect of this action Rd ZEd, when both are calculated with the prescribed partial factors for different limit states and different combinations. EN 1997-1 [21] distinguishes five different ultimate limit states. Those relevant for the static design of the gravity walls, thus considered in the present research, are the EQU limit state (loss of equilibrium of the structure or the ground) and the GEO limit state (failure or excessive deformation of the ground). Each limit state consists of three different Design Approaches with their own combinations of partial factors on actions or effects (favourable/unfavourable), partial factors on soil parameters (on the tangent of the angle of shear resistance φ') and partial resistance factors (on the bearing capacity γR,v and on the sliding resistance γR,h). By considering all the prescribed partial factors for the corresponding Design Approach and limit states, each case was checked for overturning, sliding and bearing capacity of the foundation. The governing limit state was sliding and the controlling Design Approach and combination varied according to the case. The two grey lines in Fig. 1 connect those wall geometries for which the ratio between the sliding forces and the resisting forces is very close to 1 (Rd/Ed ¼1) for the design-governing combination for the two different backfill types. This ratio is denoted in the figure as the over-design factor for sliding ODFsliding ¼ Rd/Ed ¼1. All combinations of width and height below the grey lines (each corresponding to the indicated backfill type) lead to over-design factor for sliding greater than one.

25.0m

grained backfill. Two shaking table tests showed that if the displacements normalised with respect to the wall height d/H are larger than 3.4–4.1%, a critical state of the internal friction resistance in the backfill soil is reached. Based on these experimental results and assuming the development of a single shear band in the backfill, the authors proposed a value for the horizontal wall displacement at failure of 5% of the wall height H. Wu et al. [39], on the other hand, recommended 10% of H as a wall failure horizontal displacement. In the present theoretical investigation the latter value was adopted as the horizontal wall displacement limit. In practice, especially when following the code guidelines like those mentioned in the introduction [20,23–25], smaller displacement limits are usually considered for a variety of functionality requirements. Yet, a higher theoretical threshold would allow for the consideration of more “non-failing” cases. The analytical relationships developed using more cases would have less uncertainties and a wider domain of applicability. Together with the sliding failure mechanism, rotation failure was also considered. It was associated with an excessive tilting of the wall due to significant soil deformation under the toe of the wall and a subsequent dynamic instability for which stable numerical solution could not be found (Fig. 9, solid and dashed red lines). In fact, design guidelines focus on performance criterion for acceptable sliding displacements and rarely state precise tilting angle thresholds, since the preferred Design Approach is to prevent rotation failure by adequate factors of safety against failure of the foundation bearing capacity.

213

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M. Deyanova et al. / Soil Dynamics and Earthquake Engineering 80 (2016) 210–224

2.3. Selection of real records

1.6

Ten real accelerograms from outcropping rock sites were selected [53]. The records were scaled to satisfy the criterion for spectrum compatibility, namely the difference between the average and the target spectrum does not exceed 10% in the period range from 0.15 s to 2 s. The target response spectrum was the elastic (5% structural damping) acceleration response spectrum on rock (Type 1) according to the European design provision for seismic actions and earthquake resistance of structures EN 1998-1 [45] with PGA ¼ 0.28 g. The spectra are shown in Fig. 3. The seismological characteristics of the events and the IMs of the records after scaling are listed in Table 4. It is evident that PGA, as an indirect selection criterion, varies in a small range, whereas the other IMs show a much greater dispersion. The vertical component of the ground motion was not considered. On one hand, accounting for it would bring additional complications in interpreting the results of the numerical analyses when compared with the predictions of the analytical methods listed in Table 1, which

1.4

Acc_i Mean spectrum EN 1998-1 spectrum Type 1, Rock, PGA=0.28g

Acceleration [g]

1.2 1.0 0.8 0.6 0.4 0.2 0 0

1

2

3

4

Period [s] Fig. 3. Target and mean acceleration response spectra from the accelerograms.

Table 3 Soil geotechnical parameters considered in the analyses [41-43].

Soil type

Layer

φ’[42]

K0

B1

[°] 27.5

B2

0.45

2000

37.5 1-sinφ

B2

0.45

2000

37.5

-

2400

Base Wall

ρ

[kg/m3] 0.65 1700 or

Backfill

Foundation

e[41]

-

G0 [43] [Pa]

ν c’

ψ [°]

Model

Non-linear hysteretic + Mohr-Coulomb 0.3 0 5 failure criterion + viscous damping 2e9

0.29

2.2e9

3e9

-

2400

-

Linear viscoelastic

Table 4 Seismological characteristics and IMs of the real records used in the analyses. Original

Scaled

Record

Earthquake

Station

Site

Fault

Mw

Epicentral distance [km]

PGA [g]

Scaling factor

PGA [g]

PGV [m/s]

Ia [m/s]

T [s]

DD [s]

Acc_1

Izmit, Turkey 1999 Umbria Marche, Italy 1997 Tabas, Iran 1978 Montenegro 1979 Campano Lucano, Italy 1980 Loma Prieta, California,U.S. 1989 Eastern Honshu, Japan 2003 Kyushu, Japan 2005 Eastern Honshu, Japan 2008 Eastern Honshu, Japan 2008

Ambarli-Termik Santrali Gubbio

Rock

Strike-slip

7.6

113

0.251

1.03

0.258

0.24

1.02

0.42

36.2

Rock

Normal

6

38.0

0.100

3.35

0.332

0.50

2.61

0.18

30.0

Dayhook

Rock

Oblique

7.3

12.0

0.355

1.16

0.412

0.21

2.14

0.16

32.4

Herceg. Novi-O.S.D. Pavicic School Torre del Grecco

Rock

Thrust

6.9

65.0

0.235

1.27

0.298

0.17

1.19

0.26

10.9

Rock

Normal

6.9

80.0

0.065

3.25

0.211

0.18

1.06

0.66

31.7

Gilory Array #1

Rock

Reverse-oblique

6.9

28.6

0.510

0.35

0.178

0.14

0.21

0.38

Ichinoseki, IWT

Rock

Uknown

7

50.5

0.168

2.44

0.411

0.37

4.24

0.10

27.6

Chinzei, SAG

Rock

Uknown

6.6

36.2

0.141

2.81

0.395

0.33

1.67

0.08

13.0

Ichinoseki, IWT

Rock

Uknown

6.9

23.1

0.228

1.16

0.265

0.26

1.74

0.36

19.9

Ichinoseki, IWT

Rock

Uknown

6.8

99.3

0.379

0.74

0.280

0.11

0.33

0.42

10.4

0.304 0.082

0.25 0.120

1.62 1.185

0.30 0.18

21.6 11.47

Acc_2

Acc_3 Acc_4 Acc_5

Acc_6

Acc_7 Acc_8 Acc_9 Acc_10

mean σ

3.45

M. Deyanova et al. / Soil Dynamics and Earthquake Engineering 80 (2016) 210–224

A general view of the numerical model of the earth-retaining structure is shown in Fig. 4. The wall and the soil layers described above were modelled using FLAC 7.0 with square zones of size smaller than 1/10 to 1/8 of the wavelength associated with the highest frequency component of the ground motion containing appreciable energy. This means that the maximum frequency that can be modelled accurately is fmax ¼ Vs/[(8C10)Δl]. As the size of the grid Δl and the shear wave velocity Vs vary among the soil layers, so does the frequency cut-off fmax. The minimum frequency cut-off corresponds to the “backfill” layers fmax ¼ (15C18) Hz and no additional filtering of the input records was needed. The analyses were performed in three stages. The first stage (Stage I) considered only the static equilibrium under the selfweight of the base layers. The second stage (Stage II) found the static equilibrium when the wall and the backfill were activated and finally, during the third stage (Stage III), the dynamic analysis was executed. As the boundaries of the grid are at a considerable and sufficient constant distance away from the expected disturbed regions around the wall, they were modelled as fixed for the static analysis. These fixities were replaced by quiet (viscous) and free-

Interface 2

Boundary

Boundary

Interface 1

Boundary Fig. 4. General view of the model. Different shades represent different layers of soil lithological units.

field boundaries for the dynamic analysis, at the base and at the two vertical edges respectively. The quiet boundaries were represented by independent dashpots in the normal and shear directions, which provided viscous normal and shear tractions applied at every time step. The free-field boundaries consisted of a one-dimensional “column” of unit width, simulating the behaviour of the extended medium and thus modelling conditions that are identical to those of an infinite model. As shown in Table 3 Mohr–Coulomb failure criterion was assigned to the backfill and to the foundation soil. It is characterised by a shear yield function with tension cut-off, a nonassociative flow rule for shear failure and an associative flow rule

Backfill height 6m, soil type B2

1 0.9 0.8 0.7

G/Gmax

2.4. Description of the numerical model

for tension failure. This failure criterion when coupled with cyclic loading originates rate-independent energy dissipation (i.e. hysteretic damping) within the soil. This phenomenon occurs any time when the current stress reaches the yield surface (elastoplastic response). For the elastic strain range the non-linear response was modelled by shear modulus decaying curves, which through the Masing rule originated another form of hysteretic damping. These G/Gmax curves were defined by setting up the coefficients (a,b,x0 and y0) of a built-in continuous function, so that the obtained curves corresponded to the soil type and the average mean effective stress of each sub-layer following the recommendations by Darendeli [44]. This strain dependent soil behaviour was set active throughout both the static and dynamic analysis. An example is shown in Fig. 5. Although, the shear

0.6 0.5 0.4 0.3 σ m 0.18

0.70

1.43 [atm]

0.2 a 1.000 1.000 1.000 b -0.473 -0.473 -0.473

0.1 x -1.713 -1.507 -1.399 0 y0 0.000

0 10-4

0.000 -0.000

10-3

10-2

10-1

10

Strain [%] Fig. 5. Example of the shear modulus decaying curves used in the FLAC 7.0 models.

modulus decaying curves together with the Mohr–Coulomb failure criterion should theoretically provide the expected level of damping during the dynamic excitation at medium-to-high strain levels, an additional small value of viscous Rayleigh damping of 0.1% was assigned to the whole model to account for the smallstrain, rate-dependent energy dissipation occurring in soils under cyclic loading and also to remove high frequency noise. As the derivation of incremental stress–strain relations from modulus decaying curves in FLAC 7.0 assumes that the stress depends only on strain and not on the number of cycles or on the past strain history, difference between the damping associated with the hysteretic rule in FLAC 7.0 and typical lab tests are expected. An example is presented in Fig. 6. The figure compares two one-dimensional site response analyses obtained through the

Damping curves from FLAC 7.0 Built-in damping curves

Accel. [m/s2]

do not consider any vertical component of the acceleration. On the other hand, this component is expected to have little influence on the permanent displacements [52]. Typically, the vertical and horizontal components of ground motions are not in phase and if their relation is considered random, the vertical acceleration is equally likely to increase or decrease the yield acceleration of the wall-backfill system. Thus, its net impact may be assumed negligible. This assumption is further supported by the results of a parametric study by Yan et al. [46]. The authors considered the vertical components of ground motions in Newmark analyses of a rigid block sliding on a plane and concluded that it can either increase or decrease the residual displacements with an average influence of 710%.

215

6 4 2 0 -2 -4 -6 4

6

Time [s]

8

10

Fig. 6. Comparison of the ground response analyses obtained with STRATA for case H5W30-B2-S3.

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M. Deyanova et al. / Soil Dynamics and Earthquake Engineering 80 (2016) 210–224

linear equivalent approach for accelerogram Acc_3 and soil profile of case H5W30-B2. The analyses were performed with the software STRATA [47], in which G/Gmax and damping curves by Darendeli [44] are built-in. The solid line presents the site response when the damping curves correspond to the experimental results, as discussed in [44], and the dashed line presents the site response when the damping curves are derived from the hysteretic rule in FLAC 7.0. For clarity, the graph shows only a time window of the largest acceleration values where the differences are more pronounced. As expected, FLAC 7.0 demonstrates slightly higher levels of this type of hysteretic damping predominantly in the large strain range and still within acceptable limits. The contact surfaces between the wall and the surrounding soil were modelled with interface elements (Fig. 4). They are characterised by Mohr–Coulomb yield criterion with a friction angle of 0.6φ' and 2/3φ' for Interface 2 and Interface 1 respectively and no effective cohesion. The selected friction angle follow a recommendation by EN 1998-5 [20], Clause 7.3.2.3, according to which the pressure distribution on the wall due to static and dynamic action should be taken to act with an inclination with respect to the wall not greater than 2/3φ'. The interfaces have also normal and shear stiffness and no tensile strength. The precise values of the normal and the shear stiffnesses were not crucial for the results, as long as they were large enough to prevent relative motion at the interfaces before failure and at the same time low enough to keep the speed of the convergence relatively large. Moreover, the behaviour of the interface elements under dynamic loading has been tested on a simple FLAC 7.0 model of a rigid block sitting on the free surface of a linear-elastic half-space subjected to a Ricker-wavelet (Fig. 7). The

[m/s ]

Analytical solution “Newmark” FLAC 7.0 6 4 2 0 -2 -4

rigid block interface 0

0.5

1 [s]

1.5

linear-elastic half-space

2

3. Results 3.1. Analysis of the results The failure patterns observed in the FLAC 7.0 models correspond very much to the behaviour observed in the experiments referenced in the Introduction. Fig. 8 presents the horizontal displacement contours at the last step of the static (Stage II) and dynamic (Stage III) analysis of the FLAC 7.0 model of case H6W30-B2-S1. The latter clearly shows that the formation of failure surfaces in the backfill is coupled with significant deformation at the base and with settlement of the soil behind the wall. For the same wall geometry and soil type conditions, the horizontal displacement of the wall base (dperm.), the tilting (tanθ) and the settlement (w) of the surface of the ground behind the wall are shown in Fig. 9. dperm. is obtained by subtracting from the horizontal wall movement the free-field displacement recorded at the top of the foundation layer away from the wall and it is positive when the wall base moves towards the backfill. The tilting (tanθ) is computed as the difference of the horizontal displacements recorded at the top and bottom of the wall, divided by the wall height and it is positive if the wall rotation is clockwise. Lastly, the downward settlement of the surface of the ground behind the wall is reported with a negative sign. An interesting observation about this settlement is that it spreads away from the wall at a distance greater than the wall height. Fig. 10 shows that the disturbed portion of the backfill is more than 10 m, while the height of the wall is 6 m. Two types of failure mechanisms were distinguished for the analysis of the results. The first represents failure due to very large deformations of the soil base under the toe of the wall which led to its excessive tilting, whereas the second failure refers to residual horizontal displacements of the wall greater than 10% of H. For the

X displacement [m] -0.003 -0.0025 -0.002 -0.0015 -0.001 -0.0005 0

Stage II - Static solution 0

10m

dperm. [m]

0.01

0 -0.01 -0.02 -0.03 -0.04 0

0.5

1.0

1.5

2.0

Time [s] Fig. 7. Relative horizontal displacement of a rigid block sitting on the free surface of a linear-elastic half-space.

residual horizontal displacements have been compared with the results from a double integration of the relative acceleration considering two-ways sliding and with the solution obtained by the Java program named “Newmark” available on the USGS web-site [48]. The comparison is presented in Fig. 7 and shows a satisfactory response of the FLAC 7.0 interface elements under dynamic excitation. Reversing the signal would lead to the same absolute residual displacement of the block but in the opposite direction, because the sliding-block model is symmetric.

X displacement [m] -0.40 -0.30 -0.20 -0.10 0 0.10

0

Stage III - Dynamic solution 10m

Fig. 8. Horizontal displacement contours at the last step of the static and dynamic analyses of the FLAC 7.0 model H6W30-B2-S1.

M. Deyanova et al. / Soil Dynamics and Earthquake Engineering 80 (2016) 210–224

Y displacement [m] -0.40 -0.30 -0.20 -0.10 0 0.10 0.20

217

Stage III - Dynamic solution 0

10m

Fig. 10. Settlement of the ground behind the wall.

Table 5 Summary of the results from the FLAC 7.0 models. Case

Number of failing cases within a record set

dperm.a

tanθ

Low bearing capacity at the base

dperm. 40.1 H μb [m] CoV

H6W50-B2 H6W40-B2 H6W30-B2 H5W35-B2 H5W30-B2 H5W25-B2 H4W30-B2 H4W25-B2 H4W20-B2

/ / 2 / / 2 / / /

/ / / / / / / / 2

 0.15  0.26  0.32  0.19  0.27  0.27  0.12  0.18  0.25

H6W60-B1 H6W50-B1 H6W40-B1 H5W40-B1 H5W35-B1 H4W35-B1 H4W30-B1 H4W25-B1

/ / 2 / / / / /

/ / / / 2 / / 2

 0.19  0.25  0.27  0.26  0.33  0.16  0.22  0.29

b c

|μ|c

CoV

0.62 0.59 0.46 0.61 0.49 0.42 0.58 0.52 0.045

0.0036 0.0099 0.0280 0.0037 0.0152 0.0290 0.0040 0.0074 0.0275

0.64 0.75 0.42 0.56 0.46 0.41 1.00 0.64 0.88

0.45 0.50 0.42 0.49 0.48 0.12 0.45 0.42

0.0024 0.0066 0.0017 0.0056 0.0134 0.0049 0.0065 0.0149

0.88 0.68 0.97 0.87 0.55 0.95 0.67 0.78

3.2. Comparison with the design methodologies by EN 1997-1 and EN 1998-5 The first comparison was made with the design requirements for static loads according to EN 1997-1 [21]. Based on Fig. 1, with the same considerations for the over-design factor for sliding due to gravity loads, as described in Section 2.1, the cases are plotted again in Fig. 11 but indicating this time the “failing” walls from the numerical analyses. The grey dots and the grey circles mark the “failing” cases respectively with backfill type B2 and B1.

The mean values are calculated only with the “non-failing” cases. The negative values denote displacements away from the backfill. The mean of the absolute values.

first type of failure the analyses could not be performed until the end of the records due to excessive deformation developed at the foundation. These cases are denoted in Fig. 9 with the word “Cut” following the number of the record. The mean values are depicted with a horizontal grey line, however they include only the non-failing records. It is worth noting that the residual horizontal displacements and tilting obtained from one set of records fall into a very wide range, which casts doubts on the relevance of using the mean value for the prediction of the final response of the wall from the spectrum-compatible set of records.

H [m]

a

a

10 9.5 9 8.5 8 7.5 7 6.5 6 5.5 5 4.5 4 3.5 3 2.5 2 1.5 1

1

Fig. 9. Results from FLAC 7.0 model for case H6W30-B2-S(1-10). (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

Table 5 summarises the results from all cases. The second and the third columns show the number of cases within a record set that failed due to one of the failure criteria described above. For each backfill type, three sets with “failing” walls exist. For example, for wall case H6W30-B2 for two of the records the deformation of the soil under the toe of the wall was very large and the wall experienced excessive tilting. For case H4W20-B2, however, the prevailing failure mechanism was due to a significant horizontal displacement which exceeded 10% of the wall height. Overall, failure is observed during six of the seventeen record sets. The last four columns contain the mean values of the residual horizontal displacement and tilting and the corresponding coefficient of variation, which shows quite significant dispersion of the results within a record set, especially for the tilting.

1

Backfill type B2 Backfill type B1 Failing walls - Backfill type B2 Failing walls - Bakcfill type B1 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 W [m]

Fig. 11. “Failing” walls according to FLAC 7.0 models against the range of admissible design for static sliding according to EN 1997-1 [21].

218

M. Deyanova et al. / Soil Dynamics and Earthquake Engineering 80 (2016) 210–224

Table 6 Comparison between the residual horizontal displacements from FLAC 7.0 and the design procedures from EN 1997-1 and EN 1998-5.

W H [m] [m]

3

5

[m]

0.27

Backfill type B2 EN 1997 1 [21] EN 1998-5 [20] ODF ODF d = [m] dr [m] Ed/Rd perm. sliding norm. dr*Ed/Rd

1.00

1.31

1.00

5

0.26

1.00

1.22

-Importance factor γI ¼ 1 -Soil factor S ¼1.15 for soil Type C -Response spectrum Type 1 with agR ¼0.28 g -Horizontal seismic coefficient calculated kh ¼0.161 -Vertical seismic coefficient assumed kv ¼ 0 -Factor r ¼ 2 and corresponding allowable displacement dr ¼0.097 m (Table 2) -Final residual displacement assumed Total horizontal design force ðEÞ dperm: ¼ dr  Resisting against sliding force ðRÞ The latter is defined assuming that if the allowable displacement for a certain factor r corresponds to an over-design factor for sliding Ed/Rd ¼1, then for greater values of this ratio the final displacement should be proportional to it. Table 6 summarises the results and clearly shows that under the above stated assumptions, the EN 1998-5 procedure tends to underestimate the residual displacements and could hardly predict failure.

Backfill type B1

4

ratio Ed/Rd for sliding and bearing for the Design Approaches. The design parameters are the following:

1.00 3.3. Comparison with the simplified Newmark’s block-on-plane models

mean

Table 6 shows the mean residual horizontal displacement (dFLAC ) from the FLAC 7.0 analyses, the corresponding ODFsliding for each type of backfill and for each geometry configuration and their norm: normalised values (dFLAC , ODFnorm.) with respect to the maximum mean horizontal residual displacement. The “failing” cases are excluded. The results lead to the conclusion that if a wall is designed for static loads with ODFsliding greater than 1.3 and 1.2 (partial factors considered according to [21]) for backfill of dense and loose sand respectively, the wall is unlikely to fail in regions of medium to high seismicity with PGA between 0.2 g and 0.35 g. However, there is no proportionality between the ODFsliding and the expected seismic residual horizontal displacement. For example, wall H6W60-B1 has a larger mean horizontal displacement when compared to H4W35-B1 but at the same time its ODFsliding for static design is also larger. In the similar manner, wall H6W40-B2 responded with much larger residual displacement than wall H4W25-B2 but they both have the same ODFsliding. The last three columns of Table 6 show the second comparison based on the design procedure for seismic loads according to EN 1998-5 [20]. The procedure follows the steps: 1) selection of the parameters to describe the seismic hazard of the site – type of response spectrum, reference peak ground acceleration agR and soil type with its soil factor S; 2) choice of importance factor γI of the structure, in order to calculate the design ground acceleration α ¼agRγI/g; 3) selection of the reduction factor r from Table 2 according to the allowable wall displacement; 4) calculation of the horizontal seismic coefficient from kh ¼ agRγIS/(gr); 5) selection of the partial factors to factorise the actions (wall weight as favourable and backfill weight as unfavourable), material properties (angle of shear resistance) and resistance (sliding and bearing capacity of the soil base) for the three different Design Approaches; 6) calculation of the earth pressure coefficient Ko (static plus dynamic) from the M–O formula for active state; 7) calculation of the total design force Ed acting on the retaining structure, together with the sliding resistance Rd at the base, taking into account the friction between the wall and the adjacent soil δ; 8) check of the

The uncertainties associated with Newmark's block-on-plane models listed in Table 1 make the comparison of the results obtained using the FLAC 7.0 model particularly difficult. One of the main reasons is that the simplified dynamic procedures need a value for the yielding acceleration ay in order to obtain the residual horizontal displacement of the wall. On the contrary, in the advanced non-linear FLAC 7.0 models the final wall displacement is a result of the accumulated non-linear response from each timestep of the input motion. For this reason, the analogy is based not on case-by-case values but rather on areas of mutual applicability of the different methods. The calculations with Newmark's models were made for each record and its corresponding PGA, PGV and T. The yielding accelerations were calculated through an iterative process for finding equilibrium with M–O soil wedge, following an example explained in [14]. The process is summarised in the next steps and the resultant yielding acceleration is given in Table 7: 1. The weight of the wall Gwall is calculated 2. Pseudo-static acceleration ah ¼kh  g is assumed 3.   kh ; θ ¼ tg  1 1  kv

ð1Þ

where kv ¼0

Table 7 Yielding acceleration for the 17 case studies from the M–O active soil wedge theory. Case

ay [m/s2]

Case

ay [m/s2]

H6W50-B2 H6W40-B2 H6W30-B2 H5W35-B2 H5W35-B2 H5W25-B2 H4W30-B2 H4W25-B2 H4W20-B2

2.37 2.11 1.74 2.195 2.01 1.78 2.31 2.11 1.84

H6W60-B1 H6W50-B1 H6W40-B1 H5W40-B1 H5W35-B1 H4W35-B1 H4W30-B1 H4W25-B1

2.065 1.815 1.49 1.79 1.60 1.95 1.74 1.49

M. Deyanova et al. / Soil Dynamics and Earthquake Engineering 80 (2016) 210–224

4. The resultant force for vertical wall and horizontal backfill surface is 1 P AE ¼ K AE;M  O  γ  H 2  ð1  kv Þ; 2

ð2Þ

where kv ¼ 0 and K AE;M  O ¼

  cos 2 φ‘  θ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2   sin ðδd þ φ‘Þ sin ðφ‘  θÞ cos θ  cos δd þ θ  1 þ cos ðδd þ θ Þ

5. The first guess for the yielding acceleration is    

P AE  cos δd  P AE  sin δd g ay ¼ tg δf  Gwall

ð3Þ

ð4Þ

6. The procedure is repeated until the yielding accelerations ay is equal to the assumed pseudo-static value ah.

Limit equilibrium slope stability methods can also be used to determine the yield acceleration. The slope stability approach is more generalised than the M–O and has been shown to be equivalent to M–O under the same loading conditions [16]. The horizontal displacements obtained through the methods in Table 1 with the procedure mentioned above are plotted in Fig. 12 with respect to ay/PGA. The displacements were calculated using all 10 records. All cases were analysed with reference to backfill type B2 and B1. The area enclosed by the maximum and minimum displacements for each simplified method is shaded in grey. The darkest area corresponds to the range of validity of all the plotted methods. The vertical line shows the minimum ay/PGA ratio among all the cases with one type of backfill. In other words, the horizontal displacements for the investigated cases obtained from the simplified methods fall in the shaded area to the right of the vertical line. The horizontal lines correspond to the maximum and minimum horizontal displacements computed in FLAC 7.0 among all the cases and all the records for each type of backfill.

Backfill B2

Backfill B1 0.7

Richard et al. [32] Newmark [30] Cai et al. [36] Nadim et al. [34]

0.6

0.6 0.5

0.2 0.1 0

dperm. [m]

min (ay /PGA) M-Owedge

dperm. [m]

0.5 FLAC dmax

0.3

0.2

0.3

0.4

FLAC dmax

0.4 0.3 0.2 0.1

FLAC dmin 0.1

Richard et al. [32] Newmark [30] Cai et al. [36] Nadim et al. [34] min (ay /PGA) M-Owedge

0.7

0.4

0.5

0.6

0

0.7

FLAC dmin 0.1

0.2

0.3

Backfill B2

0.1

0.4 0.3

0.1

0 0.2

0.3

0.4

ay/PGA

FLAC dmax

0.2

FLAC dmin 0.1

0.7

min (ay /PGA) M-Owedge

0.2

dperm. [m]

min (ay /PGA) M-Owedge

dperm. [m]

0.5

0.3

0.6

Whitman et al. [35] Zarrabi [33] ay<0.3 Sarma [31] 5%PE, a=1

0.6

0.5 FLAC dmax 0.4

0.5

Backfill B1 0.7

Whitman et al. [35] Zarrabi [33] ay<0.3 Sarma [31] 5%PE, a=1

0.6

0.4

ay/PGA

ay/PGA

0.7

219

0.5

0.6

0.7

FLAC dmin

0 0.1

0.2

0.3

0.4

ay/PGA

Fig. 12. Comparison between 7 Newmark-based methods and FLAC 7.0 numerical models.

0.5

0.6

0.7

220

M. Deyanova et al. / Soil Dynamics and Earthquake Engineering 80 (2016) 210–224

Thus, the intersection between the hatched and the shaded areas shows the range of ay/PGA ratio in which Newmark's methods give horizontal displacements close to the ones obtained from the advanced numerical non-linear models. Apparently, the residual horizontal wall displacements calculated using the Newmark-based procedures with the yielding acceleration calculated from the M–O active wedge assumption could be significantly underestimated. More reliable predictions would be obtained with ay less than 40% of PGA. The method by Nadim [34] with large values of PGA (Fig. 12(a) and (b)) yields results closer to the FLAC 7.0 models, whereas the methods by Whitman et al. [35], Zarrabi [33] and Sarma [31] (Fig. 12(c) and (d)) provide lower displacements than those calculated from the numerical simulations. Fig. 13 shows the relative difference in percentage between the horizontal displacements calculated with FLAC 7.0 and those predicted using the simplified dynamic methods [32,30,36] and [34]. The values of yielding acceleration used are listed in Table 7. The results plotted in Fig. 13(a) show the relative difference in the displacements with respect to the examined cases. The vertical lines show the range of variation for each case study subjected to the whole set of records. In the same manner, the relative difference in the displacements with respect to each record is plotted in Fig. 13(b) where each vertical line refers to all the cases subjected to a certain record. In the figure the jagged lines across the plots connect the mean values. The closer the mean value is to the horizontal axis, the better the correspondence between the

predictions of the simplified methods and the FLAC 7.0 results. If the results are compared with respect to each case (Fig. 13(a)) there is a trend of gradual decrease of the relative difference with the decrease of the wall size. However, if the comparison is carried out with respect to the different ground motions (Fig. 13 (b)), some of the records are noticed to provide the smallest relative error. This leads to the conclusion that the simplified procedures are much more sensitive to the IMs than to the wall geometry and that amax, as adopted by all methods, is not the most influential IM. 3.4. Relationship between the residual wall horizontal displacement (dperm.) and Arias Intensity (Ia) As already mentioned, the residual horizontal displacements obtained from FLAC 7.0 analyses show a wide dispersion within the set of spectrum compatible records (Fig. 9 and Table 5). This implies that the mean value is not a reliable parameter for the prediction of the residual displacements. Using the maximum wall displacement is also not justifiable because it could lead to a significant overestimation of the response. Having in mind that the selection of records was based on spectrum compatibility and PGA, together with the conclusions from Fig. 13, a logical question arises about the influence of different IMs. In Fig. 14 the relative horizontal displacements of the walls with height 6 m and backfill type B2 are plotted against several IMs. Models that fail due to excessive deformation of the base below the wall are excluded.

120

80 60 40 20

H4W25 - B1

H6W60 - B1

Acc_10

H4W30 - B1

H4W20 - B2

Acc_9

H4W35 - B1

H4W25 - B2

Acc_8

H5W35 - B1

H4W30 - B2

Acc_7

H5W40 - B1

H5W25 - B2

Acc_6

H6W40 - B1

H5W30 - B2

Acc_5

H6W50 - B1

H5W35 - B2

Acc_4

120

H6W30 - B2

-80

Acc_3

-60

H6W40 - B2

-40

Acc_2

-20

H6W50 - B2

0

Acc_1

(dFLAC-danalytical)/dFLAC [%]

100

(dFLAC-danalytical)/dFLAC [%]

100 80 60 40 20 0 -20

Richard et al. [32] Newmark [30] Cai et. [36] Nadim et al. [34]

-40 -60 -80

Fig. 13. Comparison between the horizontal displacements obtained from the Newmark-based methods with the yielding acceleration in Table 7 and FLAC 7.0 results.

0.7

0.6

0.6

0.5

0.5

0.4 0.3

0.43

χ2 =0.48

χ2 =

0.2

8

0.4

0.1

χ2 = 1

0.6

.69

0 χ= 2

0.4

0.5

.72

0.3

χ2 =0

0.13

χ2 =

0.2 0.1

0 2

3

4

26

χ2 =0.

0.4 0.3

43

χ2 =0.

0.2

χ2 =0.23

0.1

0 0.0

5

221

0.7

dperm. [m]

0.7

dperm. [m]

dperm. [m]

M. Deyanova et al. / Soil Dynamics and Earthquake Engineering 80 (2016) 210–224

0.1

0.2

PGA [m/s2]

0.3

0.4

0.5

0

0.6

0

10

20

PGV [m/s]

30

40

DD [s]

0.7

dperm. [m]

0.6 0.5

H6W30-B2 (2 fail)

1

0.9

9

χ2 =

.5 2 0 χ=

H6W40-B2

H6W50-B2

.92

0.4

χ2 =0

0.3 0.2 0.1

a∙Ia+b=dperm.

0 0

1

2

3

4

5

Ia [m/s] Fig. 14. Relationship between different IMs and dperm. for the cases of wall height 6 m and backfill B2.

0.6 0.5

.62

χ

0 χ= 2

0.4

.9 =0

0.86

χ2 =

0.3

0.1

79 χ2 =0.

1

H5W35-B2

0.7

2

3

Ia [m/s] H5W30-B2

a∙Ia+b=dperm.

4

5

0.4

0.7

2

0.6 0.5

dperm. [m]

0.4 0.3

3

Ia [m/s] H4W25-B2

0.3 0.2

0.1

5

0

1

H6W60-B1

H4W20-B2

2

3

4

Ia [m/s] H6W50-B1

5

H6W40-B1 (2 fail)

1

0.8

.86

χ2 =0

0.3 0.2

4

χ2 =

0.4

0.1 0

2

0.5

0.2

.71

χ2 =0

a∙Ia+b=dperm.

0.6

0 0.9 χ2 =

2

0 1

H4W30-B2

.8 2 0 χ=

4

0.8

χ=

.6 2 0 χ=

0.1 0

H5W25-B2 (2 fail)

4

0.5

0 0

dperm. [m]

.75

χ2 =0

0.3 0.2

7

.7 χ =0 2

0.4

0.1

a∙Ia+b=dperm.

0.6

0.5

0.2

0

0.7

a∙Ia+b=dperm.

0.6

1

2

dperm. [m]

dperm. [m]

0.7

a∙Ia+b=dperm.

dperm. [m]

0.7

.93

χ2 =0

0 0

1

2

3

4

Ia [m/s] H5W40-B1 H5W35-B1

5

0

1

H4W35-B1

2

3

Ia [m/s] H4W30-B1

4

5

H4W25-B1

Fig. 15. Relationship between Ia and dperm. for the different cases.

The graphs clearly show that a relationship between the horizontal permanent displacements and PGA could hardly be established but a noticeable pattern exists with the Arias Intensity (Ia). As shown in Fig. 15, this linear relationship between dperm. and Ia is also observed for the other cases. The linear regressions shown in Figs. 14 and 15 were combined altogether in a single relationship which expresses the residual horizontal displacement as a function of the wall height (H), wall

Table 8 Values for the coefficients in Eq. (5). Coefficient for Eq. (5)

c1 c2 c3 m

Mean

Upper bound

Backfill B1

Backfill B2

0.24 0.32 1.23 0.75

0.24 0.27 1.15 1.0

Backfill B1

Backfill B2

0.25 1.0 0.7

1.0

M. Deyanova et al. / Soil Dynamics and Earthquake Engineering 80 (2016) 210–224

base width (W), Arias Intensity (Ia) and type of backfill. This relationship is based on the observation that the gradient a of the linear representation dperm.  Ia could be expressed as a linear function of W/H2 for the backfill type B2 and W/H1.5 for the backfill type B1, as shown in Fig. 16. Similar conclusions were achieved for the constant b. The relationship is as follows: W  Ia H

2m

  h mi Wm  1 ½m; þc1 I a  c2 c3 s H

0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

Backfill type B2

Parameter b

Parameter b

dperm: ¼  m3

χ² = 0.93

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Parameter a

W/H 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

ð5Þ

0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

The coefficients c1, c2, c3 and m were found consistent with the values reported in Table 8. The coefficients corresponding to the upper bound can be used for design purposes, as they represent an envelope of the results obtained from the numerical analyses. The displacements obtained from Eq. (5) for the “non-failing” walls were compared in Fig. 17 with the results from the timehistory analyses (Table 5) using as predictors the mean and the upper bound. The graph shows a good agreement between the predictions of the simplified methods and the FLAC 7.0 results. It

Backfill type B1 χ² = 0.66

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Parameter a

222

0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

Backfill type B2 χ² = 0.88

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

W0.75/H

Backfill type B1 χ² = 0.76

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

W/H1.5 Fig. 16. Relationship between the coefficients a and b and the wall geometry.

Fig. 17. Comparison between results from Eq. 5 and FLAC 7.0 models.

W/H2

M. Deyanova et al. / Soil Dynamics and Earthquake Engineering 80 (2016) 210–224

also shows that Eq. (5) tends to overestimate the horizontal wall displacements for very small values of Arias Intensity. It should be remarked that Eq. (5) is valid within the limitations and assumptions of the present study. More numerical analysed and investigations are needed to confirm its validity, especially with regards to the values of the coefficients in Table 8. Yet, this relation clearly shows the influence of the wall geometry and Arias Intensity in determining the residual displacement of the wall. EN 1998-5 [20] provisions do not give any recommendations for using Arias Intensity as a ground motion intensity measure. For design purposes to estimate the displacement of the wall Eq. (5) could be used together with an appropriate attenuation relationship for Ia [49–51].

4. Conclusions and recommendations for further research The paper presents the results of a parametric study aimed to assess the dynamic response of trapezium earth-retaining gravity walls. The analyses were carried out using advanced numerical modelling. Seventeen sets of ten fully non-linear time-history analyses were performed using different wall dimensions, 10 spectrum compatible records, 2 types of backfill (loose sand and dense sand) and one type of soil for the foundation (dense sand). The results were compared with well-known simplified procedures for design of gravity retaining walls. Furthermore, the sensitivity of the residual displacements to different IMs was assessed. The following conclusions were reached: 1. The residual horizontal displacements from a set of 10 spectrum-compatible records vary significantly (one order of magnitude, just from 5 cm to 55 cm), which raises questions about what is the controlling ground motion intensity measure. 2. The ground settlement behind the wall is not proportional to the amount of horizontal displacement or tilting, and the disturbed portion of the backfill spreads away from the wall at a distance greater than the height of the wall. 3. Within the frame of the present study, the following design recommendation are proposed: if the wall with backfill of dense sand is designed for static loads with an over-design factor (ODFsliding) for sliding greater than 1.3 or the wall with backfill of loose sand is designed with an ODFsliding 41.2 (according to [21]), the wall is unlikely to fail in regions of medium-high seismicity with PGA ranging from 0.2 g to 0.35 g. Failure means either loss of bearing capacity under the toe of the wall or a residual horizontal displacements greater than 10% of the height of the wall. 4. If it is assumed according to EN 1998-5 [20] that the residual horizontal displacements can be calculated as the product of the allowable displacement dr and the ratio between the design horizontal forces and the resisting forces against sliding Ed/Rd, EN 1998-5 procedure tends to underestimate the residual displacements. 5. A comparison of the advanced numerical analyses with some wellknown Newmark's block-on-plane methods shows that when the yielding accelerations is calculated from static equilibrium using the M–O soil wedge, these methods tend to underestimate the permanent horizontal displacement of the wall. One of the reasons is that the simplified procedures do not take into account deformation of the base soil and as such they tend to underestimate the amount of soil mobilised during the seismic excitation. 6. Among the discussed Newmark's block-on-plane methods, the residual displacements calculated using [34] with the records of the highest PGA are closest to the displacements obtained from the FLAC 7.0 numerical analyses.

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7. The results from advanced numerical modelling show that a relationship between the horizontal residual displacements and PGA could hardly be established. However, a good correlation pattern was observed between permanent displacement and Arias Intensity (Ia). 8. From a number of regression analyses, an empirical relationship was proposed, which allows to compute the residual horizontal displacement as a function of the wall height, wall base width, type of backfill and Arias Intensity. This relation could be used for preliminary design although further analyses and investigations are needed to confirm its validity. Specifically it is of foremost importance to enlarge the parametric study to include walls having different geometries, founded on different soil types and subjected to a broader range of ground motions. The influence on the wall response of soil cohesion, polarity of the seismic signal, possible sloping of the backfill and inclined wall backs is worth investigation. An important outcome from the present study is that a gravity earth-retaining wall behaves under seismic loading as an integrated system of three parts – the rigid wall, the non-linear soil wedge and the adjacent foundation soil. Good understanding of the deformability and capacity of these three components, their interaction, how different intensity measures influence their relative importance also in relation to the occurrence of different failure modes, is the key to displacement-based design of gravity retaining walls.

Acknowledgements The work presented in this paper was partially supported by the financial contribution of the Italian Department of Civil Protection. This support is greatly acknowledged by the authors.

References [1] Fang YS, Yang YC, Chen TJ. Retaining walls damaged in the Chi-Chi earthquake. Can Geotech J 2003;40:1142–53. [2] Zhang J, Qu H, Liao Y, Ma Y. Seismic damage of earth structures of road engineering in the 2008 Wenchuan earthquake. Environ Earth Sci 2012;65:987–93. [3] Andersen GR, Whitman RV, Germaine JT. Tilting response of centrifugemodeled gravity retaining wall to seismic shaking: description of tests and initial analysis of results. Cambridge, MA: Department of Civil Engineering, Massachusetts Institute of technology; 1987. [4] Cascone E, Maugeri M. Saking table tests of gravity retaining walls. In: Cakmak AS, Brebbia CA, (Eds.), Proceedings of the seventh international conference on soil dynamics and earthquake engineering (SDEE 95); 1995 p. 567– 574. [5] Simonelli AL. Retaining walls under seismic actions: shaking table testing and numerical approaches. In: Proceedings of 12WCEE. Auckland, New Zealand; 2000. [6] Carafa P, Simonelli AL, Crewe AJ, Shaking table tests of gravity retaining walls to verify a displacement based design approach. In: Proceedings of the 11th european conference on earthquake engineering. Rotterdam: Balkema; 1998. [7] Watanabe K, Munaf Y, Koseki J, Tateyama M, Kojima K. Behaviors of several types of model retaining walls subjected to irregular excitation. J Jpn Geotech Soc: Soils Found 2003;43:13–27. [8] Grasso ASL, Maugeri M, Motta E. Experimental results on earth pressures on rigid wall under seismic condition. In: Proceedings of 13WCEE. Vancouver, Canada; 2004. [9] Al-Homoud A, Whitman R. Seismic analysis and design of rigid bridge abutments considering rotation and sliding incorporating non-linear soil behavior. Soil Dyn Earthq Eng 1999;18:247–77. [10] Psarropoulos P, Klonaris G, Gazetas G. Seismic earth pressures on rigid and flexible retaining walls. Soil Dyn Earthq Eng 2005;25:795–809. [11] Corigliano M, Lai CG, Pasquali R. Seismic response of gravity walls using Newmark-type modelling with double-support excitation. In: Pender MichaelJ, editor. Soil-Foundation-Structure Interaction. CRC Press; 2010. p. 141–8. [12] Tiznado JC, Rodríguez-Roa F. Seismic lateral movement prediction for gravity retaining walls on granular soils. Soil Dyn Earthq Eng 2011;31:391–400.

224

M. Deyanova et al. / Soil Dynamics and Earthquake Engineering 80 (2016) 210–224

[13] Al-Homoud A, Tahtamoni W. Comparison between predictions using different simplified Newmarks' block-on-plane models and field values of earthquake induced displacements. Soil Dyn Earthq Eng 2000;19:73–90. [14] Kramer SL. Geotechnical earthquake engineering. Upper Saddle River, NJ: Prentice Hall; 1996. [15] Matasovic N, Kavazanjian Jr E, Yan L. Newmark deformation analysis with degrading yield acceleration. In: Proceedings of Geosynthetics'97. Long Beach California, USA; 1997. p. 989–1000. [16] Anderson DG, Martin GR, Lam IP, Wang J. Seismic analysis and design of retaining walls, buried structures, slopes, and embankments (NCHRP report 611). Washington, D.C.: Transportation Research Board; 2008. [17] Chen W-F, Liu X. Limit analysis in soil mechanics. The Netherlands: Elsevier, Amsterdam; 2012. [18] Kramer S, Smith M. Modified Newmark model for seismic displacements of compliant slopes. J Geotech Geoenviron Eng 1997;123:635–44. [19] Gazetas G, Uddin N. Permanent deformation on pre-existing sliding surfaces in dams. J Geotech Eng 1994;120:2041–61. [20] European Committee for Standardization. EN 1998-5:2004: E. Eurocode 8: design of structures for earthquake resistance. Part 5: foundations, retaining structures and geotechnical aspects. Brussels: CEN; 2004. [21] European Committee for Standardization. EN 1997-1:2004: E. Eurocode 7: geotechnical design. Part 1: general rules. Brussels: CEN; 2004. [22] Guidance on the seismic design of retaining structures for residential sites in Greater Christchurch. Supplementary guidance. Version 2. Ministry of business, innovation & employment; 2014. [23] Il Ministro delle infrastrutture, Il Ministro dell'interno, Il capo del dipartimento della protezione civile, NTC 2008. Norme tecniche per le construzioni, in, Il MInistro dei Lavori Pubblici, 14/01/2008. [24] Rampello S, Silvestri F. Force-based pseudo-static methods versus displacement-based methods for slope stability. Analysis, edited by Cosenza, Eurocode 8; 2009. p. 249–262. [25] LRFD Seismic analysis and design of transportation geotechnical features and structural foundations. U.S. Department of Transportation. Federal Highway Administration; 2011. [26] Dobry R, Idriss I, Ng E. Duration characteristics of horizontal components of strong-motion earthquake records. Bull Seismol Soc Am 1978;68:1487–520. [27] Smith W. Database of Newmark displacements. Arias intensities, and Dobry durations for selected horizontal-component strong-motion records. US Geological Survey; 1994. [28] Jibson RW. Predicting earthquake-induced landslide displacements using Newmark's sliding block analysis. Washington D.C.: Transportation Research Board, National Research Council; 1993. p. 9–17. [29] Garini E, Gazetas G, Anastasopoulos I. Asymmetric'Newmark' sliding caused by motions containing severe'directivity' and'fling' pulses. Geotechnique 2011;61:733–56. [30] Newmark NM. Effects of earthquakes on dams and embankments. Géotechnique 1965;15:139–60. [31] Sarma SK. Seismic stability of earth dams and embankments. Geotechnique 1975;XXV:743–61. [32] Richards Jr. R, Elms DG. Seismic behavior of gravity retaining walls. J Geotech Eng Div, ASCE, 105; 1979. p. 449–64.

[33] Zarrabi-Kashani K. Sliding of gravity retaining wall during earthquakes: considering vertical acceleration and changing inclination of failure surface. Cambridge: Massachusetts Institute of Technology; 1979. [34] Nadim F, Whitman R. Seismically induced movement of retaining walls. J Geotech Eng 1983;109:915–31. [35] Whitman RV, Liao S. Seismic design of gravity retaining walls. Cambridge, Massachusetts: Massachusetts Institute of Technology Cambridge, Department of Civil Engineering; 1985. [36] Cai Z, Bathurst RJ. Deterministic sliding block methods for estimating seismic displacements of earth structures. Soil Dyn Earthqu Eng 1996;15:255–68. [37] Wong CP. Seismic analysis and an improved seismic design procedure for gravity retaining walls. Massachusetts: Department of Civil Engineering, Massachusetts Institute of Technology Cambridge; 1982. [38] Huang CC, Wu SH, Wu HJ. Seismic displacement criterion for soil retaining walls based on soil strength mobilization. J Geotechn Geoenviron Eng 2009;135:74–83. [39] Wu Y, Prakash S. On seismic displacements of rigid retaining walls. Geotech Spec Publ 1996(Issue 60):21–37. [40] FLAC. Fast Lagrangian analysis of Continua. User's manual. [41] Bowles JE. Physical and geotechnical properties of soils; 1979. [42] Kulhawy FH, Mayne PW. Manual on estimating soil properties for foundation design; 1990. Medium: X; Size: p. 298. [43] Ishihara K. Soil behaviour in earthquake geotechnics. Oxford; New York: Clarendon Press; 1996. [44] Darendeli MB. Development of a new family of normalized modulus reduction and material damping curves. US: ProQuest, UMI Dissertations Publishing, University of Texas at Austin; 2001. [45] European Committee for Standardization. EN 1998-1:2004: E. Eurocode 8: design of structures for earthquake resistance. Part 1: General rules, seismic actions and rules for buildings. Brussels: CEN; 2004. [46] Yan L, Matasovic N, KavazanjianJr E. Seismic response of rigid block on inclined plane to vertical and horizontal ground motions acting simultaneously. In: Proceedings of the Eleventh ASCE Engineering Mechanics Conference; 1996. p. 1110–1113. [47] Kottke AR, Rathje EM. Technical manual for Strata. Berkeley: University of California; 2008. [48] Jibson RW, Jibson MW. Java Programs for using Newmark's method and simplified decoupled analysis to model slope performance during earthquakes. In: Geological Survey (U.S.); 2003. Available on CD-ROM or Online. [49] Travasarou T, Bray JD, Abrahamson NA. Empirical attenuation relationship for Arias Intensity. Earthq Eng Struct Dyn 2003;32:1133–55. [50] Martinez-Rueda JE, Moutsokapas G, Tsantali E. Predictive equations to estimate arias intensity and cumulative absolute velocity as a function of housner intensity. In: Proceedings of AIP Conference; 2008. 1020 p. 309–316. [51] Foulser‐Piggott R, Stafford PJ. A predictive model for Arias intensity at multiple sites and consideration of spatial correlations. Earthq Eng Struct Dyn 2012;41:431–51. [52] Matasovic N, Kavazanjian E, Giroud J. Newmark seismic deformation analysis for geosynthetic covers. Geosynth Int 1998;5:237–64. [53] Zuccolo E. Personal communications; 2013.