Evaluation of active and passive seismic earth pressures considering internal friction and cohesion

Soil Dynamics and Earthquake Engineering 70 (2015) 30–47

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Soil Dynamics and Earthquake Engineering journal homepage: www.elsevier.com/locate/soildyn

Evaluation of active and passive seismic earth pressures considering internal friction and cohesion Shi-Yu Xu a,b, Anoosh Shamsabadi c, Ertugrul Taciroglu a,n a

Civil & Env. Engineering Department, University of California, Los Angeles, CA 90095, USA Architecture & Civil Engineering Department, City University of Hong Kong, Hong Kong c Caltrans, Office of Earthquake Engineering, Sacramento, CA 95816, USA b

art ic l e i nf o

a b s t r a c t

Article history: Received 19 March 2013 Received in revised form 15 August 2014 Accepted 10 November 2014

The Log-Spiral-Rankine (LSR) model, which is a generalized formulation for assessing the active and passive seismic earth pressures considering the internal friction and cohesion of backfill soil, is reviewed and improved in this study. System inconsistencies in the LSR model are identified, which result from an inaccurate assumption on the vertical normal stress field (σz ¼ γz) in a general c–ϕ soil medium, and from omitting the effect of soil cohesion when solving for the stress states along the failure surface. The remedies to the said inconsistencies are presented, and local and global iteration schemes are introduced to solve the resulting highly coupled multivariate nonlinear system of equations. The modified LSR model provides a more complete and accurate solution for earth retaining systems, including the geometry of the mobilized soil body, the stress state along the failure surface, as well as the magnitude and the point of application of the resultant earth thrust. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Earth pressure Earth force Retaining wall Limit equilibrium Method of slices Cohesion c–ϕ soil Log-Spiral-Rankine model Mononobe–Okabe model

1. Introduction Earth retaining systems are widely used in the construction of civil structures, primarily to maintain the stability of soils adjacent to excavation sites. An earth retaining system must possess sufficient capacity against various failure modes including sliding, bearing, eccentricity, as well as structural failure modes [1,4,5]. When proportioning the structural elements of earth retaining systems, the most important factor considered is the magnitude of the lateral earth thrust. This lateral force is termed passive if the wall moves toward to soil mass, and active if the soil mass moves toward to wall. An accurate estimation of the earth force is essential to achieve a safe and economical earth retaining system design. The prevailing analytical models for evaluation of active and passive earth pressures can be classified into two categories based on the presumed geometries of their failure surfaces. The models in the first category ([6,12,14,16]; and the trial wedge method outlined in [1] and [5]) assume a planar sliding surface in the backfill material when the earth retaining system fails. This

n

Corresponding author. E-mail addresses: [email protected] (S.-Y. Xu), [email protected] (A. Shamsabadi), [email protected] (E. Taciroglu). http://dx.doi.org/10.1016/j.soildyn.2014.11.004 0267-7261/& 2014 Elsevier Ltd. All rights reserved.

assumption dictates that the rupture plane of the soil elements located anywhere along the failure surface is the same plane. As a consequence of this assumption—and if the soil obeys the Mohr– Coulomb failure criterion—the stress state of the soil (i.e., σx, σz, and τxz) at any point along the failure surface may be represented by a single Mohr circle. In other words, the assumption of a planar failure surface in backfill soil implies that the behavior of the mobilized soil body resembles that of a deformable solid subject to a single stress field. This is clearly inadequate, as it has been disclosed by Mylonakis et al. [13]. Not surprisingly, these models have been reported to yield non-conservative or inaccurate estimations [3,7,18], especially in the passive case, or when the wall– soil interface roughness is taken into account. The error associated with these theories is primarily attributable to the fact that the actual sliding surface in the backfill soil medium is a curve [21] rather than a plane as assumed. The models in the second category [10,11,13,15,23,26] mend this drawback either explicitly by adopting a curvilinear sliding surface, or implicitly, by introducing multiple stress fields in the backfill soil. They significantly improve the prediction on the passive earth force. Nevertheless, the adoption of a curvilinear failure surface in general will render the problem statically indeterminate, and consequently many of these models are derived using various auxiliary assumptions that either require further investigation, or are valid only under limited conditions (for a more

S.-Y. Xu et al. / Soil Dynamics and Earthquake Engineering 70 (2015) 30–47

detailed exposition see, for example, [22]). More importantly, most of the prevailing analytical models are applicable only to retaining structures backfilled with granular soils with no cohesion. Yet, a geotechnical investigation [9], conducted in 2005 to characterize the structural backfills behind existing bridge abutments in the State of California, concluded that the existing backfills soils are predominantly c–ϕ materials (silty to clayey sands with gravel) that exhibit some to significant amount of cohesion. It has been reported that with the presence of soil cohesion—even if it is small in magnitude—the passive force will be considerably increased and the active drastically decreased [22]. Therefore, by treating backfills as cohesionless soils, earth retaining structures are nominally over-designed—a conclusion that may be drawn from the observations that retaining walls seldom failed in the past earthquake events, even though those walls were not explicitly designed for seismic loading [24]. Shamsabadi et al. [22] proposed a generalized Log-SpiralRankine (LSR) model for evaluation of seismic earth forces. They adopted a composite failure surface comprising a logarithmic spiral curve and a linear segment whose specific geometry is derived based on the stress states at the two boundaries of failure surface— i.e., in the regions next to the back-face of the retaining structure and that close to the ground surface. The model was originally proposed for general c–ϕ backfill soils in passive case under static conditions [19,20], and it had been experimentally validated with data from several field-test programs [21]. This static model was later extended by the authors [22] to accommodate both the passive and active cases under pseudo-static earthquake loads. This dynamic LSR model had also been verified numerically against six existing analytical models, and was demonstrated to yield reliable predictions for both passive and active cases. The model, however, has been found to produce an inconsistent system whose force polygon does not close if the resultant forces acting along the failure surface is back-calculated from the solved stress states. The inconsistency is primarily due to various simplifying assumptions used in the derivation of the model. In this paper, the causes to the system inconsistencies of the LSR model are investigated, their effects are quantified, and remedies are provided. The said inconsistencies are attributed primarily to the following three sources: (1) the adoption of a simplified vertical normal stress field—i.e., σz ¼γz—in the c–ϕ soil; (2) the indeterminacy of the geometry of the mobilized soil body when the earthquake induced inertial forces are present; and (3) omission of the soil cohesion and consideration of only the soil friction when solving for the shear stress field. Three local/global iterative schemes will be introduced herein to address the aforementioned issues. Following the proposed modified approach, accurate stress states along the failure surface can now be fully defined, and the stress concentration phenomenon at the bottom of retaining wall can be captured. The point of application of the earth force is also examined, and the factors affecting the location of this point are studied.

2. A review of the Log-Spiral-Rankine (LSR) model In the generalized LSR model [22], the failure surface behind a retaining wall subject to passive or active forces in a homogeneous backfill soil medium is assumed to be composed of two regions— viz., the logarithmic spiral region and the Rankine zone, as illustrated in Fig. 1. 2.1. Stress state in Rankine zone and its associated failure surface The triangular region of the mobilized soil body is referred to as the Rankine zone because the shear stress (τxz) in this region is induced solely by the body forces (including the gravitational and the seismic/inertial forces) without any contribution from the

31

inter-particle friction or cohesion—a stress state similar to that of the classical Rankine Theory [16]. The stress equilibrium equations of the soil medium in this region are identical to those of the deformable solid described as [17] ∂σ x ∂τxz þ ¼ ρ ð  ah Þ ¼ kh γ ∂x ∂z

ð1Þ

∂τxz ∂σ z þ ¼ ρðg  av Þ ¼ ð1  kv Þγ ∂x ∂z

ð2Þ

where γ is the soil unit weight; kh and kv are the coefficients of the horizontal and vertical ground accelerations, defined as kh  ah/g and kv  av/g shown in Fig. 1. The stress fields conforming to the governing equations are p¼0

σ z ¼ ð1  kv Þγz þp - σ z ¼ ð1  kv Þγz

ð3Þ

p¼0

σ x  K R σ z ¼ K R ½ð1  kv Þγz þ p - σ x ¼ K R ð1  kv Þγz s¼0

τxz ¼ kh γz þs - τxz ¼

kh σz ð1  kv Þ

ð4Þ ð5Þ

where p and s are the normal and shear surface tractions applied on top of the Rankine zone; KR is the coefficient of lateral earth pressure defined as KR  σx/σz. In order to keep the derivation brief here, p and s are henceforth set to zeros. If the soil follows the Mohr–Coulomb failure criterion, then the normal stress (σf) and shear stress (τf) pair acting on the failure surface of Rankine zone must satisfy τf ¼ σ f tan ϕ þ c:

ð6Þ

The corresponding stress state is demonstrated on the Mohr circle shown in Fig. 2. Constants c and ϕ in Eq. (6) are the cohesion and the frictional angle of the soil, respectively. Eq. (6) implicitly defines the relationship among σx, σz, and τxz of the soil at failure, and it can be reinterpreted as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ  σ 2  σx þ σz  x z : ð7Þ sin ϕ ¼ τ2xz þ = c cot ϕ þ 2 2 Substituting Eqs. (3)–(5) into Eq. (7), the lateral earth pressure coefficient at the Rankine zone, KR, can be solved, as [17] KR 

σ x 1 þ sin 2 ϕ c þ 2 tan ϕ ¼ σz σz cos 2 ϕ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    2 2 c 2 kh tan ϕ þ þj  cos ϕ σz 1  kv

;

j ¼ 71

ð8Þ

where σz ¼ 1/2(1  kv)γhR is the average vertical normal stress along the segment BE,1 and hR is the length of segment BE. For KR to be a real number, it is required that [tan(ϕ)þ c/σz]2 4 [kh/(1  kv)]2; otherwise, it is impossible to develop the prescribed earthquake induced stress fields (i.e., Eqs. (3)–(5)) within the soil medium. In Eq. (8), when j¼ þ 1, σx is greater than σz and this corresponds to the stress state for the passive case; when j ¼  1 (i.e., the active case), the converse is true. This convention on j will be maintained throughout the paper. In addition, following the sign convention (which must be enforced for Eq. (1)) defined in Fig. 1, one should assign a negative value to kh when computing the active pressures, as it is the critical condition in the active cases. By applying the Pole Method, the failure surface can be identified on the Mohr circle as illustrated in Fig. 2. The inclined angle of the failure surface of the Rankine zone (αR) can then be calculated 1 As evidenced by Eq. (8), the stress distribution σx along BE is not a linear function of depth z. Therefore, the average stress along face BE is not exactly equal to half of the stress at point E.

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individual slice. Thus, the stresses used in Eq. (8) are the average stresses acting over the vertical face BE.

from tan ð2αRp Þ ¼

αRp

τxz 2τxz ¼ ðσ x  σ z Þ=2 ðK R  1Þσ z

ð9aÞ

    1 2ðkh γ hR =2Þ 1 2 kh ¼ tan  1 ¼ tan  1 2 ðK R  1Þð1  kv ÞγhR =2 2 ðK R  1Þ ð1  kv Þ ð9bÞ

αR ¼ ð451  j ϕ=2Þ  αRp

ð10Þ

Under the static condition, kh is zero and so is αRp . This implies that the inclination of the failure surface under the seismic condition is rotated by an additional angle αRp from its orientation under the static condition. Remark: Although the active and passive limit stress states are very similar from a mathematical point of view, they differ considerably in their mechanical features. In the active case, the upper part of the Rankine wedge is subjected to tractions and this, in reality, leads to the opening of tension cracks. These cracks do not occur in the passive case. The proposed method is unable to capture this phenomenon because of the constraints adopted in the method of slices. In this approach, the mobilized soil mass is discretized into a few vertical soil slices. This implies that the best the approach can provide is to predict the stress distribution along the horizontal direction. Unless the slices are further discretized into dices (i.e., full spatial discretization), it is not possible to capture the stress distribution along the vertical face of an

2.2. Geometry of the log-spiral region and the stress state at the bottom of the wall–soil interface When the rupture plane develops within the soil medium behind the retaining wall, the frictional and cohesive forces acting along the rupture plane tend to resist the relative movement between the mobilized soil body and the resting soil medium. Furthermore, the resultant of the frictional and normal forces (i.e., Ri in Fig. 3) acting on any point along the failure surface must deviate at an angle ϕ from the normal direction of the failure surface so that—together with the cohesive forces (i.e., cLi in Fig. 3) —the produced shear and normal stress pair will comply with the Mohr–Coulomb failure criterion defined by Eq. (6). Therefore, the directions of these forces can be determined depending on whether the retaining wall is pushed toward (i.e., passive case, Fig. 3a) or move away from (i.e., active case, Fig. 3b) the soil medium. If the resultant forces Ri—excluding the contribution from cohesion—are assumed to point to the same locus, then the curvilinear portion of the failure surface (see, Fig. 1) can be described by the logarithmic spiral function [25] given by r i ¼ r 0 eΔθi

tan ðj ϕÞ

:

ð11Þ

Likewise, the directions of shear forces acting on the back-face of the retaining wall (including the wall–soil interface frictional

Fig. 1. Geometry of the mobilized soil body. (a) Under passive force; (b) under active force.

Fig. 2. Stress state of soil at failure in Rankine zone. (a) Under passive force; (b) under active force.

S.-Y. Xu et al. / Soil Dynamics and Earthquake Engineering 70 (2015) 30–47

and adhesive forces) can also be determined based on their tendency to resist the relative movement (see Fig. 3). Assume that the frictional and adhesive forces at the wall–soil interface are fully mobilized when the log-spiral-linear failure surface forms, and that the normal and shear stresses acting on the vertical face of an infinitesimal element located at the bottom of the retaining wall can be approximated by the average normal and shear stresses acting on the wall–soil interface as given in τxz ¼ ca þ σ x tan ðδw Þ:

ð12Þ

It follows that at failure, the stress state of the soil element located at the bottom of the wall can be plotted on the Mohr circle as displayed in Fig. 4. Substituting Eq. (12) and σx ¼Kwσz into Eq. (7), the lateral earth pressure coefficient at the bottom of wall– soil interface (Kw) is obtained, as in Kw 

σ x 1 þ sin 2 ϕ þ ðc=σ z Þ sin ð2ϕÞ  4ðca =σ z Þ tan ðδw Þ þ j 2 cos ϕ ¼ σz cos 2 ϕ þ 4 tan 2 ðδw Þ

pffiffiffiffi Δ

ð13aÞ    c 2 ca 2  tan δw þ σz σz    c c ca þ 4 tan δw tan ϕ þ tan δw  tan ϕ σz σz σz

is considered as the static solution; therefore, the static vertical normal stress, σz ¼γH, is applied. Inspired by the graphical solution approach [12], the inclination of the failure surface in this region under seismic condition should be adjusted by the same additional rotation from its orientation under static condition as observed in the Rankine zone. The inclined angle of the failure surface at the bottom of wall–soil interface, αw, can thus be calculated from   1  1 2K w tan δw tan ; ð14Þ ¼ αw p 2 j ðK w  1Þ αstatic ¼ ð451  j ϕ=2Þ  αw w p;

αseismic ¼ αstatic  αRp w w

ð15a; bÞ

Once the subtended angle (i.e., θm in Fig. 1) of the logarithmic spiral curve DE is determined from the inclined angle of the failure surface at Rankine zone (αR) along with the takeoff angle at the bottom of the retaining wall (αw) as given in Eq. (16), the geometry of the entire mobilized soil body can be established through Eq. (11) θm ¼ ηw  ηR ¼ αR αw

 Δ¼

33

ð16Þ

tan ϕ þ

ð13bÞ

For Kw to be a real number, it requires the value of Δ (as in Eq. (13b)) to be positive. Since Eq. (13) is the exact solution of σx/σz ratio at point D when the seismic force is absent (i.e., kh ¼kv ¼0), it

2.3. Stress state of soil along failure surface of log-spiral region The stress states of soil located at the failure surface of Rankine zone and at the bottom of the wall–soil interface are examined in Sections 2.1 and 2.2 in order to gather the required information to determine the geometry of the mobilized soil body. In this section,

Fig. 3. Equilibrium of the mobilized soil body. (a) Under passive force; (b) under active force.

Fig. 4. Stress state of soil at failure at the bottom of wall–soil interface. (a) Under passive force; (b) under active force.

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S.-Y. Xu et al. / Soil Dynamics and Earthquake Engineering 70 (2015) 30–47

Fig. 5. Typical stress state of soil along the failure surface. (a) Under passive force; (b) under active force.

the log-spiral region is discretized into vertical slices as illustrated in Fig. 3. The stress state at the bottom of each individual slice will have to be inspected to derive the additional equations needed in the estimation of the inter-slice normal and shear forces. The typical stress state of the soil at the failure surface of the log-spiral region is drawn in Fig. 5. It is very similar to that at the bottom of the wall–soil interface except now the inclined angle of failure surface (αi) is known and the shear stress acting on the vertical plane (τxz) is what yet to be solved. The shear stress acting on the vertical plane includes the contributions from both the inter-particle friction and cohesion, and it is expressed in terms of the product of the normal stress times tanδi, or τxz ¼σxtanδi. The angle δi is known as the inter-slice shear angle and is annotated in Mohr's circle shown in Fig. 5. It is related to the rotation of the principal planes αp through an angle ω, as given in δi ¼ j ð2αp  ωÞ:

ð17Þ

The rotation of the principal planes αp can be solved from αp ¼ ð451  j ϕ=2Þ  αi ; and the angle ω can be calculated through

ðσ x þ σ z Þ=2 sin ðδi Þ λ

sin ðωÞ ¼ ¼  r ðσ x þ σ z Þ=2 þ c cotðϕÞ sin ðϕÞ sin ðδi Þ

¼ sin ðϕÞ þ c=ððσ x þ σ z Þ=2Þ cos ðϕÞ λ sin ðδi Þ sin ðωÞ ¼  r sin ðϕÞ

ð18Þ

ð19aÞ

ð19bÞ

For a c–ϕ backfill soil, the value of the second term in the denominator of Eq. (19a) is approximately equal to zero [19]; thus Eq. (19b) is obtained. By substituting Eqs. (18) and (19b) into Eq. (17), a nonlinear equation of the inter-slice shear angle for noncohesive soil is derived, which is shown in Eq. (20). This equation can be solved iteratively through Newton's method [22]    sin ðδi Þ δi ¼ j ð901  j ϕÞ  2αi  sin  1 ðfor ϕ a 0Þ ð20Þ sin ðϕÞ Finally, the horizontal normal stress, σx, can be calculated from Eq. (21), which is derived by substituting τxz ¼σxtan(δi) and σx ¼Kiσz into Eq. (7) n σx 1 2 1 þ sin ϕ þ σcz sin ð2ϕÞ þ ::: ¼ 2 2 σz cos ϕ þ4 tan ðδi Þ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi "   #9 u  = u c 2 c 2 c 2 t þ tan ðδi Þ 4 þ 4 tan ϕ  1 :::j 2 cos ϕ tan ϕ þ ; σz σz σz

Ki 

ð21Þ

As it can be observed from the Mohr circles shown in Figs. 2, 4, and 5, as well as the equations for the coefficient of lateral earth pressure given in Eqs. (8), (13), and (21), it is clear that the stress states along the failure surface of the mobilized soil body cannot simply be represented by a single stress field. Consequently, the models in the first category that were mentioned in the introduction suffer from their very basic assumption of a planar failure surface, and are, therefore, unable to realistically capture the stress distribution within the mobilized soil body. 2.4. Inter-slice normal and shear forces For an intermediate slice labeled as the ith slice (see Fig. 3), the force equilibrium of the slice in the horizontal and vertical directions are ∑F x ¼ 0 ) dEi  Ei Ei  1 ¼ F ix þ j c Li cos ðαi Þ  kh W i

ð22Þ

∑F z ¼ 0 ) dT i  T i  T i  1 ¼ j F iz  j ð1  kv ÞW i  c Li sin ðαi Þ

ð23Þ

where Ei and Ti are the inter-slice normal and shear forces acting on the left face of the ith slice; Ei 1 and Ti 1 are those acting on the right face; F ix and F iz are the horizontal and vertical components of the resultant force Ri (of normal and frictional forces combined) acting at the bottom of the ith slice; cLi is the cohesive force acting along the failure surface at the bottom of the ith slice; Wi is the self-weight; and αi is the inclined angle of the failure surface at the bottom of the ith slice. The increments in the inter-slice normal and shear forces from the right face of the ith slice to its left face are denoted as dEi  Ei  Ei 1 and dTi  Ti  Ti 1, respectively. By relating F iz to F ix through the angle (αi þjϕ), Eq. (23) can be rearranged to yield ∑F z ¼ 0 ) dT i þ j ð1  kv Þ W i þc Li sin ðαi Þ ¼ j F iz ¼ j F ix = tan ðαi þ j ϕÞ: ð24Þ 2

An approximation is then made to associate dEi in Eq. (22) with dTi in Eq. (24), as in dT i ¼ dEi tan ðδi Þ:

ð25Þ F ix

and Substituting Eqs. (22) and (25) into Eq. (24) to eliminate dTi, the formula for the incremental normal force dEi can be 2 Eq. (25) is derived from the equation, Ti ¼ Eitan(δi) (see [22] for details). Realistically, δi is not a constant along a vertical line. Nonetheless, unless the vertical slices are further cut into dices, the proposed method cannot capture the stress distribution along the vertical faces of the slices. As such, it is necessary for the proposed method to keep the simplification assuming δi to be a constant. Although imperfect, this is a reasonable approximation if the wall is not too flexible.

S.-Y. Xu et al. / Soil Dynamics and Earthquake Engineering 70 (2015) 30–47 n1 

obtained, as given in 

ð1  kv ÞW i tan ðαi þj ϕÞ ðkh =ð1 kv ÞÞ þ jc dxi ½1 þ tan ðαi Þ tan ðαi þj ϕÞ dEi ¼ 1 j tan ðδi Þ tan ðαi þjϕÞ

CT ¼ ∑

i¼1

 j dT i ½ tan ðαi þ j ϕÞ  tan ðαw þj ϕÞ :

35

ð29eÞ

where dxi ( ¼Licosαi) is the width of the ith slice. The force F ix can then be calculated via Eq. (22), and the resultant force Ri via Ri ¼F ix / sin(αi þ jϕ). Remark: Note that Eq. (26) is not applicable to the slice adjacent to the retaining wall (i.e., the nth slice in Fig. 3) because the friction angle on the wall–soil interface (δw) can vary significantly from the inter-slice shear angle on its right face, δn  1, due to the abrupt change in the material properties. In other words, Eq. (25) is invalid at this slice.

The normal contact force Ph—also known as the horizontal passive or active earth force (when j¼ þ 1 or j ¼  1)—is contributed by four sources. These are the self-weight of the mobilized soil body, the adhesive force acting on the wall–soil interface, the cohesive forces acting along the failure surface, and the inter-slice shear forces due to inter-particle friction and cohesion. If the adhesive, cohesive, and inter-slice shear forces are neglected, then Shields & Tolunay's [23] solution will be recovered. The total earth force acting on the retaining wall is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P total ¼ P 2h þðP h tan δw Þ2 ¼ P h = cos ðδw Þ: ð30Þ

2.5. Contact forces at the wall–soil interface

The seismic earth pressure coefficient (KE) is, then, obtained by normalizing3 Ptotal with 1/2γH2, as in

ð26Þ

The force equilibrium equations of the nth slice—Eqs. (27a) and (28a)—can easily be derived from its free body diagram as illustrated in Fig. 3. Applying the relations Ei ¼Ei 1 þdEi and Ti ¼ Ti 1 þdTi to Eqs. (27a) and (28a) to eliminate the En 1 and Tn 1 terms, Eqs. (27b) and (28b) can be obtained. These two equations can be further rewritten to yield Eqs. (27c) and (28c) by utilizing the relationship given in Eqs. (22) and (23) ∑F x ¼ 0 : P h ¼ En  1 þ F nx þ j c Ln cos ðαn Þ  kh W n

ð27aÞ

or n1

P h ¼ ∑ dEi þ F nx þj c Ln cos ðαn Þ  kh W n i¼1

ð27bÞ

or i n h P h ¼ ∑ F ix þ j c Li cos ðαi Þ  kh W i

ð27cÞ

i¼1

∑F z ¼ 0 : P h tan δw þ ca H ¼ T n  1 þ j F nz  j ð1  kv Þ W n  c Ln sin αn ð28aÞ or n1

P h tan δw þ ca H ¼ ∑ dT i þj F nz  j ð1 kv ÞW n  c Ln sin αn

ð28bÞ

i

or i n h P h tan δw þ ca H ¼ ∑ j F iz  j ð1  kv ÞW i  c Li sin αi i¼1

ð28cÞ

It is useful to note that Eqs. (27c) and (28c)—although they are deduced from the equilibrium of the nth slice—essentially depict the global limit equilibrium conditions for the entire soil mass (see Fig. 3). Recalling that F nz is equal to F nx /tan(αw þjϕ) where F nx can be obtained from Eq. (27b), and that dEi in Eq. (27b) is related to dTi through Eq. (25), Eq. (28b) can be rearranged to produce the formula for the normal contact force Ph at the wall–soil interface, as given in Ph ¼

C W þC a þ C c þ C T ; 1  j tan ðδw Þ tan ðαw þ j ϕÞ

n

CW ¼ ∑

i¼1

 ð1  kv Þ W i tan ðαi þ j ϕÞ 

C a ¼ j ca H tan ðαw þ j ϕÞ ; n

Cc ¼ ∑

i¼1

ð29aÞ  kh ; ð1  kv Þ

j ¼ 71;

  j c dxi ½1 þ tan ðαi Þ tan ðαi þ j ϕÞ ;

ð29bÞ ð29cÞ ð29dÞ

KE ¼

P total ð0:5γH 2 Þ

¼

CW þ Ca þ Cc þ CT 2 : 1  j tan ðδw Þ tan ðαw þ j ϕÞ cos ðδw ÞγH 2

ð31Þ

2.6. Stress inconsistencies due to simplifying assumptions The LSR model had been validated against several quasi-static retaining wall test programs and showed very good agreement in the passive case [19–21]. It had also been verified numerically [22] against six existing analytical models, and displaying significant improvements over them—including the Rankine [16], Coulomb [6], Mononobe–Okabe [12,14], Mylonakis [13], Trial Wedge, and Shields and Tolunay [23] models. Nevertheless, it produces an inconsistent system in two respects, as described below. First, the stress state that the original LSR model yields at the failure surface of Rankine zone is inconsistent. In Eq. (8), an initial estimation of the height of the Rankine zone (hR) is required to compute KR. This information, however, is unavailable because at this point, the geometry of the mobilized soil body is unknown. After the geometry of the mobilized soil body is determined (see Section 2.2), a new value for hR will be obtained, which, in general, is different from the one originally assumed. This inconsistency in hR results into an inconsistency in the solution of the vertical normal stress (σz) at the Rankine zone, and the inconsistency in σz, in turn, produces inconsistencies in σx (i.e., that which is obtained from Eq. (8) vs. Eq. (21)) and in τxz (i.e., that which is computed from Eq. (5) vs. the approach introduced in Section 2.3) at the Rankine zone. Second, the inter-slice shear angles (Eq. (20)), derived based on the simplified equation as given in Eq. (19b) by dropping the effect of soil cohesion, do not reflect the real stress states along the failure surface. Consequently, the calculated incremental interslice normal and shear forces—dEi and dTi in Eqs. (26) and (25)— are inaccurate, leading to an incorrect solution of the resultant reaction forces, Ri. A similar situation also takes place at the bottom of the wall–soil interface. σz and τxz used in Eq. (13), together with the computed σx, do not produce a resultant reaction force, Rn (or equivalently, F nx and F nz ) that satisfies the equilibrium conditions as required by Eqs. (27) and (28). When these resultant reaction forces are drawn along with the inertial forces, the cohesive and adhesive forces, and the earth pressure forces, the summation of all the force vectors does not produce a closed polygon, as demonstrated in Fig. 6. As it can be observed from this figure, the magnitude of the unbalanced residual compared to that of the earth-pressure force is significant. 3 Ptotal is normalized by 0.5γH2 only to derive an earth pressure coefficient compatible with other analytical models; it does not mean that the stress distribution along the wall–soil interface is linear.

36

S.-Y. Xu et al. / Soil Dynamics and Earthquake Engineering 70 (2015) 30–47

0

0 -10

-5

-20

Vertical Inertia Force Cohesion along Failure Surface Reaction along Failure Surface

-40 -50

-10

Horizontal Inertia Force

-30

-15

Earth Force Unbalanced Residual= 14.9935 kips -80

-70

-60

-50

-40

-30

Unbalanced Residual= 1.0528 kips -20

-10

0

10

-10

-5

0

5

Fig. 6. Force polygon of the mobilized soil body (H¼ 150 , ϕ¼ 301, δw ¼ 201, c ¼0.1γH, kh ¼ j0.3). (a) Under passive force; (b) under active force.

A more precise, and inevitably more complex, solution approach that eliminates the aforementioned system inconsistencies is introduced in the following section.

3. Remedies to the system inconsistencies 3.1. Local iterations for searching the failure surface geometry The first source of the system inconsistency is due to the indeterminate height of Rankine zone, hR, when seismic forces are present. This error can be fixed by a simple local iteration scheme as explained below. Recall that in order to establish the geometry of the failure surface in the log-spiral region, the inclinations of the curve at its two boundaries—namely, the angles αR and αw—must be obtained a priori. To determine αR, an estimation of hR is required. In other words, for any trial hR with a given αw, one can obtain a corresponding log–spiral curve, which will yield a new hR. This new hR can be used to generate a new log–spiral curve, producing yet another hR. This procedure can be repeated until hR converges, which is a fixed-point iteration scheme [8]. 3.2. Local iterations for determining the correct stress state along the failure surface In a plane strain problem, the stress tensor has only three independent components (σx, σz, and τxz). The dynamic stress equilibrium equations and the associated stress fields within the solid when subjected to body forces are given in Eqs. (1)–(5). In geotechnical engineering, the typical simplifying assumption is that the vertical normal stress within any soil medium subject to gravitational and the seismic force fields is proportional to depth, or σz ¼(1 kv)γz. This vertical normal stress field is identical to that of a deformable elastic solid, and it may be appropriate for the Rankine zone, because the internal friction and cohesion of soil are not mobilized in this region. However, this is not the case for the soil in the log-spiral region. Because the second source of system inconsistencies described in Section 2.6 originates from inaccurate computation of the stress state of soil along the failure surface of the log-spiral region, we hereby remove the presumption of σz ¼(1 kv)γz, and treat the vertical normal stress as an independent variable. This removal eradicates all possible causes that result into inconsistent stress states. In other words, all the three stress components along the failure surface of the log-spiral region are now considered as independent unknown variables; three additional independent equations will, therefore, be required in order to solve them. The first equation comes from the Mohr–Coulomb failure criterion, which has been given through Eqs. (6) and (7). It is

rewritten in a quadratic form as follows:  σ  σ 2 σ x þσ z 2 x z γ 1  c cot ϕ þ sin 2 ϕ   τ2xz ¼ 0: 2 2

ð32Þ

After the geometry of the log–spiral curve is established, the inclined angle of the failure surface provides a second equation. To wit, if the radius of the Mohr circle be denoted as R (Fig. 5), then the coordinates of the failure point (σf ,  τf) can be expressed in terms of R as in ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r σx þ σz σ x  σ z 2  R sin ϕ ; where R ¼ þ τ2xz ; σf ¼ ð33aÞ 2 2 τf ¼ j R cos ϕ:

ð33bÞ

Since the coordinates of the pole is (σx,  τxz), the following expression can be derived from the geometry of the Mohr circle tan αi ¼

j R cos ϕ τxz : R sin ϕ þ ððσ x  σ z Þ=2Þ

ð34Þ

This equation can also be restated in quadratic form as in   2 σ  σ 2 σx  σz x z γ 2  τxz þ þ τ2xz ð cos ϕ  j tan αi sin ϕÞ2 ¼ 0: tan αi  2 2

ð35Þ The third equation can be constructed by examining the force equilibrium equations of the intermediate slices (see, Fig. 7a) in vertical and horizontal directions, which are ∑F x ¼ 0 ) dEi þkh W i ¼ σ x Li sin ðαi Þ þ τxz Li cos ðαi Þ;

ð36Þ

∑F z ¼ 0 ) dT i þ ð1  kv ÞW i ¼ σ z Li cos ðαi Þ þ τxz Li sin ðαi Þ:

ð37Þ

By utilizing the relationship dTi/dEi ¼tan(δi) ¼τxz/σx, Eqs. (36) and (37) can be combined to yield  ð38Þ γ 3  τ2xz  kh γhi τxz  σ z ð1 kv Þ γ hi σ x ¼ 0 where hi is the average height of the ith slice (see Fig. 7). The full stress state of soil at the failure surface can now be retrieved by solving Eqs. (32), (35), and (38), simultaneously. The coupled nonlinear system of equations obtained above can be solved using Newton's method. Let X  {σz, σx, τxz}T be the independent vector of the system, and Δγ  {γ1, γ2, γ3}T the unbalanced residual vector of the system evaluated at X. The L2-norm of the unbalanced residual vector (i.e., the magnitude of the Δγ vector) is equal to zero if and only if all the following three conditions are met: (i) the Mohr circle defined by σx, σz, and τxz touches the Mohr–Coulomb failure envelop; (ii) the inclined angle of the rupture plane in the soil caused by σx, σz, and τxz is equal to the given inclined angle of the failure surface, αi, and (iii) the stress resultant produced by σx, σz, and τxz is in equilibrium with all the external forces acting on the soil slice. The Jacobian of the coupled

S.-Y. Xu et al. / Soil Dynamics and Earthquake Engineering 70 (2015) 30–47

37

Fig. 7. Free body diagrams of the slices, and the stress states at bottom of slices. (a) Intermediate slices; (b) slice next to wall, under passive force; (c) slice next to wall, under active force.

nonlinear 2 ∂γ 1 6 ∂σ z 6 6 ∂γ 6 2 J6 6 ∂σ z 6 4 ∂γ 3 ∂σ z

system of equations is simply 3 ∂γ 1 ∂γ 1 ∂σ x ∂τxz 7 7 ∂γ 2 ∂γ 2 7 7 ∂σ x ∂τxz 7 7 7 ∂γ 3 ∂γ 3 5 ∂σ x ∂τxz

with components given by σ  σ  ∂γ 1  σ x þσ z  x z 2 sin ϕ þ ; etc: ¼ c cot ϕ þ ∂σ z 2 2

ð39Þ

will no longer be applied, because the σ ðvÞ z here is defined as a trial solution that already takes the seismic effects into account. Following the rest of the procedures introduced in Section 2, as well as those in Sections 3.1 and 3.2, the solution of the passive/ active earth force, P ðvÞ , can be obtained. h The force equilibrium equations between the external loads and the stress resultants at the bottom of the slice adjacent to the wall are then examined, which are given in Eqs. (42) and (43) below, and illustrated in Fig. 7b and c

ð40Þ

∑F x ¼ 0 ) P h  ∑ dEi þkh W n ¼ σ x dxn tan αw þj τxz dxn

The solution to the nonlinear system can then be obtained through the sequence h i1 X ðν þ 1Þ ¼ X ðνÞ  J ðνÞ Δγ ðνÞ ð41Þ

n1

ð42Þ

i¼1

n1

∑F z ¼ 0 ) j ðP h tan δw þ ca H  ∑ dT i Þ þð1 kv ÞW n ¼ σ z dxn þj τxz dxn tan αw i

ð43Þ

where superscript (ν) denotes the iteration number.

By singling out the jτxzdxn term from Eq. (42), and then by substituting it into Eq. (43), the formulas for the unbalanced residuals (γn) of the system are obtained as

3.3. Global iterations searching for the stress state at the bottom of the retaining wall

γ n ðσ z Þ  P h ðj tan δw  tan αw Þ þ j ca H  j ∑ dT i  tan αw ∑ dEi þ :::

In the original LSR model [22], the stress state at the bottom of the wall–soil interface under seismic condition was not scrutinized. The takeoff angle, αw, was computed based on a static stress state for the solid, σz ¼γH, and then adjusted by an additional rotation, αRp , to account for seismic effects. As a result, the resultant reaction force back calculated from the corresponding stress state at the failure surface did not satisfy the equilibrium equations of the slice next to the retaining wall. A global iteration scheme is developed next to rectify this last source of system inconsistency. Let σ ðvÞ z denote the trial solution of the vertical normal stress of the soil element located at the bottom of the wall–soil interface when subject to vertical and horizontal ground accelerations, where superscript (ν) is the iteration number. Assuming that at the instance of the retaining system failure, the interface friction and adhesion are fully mobilized. The average shear stress acting on the wall–soil interface will then be governed by Eq. (12). For a ðvÞ trial solution of σ ðvÞ z , the horizontal normal stress, σ x , can be calculated through Eq. (13). The takeoff angle, αðvÞ , can then be w solved from Eqs. (14) and (15a). It is useful to note that Eq. (15b)

n1

n1

i

i¼1



::: ð1  kv Þ kh tan αw W n  σ z dxn 1  K w tan 2 αw ¼ 0: A new trial solution,   γ n σ zðνÞ þ 1Þ ¼ σ zðνÞ   ; σ ðν z γ 0n σ zðνÞ

þ 1Þ , σ ðν z

!

ð44Þ

can be calculated through

where the derivative of γn with respect to σz is

γ 0n ðσ z Þ ¼ dxn K w tan 2 αw  1 :

ð45Þ

ð46Þ

By repeating this global iteration until σz converges, a consistent system will finally be achieved, as demonstrated in Fig. 8. The flow chart of the proposed iteration schemes is shown in Fig. 9. 4. Stress distribution in the mobilized soil body As pointed out in Section 3.2, when solving for the stress state at the failure surface of the intermediate slices, the presumption of the vertical normal stress field of deformable solid being

38

S.-Y. Xu et al. / Soil Dynamics and Earthquake Engineering 70 (2015) 30–47

20 0 10 0 -10 -20 -30

(1-kv)Wi in log-spiral region

-2

R1 in Rankine Zone

-4 Vertical Inertia Force Horizontal Inertia Force Cohesion along Failure Surface Reaction along Failure Surface Ri in log-spiral Earth Force region

-40 -100

R1 in Rankine Zone

Unbalanced Residual= 0.0000 kips

-80

-60

-40

-20

(1-kv)W1

in 0 Rankine Zone

(1-kv)Wi in

log-spiral-6 region -8

Rn in -10 Slice n -12

Rn in Slice n Ri in log-spiral region

Forces in Rankine Zone

-14 Unbalanced Residual= 0.0001 kips -10 -5 0

Fig. 8. Force polygon of the mobilized soil body (H¼ 150 , ϕ¼ 301, δw ¼ 201, c ¼0.1γH, kh ¼ j0.3). (a) Under passive force; (b) under active force.

Fig. 9. Flow chart of the proposed iteration scheme.

S.-Y. Xu et al. / Soil Dynamics and Earthquake Engineering 70 (2015) 30–47

Figs. 10c and 11c) and τxz (see Figs. 10b and 11b). The estimations made based on the simplified method can either underestimate or overestimate the solution; no clear trend can be identified. If τxz is normalized by σx, then the inter-slice shear angle, δi, can be obtained, as shown in Figs. 10d and 11d. It is noticed that the inter-slice shear angles in some slices under the active case are greater than the soil friction angle, ϕ. This result may seem to be unreasonable at first glance, but it, in fact, is correct, because the inter-slice shear force in a c–ϕ soil is due to two different sources— namely, the inter-particle cohesion and friction. This fact will inevitably pose considerable difficulty on many of the log-spiral

5000

4000

4000

=(1-kv) z Simplified z

3000

Rigorous is solved z

(psf)

5000

3000

xz

z

(psf)

σz ¼(1  kv)γz is removed. It is of great interest to known how the solution of σz solved from the coupled nonlinear equation system— i.e., Eqs. (32), (35), and (38)—with local and global iterations on σz,R and σz,w (hereafter referred to as the “rigorous method”) varies from that of the conventional assumption (hereafter referred to as the “simplified method”). Figs. 10a and 11a display the distribution of σz along the failure surface of the mobilized soil body in a general c–ϕ soil (c ¼0.05γH ¼95 psf, ϕ ¼301, δw ¼ 201) for the passive and active cases with kh ¼j0.3, respectively. It can be observed that the two solutions of σz are different, and so are the solutions of σx (see

39

2000 1000 0 30

2000 1000

20 10 Slice Number

0 30

0

20 10 Slice Number

0

20 10 Slice Number

0

30

2000

i

x

(psf)

(degree)

4000

0 30

20 10 Slice Number

0

20 10 0 30

Fig. 10. Stress distribution in the mobilized soil body subject to passive earth force (H¼ 150 , ϕ¼ 301, δw ¼ 201, c ¼0.05γH, kh ¼ þ 0.3).

Fig. 11. Stress distribution in the mobilized soil body subject to active earth force (H¼ 150 , ϕ ¼301, δw ¼201, c¼ 0.05∙γH, kh ¼  0.3). (a) Distribution of vertical normal stress along failure surface; (b) Variation in interslice shear stress; (c) Variation in horizontal interslice normal stress and (d) Variation in interslice shear angle.

40

S.-Y. Xu et al. / Soil Dynamics and Earthquake Engineering 70 (2015) 30–47

type analytical models in which the inter-slice force function must be pre-determined (e.g., [15] and [26]). Shamsabadi et al. [19] proposed a Log-Spiral-Hyperbolic approach to calculate the displacement at the top of retaining wall, utilizing the relationship between the normal strain and shear strain in soil [2]. The total displacement at the top of wall is equal to the summation of the local displacement at each intermediate slice. The key variable affecting the magnitude of local displacement is the radius of the strain Mohr circle associated with the slice. Since the strain Mohr circles are constructed from the stress Mohr circles, the sizes of the stress Mohr circles of the intermediate slices will indirectly control the total displacement of the retaining wall. The stress Mohr circles of all the intermediate slices of a typical earth retaining system at failure are plotted in Figs. 12 and 13 for the passive and the active cases, respectively. The hollow circles on

the figures denote the stress states of soil computed based on the simplified method, while the solid circles the stress states derived by the rigorous method. As can be seen from the figures, the radii of the former in average are smaller than those of the latter for the passive case and greater for the active. The Mohr circle for the slice next to the wall (i.e., the nth slice in Fig. 3) is not drawn on Figs. 12 or 13, for its Mohr circle is located far away from those of the intermediate slices. The stress states at this specific slice derived based on the aforementioned two approaches are listed in Tables 1 and 2 for the passive and active cases, respectively. It needs to be pointed out that due to the large external earth force suddenly imposed on the left face of this slice, the normal and shear stresses at the failure surface of this slice increase drastically from those of the neighboring slice. This can be interpreted as a stress concentration phenomenon at this slice, and so its solution depends on the selection of the width of the slice. It is obvious that the correct

By rigorous method

1500

1000 By simplified method

500

R R 0

-500

-1000

-1500

Pole 0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Fig. 12. Stress state along the failure surface of the mobilized soil body subject to passive earth force (H¼150 , ϕ¼ 301, δw ¼201, c ¼ 0.05γH, kh ¼ þ 0.3).

600

400

200

0

R -200

R -400

-600

( x,

xz

( z,

) zx

)

Pole 0

200

400

600

800

1000

1200

1400

1600

1800

2000

Fig. 13. Stress state along the failure surface of the mobilized soil body subject to active earth force (H¼150 , ϕ ¼301, δw ¼201, c¼ 0.05γH, kh ¼  0.3).

S.-Y. Xu et al. / Soil Dynamics and Earthquake Engineering 70 (2015) 30–47

Table 1 Stress concentration at the slice next to the retaining wall (under passive force). ♯ of slices

σz (psf)

σx (psf)

τxz (psf)

δw (deg.)

KE

Simplified

5 30 100

1905.0a 1905.0a 1905.0a

3283.5 3283.5 3283.5

1195.1 1195.1 1195.1

20.0 20.0 20.0

4.4260 4.4628 4.4637

Rigorous

5 30 100

7232.4b 32,815.0b 104,542.9b

11,722.4 52,224.9 165,780.6

4266.6 19,008.3 60,339.2

20.0 20.0 20.0

4.4182 4.4114 4.4112

a In the simplified method, the vertical normal stress (σz) is always computed using the assumption σz ¼γH. As such, the value of σz is independent of the refinement of mesh. Any analytical model that adopts this simplified equation for σz will not be able to capture the stress concentration phenomenon. b For a given retaining wall system, the shear force acting on the left face of the slice n (i.e., Ph  tanδw) is approximately a constant, regardless of the number of slices (N) used to discretize the log-spiral region. Assuming c ¼ 0, this force must be balanced by the self-weight of slice n, the inter-slice shear force acting on the right face of this slice (i.e., Tn  1), and the vertical force acting at the slice's bottom (i.e., F nz ). As shown in Fig. 8, the inertial and the cohesive forces are distributed almost evenly in the log-spiral region. However, the reaction force (i.e., Ri) along the failure surface is not. A very large amount of the reaction force is concentrated at the slice n. If N is large, then the width of slice n will be small. Dividing the reaction force concentrated at slice n by its small width will result in very large stresses in this slice.

Table 2 Stress concentration at the slice next to the retaining wall (under active force). ♯ of slices

σz (psf)

σx (psf) a

τxz (psf)

δw (deg.)

KE

Simplified

5 30 100

1905.0 1905.0a 1905.0a

566.4 566.4 566.4

206.1 206.1 206.1

20.0 20.0 20.0

0.4010 0.4015 0.4015

Rigorous

5 30 100

8250.9b 42,160.9b 137,021.7b

2910.2 15,460.2 50,571.8

1059.2 5627.1 18,406.6

20.0 20.0 20.0

0.4514 0.4504 0.4502

a,b

See Table 1 footnotes.

stress state of this slice cannot be adequately captured if the presumption that σz ¼(1 kv)γz (i.e., the simplified method) is applied. Although the stress state at this slice is sensitive to the refinement of the number of slices, the overall seismic earth pressure coefficient is not, as it can be observed from the variation in KE values with respect to the number of slices listed in Tables 1 and 2. Finally, it needs to be pointed out that the stress states obtained through the proposed rigorous method (as the number of slices is increased indefinitely) are not necessarily physically possible (see, the stress values provided in Tables 1 and 2), because those stress concentrations may imply material failure beyond those within the realm of the basic Mohr–Coulomb failure envelope. These concentrations should be treated in the same manner as those encountered in almost any numerical approximation method (e.g., the finite element method): the particular values of the stresses are immaterial4, as they would not affect the global results; furthermore, they will always be confined to the last slice here, no matter how much the model is refined.

5. Failure surface and influence of ground accelerations The stress states of soil at the two boundaries of the log-spiral region have been rectified above after the system inconsistency is 4 On the other hand, the location and the order of singularity are often important.

41

eliminated. Incidentally, the geometry of the mobilized soil body is altered. The geometry of the mobilized soil body of a general c–ϕ soil (H ¼150 , c¼0.05γH, ϕ¼301, δw ¼2/3ϕ) when subject to horizontal ground acceleration is illustrated in Fig. 14. The mobilized soil mass derived from the rigorous method is less than that obtained from the simplified method under either the passive or the active force. The horizontal failure region is reduced by approximately 10% for the case shown.

6. Earth pressure coefficient and point of application The point of application of the resultant earth-pressure force (Lw in Fig. 3) is a variable of great interest to the practical engineers, for it determines the magnitude of the resisting and overturning moments the earth pressures provide and impose on the base of retaining wall. Conventionally, based on the assumption of linearly distributed normal and shear stress fields along the wall–soil interface, the resultant earth force is often considered to be exerted at the third point of the height of wall measured from its base (i.e., Lw ¼ 1/3H). This assumption will be examined in this section. Following either the simplified or the rigorous LSR model, one can solve the incremental inter-slice normal and shear forces (dEi and dTi) using Eqs. (25) and (26). The resultant reaction forces along the failure surface of the mobilized soil body (F ix and F iz ) can be computed through force equilibrium by adopting Eqs. (22) and (23). It is assumed that for the intermediate slices, F ix and F iz act at the midpoint of the failure surface at the bottom of each slice. As for the Rankine zone, the location of application of F 1x and F 1z is unknown, yet it can easily be derived via the moment equilibrium condition, assuming that the location of application of the interslice normal and shear forces acting on the left face of this triangular Rankine zone (E1 and T1) is at the third point measured from the bottom.5 The point of application of the total earthpressure force at the wall–soil interface can then be backcalculated by checking the global moment equilibrium of the entire mobilized soil body. The locations of application of all the other inter-slice normal and shear forces (Ei and Ti) can similarly be determined. The variation in the seismic earth pressure coefficient (KE), the normalized location of application of the total earth force (Lw/H), and the normalized moment (KE  Lw/H) acting at the base of retaining wall predicted by the simplified LSR, the rigorous LSR, and the Mononobe–Okabe (MO) models are plotted against the horizontal ground acceleration (kh), soil friction angle (ϕ), and wail–soil interface friction angle (δw) in Figs. 15–17, respectively. It is observed that the earth pressure coefficients predicted by the two LSR approaches in the passive case are essentially identical, and are more conservative than those of the MO model. The simplified LSR model—compared to the rigorous LSR model— underestimates the active force as the ground acceleration increases, and as the roughness of the wall decreases. As for the point of application of the earth thrust, it is found to be a function of kh, ϕ, and δw. The predictions—and even the trends —that the rigorous model yields do not agree with those from the simplified model, in most cases. Moreover, the assumption as 5 This assumption is not a rigorous simplification. As observed in Eq. (8), the stress distribution σx along BE is not a linear function of depth z. Therefore, the point of application of the inter-slice forces on BE will not locate exactly at the onethird point from the base. Theoretically, we can integrate the stress distribution σx along BE to derive the magnitude and point of application of the force E1. This approach, however, will result in an inconsistent solution, because the Rankine wedge theory does not concurrently satisfy equilibrium and Mohr–Coulomb failure criteria, due to the adoption of a linear failure surface, which is not true when both c and kh are non-zeros.

42

S.-Y. Xu et al. / Soil Dynamics and Earthquake Engineering 70 (2015) 30–47

0

0

-5

-5

-10

-10 Simplified

-15

Simplified

-15

Rigorous

-20

0

10

20

30

Rigorous

-20

40

0

5

10

15

20

25

0

Fig. 14. Geometry of mobilized soil body (H ¼15 , ϕ¼ 301, δw ¼ 201, c ¼ 0.05γH). (a) Under passive force (kh ¼ þ0.3); (b) under active force (kh ¼  0.3).

6

0.6 0.5 0.3

Simplified method

Rigorous method

0.2

Mononobe-Okabe

0.1

Rigorous method Mononobe-Okabe

Simplified method 2

0

0.05

0.1

0.15

0.2

0.25

-0.25

0.5

0.5

0.4

0.4

0.3

0.3

Lw / H

Lw / H

KE

0.4

KE

4

0.2 0.1

-0.15

-0.1

-0.05

0

-0.2

-0.15

-0.1

-0.05

0

-0.2

-0.15

-0.1

-0.05

0

0.2 0.1

0

0.05

0.1

0.15

0.2

0.25

-0.25 0.2

KE * (Lw / H)

3

KE * (Lw / H)

-0.2

2 1

0

0.05

0.1

0.15

0.2

0.25

0.1

-0.25

Fig. 15. Earth pressure coefficient, point of application, and normalized moment vs. kh.

adopted by the MO model that the earth force always acts at the third point of the height of wall is not a good approximation in most cases. This assumption may result in either the over- or the underestimation of the resisting and/or overturning moment(s) at the base of retaining wall. Since the error in the point of application of earth thrust—i.e., the moment arm of the earth thrust—is directly proportional to the error in the moment the earth force imposes on the wall, the rigorous method is recommended to achieve economical and safe designs (see Figs. 15–17).

7. The deficiency of the Mononobe–Okabe model It is observed in Fig. 17 that in the active case, as the wall–soil interface friction angle decreases, the earth pressure coefficients that the rigorous method yields become larger, and diverge more and more from those predicted by the MO model. This unwelcome trend dictates that a stronger retaining wall is needed if the wall

surface is smooth—a trend that is not captured by the MO solution. A detailed investigation on this trend will be provided next. The variations in the earth pressure coefficient predicted by the simplified and rigorous LSR methods, the MO model, and the Mylonakis model under static (kh ¼0) and seismic (kh ¼  0.25) conditions are respectively displayed in Fig. 18a and b. A comparison of these results indicates that the disagreement between the rigorous LSR method and the MO model for cohesionless soil (c¼ 0, ϕ ¼401) next to a smooth wall surface emerges only when the ground acceleration is non-zero. The inconsistent results between the models are primarily due to the presence of earthquake-induced inter-particle shear stresses (cf. Eq. (5)). If kh a0, then the average shear stress acting on the left face of Rankine zone will not be zero—meaning that the inter-slice frictional angle (δi) on the left face of the Rankine zone is not zero under the seismic condition. However, at the wall–soil interface, a zero wall–soil interface frictional angle must be enforced. In other words, a delicate stress re-distribution in shear has to take

S.-Y. Xu et al. / Soil Dynamics and Earthquake Engineering 70 (2015) 30–47

Active Case, c= 0.001* H, = 0.67* , kh= -0.25, kv= 0.0

10

Simplified method

0.8

8

Rigorous method Mononobe-Okabe

0.6

KE

KE

Passive Case, c= 0.001* H, = 0.67* , kh= 0.25, kv= 0.0

6 4

30

35

40

Simplified method Rigorous method Mononobe-Okabe

20

0.5

0.5

0.4

0.4

0.3

0.3

Lw / H

Lw / H

25

0.2 0.1

25

30

35

40

25

30

35

40

30

35

40

0.2 0.1

20

25

30

35

40

20

10

0.3

8

KE * (Lw / H)

KE * (Lw / H)

0.4 0.2

2 20

43

6 4 2 20

25

30

35

40

0.2 0.1

20

25

, degree

, degree

Fig. 16. Earth pressure coefficient, point of application, and normalized moment vs. ϕ.

10

0.5

8

0.4

6 Simplified method

4

Rigorous method

2

Mononobe-Okabe 0

10

20

0.2

Simplified method

0.1

Rigorous method Mononobe-Okabe

0.5

0.5

0.4

0.4

0.3

0.3

0.2 0.1

0

10

20

30

0

10

20

30

0

10

20

30

0.2 0.1

0

10

20

30

10

0.2

8

KE * (Lw / H)

KE * (Lw / H)

0.3

30

Lw / H

Lw / H

Active Case, c= 0.000* H, = 40o, kh= -0.25, kv= 0.0

KE

KE

Passive Case, c= 0.000* H, = 40o, kh= 0.25, kv= 0.0

6 4 2 0

10

20 , degree

30

0.1

, degree

Fig. 17. Earth pressure coefficient, point of application, and normalized moment vs. δw.

44

S.-Y. Xu et al. / Soil Dynamics and Earthquake Engineering 70 (2015) 30–47

place from the left-face of Rankine zone to the back-face of the retaining wall. The MO model—and essentially all the other models that presume a planar failure surface—cannot take this stress redistribution phenomenon into account; therefore, it fails to capture the trend predicted by the rigorous method as shown in Fig. 18b. In fact, the stress state at the failure surface of the MO model back-calculated through force equilibrium of the mobilized soil wedge does not satisfy the Mohr–Coulomb failure criteria, as demonstrated in Fig. 19. Assuming that the inclined angle of failure surface at the bottom of retaining wall (αw in Eq. (15)) rotates at the same additional rotation angle (αRp in Eq. (9)) from its static configuration as in the Rankine zone, the simplified LSR method will result in a planar sliding surface when c¼0 and δw ¼0—i.e., αw ¼αR per Eqs. (10), (14) and (15). As a consequence, this model is also unable to capture the aforementioned behavior. The Mylonakis model, on the other hand, draws a similar conclusion as the rigorous LSR model—owing to its explicit consideration of multiple stress fields for soils located in different regions of backfill—and allows the stress re-distribution to develop in the transition zone; and the effects of this re-distribution are embedded implicitly in its solution (H¼ 100 , γ¼100 pcf, ϕ¼301, δw ¼0, kh ¼  0.25-α¼46.281, Pa ¼ 2.592 kips, σx ¼ 139.7 psf, σz ¼ 500 psf, τxz ¼0). Upon acknowledging the existence of the stress redistribution phenomenon, the question that follows is why the MO equation underestimates the earth thrust in the active case when δw ¼0 but kh a0. To answer this question, the stress state at the bottom of

wall–soil interface and that at the Rankine zone is examined. At the wall–soil interface, τxz ¼0 since δw ¼ 0; while at the Rankine zone, τxz ¼khσz/(1  kv) per Eq. (5). By plotting these stress states on Mohr circles, as shown in Fig. 20a and b, one can see that the inclined angle of the failure surface (the dash-dot pink line in the figures) at the bottom of the wall–soil interface is greater than that at the bottom of Rankine zone. This result indicates that—with δw ¼0 and kh pointing to the retaining wall—the failure surface of the mobilized soil body in the active case is a convex composite log-spiral. The vertical component of the reaction force (F nz in Fig. 7c) at the bottom of the soil slice adjacent to the wall is required to equilibrate the vertical inertial force and the inter-slice shear force —i.e., (1 kv)Wn and Tn  1 in Fig. 7c—acting on this soil slice. Therefore, with a larger αw, a larger reaction force, Rn, is required for this slice to remain in equilibrium, given the same (1  kv)Wn and Tn  1. This, in turn, increases the horizontal component (F nx ) of the reaction force, which ultimately results in a larger earth thrust. Since, physically, the largest possible αw in the active case is when δw equal to zero, the active earth thrust reaches its maximum under this condition in the active case. The trend can be seen in all models under the static condition (cf. Fig. 18a), but can only be captured by the rigorous LSR and the Mylonakis models under the seismic condition. It is well acknowledged in the geotechnical engineering community that the MO equation overestimates, in general, the

0.5

0.3

0.4 0.2

KE

KE

0.3

0.1

Simplified method

0.2

Simplified method

Rigorous method Mononobe-Okabe

0.1

Rigorous method Mononobe-Okabe

Mylonakis 0

5

10

Mylonakis 15

20

25

30

0

5

10

15

, degree w

20

25

30

, degree w

Fig. 18. Variation in earth pressure coefficient against δw. (a) Static condition (kh ¼0); (b) seismic condition (kh ¼  0.25).

0

200

-2 100 -4

1

0 -6 Earth Force

-8

(1-kv)*W kh*W

-10

Reaction -12

-100

-2

0

2

4

6

8

-200

( x,

xz

)

( z,

zx

)

Pole 0

100

200

300

400

500

Fig. 19. Free body diagram of MO soil wedge and the stress state at its sliding surface. (a) Free body diagram of the mobilized soil wedge; (b) stress state at sliding surface.

S.-Y. Xu et al. / Soil Dynamics and Earthquake Engineering 70 (2015) 30–47

400

100

200

50

0

45

0

-200

( x,

xz

)

( z,

zx

)

Pole

-400 0

500

-50

xz

)

( z,

zx

)

Pole

-100

1000

( x,

0

100

200

300

Fig. 20. Stress states at failure surface in active case (c¼ 0, ϕ ¼301, δw ¼ 0, kh ¼  0.15). (a) At the bottom of wall–soil interface; (b) at the bottom of Rankine zone. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

seismic demands (i.e., the active earth force) on retaining walls—a consensus that is reached through field observations, which indicated that the retaining walls seldom failed in past earthquake events (see, for example, [24]). The comparison shown in Fig. 18 indicates that the rigorous method produces even larger predictions than the MO solution in the active case; thus the accuracy of the rigorous LSR Method may appear questionable at first sight. However, this result can be explained from the following three points of view: 1. No retaining wall is frictionless, unless a specific effort is made to achieve such a condition. Therefore, the fact that retaining walls seldom failed in the past earthquake events does not necessarily invalidate the prediction of the rigorous method at small wall–soil frictional angles. In fact, as the wall–soil interface friction angle increases, the prediction of the rigorous LSR Method gradually reverts to the MO solution. 2. While the MO model considerably overestimates the moment arm under this special condition, the rigorous LSR Method still yields a smaller overturning moment than the MO solution in the active case (cf. the active case in Fig. 17). Hence, retaining walls designed following the proposed LSR model will still be more economical than those designed based on the MO equation. 3. Last but not the least, most of the backfills are not cohesionless as they might be incorrectly assumed in the design practice. A geotechnical investigation [9] was conducted in 2005 to characterize the structural backfills behind existing bridge abutments in the State of California. It was concluded that the existing backfills soils are predominantly c–ϕ materials (silty to clayey sands with gravel) that exhibit some to significant amounts of cohesion. By treating the backfills as cohesionless soils, the earth retaining structures are, thus, nominally overdesigned, since the soil cohesion can drastically decrease the active earth-pressure force the soil medium imposes on the retaining wall [22]. This is likely the major reason as to why retaining walls that are designed based on the MO equation are considerably over-strength.

the coordinate of point O can be computed using y0 ¼

tan ðηR Þ H tan ðηw Þ  tan ðηR Þ

x0 ¼  y0 cotðηR Þ ¼

H tan ðηw Þ  tan ðηR Þ

ð47Þ ð48Þ

In the passive case, this procedure always works. However, in the active case, if the coefficient of horizontal ground acceleration (i.e., kh) exceeds the threshold value such that (αR –ϕ) is negative, then the locus will deviate from line AE (cf. Fig. 21d); thus, its location becomes indeterminate, because Eqs. (47) and (48) are no longer applicable. Under this condition, neither the simplified nor the rigorous LSR model can be used to compute the active earth force.

9. Conclusions

8. Limitation of the LSR model

The Log-Spiral-Rankine (LSR) model, which is a robust analytical method for evaluating the active and passive earth pressure coefficients for earth retaining systems in a general c–ϕ soil medium, is rigorously reviewed in this paper. The original LSR model (herein referred to as the “simplified method”) is found to produce an inconsistent system whose force polygon does not close at retaining wall failure. Remedies to system inconsistencies of the simplified method are proposed. The conventional assumption that the vertical normal stress acting on any horizontal plane in soil is proportional to the depth of the plane (i.e., σz ¼ (1  kv)γz) is removed, since this assumption may not be true when the soil concurrently possesses the inter-particle friction and cohesion. A new model is thus developed (herein referred to as the “rigorous LSR method”), which relies on local and global iteration schemes that search for the stress states along the failure surface of the mobilized soil body while simultaneously satisfying (1) the Mohr–Coulomb failure criterion, (2) the equilibrium conditions between external forces and the internal stresses, and (3) the specified inclined angle of the rupture plane in soil. The newly devised approach is shown to achieve a self-consistent system/ solution. Comparison of the predictions yielded by the simplified method, the rigorous method, and the Mononobe–Okabe (MO) model reveals the following findings:

To construct the log-spiral portion of the failure surface through Eq. (11), the locus of the log–spiral curve (i.e., point O in Fig. 1) must first be defined. As long as point O falls on line AE or on its extension (cf. Fig. 21a, b, and c), from the geometry shown,

(1) Seismic earth pressure coefficients obtained via the simplified method are very close to those by the rigorous method in the passive case. However, in the active case, the simplified method tends to underestimate the seismic earth pressure

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S.-Y. Xu et al. / Soil Dynamics and Earthquake Engineering 70 (2015) 30–47

x y x y

H

r r

w w R

j j

j

w

R

Fig. 21. Possible shapes of the composite log-spiral failure surface. (a) x0 is negative, y0 is positive; (b) x0 is positive, y0 is negative; (c) x0 is positive, y0 þH is negative; (d) both x0 and y0 are positive.

coefficients, as the ground acceleration increases or as the roughness of the wall decreases. (2) Stress states along the failure surface of the mobilized soil body obtained via the rigorous method indicate that the conventional assumption of σz ¼(1  kv)γz will result in an inaccurate estimation of the soil failure surface, because the solution of σx and τxz are coupled with σz, and the rupture plane of a soil element is defined by these stress components. This, in turn, affects the accuracy of the predicted earth thrust. (3) The stress concentration phenomenon can only be observed in the rigorous method. It changes the predicted stress distribution significantly, which, in turn, alters the associated strain distribution within the mobilized soil mass, and consequently the tip displacement of the earth retaining structure. (4) The location of application of the resultant earth pressure force is found to be a function of the soil friction angle (ϕ) and the wail–soil interface friction angle (δw); but surprisingly it shows little correlation with the horizontal ground acceleration (kh). In general, it oscillates around the third point of the wall height measured from the base. The simplified method may

either overestimate or underestimate the location, making it difficult to determine whether the prediction on the overturning moment generated by the earth force imposed on the base of wall is conservative or not. (5) The stress state at the failure surface of MO model backcalculated from the force equilibrium of the mobilized soil wedge does not satisfy the Mohr–Coulomb failure criterion. This result is due to the fact that the MO model cannot capture the shear stress redistribution phenomenon in the mobilized soil body. (6) The MO equation underestimates the active earth force when the wall is smooth. On the other hand, it overestimates the moment-arm of the earth thrust. To design a retaining wall, one needs to know the demands that the soil imposes on the wall, which include the shear force demand, the bending moment demand, and the displacement demand at the top of the retaining wall. The improvement to the predicted resultant earth thrust in the present study appears to be limited, even though various deficiencies of the simplified LSR model are remedied and a relatively complex solution procedure is adopted. Nevertheless, the proposed method is

S.-Y. Xu et al. / Soil Dynamics and Earthquake Engineering 70 (2015) 30–47

currently the most consistent approach within the realm of limit equilibrium approaches, and it offers complete information required in the design practice. Moreover, it proves that the simplified method produces fairly good results (in terms of force demand), despite various theoretical defects being identified. That said, one should always adopt the rigorous method, if the bending moment demand or the tip displacement of retaining wall is of interest.

[13]

Acknowledgments

[14]

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