Analytical solution of seismic pseudo-static active thrust acting on fascia retaining walls

Soil Dynamics and Earthquake Engineering 57 (2014) 25–36

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Analytical solution of seismic pseudo-static active thrust acting on fascia retaining walls Venanzio R. Grecon Department of Civil Engineering, University of Calabria, 87036 Rende (Cs), Italy

art ic l e i nf o

a b s t r a c t

Article history: Received 9 April 2013 Received in revised form 26 September 2013 Accepted 27 September 2013 Available online 20 November 2013

This paper presents a limit equilibrium method, based on the approach of Mononobe and Okabe, for calculating the active thrust on fascia retaining walls, where common methods cannot be used owing to the narrowness of the backfill which does not permit the development of the thrust wedge in the shape and sizes predicted by these methods. The proposed method examines three distinct failure mechanisms, called Mechanism 1, Mechanism 2 and Mechanism 3, where the thrust wedge is formed by one, two or three blocks, respectively; separated by plane slip surfaces. The seismic forces have been simulated with the pseudo-static method. For all three mechanisms, the active thrust is obtained in closed form: in particular, with a cubic equation for Mechanism 2, and with a system of two equations, one cubic and the other quartic, for Mechanism 3. Mechanisms with more than three blocks cannot have analytical solutions. The study is completed by an examination of some significant cases from which the higher attenuation of the seismic thrust, with respect to the static, emerges as the backfill width reduces. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Lateral earth pressure Fascia retaining walls Analytical solutions

1. Introduction Fascia retaining walls are built near rockfaces, as a damage hazard reduction measure, or in front of existing stabilized walls for enlarging roads and highways [1,2]. This wall type cannot be analyzed with the classical methods, such as those of Coulomb, Rankine or Terzaghi [3–5], because the thrust wedge cannot develop freely, due to the narrowness of the backfill. Spangler and Handy [6] proposed estimating the horizontal pressure using Jansen's arching theory [7], formulated to assess the lateral pressure within silos and also used by Marston [8] to calculate the lateral pressure on buried vertical pipes. The experimental investigations of Frydman and Keissar [9] and Take and Valsangkar [10] confirm the reliability of this theory for unyielding retaining walls. However for walls in conditions of active thrust there is less agreement between the experimental data and theoretical predictions [9]. A similar theoretical approach, in term of stress analysis, has been used in the numerous papers by Li and Aubertin [11–16], devoted to the arching effects in narrow backfilled stopes, and Thing et al. [17]. Results sufficiently close to the experimental data are obtained using the method of characteristics [9], which, unfortunately, is little used by practicing engineers because of its complexity, while the finite element method

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with the Plaxis code has been used by Yang and Liu [18] and Fan and Fang [19]. The limit equilibrium method is simple in its formulation and very widespread in practical applications. It was employed by Leshchinsky et al. [20] using the simplified method of Bishop [21]. In agreement with the centrifuge model tests of Woodruff [22], Lawson and Yee [23] and James et al. [24] assumed a thrust wedge limited by a bilinear slip surface (Fig. 1b), assigning to the angle α the value π/4 þϕ′/2. On the contrary, Yang and Zornberg [25] recognize that the value of α must be that maximizing the active thrust; it is obtained by a trial and error procedure. More recently, three failure mechanisms have been analyzed [26] characterized by a thrust wedge formed by 1 to 3 rigid blocks depending on the backfill width (Fig. 1). The blocks are limited by slip planes inclined at angles α, ρ and λ, whose values are chosen with the criterion of maximizing the active thrust acting on the wall. The solutions are obtained in closed form and this is an advantage because these solutions are less time-consuming [27–29]. However the study is limited to the static condition, while many countries of the world are subject to earthquakes and seismic actions must be considered in building design. Therefore, this paper extends the previous study for evaluating the active thrust exerted by narrow backfill [26] to seismic conditions. The paper follows the approach of Mononobe and Okabe [30,31], in terms of the limit equilibrium method [1]. It is assumed that the thrust wedge is formed by one, two or three rigid blocks, bounded by plane surfaces, as shown in Fig. 1. The solution is obtained in closed form. The stress distribution on

26

V.R. Greco / Soil Dynamics and Earthquake Engineering 57 (2014) 25–36

Fig. 1. Thrust wedges and slip planes (in red) for Mechanism 1 (a), Mechanism 2 (b), and Mechanism 3 (c). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

the wall backface is, moreover, obtained through a semi analytical solution. The analyses contained in this paper have been developed under the following simplifying hypotheses:

 the soil is cohesionless, without pore pressure and obeys the Mohr–Coulomb failure criterion;

 the problem can be studied in plane strain conditions; this is  

possible if the wall is long enough and the cross sections are all the same; the wall movement is sufficient to induce failure along planes inside the backfill behind the wall, in accordance with Coulomb's approach: the seismic forces acting on the thrust wedge are calculated using the pseudo-static approach [30,31], by subjecting the thrust wedge to two forces, one horizontal and the other vertical, both proportional to the thrust wedge weight through two seismic coefficients, kh and kv respectively.

The study is developed with reference to a backfill limited above by a linear profile inclined at ε and formed by soil with friction angle ϕ′ and unit weight γ. The contact plane between wall and backfill is inclined at an angle β with respect to the horizontal and along it the friction angle is δ. The rockface is inclined at an angle η and the friction angle between the rock and soil is ψ. Finally, the wall is h high and the rockface is at a horizontal distance b from its heel.

2. Thrust wedge configurations Consider the schema of a fascia retaining wall reported in Fig. 1a showing the failure mechanism of Coulomb, where the thrust wedge is formed by a unique block limited by slip planes CA, inclined at an angle α, and CB, which separates wall and backfill. The inclination of the plane CA is that which maximizes the active thrust on the wall and point T is the limit position for A. This mechanism is called free Mechanism 1 if point A is between B and T, and forced Mechanism 1 if point A coincides with T. Fig. 1b displays a failure mechanism with a bilinear slip surface CDT [22–24,32,33], formed by the segments CD, internal to the backfill and inclined at α, and DT, along the rockface inclined at η. The condition that the wall displacement, due to the backfill thrust, causes no detachment of the thrust wedge from the rockface, requires the assumption of a further failure plane DE inclined at ρ [34]. Angles α and ρ must be selected so that they maximize the active thrust on the retaining wall. Because the thrust wedge ABCD is composed of two blocks, this failure mechanism is called Mechanism 2, and more specifically free

Fig. 2. Mechanism 1: forces acting on the thrust wedge ABC. Line AC (inclined at α) is the failure plane bounding the thrust wedge ABC. Line CT (inclined at αmin) is the limit position for AC.

Mechanism 2 if point E is between A and B and forced Mechanism 2 if point E coincides with B being its limit position. Fig. 1c shows a failure mechanism with three blocks (thus called Mechanism 3) where point E is located on BC and from it a further failure plane starts with inclination λ. In this case also, the values of the angles α, ρ and λ are chosen so that the thrust on the wall is maximized. If point F falls between A and B the mechanism is free, while if F is on its limit position (i.e. it coincides with A) the mechanism is forced. Obviously in principle mechanisms with higher number of blocks exist and their analyses could be made analogously to those formulated for Mechanisms 2 and 3. However they will not be treated in this paper, because, as will be shown later, they cannot be solved analytically.

3. Analysis of Mechanism 1 The forces acting on the thrust wedge ABC in Mechanism 1 are shown in Fig. 2. They are: W1 is the weight of the thrust wedge ABC; H1 is the horizontal seismic pseudo-static force (given by H1 ¼khW1); V1 is the vertical seismic pseudo-static force (given by V1 ¼kvW1); S1 is the active thrust, reactively exerted by the wall and; R1 is the reaction exerted on the surface CA by the backfill. The directions of these force are shown in the same Fig. 2. The equilibrium conditions of forces acting on the thrust wedge

V.R. Greco / Soil Dynamics and Earthquake Engineering 57 (2014) 25–36

27

ABC permit us to obtain the thrust S1 [30,31]: S1 ¼

1 1 þ kv 2 sin ðβ αÞ sin ðβ  εÞ sin ðα  ϕ′ þθÞ γ h 2 cos θ sin 2 β sin ðα  εÞ sin ðβ þδ þϕ′ αÞ

where θ ¼ tan  1




kh 1 7kv

ð1Þ

 ð2Þ

The thrust S1 is maximized by the angle α1 ¼max(αc,αmin) with [35] ! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pðp þ qÞð1 þ qrÞ  p ð3aÞ αc ¼ ϕ′ θ þ tan  1 1 þ rðp þ qÞ αmin ¼ tan  1




h sin ðβ εÞ þ b sin β sin ε h sin ðβ  εÞcot η þ b sin β cos ε

 ð3bÞ

where p ¼ tan ðϕ′  θ  εÞ q ¼ tan ðβ ϕ′þ θÞ r ¼  cotðβ þ δ  θÞ

ð4Þ

As is evident in Fig. 2, if αc Zαmin a free Mechanism 1 occurs, while a forced Mechanism 1 occurs if αc oαmin. For free Mechanism 1, the active thrust S1 is also given by S1 ¼

1 2 γ h K a1 2 s

where [36] 0

ð5Þ 12

sin ðβ  ϕ′s Þ

B C sin β K a1 ¼ @pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA sin ðϕ′s þ δs Þ sin ðϕ′s  εÞ sin ðβ þδs Þ þ sin ðβ  εÞ

ð6Þ

with [37] γs ¼ γ

1 þ kv ; cos θ

ϕ′s ¼ ϕ′  θ;

δs ¼ δ þ θ

ð7Þ

4. Analysis of Mechanism 2 Fig. 3a shows the thrust wedge ABCD and the forces acting on it, which are: W2 is the weight of the thrust wedge ABCD; H2 is the horizontal seismic pseudo-static force (given by H2 ¼khW2); V2 is the horizontal seismic pseudo-static force (given by V2 ¼kvW2); S2 is the active thrust; R2 is the surface force acting on the plane DC; P2 is the surface force acting on the plane AD; their directions are illustrated in the same figure. The equilibrium equations of these forces allows the seismic active thrust S2 to be obtained S2 ¼ W 2

1 þ kv sin ðα  ϕ′ þθÞ sin ðη  ψ þϕ′ αÞ þ P2 sin ðβ þ δ þ ϕ′  αÞ cos θ sin ðβ þδ þϕ′ αÞ

ð8Þ

sin β cos ε

3

cos η cos ε sin ðη  εÞ

sin ðβ  εÞ sin β cos ε

þ

b h

7  5 b by tan ε h  h h ð9Þ

The force P2 can be found through the equilibrium of forces acting on the subwedge AED, which is also subject to the forces (Fig. 3c): W′2 is the weight of the thrust wedge AED; H′2 is the horizontal seismic pseudo-static force (given by H′2 ¼ khW′2); V′2 is the vertical seismic pseudo-static force (given by V′2 ¼ kvW′2); R′2 is the surface force on the plane ED: 1 þ kv sin ðρþ ϕ′þ θÞ W′2 P2 ¼ sin ðρ þϕ′ þψ  ηÞ cos θ

with 1 sin ðρ  ηÞ sin ðη  εÞ γðymax  yÞ2 ð11Þ 2 2 sin η sin ðρ  εÞ The thrust S2 therefore depends on the unknown angles α and ρ. For a free Mechanism 2, S2 is maximized by the two conditions

 ∂S2 ∂S2 ¼0 ¼0 ð12Þ ∂ρ ρ ¼ ρ2 ∂α α ¼ α2

W′2 ¼

where 2 2 sin ðβ  εÞ b 1 2 6 sin β cos ε þ h tan ε  W 2 ¼ γh 4 2  sin ðβ  εÞ cot β þ 2

Fig. 3. Mechanism 2: (a) geometry of the thrust wedge ABCD; (b) forces acting on the wedge ABCD; and (c) forces acting on the subwedge ADE.

ð10Þ

having assumed S2(α2,ρ2)¼ max S2(α,ρ). The first of the conditions (12) is satisfied for ! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pðp þ qÞð1 þqrÞ  p ρ2 ¼ π  ϕ′  θ  tan  1 1 þ rðp þ qÞ

ð13Þ

where p ¼ tan ðϕ′ þθ þ εÞ q ¼  tan ðη þ ϕ′ þ θÞ r ¼ cotðη  ψ þ θÞ

ð14Þ

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V.R. Greco / Soil Dynamics and Earthquake Engineering 57 (2014) 25–36

Fig. 4. Mechanism 3: (a) geometry of the thrust wedge ABCD; (b) forces acting on the wedge ABCD; (c) forces acting on the subwedge ABED; and (d) forces acting on the subwedge BEF.

For ρ¼ ρ2 the thrust S2 can be rewritten as 1 P 2 ¼ γ s ðymax  y2 Þ2 K a2 2 where 0

ð15Þ

sin ðη þ ϕ′ þ θÞ

12

B C sin η K a2 ¼ @pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA sin ðϕ′ þ ψ Þ sin ðϕ′ þ θ þ εÞ sin ðη  ψ þ θÞ þ sin ðη  εÞ The second condition (12) is equivalent to
 ∂S2 ¼0 ∂y y ¼ y2

ð16Þ 5. Analysis of Mechanism 3

ð17Þ

For Mechanism 3, the active thrust S3, acting on the wall, is obtained from the equilibrium conditions of forces acting on the wedge ABCD, reported in Fig. 4b, whose symbols with subscript 3 have an analogous meaning to those in Fig. 3a with subscript 2.

ð18Þ

S3 ¼ W 3

where y2 is the value of y corresponding to α2 with tan α ¼

y sin η b sin η þ y cos η

The condition (17) leads to the following cubic equation  A3 y3 þ A2 y2 þA1 y þ A0 y ¼ y ¼ 0 2

ð19Þ

where coefficients A3, A2, A1 and A0 are reported in Appendix A. Note that this cubic equation becomes a quadratic equation if ψ¼ ϕ′. The value y2 gives the active thrust S2  1 n0 þ n1 y þ n2 y2 þ n3 y3  S2 ¼ γ s ð20Þ  2 d0 þ d1 y y ¼ y2 whose coefficients are also reported in Appendix A. The free Mechanism 2 is possible only if yG oy2 oymax, where yG is the height of point G, placed on the rockface at the intersection of the line starting from B and inclined at π ρ2 below the horizontal. yG ¼ h

sin ðρ2  βÞ sin η sin ρ2 sin η b sin ðρ2  ηÞ sin β sin ðρ2 ηÞ

transition from Mechanism 1 to free Mechanism 2, so S1(αmin) ¼ S2(αmin). However, if S2 attains its maximum for y¼ y2 oymax (i.e. for α ¼α2 oαmin), then S2(α2) 4S2(αmin). Therefore, if Mechanism 1 can occur only as a forced mechanism (because αmin 4αc), then the free Mechanism 2 is more critical than the forced Mechanism 1.

ð21Þ

If y2 oyG, Mechanism 2 is still possible as a forced mechanism, provided point E is made to coincide with B. In this case, however, the angle ρ2 is not given by Eq. (13), but depends on y:
 π b þ y cot η  h cot β ð22Þ ρ2 ¼ þ tan  1 2 hy Consequently the condition (17) leads to an equation of seven degrees in unknown y. Finally it remains to clarify what the most critical mechanism between the forced Mechanism 1 and the free Mechanism 2 is, when αmin 4αc (Eqs. (3a) and (3b)). Because α¼ αmin is the value of

1 þ kv sin ðα  ϕ′ þ θÞ sin ðη ψ þ ϕ′  αÞ þP 3 sin ðβ þ δ þ ϕ′  αÞ cos θ sin ðβ þ δ þ ϕ′  αÞ

ð23Þ

where W3 has the same expression as W2 in the Mechanism 2 previously examined, while the thrust P3 is obtained from the equilibrium conditions of forces acting on the subwedge ABED, which is also subject to the forces (Fig. 4c): W′3 is the weight of the subwedge; H′3 is the horizontal seismic pseudo-static force (given by H′3 ¼khW′3); V′3 is the vertical seismic pseudo-static force (given by V′3 ¼kvW′3); R′3 is the surface force on the plane ED and Q3 surface force on the plane BE. P 3 ¼ W′3

1 þkv sin ðρ þ ϕ′ þ θÞ sin ðρ  δ þ ϕ′  βÞ þQ3 sin ðρ þψ þ ϕ′  ηÞ cos θ sin ðρ þ ψ þ ϕ′ ηÞ

ð24Þ

where W′3 ¼ W 3 

1 sin ðβ  αÞ γyY 2 sin β sin α

ð25Þ

and Q3 is furnished by the equilibrium of forces on the subwedge FBE (Fig. 4d), which is also subject to the forces W″3 weight of the subwedge; H″3, horizontal seismic pseudo-static force (given by H″3 ¼khW″3); V″3 is the vertical seismic pseudo-static force (given by V″3 ¼ kvW″3) and R″3 is the surface force on the plane EF. Q3 ¼

1 1 þ kv sin ðβ  λÞ sin ðβ  εÞ sin ðλ  ϕ′ þ θÞ γ ðh  YÞ2 2 cos θ sin 2 β sin ðλ εÞ sin ðβ þ δ þϕ′ λÞ

ð26Þ

As a result, S3 depends on the unknown angles λ, ρ and α, and its maximum is attained for the values λ3, ρ3 and α3 given by the following conditions


 ∂S3 ∂S3 ∂S3 ¼0 ¼0 ¼0 ð27Þ ∂λ λ ¼ λ3 ∂ρ ρ ¼ ρ3 ∂α α ¼ α3

V.R. Greco / Soil Dynamics and Earthquake Engineering 57 (2014) 25–36

The first of Eq. (27) is equivalent to
 ∂Q 3 ¼0 ∂λ λ ¼ λ3

6. Mechanisms with more than three subwedges ð28Þ

whose solution is λ3 ¼ max(αc,λmin), where λmin ¼ tan

1



h sin ðβ  εÞ sin η þ b sin η sin ε sin β  Y sin ðη  εÞ sin β h sin ðβ  εÞ cos η þ b sin η cos ε sin β  Y sin ðη  εÞ cos β



Three basic considerations can be formulated by generalization of that above obtained.

 A forced mechanism with n subwedges is less critical for the successive free mechanism with n þ1 subwedges;

 As the backfill width becomes narrower, and the distance b gets

ð29Þ is the inclination angle of the line ET (Fig. 4a). If αc 4λmin point F is between A and B a free Mechanism 3 is the most critical; in this case, the angle λ3 is independent of angles α and ρ, and the thrust Q3 is given by Q 3 ¼ 12 γ s ðh  YÞ2 K a1


As concerns the second of Eq. (27), it is equivalent to  ∂P 3 ¼0 ∂Y Y ¼ Y 3

ð30Þ

ð31Þ

y being linked to angle ρ by (Fig. 4a) Y ¼y

sin β sin ðρ ηÞ sin β sin ρ þb sin η sin ðρ  βÞ sin ðρ βÞ

ð32Þ

and Y3 is the value of Y maximizing the thrust P3 in Mechanism 3 for an assigned value of y. Condition (31) leads to the following cubic equation ðB3 Y 3 þB2 Y 2 þ B1 Y þ B0 ÞY ¼ Y 3 ¼ 0

ð33Þ

whose coefficients B3, B2, B1 and B0 are functions of y and are given in Appendix B. Note that this equation becomes a quadratic equation if δ¼ϕ′. Finally, as concerns the third condition (27), it can be rewritten
 ∂S3 ¼0 ð34Þ ∂y y ¼ y3 where y is linked to α by Eq. (18) and y3 is the value of y maximizing S3 in Mechanism 3 for an assigned value of Y. From condition (34) the following quartic equation is obtained 4

3

2

ðC 4 y þ C 3 y þ C 2 y þ C 1 yþ C 0 Þy ¼ y3 ¼ 0

29

ð35Þ

whose coefficients C4, C3, C2, C1 and C0 are functions of Y and are reported in Appendix C. Therefore, the free Mechanism 3 is governed by a system of two simultaneous Eqs. (33) and (35), which can be solved alternatively with an iterative procedure. If αc Z λmin Mechanism 3 is really free, once the critical values Y3 and y3 are determined, the thrust S3 can be found with the relation;  1 c0 þ c1 y þ c2 y2 þc3 y3  S3 ¼ γ s ð36Þ  2 2 k0 þ k1 y þk2 y y ¼ y3 whose coefficients at numerator and denominator are given in Appendix C. It can be shown, by analogy with that shown in the previous section, that, if Mechanism 2 is only possible as a forced mechanism (i.e. y2 oyG), then the free Mechanism 3 is more critical than the forced Mechanism 2. This is very important, because while the free Mechanism 3 has an analytical solution, the forced Mechanism 2 does not, because it is governed by an equation of 71. If λmin 4αc a forced Mechanism 3 is the most critical, but it does not have an analytical solution because it is governed by two equations of degrees higher than four.



smaller, a free mechanism can occur only with an increasing number of subwedges. A free mechanism with n subwedges is governed by a quadratic equation added to a system of n  1 simultaneous equations with degrees from 3 to n þ1.

As a practical consequence, failure mechanisms with more than three subwedges do not admit an analytical solution because the governing system has at least a quintic equation and no formula exists for solving such types of equation. Problems with more than three subwedges can then only be solved by numerical solutions, which has a higher cost in terms of computing time. 6.1. Parametric studies To investigate the effects of the various parameters, and especially the seismic coefficients, on the thrust acting on the fascia retaining walls, some numerical applications have been developed with reference to the following geometrical configuration: β¼901, η¼801, ε¼101 and b variable. Two values of the friction angle have been used, ϕ′¼361 and ϕ′¼241. The first corresponds to a typical peak value of sands, while the second can be considered as a sand value obtained by a reduction by means of a safety factor application. Various angles δ and ψ have been considered to study the influence of these two parameters. Finally, the values of kh between 0 and 0.2 and ratios kv/kh between  0.5 and 1 were considered. Positive values of this ratio imply vertical seismic forces downwards, while negative values are connected with seismic forces upwards. In the graphs, the abscissae use the ratio b/h for studying the influence of the backfill width, while the ordinates display the values of the active thrust coefficients Ka which is given by Ka ¼

2Sa 2

γh

ð37Þ

where Sa is equal to S1, S2 or S3, according to whether the most critical mechanism is a Mechanism 1, Mechanism 2 or Mechanism 3, respectively. As can be seen in all the figures, if the backfill width b is higher than a value b1, the critical mechanism is Mechanism 1. If it is between b1 and b2 Mechanism 2 is the most critical, while if it is between b2 and b3 Mechanism 3 is it. For bob3 the critical condition is given by mechanisms of a higher order, for which an analytical solution cannot be found. It can be seen that the distance b1 increases significantly with the increase in kh and the decrease in ϕ′. The same b1 increases slightly with the increase in δ and the decrease in kv, while it is independent of angle ψ. Moreover kh and kv are more influential for low values of ϕ′. The distance b2 is less variable than b1; with the tested values of the parameters it is in the range [0.08,0.21]h. The possibility of having Mechanism 2 as the most critical increases for increasing values of kh and δ or decreasing values of kv and ϕ′, while the angle ψ has a negligible influence. Mechanism 3 has a smaller range b2–b3 with respect to b1–b2, as is evident in all the figures. In some cases, it probably does not exist and the critical mechanism jumps from Mechanism 2 to a mechanism higher than 3: unfortunately it has not been possible to analyze this question from a theoretical point of view; therefore

30

V.R. Greco / Soil Dynamics and Earthquake Engineering 57 (2014) 25–36

Fig. 5. Active thrust coefficient Ka in terms of the ratio b/h for β ¼ 901, η ¼ 801, ε¼ 101, ψ¼ ϕ′, δ ¼ ⅔ϕ′ and different values of kh (with kv ¼ kh/2).

Fig. 6. Active thrust coefficient Ka in terms of the ratio b/h for β ¼901, η ¼801, ε¼ 101, ψ¼ ⅔ϕ′, δ¼ ⅔ϕ′ and different values of kh (with kv ¼kh/2).

Fig. 7. Active thrust coefficient Ka in terms of the ratio b/h for β ¼ 901, η ¼ 801, ε¼ 101, ψ¼ 0, δ ¼ 0 and different values of kh (with kv ¼ kh/2).

our considerations are of an empirical nature based on the results obtained for some specific cases. For example, in the case reported in Fig. 5, for ϕ′¼ 241, kh ¼ 0.20 and kv ¼0.10, Mechanism 3, if it exists, should have a range (b2–b3)/h less than 1.7  10  9; this means that for a wall 6 m high the range of Mechanism 3, if it exists, is less than 1 μm. Therefore, in practical terms, in this case it does not exist (Fig. 6). The active thrust coefficient Ka is highly dependent on the ratio b/h. In fact as b/h decreases, the critical mechanisms proceed towards higher mechanisms, characterized by different values of Ka. For Mechanism 1, the active thrust coefficient is independent of b/h, because the rockface is too far from the wall to affect the backfill thrust. If b ob1 the greater proximity of the wall to the rockface causes a reduction in Ka, unless ψ is zero or very close to

zero and the seismic forces are not considered or are very low (Figs. 7 and 8). In the latter case, the reduction in b/h determines an increase which can be significant especially for high values of ϕ′. Except these particular cases, mechanisms of higher degree are characterized by lower values of Ka with respect to lower mechanisms: as a consequence the use of Coulomb's method (for the static condition) and the M–O method (for seismic conditions), which correspond to Mechanism 1, give conservative values of Ka. However, for severe seismic conditions the M–O method can lead to a overestimation of Ka up to over 100%, as can be seen in Fig. 10. The seismic actions produce a notable increment in the active thrust, which for kh ¼0.2 almost doubles. The effect of kv is contrasting: in the examined cases increasing kv leads to an increase in Ka, but it is not possible to exclude that in some

V.R. Greco / Soil Dynamics and Earthquake Engineering 57 (2014) 25–36

31

Fig. 8. Influence of the angle ψ on Ka in terms of the ratio b/h for the conditions static (kv ¼ kh ¼0) and seismic (kv ¼ 0.1, kh ¼ 0.2) and β¼ 901, η ¼ 801, ε ¼101, δ ¼ 2/3ϕ′.

Fig. 9. Influence of the angle δ on Ka in terms of the ratio b/h for the conditions static (kv ¼ kh ¼ 0) and seismic (kv ¼0.1, kh ¼ 0.2) and β ¼901, η ¼801, ε¼ 101, ψ¼ 2/3ϕ′.

Fig. 10. Influence of the angle δ on Kah in terms of the ratio b/h for the conditions static (kv ¼ kh ¼ 0) and seismic (kv ¼ 0.1, kh ¼0.2) and β ¼901, η ¼801, ε¼ 101, ψ¼ 2/3ϕ′.

circumstances the vertical seismic force could be more critical if upward instead of downward [38]. As can be seen, the relationship between the angle δ and the thrust coefficient Ka is not monotone and the value of δ maximizing Ka also depends on the ratio b/h, both in condition static [26] and seismic (Fig. 9). The relation between δ and Kah is on the contrary clearly monotone (Fig. 10) with Kah increasing with δ decreasing (Figs. 11–14).

6.2. Stress distribution on the wall Working with forces only, the limit equilibrium method is unable to determine the stress distribution on the wall backface.

This shortcoming can be overcome by following the Huntington procedure [37], i.e. admitting that the thrust wedge also fails internally along planes PQ starting from the thrust plane BC and having a inclination α, which depends on the depth z of point P below point B (Fig. 15). In this way, it is possible to calculate the active thrust on a part BP, z high, of the wall backface; let us call this thrust Sa(z), which can be S1, S2 or S3, according to whether the most critical mechanism connected to the depth z is Mechanism 1, Mechanism 2 or Mechanism 3, respectively. The stress sa acting on a generic point P at depth z of the wall backface BC can then be obtained as sa ðzÞ ¼ γzka ðzÞ ¼

Sa ðz þ Δz=2Þ  Sa ðz  Δz=2Þ Δz= sin β

ð38Þ

32

V.R. Greco / Soil Dynamics and Earthquake Engineering 57 (2014) 25–36

Fig. 11. Active thrust coefficient Ka in terms of the ratio b/h for β ¼ 901, η ¼801, ε¼ 101, ψ¼ ϕ′, δ ¼⅔ϕ′ and different values of kh (with kv ¼ kh).

Fig. 12. Effects of the value of kv on the thrust coefficient Ka in terms of the ratio b/h, for β ¼ 901, η ¼ 801, ε¼ 101, ψ¼ ϕ′, δ ¼ ⅔ϕ′ and kh ¼ 0.2.

Fig. 13. Effects of the value of kv on the thrust coefficient Ka in terms of the ratio b/h, for β ¼901, η ¼ 801, ε ¼101, ψ ¼⅔ϕ′, δ ¼ ⅔ϕ′, and kh ¼ 0.2.

where ka(z) is the active pressure coefficient (which must not be confused with the active thrust coefficient Ka) [39,40]. Figs. 16 and 17 report the stress distribution sa(z) and the earth pressure coefficient Ka along the wall backface in terms of the aspect ratio b/h for two values of ϕ′ (361 and 241) and two values of ψ (0 and ⅔ϕ′), with δ ¼⅔ ϕ′; the seismic condition (with kh ¼ 0.2 and kv ¼1/2kh) has been compared with the static condition. The values of the other parameters are reported in the same figures. It is evident that above a depth z1 the failure mechanism is the free Mechanism 1. Here thus the earth pressures can be calculated using the method of Mononobe

and Okabe, and ka(z) ¼Ka1 is independent of z. Moreover, z1 decreases as kh increases. Below z1 and up to z2 the most critical mechanism is Mechanism 2. Here the earth pressure coefficient varies with z. In static conditions, ka(z) generally decreases with z, but for ψ ¼0 (or other very small values of ψ) it can increase with z, especially for high values of ϕ′. This effect is much less evident with an increase in kh, and above a certain value always there is a reduction in ka(z) with an increase in z, independently of the value of ψ. The increase in kh reduces z1 and increases z2, with consequent enlargement in the range z2–z1 where Mechanism 2 is the most critical mechanism.

V.R. Greco / Soil Dynamics and Earthquake Engineering 57 (2014) 25–36

33

Fig. 14. Effects of the value of kv on the thrust coefficient Ka in terms of the ratio b/h, for β ¼901, η ¼801, ε ¼101, ψ ¼0, δ¼ 0 and kh ¼ 0.2.

Fig. 15. Subwedge ABPQ with height z; (a) geometry of the subwedge; and (b) forces acting on the subwedge.

Below z2, Mechanism 3 should be the most critical mechanism, at least up to depth z3. In reality, it is not so: in some cases (and especially for high values of kh) just below z2 the most critical mechanism can be a mechanism higher than three. In these cases, an analytical solution of the thrust is available up to depth z2 only.

7. Conclusions In this paper the method of Mononoke and Okabe [30,31], which calculates the seismic active thrusts on walls using the limit equilibrium approach and the pseudo-static method, has

been extended to calculate the seismic active thrust on fascia retaining walls, where traditional methods cannot be used because of the narrowness of the backfill. Three failure mechanisms have been considered, called Mechanism 1, Mechanism 2 and Mechanism 3, where the thrust wedge is formed by one, two or three blocks, respectively, which are separated from one another by plane failure surfaces. For the three examined cases, the seismic active thrust has been calculated analytically, while possible mechanisms with more than three blocks cannot have an analytical solution. Although free and forced failure mechanisms are both examined, each forced mechanism proved to be less critical than the successive free mechanism. This is important because, while the forced Mechanism 2 has no solution in terms of roots, free Mechanism 3 has such a solution. Analogously, it is reasonable to investigate the Mechanism 3 as long as the mechanism is free, while the solution obtained with a forced Mechanism 3 does not correspond to the most critical mechanism. The study shows that, if b reduces, the most critical mechanisms pass from Mechanism 1 to higher mechanisms. For high values of k h , Mechanism 3 can miss, and with b decreasing the most critical mechanism can pass from Mechanism 2 to a mechanism higher than 3, without passing through Mechanism 3. The thrust coefficient Ka is insensitive to the width b in Mechanism 1, while for the successive mechanisms it varies with b. It generally decreases when b decreases, with a smaller reduction for static conditions and becomes higher as kh increases. For kh ¼0.2, this reduction can be over 50 %. Therefore if the method of Mononobe and Okabe is inappropriately used for calculating thrusts on fascia retaining walls, there can be a significant overestimation of the thrust. However, for the lowest value of ψ and kh an increase in Ka can also occur as b reduces. In these cases the use of Mononobe and Okabe's methods can lead to unconservative estimates of the thrust. The lateral pressure distribution on the wall has been obtained with the procedure of Huntington. The lateral pressure coefficient ka(z) is constant and equal to Ka up to a dept z1, where the failure occurs with Mechanism 1. The depth z1 decreases sensibly kh increasing. Below z1 and up to a depth z2 the most critical mechanism is the free Mechanism 2 with a coefficient ka(z) which decreases with z, provided that ψ and kh are not lowest. The free Mechanism 3, extended between depths z2 also and z3, is characterized by a further reduction of ka(z) with the increase of z. However, for high values of kh, the Mechanism 3 may not have lieu and at depth z2 the most critical mechanism can suddenly pass from the free Mechanism 2 to a mechanism higher than 3.

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Fig. 16. Distributions of the dimensionless lateral earth pressure on the wall and the active earth pressure coefficient in terms of the dimensionless depth b/h for β ¼ 901, η ¼ 801, ε¼ 101, ϕ′¼ 361, and δ ¼ ⅔ϕ′.

Fig. 17. Distributions of the dimensionless lateral earth pressure on the wall and the active earth pressure coefficient in terms of the dimensionless depth b/h for β¼ 901, η ¼ 801, ε¼ 101, ϕ′¼ 241, and δ ¼⅔ϕ′.

w0 ¼ ðH þ b tan εÞ2

Appendix A. Coefficients of the cubic Eq. (19) for free Mechanism 2

cos η cos ε cos β cos ε  H2 þ ð2H þ b tan εÞb sin ðη  εÞ sin ðβ  εÞ

w1 ¼  b A3 ¼ 2n3 d1 A2 ¼ 3n3 d0 þ n2 d1 A1 ¼ 2n2 d0 A0 ¼ n1 d0  n0 d1

ðA:4Þ

a0 ¼  b sin η sin ϕ′s ðA:1Þ

a1 ¼ sin ðη ϕ′s Þ

ðA:5Þ

p0 ¼ K a2 ðH þ b tan εÞ2 n0 ¼ w0 a0 þ p0 q0 n1 ¼ w0 a1 þ w1 a0 þ p0 q1 þ p1 q0 n2 ¼ w1 a1 þ p1 q1 þp2 q0 n3 ¼ p2 q1

ðA:2Þ

d0 ¼ b sin η sin ðβ þ ϕ′ þ δÞ d1 ¼ sin ðβ þδ þ ϕ′ ηÞ

ðA:3Þ

p1 ¼  2K a2 ðH þ b tan εÞ
p2 ¼ K a2

sin ðη  εÞ sin η cos ε

2

sin ðη εÞ sin η cos ε ðA:6Þ

q0 ¼ b sin η sin ðη  ψ þϕ′Þ q1 ¼ sin ðϕ′ ψÞ

ðA:7Þ

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35

v1 ¼ ½w1 þ Yðcot β  cot ηÞ

ðC:4Þ

Appendix B. Coefficients of the cubic Eq. (33) for free Mechanism 3

sin ðβ þϕ′ þθÞ  b sin ðϕ′þ θÞ sin β sin ðη þ ϕ′ þ θÞ l1 ¼  sin η l0 ¼ Y

B3 ¼ 2m3 e1 B2 ¼ 3m3 e0 þ m2 e1 B1 ¼ 2m2 e0 B0 ¼ m1 e0  m0 e1

ðB:1Þ

where

j0 ¼ b sin ðβ þ δ  ϕ′Þ þ Y j1 ¼

m0 ¼ g 0 f 0 þ s 0 r 0 m1 ¼ g 0 f 1 þ g 1 f 0 þs0 r 1 þ s1 r 0 m2 ¼ g 1 f 1 þ s1 r 1 þs2 r 0 m2 ¼ g 1 f 1 þ s1 r 1 þs2 r 0 m3 ¼ s2 r 1

ðC:5Þ

sin ðϕ′  δÞ sin β

sin ðβ þδ  ϕ′ þηÞ sin η

sin ðβ þ ψ þ ϕ′  ηÞ  b sin ðψ þϕ′ ηÞ sin β sin ðψ þ ϕ′Þ z1 ¼  sin η

ðC:6Þ

z0 ¼ Y

ðB:2Þ

ðC:7Þ

where sin ðη þ ϕ′ þ θÞ  b sin ðϕ′ þ θÞ sin η sin ðβ þ ϕ′þ θÞ f1 ¼ sin β

References

f 0 ¼ y

e0 ¼  b sin ðψ þϕ′ ηÞ  y e1 ¼

sin ðβ þ ψ þ ϕ′ ηÞ sin β

ðB:3Þ

sin ðψ þ ϕ′Þ sin η ðB:4Þ

sin ðβ þ δ ϕ′þ ηÞ þb cos ðβ þ δ  ϕ′Þ sin η sin ðϕ′ δÞ r1 ¼ sin β

r0 ¼ y

g 0 ¼ Hðb þ y cot η  h cot βÞ þ

ðB:5Þ

 H cos εþ b sin ε sin η  y ðH cot η þ bÞ sin ðη  εÞ

g 1 ¼  yðcot η  cot βÞ  b

ðB:6Þ

2

s0 ¼ h K a1 s1 ¼  2hK a1 s2 ¼ K a1

ðB:7Þ

Appendix C. Coefficients of the quartic Eq. (35) for free Mechanism 3

C 0 ¼ c 1 k0  c 0 k1 C 1 ¼ 2ðc2 k0  c0 k2 Þ C 2 ¼ 3c3 k0 þc2 k1 c1 k2 C 3 ¼ 2c3 k1 C 4 ¼ c 3 k2

ðC:1Þ

where c0 ¼ w0 a0 z0 þ v0 l0 q0 þ j0 q0 K a1 ðh  YÞ2 c1 ¼ w1 a0 z0 þ w0 a1 z0 þ w0 a0 z1 þ v1 l0 q0 þ v0 l1 q0 þ v0 l0 q1 þ ðj0 q1 þj1 q0 ÞK a1 ðh  YÞ2 c2 ¼ w1 a1 z0 þ w0 a1 z1 þ w1 a0 z1 þ v1 l1 q0 þ v0 l1 q1 þ v1 l0 q1 þ j1 q1 K a1 ðh  YÞ2 c3 ¼ w1 a1 z1 þ v1 l1 q1

ðC:2Þ

k 0 ¼ z 0 d0 k 1 ¼ z 0 d1 þ z 1 d0 k 2 ¼ z 1 d1 ;

ðC:3Þ

v0 ¼ w0  Yb

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