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The paper was published in the proceedings of the 9th International Symposium on Geotechnical Aspects of Underground Construction in Soft Ground (IS – Sao Paulo 2017) and was edited by Arsenio Negro and Marlísio O. Cecílio Jr. The conference was held in Sao Paulo, Brazil in April 2017.

Analytical calculation of lateral earth pressure in finite soils considering displacement of retaining wall

X. Yang & G. Liu Department of Geotechnical Engineering, Tongji University, Shanghai, China

ABSTRACT: Based on the relationship between lateral earth pressure and displacement of retaining wall, a formula has been introduced for the calculation of lateral earth pressure. Then the formula was tes- tified by monitored data from centrifuge test and realistic project. Furthermore, the formula was applied to the calculation of lateral earth pressure for finite soils where the width of soils behind retaining wall is limited. With field data from a deep excavation in Shanghai, the characteristics of lateral earth pressure for finite soils were acquired in consideration of wall displacement s, width of finite soils, d, and friction angle between soils and wall, δ. The results show that (1) the lateral earth pressure in finite soils was smaller than Rankine active earth pressure; (2) the influence of wall displacement on lateral earth pressure varied in different soils; (3) both d and δ played a key role in the distribution of lateral earth pressure but within a certain scope.

1 INTRODUCTION calculation and model test of lateral earth pressure considering wall displacement (Milligan 1983, With the fast development of underground engi- Sherif et al. 1984, Bang 1985, Handy 1985, Fang neering in Shanghai, more and more excavations et al. 1986, 1993, Hua & Shen 1987, Matsuzawa have to go deeper and larger in limited spaces. A & Hazarika 1996, Chang 1997, Zhang et al. 1998, phenomenon has appeared that several excavations Huang et al. 1999, Paik & Salgada 2003, Xu et al. were constructed in the same field and had to be 2000, 2005, Mei 2001, Jiang et al. 2005, Lu & Yang faced with numerous underground structures in 2009, 2010, Li et al. 2012, Ying & Zheng 2012). proximity. It is well known that the lateral earth Their achievements have provided instructive ref- pressure plays significant impacts on retaining walls erences for the design and construction of excava- and underground structures. When excavations tions. However, all these works are finished with were constructed with pits or underground struc- the well-known assumption of semi-infinite soils. tures nearby, the distribution of lateral earth pres- The retaining wall of aforementioned excava- sure behind retaining wall showed different from tions in limited ground is faced with a different the classical Coulomb and Rankine earth pressure, state of lateral earth pressure when compared to just because of the soils in limited ground. semi-infinite soils. Only limited work had been pub- According to the classical earth pressure theory, lished for the calculation of lateral earth pressure in the relationship between lateral earth pressure and finite soils. Frydman & Keissar (1987) and Take & retaining wall movement has been explored (Terza- Valsangkar (2001) obtained the characteristics of ghi 1943). In engineering design, the calculation of earth pressure at rest in finite soils through centri- lateral earth pressure is simplified. Only two limit fuge test. Relevant analytical studies and numerical pressures are focused on, the active limit earth simulations were carried out to compute the limit pressure and passive limit earth pressure. However, states of lateral earth pressure as a modification it cannot reflect the variation of lateral earth pres- of the classical earth pressure theory (Handy 1985, sure induced by the displacement of retaining wall. Gao 2000, 2001, Fan & Fang 2010). Unfortunately, On the one hand, the underestimation of lateral few attentions have been paid to the variation of earth pressure may lead to irreversible engineering lateral earth pressure in finite soils when taking the disasters and loss of lives. On the other hand, the displacement of retaining wall into account. too conservative calculation of lateral earth pres- So, it is of great necessity to investigate the stress sure can ensure the safety of structures but may state of finite soils and develop relevant calcula- result in large construction cost. Therefore, on tion methods of lateral earth pressure. This paper the basis of Coulomb and Rankine earth pressure put forward a formula for the calculation of lateral theory, many researchers have contributed to the earth pressure with an arbitrary wall displacement.

81 And monitored data from field measurements and develops; (2) p(s) is a bounded function of s, which centrifuge test were used to testify the formula. is in accordance with the two limit states of lat- Afterward, the formula was applied to the study eral earth pressure; (3) the inflexion point appears on the characteristics of lateral earth pressure in when s = 0, i.e. when s > 0, the second derivative is finite soils. negative (passive state), and when s < 0, the second derivative is positive (active state); (4) the values of lateral earth pressure at rest, active limit earth pres- 2 CALCULATION FORMULA OF sure and passive limit earth pressure are displayed

LATERAL EARTH PRESSURE in decrease order as pp, p0, pa; and (5) the displace- CONSIDERING WALL DISPLACEMENT ment of passive limit earth pressure is larger than

that of active limit earth pressure, i.e. |sp| > |sa|. 2.1 Introduction of calculation formula In view of the two limit states of lateral earth pressure, the above parameters k , k , m, and n can Figure 1 shows the relationship between lateral 1 2 be obtained by Equation 2: earth pressure, p and movement of retaining wall, s. For wall movement, s, the positive values rep-  λ λ λ resent movements toward soils and the negative 1() 1+ 2(KKp − 0 )  values represent movements backward soils. The k1 =  λ2K 0 specific relationship can be referred to Terzaghi k= 2 c K (1934) and Matsuo et al. (1978).  2 0 λ On account of the characteristics of p-s curve  1(KKp − 0 ) (2) (Fig. 1), a suggested formula for calculation of lat- m = λ2()KK 0 − eral earth pressure is shown as Equation 1:  a  λ2k 1 k1 n = +1 1 − λ1 ()λλ1+ λ 2 m() λ1+ λ 2  k1 k1   +1 − σ 0  λ λ−as λ λ   1+ 2e 1+ 2  where K0, Ka, and Kp denote the coefficients of  λ2k 1 +k +1 ⋅ 1 − e−bss s > 0 earth pressure at rest, active limit earth pressure  2 λ λ λ () ()  1() 1+ 2 and passive limit earth pressure, respectively. p()s =  (1) k k The parameters λ , λ , a, and b can be acquired  1 1 1 σ 1 2  + −  0 by inversion calculation of four monitored data,  mλ+ λ e−mas m λ+ λ  ()()1 2 1 2  (p , s ), i = 1 ∼ 4.  k i i 1 1 1nbs 0 −k2 − ⋅() −e () s ≤  m λλ λ  ()1+ 2 2.2 Verification of calculation formula where σ = lateral earth pressure at rest. The param- To testify the reliability of calculation formula 0 (Eq. 1), a set of data is quoted from a centrifuge eters k1, k2, m, n, λ1, λ2, a and b are associated with the internal friction angle ϕ and the cohesive model test for compacted clayey backfill (Yue et al. strength c of soils. The values of these parameters 1992). are all positive. The units of pressure and displace- Figure 2 shows the calculation results by Equa- ment in Equation 1 are kPa and m, respectively. tion 1, along with the monitored data from the test. The given Equation 1 displays the main features The calculation was based on the inversion analysis of p-s curve (Fig. 1). In detail, (1) lateral earth pressure increases monotonically as displacement

Figure 2. Verification of the formula by centrifuge Figure 1. The curve of p – s. model test.

82 theory of soil mechanics. Based on Marston’s theory (Marston 1930), the vertical pressure in finite soils is less than the overburden weight due to the horizontal transfer of pressure. The pressure transfer is primarily associated with the frictional interaction between soils and retaining wall. Figure 4 plots the model for calculation of the

lateral earth pressure at rest, σ0, in finite soils. An infinitesimal element of soils, dz, is cut out for analysis. On the basis of static equilibrium condi- tion, the stress state of dz between the two walls is shown by Equation 3 below:

d d2 tan d Figure 3. Verification of the formula by field measure- γd⋅ z + σz d =() σ z + σ z d+ σ0 δ z (3) ments. where γ = unit weight of soils; d = width of finite soils; δ = friction angle between soils and retaining of parameters λ1, λ2, a, and b. In general, relative deviations of calculation results and monitored wall; and σz = the vertical earth pressure. data were very small, most of which fell within 5% Through an integral of Equation 3, Equation 4 except for some scattered points less than 10%. The can be concluded as: curves plotted by calculation results exhibited the z σ z dσ convergence of lateral earth pressure with the devel- dz = z (4) opment of wall displacement. By taking a close ∫0 ∫q γ− A σ 0 look, the displacement of retaining wall for active limit earth pressure were approximately located where A = 2tanδ/d, σ0 = K0σz, and q represents the in range of −150 mm to −200 mm, nearly −1%D overcharge on ground. (D represents the length of retaining wall), where Then the vertical earth pressure σz can be calcu- D = 20 m in the test. The values of |sa| showed larger lated by Equation 5: than those of relevant studies for clayey soils (Tien 1966), which might be ascribed to the soil condi- σ= 1 AK γ− γ − qAK e−AK0 z  (5) tions and depths of measuring points in the test. z ()0 () 0  In addition, the data from field measurements in a project (Wang & Liu 2003) provided a rare So, the lateral earth pressure at rest in finite opportunity for comparison. Both the calculation soils, σ0, is obtained by Equation 6: results and field data are incorporated in Figure 3. It can be noticed that the calculation results con- formed well to the field data when the retaining wall moved slightly. Yet as the wall displacement developed, field data commenced to fall just below the corresponding curve. It signified that the calculated earth pressure was a bit larger, which proved conservative for the design of retaining wall. More- over, almost all of the errors were within 10%. Thus, the calculation by Equation 1 can provide accurate results for the lateral earth pressure with an arbitrary displacement of retaining wall. As the monitored data of passive earth pressure were limited, relevant study was not able to be conducted.

3 CALCULATION OF LATERAL EARTH PRESSURE FOR FINITE SOILS

3.1 Calculation formula As mentioned above, the soils behind retaining wall in Shanghai excavations are finite soils that are different from the semi-finite soils in the classical Figure 4. Calculation model for finite soils.

83  −AK0 z  σ0=  γ−() γ − qAK 0 e A (6)

Considering Equation 1, the lateral earth pressure, σx can be expressed as:

σx =f1 σ 0 + f 2 (7)

where f1 is dimensionless and the unit of f2 is kPa. Substituting σ0 into Equation 7, the lateral earth pressure under any wall displacement can be calculated.

3.2 Distribution of lateral earth pressure in finite soils with wall displacement As an excavation progresses, the lateral earth pressure varies due to the development of wall deflection. Figure 5 shows the wall deflections from a Shanghai excavation (Tan & Wang 2013). Figure 6 presents the active earth pressures that were calculated under the following conditions: (1) the width of finite soils was 17 m (i.e. d = 0.5D), and (2) the friction angle between soils and wall Figure 6. Lateral earth pressure considering wall was 15° (i.e. δ = 15°). For comparison, Rankine deflection. active earth pressure was also included. The soil properties of this project are shown in Table 1. Geotechnical parameters of soils. Table 1. The project was supported by 34 m deep Soils diaphragm walls and excavated to 17.85 m below c′ (kPa) ϕ′ (°) K0 (= 1 − sinϕ′) the ground surface. On the basis of wall deflections, the characteristics of lateral earth pressure in Very soft clay 10 15 0.741 Firm clay 15 18 0.691 finite soils were studied. Table 2 presents the neces- Firm silty clay 20 20 0.658 sary parameters λ1, λ2, a, and b that were obtained through back analysis of field data. In Figure 6, σ represents the lateral earth pres- 0 Table 2. Calculation parameters of soils. sure at rest in finite soils. When no displacement

Soils λ1 λ2 a b

Very soft clay 9.98 0.11 0.18 2.53 Firm clay 9.89 0.10 0.14 9.86 Firm silty clay 9.93 0.09 0.16 9.97

happened on diaphragm wall, larger lateral earth pressures appeared in comparison with Rankine active earth pressures. Similar phenomenon was found at stage I when diaphragm wall was subjected to slight deformations. However, under significant deflections (stage II), the lateral earth pressure showed smaller than Rankine active earth pressure. This is benefitting from the decrease effect of active wall movement on the lateral earth

pressure (Fig. 1). Compared to σ0, σx at stage II dropped a lot. The maximum value of was up to 40% at the depth of 21 m, located in firm silty clay. Considering wall deflections in Figure 5, the maximum value of deflections appeared at the depth of Figure 5. Wall deflection along the depth. 15 m, located in very soft clay. This contradictory

84 may be induced by the different properties of soils. earth pressure, within the upper 25 m of the wall. It It seemed that the lateral earth pressure in stiffer meant that the wall deflections there had reached soils was more susceptible to the variation of wall the displacement of active limit earth pressure, displacement. sa. For the rest of the wall, the lateral earth pressure showed larger than Rankine’s pressure. That is to say, the wall displacement had not developed 3.3 Development of lateral earth pressure with enough. Yet, it turned different while the interface friction angle between soils and wall between soils and wall was frictional. In the upper

In classical Rankine earth pressure theory, the 25 m of the wall, σx showed less than Rankine back of retaining wall is supposed to be ideally active earth pressure. As to the deeper section of smooth. For Shanghai excavations, diaphragm the wall, σx gradually became larger. Because the wall is commonly used in the retaining system. deflections at the bottom of the wall were quite Since diaphragm wall is often constructed in situ, small (Fig. 5), the decline of lateral earth pres- the interface between soils and wall is frictional. sure in active state were not mobilized adequately It cannot be overlooked in the analysis of soil- according to Figure 1. structure interaction. To calculate earth pressure in finite soils, the frictional interface between soils 3.4 Development of lateral earth pressure and wall had to be considered. In view of the wall with width of finite soils deflections at stage II (Fig. 5), the development of lateral earth pressure with different friction angles As described above, finite soils are restricted by are given by Figure 7. The calculation was carried two walls in plane strain analysis (Fig. 4). It is out with the assumption of 17 m width finite soils believed that the stress state in finite soils is closely (i.e. d = 0.5D). Rankine active earth pressure was related to the width of soils, d. To explore the effect also incorporated in Figure 7 for comparison. of d, development of the lateral earth pressure is In Figure 7, δ represents the friction angle shown in Figure 8. The calculation was under the between finite soils and wall. Based on Equa- assumption of δ = 15°. tion 6–7, the frictional interface between soils and In Figure 8, D denotes the length of retaining wall plays a decrease effect on the lateral earth pres- wall. It proved that the lateral earth pressures in sure. From Figure 7, the decrease effect turned out finite soils were smaller in comparison to Rankine more significant as δ increased. When the back of active earth pressure, except for those at the deeper retaining wall is ideally smooth (i.e. δ = 0°), the lat- section of the wall (i.e. when z ≥ 25 m). With the eral earth pressure coincided with Rankine active increase of d, the curve of σx approached that of

Figure 7. Development of lateral earth pressure with δ. Figure 8. Development of lateral earth pressure with d.

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