Computers and Geotechnics 104 (2018) 54–68 Contents lists available at ScienceDirect Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo Research Paper Predicting deformation of multipropped excavations in soft clay with a modiﬁed mobilizable strength design (MMSD) method T ⁎ L.Z. Wanga,b, Y.J. Liub, Y. Hongb, , S.M. Liuc a City College, Key Laboratory of Oﬀshore Geotechnics and Material of Zhejiang Province, Zhejiang University, China Key Laboratory of Oﬀshore Geotechnics and Material of Zhejiang Province, College of Civil Engineering and Architecture, Zhejiang University, China c PowerChina Huadong Engineering Corporation, China b A R T I C LE I N FO A B S T R A C T Keywords: Modiﬁed mobilizable strength method Multipropped excavation Deformation Soft clay Field validation Design charts Although serviceability (i.e., deformation) governs the performance of deep excavations in urban areas with soft clay deposits, the common design methods are only able to check the potential of base stability, rather than the deformation. This limitation has led to the development of an analytical approach entitled the mobilizable strength design (MSD) method, which can predict both serviceability and safety in a single step of calculation. The modiﬁcation of the MSD method (i.e., the MMSD method) was recently made by the authors, by implementing a more realistic plastic deformation mechanism into the calculation. The objectives of this study are (a) to validate the MMSD method by comparing its predicted deformation with eight well-documented case histories in Shanghai and Hangzhou soft clay and (b) to quantify the inﬂuence of some potentially important factors on the deformation of deep excavations in Shanghai and Hangzhou. The accuracy of the MMSD predictions for the maximum lateral wall displacement (δh-max) have been found to fall within 35% of the ﬁeld data in most of the cases. This accuracy is more satisfactory than the accuracy (i.e., 120%) of the MSD predictions for the corresponding case histories, suggesting the eﬀectiveness of the modiﬁed plastic deformation mechanism considered in the MMSD calculation. Based on the veriﬁed MMSD method, preliminary design charts that predict δh-max of narrow excavations (speciﬁcally for a metro station) considering diﬀerent geometries and strength mobilization characteristics have been developed. 1. Introduction Th unsatisfactory performance of deep excavations in urban areas with soft clay deposits is usually attributed to excessive deformation, which could impose damage on the fragile neighboring infrastructures. In contrast, the conventional limit analysis methods based on the plastic limit equilibrium analysis [61,2,62,21] only check against collapse, which is associated with large deformations far exceeding the limits permitted in practice. This conventional limit analysis has necessitated the development of methods for excavation-induced deformation in the past decades. The commonly adopted methods for predicting excavation-induced deformation involve the semi-empirical approach that generalizes trends from similar types of case histories or model tests [52,9,50,37,44,23,64,31,59,60,10,22,8,33], numerical analysis [54,55,46,28,41,45,18,53,57] and probabilistic methods [39,51,7,74,13]. One major deﬁciency of the semi-empirical approach is that the expansion of the database may even lead to contradictions ⁎ [44], rather than a more generalized trend in deformation, because the semi-empirical approach cannot inherently account for the non-linear soil behavior and construction sequences. These case-speciﬁc conditions can be inherently considered in a numerical analysis. Satisfactory numerical predictions for excavation-induced deformation could be achieved with the use of sophisticated constitutive models considering the non-linearity of soil stiﬀness at small strains, such as the Brick model [55], MIT-E3 model [15], hardening soil small-strain model [1], and the hypoplastic models with the intergranular strain concept [42]. Most of these sophisticated constituted models contain quite a few parameters, which are sometimes not easy to calibrate [53,18]. The successive use of numerical analysis also requires the designer having a solid base in soil mechanics and ﬁnite elements. In view of the over-simpliﬁcation of the semi-empirical approach and the relative complication of the numerical analysis, Osman and Bolton [49] proposed a relatively simple analytical approach (mobilizable strength method, or MSD) for predicting both deformation and stability of multipropped wide excavations (B > 2 2 (l−h), where Corresponding author. E-mail addresses: wanglz@zju.edu.cn (L.Z. Wang), yajing_liu@zju.edu.cn (Y.J. Liu), yi_hong@zju.edu.cn (Y. Hong), liu_sm@ecidi.com (S.M. Liu). https://doi.org/10.1016/j.compgeo.2018.07.018 Received 31 January 2018; Received in revised form 8 July 2018; Accepted 16 July 2018 0266-352X/ © 2018 Elsevier Ltd. All rights reserved. Computers and Geotechnics 104 (2018) 54–68 L.Z. Wang et al. B = excavation width, l = wavelength of the deformation and h = excavation depth below the lowest prop) in clay. The MSD method can consider the case-speciﬁc construction sequences and non-linear soil behavior without the use of a constitutive model by assuming a plastic deformation mechanism and a working stress ﬁeld around a multipropped excavation. Based on the equilibrium at any given excavation stage, mobilized average shear strength can be deduced and converted to mobilized average shear strain through the measured represented stress-strain relationship of the in situ soil. The incremental deformation can then be calculated based on the mobilized average strain and the assumed plastic deformation mechanism (details to be given later). The applicability of Osman and Bolton’s [49] MSD method was later extended for predicting the deformation of narrow excavations (B < 2 2 (l−h)) by Lam and Bolton [26]. The authors [65,36] recently proposed a modiﬁed MSD method (i.e., MMSD method) for both wide and narrow excavations, by implementing a more realistic ground deformation mechanism into the MSD methods [49,26]. Although the MMSD method has been shown to reasonably predict the base instability in three case histories [65], the predictive capability of the MMSD method for excavation-induced deformation has not been examined. This study aims to validate the MMSD method by comparing its predicted deformations with seven well-documented case histories in Shanghai soft clay and one case history in Hangzhou soft clay and the corresponding predictions made by the MSD method. Parametric studies using the veriﬁed MMSD method are also performed to quantify the inﬂuence of excavation geometries and strength mobilization characteristics on the deformation of narrow excavations (speciﬁcally for metro stations) in Shanghai. Preliminary design charts considering these factors are then developed with the aim of assisting decisionmaking prior to any detailed analysis in the future design. Δδ h Δδ h Δδ h Δδ h (a) Δδ h 2. A modiﬁed mobilizable strength design (MMSD) method Δδ h 2.1. Modiﬁed incremental wall displacement proﬁle Δδ h In the MSD method, the incremental proﬁle of the ground displacement below the lowest prop is idealized to be a cosine function (see Fig. 1), as proposed by O’Rourke [48]. The cosine function is formulated by Eq. (1): 2πy ⎤ ⎞ Δδh = Δδh-max ⎡1−cos ⎛ ⎝ l ⎠⎦ ⎣ (b) (1) Fig. 2. Plastic ground deformation mechanism in the MSD method: (a) wide excavation [49]; (b) narrow excavation [26]. where Δδh-max and Δδh denote the maximum incremental displacement and the incremental displacement at any distance y underneath the lowest prop, respectively. The parameter l represents the full wavelength of the deformation proﬁle, which is correlated to the length of the wall below the lowest prop (s), as shown in Fig. 1, as follows: l = αs (2) The parameter α in Eq. (2) depends on the ﬁxity condition at the Fig. 1. Incremental displacement proﬁle for a multipropped excavation assumed in the MSD [49] and MMSD [65] methods. 55 Computers and Geotechnics 104 (2018) 54–68 L.Z. Wang et al. wall toe. It is bounded by a lower limit of 1 for a short wall embedded in stiﬀ clay [49] and an upper limit of 2 for a short wall embedded in very soft clay [14]. For simplicity, the value of α was usually taken as a constant during the entire excavation process [49,5]. In this study, the α was optimized after each increment of calculation to satisfy the principle of minimum energy dissipation, as suggested by Wang and Long [65]. After re-visiting the typical deformation proﬁle of multipropped excavations in clay, Wang and Long [65] found that the incremental proﬁle of lateral wall displacement can be more realistically represented by an exponential family distribution. The following exponential family function (Eq. (3)) is, therefore, adopted by the authors to replace the cosine function: Δδ h Δδ h Δδ h Δδh = Δδh-max 4y 1 8y 2 exp ⎛ − 2 ⎞ l ⎝2 l ⎠ ⎜ ⎟ (3) The newly proposed distribution in Eq. (3) forms the basis for modifying the ground deformation mechanism of the MSD method (to be presented in the following section). (a) 2.2. Modiﬁed ground deformation mechanism Fig. 2(a) shows the plastic ground deformation mechanism assumed in the MSD method for a multipropped wide excavation [49]. In the mechanism, the incremental proﬁle of the soil deformation also follows the cosine function deﬁned in Eq. (1), to ensure the deformation compatibility between the soil and the wall. The plastic deformation mechanism consists of four zones of distributed shear, namely, a rectangular zone (ABCD) on the retained side above the lowest prop, a circular fan zone (CDE) centered at the lowest prop, a circular fan zone (FEH) centered at the interception between the wall and the excavation bottom, and a triangular zone (FHI) under the excavation bottom. Each dotted line with arrows, which passes across the four zones of distributed shear, represents an equal-deformation line. By doing so, the soil within the four zones deforms continuously and compatibly, with no occurrence of sliding between each pair of the neighboring zones. Lam and Bolton [26] further extended the applicability of Osman and Bolton’s [49] MSD method for wide excavations to the scenario of narrow excavations by replacing the mechanism in the passive zones (FEH and FHI, see Fig. 2(a)) with a rectangular one (FEHI, see Fig. 2(b)). Based on the aforementioned framework, the authors [65,36] modiﬁed the ground deformation mechanisms by implementing a more realistic incremental deformation proﬁle (see Eq. (3)). Fig. 3(a) and (b) illustrates the modiﬁed ground deformation mechanisms for wide and narrow multipropped excavations in clay, respectively. Δδ h Δδ h Δδ h (b) Fig. 3. Plastic ground deformation mechanism in the MMSD method: (a) wide excavation [65]; (b) narrow excavation [36]. Table 1 Comparison of engineering shear strain increment of a wide excavation calculated by the MMSD and MSD methods. Zone ABDC Engineering shear strain increment calculated by MMSD method for a wide excavation δγ = Δδh-max 4 l (1− x ) exp ⎛⎝ − 16 2 l2 1 2 Engineering shear strain increment calculated by MSD method for a wide excavation π l 8x 2 ⎞ l2 δγ = −Δδh-max sin ⎠ x = distance from the wall CDE δγ = Δδh-max 64r 2 1 8(r + h)2 ⎞ exp ⎛ − 2 l3 l2 π l δγ = −Δδh-max sin ⎝ ⎠ r = radial distance from the center of the circular arc (D) EFH FIH 64 −Δδh-max 3 (r l 1 8(r + h) h)2 exp ⎛ − l2 ⎝2 4 + (1− 16 (r l2 ) 1 8(r + h)2 ⎞ + h)2 exp ⎛ − l2 ⎝2 ⎠ r = distance along FH between point F and the intersection with a plastic ﬂow line δγ = −Δδh-max l3 2πr l + Δδh-max 2r (1−cos ) 2πr l r = radial distance from the center of the circular arc (D) 2 ⎞ ⎠ r = radial distance from the center of the circular arc (F); and h = distance between the lowest support and the excavation level δγ = 2πx l x = distance from the wall 56 π l δγ = −Δδh-max sin 2π (r + h) l + Δδh-max 2r (1−cos 2π (r + h) l ) r = radial distance from the center of the circular arc (F); and h = distance between the lowest support and the excavation level π l δγ = −Δδh-max sin 2π (r + h) l r = distance along FH between point F and the intersection with a plastic ﬂow line Computers and Geotechnics 104 (2018) 54–68 L.Z. Wang et al. Table 2 Comparison of engineering shear strain increment of a narrow excavation calculated by the MMSD and MSD methods. Zone ABDC Engineering shear strain increment calculated by MMSD method for a narrow excavation δγ = Δδh-max 4 l (1− x ) exp ⎛⎝ − 16 2 l2 1 2 π l 8x 2 ⎞ l2 δγ = −Δδh-max sin ⎠ δγ = 1 8r 2 64 Δδh-max 3 r 2 exp ⎛ − 2 ⎞ 2 l l δγ = 16y 2 4 π 2l 1 8y 2 Δδh-max ⎧⎛ 3 ⎛1− 2 ⎞− 2 ⎞ exp ⎛ − 2 ⎞ 2 l 4B l ⎨ l π l δγ = −Δδh-max sin ⎝ ⎠ r = radial distance from the center of the circular arc (D) FIEH ⎟ 2πr l + Δδh-max 2r (1−cos ) 2πr l r = radial distance from the center of the circular arc (D) 2 2 π 2le−7.5 ⎫ πx cos B 4B2 ⎬ ( ) + ⎝ ⎠ ⎠ ⎠ ⎩⎝ ⎝ ⎭ x = distance from the wall; y = vertical distance from the interception between the wall and the excavation bottom (F) ⎜ 2πx l x = distance from the wall x = distance from the wall CDE Engineering shear strain increment calculated by MSD method for a narrow excavation DSS: direct simple shear PSA: plain strain active PSP: plain strain passive ( ) 2π y 2πy π l πl π πx δγ = Δδh-max ⎡ 2 + 2 + − sin ⎤ cos l ⎦ B 4B l 4B2 l ⎣ 4B x = distance from the wall; y = vertical distance from the midpoint of the FE ( ) normalized maximum incremental lateral wall displacement Δδh-max / l . It is therefore anticipated that the shear strain increment (δγ) in the solution of the MMSD method diﬀers from that in the MSD method due to the modiﬁed ground deformation mechanism. Table 1 compares the equations of the incremental shear strains that are compatible with the deformation mechanisms deﬁned in the MSD and MMSD methods for a wide excavation. The shear strain increments in the MSD and MMSD methods for a narrow excavation are compared in Table 2. 2.4. Shear mode and strength mobilization within the mechanism The incremental mobilization of shear strain (as formulated in Tables 1 and 2 is accompanied by mobilization of shear strength, which depends on the mode of shearing. Fig. 4 illustrates the shearing modes (indicated by the orientation of the major principal stress) of the soil within the assumed plastic mechanism in the MMSD method for wide excavations. The soil at diﬀerent locations around the wide excavation is subjected to three distinct shearing modes, i.e., direct simple shear (DSS), plain-strain active (PSA) and plain-strain passive (PSP). To account for the degree of strength mobilization at each excavation stage, a parameter β is deﬁned as follows: Fig. 4. Shear modes (directions of major principal stress) compatible with the plastic deformation mechanism in the MSD and MMSD methods for a wide excavation. β = su_ mob/ su (4) where su_mob and su denote the mobilized and the undrained shear strength of soil. The value of β is deduced based on the principle of energy conservation, as given in the following section. 2.3. Modiﬁed distribution of shear strain 2.5. Energy conservation Since the soil deformation mechanism is compatible with the lateral wall displacement, the incremental shear strain (due to excavation) in each zone of the proposed mechanism should be proportional to the Based on the principle of energy conservation, the virtual loss of potential energy resulting from the incremental ground settlement (Δδ v ) Table 3 Comparison of incremental vertical displacement of a wide excavation calculated by the MMSD and MSD methods. Zone ABCD Incremental vertical displacement calculated by MMSD method for a wide excavation Δδ v = Δδh-max 4x 1 8x 2 exp ⎛ − 2 ⎞ l ⎝2 l ⎠ Δδ v = Δδ v = Δδh-max 4r cosw 1 8r 2 exp( − 2 ) 2 l l Δδ v = Δδ v = −Δδh-max 4(r + h) 1 8(r + h)2 ⎞ sinw exp ⎛ − l l2 ⎝2 ⎠ Δδ v = −Δδh-max Δδh-max cosw 2πr ⎡1−cos 2 l ⎣ Δδ v = − ( ) ( Δδh-max sinw ⎡1−cos 2π (r + h) 2 l ⎣ ) ⎤⎦ r = radial distance from the center of the circular arc (F) r = radial distance from the center of the circular arc (F) FIH ( ) ⎤⎦ ⎤ ⎦ r = radial distance from the center of the circular arc (D) r = radial distance from the center of the circular arc (D) EFH Δδh-max 2πx ⎡1−cos l 2 ⎣ x = distance from the wall x = distance from the wall CDE Incremental vertical displacement calculated by MSD method for a wide excavation 4(r + h) 1 8(r + h)2 ⎞ sin π exp ⎛ − l l2 ⎝2 ⎠ 4 Δδ v = − ( Δδh-max ⎡1−cos 2π (r + h) l 2 ⎣ ) ⎤⎦ sin π 4 r = radial distance from the center of the circular arc (F) r = radial distance from the center of the circular arc (F) 57 Computers and Geotechnics 104 (2018) 54–68 L.Z. Wang et al. Table 4 Comparison of incremental vertical displacement of a narrow excavation calculated by the MMSD and MSD methods. Zone ABCD Incremental vertical displacement calculated by MMSD method for a narrow excavation Δδ v = Δδh-max 4x 1 8x 2 exp ⎛ − 2 ⎞ l ⎝2 l ⎠ Δδ v = Δδh-max 4r cosw 1 8r 2 exp ⎛ − 2 ⎞ l ⎝2 l ⎠ Δδ v = − Δδh-max 2πx ⎡1−cos 2 l ⎣ Δδ v = Δδh-max cosw 2πr ⎡1−cos 2 l ⎣ ( ) ( ) ⎤ ⎦ r = radial distance from the center of the circular arc (D) r = radial distance from the center of the circular arc (D) FIEH Δδ v = ⎤ ⎦ x = distance from the wall x = distance from the wall CDE Incremental vertical displacement calculated by MSD method for a narrow excavation πlΔδh-max ⎡ πx 1 8y 2 exp ⎛ − 2 ⎞−e−7.5⎤ sin B 4B ⎢ ⎥ ⎝2 l ⎠ ( ) Δδ v = − ⎣ ⎦ x = distance from the wall; y = vertical distance from the interception between the wall and the excavation bottom (F) lΔδh-max ⎡π 4B ⎣ + 2πy l + sin ( ) ⎤⎦ sin ( ) 2πy l πx B x = distance from the wall; y = vertical distance from the midpoint of the FE Beijing Shanghai (a) (b) Fig. 5. (a) The location of Shanghai in the People’s Republic of China; (b) the locations of the seven case histories and soil sampling for the validation proposed in Shanghai. 58 Computers and Geotechnics 104 (2018) 54–68 L.Z. Wang et al. Fig. 6. A simpliﬁed soil proﬁle and geotechnical parameters in Shanghai [5]. is assumed to be balanced by the virtual plastic work done in shearing the soil to a certain degree of strength mobilization (β): ∫v γsat Δδv dv = ∫v βsu δγdv (5) where γsat and v denote the saturated weight of soil per unit volume and the whole volume of the deformed zones, respectively. Numerically, β is calculated using Eq. (6) derived from the principle of virtual work, i.e., virtual loss of potential energy (∫v γsat Δδ v dv ) balancing the virtual plastic work in distributed shearing (∫v βsu δγdv ) . β= ∫v γsat Δδv dv ∫v su δγdv (6) In Eq. (6), the incremental exponential shear strains (δγ ) are calculated based on the equations summarized in Tables 1 and 2 for wide and narrow excavation, respectively. Additionally, the incremental vertical displacement (Δδ v ) of wide excavation and narrow excavation in Eq. (6) is calculated based on the equations summarized in Tables 3 and 4, respectively. By substituting the equations in Tables 1–4 into Eq. (6), the β value for wide and narrow excavations can be calculated numerically using Eqs. (7) and (8), respectively. ( ) exp ( − ) dxdy exp ( − ) rdrdw exp ( − ) rdrdw (1− ) exp ( − ) rdrdw⎞⎠ exp ( − ) dxdy exp ( − ) coswdrdw exp ( − ) sinwdrdw exp ( − ) sin dr ⎞⎠ H−h l 4 16x β = ⎛∫0 ∫0 su l 1− l2 ⎝ π l π l + ∫0 2 ∫0 su 64r 2 l3 + ∫0 4 ∫0 −su π l / ⎛∫0 ⎝ 4 ∫ π l 4r 2 l l 1 2 − ∫0 γsat 4r (r + h) l 2 1 2 ⎜ ⎝ 0 π 4x 1 8x 2 exp ⎛ − 2 ⎞ dxdy + ∫0 2 0 sat l ⎝2 l ⎠ 1 πl ∫hl ∫0B γsat 4B ⎡⎢exp ⎛ 2 − ⎣ ⎝ ∫0l γsat π 2le−7.5 ⎤ πx cos B 4B2 ⎥ ⎦ ( ) ⎞ dxdy ⎟/ ⎠ 4r 2 1 8r 2 exp ⎛ − 2 ⎞ coswdrdw− l ⎝2 l ⎠ 8y 2 ⎞ −7.5⎤ πx −e sin dxdy ⎟⎞ B ⎥ l2 ⎠ ⎦ ( ) ⎠ (8) The maximum incremental wall deﬂection is calculated by substituting the equations in Tables 1 and 2 into Eq. (9). 8(r + h)2 l2 π 4 ⎟ ⎛⎜∫ H − h ∫l γ 8(r + h)2 l2 8r 2 l2 8(r + h)2 l2 2 2 16y 4 1 8y π l ∫0H − h ∫0l su ⎡⎢ ⎛ 3 ⎛1− 2 ⎞− 2 ⎞ exp ⎛ 2 − 2 ⎞ + l ⎠ 4B ⎠ l ⎠ ⎝ ⎣⎝ l ⎝ 8(r + h)2 l2 8x 2 l2 1 2 1 2 π 4 16x 2 1 8x 2 64r 2 1 8r 2 H−h l β = ⎜⎛∫0 ∫0 su l ⎛1− 2 ⎞ exp ⎛ 2 − 2 ⎞ dxdy + ∫0 2 ∫0l −su 3 exp ⎛ 2 − 2 ⎞ rdrdw+ l ⎠ l ⎠ l l ⎠ ⎝ ⎝ ⎝ ⎝ 8(r + h)2 l2 1 2 1 2 l 4r (r + h) γ l 0 sat − ∫0 ∫ 8x 2 l2 16(r + h)2 l2 l 4x γ 0 sat l + ∫0 2 ∫0 γsat π 4 1 2 1 2 64(r + h)2 l3 + ∫0 4 ∫0 −su 3 l H−h 2 Fig. 7. Undrained soil strength proﬁle adopted for MMSD and MSD calculations (a) Shanghai soft clay; (b) Hangzhou soft clay. (See above-mentioned references for further information.) δγmob (β ) = (7) ∫v δγ (Δδh-max ) dv ∫v dv (9) where the mobilized shear strain increment (δγmob ) can readily be deduced based on the known β value and a given stress-strain curve. The 59 Computers and Geotechnics 104 (2018) 54–68 L.Z. Wang et al. Table 5 Geometrical properties of the eight case histories analyzed in this study. Station Thickness of diaphragm wall (mm) Wall depth (m) Maximumexcavationdepth (m) Excavation length on plane (m) Excavation width on plane (m) Location Yishan Station South Xizan Station South Pudong Station Pudongdadao Station Pudian Station Yanchang Station Gubei Station West Wenyi Station 600 800 600 600 600 600 600 800 28.0 38.0 27.0 27.0 26.5 27.0 26 38.5 15.5 20.6 17.3 16.0 16.5 15.2 14.5 15.0 335.0 169.0 196.0 200.0 194.0 226.0 149.5 151.0 17.3 22.8 20.8 18.5 20.4 18.1 17.5 24.1 Shanghai Shanghai Shanghai Shanghai Shanghai Shanghai Shanghai Hangzhou the four construction sites is thought to be representative for the seven case histories. As shown in Fig. 7(a), the vane shear strength measured within a depth of 30 m below the ground surface can be ﬁtted by the following linear function: only unknown (Δδh-max ), which is a variable of δγ (see Tables 1 and 2, can then be solved using Eq. (9). 3. Validation against case histories in soft clay The validation of the MMSD method is examined by comparing its predictions with the measured deformations from eight well-documented multipropped excavations in Shanghai soft clay [68,47] and Hangzhou soft clay [67]. The predictive capability of the MSD method for the eight case histories is also evaluated. This section reviews the ground conditions and the eight selected case histories in Shanghai and Hangzhou and then validates the MMSD method against the case histories. 15 + 1.7Z Z ⩽ 15m su = ⎧ ⎨ ⎩− 49.5 + 6.0Z 15m < Z ⩽ 30m (10) where Z represents the depth below the ground surface. Considering the absence of ﬁeld data concerning the strength of the layer below the depth of 30 m (i.e., silty clay with sand), the corresponding su proﬁle was deduced with a theoretical undrained strength ratio (su/ σ ′v ) of the layer. According to Wang et al. [66], the su/ σ ′v value for clay has an eﬀective friction angle ϕ′ of 30°–35° (measured from the triaxial compression test) falling in the range between 0.240 and 0.298. Wang’s [69] triaxial compression test on the silty clay with the sand layer suggests an eﬀective friction angle ϕ′ of 35°. The su/ σ ′v value of the layer is, therefore, taken as 0.298. Since the excavations analyzed in this study were carried out under the plane strain condition, a strength anisotropy factor of 1.25 [24,25] was applied to convert the value of su/ σ ′v under the triaxial condition (i.e., 0.298) to that under the plane strain condition (i.e., 0.373). Correspondingly, the su proﬁle of the layer at the depth between 30 and 50 m was semi-empirically determined by the following equation, given that the average soil unit eﬀective weight is 8.5 kN/m3: 3.1. Geotechnical proﬁles and soil properties of Shanghai and Hangzhou soft clay Shanghai is located on the west bank of the East China Sea (see Fig. 6(a)), while the seven selected case histories are positioned around the Pearl 2 circle line of Shanghai (see Fig. 5(b)). The top alluvial sediments in Shanghai, which were deposited during the Quaternary Period, are present to a depth up to 400 m [27]. The soil proﬁles in Shanghai are found to be relatively uniform [27,17] and can be divided into many layers. Fig. 6 illustrates a simpliﬁed soil proﬁle for Shanghai, which was generalized by Bolton et al. [5] based on the borehole information of 249 sites in Shanghai (as documented by Xu [72]. Within the depth below 50 m, which is of engineering signiﬁcance to the seven excavations analyzed herein (excavation depth from 14.5 to 20.6 m), the ground consists of six layers, i.e., a top ﬁll layer underlain with ﬁve ﬁne-grained layers. The ground water table is located at approximately 1 m below the ground surface [47]. Therefore, the great majority of the six soil strata are fully saturated with water, except the ﬁll within the top 1 m. The top two layers (i.e., ﬁll and silty clay with sand) are slightly over-consolidated due to the desiccation in the stress history [29], while the underlying four layers are normally consolidated [5]. There may be an over-consolidated layer at depths between 25 and 30 m below the ground surface, as implied by the proﬁle of cone penetration resistance (qc) of a speciﬁc site near the underground railway line 4 in Shanghai [73,56]. Fig. 7(a) shows the in situ su proﬁles, which were measured by the vane shear apparatus at four typical construction sites in Shanghai. According to Bjerrum [4], the ratio of vane shear strength su_vane to triaxial undrained compressive strength su ranges from 0.95 to 1.05, which gives a mean value of 1.0 (i.e., su_vane = su) for clay with a plasticity index (Ip) of 10–24. Since the Ip values of the soil in the case histories analyzed in this study fall within the range between 10 and 24, the assumption of su_vane = su is adopted in the MMSD and MSD calculations reported herein. The locations of these vane shear measurements, as shown in Fig. 5(b), are in close vicinity to the seven case histories analyzed in this study. Considering the reasonable uniformity of Shanghai soil proﬁles [27], the vane shear strength measured from Su = 37.5 + 3.1Z (11) The vertical permeability of the ﬁve ﬁne-grained layers (from the second to the sixth layers below the ground surface) is in the range between 2.4 × 10−9 and 7.7 × 10−9 m/s, while the horizontal permeability is in the range between 1.2 × 10−8 and 7.6 × 10−8 m/s [19], which leads to approximately undrained behavior of the conventional excavations in Shanghai [47]. Hangzhou is located in the east of China, where the Qiantang River ﬂows. The selected case history in Hangzhou is situated in the West Wenyi Road, where the soil proﬁle near the ground surface consists of six layers. The top layer is 1.5 m thick ﬁll, whereas the second layer is a 2.5–3.0 m thick silt clay. Beneath these layers is a 25–32 m thick soft silt layer, which has a high void ratio and low shear strength. The saturated permeability of this layer is of the order of 10−9 m/s. The fourth and ﬁfth layers are gravel silt clay and gravel containing mica, oyster shells, and other organic material, whereas the sixth layer is weathering breccia. The ground water table in this case history is found to be approximately 1.5 m below the ground surface. Fig. 7(b) shows the in situ shear strength proﬁle measured by a vane shear in the case history in Hangzhou. This proﬁle can best be ﬁtted by the following equation, given that the average soil unit eﬀective weight is 8.5 kN/m3: su = 3.1Z 60 (12) Computers and Geotechnics 104 (2018) 54–68 L.Z. Wang et al. 600 km2, within which the ground conditions are similar. The length (L) and width (W) of the seven station excavations in Shanghai, which are rectangular-shaped on the plane, fall in the ranges of 169–335 m and 17–23 m, respectively. The aspect ratio of excavation in Hangzhou is approximately 6.2, which is slightly lower than that in Shanghai (i.e., L/W = 7.4–19.4). Due to the large aspect ratios (i.e., L/W = 6.20–19.4) of the eight-station excavations, the ground deformations near the middle span along the longitudinal axis of each excavation were found to exhibit a plane-strain behavior [68,17]. The eight excavations were all retained by concrete diaphragm walls and multi-levels of prop, due to the large excavation depths (15 ∼ 23 m) and the thick soft clay deposit. The height between each level of the excavation typically ranged between 3 and 4 m. After each level of excavation, a level of prop was installed at a height of approximate 0.3 m above the excavation bottom. For ground improvement purposes, compaction grouting was installed inside four of the eight excavations (i.e., Yishan Station, South Xizan Station, Gubei Station and West Wenyi Station). The excavation rate of the eight case histories varies from 0.33 to 0.61 m/day. The previous ﬁeld measurements suggest that at an excavation rate of 0.4 m/day in soft clay, the excess pore water pressures on both sides of the wall merely dissipated during the excavation [16]. According to the coupled-consolidation ﬁnite element analysis of typical excavations in soft clay [16], an excavation rate of 0.3 m/day still results in the undrained soil response, implying that all the case histories reported herein were likely to be excavated under the undrained condition. Considering that the eight excavations were carried out under the undrained condition and deformed in a plane-strain manner, they are identiﬁed as ideal case histories for validating the MMSD method. Despite the grouting installed inside four of the eight excavations, the beneﬁcial eﬀect of this grouting on the ground deformation was not evident [68]. The presence of the compaction grouting was, therefore, not considered in the MMSD and MSD predictions. 3.3. Input parameters and calculation procedure of the MMSD method The input soil parameters include mainly su proﬁles (as illustrated in Fig. 8 and quantiﬁed by Eqs. (10)–(12)) and strength mobilization curves of Shanghai and Hangzhou clay. Fig. 8(a) shows relationships between the degree of mobilized undrained shear strength (β = su_mob/su) and the mobilized shear strain of Shanghai clay. These data resulted from Li’s [34] undrained triaxial tests and from the authors’ direct simple shear test on intact K0-consolidated clay sample [67]. The soil samples tested by Li [34] and in this study were cored from the clay layer at a depth of 8 and 14.5 m in the construction sites for Longhua Station and Guiqiao Station, respectively. Considering the nearness of the soil sampling locations to the seven selected case histories and the uniformity of the Shanghai soil proﬁle [27], the measured strength mobilization curves in Fig. 8(a) are likely to be representative for analyzing the seven case histories. The quality of these soil samples was evaluated using the criterion of Lunne et al. [38], which suggested the sample used in the triaxial tests [34] and the direct simple shear test (this study) are of rate 2 (good to fair). Despite the diﬀering depths and locations of the clay sample tested, the three measured strength mobilization curves exhibit very similar trends, conﬁrming the reasonable uniformity of Shanghai clay. As anticipated, the strength mobilization curve resulting from the direct simple shear test is bounded by that from the triaxial compression and triaixal extension tests. The measured strength mobilization curves were ﬁtted by a power function, which was shown to broadly ﬁt the data of 19 types of clay [63]. As far as the Shanghai clay is concerned, however, the power function generally underestimates the undrained strength of the clay and overestimates the initial stiﬀness of the clay. The overestimation of the initial stiﬀness by the power function may be due to the use of fair to good quality samples. Stress-strain curves from Fig. 8. (a) Normalized stress-strain relationship of intact K0-consolidated Shanghai clay samples obtained from undrained shearing test; (b) comparisons of the strength mobilization curve between Shanghai clay and soft clays worldwide; (c) normalized stress-strain relationship of intact K0-consolidated Hangzhou clay samples obtained from undrained shearing test. (See abovementioned references for further information.) 3.2. Construction details and geometrical properties of the case histories Table 5 summarizes construction details and geometrical properties of the eight multipropped excavations analyzed in Shanghai and Hangzhou, which were all for metro stations. As shown in Fig. 5(b), the seven stations in Shanghai are in a plane area of approximately 61 Computers and Geotechnics 104 (2018) 54–68 L.Z. Wang et al. Fig. 9. Comparisons between the measured and calculated (by the MMSD and MSD methods) lateral wall displacement in the seven case histories: (a) Yishan Station; (b) South Xizan Station; (c) South Pudong Station; (d) Pudongdadao Station; (e) Pudian Station; (f) Yanchang Station; (g) Gubei Station. the geotechnical parameters are unavoidably uncertain, which can be, rationally, treated under a probabilistic framework [6,32]. Fig. 8(b) compares the strength mobilization curve of Shanghai clay with that of eight types of clays worldwide. The stiﬀness of Shanghai clay under relatively small strains (shear strain < 1%) is higher than the stiﬀness of most of the clays in other areas around the world, partly explaining why the observed excavation-induced ground deformations in Shanghai clay are relatively small compared with similar case histories worldwide [29]. This observation also suggests that it could be non-conservative to extrapolate the calculation chart developed for predicting deformation of excavation in Shanghai (presented in latter part of this study) to similar case histories in softer soil around the world. Fig. 8(c) shows the relationships between the degree of mobilized high quality samples will be required for better calibration of the strength mobilization curve in the future. Alternatively, the strength mobilization curves of Shanghai clay were found to be better ﬁtted by a rational function, as follows: β= γ ⎧ a + γ − (1 + a) γ 2 (γ ⩽ b) ⎨1 (b < γ ⩽ 1) ⎩ (13) where a and b are ﬁtting parameters. The best ﬁtting parameters for a and b are 0.0035 and 0.06, respectively, which are adopted as the input of the MMSD and MSD predictions for the seven case histories in Shanghai. The other two variants of “a” used in the parametric study (i.e., a = 0.002 and 0.005) represent the upper and lower bound parameters that can ﬁt the test data of strength mobilization curves for Shanghai clay, as shown in Fig. 8(a). With a limited number of test data, 62 Computers and Geotechnics 104 (2018) 54–68 L.Z. Wang et al. (e) (g) (f) Fig. 9. (continued) undrained shear strength (β = su_mob/su) and the mobilized shear strain of the Hangzhou clay. These data were obtained from the undrained triaxial compression tests performed by the authors on two intact K0consolidated clay samples. The two soil samples tested in this study were cored from the construction sites of West Wenyi Station at depths of 9 and 19 m below the ground surface. The strength mobilization curves of the Hangzhou clay are reﬁtted by a rational function (see Eq. (13)). (b) Read the corresponding mobilized shear strain (δγmob) from the strength mobilization curve of a representative intact sample, using Eq. (13). (c) Deduce the maximum incremental wall movement Δδh-max by substituting the δγ in Tables 1 or 2 and δγmob into Eq. (9). (d) Calculate the incremental displacement proﬁle, based on Eq. (3). (e) Obtain the cumulative deformation proﬁle by accumulating the incremental movement proﬁles within the proposed mechanism. 3.3.1. Calculation procedure for the MMSD method Based on the ﬁtted su proﬁle (Eqs. (10)–(12)) and strength mobilization curve (Eq. (13)) described above, the following calculation procedure is programmed in MATLAB to calculate the excavation-induced lateral wall displacement in the eight selected case histories: 3.4. Validation of the MMSD method against the case histories Fig. 9(a)–(g) compare the measured and predicted lateral wall displacement from the seven selected excavations in Shanghai at two typical stages, namely, one intermediate stage and the ﬁnal excavation stage. In the ﬁgure, H and He denote the excavation depth at the intermediate stage and at the ﬁnal stage, respectively. The predictions in the ﬁgure are made based on the MMSD and MSD methods, which diﬀer only in the form of incremental displacement proﬁles (see Figs. 2 and 3. (a) Calculate the average mobilized strength (β = su_mob/su) within the proposed mechanism based on equilibrium and energy conservation after each excavation stage according to Eq. (6). 63 Computers and Geotechnics 104 (2018) 54–68 L.Z. Wang et al. Fig. 10. Comparison between the measured and calculated (by the MMSD and MSD methods) lateral wall displacement in West Wenyi Station in Hangzhou soft clay. In both methods, the kinematic mechanism for a narrow excavation (B < 2 2 (l−h), see Figs. 2(b) and 3(b)) is considered. Calculated deformation wall proﬁles by both MMSD and MSD methods are similar to the measured data in each case history, with some certain extent of overestimation in the maximum lateral wall displacement. The overestimation is probably associated with the use of strength and stiﬀness parameters obtained from only fair to good quality samples. As suggested by Osman and Bolton [49], high-quality samples will be required to better calibrate the stress-strain curves for improving the MMSD prediction in the future. Comparatively, the MMSD method yields a better prediction than the MSD method. Fig. 10 compares the performance of the MSD and MMSD methods for predicting the lateral wall displacement (δh) of the excavation in Hangzhou. Compared to the MSD method, the MMSD yields a better prediction for δh at the two typical stages. Qualitatively, the MMSD method overestimates the measured δh at the last stage of excavation by 10%, which is smaller than the overestimation by the MSD method (i.e., 15%). To quantify the accuracy of the MMSD method, the measured and predicted maximum lateral displacement (using the MMSD method) after each excavation is extracted from Figs. 9 and 10 and presented in Fig. 11(a) to make a direct comparison. As illustrated, 85% of the MMSD predictions (i.e., 7 out of the 8 excavations) fall within 35% of the corresponding measurements. Considering the ignorance of the bending elastic energy stored in the wall in the current MMSD method [26], its predictive capability is considered fairly satisfactory. The accuracy of the MSD method for the same excavations is examined in Fig. 11(b), which compares the MSD predictions with the measured data. The comparison suggests that the results predicted by MSD fall within 120% of the ﬁeld observations. This error band given by the MSD method for predicting the maximum lateral wall displacement (i.e., 120%) is obviously larger than the error band of the MMSD predictions, i.e., 35% (see Fig. 11(a)). In other words, the accuracy of the MMSD method for predicting the maximum lateral wall displacement has been substantially improved by implementing a more realistic incremental deformation proﬁle than that assumed in the MSD method. Fig. 12(a) and (b) examine the error bands of the MMSD and MSD Fig. 11. Accuracy of the calculated maximum lateral wall displacement by: (a) MMSD and (b) MSD methods. methods in predicting the depth of the maximum lateral wall displacement, respectively. As anticipated, the MMSD method (error band = 30%) yields a better prediction in the depth of the maximum lateral wall displacement than the MSD method (error band = 60%), due to the acceptance of a more realistic incremental wall deformation proﬁle. 4. Parametric study and design charts of narrow excavations (for metro station) in Shanghai soft clay Having veriﬁed the MMSD method against eight well-documented multipropped excavations, a series of parametric studies is performed to quantify the eﬀects of three potentially important variables on the deformation induced by narrow excavations speciﬁcally for metro stations in Shanghai soft clay. The three variables include excavation width (B), embedment depth of the wall (D) under the base and the height of the lowest prop above the excavation bottom (h). These variables are normalized by the ﬁnal excavation depth (He). By examining the geometrical properties of the 23 metro station excavations out of more than 300 Shanghai case histories documented 64 Computers and Geotechnics 104 (2018) 54–68 L.Z. Wang et al. by Liu et al. [29] and Wang et al. [64], the dimensionless groups B/He, D/He and h/He in this parametric study are determined to be in the ranges of 0.50–1.50, 0.60–1.20 and 0.235–0.365, respectively. In each analysis, the value of He is kept as a constant, i.e., 15 m. In each analysis, the undrained shear strength proﬁle (see Eqs. (10) and (11)) and the strength mobilization curve (see Eq. (13)) are identical to those adopted in the analyses of the seven case histories in Shanghai. Fig. 13 shows the calculated δh-max due to multipropped excavations with diﬀerent geometrical properties in Shanghai soft clay, which has ﬁtting parameters a of 0.0035 and b of 0.06 (see Fig. 9(a)). It is anticipated that for each given D/He and h/He, the value of δh-max reduces as the excavation becomes narrower (i.e., smaller B/He), with a maximum percentage reduction of 72.0%. The ﬁgure also shows that for each given combination of B/He and h/He, the value of δh-max declines as the wall penetrations go more into the ground (i.e., larger D/He), with a maximum percentage diﬀerence of 67.7%. This ﬁnding is attributed to a larger D/He resulting in a longer wavelength, which increases the integral domain for the soil in shearing to resist the potential energy loss and therefore a declined mobilization of shear strain. Although a similar trend in δh-max is also observed for excavations with a decreasing h/He at a given combination of B/He and D/He, the variation in h/He leads to a much smaller percentage reduction (i.e., 0.2%∼9.8%) than those caused by changing B/He and D/He (i.e., 72.0% and 67.7%, respectively) because the h/He in the range of engineering interest (0.235–0.365) can only alter the wavelength by up to 14.8% (i.e., D/He = 0.6, α = 1), leading to a very limited change (up to 9.8%) in the mobilization of shear strain. Based on the two most inﬂuential dimensionless forms (i.e., B/He and D/He), as identiﬁed above, a more detailed parametric study was performed to develop preliminary design charts, with the aim of assisting decision-making prior to any detailed analysis in design. In the detailed parametric study, diﬀerent rates of strength mobilization with strain are considered by adopting three diﬀerent groups of ﬁtting parameters (as deﬁned in Eq. (13)), namely, a = 0.0020, b = 0.045; a = 0.0035, b = 0.06; and a = 0.0050, b = 0.09. In the meantime, the strength proﬁles of the three soils are identical (Fig. 7(a)). For the soil with each given strength mobilization factor, 121 analyses considering diﬀerent combinations of D/He and B/He are performed. Fig. 14(a)–(c) shows the design charts for estimating the normalized maximum lateral wall displacement (δh-max/He) of excavation in clay with three diﬀerent groups of ﬁtting parameters, namely, a = 0.0020, b = 0.045; a = 0.0035, b = 0.06; and a = 0.0050, b = 0.09. In each chart, the deformation control criteria for deep excavations with diﬀerent protective grades (i.e., I, II and III) in Shanghai [30] are also marked to gauge the environmental eﬀects due to the wall deformation. The three ﬁgures show that the lateral wall displacement can be reduced by either decreasing the excavation width (B/He) or increasing the embedment depth of the wall (D/He), as anticipated. A comparison between the three ﬁgures also suggests that for a given geometrical property of an excavation (i.e., B/He and D/He), smaller lateral wall displacement results in a relatively stiﬀer ground (i.e., with a smaller value of parameter a). Fig. 14(a) and (b) shows that within the geometrical properties of interest, the resulted maximum lateral wall displacements fall within the Grade III deformation control criterion (δhmax/He < 0.7%) for excavations in the soil with the ﬁtting parameter a of 0.0020 and 0.0035. When the parameter a is larger than 0.0050 (see Fig. 14(c)), however, the maximum lateral wall displacement can exceed the Grade III deformation control criterion (δh-max/He > 0.7%). Despite the inﬂuence of parameter a (governing the strength mobilization rate) and b on the wall deformation, the factor of safety (FOS) against basal heave should not be altered provided that an identical strength proﬁle is adopted. Fig. 15 shows the design chart for estimating the FOS of the cases analyzed in this parametric study. Within the range of geometries considered, the FOS falls within the range between 1.3 and 2.2, implying that all the excavations considered in this parametric study have suﬃcient resistance against base heave failure, Fig. 12. Accuracy of the calculated depth of the maximum lateral wall displacement by: (a) MMSD and (b) MSD methods. Fig. 13. Inﬂuence of normalized excavation width, wall embedded depth and height of the lowest prop on the maximum lateral wall displacement. 65 Computers and Geotechnics 104 (2018) 54–68 L.Z. Wang et al. (a) (b) (c) Note: Grade I, Grade II and Grade III are based on the deformation control criteria for deep excavations in Shanghai (Liu and Hou, 1997). Fig. 14. Design charts for the normalized maximum lateral wall displacement (δh-max/He) of Shanghai excavations considering diﬀerent geometries and strength mobilization factor a (as deﬁned in Eq. (13)): (a) a = 0.0020, b = 0.045 (b) a = 0.0035, b = 0.060 (c) a = 0.0050, b = 0.090. eight carefully selected and well-documented deep excavations in Shanghai and Hangzhou soft clay. A comparison was also made between the calculated results of both MMSD and MSD methods for the seven case histories. The accuracy of the MMSD predictions for the maximum lateral wall displacement (δh-max ) falls within 35% of the measurements in most of the eight case histories (i.e., 7 out of the 8 excavations). This accuracy is obviously more satisfactory than the accuracy (i.e., 120%) of the MSD method for the corresponding case histories. Additionally, the MMSD yields a slightly better prediction for the depth where the δh-max occurs than the MSD method. These comparisons have demonstrated the improvement made by the MMSD method, due to the implementation of a more realistic plastic deformation mechanism in the latter. With the validated MMSD method, a series of parametric studies is carried out for developing calculation charts to predict the values of δh-max and the factor of safety (FOS) against basal heave for typical metro station excavations (narrow excavations) in Shanghai soft clay. Various excavation geometries and strength mobilization characteristics of soil are considered in the 363 runs of the parametric study. By considering the excavation geometries of common engineering practice in Shanghai, the parametric study reveals that the most effective way to reduce the excavation-induced deformation is to either narrow the excavation or increase the wall embedded depth. The results of the parametric study also conﬁrm that there is no one-to-one correlation between the deformation and FOS. For example, at a given FOS of 1.5, the induced δh-max can vary between 0.14% and 0.60% of the ﬁnal excavation depth, due to excavations in soils with diﬀerent strength mobilization characteristics but having the same strength, practically implying that the existing semi-empirical methods that relate the values of δh-max to the FOS may be treated with caution. Fig. 15. Design chart for factor of safety (FOS) against basal heave considering diﬀerent excavation geometries. although excessive deformation (see Fig. 14(c)) can occur. A comparison between Figs. 14 and 15 suggests that there is no one-to-one correlation between deformation and FOS. To be more speciﬁc, the value of δh-max/He can vary between 0.14% and 0.60% at a given FOS of 1.5. In other words, semi-empirical approaches (such as the chart of Mana and Clough [40], which correlate deformation to FOS, may, therefore, be treated with caution, highlighting the usefulness of the mobilization strength methods that are capable of predicting both deformation and FOS in a single step of calculation. Acknowledgements 5. Summary and conclusions The work presented in this paper is supported by National Key Research and Development Program (Grant No. 2016YFC0800204), National Natural Science Foundation of China (51338009), Zhejiang Provincial Key Research and Development Program (2018C03031), Zhejiang Provincial Natural Science Foundation of China (Y17E090016, LQ18E080004), Shanghai Science and Technology Development Funds (16QB1403400). Despite the recent development of the modiﬁed mobilizable strength design (MMSD) method by the authors, its predictive capability for excavation-induced deformation in undrained soft clay has not been evaluated. 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