The disturbance forces in the surrounding soil during the curved shield construction can be divided into three types: the uneven thrust force q at the excavation surface of the shield, the friction force f between the shield shell and surrounding soil, and the grouting pressure p at the end of the shield. Based on the characteristics of the three forces, a shield excavation mechanics model is established, where the excavation direction of the shield machine is in the positive direction of the x-axis, and the excavation surface is located in the yoz plane at x = 0.

For convenience, subsequent calculations satisfy the following assumptions:

1) The soil has two layers with different elastic parameters, where the thickness of the upper layer is H ₁, and the lower layer is an elastic semi-infinite space. Both the upper and lower layers are undrained.

Because the Mindlin solution is used in an isotropic linear elastic semi-infinite space, when dealing with complex soil conditions, the weighted average method can be used to simplify different soil layers with approximate mechanical parameters into a single uniform soil layer, Subsequently, for calculations, the layer method is used to convert the upper soft and lower hard soil layers with a large difference in mechanical parameters into a uniform soil layer with unified parameters (Zhou et al., 2020). The soil transformation model is shown in Figure 1, the upper soil layer thickness is H ₁, and the elastic modulus, Poisson’s ratio, and internal friction angle are E ₁, μ ₁, and φ ₁, respectively. The modulus of elasticity, Poisson’s ratio, and angle of internal friction of the soil layer below are E ₂, μ ₂, and φ ₂, respectively. The buried depth of the shield axis is H, and the shield radius is R. After transformation, the mechanical parameters of the upper soil layer with thickness H ₁′ become E ₂, μ ₂, and φ ₂. The buried depth of the shield axis is H’.

The height of the upper layer of soil after conversion is H 1 ′ = H 1 E 1 / E 2 a . (1)

When the mechanical parameters of the upper and lower soil layers differ greatly, a = 0.33, and when the mechanical parameters of the upper and lower soil layers have a small difference, a = 0.5. Assuming that the coordinates of any point in the current soil layer are (x, y, z), a new coordinate system (x′, y′, z′) after transformation of the upper soft and lower hard soil layers is established, where x ′ = x y ′ = y z ′ = z E 1 / E 2 a ,   z ≤ H 1 z ′ = H 1 ′ + z − H 1 ,   z > H 1 } . (2)

Mindlin (1936) derived a point (0, 0, c) in a semi-infinite elastic space (x′, y′, z′) under the action of the vertical concentrated force P _v and horizontal concentrated force P _h. The vertical displacements w _v and w _h are obtained using Eqs 3, 4, respectively. w v = P v 16 π G ( 1 − μ ) [ 3 − 4 μ R 1 + 8 ( 1 − μ ) 2 − ( 3 − 4 μ ) R 2 + ( z ′ − c ) 2 R 1 3 + ( 3 − 4 μ ) ( z ′ + c ) 2 − 2 c z ′ R 2 3 + 6 c z ′ ( z ′ + c ) 2 R 2 5 ] , (3) w h = P h x ′ 16 π G ( 1 − μ ) [ z ′ − c R 1 3 + ( 3 − 4 μ ) ( z ′ + c ) 2 − 2 c z ′ R 2 3 − 6 c z ′ ( z ′ + c ) 2 R 2 5 + 4 ( 1 − μ ) ( 1 − 2 μ ) R 2 ( R 2 + z ′ + c ) ] , (4) where G is the soil shear modulus (kPa) and μ is Poisson’s ratio. R 1 = x ′ 2 + y ′ 2 + ( z ′ − c ) 2 R 2 = x ′ 2 + y ′ 2 + ( z ′ + c ) 2 } . (5)

The soil deformation calculation model obtained by the additional thrust on the excavation surface when the shield machine is driving in the curved section of the upper soft and lower hard soil layer is shown in Figure 2. The additional thrust q is assumed to be unevenly distributed at the cut, and the inner and outer thrusts of the curve are q ₁ and q ₂, respectively; this is shown by the division of the tunneling soil layer into upper and lower layers.In the area dA = rdrdθ of any micro-element in the excavation surface, r is the distance from the micro-element to the center of the excavation surface, and θ is the angle between the micro-element and the center horizontal plane of the excavation surface.

Based on the coordinate transformation, the general analytical Mindlin equation at any point in the space coordinates can be derived. We assume that the space coordinate xyz is parallel to the corresponding coordinate axes of the local coordinate x′y′z′. The distances from the coordinate origin of the offset space coordinate system xyz are l, m, and n, respectively, and the following relationship holds: x ′ = x − l y ′ = y − m z ′ = z E 1 / E 2 a − n ,   z ≤ H 1 z ′ = H 1 ′ + z − H 1 − n ,   z > H 1 } . (6)

Substituting Eq. 6 into Eqs 3–5, the vertical displacement at any point (x, y, z) in the space coordinates of the actual soil mass can be obtained. Substituting l = 0, m = rcosθ, and n = 0 into Eq. 4, and calculating for the uneven thrust on both sides of the shield curve tunneling construction, we obtain d w q = d w q 1 + d w q 2 = x ′ q 1 r d r d θ 16 π G ( 1 − μ 2 ) { α [ z ′ − c R 1 3 + ( 3 − 4 μ 2 ) ( z ′ − c ) R 2 3 − 6 z ′ c ( z ′ + c ) R 2 5 + 4 ( 1 − μ 2 ) ( 1 − 2 μ 2 ) R 2 ( R 2 + z ′ + c ) ] + z ′ − c R 3 3 + ( 3 − 4 μ 2 ) ( z ′ − c ) R 4 3 − 6 z ′ c ( z ′ + c ) R 4 5 + 4 ( 1 − μ 2 ) ( 1 − 2 μ 2 ) R 4 ( R 4 + z ′ + c ) } , (7) where α is the thrust difference coefficient and R 1 = [ x ′ 2 + ( y + r ⁡ cos ⁡ θ ) 2 + ( z ′ − c ) 2 ] R 2 = [ x ′ 2 + ( y + r ⁡ cos ⁡ θ ) 2 + ( z ′ + c ) 2 ] R 3 = [ x ′ 2 + ( y − r ⁡ cos ⁡ θ ) 2 + ( z ′ − c ) 2 ] R 4 = [ x ′ 2 + ( y − r ⁡ cos ⁡ θ ) 2 + ( z ′ + c ) 2 ] } . (8)

The soil conditions that the shield machine may encounter during the actual construction process, as shown in Figure 3, can be divided into the following four situations: 1) the shield machine in the hard soil layer, 2) most of the shield machine in the hard soil layer, 3) most of the shield machines in the soft soil layer, and 4) shield machines in the soft soil layer. The cross-sectional area of the weak soil layer, marked by the red dashed line, is determined by the ellipse and soil layer boundaries, and the integration area of the hard soil layer, marked by the black solid line, is determined by the original circle and soil layer boundaries.

In addition, owing to the different positions of the soil layer in the tunnel section, the calculation parameter c in Eq. 7 can be expressed as a geometric relationship as follows: H ′ = z E 1 / E 2 a , H ≤ H 1 H ′ = H − H 1 + H 1 ′ , H ≥ H 1 c = H ′ − R ⁡ sin ⁡ θ } . (9)

1) When the shield tunnels are in the lower soil layer, substituting Eqs. 6, 8, and 9 into Eq. 7, the vertical displacement of the soil layer caused by the unbalanced force at the excavation surface can be obtained as

1) Upper ellipse area integral, w _qt:

The upper and lower limits of θ are θ 1 = a r c s i n ( H − H 1 ) E 1 / E 2 a R 2 − [ ( E 1 / E 2 ) 2 a − 1 ] ( H − H 1 ) 2 θ 2 = π 2 } . (11)

The upper and lower limits of r′ are r 1 = ( H − H 1 ) E 1 / E 2 a s i n θ r 2 = R c o s 2 θ + ( E 2 / E 1 ) 2 a s i n 2 θ } . (12)

Substituting Eqs. 6, 8, 9, 11, and 12 into Eq. 7, we derive w q t = ∫ r 1 r 2 ∫ θ 1 θ 2 d w q . (13)

2) Area integral w _qy of the lower part of the circle:

The upper and lower limits of θ are θ 1 = − π 2 θ 2 = arcsin ( H − H 1 R ) } . (14)

Substituting Eqs. 7, 9, 10, and 14 into Eq. 7, we obtain w q y = ∫ 0 R ∫ θ 1 θ 2 d w q . (15)

3) Area integral w _qs of the lower triangle:

The upper and lower limits of θ are θ 1 = arcsin ( H − H 1 R ) θ 2 = π 2 } . (16)

The upper and lower limits of r′ are r 1 = 0 r 2 = H − H 1 sin ⁡ θ } . (17)

Substituting Eqs. 6, 8, 9, 16, and 17 into Eq. 7, we derive w q s = ∫ r 1 r 2 ∫ θ 1 θ 2 d w q . (18)

The total vertical displacement is w q = w q t + w q y + w q s . (19)

3) Most sections of the shield tunnel were in the upper soil.

1) Upper ellipse area integral, w _qt:

The upper and lower limits of θ are θ 1 = a r c s i n ( H − H 1 ) E 1 / E 2 a R 2 − [ ( E 1 / E 2 ) 2 a − 1 ] ( H − H 1 ) 2 θ 2 = π 2 } . (20)

The upper and lower limits of r′ are r 1 = 0 r 2 = R c o s 2 θ + ( E 2 / E 1 ) 2 a s i n 2 θ } . (21)

Substituting Eqs. 6, 8, 9, 20, and 21 into Eq. 7, we derive w q t = ∫ r 1 r 2 ∫ θ 1 θ 2 d w q . (22)

2) Area integral w _qy of the lower part of the circle:

The upper and lower limits of θ are θ 1 = − π 2   θ 2 = arcsin ( H − H 1 R ) } . (23)

The upper and lower limits of r′ are r 1 = H − H 1 sin ⁡ θ r 2 = R } . (24)

Substituting Eqs. 6, 8, 9, 23, and 24 into Eq. 7, we derive w q t = ∫ r 1 r 2 ∫ θ 1 θ 2 d w q . (25)

3) Integral w _qs of the lower part of the triangular area:

The upper and lower limits of the θ are θ 1 = π 2 θ 2 = a r c s i n ( H − H 1 ) E 1 / E 2 a R 2 − [ ( E 1 / E 2 ) 2 a − 1 ] ( H − H 1 ) 2 } . (26)

The upper and lower limits of r′ are r 1 = 0 r 2 = H − H 1 sin ⁡ θ } . (27)

Substituting Eqs. 6, 8, 9, 26, and 27 into Eq. 7, we derive w q s = ∫ r 1 r 2 ∫ θ 1 θ 2 d w q . (28)

4) When the shield tunnels are in the upper soil, the upper and lower integral limits of r are r 1 = 0 r 2 = R c o s 2 θ + ( E 2 / E 1 ) 2 a s i n 2 θ } . (29)

Substituting Eqs. 6, 8, 9, and 29 into Eq. 7, we derive w q = ∫ r 1 r 2 ∫ − π 2 π 2 d w q . (30)

The calculation model of the ground uplift caused by the friction f between the shield shell and surrounding soil is shown in Figure 4. The calculation model can be regarded as the horizontal load acting along the cylinder axis on the side surface of the cylinder. The friction force f is assumed to be unevenly distributed along the central axis. For f ₁ and f ₂, the tunneling soil layer is divided into upper and lower layers. The inner and outer thrusts of the curve are q ₁ and q ₂, respectively. Suppose the length of the shield from the excavation cutter head to the tail of the shield is L. Then, the area of any infinite element on the surface of the shell is given by dA = Rdθdl, where l is the axial distance from the infinite element to the excavation surface, and the concentrated force received is dP _h = fR′dθdl. We substitute l = − l, m = R′cosθ, and n = 0 in Eq. 6 to transform the coordinate system as follows.

Because of the unequal frictional resistance of the inner and outer shells of the curved shield tunnel and the integral between the partitions, the vertical displacement of the soil layer caused by the frictional distribution force of the infinitesimal element can be obtained from Eq. 4 as follows: d w f = d w f 1 + d w f 2 = x ′ f 2 R d l d θ 16 π G ( 1 − μ 2 ) { β [ z ′ − c R 1 3 + ( 3 − 4 μ 2 ) ( z ′ − c ) R 2 3 − 6 z ′ c ( z ′ + c ) R 2 5 + 4 ( 1 − μ 2 ) ( 1 − 2 μ 2 ) R 2 ( R 2 + z ′ + c ) ] + z ′ − c R 3 3 + ( 3 − 4 μ 2 ) ( z ′ − c ) R 4 3 − 6 z ′ c ( z ′ + c ) R 4 5 + 4 ( 1 − μ 2 ) ( 1 − 2 μ 2 ) R 4 ( R 4 + z ′ + c ) } , (31) where β is the thrust difference coefficient and R 1 = [ x ′ 2 + ( y ′ + R ′ ⁡ cos ⁡ θ ) + ( z ′ − c ) 2 ] R 2 = [ x ′ 2 + ( y ′ + R ′ ⁡ cos ⁡ θ ) + ( z ′ + c ) 2 ] R 3 = [ x ′ 2 + ( y ′ − R ′ ⁡ cos ⁡ θ ) + ( z ′ − c ) 2 ] R 4 = [ x ′ 2 + ( y ′ − R ′ ⁡ cos ⁡ θ ) + ( z ′ + c ) 2 ] } . (32)

Because there is no need to integrate in the triangular area, the second and third cases shown in Figure 3 can be regarded as the same. Therefore, the integration cases were as follows: 1) the full section of the tunnel is located in the lower soil; 2) the tunnel section spans the upper and lower layers; and 3) the full section of the tunnel is located in the upper soil. Because of the different positions of the soil layer in the section, the calculation parameters R and c in Eq. 31 are different. According to the geometric relationship, it can be expressed as follows: R ′ = R c o s 2 θ + ( E 2 / E 1 ) 2 a s i n 2 θ c = H E 1 / E 2 a − R s i n θ } ( U p p e r   s o i l ) R ′ = R c = H − H 1 + H 1 E 1 / E 2 a − R s i n θ } ( L o w e r   s o i l ) } . (33)

1) When the full section of the tunnel is in the lower layer of soil, substituting Eqs. 6, 32, and 33 into Eq. 31, the vertical displacement of the soil layer caused by friction can be obtained as follows:

1) Upper ellipse area integral w _ft:

The upper and lower integral limits of θ are θ 1 = arcsin ( H − H 1 ) E 1 / E 2 a R 2 + [ ( E 1 / E 2 ) 2 a − 1 ] ( H − H 1 ) 2 θ 2 = π 2 } . (35)

Substituting Eqs. 6, 32, 33, and 35 into Eq. 31, we derive w f t = ∫ 0 L ∫ θ 1 θ 2 d w f . (36)

2) Area integral of the lower part of the circle w _fy:

The upper and lower integral limits of θ are θ 1 = − π 2 θ 2 = 2 π + arcsin ( H − H 1 R ) } . (37)

Substituting Eqs. 6, 32, 33, and 37 into Eq. 31, we derive w f y = ∫ 0 L ∫ θ 1 θ 2 d w f . (38)

The total vertical displacement is w f = w f t + w f y . (39)

3) When the full section of the tunnel is in the upper soil, substituting Eqs. 6, 32, and 33 into Eq. 31, we derive

As shown in Figure 5, the length of the shield from the shield tail to the grouting end is s, and the grouting pressure at the shield tail is p. The grouting pressure decreases along the grouting length, reaching 0 at point S, far from the shield tail. For any infinite element dA = Rdθds, the concentrated force dp can be decomposed into the horizontal and vertical components dp _h and dp _v , respectively. This paper only considers the ground displacement caused by the vertical component of the force, dp _h = p(1 − s/S)·Rdθds. In this section, we do not consider the influence of curved shield tunneling on the grouting pressure. Therefore, we set l = −L−s, m = Rcosθ, and n = 0, and substitute these in Eq. 6 to transform the coordinate system as follows:

From Eq. 2, the vertical displacement of the soil layer caused by the vertical component of the additional pressure of the synchronous grouting of the micro-element shield tail is obtained as d w p = p R ⁡ sin ⁡ θ d θ d s 16 π G ( 1 − μ 2 ) [ 3 − 4 μ 2 R 1 + 8 ( 1 − μ 2 ) 2 − ( 3 − 4 μ 2 ) R 2 + ( z ′ − c ) 2 R 1 3 + ( 3 − 4 μ 2 ) ( z ′ + c ) 2 − 2 c z ′ R 2 3 + 6 c z ′ ( z ′ + c ) 2 R 2 5 ] , (41) where S is the length of the grouting section of the shield tail (m). Generally, the width of a ring segment is 1.2 m. R 1 = [ ( x ′ + L + s ) 2 + ( y ′ − R ⁡ cos ⁡ θ ) 2 + ( z ′ − c ) 2 ] R 2 = [ ( x ′ + L + s ) 2 + ( y ′ − R ⁡ cos ⁡ θ ) 2 + ( z ′ + c ) 2 ] } , (42)

Considering similar cases as in Section 3.2, we derive the following:

1) When the full section of the tunnel is in the lower soil, substituting Eqs. 6, 32, and 42 into Eq. 41, the vertical displacement of the soil layer caused by the shield tail grouting pressure can be obtained as

1) Upper ellipse area integral w _pt:

The upper and lower integral limits of θ are θ 1 = arcsin ( H − H 1 ) E 1 / E 2 a R 2 + [ ( E 1 / E 2 ) 2 a − 1 ] ( H − H 1 ) 2 θ 2 = π − arcsin ( H − H 1 ) E 1 / E 2 a R 2 + [ ( E 1 / E 2 ) 2 a − 1 ] ( H − H 1 ) 2 } . (44)

Substituting Eqs. 6, 32, 42, and 44 into Eq. 41, we derive w p t = ∫ 0 S ∫ θ 1 θ 2 d w p . (45)

2) Area integral of the lower part of the circle w _py:

The upper and lower integral limits of θ are θ 1 = π − arcsin ( H − H 1 R ) θ 2 = 2 π + arcsin ( H − H 1 R ) } . (46)

Substituting Eqs. 6, 32, 42, and 46 into Eq. 41, we derive w p y = ∫ 0 S ∫ θ 1 θ 2 d w p . (47)

The total vertical displacement is w p = w p t + w p y . (48)

3) When the full section of the tunnel is in the upper soil, substituting Eqs. 6, 32, and 42 into Eq. 41, we derive

During shield tunneling, over-excavation of the cutter head creates a gap between the shield shell and the surrounding soil, resulting in the loss, loosening, and deformation of the soil layer (Zou and Zuo, 2017). When calculating the surface deformation caused by the loss of the soil layer, as the curved shield will over-excavate the inner soil, the influence of the over-excavated part on the excavation boundary needs to be considered. When the upper soft and lower hard soil layers are equivalent to a single soil layer, the excavation and convergence boundary will change after convergence, that is, from a circle to a combination of an ellipse at the top and a circle at the bottom (Cao et al., 2020), as shown in Figure 6. The length of the semi-horizontal axis of the ellipse is the same as the radius of the original circle, the length of the semi-longitudinal axis is E 1 / E 2 times the radius of the original circle, and the center of the ellipse is below the center of the circle.

For curved shields, owing to the large over-excavation on the inner side of the curved shield tunneling route, the equivalent soil loss parameter g on the inner side is larger than that on the outer side. The extra over-excavation amount on the inner side of the line is set as δ , and the soil loss of the curved shield can be obtained as ε = π R 2 + 1 2 π R δ − π ( R − g 2 ) 2 π R 2 + 1 2 π R δ . (50)

According to Sun et al. (2019), the over-excavation amount on the inner side of the line in the curve shield construction is δ = ( 2 R c + 2 R ) 2 + L 2 − ( 2 R c + 2 R ) 2 , (51) where R _c is the curve radius of the curved shield tunnel and L is the length of the single-section shield shell. If the shield machine is equipped with a hinge device, L is half the length of the single-section shield shell.

The changes in the excavation boundary and convergence boundary are calculated using the random medium theory method. The influence of tunnel excavation on the ground surface can be regarded as the sum of the influences of an infinite number of unit excavation ( ζ , η ) below the ground. In infinitely small micro-element excavations, when the excavation unit completely collapses, the soil settlement at the coordinate ( y ' , z ' ) can be calculated using Eq. (52). u z = tan ⁡ β η − z ′ e x p [ − π t a n 2 β ( η − z ′ ) 2 ( y ′ − ζ ) 2 ] d ζ d η , (52) where β is the main influence angle of the overburden of the tunnel and is calculated as follows (Wei et al., 2006): tan ⁡ β = H ′ − z ′ k H ′ ( 1 − z ′ / H ′ ) 0.3 2 π , (53) where k is the width parameter of the ground settlement trough. According to Han and Li (2007), k = 1 − 0.02 φ and φ is the internal friction angle of the soil.

According to Cao et al. (2019), the value of β in Eq. 53 varies with the location of the tunnel and can be divided into the following two cases:

1) When the tunnel axis is located in the upper soil, i.e., when H ≤ H ₁, there is

Equation 52 can only calculate the final stable settlement under the plane strain state and cannot calculate the three-dimensional settlement caused by soil loss. Cao et al. (2020) extended Eq. 52 to a three-dimensional state to determine the change in soil loss along the tunnel excavation direction as follows:   u z ¯ ¯ = u z 2 ( 1 − x ′ + L ( x ′ + L ) 2 + ( y ′ − ζ ) 2 + η 2 ) . (56)

Then, the vertical displacement of the soil layer can be obtained by the integral method, as follows: w z = ∬ Ω ′ − ω ′ u z ¯ ¯ d ζ d η . (57)

According to Figure 6, when the excavation surface is located in the upper soft and lower hard soil layers, Eq. 57 can be transformed into the following equation: w z = ( ∫ a 1 b 1 ∫ c 1 d 1 u z ¯ ¯ d ζ d η − ∫ e 1 f 1 ∫ g 1 h 1 u z ¯ ¯ d ζ d η ) + ( ∫ a 2 b 2 ∫ c 2 d 2 u z ¯ ¯ d ζ d η − ∫ e 2 f 2 ∫ g 2 h 2 u z ¯ ¯ d ζ d η ) , (58) where a i , b i , c i , d i (i = 1,2) are the upper and lower integral limits of the boundary of the excavation in the ith layer of soil, and e i , f i , g i , h i (i = 1,2) are the upper and lower integral limits of the convergent boundary in the ith layer of soil, respectively. To obtain the upper and lower limits of the above integral boundary, it is necessary to obtain the similarity ratio Mag 1 2 between the excavation and the convergence boundary after elastic transformation. According to the geometric relationship between the two figures, Mag 1 2 is determined as follows: M a g 1 2 = 1 − g ( H 1 − H + R ) E 1 / E 2 a + ( H + R − H 1 ) . (59)

According to the ellipse equation, the upper and lower limits of the integral can be expressed by { a 1 = ( H − R ) E 1 / E 2 a , b 1 = H 1 ′ , e 1 = ( H − R + g ) E 1 / E 2 a , f 1 = H 1 ′ a 2 = H 1 ′ , b 2 = H 1 ′ + H + R − H 1 , e 2 = H 1 ′ , f 2 = H 1 ′ + H + R − H 1 c 1 = − ( R + γ ) 1 − ( η − H E 1 / E 2 a ) 2 / ( R E 1 / E 2 a ) 2 d 1 = R 1 − ( η − H E 1 / E 2 a ) 2 / ( R E 1 / E 2 a ) 2 g 1 = − M a g 1 2 ∗ R 1 − ( η − H E 1 / E 2 a ) 2 / ( M a g 1 2 E 1 / E 2 a ) 2 = − h 1 c 2 = − ( R + γ ) 1 − ( η − H 1 ′ − H + H 1 ) 2 / R 2 d 2 = R 1 − ( η − H 1 ′ − H + H 1 ) 2 / R 2 g 2 = − M a g 1 2 ∗ R 1 − ( η − H 1 ′ − H + H 1 ) 2 / ( M a g 1 2 ∗ R ) 2 = − h 2 . (60)

The total length of the shield tunnel was 3,257.36 m. The excavation diameter of the shield is 9.14 m, the length is 7.1 m, the outer diameter of the segment is 8.8 m, the inner diameter is 8.0 m, the ring width is 1.8 m, and the thickness is 0.4 m. The cover of the shield tunnel is 2.98–34.45 m. The maximum slope of the interval tunnel is 3%, and the minimum curve radius is 800 m. The crossing soil layers are mainly mixed soil layers of silty clay and fully weathered granite and are a mix of fully weathered, strongly weathered, and moderately weathered granite soil layers. The average pressure of the shield soil bin is maintained at 240–300 kPa, which is slightly higher than the static soil pressure at the center of the tunnel. Because the soil layer before the excavation slightly swells, a shield is used to compensate for the soil loss caused by the shield tail falling out. According to existing research (Sun et al., 2019; Deng et al., 2022) and engineering experience, the additional thrust on the inner side of the excavation surface is 60 kPa, the difference coefficient α = 1.05 , friction force = 50 kPa, the difference coefficient β = 0.95 , and shield tail grouting pressure = 200 kPa. The grouting position is at the shield tail and has a length of one segment. In this study, mileages ydk16 + 403 (Section 1) and ydk16 + 458 (Section 2) in the construction section are selected for analysis. The measured and calculated values are compared, analyzed, and then verified using the numerical model, as shown in Figure 7.

To verify the accuracy of the calculation results of the prediction equation (Likitlersuang et al., 2019; Petchkaew et al., 2022), the construction process of the shield machine curve tunneling in the upper soft and lower hard strata is simulated using the Midas GTS finite element software. The dimensions of the model are taken as 60, 100, and 40 m along the X, Y, and Z directions, respectively. The boundary conditions of the model are set to default. The bottom surface is fully constrained, the top surface is free, and normal constraints are applied on four sides. The numerical calculation model is shown in Figure 8.

The constitutive model of the rock and soil mass adopts the Mohr column model. The main structure adopted the elastic model, and the physical and mechanical parameters of the rock, soil mass, and main structural materials are listed in Table 1 γ, E, μ, φ, e, and c are the characteristic values of gravity, elastic modulus, Poisson’s ratio, internal friction angle, porosity, and cohesion, respectively.

The surface deformations of Sections 1, 2 are calculated using the revised theoretical formula. The calculated results are compared with the measured, numerical simulation, and (original) theoretical formula values, as shown in Figures 9–12. The soil layer conditions and tunnel burial depth of Sections 1, 2 are shown in Figure 7, and the equivalent soil loss parameters are equal to 9.5 and 9.7 mm, respectively.

Figures 9, 11 shows that the longitudinal deformations of the surfaces of Sections 1, 2 are consistent. The longitudinal deformation of the surface caused by the additional thrust during shield construction is antisymmetrically distributed at the excavation face, and the maximum uplift value appears in front of the excavation face. At distances in the range of 5–10 m, the surface deformation caused by the frictional force of the shield shell reaches its maximum value along the rear of the excavation face at 0.5 L, it is uplifted in front of it, and subsidence is formed behind it. The grouting pressure of the shield tail caused the surface uplift. Moreover, the maximum uplift value appeared at 1 L behind the excavation face, that is, at the position at which the segment came out of the shield shell. The total surface deformation exhibited a rapid subsidence stage (5 m in front of the cutterhead to 25 m behind the cutterhead), and the uplift was larger within the range of 5–15 m in front of the cutterhead. At distances >25 m, the deformation was stable behind the cutterhead.

For analysis the lateral deformation of the surface of Section 1 at which the excavation surface is located (x = −14.2 m) and the lateral deformation of Section 2 at which the excavation surface is located (x = −5.1 m) were selected for analysis (Figures 10, 12). The lateral deformation laws of the surfaces of the two sections were the same. Due to the difference in friction between the two sides of the shield shell, the uneven additional thrust on the excavation surface, and the over-excavation inside the shield turning line, the soil loss inside the turning was large and led to the formation of the surface settlement tank. The lateral deformation of the total surface presents a specific asymmetry. However, owing to the high proportion of weak soil in Section 2, the peak value of the surface deformation curve was large.

Figures 9–12 show that the lateral and longitudinal deformations of Sections 1, 2 calculated by the modified theoretical formula are in good agreement with the measured deformation values, and the variation trend of the surface soil before and after excavation is consistent. In addition, the deformation curve of the simulation results obeys the same law as that obeyed by the measured results. This curve can also represent more accurately the process responsible for the changes in the surface deformation (which increases slowly and then falls rapidly) in front of the excavation. However, compared with the revised theoretical formula and the measured results, the numerical simulation showed that the surface settlement value behind the excavation face was larger, and the increase of the soil mass in front of the excavation face was smaller. This is mainly attributed to the fact that the numerical simulations do not consider the friction between the shield shell and the soil during excavation, or the influence of the shield shell side on the surrounding soil during shield turning.

The deformation curve of the original theoretical formula has values that are smaller than the measured results, and the calculated lateral settlement curve is centrosymmetric and has values that are different from the measured result. This is because the original theoretical formula did not consider the influences of the upper soft soil and the lower hard soil layers on the construction parameters, or the offset, or the increase in the settlement tank caused by the over-excavation of the shield machine inside the curve during tunnel construction.

In summary, the calculation method in this study can more accurately represent the surface deformation caused by the curved construction of the shield machine in the upper soft and lower hard soil layers. In conclusion, all construction parameters are affected by the changes in different soil layers and the construction process. Collectively, these parameters can provide a theoretical basis for the curved shield construction in the upper soft and lower hard soil layers.

When the shield is changed from straight excavation to curved excavation, as shown in Figure 2, the shield will apply an uneven additional thrust on the excavation surface. As the radius of the curve increases, the difference coefficient α increases accordingly. Differences in the friction between the inner and outer sides of the shield shell and the soil are observed during curve shield excavation. As the radius of the curve increases, the difference coefficient β decreases. During shield tunneling, over-excavation appears on the inner side of the curve, and the soil loss on the inner side is greater than that on the outer side of the curve. The uplift curve of the surface in soft soil at x = −50 m under different turning radius conditions is shown in Figure 13. As the turning radius decreases, the amount of over-excavation inside the shield increases, the soil loss increases, and the final deformation of the ground surface becomes larger. The offset value of the settlement trough decreases as the turning radius increases.

For comparing the uplift characteristics of the ground surface when the shield is driven in a straight line under different soil conditions, we assume that the buried depth of the tunnel is 4D. Four soil conditions were considered according to the position of the shield: 1) the overlying soil is all hard soil ( H 1 = 0 ) ; 2) the overlying soil is partially soft and partially hard soil ( H 1 = 2 D ) ; 3) all overlying soil is soft soil ( H 1 = 4 D ) ; and 4) all sections are soft soil ( H 1 = 6 D ) . The uplift curve of the ground surface at x = − 50  m under different soil layer conditions is shown in Figure 14. When the proportion of the weak bottom layer in the excavation surface of the shield is higher, the deformation of the ground surface is greater, and the width of the settlement trough is smaller.