According to actual site conditions, if L represents the span between the karst and the tunnel, d denotes the distance between them, E is the elastic modulus of the formation after grouting reinforcement, P _a is the weight of the reinforced structure following cave treatment, P _w indicates the water pressure exerted by groundwater, P be the weight of the surrounding rocks, and q represents the support force provided by the shield tunnel segments. The following assumptions are made:

The formation is considered intact after grouting reinforcement, and the surrounding rocks above the shield tunnel are simplified as a beam structure for stress analysis. 2) The above loads act on the shield tunnel in the form of uniformly distributed loads. 3) The stress at both ends of the tunnel is simplified as a horizontal load F.

Based on these assumptions, the mechanical model depicted in Figure 5 can be established.

According to Reference (Jiang et al., 2005), the deflection of the beam axis shown in Figure 5 can be expressed as:m f = ∑ i = 1 ∞ w i ⁡ sin i π l L (13)

According to Equation 13, the initial deflection of the beam axis is: f = w ⁡ sin π d l L (14) where w represents the deflection of the beam at L/2, and l denotes the arc length from the endpoint of the beam to any position along the axis.

Based on the cusp catastrophe theory of grouting reinforcement, the potential function of the surrounding rocks is: Π = U − W x − W y (15) where U represents the strain energy of the beam, W _x denotes the work done in the horizontal direction, and W _y is the work done by vertical forces.

According to the theories of material mechanics and elasticity, based on Equation 14, the terms in Equation 15 can be expressed as: U = EI 2 ∫ 0 L d 2 f d l 2 2 1 − d f d l 2 − 1 ds W x = F ∫ 0 L 1 − d f d l 2 1 2 1 2 W y = − Q − q 2 L w π (16)

By combining Equations 15, 16, the potential function of the surrounding rocks can be obtained. After simplification, it is expressed as: ∏ = E I π 6 16 L 5 w 4 + π 2 4 L E I π 2 L 2 − F w 2 + 2 L Q − q π w (17)

Let w = L π 4 L E I π 2 1 4 x A = L π L E I 1 2 E I π 2 L 2 − F B = L π 4 L E I π 2 1 4 Q − q (18)

Combining Equations 17, 18 yields: Π x = 1 4 x 4 + 1 2 A x 2 + B x (19)

Equation 19 represents the standard form of cusp catastrophe, which is consistent with Equation 12. When the bifurcation set satisfies Δ > 0, the surrounding rocks reinforced by grouting remain stable during shield tunneling. Thus, the following condition holds: Δ = 4 A 3 + 27 B 2 > 0 (20)

By substituting the terms from Equation 18 into Equation 20 and considering I = d ³ /12 and Q = Pa + Pw + P, the elastic modulus E of the surrounding rocks after grouting reinforcement can be determined. This is expressed in Equation 21, which serves as the grouting reinforcement standard for underwater karst formations located above the shield tunnel: E > 12 L 2 ( − F λ 2 + F 2 λ 2 4 + λ 3 27 1 2 1 3 − F λ 2 + F 2 λ 2 4 + λ 3 27 1 2 1 3 + F d 3 π 2 (21) where λ = 27 2 π 2 P a + P w + P − q 2

According to Equation 21, the control standard for grouting reinforcement depends on factors such as the karst distance, the properties of the surrounding rocks after reinforcement, the material performance following karst reinforcement, and the influence of groundwater.

Based on the actual site conditions, we assume that the span between karst and shield tunnel is L, the distance between the two is d, the elastic modulus of the formation after grouting reinforcement is given as E, the load of the shield machine is P ₁, the weight of the surrounding rocks is P r ′ , and the water pressure is P w ′ . The following assumptions are also made: The formation remains intact after grouting reinforcement, and the surrounding rocks above the shield are simplified as a fixed support beam plate for stress analysis. The above loads act on the surrounding rock mass in the form of uniformly distributed loads.

Based on these assumptions, the mechanical model illustrated in Figure 6 can be established.

According to References (Chen, 2002; Xu, 2024), the deflection of the fixed support beam plate shown in Figure 6 is expressed as: w x , y = 12 1 − μ 2 E d 3 Q 8 3 a 4 + 2 a 2 b 2 + 3 b 4 x 2 a 2 + y 2 b 2 − 1 2 (22) where E represents the elastic modulus of the surrounding rocks after grouting reinforcement, and μ is the corresponding Poisson’s ratio.

If we let K = 12 1 − μ 2 E d 3 Q 8 3 a 4 + 2 a 2 b 2 + 3 b 4 , then Equation 22 can be rewritten as: w x , y = K x 2 a 2 + y 2 b 2 − 1 2 (23)

Converting Equation 23 into a polar coordinate form, we have w x , y = K r 2 R 2 − 1 2 (24) where R = a b a 2 s i n 2 θ + b 2 c o s 2 θ .

Additionally, based on Equation 24, considering the boundary conditions and deterministic laws, the radial displacement of the surrounding rocks is given by: S r = r R 1 − r R 1.206 K 2 R − 1.785 K 2 R r R (25)

According to the cusp catastrophe theory and the established model, combining Equation 25 for radial displacement of surrounding rock, the potential function of the surrounding rocks after grouting reinforcement is: Π = U 1 + U 2 − W 1 − W 2 (26) where U ₁ represents the bending deformation potential energy of the surrounding rocks after grouting reinforcement, U ₂ denotes the surface strain potential energy, W ₁ is the axial work done by external loads, and W ₂ represents the radial work done by external loads, respectively.

Based on elastic mechanics and the constructed model, it follows that: U 1 = K 2 ∫ 0 2 π d θ ∫ 0 R d 2 K d r 2 2 + 1 r 2 d K d r 2 r d r U 2 = E d 2 1 − μ 2 ∫ 0 2 π d θ ∫ 0 R d S r d r + 1 2 d w d r 2 2 + S r 2 r 2 + 2 μ S r r d S r d r + 1 2 d K d r 2 r d r W 1 = ∬ Q w x , y d x d y W 2 = ∬ Q S r r d θ d w (27)

By combining Equations 26, 27 and performing term transposition, the potential function of the surrounding rocks is obtained as: ∏ = 0.603 π E d 1 − μ 2 1 a 2 + 1 b 2 w 4 + 0.042 Q π w 3 + 4 π E h 3 9 1 − μ 2 1 a 2 + 1 b 2 w 2 − a b q π 2 w (28)

Let w = x + 0.042 Q π 4 1 a 2 + 1 b 2 1 − μ 2 0.603 π E d A = 4 h 0.603 − 3 8 0.042 Q π 0.603 π E d 1 − μ 2 1 a 2 + 1 b 2 2 B = − a b q π 0.1206 π E d 1 − μ 2 1 a 2 + 1 b 2 − 0.084 Q h 3 1 − μ 2 0.603 2 1 a 2 + 1 b 2 9 E d 3 + 3 8 0.042 Q π 0.603 π E d 1 − μ 2 1 a 2 + 1 b 2 2

Then Equation 28 can be rewritten as, Π x = 1 4 x 4 + 1 2 A x 2 + B x (29)

Equation 29 represents the standard form of cusp catastrophe, which aligns with Equation 12. When its bifurcation set satisfies Δ > 0, the terms of w, A, and B are substituted into Equation 20, and considering Q = P 1 + P r ′ − P w ′ . The elastic modulus E of the surrounding rocks after grouting reinforcement is then determined, as expressed in Equation 30, which serves as the grouting reinforcement standard when the underwater karst is located beneath the shield tunnel. E > 1 − μ 2 d 2 0.33 P 1 + P r ′ − P w ′ 67 1 a 2 + 1 b 2 (30)

According to Equation 30, the control standard for grouting reinforcement depends on factors such as the karst distance, properties of the surrounding rocks after reinforcement, material performance following karst reinforcement, karst size, and the influence of groundwater.

Based on actual site conditions, let d represent the safe distance between the karst and the shield tunnel, r denote the tunnel radius, E be the elastic modulus of the formation after grouting reinforcement, P indicate the horizontal load exerted by the karst on the tunnel face, and F represent the shield support force. The following assumptions are considered: the formation remains intact after grouting reinforcement, and the karst is positioned directly in front of the tunnel axis. Horizontal loads act uniformly on the tunnel face.

Based on these assumptions, the mechanical model illustrated in Figure 7 can be established.

According to the cusp catastrophe theory and the mechanical model in Figure 7, the potential function of the tunnel face after grouting reinforcement is expressed as: Π = U 1 + U 2 − W 1 − W 2 (31) where U ₁ and U ₂ represent the bending deformation potential energy and surface strain potential energy of the tunnel face after grouting reinforcement, respectively, and W ₁ and W ₂ are the axial and radial work done by external loads, respectively.

According to elastic mechanics and the constructed model, the terms in Equation 31 can be expressed as: U 1 = π E d 3 12 1 − μ 2 ∫ 0 r d 2 u x d y 2 2 + 1 y 2 d u x d y 2 y d y U 2 = π E d 1 − μ 2 ∫ 0 r d u y d y + 1 2 d u x d y 2 2 + u y 2 y 2 + 2 μ u y y d u y d y + 1 2 d u x d y 2 y d y W 1 = ∫ 0 r ∫ 0 2 π P − F u x y d θ d y W 2 = ∬ P − F u x y d θ d u x (32) where u x and u y represent the axial and radial displacements of the tunnel face in the grouting-reinforced structure, and μ is Poisson’s ratio of the grouting-reinforced structure.

Based on elastic mechanics and the mechanical model in Figure 7, the following expression holds: u x = K 1 − y 2 r 2 2 u y = K 2 r 2 1.206 − 1.785 y r 1 − y r y (33)

By combining Equations 32, 33 and performing term transposition, the potential function of the surrounding rocks is obtained as: ∏ = 0.216 π E d 1 − μ 2 r 2 K 4 + 0.042 F − P π K 3 + 8 π E h 3 9 r 1 − μ 2 K 2 − F − P a 2 π 3 K (34)

Let A = − 3 0.046 1 − μ 2 r 2 F − P E h 2 + 2.058 h 2 B = − 2 0.046 1 − μ 2 r 2 F − P E h 3 − 8.33 r 2 + 2.058 h 2 0.046 1 − μ 2 r 2 F − P E h (35)

Then Equation 34 can be rewritten as: Π x = 1 4 x 4 + 1 2 A x 2 + B x (36)

Equation 36 represents the standard form of cusp catastrophe, which is consistent with Equation 12. When the bifurcation set satisfies Δ > 0, substituting Equation 35 into Equation 20 again yields the elastic modulus E of the surrounding rocks after grouting reinforcement. This defines the grouting reinforcement standard when the underwater karst is positioned in front of the shield tunnel, as expressed in Equation 37. E > 0.2357 r 2 1 − μ 2 F − P d 2 (37)

According to Equation 37, the control standard for grouting reinforcement depends on factors such as the karst distance, the performance of the reinforced surrounding rocks, the support force of the shield machine, the material properties following karst reinforcement, and the karst size.