As shown in Figure 1A, the umbrella-shaped bolt consists of the top hinge, the rigid rod, the linking rod, the anchoring tray, the bottom sliding block, the main bolt, the middle sliding block, and the middle hinge. Many rigid rods are connected to the main bolt through the top hinge, and they are also connected with the corresponding linking rods with the middle hinge. The linking rod can move with the middle sliding block along the main bolt. In addition, there is also a bottom sliding block adhering to the anchoring tray. The bolt in the packing state (Figure 1B) and opening state (Figure 1C) is also illustrated.

During installation (as illustrated in Figure 2A), a bore hole with a reasonable dimension is drilled at first and the U-bolt is then inserted into the hole of the rock mass. Both the rigid rods and linking rods shrink together, which are close to the main bolt in this process. The bottom sliding block is adhered to the rock mass with glue, and the anchoring tray is fixed on the surface of the internal tunnel free face.

When the rock mass initiates dilation with many excavation-induced fractures (as shown in Figure 2B), the whole section begins to expand (Wang et al., 2020). The main bolt then starts to extend, and the rigid rods initiate expanding. They squeeze the internal face of the bore hole, and friction emerges between the internal face and the rigid rod. This friction and the tensile strength inhibit the dilation of the rock mass. Generally speaking, the tension of the bolt is converted into the extrusion force on the rock mass, and the higher the compressive strength of the rock mass is, the greater the friction force and anchoring force the bolt can obtain.

The mechanical analysis is studied in this section. As shown in Figure 3A, when the U-bolt is totally expanding, we define the distance between the top and middle hinges as width w, and the distance between the top hinge and the end of the rigid rod as distance d. The angle between the main bolt and the linking rod is α , and the length of the linking rod can be calculated as follows: l = w s i n α . (1)

As illustrated in Figure 3B, when the U-bolt is working, the opening angle is defined as Δθ , and the expanding length can be acquired as follows: Δ d = d · s i n Δ θ . (2)

In Figure 3C, we can find that the initial position distance between the top hinge and middle sliding block can be obtained as follows: D = l + w . (3)

The law of cosines (Lee, 1997) will be satisfied (as shown in Figure 3D), and the relation between the opening angle Δθ and the extension length ΔL is shown as follows: cosΔθ = ( D − Δ L ) 2 + d 2 − l 2 2 d ( D − Δ L ) . (4)

Combining Eqs 1–4, we get Δ d = d · 1 − ( D − Δ L ) 2 + d 2 − l 2 2 d ( D − Δ L )   . (5)

In the reasonable range of ΔL , the expanding length Δ d monotonically increases with the extension length ΔL .

When the U-bolt is working (Figure 4A), the additional tensile stress is provided with two parts, including the tensile strength σ t b of the main bolt and the friction f between the bore hole inner face and the rigid rod, which indicates that Δ σ t = σ t b + f . (6)

As illustrated in Figure 4B, the friction can be calculated according to the Coulomb criterion (Renard, 2006), which means that f = μ σ N   , (7) where μ is the friction coefficient and σ N is the normal stress.

The expanding length Δ d in Figures 3B, 4B is also known as the penetration; therefore, the normal force σ N can be defined by a linear relation (1) as follows: σ N = K Δ d   , (8) where K is the contact stiffness, and relation between penetration and the point normal stress will be discussed in the next section.

As pointed out in Eq. 8, in order to study the relation between the penetration and the point normal stress, it is better to establish the relation between the stiffness and the penetration in Eq. 8. Here, a numerical approach is provided, and the analysis zone is considered as an elastic body.

As illustrated in Figure 5, four simulation grids are set, including 0.5 mm (Figure 5A), 1 mm (Figure 5B), 2.5 mm (Figure 5C), and 5 mm (Figure 5D). The size of all these four grids is 0.5 mm × 0.1 mm, and the aspect ratios are all 2 to eliminate the influence of the aspect ratio effect. The bottom edge of these simulation specimens is fixed at the vertical direction, and the middle of the bottom edge is also fixed at the horizontal direction. Only the middle of the top edge is vertical displacement loading representing the point loading.

At least 18 simulation tests were conducted for each grid, and the mechanical properties are listed in Table 1. It should be noted that 45 tests were conducted for 0.05 mm element size, and 18 tests (Nos. 1∼18 in Table 1) were conducted for other element sizes.

A fitting function between the normal stiffness K n against the elastic modulus E and Poisson’s ratio v is established as follows: K n = a E 1 − b v   , (9) where a and b are the fitting coefficients. The fitting coefficients and R square are provided in Table 2. As the simulation is conduction for the elastic body, the R squares for all the element sizes are close to 1, and the fitting function (Figures 6A–D) performs well.

The element size effect is also considered, an exponent function is proposed, and the following relation is obtained: a = 524.3 e − 0.7489 s , (10.1) b = 0.3808 e − 0.0099 s , (10.2) where s is the element size. The fitting results are illustrated in Figures 7A,B. The R square for the coefficient a is 93.96%, and that for b is 97.46%, and they all have relatively good fitting performances.

Combing all these equations, the enhanced tensile strength provided by the U-bolt can be calculated as follows: Δ σ t = σ t b + μ 524.3 e − 0.7489 s E 1 − 0.3808 e − 0.0099 s v d · 1 − ( w + w s i n α − Δ L ) 2 + d 2 − ( w s i n α ) 2 2 d ( w + w s i n α − Δ L ) . (11)

In Eq. 11, there are only 8 parameters that need to be determined, of which E , v , and μ are the mechanical properties of the rock mass, which is known before simulation. σ t b is the tensile strength of the main bolt; d , w , and α are the structural parameters of the U-bolt, and they determine the reinforcement ability of the U-bolt. Only s and Δ L are the variables which need to be determined during the simulation. For the structural parameters d , w , and α , the sensibility analysis has been conducted, as illustrated in Figure 5, in order to study the influence of them. The results indicate that the enhanced tensile strength increases with the increase in d and decreases with the increase in α . With the increase in w , the enhanced tensile strength increases at first and then decreases.

The numerical implementation of the U-bolt follows a simple framework, which is finding the element where bolts go through (they are defined as the bolt elements), then calculating the enhanced tensile strength based on the extension length Δ L , and finally adding the enhanced tensile strength to the current strength of the bolt elements. These processes will be introduced in detail in the next sections.

The bolt elements are identified at the beginning of the simulation. There are three cases that emerge during the identification of bolt elements, as shown in Figure 8, including case A where the bolt does not go through any node of the bolt element, case B where the bolt goes through 1 node of the bolt element, and case C where the bolt goes through 2 nodes of the bolt element.

For cases A and B, it is easy to point out the bolt element through elements where the bolt cut across. For case C, it may be a little different, in which the 2 elements with opposite positions where the bolt goes through are all treated as the bolt elements.

Figure 9 shows two moments of the U-bolt operation in the simulation. For Figure 9A, when the bolt is installed, the initial length of the U-bolt will be calculated using the following equation: L = | [ ( x s t a r t + u x s t a r t ) − ( x e n d + u x e n d ) ] c o s β + [ ( y s t a r t + u y s t a r t ) − ( y e n d + u y e n d ) ] s i n β |   , (12) where x s t a r t ,   y s t a r t ,   x e n d , and y e n d are the horizontal and vertical positions of the two ends of the U-bolt, and u x s t a r t , u y s t a r t ,   u x e n d , and u y e n d are the horizontal and vertical displacements of the two ends of the U-bolt. β is the installation angle which is the angle between the U-bolt and the horizontal line, and it can be obtained as follows: β = atan ( ( y s t a r t + u y s t a r t ) − ( y e n d + u y e n d ) ( x s t a r t + u x s t a r t ) − ( x e n d + u x e n d ) )   . (13)

Using Eqs 12, 13, the current length of the U-bolt L + Δ L can also be calculated, and the extension length Δ L can be acquired easily. It should be pointed out that Δ L is calculated in each iteration and updated all the time.

Once the extension length Δ L is acquired in each simulating iteration, the enhanced tensile strength is calculated using Eq. 11. Then, it is added to the current tensile strength and cohesion of the bolt elements. This process occurs all the time for all the U-bolt elements, which indicates that the tensile strength and cohesion of the U-bolt element vary with the calculation.