Analysis of Inflatable Rock Bolts

Abstract

An inflatable bolt is integrated in the rock mass through the friction and mechanical interlock at the bolt–rock interface. The pullout resistance of the inflatable bolt is determined by the contact stress at the interface. The contact stress is composed of two parts, termed the primary and secondary contact stresses. The former refers to the stress established during bolt installation and the latter is mobilized when the bolt tends to slip in the borehole owing to the roughness of the borehole surface. The existing analysis of the inflatable rock bolt does not appropriately describe the interaction between the bolt and the rock since the influence of the folded tongue of the bolt on the stiffness of the bolt and the elastic rebound of the bolt tube in the end of bolt installation are ignored. The interaction of the inflatable bolt with the rock is thoroughly analysed by taking into account the elastic displacements of the rock mass and the bolt tube during and after bolt installation in this article. The study aims to reveal the influence of the bolt tongue on the contact stress and the different anchoring mechanisms of the bolt in hard and soft rocks. A new solution to the primary contact stress is derived, which is more realistic than the existing one in describing the interaction between the bolt and the rock. The mechanism of the secondary contact stress is also discussed from the point of view of the mechanical behaviour of the asperities on the borehole surface. The analytical solutions are in agreement with both the laboratory and field pullout test results. The analysis reveals that the primary contact stress decreases with the Young’s modulus of the rock mass and increases with the borehole diameter and installation pump pressure. The primary contact stress can be easily established in soft and weak rock but is low or zero in hard and strong rock. In soft and weak rock, the primary contact stress is crucially important for the anchorage of the bolt, while in hard and strong rock it is the secondary contact stress that plays a vital role.

Appendix: Analysis of the Folded Steel Tube

A cross section of an inflatable bolt during bolt installation is illustrated in Fig. 20. The folded tongue is simplified to a half ellipse whose half axis lengths are represented by a and b, respectively, in the figure. It is loaded by an external pressure p _i which is caused by the contraction of the borehole and an internal water pressure p _p. Only half of the cross section is taken into account in the analysis below because the cross section is symmetric about the vertical line in the sketch. The half cross section is further decomposed into two free bodies: a circular cantilever beam representing the tube that makes contact with the rock and an elliptic cantilever beam representing the folded tongue (Fig. 20). The circular beam is cantilevered at the lower end and supported by a reaction force F at the upper end. In addition, it is loaded by uniformly distributed stresses p _i and p _p on the external and internal sides, respectively. The elliptic beam is also cantilevered at the lower end and supported at the upper end by a force F having the same value as the one on the circular beam but in the opposite direction. The force F is an internal force within the folded tube.

Fig. 20

Configuration of the cross section of a partly inflated bolt tube: a cross section of the folded tube under inflation; b free-body sketches

The width of the tongue opening, b, is ignored in the calculation of the force F since it is small compared with the radius of the circular beam and has a minor influence on F. A cantilever beam is usually an indeterminate problem, but the circular cantilever beam in question is determinate because of the symmetric loading by p _i and p _p. The horizontal reaction forces are equal at the lower and upper ends of the beam. The force F is thus obtained as follows:

$$F = r_{\text{i}} \left( {p_{\text{p}} - p_{\text{i}} } \right)$$

(24)

where r _i is the external radius of the circular beam. It should be the median radius that is used in the above expression, but the use of the external radius would not cause a large error since the thickness of the beam, t, is small compared with the radius of the beam. Such a treatment makes the subsequent analysis easy.

The displacement of the upper ends of the circular and elliptic beams in the direction of force F will be determined below through the analysis of the elliptic tongue with the energy method. During bolt installation, the tongue is loaded by the internal water pressure and the force F. The borehole diameter is usually close to the original diameter of the bolt tube. When installed in such a borehole, the tongue length, that is, a, is very small. The influence of the water pressure on the tongue would be not profound. For the sake of simplicity, the water pressure on the tongue is ignored in the analysis below. With such a simplification, the tongue is only loaded by force F. An ellipse can be described by its parametric equations as:

$$\begin{aligned} x & = a\cos \theta \\ y & = b\sin \theta \\ \end{aligned} .$$

(25)

The bending moment at point (x, y, θ) on the elliptic tongue is:

$$M = Fx = Fa\cos \theta .$$

(26)

The strain energy stored in the elliptical beam is expressed by:

$$W = \int {\frac{{M^{2} }}{{2E_{\text{s}} I}}{\text{d}}s}$$

where $ds = \sqrt {dx^{2} + dy^{2} } = b\sqrt {\rho^{2} \sin^{2} \theta + \cos^{2} \theta } d\theta$ and ρ is the ratio of the ellipse axes, defined as ρ = a/b. This ratio can be approximately related to the original diameter of the bolt tube, D, and the borehole diameter, d, as follows:

$$\rho^{2} = 2\left( {\frac{D - d}{b} + \frac{1}{\sqrt 2 }} \right)^{2} - 1 .$$

(27)

The displacement δ _F of the upper end of the beam in the direction of F, according to the energy method in the beam theory, is obtained as:

$$\delta_{\text{F}} = \frac{\partial W}{\partial F} = \frac{1}{{E_{\text{s}} I}}\int_{{\frac{\pi }{2}}}^{0} {M\frac{\partial M}{\partial F}{\text{d}}s} = \frac{1}{{E_{\text{s}} I}}\int_{{\frac{\pi }{2}}}^{0} {Fa^{2} \cos^{2} \theta {\text{d}}s}$$

Substituting Eq. (24) and the expression of ds into the above equation and also noticing that the moment of inertia I = t ³/12, where t is the thickness of the beam, we then obtain:

$$\delta_{\text{F}} = \frac{\xi }{{\eta_{1} }}\frac{12}{{E_{\text{s}} }}\frac{{b^{3} }}{{t^{3} }}p_{\text{i}} r_{\text{i}}$$

(28)

where E _s is the Young’s modulus of the bolt steel. The coefficient ξ is a function of the ratio of the internal and external pressures p _p/p _i:

$$\xi = \frac{{p_{\text{p}} }}{{p_{\text{i}} }} - 1 .$$

(29)

The coefficient η ₁ is the integration along the perimeter of the elliptical tongue in the interval [0^o, 90^o]:

$$\frac{1}{{\eta_{1} }} = \rho^{2} \int_{{\frac{\pi }{2}}}^{0} {\cos^{2} \theta \sqrt {\rho^{2} \sin^{2} \theta + \cos^{2} \theta } {\text{d}}\theta } .$$

(30)

This tangential displacement δ _F causes a radial displacement, denoted as u ₁. They are related by:

$$\delta_{\text{F}} = \pi u_{1} .$$

The relationship between the radial displacement u ₁ and the external pressure p _i on the circular cantilever beam is obtained by substituting Eq. (28) into the above expression:

$$u_{1} = \frac{\xi }{{\eta_{1} }}\frac{{p_{\text{i}} r_{\text{i}} }}{{K_{1} }}$$

(31)

where coefficient K ₁ has the form:

$$K_{1} = \frac{{\pi E_{\text{s}} }}{12}\frac{{t^{3} }}{{b^{3} }} .$$

(32)

The tangential displacement δ _F is purely caused by the deflection of the tongue. There exists another displacement component that is due to the tangential deformation of the circular beam under the load F. This second displacement component, denoted as u ₂, can be approximately obtained by treating the partially inflated bolt tube as a circular ring (notice that the tongue is short after installation). The relationship between the displacement u ₂ and the applied stresses can be obtained, from the theory of the thick-wall cylinder (Boresi and Schmidt 2003), as:

$$u_{2} = \frac{{p_{\text{i}} r_{\text{i}} }}{{\eta_{2} K_{2} }}$$

(33)

where

$$K_{2} = \frac{{E_{\text{s}} }}{{1 - \nu_{\text{s}}^{2} }}\frac{t}{{r_{i} }}$$

(34)

$$\frac{1}{{\eta_{2} }} = \frac{1}{{\left( {1 - \nu_{\text{s}} } \right)\left( {2r_{\text{i}} - t} \right)r_{\text{i}} }}\left\{ {2\left( {r_{\text{i}} - t} \right)^{2} \left( {1 - \nu_{\text{s}} } \right)\frac{{p_{\text{p}} }}{{p_{\text{i}} }} - \left[ {\left( {1 - 2\nu_{\text{s}} } \right)r_{\text{i}}^{2} + \left( {r_{\text{i}} - t} \right)^{2} } \right]} \right\}$$

(35)

where ν _s is the Poisson’s ratio of the bolt steel. The second order of the thickness t is small compared with the radius r _i. An approximation to η ₂ is obtained by neglecting the term t ² in the above expression:

$$\frac{1}{{\eta_{2} }} \approx \frac{2}{{2r_{\text{i}} - t}}\left[ {\left( {r_{\text{i}} - 2t} \right)\frac{{p_{\text{p}} }}{{p_{\text{i}} }} - \left( {r_{\text{i}} - \frac{t}{1 - \nu_{\text{s}} }} \right)} \right]$$

(36)

The total elastic radial displacement is the sum of u ₁ and u ₂, that is,

$$u = u_{1} + u_{2} = \frac{{p_{\text{i}} r_{\text{i}} }}{{K_{\text{b}} }}$$

(37)

where K _b is the radial stiffness of the partly inflated bolt tube, which is expressed by

$$K_{\text{b}} = \frac{{\eta_{1} \eta_{2} K_{1} K_{2} }}{{\eta_{1} K_{1} + \xi \eta_{2} K_{2} }} .$$

(38)