Inuence of Poisson Effect of Compression Anchor Grout on Interfacial Shear Stress

: The distribution and magnitude of the shear stress at the interface between the grout of 12 a compression anchor rod and rock are strongly affected by the Poisson effect. To quantitatively 13 analyze the influence of the Poisson effect on the interfacial shear stress of compression anchor 14 rods, the equations for calculating the axial force and interfacial shear stress at the grout cross 15 section in the anchorage section are derived in this paper, accounting for the Poisson effect of the 16 grout. Based on the analytical solution, a new equation of the influence coefficient of the Poisson 17 effect is proposed to quantitatively evaluate the influence of the Poisson effect on the interfacial 18 shear stress. Distributions of the interfacial shear stress and the influence coefficient of the 19 Poisson effect are analyzed with different parameter values. There is a neutral point in the 20 anchorage section near the bearing plate, at which the magnitude of the shear stress is not 21 affected by the Poisson effect. When the Poisson effect is considered, the interfacial shear stress 22 from the neutral point to the bearing plate increases, and the distribution curve becomes steep. 23 However, the interfacial shear stress far from the neutral point is low, and the distribution curve 24 becomes smooth. Overall, the Poisson effect leads to a more nonuniform distribution of the shear 1 stress at the interface of the compression anchor rod. A larger Poisson's ratio, smaller elastic 2 modulus, and smaller diameter of the grout lead to a greater influence of the Poisson effect. 3 Furthermore, a larger elastic modulus of rock leads to a greater influence of the Poisson effect. 4 The Poisson's ratio of rock and that of grout both affect the Poisson effect greatly, but the 5 influence of the variation in the Poisson’s ratio of rock on the Poisson effect is negligible. A 6 larger interface friction angle leads to a greater influence of the Poisson effect. 7


Introduction
Various types of anchors are frequently used in civil engineering, such as retaining walls  21 testing and numerical modeling of the pullout resistance of granular anchor installations in 22 over-consolidated clay for an undrained condition. Merifield (2011) used numerical modeling 23 techniques to analyze multiplate circular anchor foundation behavior in clay soil, and studied the 1 undrained uplift behavior of helical anchors in clays using the centrifuge model test and a 2 "large-deformation, finite-element" approach (Wang et al, 2013). In contrast, many researchers 3 have investigated the performance of tension anchors because of their wide use as foundations to 4 provide uplift or lateral resistance. Su and Fragaszy (1988) conducted comparison tests of 18 5 ground anchors vertically buried in sand to determine the influence of factors such as diameter, 6 fixed anchor length, and buried depth on the uplift capacity of anchors. Serrano and Olalla (1999) 7 obtained the tensile resistance of rock anchors using the Euler's variational method and assuming 8 a rock mass failure criterion of Hoek and Brown type. Zhang et al. (2001) investigated the tensile 9 behavior of fiber-reinforced polymer (FRP) ground anchors. Xiao and Chen (2008) studied the 10 load transfer mechanism of the tension-type anchor and analyzed the mechanical characteristic of 11 an anchorage segment based on elasto-plastic theory. Ivanovic and Neilson (2009) presented a 12 study in which the dynamic modeling of the debonding of the proximal end of the fixed anchor 13 length of an anchorage was considered. Additionally, Liao et al. (1994) and Liu et al (2017) 14 conducted full-scale pullout tests to focus on the behavior of ground anchors under ultimate load 15 conditions. 16 In recent years, for corrosion protection of permanent anchors, compression anchors have 17 been used due to their better corrosion protection, because whole strands are covered by sheaths 18 filled with grease and they have less susceptibility to creep than tension anchors. In addition, in 19 cases where anchors are installed adjacent to existing buildings or planned subway lines, there is 20 an increasing use of compression anchors, which can be removed from the ground after 21 construction to avoid forming underground obstacles. 22 Few in-depth studies have been done on the performance of compression anchors. Kim 23 1 anchors installed in weathered soil for three tension type anchors and four compression type 2 anchors, which were 165 mm in diameter and embedded at a depth of 9 to 12 m. Based on the 3 measurements, a load transfer mechanism for tension and compression ground anchors was pull-out resistance and the group effect of the compression ground anchor by performing 12 pilot-scale laboratory chamber tests and field tests. Although several previous studies of 13 compression anchors have been conducted, the Poisson effect of compression anchor grout has 14 not been properly taken into consideration in the aforementioned studies. However, due to the 15 Poisson effect of the grout of the compression anchor rod, the grout near the bearing plate 16 undergoes radial expansion under the compression of the bearing plate, causing the grout and 17 rock mass to be squeezed at the interface, the normal stress at the interface to increase, and the 18 interfacial shear stress to increase within this range. Therefore, the calculated bearing load of the 19 grout near the bearing plate is larger than that when the Poisson effect is not considered, and the 20 calculated bearing load of the grout far from the bearing plate becomes lower, thereby causing 21 the shear stress at the interface between the grout and the rock mass far from the bearing plate to 22 decrease. 23 In this paper, the formulas of the axial force on the cross-section of the anchorage body and 1 the interfacial shear stress considering the Poisson effect are first derived. Next, a new equation    17 Taking the location of the bearing plate of the compression anchor as the origin of the 18 coordinates, a one-dimensional rectangular coordinate system is established along the direction 19 of the anchor head, as shown in Figure 1. 20 Since the grout is not an ideal rigid body, the grout will undergo radial expansion within a 21 certain range due to the Poisson effect when its bottom end is squeezed by the bearing plate. 22 Hence, a radial stress σr will be generated at the interface between the grout and the rock, 23 improving the interfacial bond strength to some extent. A micro-element from the grout of the 1 anchorage body is shown in Figure 1, and a corresponding stress analysis diagram is established, 2 as shown in Figure 2. 3 Based on static equilibrium, the following equation is satisfied:

Theoretical Solution
where τ1(x) is the shear stress at the interface between the grout and the rock (kPa), P1(x) is the 6 axial force on the cross-section of the grout (kN), and D is the diameter of the grout (m).

7
For a tension anchor, the relation between the shear stress at the grout-rock interface and 8 the composite shear stiffness of the interface can be established using the shear force intensity as 9 follows: where q(x) is the shear force per unit length of the anchorage body of the tension anchor (kN/m), 12 w(x) is the interfacial shear displacement of the anchorage body of the tension anchor rod (m), 13 and Ks is the composite tangent stiffness of the interface between the grout body and the rock 14 (kPa). The physical meaning of Ks is the shear force per unit length required at the interface to 15 produce a unit shear displacement on the corresponding interface, which can be calculated using 16 the shear stiffnesses of the grout and the rock proposed by Oda et al. (1997): where Kb is the shear stiffness of the grout, and Kr is the shear stiffness of the rock. 19 As Chou and Pagano (1992) proposed, the shear stiffness Kb of the grout can be obtained by 20 considering the equation for a thick-walled cylinder from the theory of elasticity:

22
where Gg is the shear modulus of the grout, which is defined herein as , Eg is the 1 elastic modulus of the grout, μg is the Poisson's ratio of the grout, and d is the diameter of the 2 strand. 3 For the compression anchor, due to the Poisson effect, the shear force intensity of the grout 4 can be decomposed into two parts: that caused by the interfacial shear displacement and that 5 caused by the interfacial radial stress. The shear force intensity can be expressed as follows: where q1(x) is the shear force per unit length of the grout of the compression anchor (kN/m), w1(x) 8 is the interfacial shear displacement of the grout at the coordinate x of the anchorage body of the 9 compression anchor (m), σr(x) is the radial stress at the interface between the grout and the rock 10 (kPa), and δ is the interface friction angle (°) between the grout and the rock.

11
For the compression anchor, the shear displacement w 1 (x) of the grout interface at the 12 coordinate x of the anchorage body in Figure 1 can be expressed by Hooke's law (Chou and 13 Pagono, 1992) as follows: where A g is the net bearing area of the grout, , and l a is the total length of the 16 anchorage body. 17 Taking the derivative of Equation (6) with respect to x yields the following: Solving Equations (1), (5), and (7) simultaneously yields the following:

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According to the physical equation in cylindrical coordinates of the space problem from 1 elasticity theory (Chou and Pagono, 1992), we have the following: According to the third assumption, φ r      , which is substituted into Equation (9), 4 resulting in the following: where ερ is the radial strain of the grout, and σx is the normal stress on the cross-section of the 7 grout. 8 According to the theory of elasticity, when a circular hole with a radius R on an infinite 9 plane is subjected to a uniform internal pressure σr, the radial displacement of the hole wall is as 10 follows: At the interface between the grout and the rock, i.e., at ρ = R, Equations (10) and (11) yield 13 the following: where μr is the Poisson's ratio of the rock, and Er is the elastic modulus of the rock. 16 Equation (12) is integrated and rearranged, yielding the following: (1 ) Substituting Equation (7) into Equation (13) yields Substituting Equation (14) into Equation (8) yields the following: The characteristic equation for Equation (15) is as follows: The discriminant of Equation (16) is If the discriminant given by Equation (17) is greater than zero, Equation (16) has two 8 unequal real roots, r1 and r2, expressed as follows: Thus, the general solution of Equation (15) is

17
Substituting Equations (20), (21), and (22) into Equation (7), we obtain the axial force 18 acting on the grout considering the  6 Assuming that the grout of the compression anchor only undergoes axial compression 7 without radial deformation, the Poisson effect will not occur when the grout is compressed. In 8 this case, it can be assumed that the Poisson's ratio of the grout is μ g = 0. Consequently, k = 0, 9 and the radial stress r 0   is obtained from Equation (13). Thus, the Poisson effect of the grout 10 of the compression anchor can be neglected. Similarly, substituting the simplified r1 and r2 corresponding to μg = 0 into Equation (24) 17 gives the interfacial shear stress of the anchorage body of the compression anchor when the 18 Poisson effect of the grout is neglected:

Evaluation of influence of Poisson effect
To quantitatively analyze and evaluate the influence of the Poisson effect of the grout of the 2 compression anchor, the influence coefficient λ of the Poisson effect is defined as the ratio of the 3 interfacial shear stress when the Poisson effect of the grout is considered to that when the 4 Poisson effect is neglected, i.e., , which is used to analyze and evaluate the  respectively, as shown in Figure 3. The distribution curve of the corresponding influence 19 coefficient of the Poisson effect along the grout is also shown in Figure 3.    Table 1.   The elastic modulus of the grout Eg was set to 5, 10, and 15 GPa, and the distribution curves  13 Figure 6a shows that when the Poisson effect is considered, the interfacial shear stress 14 increases first and then decreases as the distance from the bearing plate increases, and the overall 15 distribution is more uneven than that of the case without the Poisson effect. A larger diameter of 16 the grout corresponds to a more gradually varying distribution curve of the interfacial shear 17 stress and a lower peak shear stress at the interface of the bearing plate. This occurs because a 18 larger diameter corresponds to a larger area of the interface between the grout and the rock and a 19 smaller load per unit interface area. Thus, the interfacial shear stress concentration near the 20 bearing plate is greatly reduced. As a result, the peak shear stress is significantly reduced, the 21 distribution curve of the shear stress along the length of the anchorage body is more uniform, and 22 the curve varies more gradually. 23 When the Poisson effect is neglected and D is taken as 0.10, 0.15, and 0.20 m, the 1 interfacial shear stresses at the bearing plate are 2.498, 1.079, and 0.607 MPa, respectively.

2
When the Poisson effect is considered, the interfacial shear stresses at the bearing plate increase   19 Figure 8a shows that, when the Poisson effect is considered, the interfacial shear stress 20 increases first and then decreases with increasing distance from the bearing plate, and the 21 distribution curve is more uneven than that with the Poisson effect neglected. At the bearing plate, 22 the increase of the interfacial shear stress caused by the Poisson effect improves with increasing 23 Er. This occurs because, under the same lateral expansion, a greater elastic modulus of the rock 1 corresponds to a stronger ability to restrain the lateral expansion deformation of the grout, 2 leading to a higher interfacial radial stress and hence a higher shear stress. In comparison, 3 beyond a certain distance from the bearing plate, the interfacial shear stress decreases with 4 increasing Er. 5 When the Poisson effect is neglected and Er is 1, 5, and 10 GPa, the interfacial shear   Figure 9a shows that when the Poisson effect is considered, the interfacial shear stress 8 increases first and then decreases as the distance from the bearing plate increases, and the overall 9 distribution is more uneven than that when the Poisson effect is neglected. Within the range of 10 the neutral point depth, the larger the interface friction angle is, the higher the interfacial shear 11 stress becomes, i.e., the greater the extent of the increase in the interfacial shear stress caused by 12 the Poisson effect is. Beyond the depth of the neutral point, the larger the interface friction angle 13 is, the lower the interfacial shear stress is, and the larger the extent of the decrease in the 14 interfacial shear stress caused by the Poisson effect becomes. 15 When the Poisson effect is neglected, the interfacial shear stress at the bearing plate is 1. effect of the grout reduces the length of the anchorage body that mainly bears the load, and the 2 extent of reduction improves as the interface friction angle increases.
3 Figure 9b shows that within the range of the neutral point depth, the influence coefficient of 4 Poisson effect λ increases with increasing δ. Beyond this range, λ decreases with increasing δ.  can be summarized as follows.

17
(1) The Poisson effect of compression anchor grout results in increased interfacial shear 18 stress between the grout and rock within the depth of the neutral point, and in reduced interfacial 19 shear stress far from the neutral point. Distribution of the interfacial shear stress becomes more 20 uneven, and the length of the anchorage body mainly bearing the load decreases significantly, in 21 contrast to the case without the Poisson effect.

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(2) A larger Poisson's ratio, smaller elastic modulus, and smaller diameter of the grout lead         In uence of interface friction angle: (a) Distribution of shear stress; (b) Distribution of in uence coe cient of Poisson effect.