A design method for the tunnel prestressed anchorage layer considering the effect of advanced support

Abstract

The present invention discloses a design method for a tunnel prestressed anchorage layer considering the action effect of advanced support. First, a load-structure calculation model for the prestressed anchorage layer is constructed. Secondly, the Mohr-Coulomb strength theory is used to derive a calculation method for the compressive strength of the prestressed anchorage layer under the ultimate equilibrium state, and a modified calculation method for the loosening pressure of the surrounding rock considering the advanced support load reduction mechanism is proposed. Finally, a support design method for the tunnel prestressed anchorage layer based on the loosening pressure of the surrounding rock is formed, providing a calculation basis for the precise design of tunnel shotcrete support. The technical solution includes the following steps: S1. Construct a load-structure calculation model for the prestressed anchorage layer; S2. Derive a calculation method for the compressive strength of the prestressed anchorage layer under the ultimate equilibrium state; S3. Propose a modified calculation method for the loosening pressure of the surrounding rock considering the advanced support load reduction mechanism; S4. Based on the modified load calculation method, use the load-structure model to calculate the design parameters of the prestressed anchorage layer.

Description

Technical Field

[0001] The invention belongs to the technical field of tunnel engineering, and particularly relates to a design method for a tunnel prestressed anchorage layer considering the action effect of advanced support. Background Technique

[0002] The statements in this part only provide background technical information related to the present invention and do not necessarily constitute prior art.

[0003] In existing research, there has been a preliminary study on the support design method for the prestressed anchorage layer. For example, CN110210069B discloses a design method for a tunnel advanced support system and a tunnel advanced support design method, aiming to improve the stability of the heading face and realize the mechanized full-face construction of tunnels under soft surrounding rock conditions. However, the influence on the prestressed anchorage layer is not considered in the calculation model during the design process of the tunnel advanced support system.

[0004] In the existing support design method for the anchorage layer, the influence of prestress on the mechanical properties of the anchorage layer is ignored, and the problem of the load borne by the anchorage layer is not combined with the loosening pressure in the existing "Code for Design of Railway Tunnels" (TB 10003 - 2016). The calculation method of this loosening pressure combines the advantages of the whole soil column theory, Xie Jiaxiu's formula, and the empirical formula based on sample statistics and surrounding rock classification. However, the influence of advanced support on the load borne by the anchorage layer is not considered either. Summary of the Invention

[0005] In view of the above problems, the present invention provides a design method for a tunnel prestressed anchorage layer considering the action effect of advanced support. First, a load - structure calculation model for the prestressed anchorage layer is constructed. Secondly, the Mohr - Coulomb strength theory is used to derive the calculation method for the compressive strength of the prestressed anchorage layer under the limit equilibrium state, and a modified calculation method for the loosening pressure of the surrounding rock considering the load reduction mechanism of advanced support is proposed. Finally, a support design method for the tunnel prestressed anchorage layer based on the loosening pressure of the surrounding rock is formed, providing a calculation basis for the precise design of tunnel shotcrete support.

[0006] In order to achieve the above object, the present invention is realized through the following technical solutions:

[0007] The present invention provides a design method for a tunnel prestressed anchorage layer considering the action effect of advanced support. The anchor rod in the advanced support is inserted into the rock formation, and the prestressed anchorage layer is formed after grouting at the end of the anchor rod. The design method includes the following steps:

[0008] S1. Construct a load - structure calculation model for the prestressed anchorage layer;

[0009] S2. Use the Mohr - Coulomb strength theory to derive the calculation method for the compressive strength of the prestressed anchorage layer under the limit equilibrium state;

[0010] S3. Propose a modified calculation method for the loosening pressure of surrounding rock considering the load reduction mechanism of the advanced support;

[0011] S4. Based on the modified calculation method for the loosening pressure of surrounding rock, use the load - structure calculation model to calculate the design parameters of the prestressed anchorage layer.

[0012] Furthermore, in the process of establishing the load - structure calculation model of the prestressed anchorage layer in S1, the prestressed anchorage layer structure body is simulated by beam elements, the interaction between the prestressed anchorage layer and the stratum is simulated by radial tensionless springs, and fixed - end constraints are used for fixation at the wall foot.

[0013] Furthermore, in the process of constructing the load - structure calculation model of the prestressed anchorage layer, the length of the prestressed anchor rod, the width of the backing plate, and the circumferential spacing of the anchor rods need to be considered in the calculation process of the thickness of the prestressed anchorage layer; the calculation formula for the thickness of the prestressed anchorage layer is as follows:

[0014]

[0015] Where, L is the thickness of the prestressed anchorage layer; R is the tunnel radius; S Tb is the circumferential spacing of the anchor rods; L fb is the free - segment length of the prestressed anchor rod; b is the width of the anchor rod gasket; α is the pressure diffusion angle.

[0016] Furthermore, the equivalent elastic modulus of the surrounding rock after anchor reinforcement is affected by both the elastic modulus of the anchor rod and the elastic modulus of the surrounding rock. According to the area - equivalence principle, the expression for the equivalent elastic modulus E * is:

[0017]

[0018] Where, E b is the elastic modulus of the anchor rod; E is the elastic modulus of the surrounding rock; S Lb is the axial spacing of the anchor rods; d is the diameter of the anchor rod.

[0019] Furthermore, the expressions for the equivalent cohesive force c * and the equivalent internal friction angle are:

[0020]

[0021] Where, λ is the friction resistance coefficient between the anchor rod and the surrounding rock; in the deformed steel bar anchor rod support system c is the original cohesive force of the surrounding rock; is the internal friction angle.

[0022] Furthermore, the calculation formula for the safety factor is as follows:

[0023] k b N b ≤μασ b bL

[0024] Among them, k b is the safety factor of the prestressed anchorage layer; N b is the axial force of the section to be checked; μ is the longitudinal bending coefficient of the section; α is the influence coefficient of axial force eccentricity; b is the calculated width of the bearing arch; L is the calculated height of the prestressed anchorage layer; σ b is the ultimate compressive strength of the prestressed anchorage layer.

[0025] Furthermore, according to the Mohr-Coulomb strength theory in S2, the original compressive strength expression of the surrounding rock is:

[0026]

[0027] Among them, σ3 is the minimum principal stress after the stress of the surrounding rock is readjusted before the installation of the bolt after the tunnel excavation.

[0028] Furthermore, the calculation formula for the compressive strength of the prestressed anchorage layer after the installation of the prestressed bolt is;

[0029]

[0030] Among them, c * is the equivalent cohesive force of the surrounding rock; is the equivalent internal friction angle; F a is the pre-tightening force applied by the bolt.

[0031] Furthermore, if the tunnel is pre-supported by advanced pipe-roof, the calculation formula for the surrounding rock pressure should be corrected, and the corrected calculation formula for the vertical uniform surrounding rock pressure is:

[0032] q l =ξγh a

[0033] h a =0.45×2 s-1 ×w

[0034] Among them, q l is the vertical uniform surrounding rock pressure; γ is the unit weight of the surrounding rock; h a is the calculated height of the surrounding rock; w is the width influence coefficient; B is the tunnel width; ξ is the reduction coefficient of the surrounding rock pressure.

[0035] Furthermore, the corrected calculation formula for the horizontal surrounding rock pressure is:

[0036] e=λq l

[0037] Among them, e is the horizontal pressure; λ is the ratio of the lateral pressure to the vertical pressure.

[0038] Compared with the prior art, the advantages and positive effects of the present invention are as follows:

[0039] The present invention constructs a load - structure calculation model for the prestressed anchorage layer. Secondly, by using the Mohr - Coulomb strength theory, the calculation method for the compressive strength of the prestressed anchorage layer under the ultimate equilibrium state is deduced, and a modified calculation method for the loosening pressure of the surrounding rock considering the load reduction mechanism of the advanced support is proposed. Finally, a design method for the tunnel prestressed anchorage layer considering the action effect of the advanced support is formed, providing a calculation basis for the precise design of tunnel shotcrete support. It solves the problem that the influence of prestress on the mechanical properties of the prestressed anchorage layer is not considered in the existing design methods of the prestressed anchorage layer, uses a relatively reliable loosening pressure of the surrounding rock as the calculation load for calculation, overcomes the defect that prestress is not involved in the design of the tunnel engineering support structure, and enriches the relevant theories of the refined design of the tunnel engineering support structure. BRIEF DESCRIPTION OF THE DRAWINGS

[0040] The attached drawings forming a part of this specification are used to provide a further understanding of the present invention. The schematic embodiments and descriptions thereof of the present invention are used to explain the present invention and do not constitute an improper limitation to the present invention.

[0041] Figure 1 It is a flowchart of the design method for the tunnel prestressed anchorage layer of the present invention;

[0042] Figure 2 It is a diagram of the load - structure calculation model for the prestressed anchorage layer of the present invention;

[0043] Figure 3 It is an analysis diagram of the reinforcement effect of the prestressed anchor bolt of the present invention;

[0044] Figure 4 It is a diagram showing the change of the compressive strength improvement ratio of the prestressed anchorage layer under different bolt spacings of the present invention;

[0045] Figure 5 It is a diagram showing the change of the compressive strength improvement ratio of the prestressed anchorage layer under different bolt parameters of the present invention. DETAILED DESCRIPTION OF THE INVENTION

[0046] It should be noted that the following detailed description is illustrative and is intended to provide further explanation of the present invention. Unless otherwise specified, all technical and scientific terms used in the present invention have the same meaning as commonly understood by those of ordinary skill in the technical field to which the present invention belongs.

[0047] It should be noted that the terms used herein are only for describing specific embodiments and are not intended to limit the exemplary embodiments according to the present invention. As used herein, unless the present invention clearly indicates otherwise, the singular form is also intended to include the plural form. In addition, it should also be understood that when the terms "comprising" and / or "including" are used in this specification, they indicate the presence of features, steps, operations, devices, components, and / or combinations thereof;

[0048] A conveyor belt device that is easy to maintain disclosed in this embodiment, as Figure 1 - Figure 2 shown, the design method of the tunnel prestressed anchorage layer considering the effect of advanced support mainly includes four steps:

[0049] S1. Construct a load - structure calculation model for the prestressed anchorage layer; according to the bearing characteristics of the prestressed anchorage layer, a corresponding structural calculation model is proposed, that is, the load - structure calculation model of the prestressed anchorage layer. After the prestressed anchor bolts are constructed, a support structure layer with a certain bearing capacity can be actively formed around the tunnel, avoiding the possibility of collapse due to the loosening of the surrounding rock within a certain range around the tunnel after excavation. For this reason, a load - structure calculation model of the prestressed anchorage layer is established, as Figure 2 shown. The structure of the prestressed anchorage layer is simulated by beam elements, the interaction between the prestressed anchorage layer and the stratum is simulated by radial tensionless springs, and fixed - end constraints are used for fixation at the wall foot; among them, the initial shotcrete is not reflected in the load - structure model of the prestressed anchorage layer for two reasons: First, the thickness of the initial shotcrete is relatively small compared to the thickness of the prestressed anchorage layer. Considering safety, its contribution to the thickness of the prestressed anchorage layer can be ignored during calculation; Second, it is considered that after the prestressed anchor bolt support, a complete prestressed anchorage layer structure can be formed under the protection of the initial shotcrete, that is, the effect of the initial shotcrete on the structural performance of the prestressed anchorage layer has been defaultly considered.

[0050] During the calculation of the thickness of the prestressed anchorage layer, the length of the prestressed anchor bolt, the width of the bearing plate, and the circumferential spacing of the anchor bolts need to be considered; the thickness calculation formula of the prestressed anchorage layer is as follows:

[0051]

[0052] Among them, L is the thickness of the prestressed anchorage layer, m; R is the tunnel radius, m; S Tb is the circumferential spacing of the anchor bolts, m; L fb is the free - segment length of the prestressed anchor bolt, m; b is the width of the anchor bolt gasket, m; α is the pressure diffusion angle (usually taken θ = 45° for calculation).

[0053] The equivalent elastic modulus of the surrounding rock after anchor reinforcement is affected by both the elastic modulus of the anchor bolt and the elastic modulus of the surrounding rock. According to the area equivalence principle, the equivalent elastic modulus E *The expression is:

[0054]

[0055] In the formula, E b is the elastic modulus of the bolt (MPa); E is the elastic modulus of the surrounding rock (MPa); S Lb is the axial spacing of the bolts (m); d is the diameter of the bolt, m.

[0056] The calculation formula for the compressive strength of the prestressed anchorage layer in the model is;

[0057]

[0058] In the formula, c * 、 are the equivalent cohesive force (MPa) and equivalent internal friction angle (°) of the surrounding rock; F a is the pre-tightening force applied by the bolt (kN).

[0059] The equivalent cohesive force c * and equivalent internal friction angle of the surrounding rock under bolt support are expressed as:

[0060]

[0061] In the formula, λ is the friction resistance coefficient between the bolt and the surrounding rock. In the deformed steel bolt support system c、 are the original cohesive force (MPa) and internal friction angle (°) of the surrounding rock.

[0062] The radial spring stiffness parameter between the prestressed anchorage layer and the surrounding rock can be selected according to the elastic reaction coefficient specified in the "Code for Design of Railway Tunnels" (TB10003 - 2016).

[0063] Safety evaluation method: The safety factor of the prestressed anchorage layer can be calculated according to the failure stage method in the current "Code for Design of Railway Tunnels" (TB10003 - 2016). Since the thickness of the prestressed anchorage layer is generally large and its stress state is generally small eccentricity, the following formula can be used for calculation according to the code.

[0064] k b N b ≤μασ b bL

[0065] Among them, k b is the safety factor of the prestressed anchorage layer, which can be evaluated according to the strength safety factor judgment criterion of the plain concrete structure in the "Code for Design of Railway Tunnels" (TB10003 - 2016); N bTo check the axial force of the cross-section, N; μ is the longitudinal bending coefficient of the cross-section; α is the influence coefficient of axial force eccentricity; b is the calculated width of the bearing arch; L is the calculated height of the prestressed anchorage layer, m; σ b is the ultimate compressive strength of the prestressed anchorage layer, MPa, and is calculated according to the calculation formula of the compressive strength of the prestressed anchorage layer in the model.

[0066] S2. Adopt the Mohr-Coulomb strength theory to derive the calculation method of the compressive strength of the prestressed anchorage layer under the ultimate equilibrium state; since in the load-structure calculation model of the prestressed anchorage layer, the compressive strength of the prestressed anchorage layer has a greater influence on the support performance and stability of the prestressed anchorage layer structure, therefore, the present invention conducts an analysis of the calculation method of the compressive strength of the tunnel prestressed anchorage layer.

[0067] After the surrounding rock is supported by prestressed anchor bolts, on the one hand, the anchor bolts and the surrounding rock in the anchorage area form a prestressed anchorage layer, and on the other hand, the pre-tightening force of the anchor bolts causes the surrounding rock in the anchorage area to be subjected to confining pressure. According to the mechanical properties of rocks such as Figure 3 shown, the mechanical properties of the surrounding rock in the anchorage area are improved by the action effects in both aspects.

[0068] Therefore, according to the Mohr-Coulomb strength theory, the original compressive strength expression of the surrounding rock is:

[0069]

[0070] Among them, σ3 is the minimum principal stress after the stress of the surrounding rock is readjusted before the installation of the anchor bolts after the tunnel excavation, MPa.

[0071] After the prestressed anchor bolts are installed, the original compressive strength of the surrounding rock is effectively improved, mainly manifested in two aspects: providing confining pressure and forming an anchor-rock complex. Therefore, the calculation formula for the compressive strength of the prestressed anchorage layer after the installation of the prestressed anchor bolts is:

[0072]

[0073] It should be noted that the original compressive strength of the surrounding rock mentioned in the present invention refers to the compressive strength value in the state of the surrounding rock before support after the tunnel excavation. Since the state of the surrounding rock at this time is highly correlated with the properties of the surrounding rock itself, construction techniques (such as blasting), the stability state of the tunnel face, and the installation of advanced pre-support, therefore, considering conservatively, the mechanical parameters of the original rock in the prestressed anchorage layer are calculated using their residual strength.

[0074] According to the calculation formula for the compressive strength of the prestressed anchorage layer after the installation of the prestressed anchor bolts, the compressive strength of the prestressed anchorage layer is related to the design parameters of the prestressed anchor bolts, such as the pre-tightening force of the anchor bolts, the bolt spacing, length, and diameter. Therefore, five gradients are selected for each variable to conduct an analysis, and the specific parameters are as follows:

[0075] Among them, the parameters of the prestressed anchor bolts are as follows:

[0076] SLb: 0.8, 1.0, 1.2, 1.4, 1.6 (m);

[0077] STb: 0.8, 1.0, 1.2, 1.4, 1.6 (m);

[0078] Lfb: 3.5 (m);

[0079] Fa: 30, 50, 70, 90, 110 (kN);

[0080] d: 0.022 (m);

[0081] The parameter c of the surrounding rock: 0.05 (MPa);

[0082] φ: 20 (°);

[0083] E: 1.0 (GPa);

[0084] Tunnel size R: 7.5 (m);

[0085] As Figure 3 shown, it is a diagram of the change in the compressive strength improvement rate of the prestressed anchorage layer under different bolt spacings. From the analysis of the reinforcement effect of the prestressed anchor bolts, the compressive strength of the prestressed anchorage layer can be divided into three categories, namely, the compressive strength σ b1 of the anchor-rock complex, the compressive strength σ b2 of the confining pressure type, and the compressive strength σ b of the anchor-rock complex + confining pressure type.

[0086] It can be seen from Figure 3 that the improvement rates of the three types of compressive strength of the prestressed anchorage layer all decrease with the increase of the bolt spacing, but generally show that σb > σb2 > σb1. For example, when the bolt spacing is 0.8 m, 1.0 m, 1.2 m, 1.4 m, 1.6 m, σ b1 are 29%, 19%, 13%, 10%, 7% respectively, σ b2 are 101%, 65%, 46%, 34%, 26% respectively, and σ b are 160%, 96%, 65%, 47%, 36% respectively. Among them, the compressive strength σ b1 of the anchor-rock complex can be approximately regarded as the compressive strength of the prestressed anchorage layer after traditional mortar bolt support. From the calculation results, it can be seen that its improvement effect on the compressive strength of the original rock is small (7% - 29%). After adopting prestressed anchor bolt support, the formed prestressed anchorage layer has a large increase in compressive strength (36% - 160%) due to the effect of the confining pressure Δσ3, and σ b is greater than σ b2 and σb1 The sum is such that the stability improvement effect of the prestressed anchorage layer is obvious, and it can effectively and stably control the stability of the surrounding rock.

[0087] In addition, as the pre-tightening force of the prestressed anchor bolt increases, the compressive strength of the prestressed anchorage layer gradually increases. For example, when the anchor bolt prestress is 30 kN, 50 kN, 70 kN, 90 kN, and 110 kN, the compressive strength improvement rates of the prestressed anchorage layer are 44%, 65%, 86%, 106%, and 127% respectively, as Figure 4 shown. In summary, when using prestressed anchor bolts to support the tunnel, the stability of the tunnel surrounding rock can be improved by forming a high-strength prestressed anchorage layer.

[0088] S3. Propose a modified calculation method for the loosening pressure of the surrounding rock considering the load reduction mechanism of the advanced support; the modified calculation method for the loosening pressure of the tunnel surrounding rock; the surrounding rock pressure is determined according to the collapse height of the surrounding rock. Therefore, for the determination of the surrounding rock pressure in the load-structure calculation model of the prestressed anchorage layer, it is approximately considered as the calculated value of the surrounding rock pressure of the deep-buried tunnel. In addition, based on the existing analysis of the reinforcement mechanism of the advanced pipe shed, when using the advanced pipe shed for pre-support, the overlying load of the surrounding rock can be reduced. Therefore, if the tunnel uses the advanced pipe shed for pre-support, the calculation formula for the surrounding rock pressure should be modified, and the modified calculation expression for the surrounding rock pressure is obtained as:

[0089] q l = ξγh a

[0090] h a = 0.45×2 s-1 ×w

[0091] e = λq l

[0092] In the formula, q l is the vertical uniform pressure of the surrounding rock (kPa); e is the horizontal pressure (kPa); γ is the unit weight of the surrounding rock (kN / m 3 ); h a is the calculated height of the surrounding rock (m); λ is the ratio of the lateral pressure to the vertical pressure; w is the width influence coefficient, w = 1 + i(B - 5), i is the increase or decrease rate of the surrounding rock pressure when B increases or decreases by 1 m. When B < 5 m, i = 0.2; when B > 5 m, i can be taken as 0.1; B is the tunnel width.

[0093] Among them, ξ is the surrounding rock pressure reduction coefficient, 0 < ξ < 1, which can be calculated by referring to the existing advanced support calculation method and selecting the final reduction coefficient. When the advanced pipe shed is not used for pre-support, ξ = 1.

[0094] S4. Based on the modified load calculation method, the load-structure model is adopted to calculate the relevant design parameters of the prestressed anchorage layer.

[0095] Although the specific implementation manners of the present invention have been described above in conjunction with the accompanying drawings, it is not a limitation on the protection scope of the present invention. Those skilled in the art should understand that various modifications or deformations that can be made without creative efforts on the basis of the technical solution of the present invention are still within the protection scope of the present invention.

Claims

1. A design method for a tunnel prestressed anchor layer taking into account the effect of advance support, wherein the anchor rods in the advance support are inserted into the rock layer, and the ends of the anchor rods are grouting to form a prestressed anchor layer, characterized in that: The design methodology includes the following steps: S1. Construct the load-structure calculation model of the prestressed anchor layer; S2. Using Mohr-Coulomb strength theory, a method for calculating the compressive strength of the prestressed anchor layer under the limit equilibrium state is derived; S3. Propose a calculation method for the correction of the surrounding rock loosening pressure taking into account the load reduction mechanism of the advanced support; S4, based on the surrounding rock loosening pressure correction calculation method, using the load-structure calculation model to calculate the design parameters of the prestressed anchoring layer; In the process of establishing the load-structure calculation model of the prestressed anchor layer in S1, the prestressed anchor layer structure is simulated by beam elements, the interaction between the prestressed anchor layer and the stratum is simulated by radial tension-free springs, and the foot of the wall is fixed by fixed end constraints; The thickness of the prestressed anchor layer needs to be calculated by considering the length of the prestressed anchor rod, the width of the pad and the annular spacing of the anchor rod; the thickness calculation formula of the prestressed anchor layer is as follows: Where L is the thickness of the prestressed anchor layer; R is the tunnel radius; S Tb L is the circumferential spacing of anchor rods; fb is the free section length of the prestressed anchor rod; b is the width of the anchor rod gasket; α is the pressure diffusion angle; The equivalent elastic modulus of the surrounding rock after anchor reinforcement is affected by the elastic modulus of the anchor and the elastic modulus of the surrounding rock. According to the area equivalence principle, the equivalent elastic modulus E is obtained. * The expression is: Among them, E b is the elastic modulus of the anchor rod; E is the elastic modulus of the surrounding rock; S Lb is the axial spacing of the anchor rods; d is the diameter of the anchor rods; Equivalent cohesion of surrounding rock c * and the equivalent internal friction angle The expression is: Among them, λ is the friction coefficient between the anchor and the surrounding rock; in the threaded steel anchor support system, c is the original cohesion of surrounding rock; is the internal friction angle; The ultimate compressive strength safety factor calculation formula of the prestressed anchor layer is as follows: k b N b ≤μασ b bL Among them, k b is the safety factor of the prestressed anchor layer; N b is the axial force of the cross section; μ is the longitudinal bending coefficient of the cross section; α is the eccentricity influence coefficient of the axial force; b is the calculated width of the bearing arch; L is the calculated height of the prestressed anchor layer; σ b is the ultimate compressive strength of the prestressed anchor layer; In S2, according to the Mohr-Coulomb strength theory, the original compressive strength of the surrounding rock is expressed as: Among them, σ3 is the minimum principal stress after the surrounding rock stress is readjusted before the anchor bolt is applied after tunnel excavation; The calculation formula of the compressive strength of the prestressed anchoring layer after the prestressed anchor rod is applied is: Among them, c * is the equivalent cohesion of surrounding rock; is the equivalent internal friction angle; F a The preload applied to the anchor bolt; If the tunnel is pre-supported with an advanced pipe shed, the surrounding rock pressure calculation formula should be corrected, and the corrected surrounding rock vertical uniform pressure calculation formula is: q l =ξγh a h a =0.45×2 s-1 ×w Among them, q l is the vertical uniform pressure of the surrounding rock; γ is the weight of the surrounding rock; h a is the height of the surrounding rock; w is the influence coefficient of width, w=1+i(B-5), i is the rate of increase or decrease of the surrounding rock pressure when B increases or decreases by 1m; B is the width of the tunnel; ξ is the surrounding rock pressure reduction coefficient; The corrected calculation formula for the surrounding rock horizontal pressure is: e=λq l Where, e is the horizontal pressure; λ is the ratio of the lateral pressure to the vertical pressure.

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