CN111428336A - Method for calculating full-length bonding type anchor rod ultimate pullout resistance of elastic-plastic model based on Mocoul yield condition - Google Patents
Method for calculating full-length bonding type anchor rod ultimate pullout resistance of elastic-plastic model based on Mocoul yield condition Download PDFInfo
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Abstract
The embodiment of the invention provides a method for calculating the ultimate pullout resistance of a full-length bonded anchor rod based on an elastoplastic model under a Moire coulomb yield condition, which belongs to the technical field of underground engineering and comprises the following steps: analyzing the anchor rod stress micro unit to obtain the relation between the displacement of the anchoring body and the shear stress on the anchoring body interface; according to the relation between the displacement of the anchoring body and the shear stress on the anchoring body interface, simultaneously considering the elastoplasticity effect, and establishing an elastoplasticity model based on the Moore coulomb yield condition; and calculating the pulling resistance of the full-length bonding type anchor rod according to the elastic-plastic model. The method can accurately estimate the ultimate pullout resistance of the full-length bonding anchor rod according to the technical data of the full-length bonding anchor rod such as the geometric dimension and material performance, the performance of grouting materials, the properties of anchoring bodies and the like, and has important significance for the design and construction of the full-length bonding anchor rod and the safety evaluation of side slopes.
Description
Technical Field
The embodiment of the invention belongs to the technical field of underground engineering, and particularly relates to a method for calculating the limit pullout resistance of a full-length bonded anchor rod based on an elastoplastic model of a Mocouulomb yield condition and an elastoplastic model of the Mocoulobmb yield condition.
Background
Geotechnical anchoring is an important branch in the field of geotechnical engineering, has been widely applied to engineering construction of side slopes, foundation pits, mines, tunnels, underground engineering, dam bodies, navigation channels, reservoirs, airports, anti-inclination structures, anti-floating structures and the like, and has a plurality of advantages. The rock-soil anchoring technology is that a structure and a rock-soil body are tightly interlocked together through an anchor rod embedded in the rock-soil body, and the tension of the structure is transmitted or the unstable part of the rock-soil body is reinforced depending on the shear strength of the anchor rod and the rock-soil body, so as to keep the stability of an anchoring system (the rock-soil body, grouting body, an interface between an anchor rod body and media). No matter in rock or soil layer, the main function of the anchor rod is to transmit load to rock and soil mass, different anchoring structures (including anchoring system and external structure connected with the anchoring system), different performance of the anchor rod, even the same anchoring system and different anchoring depths, different anchoring effects can be generated, how to correctly select a load transmission model, and how to accurately estimate the ultimate pullout resistance of the anchor rod is very important.
At present, the overall length of the adhesive force is uniformly distributed in the design of the anti-pulling force of the anchor rod, or the elastic assumption is adopted in the theoretical research of an elastic-plastic model, so that the working capacity of the anchor rod in a plastic state cannot be reflected.
The elastic theory method is to research the relation between acting force and displacement between anchoring bodies by using an elastic theory method (namely a Mindlin solution of bearing force at one point in a half space) for an anchoring system so as to obtain a mode of mutual load transmission action. The anchoring systems in the study all make the following assumptions: the anchor is an elastic, uniform, continuous isotropic semi-infinite body with an elastic constant Er、urIs not changed by the insertion of the anchor rod. Many scholars work out the shear stress distribution around the rod body by using an elastic theory method, and the results show that the shear stress of the anchoring section is maximum at the top end, gradually decreases inwards, gradually decreases in speed reduction, and finally approachesAt 0, the distribution appears as a descending curve with 0 as an asymptote, and the calculated ultimate pullout resistance is larger, which is inconsistent with the actual test result on site.
Disclosure of Invention
In view of this, the embodiment of the invention provides a method for calculating the limit pullout force of a full-length bond type anchor rod based on an elasto-plastic model under a Moire Coulomb yield condition.
The embodiment of the invention provides a method for calculating the limit uplift force of a full-length bonded anchor rod based on an elastoplastic model under a Moire coulomb yield condition, which comprises the following steps of:
analyzing the anchor rod stress micro unit to obtain the relation between the displacement of the anchoring body and the shear stress on the anchoring body interface;
according to the relation between the displacement of the anchoring body and the shear stress on the anchoring body interface, simultaneously considering the elastoplasticity effect, and establishing an elastoplasticity model based on the Moore coulomb yield condition;
and calculating the pulling resistance of the full-length bonding type anchor rod according to the elastic-plastic model.
Further, the analysis is carried out to stock atress microcell, obtains the relation of the displacement of anchor and the shear stress on the anchor interface, includes:
taking a micro-unit section of the anchoring body for stress analysis;
from the equilibrium conditions, it follows:
N(z)-(N(z)+dN(z))-2πaτ(z)dz＝0 (1)
obtaining:
wherein, a-anchor radius; n (z) -anchor axial force; τ (z) -shear stress on the anchor interface; z-distance from anchor port;
additionally, the displacement of the anchor w (z) may be determined by the anchor axial force n (z):
namely:
wherein E-modulus of elasticity of the anchor; a-cross-sectional area of anchor;
substituting formula (4) into formula (2), and noting that A ═ pi a2Obtaining:
further, according to the relation between the displacement of the anchor body and the shear stress on the interface of the anchor body, and simultaneously considering the elastoplasticity effect, an elastoplasticity model based on the Moore Coulomb yield condition is established, and the method comprises the following steps:
according to the non-associative flow law, we obtain:
in the formula (I), the compound is shown in the specification,
-the axial plastic linear strain rate of the interface layer;
-a plastic product factor; f is the potential function of the interface layer; sigman-normal stress of the anchor interface;
-the radial plastic linear strain rate of the interface layer;
the Moore coulomb yield condition is satisfied in the interface plastic flow state:
F＝τ-σntanφ-c＝0 (7)
wherein c and phi are respectively the cohesive force and the internal friction angle of the interface layer;
therefore, formula (7) is substituted for formula (6) to obtain:
the above formula indicates that the interfacial layer has a bulk expansion phenomenon; in the interface layer, there is formula (9):
in the formula ur-radial displacement of the interface layer; Δ a-radial variation in interfacial layer radius;
radial displacement u of the interface layerrThe grouting material is composed of two parts, namely a hole wall generating radial deformation under the action of radial stress and a grouting body generating radial deformation; the radial deformation generated by the deformed drill hole wall meets the Winkler assumption, and the deformation of the grouting body meets the Hooke law, namely: in the formula, k is the resistance coefficient of the surrounding rock mass, and can be determined by the sum of the elastic solution of the plane strain problem of the infinite plane circular hole under the action of internal pressure and the deformation of the cylinder under the action of uniform side pressure:
wherein E ', mu ' -the elastic modulus and Poisson's ratio of the surrounding rock-soil mass;
the radial stress can be expressed as:
σn＝Kur(10)
in the formula, K-the deformation coefficient of the interface layer:
therefore, from equations (8) and (9):
σn＝Kw(z)tanφ (11)
substituting into Moire coulomb yield condition (7), and combining with formula (5):
equation (12) is the differential equation of the displacement of the anchoring interface layer; the equation is a second-order linear non-homogeneous ordinary differential equation, and the general solution of the equation is as follows:
in the formula (I), the compound is shown in the specification,
C1、C2-a coefficient to be determined;
respectively substituting formula (13) into formula (5) to obtain an elastic-plastic model:
further, according to the elastic-plastic model, the full-length bond type anchor rod pulling resistance is calculated, and the method comprises the following steps:
equation (14) is substituted for equation (2) and integrated to yield:
in which the constant C is to be determined1And C2According to the boundary conditions, the following steps are obtained:
① when z is equal to l, n (z) is equal to P, the substitution of formula (15) gives:
② when z is 0, n (z) is 0, and formula (15) is substituted by:
③ when z is equal to l, τ (z) is equal to 0, substituting formula (14):
simultaneous (16) and (17) gives:
since it is assumed that the anchor body and the rock-soil body interface yields and enters into plastic flow in the former push-type (14), n (l) of formula (19) represents the ultimate pullout resistance of the anchor rod.
The method has the advantages that the ultimate pullout resistance of the full-length bonding anchor rod can be accurately estimated according to the technical data such as the geometric dimension and material performance of the full-length bonding anchor rod, the performance of grouting materials, the properties of an anchoring body and the like, and the method has important significance for design and construction of the full-length bonding anchor rod and safety evaluation of a side slope.
Drawings
The accompanying drawings, which are included to provide a further understanding of the invention and are incorporated in and constitute a part of this specification, illustrate embodiments of the invention and together with the description serve to explain the invention and not to limit the invention. In the drawings:
FIG. 1 is a flow chart of a method for calculating the ultimate pullout resistance of a full-length bond type anchor rod based on an elasto-plastic model of a Moire Coulomb yield condition, which is provided by the embodiment of the invention;
FIG. 2 is a force diagram of an anchor infinitesimal segment in an embodiment of the invention;
FIG. 3 is a diagram of a variation of an interface layer in an embodiment of the invention;
FIG. 4 is a graph of radial deformation and resistance of a surrounding rock and grout body, wherein (a) the borehole wall is radially deformed; (b) the grout body deforms radially.
Detailed Description
In order to make the objects, technical solutions and advantages of the embodiments of the present invention clearer, the following will clearly and completely describe the technical solutions of the embodiments of the present invention with reference to specific embodiments of the present invention and corresponding drawings. It is to be understood that the described embodiments are only some, and not all, embodiments of the invention. All other embodiments obtained by persons of ordinary skill in the art based on the embodiments of the present invention without any creative efforts belong to the protection scope of the embodiments of the present invention.
Fig. 1 is a flowchart of a method for calculating a full-length bond type anchor rod ultimate pullout resistance based on an elasto-plastic model of a moire coulomb yield condition, which includes:
s101, analyzing a stress micro unit of the anchor rod to obtain the relation between the displacement of the anchoring body and the shear stress on the interface of the anchoring body;
specifically, one microcell segment of the anchor is taken and subjected to a force as shown in fig. 2 and 3.
From the equilibrium conditions, it follows:
N(z)-(N(z)+dN(z))-2πaτ(z)dz＝0 (1)
obtaining:
wherein, a-anchor radius; n-anchor axial force; τ (z) -shear stress on the anchor interface; z-distance from anchor port;
additionally, the displacement w (z) of the anchor may be determined by the axial force n (z) of the anchor:
namely:
wherein E-modulus of elasticity of the anchor; a-cross-sectional area of anchor.
Substituting formula (4) into formula (2), and noting that A ═ pi a2Obtaining:
step S102, establishing an elastoplasticity model based on Moore coulomb yield conditions according to the relation between the displacement of the anchoring body and the shear stress on the anchoring body interface and considering the elastoplasticity effect, and specifically:
assuming that the interface of the anchoring body and the rock-soil body has a certain thickness and enters a plastic flow state, the method is obtained according to a non-related flow rule:
in the formula (I), the compound is shown in the specification,
-the axial plastic linear strain rate of the interface layer;
-a plastic product factor; f is potential function of the interface layer; sigman-normal stress of the anchor interface;
-the radial plastic linear strain rate of the interface layer;
the Moore coulomb yield condition is satisfied in the interface plastic flow state:
F＝τ-σntanφ-c＝0 (7)
wherein c and phi are cohesive force and internal friction angle of the interface layer;
therefore, formula (7) is substituted for formula (6) to obtain:
the above formula indicates that the interfacial layer has bulk. In the interface layer, there is formula (9):
in the formula ur-radial displacement of the interface layer; Δ a-radial variation in interfacial layer radius;
radial displacement u of the interface layerrThe radial deformation of the hole wall and the radial deformation of the grout body under the action of radial stress. The radial deformation generated by the deformed drill hole wall meets the Winkler assumption, and the deformation of the grouting body meets the Hooke law, namely: σ ═ ku. In the formula, k, the resistance coefficient of the surrounding rock-soil mass, can be determined by the sum of the elastic solution of the plane strain problem of the infinite plane circular hole under the action of the internal pressure and the deformation of the cylinder under the action of the uniform side pressure (as shown in fig. 4):
wherein E ', mu ' -the elastic modulus and Poisson's ratio of the surrounding rock-soil mass.
The radial stress can be expressed as:
σn＝Kur(10)
in the formula, K-the deformation coefficient of the interface layer:
therefore, from equations (8) and (9):
σn＝Kw(z)tanφ (11)
substituting into Moire coulomb yield condition (7), and combining with formula (5):
equation (12) is the differential equation for the displacement of the anchoring interface layer. The equation is a second-order linear non-homogeneous ordinary differential equation, and the general solution of the equation is as follows:
in the formula (I), the compound is shown in the specification,
C1、C2-a pending coefficient.
The following formula (13) is substituted for each of formula (5):
step S103, calculating the pulling resistance of the full-length bonded anchor rod according to the elastic-plastic model, specifically:
equation (14) is substituted for equation (2) and integrated to yield:
in which the constant C is to be determined1And C2According to the boundary conditions, the following steps are obtained:
① when z is equal to l, n (z) is equal to P, the substitution of formula (15) gives:
② when z is 0, n (z) is 0, and formula (15) is substituted by:
③ when z is equal to l, τ (z) is equal to 0, substituting formula (14):
simultaneous (16) and (17) gives:
since it is assumed that the anchor body and the rock-soil body interface yields and enters into plastic flow in the former push-type (14), n (l) of formula (19) represents the ultimate pullout resistance of the anchor rod.
The above description is only for the preferred embodiment of the present invention and is not intended to limit the scope of the present invention, and all equivalent flow transformations made by the present specification and drawings, or applied directly or indirectly to other related technical fields, are included in the scope of the present invention.
Claims (4)
1. A method for calculating the limit pullout force of a full-length bonded anchor rod based on an elastoplastic model of a Moire Coulomb yield condition is characterized by comprising the following steps of:
analyzing the anchor rod stress micro unit to obtain the relation between the displacement of the anchoring body and the shear stress on the anchoring body interface;
according to the relation between the displacement of the anchoring body and the shear stress on the anchoring body interface, simultaneously considering the elastoplasticity effect, and establishing an elastoplasticity model based on the Moore coulomb yield condition;
and calculating the pulling resistance of the full-length bonding type anchor rod according to the elastic-plastic model.
2. The method for calculating the limit pullout force of the full-length bonded anchor rod based on the elasto-plastic model under the Moire Coulomb yield condition as claimed in claim 1, wherein the analysis is performed on the anchor rod stress microcell to obtain the relationship between the displacement of the anchoring body and the shear stress on the anchoring body interface, and comprises the following steps:
taking a micro-unit section of the anchoring body for stress analysis;
from the equilibrium conditions, it follows:
N(z)-(N(z)+dN(z))-2πaτ(z)dz＝0 (1)
obtaining:
wherein, a-anchor radius; n (z) -anchor axial force; τ (z) -shear stress on the anchor interface; z-distance from anchor port;
additionally, the displacement of the anchor w (z) may be determined by the anchor axial force n (z):
namely:
wherein E-modulus of elasticity of the anchor; a-cross-sectional area of anchor;
substituting formula (4) into formula (2), and noting that A ═ pi a2Obtaining:
3. the method for calculating the limit pullout force of the full-length bonded anchor rod based on the elasto-plastic model under the Moore Coulomb yield condition as claimed in claim 2, wherein the establishment of the elasto-plastic model under the Moore Coulomb yield condition according to the relation between the displacement of the anchor body and the shear stress on the anchor body interface and considering the elasto-plastic effect comprises the following steps:
according to the non-associative flow law, we obtain:
in the formula (I), the compound is shown in the specification,
-the axial plastic linear strain rate of the interface layer;
-a plastic product factor; f is potential function of the interface layer; sigman-normal stress of the anchor interface;
-the radial plastic linear strain rate of the interface layer;
the Moore coulomb yield condition is satisfied in the interface plastic flow state:
F＝τ-σntanφ-c＝0 (7)
wherein c and phi are respectively the cohesive force and the internal friction angle of the interface layer;
therefore, formula (7) is substituted for formula (6) to obtain:
the above formula indicates that the interfacial layer has a bulk expansion phenomenon; in the interface layer, there is formula (9):
in the formula ur-radial displacement of the interface layer; Δ a-radial variation in interfacial layer radius;
radial displacement u of the interface layerrThe grouting material is composed of two parts, namely a hole wall generating radial deformation under the action of radial stress and a grouting body generating radial deformation; the radial deformation generated by the deformed drill hole wall meets the Winkler assumption, and the deformation of the grouting body meets the Hooke law, namely: in the formula, k is the resistance coefficient of the surrounding rock mass, and can be determined by the sum of the elastic solution of the plane strain problem of the infinite plane circular hole under the action of internal pressure and the deformation of the cylinder under the action of uniform side pressure:
wherein E ', mu ' -the elastic modulus and Poisson's ratio of the surrounding rock-soil mass;
the radial stress can be expressed as:
σn＝Kur(10)
in the formula, K-the deformation coefficient of the interface layer:
therefore, from equations (8) and (9):
σn＝Kw(z)tanφ (11)
substituting into Moire coulomb yield condition (7), and combining with formula (5):
equation (12) is the differential equation of the displacement of the anchoring interface layer; the equation is a second-order linear non-homogeneous ordinary differential equation, and the general solution of the equation is as follows:
in the formula (I), the compound is shown in the specification,
C1、C2-a coefficient to be determined;
respectively substituting formula (13) into formula (5) to obtain an elastic-plastic model:
4. the method for calculating the limit pullout force of the full-length bonded bolt based on the elasto-plastic model of the Moore Coulomb yield condition in claim 3, wherein the step of calculating the limit pullout force of the full-length bonded bolt according to the elasto-plastic model comprises the following steps:
equation (14) is substituted for equation (2) and integrated to yield:
in which the constant C is to be determined1And C2According to the boundary conditions, the following steps are obtained:
① when z is equal to l, n (z) is equal to P, the substitution of formula (15) gives:
② when z is 0, n (z) is 0, and formula (15) is substituted by:
③ when z is equal to l, τ (z) is equal to 0, substituting formula (14):
simultaneous (16) and (17) gives:
since it is assumed that the interface between the anchor body and the rock-soil body yields and enters into plastic flow in the former push-type (14), n (l) of formula (19) represents the ultimate pullout resistance of the anchor rod.
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Cited By (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN113569416A (en) * | 2021-08-02 | 2021-10-29 | 中国地质科学院探矿工艺研究所 | Method for calculating limit bearing capacity of multi-section hole expanding type anchor rod in soil body |
| CN114841009A (en) * | 2022-05-19 | 2022-08-02 | 江苏南京地质工程勘察院 | Method for predicting side friction resistance of anchoring section of anchor rod in plastic stage |
| CN114841009B (en) * | 2022-05-19 | 2024-06-28 | 江苏南京地质工程勘察院 | Prediction method for side friction resistance of anchor rod anchoring section in plastic stage |
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| CN104794365A (en) * | 2015-05-06 | 2015-07-22 | 南华大学 | Computation method for predicting ultimate bearing capacity of anchor rod based on mathematical model |
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| CN104794365A (en) * | 2015-05-06 | 2015-07-22 | 南华大学 | Computation method for predicting ultimate bearing capacity of anchor rod based on mathematical model |
| CN108716227A (en) * | 2018-05-28 | 2018-10-30 | 青岛理工大学 | A kind of analysis method of full grouted GFRP anti-float anchor rods axle power and Displacements Distribution |
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Cited By (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN113569416A (en) * | 2021-08-02 | 2021-10-29 | 中国地质科学院探矿工艺研究所 | Method for calculating limit bearing capacity of multi-section hole expanding type anchor rod in soil body |
| CN113569416B (en) * | 2021-08-02 | 2023-12-01 | 中国地质科学院探矿工艺研究所 | Method for calculating ultimate bearing capacity of multistage hole-expanding anchor rod in soil body |
| CN114841009A (en) * | 2022-05-19 | 2022-08-02 | 江苏南京地质工程勘察院 | Method for predicting side friction resistance of anchoring section of anchor rod in plastic stage |
| CN114841009B (en) * | 2022-05-19 | 2024-06-28 | 江苏南京地质工程勘察院 | Prediction method for side friction resistance of anchor rod anchoring section in plastic stage |
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