CN108830022B - Steel strand bonding strength prediction method based on rotation and protective layer cracking failure - Google Patents
Steel strand bonding strength prediction method based on rotation and protective layer cracking failure Download PDFInfo
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Abstract
The invention discloses a steel strand bonding strength prediction method based on rotation and protective layer cracking failure, which comprises the steps of establishing a relational expression of ultimate bonding strength and bonding failure surface radial stress, determining the radial stress of a bonding failure surface when concrete splitting bonding failure occurs, determining the critical radial stress of the bonding failure surface when concrete splitting failure is converted into drawing rotation failure, determining the minimum concrete protective layer thickness under the critical radial compressive stress, carrying out comparative evaluation on the method through the existing drawing experiment data related to steel strands, and verifying the precision of the method by combining two failure modes. Considering the screw structure for expelling the steel strand and the rotary bonding mechanism of the steel strand, the ultimate bonding strength of the single prestressed steel strand and the concrete under two failure modes, namely the splitting failure of the concrete and the drawing rotary failure, can be effectively simulated and predicted, and the method has better precision; the defects of research on theoretical bonding models of the prestressed steel strands and the concrete are overcome.
Description
Technical Field
The invention relates to the technical field of steel strand ultimate bonding strength calculation methods, in particular to a steel strand bonding strength prediction method based on rotation and protective layer cracking failure.
Background
As prestressed concrete structures are widely used in various large projects, safety problems thereof are more and more prominent. The bonding performance between the prestressed tendons and the concrete is an important factor influencing the safe use of the structure. The bonding between the steel strand and the concrete is mainly realized by the shearing force between the contact surfaces of the steel strand and the concrete. The interface is complex in stress and is influenced by concrete strength, protective layer thickness, steel bar diameter, steel strand surface characteristics and the like. At present, few researches are conducted on a bonding model of a steel strand and concrete, and most of the researches are concentrated on common steel bars. The prestressed steel strand is mostly in a twisted spiral structure, and the bonding performance of the twisted steel strand is inevitably different from that of the common steel bar.
Most of the existing research on the bonding strength model between the steel strand and the concrete is developed based on experiments, the prediction result has high dependence degree on the experiment condition, and the application in the actual engineering is limited. Some scholars have theoretically studied through simplification of the steel strand structure and through finite element simulation and the like. The traditional technology simplifies the steel strand outer wire into a rib spirally surrounding the inner wire surface at a certain angle, and establishes a bonding strength model of the steel strand and cement paste under different side pressures in the ground anchor structure. However, no bonding strength model can consider the influence of a rotary bonding mechanism caused by a twisting structure of the steel strand on the ultimate bonding strength when the steel strand is stressed, and a corresponding bonding strength mechanical model is yet to be developed.
Disclosure of Invention
The invention aims to provide a method for predicting the bonding strength of a steel strand based on rotation and failure of cracking of a protective layer, and the technical problems are effectively solved.
In order to effectively solve the technical problems, the technical scheme of the invention is as follows:
the method for predicting the bonding strength of the steel strand based on rotation and failure of protective layer cracking comprises the following steps:
(1) Establishing a relational expression of the ultimate bonding strength and the radial stress of a bonding failure surface: considering the spiral structural characteristics of the surface of the steel strand, and deducing the relationship between the ultimate bonding strength and the radial stress of a bonding failure surface under two failure modes of concrete splitting failure and drawing rotation failure based on the stress balance of the surface of the outer wire rib in the drawing process;
(2) Determining the radial stress of a bonding failure surface when the concrete is split and bonded to fail: according to the relation between the radial displacement of the concrete at the contact surface caused by the sliding of the steel strand ribs and the radial stress in the stress process of the steel strand, regarding the concrete outside the steel strand as a thick-wall cylinder, and determining the radial stress of the bonding failure surface when the concrete is cracked and bonded and fails in consideration of three stages of non-cracking, partial cracking and complete cracking of the concrete around the steel strand caused by bonding;
(3) Determining the critical radial stress of a bonding failure surface when the concrete splitting failure is converted into the drawing rotation failure: according to the stress characteristic of the steel strand when the rotation failure occurs, establishing a bending moment stress balance relation based on the center of the steel strand, and deducing the critical radial stress of a bonding failure surface when the concrete splitting failure is converted into the drawing rotation failure;
(4) Determining the minimum concrete protective layer thickness under the critical radial compressive stress: according to the relation between the contact surface concrete radial displacement caused by the steel strand ribs and the corresponding radial stress in the steel strand stress process, establishing the relation between the maximum radial stress of the steel strand surface ribs and the thickness, strength and the like of the concrete protective layer, and further deducing to obtain the minimum thickness of the concrete protective layer under the fixed concrete strength;
(5) And (3) model verification: the method is compared and evaluated through the existing drawing experiment data related to the steel strand, and the precision of the method is verified by combining two failure modes.
In particular, the step (1) further comprises the following steps:
firstly, dividing a steel strand into a plurality of longitudinal differential units with the same length according to any lay length of the steel strand, then analyzing mechanical engaging force acting on the surface of the unit steel strand, integrating the effective stressed area of the mechanical engaging force on the surface of the steel strand according to a stress balance principle, deducing the relation between the drawing force provided by the length of the differential unit of the single prestressed steel strand and the radial stress of a bonding failure surface according to the effective stressed area, and finally, uniformly distributing the drawing force on a stressed surface to obtain a relational expression of the ultimate bonding strength and the radial stress of the bonding failure surface;
the derivation about the relation between the bonding strength and the radial stress of the bonding failure surface in the step (1) is as follows:
the bonding nature of the steel strand to the concrete is the shear force between the contact surfaces, and the bonding stress τ b can be expressed as formula (1):
τ b ＝F b /(k·π·d·l b )
wherein Fb and d are respectively the tension and nominal diameter of the steel strand; lb is the bond length; k is a perimeter expansion coefficient introduced by considering the outer surface structure of the steel strand, and can be 4/3;
taking the longitudinal dz length of the steel strand, wherein the corresponding cross section rotation angle is d alpha, the included angle between the outer wire rib and the longitudinal direction is delta, analyzing the stress area of the steel strand rib for providing the occlusion force, and can be approximately regarded as six incomplete crescent moon shapes, assuming that the stress of the outer wire of each steel strand is the same, taking any one outer wire crescent moon shape for analysis, wherein the effective coverage angle of the incomplete crescent moon shape is theta, the value range is [0,2 pi/3 ], and the area dA on the steel strand rib in the d theta region can be expressed as a formula (2):
dA＝h r /sinδ·d b /2·dθ
wherein hr is the height of the transverse rib; db is the diameter of the outer wire of the steel strand, and stress analysis is carried out on the inclined plane dA;
the adhesive force acting on the rib surface dA is composed of a shearing force dFv and a frictional force dFf;
wherein the shearing force dFv is parallel to the rib surface, a friction angle phi exists between the friction force dFf and the normal direction of the rib surface, and if the unit bonding force between the steel strand and the concrete is fcoh and the normal stress acting on the shearing failure plane is fn, then the dFv and the dFf can be respectively expressed as formulas (3) and (4):
dF v ＝f coh dA
the shearing force dFv and the friction force dFf can be decomposed into a parallel force dFb parallel to the axial direction of the steel strand and a radial force dFsp perpendicular to the axial direction, as shown in the following formulas (5), (6):
each strand has a total of six faces providing parallel and hoop forces, and therefore the longitudinal drawing force Fb in the dz range is expressed by the equations (7), (8):
combining formula (7) and formula (8) to obtain expression (9) of Fb in the dz range:
substituting Fb into formula (1) to obtain the expression (10) for the adhesion strength τ b in the dz range:
in particular, the step (2) further comprises the following steps:
in the step (2), the calculation of the radial stress of the bonding failure surface when the concrete splitting bonding fails specifically comprises the following steps:
regarding concrete outside the steel bar as a thick-wall cylinder, considering three stages of non-cracking, partial cracking and complete cracking of the concrete in the drawing process, and combining a bilinear tensile stress degradation model after the concrete is subjected to tensile cracking to obtain expressions of different stages of fn:
the relation expression (11) of tangential stress at radius r, t, r and fn at the uncracked stage:
in the formula, ri and re are respectively the inner radius and the outer radius of the cylinder;
fn in the partial cracking stage consists of two parts, including the uncracked part
And a cleavage part
For the concrete of the uncracked part, the concrete is regarded as a linear elastic material, and for the concrete of the cracked part, the softening behavior of the concrete in a tensile state is considered, and a virtual crack model is introduced. The expression for the partial cracking stage fn is obtained as follows (12):
in the formula, rcr is the radius of the crack front edge; rs = d/2 is the nominal radius of the steel strand; ec is the modulus of elasticity of concrete; epsilon cr = fct/Ec represents the concrete cracking strain; n is the number of virtual fractures, assuming n =3 herein; a and b are constants representing the cracking and softening behavior of the concrete; w0 is the maximum crack width when the concrete tensile stress fails, and the radial stress at the complete cracking stage is consistent with the stress of the cracked part at the partial cracking stage.
In particular, the step (3) further comprises the following steps:
in the step (3), the calculation of the critical radial stress of the bonding failure surface when the concrete splitting failure is converted into the drawing rotation failure specifically comprises the following steps:
the longitudinal component force Frib of the steel strand can be decomposed into a force Fribv vertical to the splitting plane and a force Fribh parallel to the splitting plane;
the force Fribv perpendicular to the cleavage plane is expressed (13) as follows:
the force Fribh parallel to the cleavage plane, expression (14) is as follows:
frib can therefore be expressed as (15):
the relational expression (16) of fn to the torque Mrib in the dz range can be obtained by calculating the torque for the neutral wire:
next, the maximum torque Mmax provided by the friction force of the concrete is obtained, and the maximum torque Mmax can be expressed as (17):
wherein μ = tan (Φ) is a friction coefficient;
neglecting the difference in the wire diameter inside and outside the steel strand, i.e. da = db, then expression (18):
when Mrib reaches Mmax, the steel strand reaches a rotation critical state, and then the steel strand starts to rotate along with the continuous increase of the drawing force to reach the maximum bonding stress; the fn, max when the maximum bonding stress is reached can be obtained by equaling the formula (17) and the formula (18), and the critical compressive stress fn when the contact interface steel strand rotates is obtained, crit is expressed by the following expression (19):
the beneficial effects of the invention are as follows: according to the method for predicting the bonding strength of the steel strand under the rotation and protective layer cracking failure, a mechanism of expelling a spiral structure of the steel strand and a mechanism of rotary bonding of the steel strand are considered, the ultimate bonding strength of the single prestressed steel strand and the concrete under two failure modes, namely the concrete cracking failure and the drawing rotary failure, can be effectively simulated and predicted, and the method has good precision; the defects of research on a theoretical bonding model of the prestressed steel strand and the concrete are overcome.
The present invention will be described in detail with reference to the accompanying drawings.
Drawings
Fig. 1 is a schematic view of the length dz.
Fig. 2 is a longitudinal direction projection view of the steel strand.
Fig. 3 is a longitudinal projection view of any outer filament of the steel strand.
Figure 4 is a force diagram on the rib surface.
Fig. 5 is a torque diagram of the gripping force on the steel strand rib.
Fig. 6 is a graph of friction versus maximum torque produced by a steel strand.
Detailed Description
Example 1:
referring to fig. 1 to 6, the main method for implementing the method of the present embodiment includes: firstly, considering the twisting structural characteristics of the steel strand, establishing a stress balance equation under the constraint of the steel strand and the surrounding concrete, and deducing the relation between the bonding strength of the steel strand and the radial stress of a bonding failure surface; then, considering concrete splitting damage and steel strand drawing rotation failure, providing a calculation expression of radial stress of a contact surface in two failure modes, and further establishing a theoretical calculation model of the ultimate bonding strength of the steel strand in the two failure modes; finally, the accuracy and feasibility of the method are verified through the existing experimental data. The method comprises the following specific steps:
(1) Establishing a relational expression of the ultimate bonding strength and the radial stress of a bonding failure surface: considering the spiral structural characteristics of the surface of the steel strand, and deducing the relationship between the ultimate bonding strength and the radial stress of a bonding failure surface under two failure modes of concrete splitting failure and drawing rotation failure based on the stress balance of the surface of an outer wire rib in the drawing process;
(2) Determining the radial stress of a bonding failure surface when the concrete is split and bonded to fail: according to the relation between the contact surface concrete radial displacement and the radial stress caused by the sliding of steel strand ribs in the stress process of the steel strand, regarding the concrete outside the steel strand as a thick-wall cylinder, and determining the radial stress of a bonding failure surface when the concrete is split and bonded and fails in consideration of three stages of non-cracking, partial cracking and complete cracking of the concrete around the steel strand caused by bonding;
(3) Determining the critical radial stress of a bonding failure surface when the concrete splitting failure is converted into the drawing rotation failure: according to the stress characteristics of the steel strand when the steel strand is in rotary failure, establishing a bending moment stress balance relation based on the center of the steel strand, and deducing the critical radial stress of a bonding failure surface when the concrete splitting failure is converted into the drawing rotary failure;
(4) Determining the minimum concrete protective layer thickness under the critical radial compressive stress: according to the relation between the contact surface concrete radial displacement caused by the steel strand ribs and the corresponding radial stress in the steel strand stress process, establishing the relation between the maximum radial stress of the steel strand surface ribs and the thickness of the concrete protective layer, the strength of the concrete and the like, and further deducing to obtain the minimum thickness of the concrete protective layer under the fixed strength of the concrete;
(5) And (3) model verification: the method is contrastively evaluated through the existing drawing experiment data related to the steel strand, and the precision of the method is verified by combining two failure modes.
The applicant states that a new method, which is generated by combining some steps of the above-mentioned embodiment with the technical solution of the summary part of the invention based on the above-mentioned embodiment, is also one of the description scope of the present invention, and other embodiments of these steps are not listed in the present application for the sake of brevity.
The step (1) further comprises the following steps:
firstly, dividing a steel strand into a plurality of longitudinal differential units with the same length according to any lay length of the steel strand, analyzing mechanical engagement force acting on the surface of the unit steel strand, integrating the effective stress area of the mechanical engagement force on the surface of the steel strand according to a stress balance principle, deducing the relation between the drawing force provided by the length of the differential unit of the single prestressed steel strand and the radial stress of a bonding failure surface according to the effective stress area, and finally, uniformly distributing the drawing force on the stress surface to obtain a relational expression of the ultimate bonding strength and the radial stress of the bonding failure surface;
the derivation about the relation between the bonding strength and the radial stress of the bonding failure surface in the step (1) is as follows:
the bonding nature of the steel strand and the concrete is a shearing force between contact surfaces, and the bonding stress τ b can be expressed as formula (1):
τ b ＝F b /(k·π·d·l b )
wherein Fb and d are respectively the tension and nominal diameter of the steel strand; lb is the bond length; k is a perimeter expansion coefficient introduced by considering the outer surface structure of the steel strand, and can be 4/3;
taking the longitudinal dz length of the steel strand, wherein the corresponding rotation angle of the cross section is d alpha, the included angle between the outer wire rib and the longitudinal direction is delta, analyzing the stress area of the steel strand rib for providing the occlusion force, wherein the stress area can be approximately regarded as six incomplete crescent moon shapes, assuming that the stress of the outer wire of each steel strand is the same, taking any outer wire crescent shape for analysis, the effective coverage angle of the incomplete crescent shape is theta, the value range is [0,2 pi/3 ], and the area dA on the steel strand rib in the d theta region can be expressed as a formula (2):
dA＝h r /sinδ·d b /2·dθ
wherein hr is the height of the transverse rib; db is the diameter of the outer wire of the steel strand, and the stress of the inclined plane dA is analyzed;
the adhesive force acting on the rib surface dA is composed of a shearing force dFv and a frictional force dFf;
wherein the shearing force dFv is parallel to the rib surface, a friction angle phi exists between the friction force dFf and the normal direction of the rib surface, if the unit cohesive force between the steel strand and the concrete is fcoh, and the normal stress acting on the shearing failure plane is fn, then the dFv and the dFf can be respectively expressed as formulas (3) and (4):
dF v ＝f coh dA
the shearing force dFv and the friction force dFf can be decomposed into a parallel force dFb parallel to the axial direction of the steel strand and a radial force dFsp perpendicular to the axial direction according to the following formulas (5) and (6):
each strand has a total of six faces providing parallel and hoop forces, and therefore the longitudinal drawing force Fb in the dz range is expressed by the equations (7), (8):
combining formula (7) and formula (8) to obtain expression (9) of Fb in the dz range:
substituting Fb into formula (1) to obtain the expression (10) for the adhesion strength τ b in the dz range:
the step (2) further comprises the following steps:
in the step (2), the calculation of the radial stress of the bonding failure surface when the concrete splitting bonding fails specifically comprises the following steps:
regarding concrete outside the steel bar as a thick-wall cylinder, considering three stages of non-cracking, partial cracking and complete cracking of the concrete in the drawing process, and combining a bilinear tensile stress degradation model after the concrete is subjected to tensile cracking to obtain expressions of different stages of fn:
the relation expression (11) of tangential stress at radius r, t, r and fn at the uncracked stage:
in the formula, ri and re are respectively the inner radius and the outer radius of the cylinder;
fn of the partial cracking stage consisting of two parts, including the uncracked part
And a cleavage part
For the uncracked portion of the concrete, it was considered as a linear elastic material, and for the cracked portion of the concrete, the softening behavior of the concrete in a tensile state was considered and a virtual crack model was introduced. The expression for the partial cracking stage fn is obtained as follows (12):
in the formula, rcr is the radius of the crack front edge; rs = d/2 is the nominal radius of the steel strand; ec is the modulus of elasticity of concrete; epsilon cr = fct/Ec represents the concrete cracking strain; n is the number of virtual fractures, assuming n =3 herein; a and b are constants representing the cracking and softening behavior of the concrete; w0 is the maximum crack width when the concrete tensile stress fails, and the radial stress at the complete cracking stage is consistent with the stress of the cracked part at the partial cracking stage.
The step (3) further comprises the following steps:
in the step (3), the calculation of the critical radial stress of the bonding failure surface when the concrete splitting failure is converted into the drawing rotation failure specifically comprises the following steps:
the longitudinal component force Frib of the steel strand can be decomposed into a force Fribv vertical to the splitting plane and a force Fribh parallel to the splitting plane;
the force Fribv perpendicular to the cleavage plane is expressed (13) as follows:
the force Fribh parallel to the cleavage plane, expression (14) is as follows:
frib can therefore be expressed as (15):
the relational expression (16) of fn to the torque Mrib in the dz range can be obtained by calculating the torque for the neutral wire:
next, the maximum torque Mmax provided by the friction force of the concrete is obtained, and the maximum torque Mmax can be expressed as (17):
wherein μ = tan (Φ) is a friction coefficient;
neglecting the difference between the inner and outer wire diameters of the steel strand, namely da = db, expression (18):
when Mrib reaches Mmax, the steel strand reaches a rotation critical state, and then the steel strand starts to rotate along with the continuous increase of the drawing force to reach the maximum bonding stress; the fn, max when the maximum bonding stress is reached can be obtained by equaling the formula (17) and the formula (18), and the critical compressive stress fn when the contact interface steel strand rotates is obtained, crit is expressed by the following expression (19):
the applicant further states that the present invention is described in the above embodiments to explain the implementation method and device structure of the present invention, but the present invention is not limited to the above embodiments, i.e. it is not meant to imply that the present invention must rely on the above methods and structures to implement the present invention. It should be understood by those skilled in the art that any modifications to the present invention, additions of equivalent or step implementations, selection of specific modes, etc., are intended to fall within the scope and disclosure of the present invention.
Claims (3)
1. The method for predicting the bonding strength of the steel strand based on rotation and protective layer cracking failure is characterized by comprising the following steps of:
(1) Establishing a relational expression of the ultimate bonding strength and the radial stress of a bonding failure surface: considering the spiral structural characteristics of the surface of the steel strand, and deducing the relationship between the ultimate bonding strength and the radial stress of a bonding failure surface under two failure modes of concrete splitting failure and drawing rotation failure based on the stress balance of the surface of an outer wire rib in the drawing process;
(2) Determining the radial stress of a bonding failure surface when the concrete is split and bonded to fail: according to the relation between the contact surface concrete radial displacement and the radial stress caused by the sliding of steel strand ribs in the stress process of the steel strand, regarding the concrete outside the steel strand as a thick-wall cylinder, and determining the radial stress of a bonding failure surface when the concrete is split and bonded and fails in consideration of three stages of non-cracking, partial cracking and complete cracking of the concrete around the steel strand caused by bonding;
(3) Determining the critical radial stress of a bonding failure surface when the concrete splitting failure is converted into the drawing rotation failure: according to the stress characteristic of the steel strand when the rotation failure occurs, establishing a bending moment stress balance relation based on the center of the steel strand, and deducing the critical radial stress of a bonding failure surface when the concrete splitting failure is converted into the drawing rotation failure;
(4) Determining the minimum concrete protective layer thickness under the critical radial compressive stress: according to the relation between the contact surface concrete radial displacement caused by the steel strand ribs and the corresponding radial stress in the steel strand stress process, establishing the relation between the maximum radial stress of the steel strand surface ribs and the thickness of the concrete protective layer and the strength of the concrete, and further deducing to obtain the minimum thickness of the concrete protective layer under the fixed strength of the concrete;
(5) And (3) model verification: the accuracy of the method is verified by carrying out comparative evaluation on the existing drawing experiment data related to the steel strand and combining two failure modes.
2. The method for predicting the bonding strength of the steel strand under the failure based on the rotation and the cracking of the protective layer as claimed in claim 1, wherein the step (1) further comprises the following steps:
firstly, dividing a steel strand into a plurality of longitudinal differential units with the same length according to any lay length of the steel strand, analyzing mechanical engagement force acting on the surface of the unit steel strand, integrating the effective stress area of the mechanical engagement force on the surface of the steel strand according to a stress balance principle, deducing the relation between the drawing force provided by the length of the differential unit of the single prestressed steel strand and the radial stress of a bonding failure surface according to the effective stress area, and finally, uniformly distributing the drawing force on the stress surface to obtain a relational expression of the ultimate bonding strength and the radial stress of the bonding failure surface;
the derivation about the relation between the bonding strength and the radial stress of the bonding failure surface in the step (1) is specifically as follows:
the bonding nature of the steel strand and the concrete is contactInterfacial shear force, adhesive stress tau b Expressed as the formula:
in the formula, F b And d is the tension and nominal diameter of the steel strand respectively; l. the b Is the bonding length; k is a perimeter expansion coefficient introduced by considering the outer surface structure of the steel strand, and is taken as 4/3;
get the vertical dz length of steel strand wires, the cross section rotation angle that corresponds this moment is d alpha, the contained angle of outer silk rib and longitudinal direction is delta, analysis steel strand wires rib provides the lifting surface area of occlusal force, every steel strand wires outer silk atress is the same, take one of them arbitrary outer silk crescent to analyze, the effective coverage angle of incomplete crescent is theta, the value range is [0,2 pi/3 ], area dA on the steel strand wires rib in the d theta region expresses as the formula:
in the formula, h r Is the height of the transverse rib; d is a radical of b Analyzing the stress of the surface of the rib for the diameter of the outer wire of the steel strand;
the adhesive force acting on the rib surface being determined by the shear force dF v And frictional force dF f Forming;
wherein the shear force dF v Parallel to the rib surface, frictional force dF f A friction angle (98111) exists in the normal direction of the surface of the rib, and the unit bonding force between the steel strand and the concrete is f coh The normal stress acting on the shear failure plane is f n Then dF v And dF f Respectively expressed as the formula:
shear force dF v And frictional force dF f Decomposed into parallel forces dF parallel to the axial direction of the steel strand b And radial force dF perpendicular to the axial direction sp The following formula:
a total of six faces per strand provide parallel and hoop forces, and therefore d z Longitudinal drawing force F within range b Expressed as the formula:
combining the two formulas to obtain F in the dz range b Expression (c):
f is to be b Substitution of bonding stress tau b The formula gives the bonding stress tau in the dz range b Expression:
3. the method for predicting the bond strength of the steel strand under the failure of the rotation and the cracking of the protective layer according to claim 2, wherein the step (2) further comprises the following steps:
in the step (2), the calculation of the radial stress of the bonding failure surface when the concrete is in the splitting bonding failure concretely comprises the following steps:
considering concrete outside the steel bar as a thick-wall cylinder, considering three stages of no cracking, partial cracking and complete cracking of the concrete in the drawing process, and combining a bilinear tensile stress degradation model after the concrete is cracked in tension to obtain f n Expressions at different stages:
at the uncracked stage, tangential stress σ at radius r t,r And f n The relational expression of (1):
in the formula, r i And r e The inner radius and the outer radius of the cylinder are respectively; partial cracking stage f n Consisting of two parts, including non-split parts
And a cleavage part
Regarding concrete of the part which is not cracked as a linear elastic material, regarding the concrete of the cracked part as a softening behavior of the concrete under a tensile state, introducing a virtual crack model, and obtaining a partial cracking stage f n The expression of (a) is as follows:
in the formula, r cr The radius of the crack front; r is a radical of hydrogen s D/2 is the nominal radius of the steel strand; ec is the modulus of elasticity of concrete; epsilon cr =f ct Concrete is/EcCracking strain; n is the number of virtual cracks, n =3; a and b are constants representing the cracking and softening behavior of the concrete; w is a 0 The maximum crack width when the concrete tensile stress fails, the radial stress at the complete cracking stage is consistent with the stress of the cracked part at the partial cracking stage.
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