CN112255099A - Prediction method of prestressed transfer length of pretensioned PC members under concrete rust expansion and cracking - Google Patents

Abstract

The method for predicting the prestressed transmission length of a pretensioned PC member under concrete rust expansion and cracking disclosed in the invention takes into account the development process of the corrosion product and combines the structural characteristics of the steel strand to analyze the effect of the corrosion of the prestressed steel strand on the transmission length of the prestressed pretensioned member. The influence of the radial stress around the steel strand within the range, and the relationship between the radial stress and the corrosion penetration depth was established. Then, based on the theory of thick-walled cylinders, the relationship between the radial displacement and the radial stress of the contact surface between the steel strand and the concrete was obtained. The relationship between the radial stress of the contact surface and the corrosion penetration depth after the component is not cracked, partially cracked or completely cracked is deduced, and the corresponding bond stress distribution is obtained. Finally, based on the bond stress distribution, the prestressed components are divided into Multiple micro-segments, and based on the bond stress and the force balance between the steel strand and the concrete, the strain distribution of the steel strand and the concrete is obtained, and finally, the corrosion is determined based on the strain distribution characteristics of the steel strand and the concrete in the transmission length range. The transmission length of the strand.

Description

Prediction method of prestressed transfer length of pretensioned PC members under concrete rust expansion and cracking

technical field

The invention belongs to the technical field of pretensioned PC components under concrete rust expansion and cracking, and particularly relates to a method for predicting the prestress transmission length of pretensioned PC components under concrete rust expansion and cracking.

Background technique

For pretensioned prestressed concrete members, sufficient transmission length is the key to whether the initial tensile stress of prestressed tendons can be fully transmitted to the concrete. When the prestressed tendons are corroded due to the external environment, the cross-sectional area of the prestressed tendons will decrease, the contact conditions between the prestressed tendons and the concrete will change, and the volume expansion of the corrosion products will cause the surrounding concrete to crack, etc., which will affect the initial prestressing in the concrete. The transmission of , resulting in the distribution of bond stress, prestressed tendon stress, concrete stress, etc. within the transmission length, and ultimately lead to the change of the transmission length.

At present, most theoretical studies on the transfer length of pretensioned prestressed members are concentrated in non-corroded prestressed members. Lee et al. ^[98] derived the slip distribution over the transfer length by assuming a linear local bond-slip relationship. Then, based on the existing test data of transmission length, through regression analysis, various coefficients related to transmission length are determined, and then the calculation formula of transmission length is obtained. Based on the thick-walled cylinder theory, Ramirez-Garcia et al. ^[99] simulated the bonding behavior of prestressed steel strands and concrete in the transfer area of pretensioned members by finite element program, including bonding stress distribution, cracking degree and prestressed steel The transmission length of the strand. Oh et al. ^[97,101] first deduced the bond stress-slip relationship of pretensioned prestressed components based on experimental data, and then deduced the calculation formula of transfer length combined with finite element. M.Ben _1'tez et al. ^[160] established the bond model of steel wire and concrete during the tensioning process of prestressed steel wire, and derived the relationship between bond stress, steel and concrete stress and slip, and obtained the relationship between the steel wire and the steel wire. Calculation formula of transfer length for notch depth and concrete cover thickness. ^[106] described local bond stress as a function of local slip and bar stress changes based on pull-out and push-in tests. Based on this model, a bilinear relationship between transit length and development length was simulated. Abdelatif et al. ^[96] adopted the thick-walled cylinder theory, assuming that both concrete and steel bars exhibit elastic material behavior. The transfer of prestress in pretensioned members is simulated by means of force balance equations, compatibility equations, and bonding boundary conditions. The distribution of longitudinal stress and radial stress of the steel bar is obtained by continuous solving, and the transmission length is obtained. Balazs ^[102] developed a nonlinear equation for the transfer length based on a specific bond stress-slip relationship, taking into account the effective and initial prestresses as well as the concrete strength and strand size at transfer. However, there are few studies on the transmission length of prestressed steel strands under corrosion.

SUMMARY OF THE INVENTION

The purpose of the present invention is to provide a method for predicting the prestressed transfer length of pre-tensioned PC members under concrete rust expansion and cracking in order to solve the above problems.

The method for predicting the prestressed transfer length of pre-tensioned PC members under concrete rust expansion and cracking includes the following steps:

S1: Determine the relevant parameters of the thick-walled cylinder and the size of the stirrup position, and determine the parameters related to the thick-walled cylinder based on the structure size of the pretensioning method, including the inner and outer radius of the cylinder, concrete compressive strength, tensile strength, concrete elastic modulus and Poisson’s ratio, related parameters of prestressed steel strand, including nominal initial radius, initial tensile prestress, nominal tensile strength, elastic modulus and Poisson’s ratio. For specimens with stirrups, stirrups need to be determined. The position of the radius in the cylinder, the radius of the stirrups, the cross-sectional area, the tensile strength, the elastic modulus, and the stirrup spacing parameters;

S2: Calculate the radius R _j under the action of the initial prestress f _pj before the prestressed steel strand is stretched;

S3: Calculate the corresponding concrete tensile stress f _cz under the action of the steel strand tensile stress f _pz within the initial prestress range;

S4: Calculate the radial compressive stress p of the contact surface between the steel strand and the concrete;

S5: Calculate the radial displacement u _j of the contact surface and obtain the radial strain ε _θ (R _j ) of the contact surface;

S6: Compare the calculated radial strain ε _θ (R _j ) of the contact surface with the cracking tensile strain ε _ct of concrete, if ε _θ (R _j )>ε _ct , go to the next step, if ε _θ (R _j )≤ ε _ct , the bonding stress τ is obtained by solving;

The radial strain ε _θ (R _j ) of the contact surface calculated in S5 is compared with the tensile strain ε _ct when the concrete reaches the ultimate tensile stress, so as to judge the cracking situation of the contact surface concrete, if the former is smaller than the latter , it means that the concrete on the contact surface is not cracked, at this time, the bond stress between the steel strand and the concrete corresponding to the strain can be directly obtained by formula (5.10);

S7: Calculate the crack front radius R _c when the contact surface radial strain ε _c (R _j ) exceeds the concrete cracking tensile strain ε _ct in the uncorroded state;

When the radial strain ε _c (R _j ) of the contact surface calculated in S5 is greater than the tensile strain ε _ct when the concrete reaches the ultimate tensile stress, it indicates that the contact surface concrete has been cracked, and the concrete protective layer is in a state of partial cracking or complete cracking , the concrete begins to show softening behavior, and the tensile behavior of concrete must be reconsidered. First, the crack front radius R _c caused by prestress relaxation is solved by formula (5.12b);

S8: Calculate the corrosion intrusion depth x _c corresponding to filling the concrete crack, and based on the crack front radius R _c caused by the tension of the prestressed steel strand obtained in S7, determine the corrosion intrusion depth x required to fill the crack _c to solve, when x ≤ x _c , the corrosion of the steel strand will not cause additional displacement on the contact surface, nor will it cause the radial stress p of the contact surface to change, the radial pressure at this time is consistent with that without corrosion, When x>x _c , the corrosion of steel strands begins to cause additional radial displacement _ur in the concrete of the contact surface;

S9: Solve the crack front radius R _c2 in the case of the corrosion depth x>x _c , and solve the relationship between the arbitrary corrosion depth x and the crack front radius R _c2 in the case of the corrosion depth x> x _c , and therefore find that the crack completely penetrates the cylinder, that is When R _c2 is equal to the critical corrosion penetration depth x _cr when the outer radius R ₀ of the cylinder, when the calculated R _c2 exceeds the outer radius R ₀ , take R _c2 =R ₀ ;

S10: Calculate the concrete strain ε _θ (r) corresponding to the radius of the crack front and the corresponding radii R ₁ and R _u when the strains ε ₁ and ε _u of the two control points are reached. Based on whether the crack front radius R _c2 exceeds the peripheral radius R ₀ It is divided into two steps. In the first step, when R _c2 <R ₀ , R ₁ and R _u are calculated by formula (5.16); in the second step, when R _c2 >R ₀ , R 1 and R u are calculated by formula (5.26). ₁ and R _u are used for calculation. During the calculation process, if the calculated value of R ₁ or R _u exceeds the outer radius R ₀ of the cylinder, the value of R ₀ is taken;

S11: Calculate the corrosion penetration depth x<x _cr , that is, the radial pressure p of the contact surface when the protective layer is partially cracked. First, calculate the concrete confinement effect p _c at the front of the cracked concrete by formula (5.21), and obtain the cracked part of the concrete along the diameter The strain distribution in the direction ε _θ (r), and the radial compressive stress p of the contact surface in the partial cracking stage is obtained;

S12: Solve the rust penetration depth x>x _cr , that is, the radial pressure p of the contact surface when the protective layer is completely cracked;

S13: Calculate the bond stress τ in the uncracked, partially cracked and completely cracked stages, calculate Δf _pz , calculate Δz when f _pz,n =f _pj , calculate f pz,n corresponding to Δε _pz,n =ε _{cz, n} , determine the corresponding value n _k , n _k <n, and obtain the transmission length l _tr =n _k ·Δz.

In particular, in steps S1-S5, due to the radial displacement u _j of the contact surface between the steel strand and the concrete caused by the expansion of the steel strand, the prestressed steel strand cuts the surrounding circumferential concrete during the unwinding process. The tangential strain ε _q =u _j /R _j , when the tangential strain ε _q exceeds the ultimate tensile strain ε _cr =f _ct /E _c of the concrete, the concrete begins to split, forming splitting cracks. The cross-section of the steel strand will be reduced due to the action of prestress, and the radius R _j of the steel strand after stress is compared with the radius R _i under the non-tensioned state has the following relationship:

Among them, f _pj : the initial tensile stress of the prestressed steel strand, which is generally 0.75 times the nominal tensile strength f _ps of the steel strand; E _p : the elastic modulus of the prestressed steel strand; v _p : the prestressed steel strand Poisson's ratio;

_Rj obtained from formula (5.1) is the radius of the steel strand in the longitudinal direction caused by the Poisson effect of the steel strand when the steel strand reaches the initial tension. Influence, it will produce displacement in the radial direction within the transmission length. The displacement at the free end of the component is the difference between the radii before and after the steel strand is tensioned, that is, R _i -R _j , and gradually decreases in the direction of the transmission length until When reaching zero at the end of the transmission length, due to the radial displacement of the steel strand, the surrounding concrete will have the same radial displacement, and therefore radial compressive stress will be generated on the contact surface between the steel strand and the concrete, and a ring will be generated in the tangential direction of the concrete. Tensile stress, according to the radial compressive stress and hoop tensile stress, the concrete thick-walled cylinder can be divided into the following three stages according to the cracking situation of the concrete: three stages: uncracked, partially cracked and completely cracked; the following The relationship between the radial displacement and the radial stress of the contact surface at different stages is evaluated under the non-corroded and corroded states of the steel strand respectively;

For the non-corroded prestressed steel strand, according to the research of Oh et al. ^[97] , the radial displacement u _j of the contact surface with the concrete in the polar coordinate system during the tensioning process of the prestressed steel strand can be expressed as:

Wherein, E _c : elastic modulus of concrete;

v _c : Poisson's ratio of concrete;

f _cz : longitudinal stress of concrete;

p: The radial pressure on the contact surface due to the expansion of the steel strand when the prestressed steel strand is stretched,

Among them, the last two items f _cz and p can be expressed as:

Among them, f _pz : the stress in the prestressed steel strand;

A _p : the cross-sectional area of the prestressed steel strand;

A: The cross-sectional area of the full section of concrete;

I: moment of inertia of concrete section;

e: The eccentric distance from the prestressed steel strand to the center of the concrete cross-section.

In particular, in steps S6-S8, under the condition that the constraining stress p of the prestressed steel strand of the pretensioned member under different corrosion degrees is known, the bonding stress τ between the prestressed steel strand and the concrete can be determined by the following basic The governing equation says:

τ=μ·p (5.29)

In the formula, μ is the friction coefficient, which can be regarded as a constant under a specific corrosion rate; p is the radial pressure acting on the surface of the steel strand, which is calculated in Section 5.2, under the premise that the friction coefficient μ and p are known , the bond stress distribution can be obtained;

μ is the friction coefficient between the prestressed steel strand and the concrete, which is determined by the surface conditions of the contact surface between the steel strand and the concrete, and changes with the corrosion degree of the steel strand. The calculation of the friction coefficient μ remains the same, that is, μ(ρ _p )=0.343-0.26(xx _cr ), where x _cr is the corresponding corrosion penetration depth when the concrete protective layer is completely cracked, and x _cr =0.031;

The relationship between the radial pressure p of the contact surface, the confinement stress p _c of the concrete at the front of the crack, the residual tensile stress σ _θ (r) of the concrete in the tangential direction, and the tensile stress of the stirrups σ _st (r) are as follows:

Assuming that the displacement in the radial direction of the concrete during the stress process is linear elasticity, the relationship between the radial displacement u(r) and the tangential strain ε _θ (r) at any radial radius r of the cylinder and the crack front radius R _c is as follows Show:

In the case of partial cracking of concrete, replace R _j in Equation (5.11a) and make it equal to Equation (5.2), the calculation expression of the radius R _c of the cracking front of the crack after the uncorroded steel strand is stretched can be obtained first :

For the additional displacement ur | _{r =Rj} of the contact surface caused by the corrosion of the steel strand, it can be expressed as:

Equation (5.10) and (5.13) are equal to obtain the relationship between the corrosion penetration depth x and the crack front radius R _c2 when the corrosion penetration depth x>x _c is as follows:

After that, the tangential strain distribution of cracked concrete can be obtained by formula (5.11b).

In particular, in steps _S9 - _S12 , the concrete cracking caused by the shrinkage of the strand develops to the surface of the component, that is, the stress condition after the concrete protective layer is completely cracked. The displacement u(r) at radius r and the corresponding shear strain ε _θ (r) are as follows:

Among them, _{ε θc} is the tangential strain of the edge of the protective layer after the concrete is completely cracked, and the radial displacement of the contact surface after the concrete is completely cracked is the radial displacement of the contact surface after the concrete is completely cracked. The displacement u _j caused by the tension and the additional displacement ur _r caused by the corrosion of the steel strand are composed together, and the following expression can be obtained:

Therefore, the tangential stress ε _θc at the edge of the concrete cover can be calculated by formula (5.23):

At this time, ur can be obtained by replacing the crack front radius R _c2 with the cylinder peripheral radius _{R 0} by formula (5.10):

The tangential strain distribution along the tangential direction of the protective layer after the concrete protective layer is completely cracked can be obtained by formula (5.22b), and the residual tensile stress σ _θ of the cracked concrete cylinder can be obtained by formula (5.19), and the critical strain ε at this time The relationship between the corresponding radii R ₁ and R _u of ₁ and ε _u and ε _θc can be obtained by substituting ε ₁ =0.0003 and ε _u =0.002 into formula (5.22b), respectively:

Since the confining stress p _c is no longer provided after the concrete is completely cracked, equation (5.18) can be simplified as:

Equation (5.27) can be used to calculate the expansion stress p of the contact surface when the concrete is fully cracked,

中:At this time, the force of the stirrup is:

middle:

The above is the calculation process of the radial pressure of the contact surface between the steel strand and the concrete under the condition of no cracking, partial cracking and complete cracking of the obtained concrete after the prestressed prestressing method is completed and the influence of the corrosion of the steel strand is considered. , based on this process, the relationship between the corrosion penetration depth x and the radial stress p at different positions within the transfer length range can be deduced, and then the transfer length under the influence of corrosion is calculated based on the derived relationship.

In particular, in step S13-S18, under the condition that the constraining stress p of the prestressed steel strand of the pretensioned member under different corrosion degrees is known, the bonding stress τ between the prestressed steel strand and the concrete can be calculated by the following basic The governing equation of :

τ=μ·p (5.29)

In the formula, μ is the friction coefficient, which can be regarded as a constant under a specific corrosion rate; p is the radial pressure acting on the surface of the steel strand, which is calculated in Section 5.2, under the premise that the friction coefficient μ and p are known , the bond stress distribution can be obtained,

μ is the friction coefficient between the prestressed steel strand and the concrete, which is determined by the surface conditions of the contact surface between the steel strand and the concrete, and changes with the corrosion degree of the steel strand. The calculation of the friction coefficient μ remains the same, that is, μ(ρ _p )=0.343-0.26(xx _cr ), where x _cr is the corresponding corrosion penetration depth when the concrete protective layer is completely cracked, and x _cr =0.031;

Assuming that the bonding stress in each Δz length range is uniformly distributed, the prestress increment Δf _pz of the prestressed steel strand accumulated by the bonding stress on each micro-segment can be expressed as:

Considering that the prestress at the end position of the prestressed member is 0, and assuming that the strain change Δε _pz of the steel strand is consistent with the initial prestrain strain ε _pi , therefore, the prestressed steel strand is prestressed at any nth Δz length. The stress f _pz,n and the strain variable Δε _pz,n can be calculated as:

Substituting the strand stress calculated by equation (5.31) into equation (5.3), the concrete strain ε _cz,n at the nth Δz length can be calculated,

When the strain variable Δε _pz,n of the steel strand calculated by the formula (5.32) is equal to the concrete strain ε _cz,n calculated by the formula (5.33), it means that there is no more displacement between the steel strand and the concrete at this time. The distance from the position to the end of the member is the transmission length l _tr of the prestressed steel strand.

The advantages and positive effects of the present invention are as follows: the present invention causes the radial displacement between the steel strand and the concrete contact surface due to the retraction of the prestressed steel strand during the unwinding process of the prestressed steel strand, resulting in the concrete Hoop tensile stress is formed around it. When the tensile stress exceeds the tensile strength of the concrete, cracking occurs in the contact surface concrete. In addition, when the prestressed strand corrodes, the expansion of the corrosion products can further change the restraint condition of the strand. This makes the previously uncracked concrete develop towards partial cracking, and the partially cracked concrete develops towards the fully cracked concrete. Therefore, this paper analyzes the stress distribution of prestressed steel bars and surrounding concrete under the influence of corrosion on the basis of the development of cracks in the concrete protective layer caused by prestressing and corrosion in pretensioned prestressed members, and deduces the transfer length under the influence of corrosion. The confinement stress distribution and bonding stress distribution of the steel strand within the range are obtained, and then the method for determining the transmission length of the corroded steel strand is obtained.

Description of drawings

Figure 1 is a schematic diagram of concrete cracking caused by the transfer of pretensioned structures.

Figure 2 is a model of a thick-walled cylinder of corroded steel strands.

Figure 3 is a schematic diagram of the stress-strain relationship after concrete cracking.

Figure 4 is the equilibrium equation for the partially cracked concrete condition.

Figure 5 is the equilibrium equation for the fully cracked concrete condition.

Figure 6 is an exploded view of the pre-tensioned prestressed structure.

Detailed ways

In order to further understand the content, characteristics and effects of the present invention, the following embodiments are listed and described in detail below with the accompanying drawings.

The present invention is described below in conjunction with Fig. 1: a method for predicting the prestressed transfer length of pre-tensioned PC members under concrete rust expansion and cracking, comprising the following steps:

S1: Determine the relevant parameters of the thick-walled cylinder and the size of the stirrup position, and determine the parameters related to the thick-walled cylinder based on the structure size of the pretensioning method, including the inner and outer radius of the cylinder, concrete compressive strength, tensile strength, concrete elastic modulus and Poisson’s ratio, related parameters of prestressed steel strand, including nominal initial radius, initial tensile prestress, nominal tensile strength, elastic modulus and Poisson’s ratio. For specimens with stirrups, stirrups need to be determined. The position of the radius in the cylinder, the radius of the stirrups, the cross-sectional area, the tensile strength, the elastic modulus, and the stirrup spacing parameters;

S2: Calculate the radius R _j under the action of the initial prestress f _pj before the prestressed steel strand is stretched;

S3: Calculate the corresponding concrete tensile stress f _cz under the action of the steel strand tensile stress f _pz within the initial prestress range;

S4: Calculate the radial compressive stress p of the contact surface between the steel strand and the concrete;

S5: Calculate the radial displacement u _j of the contact surface and obtain the radial strain ε _θ (R _j ) of the contact surface;

S6: Compare the calculated radial strain ε _θ (R _j ) of the contact surface with the cracking tensile strain ε _ct of concrete, if ε _θ (R _j )>ε _ct , go to the next step, if ε _θ (R _j )≤ ε _ct , the bonding stress τ is obtained by solving;

The radial strain ε _θ (R _j ) of the contact surface calculated in S5 is compared with the tensile strain ε _ct when the concrete reaches the ultimate tensile stress, so as to judge the cracking situation of the contact surface concrete, if the former is smaller than the latter , it means that the concrete on the contact surface is not cracked, at this time, the bond stress between the steel strand and the concrete corresponding to the strain can be directly obtained by formula (5.10);

S7: Calculate the crack front radius R _c when the contact surface radial strain ε _c (R _j ) exceeds the concrete cracking tensile strain ε _ct in the uncorroded state;

When the radial strain ε _c (R _j ) of the contact surface calculated in S5 is greater than the tensile strain ε _ct when the concrete reaches the ultimate tensile stress, it indicates that the contact surface concrete has been cracked, and the concrete protective layer is in a state of partial cracking or complete cracking , the concrete begins to show softening behavior, and the tensile behavior of concrete must be reconsidered. First, the crack front radius R _c caused by prestress relaxation is solved by formula (5.12b);

S8: Calculate the corrosion intrusion depth x _c corresponding to filling the concrete crack, and based on the crack front radius R _c caused by the tension of the prestressed steel strand obtained in S7, determine the corrosion intrusion depth x required to fill the crack _c to solve, when x ≤ x _c , the corrosion of the steel strand will not cause additional displacement on the contact surface, nor will it cause the radial stress p of the contact surface to change, the radial pressure at this time is consistent with that without corrosion, When x>x _c , the corrosion of steel strands begins to cause additional radial displacement _ur in the concrete of the contact surface;

S9: Solve the crack front radius R _c2 in the case of the corrosion depth x>x _c , and solve the relationship between the arbitrary corrosion depth x and the crack front radius R _c2 in the case of the corrosion depth x> x _c , and therefore find that the crack completely penetrates the cylinder, that is When R _c2 is equal to the critical corrosion penetration depth x _cr when the outer radius R ₀ of the cylinder, when the calculated R _c2 exceeds the outer radius R ₀ , take R _c2 =R ₀ ;

S10: Calculate the concrete strain ε _θ (r) corresponding to the radius of the crack front and the corresponding radii R ₁ and R _u when the strains ε ₁ and ε _u of the two control points are reached. Based on whether the crack front radius R _c2 exceeds the peripheral radius R ₀ It is divided into two steps. In the first step, when R _c2 <R ₀ , R ₁ and R _u are calculated by formula (5.16); in the second step, when R _c2 >R ₀ , R 1 and R u are calculated by formula (5.26). ₁ and R _u are used for calculation. During the calculation process, if the calculated value of R ₁ or R _u exceeds the outer radius R ₀ of the cylinder, the value of R ₀ is taken;

S11: Calculate the corrosion penetration depth x<x _cr , that is, the radial pressure p of the contact surface when the protective layer is partially cracked. First, calculate the concrete confinement effect p _c at the front of the cracked concrete by formula (5.21), and obtain the cracked part of the concrete along the diameter The strain distribution in the direction ε _θ (r), and the radial compressive stress p of the contact surface in the partial cracking stage is obtained;

S12: Solve the rust penetration depth x>x _cr , that is, the radial pressure p of the contact surface when the protective layer is completely cracked;

S13: Calculate the bond stress τ in the uncracked, partially cracked and completely cracked stages, calculate Δf _pz , calculate Δz when f _pz,n =f _pj , calculate f pz,n corresponding to Δε _pz,n =ε _{cz, n} , determine the corresponding value n _k , n _k <n, and obtain the transmission length l _tr =n _k ·Δz.

In step S1, the plastic sheet is prepared by printing out small squares on the plane of the expanded view through a transparent material, or directly using a transparent sheet and a blister mold to shape the outline of the product.

In this embodiment,

During the tensioning process of the prestressed steel strand, the radial displacement between the steel strand and the concrete contact surface is caused by the retraction of the prestressed steel strand, resulting in the formation of hoop tensile stress around the concrete. When the tensile stress exceeds the tensile strength of the concrete, cracking occurs in the contact surface concrete. In addition, when the prestressed strand corrodes, the expansion of the corrosion products can further change the restraint condition of the strand. This makes the previously uncracked concrete develop towards partial cracking, and the partially cracked concrete develops towards the fully cracked concrete. Therefore, this paper analyzes the stress distribution of prestressed steel bars and surrounding concrete under the influence of corrosion on the basis of the development of cracks in the concrete protective layer caused by prestressing and corrosion in pretensioned prestressed members, and deduces the transfer length under the influence of corrosion. The confinement stress distribution and bonding stress distribution of the steel strand within the range are obtained, and then the method for determining the transmission length of the corroded steel strand is obtained.

In the pre-tensioned prestressed member, the longitudinal cross-sectional area of the steel strand is reduced due to the Poisson effect under the action of prestressing. During the prestress release process, with the release of the prestress, the cross section of the prestressed steel strand will recover to the original cross section. However, this expansion is limited by the confinement of the concrete, creating hoop compressive stress around the prestressed strands. Considering the surrounding concrete of the prestressed steel strand as a thick-walled cylinder, the development of concrete cracking caused by the prestressing in the concrete transfer process during the release process is shown in Figure 1.

R _i : initial radius of untensioned prestressed steel strand;

R _j : the radius of the prestressed steel strand before it is stretched;

R _c : the distance from the center of the prestressed steel strand to the front of the crack;

R ₀ : the distance from the center of the prestressed steel strand to the thickness edge of the protective layer (outer edge of the cylinder);

u _j : the deformation of the prestressed steel strand in the radial direction when the prestress is released.

Due to the radial displacement u _j of the contact surface between the steel strand and the concrete caused by the expansion of the steel strand, the prestressed steel strand produces a tangential strain ε _q = u _j /R to the surrounding circumferential concrete during the unwinding process _j , when the tangential strain ε _q exceeds the ultimate tensile strain of concrete ε _cr =f _ct /E _c , the concrete begins to split, forming splitting cracks. At this time, the cross section of the steel strand will be reduced due to the effect of prestressing, and the radius R _j of the steel strand after stress is compared with the radius R _i in the non-tensioned state has the following relationship:

Among them, f _pj : the initial tensile stress of the prestressed steel strand, which is generally 0.75 times the nominal tensile strength f _ps of the steel strand;

E _p : elastic modulus of prestressed steel strand;

v _p : Poisson's ratio of prestressed steel strand;

_Rj obtained from formula (5.1) is the radius of the steel strand in the longitudinal direction caused by the Poisson effect of the steel strand when the steel strand reaches the initial tension force. During the unwinding of the steel strand, due to the influence of the expansion of the steel strand, there will be displacement in the radial direction within the transmission length. The displacement at the free end of the member is the difference between the radii of the steel strand before and after tensioning, that is, R _i -R _j . And gradually decrease in the direction of the transfer length until it reaches zero at the end of the transfer length. Due to the radial displacement of the steel strand, the same radial displacement of the surrounding concrete occurs, and therefore radial compressive stress is generated on the contact surface between the steel strand and the concrete. And the hoop tensile stress is generated in the tangential direction of the concrete. According to the magnitude of the radial compressive stress and the hoop tensile stress, the concrete thick-walled cylinder can be divided into the following three stages according to the cracking situation of the concrete: three stages: uncracked, partially cracked and completely cracked. In the following, the relationship between the radial displacement and radial stress of the contact surface at different stages is evaluated in the non-rusted and rusted state of the steel strand.

For the non-corroded prestressed steel strand, according to the research of Oh et al. ^[97] , the radial displacement u _j of the contact surface with the concrete in the polar coordinate system during the tensioning process of the prestressed steel strand can be expressed as:

Wherein, E _c : elastic modulus of concrete;

v _c : Poisson's ratio of concrete;

f _cz : longitudinal stress of concrete;

p: The radial pressure on the contact surface due to the expansion of the steel strand when the prestressed steel strand is stretched.

Among them, the last two items f _cz and p can be expressed as:

Among them, f _pz : the stress in the prestressed steel strand;

A _p : the cross-sectional area of the prestressed steel strand;

A: The cross-sectional area of the full section of concrete;

I: moment of inertia of concrete section;

e: The eccentric distance from the prestressed steel strand to the center of the concrete cross-section.

For corroded steel strands, the additional rust expansion stress generated by the corrosion of the steel strand must be considered. Due to the increase in the volume of the corrosion products produced by the corrosion of the steel strand, an additional radial displacement _ur is generated at the contact surface, resulting in a change in the restraint stress around the steel strand. In the previous section on bond strength calculation, it is assumed that the outer wire and inner wire of the steel strand are uniformly corroded, and the corrosion penetration depth or radius loss of a single steel wire is x. Ignore the diameter difference between the inner and outer wires of the steel strand, that is, da = _{d b .} The radius of a single steel wire after uniform corrosion R _bs =R _b -x, R _b is the radius of the uncorroded outer wire. The relationship between the rust rate ρ _p and the rust depth x is obtained as:

为简化计算，假设外丝锈蚀产物均匀覆盖在钢绞线名义直径周长周围，如图2所示。According to the corrosion characteristics of the steel strand, it is assumed that the gaps between the inner and outer wires of the steel strand are filled by the corrosion products of the inner wires, and do not spread to the outside of the steel strand. Expansion effect to obtain volume loss per unit length of corroded strand

To simplify the calculation, it is assumed that the outer wire corrosion products are uniformly covered around the perimeter of the nominal diameter of the steel strand, as shown in Figure 2.

The development of strand corrosion products here differs from that in the previous chapters. Since the prestressed steel strands cause some concrete cracks after the prestress transfer is completed, it is assumed here that the corrosion products of the prestressed tendons will first fill the concrete cracks caused by the expansion of the prestressed steel strands, so the following expression can be obtained:

In the formula, ∑w is the total width of concrete cracks at the radius R _j caused by the unwinding of steel strands, ∑w=2π·u _r | _r=Rj =2π·u _j . u _j is calculated by formula (5.2). m is the expansion coefficient of rust products, and the value is different according to different rust products ^[142] . R _c is the radius of the crack front in the concrete caused by the uncorrosion of the strand in the non-corroded state.

Assuming the critical corrosion depth x _c required to fill the initial crack width, equation (5.6) can be further simplified as:

Equation (5.7) is a quadratic function about the corrosion penetration depth x _c , and the calculation expression of x _c is:

At this time, according to the relationship between the corrosion depth x and x _c , it can be judged whether there is additional displacement between the steel strand and the concrete due to the corrosion of the steel strand. When the rust depth x does not exceed x _c , that is, x≤x _c , it means that the contact surface between the steel strand and the concrete does not generate additional displacement due to corrosion, and therefore does not generate additional rust expansion stress. When the rust depth x exceeds x _c , that is, x>x _c , additional displacement _ur begins to occur, and therefore the concrete cracks caused by the previous expansion continue to develop outward. Assuming that the corresponding crack front radius at this time is R _c2 , the relationship between the volume caused by the corrosion of the steel strand and the corrosion invasion depth x and the crack front radius R _c2 is as follows:

In the formula, R _r is the radius of the outer edge corrosion product after the steel strand is corroded; ∑w is the total width of the rust expansion crack at the radius R _r , ∑w=2π·u _r | _r=R0 =2π(R _r -R _j ). R _r = _{R j} +ur . Solving equation (5.9), the displacement ur | _{r =Rj at} the contact surface radius Rj can be obtained as:

where R _j is the nominal radius of the uncorroded steel strand.

Assuming that the displacement in the radial direction of the concrete during the stress process is linear elasticity, the relationship between the radial displacement u(r) and the tangential strain ε _θ (r) at any radial radius r of the cylinder and the crack front radius R _c is as follows Show:

In the case of partial cracking of concrete, replace R _j in Equation (5.11a) and make it equal to Equation (5.2), the calculation expression of the radius R _c of the cracking front of the crack after the uncorroded steel strand is stretched can be obtained first :

For the additional displacement ur | _{r =Rj} of the contact surface caused by the corrosion of the steel strand, it can be expressed as:

Equation (5.10) and (5.13) are equal to obtain the relationship between the corrosion penetration depth x and the crack front radius R _c2 after corrosion when the corrosion penetration depth x > x _c , as shown below:

After that, the tangential strain distribution of cracked concrete can be obtained by formula (5.11b).

For the cracked concrete cylinder, the concrete will soften after the cracking. The concrete performance is that the concrete stress σ _θ (r) decreases rapidly with the increase of the concrete strain ε _θ (r), and the decrease becomes slower when it exceeds a certain value, until the concrete loses its tensile capacity completely when it exceeds a certain critical strain. In the existing research, by taking the concrete tangential tensile strain ε _θ (r) as a variable, the expression of the stress-strain relationship of cracked concrete is as follows:

Among them, ε ₁ and ε _u are respectively 0.0003 and 0.002.

Assuming that the radii of the concrete strain reaching the critical strain ε ₁ and ε _u are R ₁ and R _u respectively, as shown in Fig. 3, then ε ₁ =0.0003 and ε _u =0.002 can be substituted into formula (5.12b) to obtain R ₁ respectively and R _u and the crack front radius R _c2 :

When the two critical radii R ₁ and R _u calculated by formula (5.16) are lower than the radius R _j after the prestressed steel strand is tensioned, it indicates that the concrete cracking strain value does not exceed the corresponding critical strain, and the R at this time ₁ and R _u are directly replaced by R _j .

In addition, for a thick-walled cylinder with stirrups, assuming that the stirrup is a circle surrounding the steel strand, the distance from the center of the stirrup to the center of the steel strand, that is, the radius of the position where the stirrup is located, is R _s . As shown in Figure 4. At this time, it is considered that the strain of the stirrup is consistent with the concrete strain here, and R _s can be substituted into r by formula (5.11b). Then the tangential tensile stress σ _st of the stirrup can be expressed as:

Then, the radial compressive stress of the contact surface after the concrete part is cracked is calculated.

As shown in the figure, the relationship between the radial pressure p of the contact surface, the confinement stress p _c of the concrete at the front of the crack, the residual tensile stress σ _θ (r) of the concrete in the tangential direction and the tensile stress of the stirrup σ _st (r) As follows:

可结合式(5.11b)，式(5.15)和式(5.16)得到，如下所示：in,

It can be obtained by combining formula (5.11b), formula (5.15) and formula (5.16), as follows:

中:and

middle:

In the formula, R _st is the radius of the stirrup; D _st is the diameter of the stirrup D _st = 2R _st ; A _st is the cross-sectional area of the stirrup A _st = πR _st ² ; D _j is the prestressed tension of the steel strand The drawn diameter D _j =2R _j ; S _v is the spacing of the stirrups; R _s is the distance from the center point of the stirrup to the center point of the steel strand;

In addition, the tensile stress σ _θ (R _c ) at the cracking front should be consistent with the concrete tensile strength f _ct , ie σ _θ (R _c )=f _ct . Therefore, the confinement stress p _c of concrete at the crack front can be expressed as:

Substituting the restraint stress p _c calculated by Equation (5.21) into Equation (5.18) can obtain the pressure p of the contact surface in the partial cracking stage.

At this time, the concrete cracking caused by the shrinkage of the steel strand develops to the surface of the component, that is, the stress condition after the concrete protective layer is completely cracked. By substituting R ₀ for R _c in equation (5.11), the displacement u(r) and the corresponding shear strain ε _θ (r) at any radius r of the concrete can be obtained as follows:

where ε _θc is the tangential strain at the edge of the cover after the concrete is fully cracked. Replacing r in formula (5.22a) with R _j is the radial displacement of the contact surface after the concrete is completely cracked. The radial displacement at this time is the displacement u _j caused by the tension and the additional displacement u _r caused by the corrosion of the steel strand. Composed together, the following expression can be obtained:

Therefore, the tangential stress ε _θc at the edge of the concrete cover can be calculated by formula (5.23):

At this time, ur can be obtained by replacing the crack front radius R _c2 with the cylinder peripheral radius _{R 0} by formula (5.10):

The tangential strain distribution along the tangential direction of the protective layer after the concrete protective layer is completely cracked can be obtained by formula (5.22b). The residual tensile stress σ _θ of the cracked concrete cylinder can be obtained by formula (5.19). At this time, the relationship between the radii R ₁ and R _u corresponding to the critical strains ε ₁ and ε _u and ε _θc can be obtained by substituting ε ₁ =0.0003 and ε _u =0.002 into equation (5.22b):

Since the confining stress p _c is no longer provided after the concrete is completely cracked, equation (5.18) can be simplified as:

Equation (5.27) can be used to calculate the expansion stress p of the contact surface when the concrete is fully cracked.

中:At this time, the force of the stirrup is:

middle:

The above is the calculation process of the radial pressure of the contact surface between the steel strand and the concrete under the condition of no cracking, partial cracking and complete cracking of the obtained concrete after the prestressed prestressing method is completed and the influence of the corrosion of the steel strand is considered. . Based on this process, the relationship between the corrosion penetration depth x and the radial stress p at different positions within the transmission length range can be derived. Next, the transfer length under the influence of corrosion is calculated based on the derived relationship.

Under the condition that the restraint stress p of the prestressed steel strand of the pretensioned member is known under different corrosion degrees, the bond stress τ between the prestressed steel strand and the concrete can be expressed by the following basic governing equation:

τ=μ·p (5.29)

In the formula, μ is the friction coefficient, which can be regarded as a constant under a specific corrosion rate; p is the radial pressure acting on the surface of the steel strand. Calculations are performed through Section 5.2. On the premise that both the friction coefficient μ and p are known, the adhesive stress distribution can be obtained.

μ is the friction coefficient between the prestressed steel strand and the concrete, which is determined by the surface conditions of the contact surface between the steel strand and the concrete, and changes with the corrosion degree of the steel strand. This is consistent with the calculation of the coefficient of friction μ in the previous chapter on the bond strength model, ie μ(ρ _p ) = 0.343-0.26(xx _cr ). Among them, x _cr is the corresponding corrosion penetration depth when the concrete protective layer is completely cracked, x _cr = 0.031;

The prestressed concrete member can be decomposed into n micro-segments of length Δz in the longitudinal direction of the steel strand, as shown in Figure 6.

Assuming that the bonding stress in each Δz length range is uniformly distributed, the prestress increment Δf _pz of the prestressed steel strand accumulated by the bonding stress on each micro-segment can be expressed as:

Considering that the prestress at the end of the prestressed member is 0, and assuming that the strain change Δε _pz of the steel strand is consistent with the initial prestrain ε _pi . Therefore, the stress f _pz,n and the strain variable Δε _pz,n of the prestressed strand at any nth Δz length can be calculated as:

By substituting the strand stress calculated by Equation (5.31) into Equation (5.3), the concrete strain ε _cz,n at the nth Δz length can be calculated.

When the strain variable Δε _pz,n of the steel strand calculated by the formula (5.32) is equal to the concrete strain ε _cz,n calculated by the formula (5.33), it means that there is no more displacement between the steel strand and the concrete at this time. The distance from the position to the end of the member is the transmission length l _tr of the prestressed steel strand.

The above is only the preferred embodiment of the present invention, and does not limit the present invention in any form. Any simple modification, equivalent change and modification made to the above embodiment according to the method of the present invention belong to the present invention. within the scope of the inventive method scheme.

Claims (5)

1. The method for predicting the prestressed transfer length of pre-tensioned PC members under concrete rust expansion and cracking is characterized in that, comprising the following steps: S1: Determine the relevant parameters of the thick-walled cylinder and the size of the stirrup position, and determine the parameters related to the thick-walled cylinder based on the structure size of the pretensioning method, including the inner and outer radius of the cylinder, concrete compressive strength, tensile strength, concrete elastic modulus and Poisson’s ratio, related parameters of prestressed steel strand, including nominal initial radius, initial tensile prestress, nominal tensile strength, elastic modulus and Poisson’s ratio. For specimens with stirrups, stirrups need to be determined. The position of the radius in the cylinder, the radius of the stirrups, the cross-sectional area, the tensile strength, the elastic modulus, and the stirrup spacing parameters; S2: Calculate the radius R _j under the action of the initial prestress f _pj before the prestressed steel strand is stretched; S3: Calculate the corresponding concrete tensile stress f _cz under the action of the steel strand tensile stress f _pz within the initial prestress range; S4: Calculate the radial compressive stress p of the contact surface between the steel strand and the concrete; S5: Calculate the radial displacement u _j of the contact surface and obtain the radial strain ε _θ (R _j ) of the contact surface; S6: Compare the calculated radial strain ε _θ (R _j ) of the contact surface with the cracking tensile strain ε _ct of concrete, if ε _θ (R _j )>ε _ct , go to the next step, if ε _θ (R _j )≤ ε _ct , the bonding stress τ is obtained by solving; The radial strain ε _θ (R _j ) of the contact surface calculated in S5 is compared with the tensile strain ε _ct when the concrete reaches the ultimate tensile stress, so as to judge the cracking situation of the contact surface concrete, if the former is smaller than the latter , it means that the concrete on the contact surface is not cracked, at this time, the bond stress between the steel strand and the concrete corresponding to the strain can be directly obtained by formula (5.10); S7: Calculate the crack front radius R _c when the contact surface radial strain ε _c (R _j ) exceeds the concrete cracking tensile strain ε _ct in the uncorroded state; When the radial strain ε _c (R _j ) of the contact surface calculated in S5 is greater than the tensile strain ε _ct when the concrete reaches the ultimate tensile stress, it indicates that the contact surface concrete has been cracked, and the concrete protective layer is in a state of partial cracking or complete cracking , the concrete begins to show softening behavior, and the tensile behavior of concrete must be reconsidered. First, the crack front radius R _c caused by prestress relaxation is solved by formula (5.12b); S8: Calculate the corrosion intrusion depth x _c corresponding to filling the concrete crack, and based on the crack front radius R _c caused by the tension of the prestressed steel strand obtained in S7, determine the corrosion intrusion depth x required to fill the crack _c to solve, when x ≤ x _c , the corrosion of the steel strand will not cause additional displacement on the contact surface, nor will it cause the radial stress p of the contact surface to change, the radial pressure at this time is consistent with that without corrosion, When x>x _c , the corrosion of steel strands begins to cause additional radial displacement _ur in the concrete of the contact surface; S9: Solve the crack front radius R _c2 in the case of the corrosion depth x>x _c , and solve the relationship between the arbitrary corrosion depth x and the crack front radius R _c2 in the case of the corrosion depth x> x _c , and therefore find that the crack completely penetrates the cylinder, that is When R _c2 is equal to the critical corrosion penetration depth x _cr when the outer radius R ₀ of the cylinder, when the calculated R _c2 exceeds the outer radius R ₀ , take R _c2 =R ₀ ; S10: Calculate the concrete strain ε _θ (r) corresponding to the radius of the crack front and the corresponding radii R ₁ and R _u when the strains ε ₁ and ε _u of the two control points are reached. Based on whether the crack front radius R _c2 exceeds the peripheral radius R ₀ It is divided into two steps. In the first step, when R _c2 <R ₀ , R ₁ and R _u are calculated by formula (5.16); in the second step, when R _c2 >R ₀ , R 1 and R u are calculated by formula (5.26). ₁ and R _u are used for calculation. During the calculation process, if the calculated value of R ₁ or R _u exceeds the outer radius R ₀ of the cylinder, the value of R ₀ is taken; S11: Calculate the corrosion penetration depth x<x _cr , that is, the radial pressure p of the contact surface in the partially cracked state of the protective layer. First, calculate the concrete confinement effect p _c at the front of the cracked concrete by formula (5.21), and obtain the cracked part of the concrete along the diameter The strain distribution in the direction ε _θ (r), and the radial compressive stress p of the contact surface in the partial cracking stage is obtained; S12: Solve the rust penetration depth x>x _cr , that is, the radial pressure p of the contact surface when the protective layer is completely cracked; S13: Calculate the bond stress τ in the uncracked, partially cracked and completely cracked stages, calculate Δf _pz , calculate Δz when f _pz,n =f _pj , calculate f pz,n corresponding to Δε _pz,n =ε _{cz, n} , determine the corresponding value n _k , n _k <n, and obtain the transmission length l _tr =n _k ·Δz. 2. The method for predicting the prestressed transfer length of pre-tensioned PC members under concrete rust expansion and cracking according to claim 1, is characterized in that: In steps S1-S5, due to the radial displacement u _j of the contact surface between the steel strand and the concrete caused by the expansion of the steel strand, the prestressed steel strand produces a tangential strain ε to the surrounding hoop concrete during the unwinding process. _q = u _j /R _j , when the tangential strain ε _q exceeds the ultimate tensile strain ε _cr =f _ct /E _c of the concrete, the concrete begins to split, forming splitting cracks. The effect of stress will cause the cross section of the steel strand to decrease, and the radius R _j of the steel strand after stress is compared with the radius R _i in the non-tensioned state, which has the following relationship:

Among them, f _pj : the initial tensile stress of the prestressed steel strand, which is generally 0.75 times the nominal tensile strength f _ps of the steel strand; E _p : the elastic modulus of the prestressed steel strand; v _p : the prestressed steel strand Poisson's ratio; _Rj obtained from formula (5.1) is the radius of the steel strand in the longitudinal direction caused by the Poisson effect of the steel strand when the steel strand reaches the initial tension. Influence, it will produce displacement in the radial direction within the transmission length. The displacement at the free end of the component is the difference between the radii before and after the steel strand is tensioned, that is, R _i -R _j , and gradually decreases in the direction of the transmission length until When reaching zero at the end of the transmission length, due to the radial displacement of the steel strand, the surrounding concrete will have the same radial displacement, and thus radial compressive stress will be generated on the contact surface between the steel strand and the concrete, and a ring will be generated in the tangential direction of the concrete. Tensile stress, according to the radial compressive stress and hoop tensile stress, the concrete thick-walled cylinder can be divided into the following three stages according to the cracking situation of concrete: three stages: uncracked, partially cracked and completely cracked; the following The relationship between the radial displacement and the radial stress of the contact surface at different stages is evaluated under the non-corroded and corroded states of the steel strand, respectively; For the non-corroded prestressed steel strand, according to the research of Oh et al. ^[97] , the radial displacement u _j of the contact surface with the concrete in the polar coordinate system during the tensioning process of the prestressed steel strand can be expressed as:

Wherein, E _c : elastic modulus of concrete; v _c : Poisson's ratio of concrete; f _cz : longitudinal stress of concrete; p: The radial pressure on the contact surface due to the expansion of the steel strand when the prestressed steel strand is stretched, Among them, the last two items f _cz and p can be expressed as:

Among them, f _pz : the stress in the prestressed steel strand; A _p : the cross-sectional area of the prestressed steel strand; A: The cross-sectional area of the full section of concrete; I: moment of inertia of concrete section; e: The eccentric distance from the prestressed steel strand to the center of the concrete cross-section. 3. The method for predicting the prestressed transfer length of pre-tensioned PC members under concrete rust expansion and cracking according to claim 1, characterized in that: In steps S6-S8, under the condition that the restraint stress p of the prestressed steel strand of the pretensioned member under different corrosion degrees is known, the bonding stress τ between the prestressed steel strand and the concrete can be expressed by the following basic control equation: : τ=μ·p (5.29) In the formula, μ is the friction coefficient, which can be regarded as a constant under a specific corrosion rate; p is the radial pressure acting on the surface of the steel strand, which is calculated in Section 5.2, under the premise that the friction coefficient μ and p are known , the bond stress distribution can be obtained; μ is the friction coefficient between the prestressed steel strand and the concrete, which is determined by the surface conditions of the contact surface between the steel strand and the concrete, and changes with the corrosion degree of the steel strand. The calculation of the friction coefficient μ remains the same, that is, μ(ρ _p )=0.343-0.26(xx _cr ), where x _cr is the corresponding corrosion penetration depth when the concrete protective layer is completely cracked, and x _cr =0.031; The relationship between the radial pressure p of the contact surface, the confinement stress p _c of the concrete at the front of the crack, the residual tensile stress σ _θ (r) of the concrete in the tangential direction, and the tensile stress of the stirrups σ _st (r) are as follows:

Assuming that the displacement in the radial direction of the concrete during the stress process is linear elasticity, the relationship between the radial displacement u(r) and the tangential strain ε _θ (r) at any radial radius r of the cylinder and the crack front radius R _c is as follows Show:

In the case of partial cracking of concrete, replace R _j in Equation (5.11a) and make it equal to Equation (5.2), the calculation expression of the radius R _c of the cracking front of the crack after the uncorroded steel strand is stretched can be obtained first :

For the additional displacement ur | _{r =Rj} of the contact surface caused by the corrosion of the steel strand, it can be expressed as:

Equation (5.10) and (5.13) are equal to obtain the relationship between the corrosion penetration depth x and the crack front radius R _c2 after corrosion when the corrosion penetration depth x > x _c , as shown below:

After that, the tangential strain distribution of cracked concrete can be obtained by formula (5.11b). 4. The method for predicting the prestressed transfer length of pre-tensioned PC members under concrete rust expansion and cracking according to claim 1, characterized in that: In steps _S9 - _S12 , the concrete cracking caused by the shrinkage of the strand develops to the surface of the component, that is, the stress condition after the concrete protective layer is completely cracked. The displacement u(r) and the corresponding shear strain ε _θ (r) are as follows:

Among them, _{ε θc} is the tangential strain of the edge of the protective layer after the concrete is completely cracked, and the radial displacement of the contact surface after the concrete is completely cracked is the radial displacement of the contact surface after the concrete is completely cracked. The displacement u _j caused by the tension and the additional displacement ur _r caused by the corrosion of the steel strand are composed together, and the following expression can be obtained:

Therefore, the tangential stress ε _θc at the edge of the concrete cover can be calculated by formula (5.23):

At this time, ur can be obtained by replacing the crack front radius R _c2 with the cylinder peripheral radius _{R 0} by formula (5.10):

The tangential strain distribution along the tangential direction of the protective layer after the concrete protective layer is completely cracked can be obtained by formula (5.22b), and the residual tensile stress σ _θ of the cracked concrete cylinder can be obtained by formula (5.19), and the critical strain ε at this time The relationship between the corresponding radii R ₁ and R _u of ₁ and ε _u and ε _θc can be obtained by substituting ε ₁ =0.0003 and ε _u =0.002 into formula (5.22b), respectively:

Since the confining stress p _c is no longer provided after the concrete is completely cracked, equation (5.18) can be simplified as:

Equation (5.27) can be used to calculate the expansion stress p of the contact surface when the concrete is fully cracked, At this time, the force of the stirrup is:

middle:

The above is the calculation process of the radial pressure of the contact surface between the steel strand and the concrete under the condition of no cracking, partial cracking and complete cracking of the obtained concrete after the prestressed prestressing method is completed and the influence of the corrosion of the steel strand is considered. , based on this process, the relationship between the corrosion penetration depth x and the radial stress p at different positions within the transfer length range can be deduced, and then the transfer length under the influence of corrosion is calculated based on the derived relationship. 5. The method for predicting the prestressed transfer length of pre-tensioned PC members under concrete rust expansion and cracking according to claim 1, characterized in that: Step S13 - is in S18, under the condition that the restraint stress p of the prestressed steel strand of the pretensioned member is known under different degrees of corrosion, the bond stress τ between the prestressed steel strand and the concrete can be controlled by the following basic control equation: express: τ=μ·p (5.29) In the formula, μ is the friction coefficient, which can be regarded as a constant under a specific corrosion rate; p is the radial pressure acting on the surface of the steel strand, which is calculated in Section 5.2, under the premise that the friction coefficient μ and p are known , the bond stress distribution can be obtained, μ is the friction coefficient between the prestressed steel strand and the concrete, which is determined by the surface conditions of the contact surface between the steel strand and the concrete, and changes with the corrosion degree of the steel strand. The calculation of the friction coefficient μ remains the same, that is, μ(ρ _p )=0.343-0.26(xx _cr ), where x _cr is the corresponding corrosion penetration depth when the concrete protective layer is completely cracked, and x _cr =0.031; Assuming that the bonding stress in each Δz length range is uniformly distributed, the prestress increment Δf _pz of the prestressed steel strand accumulated by the bonding stress on each micro-segment can be expressed as:

Considering that the prestress at the end position of the prestressed member is 0, and assuming that the strain change Δε _pz of the steel strand is consistent with the initial prestrain strain ε _pi , therefore, the prestressed steel strand is prestressed at any nth Δz length. The stress f _pz,n and the strain variable Δε _pz,n can be calculated as:

Substituting the strand stress calculated by equation (5.31) into equation (5.3), the concrete strain ε _cz,n at the nth Δz length can be calculated,

When the strain variable Δε _pz,n of the steel strand calculated by the formula (5.32) is equal to the concrete strain ε _cz,n calculated by the formula (5.33), it means that there is no more displacement between the steel strand and the concrete at this time. The distance from the position to the end of the member is the transmission length l _tr of the prestressed steel strand.

Images