CN112255099A - Method for predicting pre-tensioning PC (polycarbonate) component prestress transmission length under concrete rust cracking - Google Patents

Abstract

The invention discloses a method for predicting the prestress transmission length of a pretensioned PC (polycarbonate) member under the condition of concrete rust expansion cracking, which considers the development process of a rust product, combines the structural characteristics of a steel strand, analyzes the influence of the rust of the prestressed steel strand on the radial stress around the steel strand in the transmission length range of the pretensioned member, establishes the relation between the radial stress and the rust invasion depth, deduces the relation between the radial stress and the rust invasion depth of a contact surface after the member is not cracked, partially cracked and completely cracked through the relation between the radial displacement and the radial stress of the contact surface of the steel strand and the concrete based on the thick-wall cylinder theory, and accordingly obtains the corresponding bonding stress distribution, finally divides the prestressed member into a plurality of micro-segments based on the bonding stress distribution, and obtains the strain distribution of the steel strand and the concrete based on the force balance relation between the bonding stress and the steel strand and the concrete, and finally, determining the transmission length of the rusted steel strand based on the strain distribution characteristics of the steel strand and the concrete in the transmission length range.

Description

Method for predicting pre-tensioning PC (polycarbonate) component prestress transmission length under concrete rust cracking

Technical Field

The invention belongs to the technical field of pre-tensioned PC (polycarbonate) members under the condition of concrete rusty cracking, and particularly relates to a method for predicting the prestress transmission length of the pre-tensioned PC members under the condition of concrete rusty cracking.

Background

For pretensioned prestressed concrete elements, the sufficient transfer length is critical to whether the initial tensile stress of the tendon can be transferred to the concrete completely. When the prestressed tendon is corroded by the external environment, the cross-sectional area of the prestressed tendon is reduced, the contact condition of the prestressed tendon and concrete is changed, the corrosion product expands in volume to cause cracking of the surrounding concrete, the initial prestress is influenced to be transmitted in the concrete, the distribution of bonding stress, prestressed tendon stress, concrete stress and the like in the transmission length is caused, and the transmission length is finally changed.

At present, most of theoretical researches on the transfer length of the pretensioned prestressing member are concentrated in the non-rusted prestressing member. Lee et al^[98]The slip distribution over the range of transfer lengths is derived by the assumed linear local stick-slip relationship. Then, on the basis of the existing transfer length test data, determining each coefficient related to the transfer length through regression analysis, and further obtaining a calculation formula of the transfer length. Ramirez-Garcia et al^[99]Based on the thick-wall cylinder theory, the bonding behavior of the prestressed steel strand and concrete in the pretensioned member transfer area is simulated through a finite element program, and the bonding behavior comprises bonding stress distribution, cracking degree and the transfer length of the prestressed steel strand. Oh et al^[97,101]Firstly, the bonding stress-slip relation of the pre-tensioned prestressed member is deduced based on experimental data, and then a calculation formula of the transfer length is deduced by combining with a finite element. Ben₁' tez etc^[160]A bonding model of the steel wire and the concrete in the prestressed steel wire releasing and tensioning process is established, the relationship among the bonding stress, the stress of the steel bar and the concrete and the slippage is obtained through deduction, and a transfer length calculation formula considering the indentation depth of the steel wire and the thickness of the concrete protective layer is obtained. den Uijl et al^[106]Based on pull and push tests, the local bond stress is described as a function of local slip and bar stress variation. Based on the model, a bilinear relationship of the transfer length and the development length is simulated. Abdelatif et al^[96]By using a thickness ofWall cylinder theory, assuming that both concrete and rebar exhibit elastic material behavior. The prestress transfer in the pretensioned element is simulated by a force balance equation, a compatibility equation, and a bonding boundary condition. And continuously solving to obtain the distribution of the longitudinal stress and the radial stress of the reinforcing steel bar and obtain the transfer length. Balazs^[102]Based on a specific bonding stress-slip relation, a nonlinear equation aiming at the transfer length is established, and effective and initial prestress, concrete strength and steel strand size during transfer are considered. However, few studies have been made on the transmission length of the prestressed steel strand under corrosion.

Disclosure of Invention

The invention aims to solve the problems and provides a method for predicting the prestress transmission length of a pretensioned PC (polycarbonate) component under the condition of concrete rust cracking.

The method for predicting the prestress transmission length of the pretensioned PC member under the condition of concrete rust cracking comprises the following steps:

s1, determining relevant parameters of the thick-wall cylinder and the position size of the stirrup, determining relevant parameters of the thick-wall cylinder based on the structural size of a pretensioning method, wherein the relevant parameters comprise the inner radius and the outer radius of the cylinder, the compression resistance and the tensile strength of concrete, the elastic modulus of concrete and the Poisson ratio, and relevant parameters of the prestressed steel strand comprise a nominal initial radius, an initial tensioning prestress, a nominal tensile strength, an elastic modulus and a Poisson ratio, and for a test piece with the stirrup, the radius position of the stirrup in the cylinder, the radius, the cross section area, the tensile strength, the elastic modulus and the stirrup spacing parameter are also required to be determined;

s2: calculating the initial prestress f before the prestressed steel strand is released_pjRadius under action R_j；

S3: calculating the stretching stress f of the steel strand in the initial prestress range_pzCorresponding concrete tensile stress f under the action of_cz；

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

s5: calculating the radial displacement u of the contact surface_jAnd calculating the radial strain epsilon of the contact surface_θ(R_j)；

S6: the calculated radial strain epsilon of the contact surface_θ(R_j) Tensile strain at crack of concrete_ctBy contrast, if ε_θ(R_j)>ε_ctGo to the next step if ε_θ(R_j)≤ε_ctSolving to obtain bonding stress tau;

the radial strain ε of the contact surface obtained by calculation in S5_θ(R_j) And tensile strain epsilon when reaching ultimate tensile stress with concrete_ctComparing, judging the cracking condition of the contact surface concrete, if the former is smaller than the latter, indicating that the contact surface concrete is not cracked, and directly obtaining the bonding stress between the steel strand and the concrete under the strain through the formula (5.10);

s7: solving the radial strain epsilon of the contact surface under the non-rusted state_c(R_j) Exceeding the cracking tensile strain epsilon of concrete_ctRadius of time fracture front_c；

When the radial strain ε of the contact surface is calculated in S5_c(R_j) Greater than the tensile strain epsilon of the concrete when the ultimate tensile stress is reached_ctWhen the concrete at the contact surface is cracked, the concrete protective layer is in a partially cracked or completely cracked state, the concrete begins to show a softening behavior at the moment, the tensile behavior of the concrete needs to be considered again, and the radius R of the front edge of the crack caused by releasing and stretching the prestress is firstly considered by the formula (5.12b)_cSolving is carried out;

s8: solving the corrosion invasion depth x corresponding to the filled concrete crack_cThe crack front radius R of the prestressed steel strand obtained in S7_cOn the basis of the depth x of corrosion penetration required to fill the crack_cSolving is carried out, when x is less than or equal to x_cIn the process, the corrosion of the steel strand cannot cause additional displacement on the contact surface, and the radial stress p of the contact surface cannot be changed, the radial pressure at the moment is consistent with that of the steel strand which is not corroded, and when x is equal to x, the radial stress p is not changed>x_cWhen the corrosion of the steel strand begins to cause the concrete of the contact surface to generate additional radial displacement u_r；

S9: solving for rust depth x>x_cCrack under the conditionRadius of leading edge R_c2To solve for the rust depth x>x_cArbitrary depth x of corrosion and radius R of crack front under the condition_c2And thus determining that the crack penetrates completely through the cylinder, i.e. when R_c2Equal to the peripheral radius R of the cylinder₀Critical rust penetration depth x_crWhen calculated R_c2Exceeds the peripheral radius R₀When, get R_c2＝R₀；

S10: solving concrete strain epsilon corresponding to crack front edge radius_θ(r) Strain ε to two control points₁And ε_uRadius R of time₁And R_uBased on the radius R of the fracture front_c2Whether or not the peripheral radius R is exceeded₀And is divided into two steps, the first step, when R is_c2<R₀When the compound is represented by the formula (5.16) to R₁And R_uCalculating; second step, when R is_c2>R₀When the compound is represented by the formula (5.26) to R₁And R_uPerforming calculation if R is present in the calculation process₁Or R_uCalculated value of (a) exceeds the cylinder peripheral radius R₀When, get R₀A value of (d);

s11: solving for rust intrusion depth x<x_crI.e. the radial pressure p of the contact surface in the partially cracked state of the protective layer, the concrete restraint p of the cracked concrete front is first calculated by the formula (5.21)_cObtaining the strain distribution epsilon of the concrete at the cracked part along the radial direction_θ(r) obtaining the radial compressive stress p of the contact surface at the partial cracking stage;

s12: solving for rust intrusion depth x>x_crI.e. the radial pressure p of the contact surface in the state of complete cracking of the protective layer;

s13: calculating the bonding stress tau at the non-cracked, partially cracked and fully cracked stages, and calculating deltaf_pzCalculating f_pz,n＝f_pjΔ z of time, calculating Δ ε_pz,n＝ε_cz,nCorresponding f_pz,nDetermining the corresponding value n_k，n_k<n, finding the transmission length l_tr＝n_k·Δz。

In particular, in steps S1-S5, the steel strand is twistedRadial displacement u generated by steel strand and concrete contact surface caused by expansion_jResulting in the tangential strain epsilon of the prestressed steel strand on the circumferential annular concrete in the releasing and tensioning process_q＝u_j/R_jWhen the tangential strain ε_qExceeding the ultimate tensile strain epsilon of the concrete_cr＝f_ct/E_cWhen the concrete is cracked, the concrete begins to be cracked to form cracked cracks, the cross section of the steel strand is reduced under the action of prestress, and the radius R of the steel strand after stress is reduced_jRadius R in the non-tensioned state_iThe following relationship is compared:

wherein f is_pj: initial tensile stress of the prestressed strand, generally the nominal tensile strength f of the strand_ps0.75 times of; e_p: the elastic modulus of the prestressed steel strand; v. of_p: the poisson ratio of the prestressed steel strands;

r obtained by the formula (5.1)_jThe 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 can generate displacement in the radial direction within the transmission length range due to the expansion effect of the steel strand in the process of releasing and tensioning the steel strand, and the displacement is the difference value of the radii of the steel strand before and after tensioning, namely R at the free end of the member_i-R_jAnd gradually reduce toward the transfer length direction, until reaching the transfer length end and being zero, because the radial displacement of steel strand wires, lead to its surrounding concrete to produce the same radial displacement, and consequently produce radial compressive stress at steel strand wires and concrete interface, and produce hoop tensile stress to concrete tangential direction, according to the size of this radial compressive stress and hoop tensile stress, this concrete thick wall cylinder can divide into following three stage according to the fracture condition of concrete: three stages of non-cracking, partial cracking and complete cracking; evaluating the relation between the radial displacement and the radial stress of the contact surface at different stages under the non-corrosion state and the corrosion state of the steel strand respectively;

for non-rusting prestressed steel strands, according to Oh et al^[97]The radial displacement u of the contact surface of the prestressed steel strand and the concrete in a polar coordinate system in the releasing and tensioning process_jCan be expressed as:

wherein E is_cThe modulus of elasticity of the concrete;

v_cthe Poisson's ratio of the concrete;

f_czlongitudinal stress of the concrete;

p, radial pressure generated on the contact surface due to the expansion of the steel strand when the prestressed steel strand is released,

wherein the last two terms f_czAnd p may be represented as:

wherein f is_pzStress in the pre-stressed steel strand;

A_pthe cross section area of the prestressed steel strand;

a is the cross sectional area of the whole concrete section;

i, moment of inertia of the concrete section;

and e, the eccentricity from the prestressed steel strand to the center of the cross section of the concrete.

In particular, in the steps S6-S8, in the case that the constraint stress p of the pretensioned steel strand of the pretensioned member is known under different corrosion degrees, 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)

wherein mu is a friction coefficient and can be regarded as a constant under a specific corrosion rate; p is the radial pressure acting on the surface of the steel strand, 5.2 sections are used for calculation, and the bonding stress distribution can be obtained on the premise that the friction coefficients mu and p are known;

mu is taken as the friction coefficient of the prestressed steel strand and the concrete, is determined by the surface condition of the contact surface of the steel strand and the concrete, and changes along with the corrosion degree of the steel strand, and is consistent with the calculation of the friction coefficient mu in the previous section of the bond strength model, namely mu (rho)_p)＝0.343-0.26(x-x_cr) Wherein x is_crIs the corresponding corrosion invasion depth x when the concrete protective layer is completely cracked_cr＝0.031；

Radial pressure p of contact surface, constraint stress p of concrete at crack front_cResidual tensile stress sigma of concrete in tangential direction_θ(r) and sigma under hoop tensile stress_stThe relationship of (r) is as follows:

assuming that the displacement of the concrete in the radial direction in the stress process is linear elasticity, the radial displacement u (r) and the tangential strain epsilon are at any radial radius r of the cylinder_θ(R) and crack front radius R_cThe relationship of (c) is as follows:

in the case of partial cracking of the concrete, R is added_jThe radius R of the crack front edge of the rustless steel strand after the unwinding is finished can be obtained firstly by replacing R in the formula (5.11a) and being equal to the formula (5.2)_cThe calculation expression of (1):

for the additional displacement u of the contact surface caused by the corrosion of the steel strand_r|_r＝RjIt can be expressed as:

when the formula (5.10) and the formula (5.13) are equal to each other, the depth x of corrosion intrusion can be obtained>x_cDepth of corrosion intrusion x and radius of crack front after corrosion R_c2The relationship of (c) is as follows:

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

Particularly, in the steps S9-S12, concrete cracks caused by strand contraction develop to the surface of the member, namely the stress condition after the concrete protective layer is completely cracked, by combining R₀R in alternative formula (5.11)_cThe displacement u (r) and the corresponding shear strain epsilon of the concrete at any radius r can be obtained_θ(r) is as follows:

wherein epsilon_θcIs the tangential strain of the edge of the protective layer after the concrete is completely cracked, R in the formula (5.22a) is R_jInstead of beingRadial displacement of contact surface after concrete is completely cracked, wherein the radial displacement is the displacement u caused by tension release_jAnd additional displacement u caused by corrosion of steel strand_rTaken together, the following expression can be obtained:

thus, the tangential stress ε of the edge of the concrete protective layer_θcThe calculation can be made by equation (5.23):

and u at this time_rThe crack front radius R can be determined by the formula (5.10)_c2By the peripheral radius R of the cylinder₀Instead, the following can be obtained:

the tangential strain distribution of the concrete protective layer along the tangential direction of the protective layer after the concrete protective layer is completely cracked and the residual tensile stress sigma of the cracked concrete cylinder can be obtained through the formula (5.22b)_θCan be obtained by the formula (5.19) where the critical strain ε₁And ε_uCorresponding radius R of₁And R_uAnd epsilon_θcCan be determined by relating epsilon to₁0.0003 and ε_uThe formula (5.22b) can be determined as 0.002:

the concrete is not provided with the constraint stress p after being completely cracked_cTherefore, the formula (5.18) can be simplified to：

The formula (5.27) can be used for calculating the expansion stress p of the contact surface in the complete cracking state of the concrete,

and the acting force of the stirrup at the moment is as follows:

the method comprises the following steps:

after the prestress releasing and tensioning of the prestress pre-tensioning method member is finished, the influence of corrosion of the steel strand is considered, the obtained concrete is not cracked, is partially cracked and is in a complete cracking state, the radial pressure of the contact surface of the steel strand and the concrete is calculated, the relation between the corrosion invasion depth x and the radial stress p at different positions in the transmission length range can be deduced based on the process, and then the transmission length under the influence of corrosion is calculated based on the deduced relation.

In particular, in step S13 — yes S18, in the case that the constraint stress p of the pretensioned steel strand of the pretensioned member is known under different corrosion degrees, 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)

wherein mu is a friction coefficient and can be regarded as a constant under a specific corrosion rate; p is the radial pressure acting on the surface of the steel strand, 5.2 sections are used for calculation, and the bonding stress distribution can be obtained on the premise that the friction coefficients mu and p are known,

mu is taken as the friction coefficient of the prestressed steel strand and the concrete, is determined by the surface condition of the contact surface of the steel strand and the concrete, and changes along with the corrosion degree of the steel strand, and is consistent with the calculation of the friction coefficient mu in the previous section of the bond strength modelI.e. mu (p)_p)＝0.343-0.26(x-x_cr) Wherein x is_crIs the corresponding corrosion invasion depth x when the concrete protective layer is completely cracked_cr＝0.031；

Assuming that the bonding stress within each range of the length of Δ z is uniformly distributed, the prestress increment Δ f of the prestressed steel strand accumulated by the bonding stress on each micro-segment_pzCan be expressed as:

considering that the prestress at the end position of the prestressed member is 0, and assuming the strain change Δ ∈ of the steel strand_pzWith initial pre-strain_piTo be uniform, the stress f of the prestressed steel strand is thus at any nth Δ z length_pz,nAnd strain variable Δ ε_pz,nCan be respectively calculated as:

substituting the steel strand stress obtained by the calculation of the formula (5.31) into the formula (5.3), and calculating the concrete strain epsilon at the nth delta z length_cz,n，

When the strain variable Delta epsilon of the steel strand is calculated by the formula (5.32)_pz,nAnd the concrete strain ε calculated by the formula (5.33)_cz,nWhen the positions are equal, the steel strand and the concrete do not move any more, and the distance from the position to the end part of the member is the transmission length l of the prestressed steel strand_tr。

The invention has the following advantages and positive effects: in the releasing and tensioning process of the prestressed steel strands, the radial displacement between the steel strands and the concrete contact surface is caused by the retraction effect of the prestressed steel strands, so that the circumferential tensile stress is formed around the concrete. When the tensile stress exceeds the tensile strength of the concrete, cracking of the interface concrete occurs. In addition, when the prestressed steel strand is corroded, the constraint condition of the steel strand is further changed by the expansion of corrosion products. So that the concrete which is not cracked before develops partial cracking, and the concrete which partially cracks develops in the direction of completely cracking the concrete. Therefore, the method for determining the transmission length of the rusted steel strand is obtained by analyzing the stress distribution of the prestressed reinforcement under the influence of the corrosion and the stress distribution of the surrounding concrete based on the development of the concrete protective layer crack caused by the relaxation and corrosion of the prestress in the pre-tensioned prestressed member, deducing the constraint stress distribution and the bonding stress distribution of the steel strand within the transmission length range under the influence of the corrosion and further obtaining the method for determining the transmission length of the rusted steel strand.

Drawings

FIG. 1 is a schematic diagram of concrete cracking caused by the transmission of pre-tensioned structural pre-stress.

Fig. 2 is a thick-walled cylinder model of a rusted steel strand.

FIG. 3 is a graph showing the relationship between stress and strain after cracking of concrete.

FIG. 4 is an equation of equilibrium for a partially cracked condition of concrete.

FIG. 5 is an equation of equilibrium for a complete concrete crack condition.

Fig. 6 is a pre-tensioned prestressed structural decomposition.

Detailed Description

In order to further understand the contents, features and effects of the present invention, the following embodiments are described in detail with reference to the accompanying drawings.

The invention is described below with reference to fig. 1: the method for predicting the prestress transmission length of the pretensioned PC member under the condition of concrete rust cracking comprises the following steps:

s1, determining relevant parameters of the thick-wall cylinder and the position size of the stirrup, determining relevant parameters of the thick-wall cylinder based on the structural size of a pretensioning method, wherein the relevant parameters comprise the inner radius and the outer radius of the cylinder, the compression resistance and the tensile strength of concrete, the elastic modulus of concrete and the Poisson ratio, and relevant parameters of the prestressed steel strand comprise a nominal initial radius, an initial tensioning prestress, a nominal tensile strength, an elastic modulus and a Poisson ratio, and for a test piece with the stirrup, the radius position of the stirrup in the cylinder, the radius, the cross section area, the tensile strength, the elastic modulus and the stirrup spacing parameter are also required to be determined;

s2: calculating the initial prestress f before the prestressed steel strand is released_pjRadius under action R_j；

S3: calculating the stretching stress f of the steel strand in the initial prestress range_pzCorresponding concrete tensile stress f under the action of_cz；

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

s5: calculating the radial displacement u of the contact surface_jAnd calculating the radial strain epsilon of the contact surface_θ(R_j)；

S6: the calculated radial strain epsilon of the contact surface_θ(R_j) Tensile strain at crack of concrete_ctBy contrast, if ε_θ(R_j)>ε_ctGo to the next step if ε_θ(R_j)≤ε_ctSolving to obtain bonding stress tau;

the radial strain ε of the contact surface obtained by calculation in S5_θ(R_j) And tensile strain epsilon when reaching ultimate tensile stress with concrete_ctComparing, judging the cracking condition of the contact surface concrete, if the former is smaller than the latter, indicating that the contact surface concrete is not cracked, and directly obtaining the bonding stress between the steel strand and the concrete under the strain through the formula (5.10);

s7: solving the radial strain epsilon of the contact surface under the non-rusted state_c(R_j) Exceeding the cracking tensile strain epsilon of concrete_ctRadius of time fracture front_c；

When the radial strain ε of the contact surface is calculated in S5_c(R_j) Greater than the tensile strain epsilon of the concrete when the ultimate tensile stress is reached_ctIndicating contact surface mixingWhen the concrete is cracked and the concrete protective layer is in a partially cracked or completely cracked state, the concrete begins to show softening behavior, the tensile behavior of the concrete needs to be considered again, and firstly, the radius R of the front edge of the crack caused by releasing and stretching prestress is determined by the formula (5.12b)_cSolving is carried out;

s8: solving the corrosion invasion depth x corresponding to the filled concrete crack_cThe crack front radius R of the prestressed steel strand obtained in S7_cOn the basis of the depth x of corrosion penetration required to fill the crack_cSolving is carried out, when x is less than or equal to x_cIn the process, the corrosion of the steel strand cannot cause additional displacement on the contact surface, and the radial stress p of the contact surface cannot be changed, the radial pressure at the moment is consistent with that of the steel strand which is not corroded, and when x is equal to x, the radial stress p is not changed>x_cWhen the corrosion of the steel strand begins to cause the concrete of the contact surface to generate additional radial displacement u_r；

S9: solving for rust depth x>x_cRadius of the leading edge of the crack R_c2To solve for the rust depth x>x_cArbitrary depth x of corrosion and radius R of crack front under the condition_c2And thus determining that the crack penetrates completely through the cylinder, i.e. when R_c2Equal to the peripheral radius R of the cylinder₀Critical rust penetration depth x_crWhen calculated R_c2Exceeds the peripheral radius R₀When, get R_c2＝R₀；

S10: solving concrete strain epsilon corresponding to crack front edge radius_θ(r) Strain ε to two control points₁And ε_uRadius R of time₁And R_uBased on the radius R of the fracture front_c2Whether or not the peripheral radius R is exceeded₀And is divided into two steps, the first step, when R is_c2<R₀When the compound is represented by the formula (5.16) to R₁And R_uCalculating; second step, when R is_c2>R₀When the compound is represented by the formula (5.26) to R₁And R_uPerforming calculation if R is present in the calculation process₁Or R_uCalculated value of (a) exceeds the cylinder peripheral radius R₀When, get R₀A value of (d);

s11: solution to rustDepth of erosion penetration x<x_crI.e. the radial pressure p of the contact surface in the partially cracked state of the protective layer, the concrete restraint p of the cracked concrete front is first calculated by the formula (5.21)_cObtaining the strain distribution epsilon of the concrete at the cracked part along the radial direction_θ(r) obtaining the radial compressive stress p of the contact surface at the partial cracking stage;

s12: solving for rust intrusion depth x>x_crI.e. the radial pressure p of the contact surface in the state of complete cracking of the protective layer;

s13: calculating the bonding stress tau at the non-cracked, partially cracked and fully cracked stages, and calculating deltaf_pzCalculating f_pz,n＝f_pjΔ z of time, calculating Δ ε_pz,n＝ε_cz,nCorresponding f_pz,nDetermining the corresponding value n_k，n_k<n, finding the transmission length l_tr＝n_k·Δz。

In step S1, the plastic sheet preparation planar development is printed with tiny squares through transparent material or is directly molded with a plastic suction mold to hook the contour line of the product.

In the present embodiment, it is preferred that,

in the releasing and tensioning process of the prestressed steel strands, radial displacement between the steel strands and the concrete contact surface is caused due to the retraction effect of the prestressed steel strands, and therefore hoop tensile stress is formed around the concrete. When the tensile stress exceeds the tensile strength of the concrete, cracking of the interface concrete occurs. In addition, when the prestressed steel strand is corroded, the constraint condition of the steel strand is further changed by the expansion of corrosion products. So that the concrete which is not cracked before develops partial cracking, and the concrete which partially cracks develops in the direction of completely cracking the concrete. Therefore, the method for determining the transmission length of the rusted steel strand is obtained by analyzing the stress distribution of the prestressed reinforcement under the influence of the corrosion and the stress distribution of the surrounding concrete based on the development of the concrete protective layer crack caused by the relaxation and corrosion of the prestress in the pre-tensioned prestressed member, deducing the constraint stress distribution and the bonding stress distribution of the steel strand within the transmission length range under the influence of the corrosion and further obtaining the method for determining the transmission length of the rusted steel strand.

In the pretensioning prestressed component, the longitudinal cross-sectional area of the steel strand is reduced under the action of prestress due to the Poisson effect. And in the prestress releasing and tensioning process, the cross section of the prestress steel strand can be recovered to the original cross section along with the releasing and tensioning of the prestress. However, this expansion is limited by the constraining action of the concrete, thereby creating hoop compressive stress around the prestressed steel strands. Considering the peripheral concrete of the prestressed steel strand as a thick-wall cylinder, the concrete cracking development caused by the prestress in the concrete transfer process in the tension releasing process is shown in fig. 1.

R_iInitial radius of the unstretched prestressed steel strand;

R_jradius before releasing the prestressed steel strand;

R_cthe distance from the center of the prestressed steel strand to the front edge of the crack;

R₀the distance from the center of the prestressed steel strand to the thickness edge (the outer edge of the cylinder) of the protective layer;

u_jthe prestressed steel strand deforms in the radial direction when the prestress is released.

Radial displacement u generated on the contact surface of the steel strand and the concrete due to the expansion of the steel strand_jResulting in the tangential strain epsilon of the prestressed steel strand on the circumferential annular concrete in the releasing and tensioning process_q＝u_j/R_jWhen the tangential strain ε_qExceeding the ultimate tensile strain epsilon of the concrete_cr＝f_ct/E_cWhen the concrete is used, the concrete begins to generate cleavage, and a cleavage crack is formed. At the moment, the cross section of the steel strand is reduced due to the action of prestress, and the radius R of the stressed steel strand is_jRadius R in the non-tensioned state_iThe following relationship is compared:

wherein f is_pj: initial tensile stress of the prestressed strand, generally the nominal tensile strength f of the strand_ps0.75 times of;

E_p: prestressed steel strandLinear modulus of elasticity;

v_p: the poisson ratio of the prestressed steel strands;

r obtained by the formula (5.1)_jThe 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 tensile force. During the process of releasing and tensioning the steel strand, the steel strand can displace in the radial direction within the transmission length range due to the influence of the expansion of the steel strand. The displacement is the difference of the radii of the steel strand before and after tensioning at the free end of the member, namely R_i-R_j. And gradually decreases in the direction of the transfer length until the end of the transfer length is zero. Due to the radial displacement of the steel strands, the concrete around the steel strands generates the same radial displacement, and therefore radial compressive stress is generated on the contact surface of the steel strands and the concrete. And generates hoop tensile stress to the concrete in the tangential direction. According to the magnitude of the radial compressive stress and the hoop tensile stress, the concrete thick-wall cylinder can be divided into the following three stages according to the cracking condition of concrete: namely, three stages of non-cracking, partial cracking and complete cracking. The relationship between the radial displacement and the radial stress of the contact surface at different stages under the non-corrosion state and the corrosion state of the steel strand is evaluated respectively.

For non-rusting prestressed steel strands, according to Oh et al^[97]The radial displacement u of the contact surface of the prestressed steel strand and the concrete in a polar coordinate system in the releasing and tensioning process_jCan be expressed as:

wherein E is_cThe modulus of elasticity of the concrete;

v_cthe Poisson's ratio of the concrete;

f_czlongitudinal stress of the concrete;

and p, generating radial pressure on the contact surface due to the expansion of the steel strand when the prestressed steel strand is released.

Wherein the last two terms f_czAnd p may be represented as:

wherein f is_pzStress in the pre-stressed steel strand;

A_pthe cross section area of the prestressed steel strand;

a is the cross sectional area of the whole concrete section;

i, moment of inertia of the concrete section;

and e, the eccentricity from the prestressed steel strand to the center of the cross section of the concrete.

For rusting the steel strand, the additional stress of the rust generated by the rusting of the steel strand must be considered. The corrosion product generated by the corrosion of the steel strand is increased in volume, so that the contact surface generates additional radial displacement u_rResulting in a change in the confining stress around the steel strand. In the section of calculating the bonding strength, the corrosion of the outer wire and the inner wire of the steel strand is uniform, and the corrosion penetration depth or the radius loss of a single steel wire is x. Neglecting the difference in diameter of the inner and outer wires of the steel strand, i.e. d_a＝d_b. Radius R of single steel wire after uniform corrosion_bs＝R_b-x，R_bThe radius of the untarnished outer wire. Obtaining the corrosion rate rho_pThe relationship to the rust depth x is:

according to the corrosion characteristics of the steel strand, assuming that gaps between the inner wire and the outer wire of the steel strand are filled with corrosion products of the inner wire and do not diffuse to the outside of the steel strand, only considering the expansion effect of the corrosion products of the outer wire of the steel strand on external concrete, and obtaining the volume loss of the rusted steel strand per unit length

To simplify the calculations, it was assumed that the outer wire rust product evenly covered the circumference of the nominal diameter of the steel strand, as shown in fig. 2.

The development of corrosion products of the steel strand here differs from that in the preceding section. Since the prestressed steel strand causes partial concrete cracking after the prestressed steel strand is completely transmitted, it is assumed here that the prestressed reinforcement corrosion product will fill the concrete crack caused by the prestressed steel strand expansion first, and therefore the following expression can be obtained:

where Σ w is the radius R_jThe total width of concrete crack caused by the tension of steel strand is controlled, sigma w is 2 pi.u_r| _r＝Rj＝2π·u_j。u_jThe calculation is performed by equation (5.2). m is the expansion coefficient of the corrosion product, and the value is different according to different corrosion products^[142]。R_cThe radius of the front edge of a crack in concrete caused by the tension of a steel strand in a non-rusty state.

Assume the critical rust depth x required to fill the initial crack width_cThen equation (5.6) can be further simplified as:

equation (5.7) relates to the depth of rust penetration x_cA unary quadratic function of, get x_cThe calculation expression of (a) is:

at this time, the depth of the rust can be determined according to x and x_cThe relationship (2) determines whether additional displacement is generated between the steel strand and the concrete due to corrosion of the steel strand. When the rust depth x does not exceed x_cWhen x is less than or equal to x_cThe steel strand and the concrete contact surface are not rusted at the momentAdditional displacements occur and therefore no additional stress of the rust is generated. When the corrosion depth x exceeds x_cWhen is x>x_cStarting to generate an additional displacement u_rAnd thus the concrete cracks previously caused by the tension release continue to develop outward. Let the radius of the corresponding crack front be R_c2Volume and rust penetration depth x and crack front radius R caused by strand rust_c2The relationship of (a) is as follows:

in the formula, R_rThe radius of the outer edge corrosion product after the steel strand is corroded; Σ w is the radius R_rThe total width of the rusty crack, ∑ w ═ 2 pi · u_r|_r＝R0＝2π(R_r-R_j)。R_r＝R_j+u_r. The radius R of the contact surface can be obtained by solving the formula (5.9)_jDisplacement u of_r|_r＝RjExpressed as:

in the formula, R_jNominal radius of the untrusted steel strand.

Assuming that the displacement of the concrete in the radial direction in the stress process is linear elasticity, the radial displacement u (r) and the tangential strain epsilon are at any radial radius r of the cylinder_θ(R) and crack front radius R_cThe relationship of (c) is as follows:

in the case of partial cracking of the concrete, R is added_jInstead of r in the formula (5.11a), and(5.2) equal radius R of crack front after the non-rusted steel strand is released and tensioned can be obtained firstly_cThe calculation expression of (1):

for the additional displacement u of the contact surface caused by the corrosion of the steel strand_r|_r＝RjIt can be expressed as:

when the formula (5.10) and the formula (5.13) are equal to each other, the depth x of corrosion intrusion can be obtained>x_cDepth of corrosion intrusion x and radius of crack front after corrosion R_c2The relationship of (c) is as follows:

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

For a cracked concrete cylinder, the concrete can be softened after cracking. Is expressed by concrete stress sigma_θ(r) Strain with concrete ∈_θ(r) increases and decreases rapidly, and above a certain value the decrease becomes gradual until the concrete loses its tensile capacity completely above a certain critical strain. In prior studies, the strain ε was determined by pulling the concrete tangentially_θ(r) is a variable, and the relation expression of stress and strain of cracked concrete is as follows:

wherein epsilon₁And ε_uThe values are 0.0003 and 0.002 respectively.

Assuming that the concrete strain respectively reaches critical strain epsilon₁And ε_uRespectively has a radius of R₁And R_uIf, as shown in FIG. 3, then ε will be₁0.0003 and ε_uR can be obtained by substituting 0.002 into the formula (5.12b)₁And R_uRadius R of front edge of crack_c2The relationship of (1):

when two critical radii R are calculated by the formula (5.16)₁And R_uIs lower than the radius R of the prestressed steel strand after being tensioned_jWhen the stress is higher than the critical stress, the cracking strain value of the concrete is not higher than the critical strain, and R is higher than R₁And R_uBy direct application of R_jAnd (4) replacing.

In addition, for the thick-walled cylinder with the stirrup, assuming that the stirrup is circular around the steel strand, the distance from the center of the stirrup to the center of the steel strand, namely the radius of the position where the stirrup is located, is R_s. As shown in fig. 4. At this time, the strain of the stirrup is considered to be consistent with the strain of the concrete, and R is expressed by the formula (5.11b)_sSubstituting into r. Then stirrupTangential tensile stress σ of_stCan be expressed as:

and then calculating the radial compressive stress of the contact surface after the concrete is partially cracked.

As shown in the figure, the radial pressure p of the contact surface and the constraint stress p of the concrete at the front of the crack_cResidual tensile stress sigma of concrete in tangential direction_θ(r) and sigma under hoop tensile stress_stThe relationship of (r) is as follows:

wherein,

obtainable by combining formula (5.11b), formula (5.15) and formula (5.16) as follows:

while

The method comprises the following steps:

in the formula, R_stIs the radius of the stirrup; d_stIs the diameter D of the stirrup_st＝2R_st；A_stIs the cross-sectional area A of the stirrup_st＝πR_st ²；D_jDiameter D of steel strand after being prestressed and tensioned_j＝2R_j；S_vThe distance between the stirrups; r_sThe distance from the center point of the stirrup to the center point of the steel strand;

in addition, the tensile stress σ of the fracture cracking front_θ(R_c) Tensile strength f to be compared with concrete_ctCoincidence, i.e. σ_θ(R_c) ＝f_ct. Thus the constraining stress p of the concrete at the crack front_cCan be expressed as:

constraint stress p calculated from equation (5.21)_cThe pressure p of the contact surface in the partial crack phase can be determined by substituting formula (5.18).

At the moment, concrete cracking caused by the shrinkage of the steel strands develops to the surface of the member, namely the stress condition after the concrete protective layer is completely cracked. By adding R₀R in alternative formula (5.11)_cThe displacement u (r) and the corresponding shear strain epsilon of the concrete at any radius r can be obtained_θ(r) is as follows:

wherein epsilon_θcIs the tangential strain at the edge of the protective layer after the concrete has completely cracked. R in the formula (5.22a) is represented by R_jThe replacement is radial displacement of the contact surface after the concrete is completely cracked, and the radial displacement at the moment is displacement u caused by tension release_jAnd additional displacement u caused by corrosion of steel strand_rTaken together, the following expression can be obtained:

thus, the tangential stress ε of the edge of the concrete protective layer_θcThe calculation can be made by equation (5.23):

and u at this time_rThe crack front radius R can be determined by the formula (5.10)_c2By the peripheral radius R of the cylinder₀Instead, the following can be obtained:

the tangential strain distribution along the tangential direction of the concrete protective layer after the concrete protective layer is completely cracked can be obtained by the formula (5.22 b). Residual tensile stress sigma of cracked concrete cylinder_θThis can be obtained by the formula (5.19). At this time, critical strain ∈₁And ε_uCorresponding radius R of₁And R_uAnd epsilon_θcCan be determined by relating epsilon to₁0.0003 and ε_uThe formula (5.22b) can be determined as 0.002:

the concrete is not provided with the constraint stress p after being completely cracked_cTherefore, equation (5.18) can be simplified as:

the formula (5.27) can be used for calculating the expansion stress p of the contact surface in the complete cracking state of the concrete.

And the acting force of the stirrup at the moment is as follows:

the method comprises the following steps:

the method is a calculation process of the radial pressure of the contact surface of the steel strand and the concrete in the states of no cracking, partial cracking and complete cracking of the obtained concrete by considering the influence of corrosion of the steel strand after the prestress releasing of the prestress pretensioning method member is finished. The relationship of rust penetration depth x to radial stress p at different locations within the range of transmission lengths can be derived based on this process. Next, the transfer length under the influence of rust is calculated based on the derived relationship.

Under the condition that the constraint stress p of the pretensioned member prestressed steel strand is known under different corrosion degrees, the bonding stress tau between the prestressed steel strand and the concrete can be expressed by the following basic control equation:

τ＝μ·p (5.29)

wherein mu is a friction coefficient and can be regarded as a constant under a specific corrosion rate; p is the radial pressure acting on the surface of the steel strand. The calculation is performed by section 5.2. The bonding stress distribution can be determined on the premise that the friction coefficients μ and p are known.

Mu is taken as the friction coefficient of the prestressed steel strand and the concrete, is determined by the surface condition of the contact surface of the steel strand and the concrete, and changes along with the corrosion degree of the steel strand. This is consistent with the calculation of the coefficient of friction μ in the previous section of the bond strength model herein, i.e., μ(ρ_p)＝0.343-0.26(x-x_cr). Wherein x is_crIs the corresponding corrosion invasion depth x when the concrete protective layer is completely cracked_cr＝0.031；

The prestressed concrete structure can be divided into n micro-segments of length Δ z in the longitudinal direction of the steel strand, as shown in fig. 6.

Assuming that the bonding stress within each range of the length of Δ z is uniformly distributed, the prestress increment Δ f of the prestressed steel strand accumulated by the bonding stress on each micro-segment_pzCan be expressed as:

considering that the prestress at the end position of the prestressed member is 0, and assuming the strain change Δ ∈ of the steel strand_pzWith initial pre-strain_piAre consistent. Thus, the stress f of the prestressed steel strand at any nth Δ z length_pz,nAnd strain variable Δ ε_pz,nCan be respectively calculated as:

substituting the steel strand stress obtained by the calculation of the formula (5.31) into the formula (5.3), and calculating the concrete strain epsilon at the nth delta z length_cz,n。

When the strain variable Delta epsilon of the steel strand is calculated by the formula (5.32)_pz,nAnd the concrete strain ε calculated by the formula (5.33)_cz,nWhen they are equal, the steel strand and the concrete are no longer displaced, and the distance from the position to the end of the memberThe separation is the transfer length l of the prestressed steel strand_tr。

The foregoing is illustrative of the present invention and is not to be construed as limiting thereof in any way, and any simple modifications, equivalent alterations and modifications of the above embodiments, which are obvious from the practice of the method of the present invention, are intended to be included within the scope of the method of the present invention.

Claims (5)

1. The method for predicting the prestress transmission length of the pretensioned PC component under the condition of concrete rust cracking is characterized by comprising the following steps of:

s1, determining relevant parameters of the thick-wall cylinder and the position size of the stirrup, determining relevant parameters of the thick-wall cylinder based on the structural size of a pretensioning method, wherein the relevant parameters comprise the inner radius and the outer radius of the cylinder, the compression resistance and the tensile strength of concrete, the elastic modulus of concrete and the Poisson ratio, and relevant parameters of the prestressed steel strand comprise a nominal initial radius, an initial tensioning prestress, a nominal tensile strength, an elastic modulus and a Poisson ratio, and for a test piece with the stirrup, the radius position of the stirrup in the cylinder, the radius, the cross section area, the tensile strength, the elastic modulus and the stirrup spacing parameter are also required to be determined;

s2: calculating the initial prestress f before the prestressed steel strand is released_pjRadius under action R_j；

S3: calculating the stretching stress f of the steel strand in the initial prestress range_pzCorresponding concrete tensile stress f under the action of_cz；

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

s5: calculating the radial displacement u of the contact surface_jAnd calculating the radial strain epsilon of the contact surface_θ(R_j)；

S6: the calculated radial strain epsilon of the contact surface_θ(R_j) Tensile strain at crack of concrete_ctBy contrast, if ε_θ(R_j)>ε_ctGo to the next step if ε_θ(R_j)≤ε_ctSolving to obtain bonding stress tau;

calculated in S5Radial strain epsilon of the contact surface_θ(R_j) And tensile strain epsilon when reaching ultimate tensile stress with concrete_ctComparing, judging the cracking condition of the contact surface concrete, if the former is smaller than the latter, indicating that the contact surface concrete is not cracked, and directly obtaining the bonding stress between the steel strand and the concrete under the strain through the formula (5.10);

s7: solving the radial strain epsilon of the contact surface under the non-rusted state_c(R_j) Exceeding the cracking tensile strain epsilon of concrete_ctRadius of time fracture front_c；

When the radial strain ε of the contact surface is calculated in S5_c(R_j) Greater than the tensile strain epsilon of the concrete when the ultimate tensile stress is reached_ctWhen the concrete at the contact surface is cracked, the concrete protective layer is in a partially cracked or completely cracked state, the concrete begins to show a softening behavior at the moment, the tensile behavior of the concrete needs to be considered again, and the radius R of the front edge of the crack caused by releasing and stretching the prestress is firstly considered by the formula (5.12b)_cSolving is carried out;

s8: solving the corrosion invasion depth x corresponding to the filled concrete crack_cThe crack front radius R of the prestressed steel strand obtained in S7_cOn the basis of the depth x of corrosion penetration required to fill the crack_cSolving is carried out, when x is less than or equal to x_cIn the process, the corrosion of the steel strand cannot cause additional displacement on the contact surface, and the radial stress p of the contact surface cannot be changed, the radial pressure at the moment is consistent with that of the steel strand which is not corroded, and when x is equal to x, the radial stress p is not changed>x_cWhen the corrosion of the steel strand begins to cause the concrete of the contact surface to generate additional radial displacement u_r；

S9: solving for rust depth x>x_cRadius of the leading edge of the crack R_c2To solve for the rust depth x>x_cArbitrary depth x of corrosion and radius R of crack front under the condition_c2And thus determining that the crack penetrates completely through the cylinder, i.e. when R_c2Equal to the peripheral radius R of the cylinder₀Critical rust penetration depth x_crWhen calculated R_c2Exceeds the peripheral radius R₀When, get R_c2＝R₀；

S10: solving concrete strain epsilon corresponding to crack front edge radius_θ(r) Strain ε to two control points₁And ε_uRadius R of time₁And R_uBased on the radius R of the fracture front_c2Whether or not the peripheral radius R is exceeded₀And is divided into two steps, the first step, when R is_c2<R₀When the compound is represented by the formula (5.16) to R₁And R_uCalculating; second step, when R is_c2>R₀When the compound is represented by the formula (5.26) to R₁And R_uPerforming calculation if R is present in the calculation process₁Or R_uCalculated value of (a) exceeds the cylinder peripheral radius R₀When, get R₀A value of (d);

s11: solving for rust intrusion depth x<x_crI.e. the radial pressure p of the contact surface in the partially cracked state of the protective layer, the concrete restraint p of the cracked concrete front is first calculated by the formula (5.21)_cObtaining the strain distribution epsilon of the concrete at the cracked part along the radial direction_θ(r) obtaining the radial compressive stress p of the contact surface at the partial cracking stage;

s12: solving for rust intrusion depth x>x_crI.e. the radial pressure p of the contact surface in the state of complete cracking of the protective layer;

s13: calculating the bonding stress tau at the non-cracked, partially cracked and fully cracked stages, and calculating deltaf_pzCalculating f_pz,n＝f_pjΔ z of time, calculating Δ ε_pz,n＝ε_cz,nCorresponding f_pz,nDetermining the corresponding value n_k，n_k<n, finding the transmission length l_tr＝n_k·Δz。

2. The method for predicting the prestress transmission length of a pretensioned PC member under concrete rust cracking according to claim 1, wherein:

in steps S1-S5, the radial displacement u generated on the contact surface of the steel strand and the concrete due to the expansion of the steel strand_jResulting in the tangential strain epsilon of the prestressed steel strand on the circumferential annular concrete in the releasing and tensioning process_q＝u_j/R_jWhen the tangential strain ε_qExceeding the ultimate tensile strain epsilon of the concrete_cr＝f_ct/E_cWhen the concrete is cracked, the concrete begins to be cracked to form cracked cracks, the cross section of the steel strand is reduced under the action of prestress, and the radius R of the steel strand after stress is reduced_jRadius R in the non-tensioned state_iThe following relationship is compared:

wherein f is_pj: initial tensile stress of the prestressed strand, generally the nominal tensile strength f of the strand_ps0.75 times of; e_p: the elastic modulus of the prestressed steel strand; v. of_p: the poisson ratio of the prestressed steel strands;

r obtained by the formula (5.1)_jThe 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 can generate displacement in the radial direction within the transmission length range due to the expansion effect of the steel strand in the process of releasing and tensioning the steel strand, and the displacement is the difference value of the radii of the steel strand before and after tensioning, namely R at the free end of the member_i-R_jAnd gradually reduce toward the transfer length direction, until reaching the transfer length end and being zero, because the radial displacement of steel strand wires, lead to its surrounding concrete to produce the same radial displacement, and consequently produce radial compressive stress at steel strand wires and concrete interface, and produce hoop tensile stress to concrete tangential direction, according to the size of this radial compressive stress and hoop tensile stress, this concrete thick wall cylinder can divide into following three stage according to the fracture condition of concrete: three stages of non-cracking, partial cracking and complete cracking; evaluating the relation between the radial displacement and the radial stress of the contact surface at different stages under the non-corrosion state and the corrosion state of the steel strand respectively;

for non-rusting prestressed steel strands, according to Oh et al^[97]The prestressed steel strand is in contact with the concrete in a polar coordinate system in the releasing and tensioning processRadial displacement u_jCan be expressed as:

wherein E is_cThe modulus of elasticity of the concrete;

v_cthe Poisson's ratio of the concrete;

f_czlongitudinal stress of the concrete;

p, radial pressure generated on the contact surface due to the expansion of the steel strand when the prestressed steel strand is released,

wherein the last two terms f_czAnd p may be represented as:

wherein f is_pzStress in the pre-stressed steel strand;

A_pthe cross section area of the prestressed steel strand;

a is the cross sectional area of the whole concrete section;

i, moment of inertia of the concrete section;

and e, the eccentricity from the prestressed steel strand to the center of the cross section of the concrete.

3. The method for predicting the prestress transmission length of a pretensioned PC member under concrete rust cracking according to claim 1, wherein:

in steps S6-S8, in the case that the constraint stress p of the pretensioned steel strand of the pretensioned member is known under different corrosion degrees, 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)

wherein mu is a friction coefficient and can be regarded as a constant under a specific corrosion rate; p is the radial pressure acting on the surface of the steel strand, 5.2 sections are used for calculation, and the bonding stress distribution can be obtained on the premise that the friction coefficients mu and p are known;

mu is taken as the friction coefficient of the prestressed steel strand and the concrete, is determined by the surface condition of the contact surface of the steel strand and the concrete, and changes along with the corrosion degree of the steel strand, and is consistent with the calculation of the friction coefficient mu in the previous section of the bond strength model, namely mu (rho)_p)＝0.343-0.26(x-x_cr) Wherein x is_crIs the corresponding corrosion invasion depth x when the concrete protective layer is completely cracked_cr＝0.031；

Radial pressure p of contact surface, constraint stress p of concrete at crack front_cResidual tensile stress sigma of concrete in tangential direction_θ(r) and sigma under hoop tensile stress_stThe relationship of (r) is as follows:

assuming that the displacement of the concrete in the radial direction in the stress process is linear elasticity, the radial displacement u (r) and the tangential strain epsilon are at any radial radius r of the cylinder_θ(R) and crack front radius R_cThe relationship of (c) is as follows:

in the case of partial cracking of the concrete, R is added_jThe radius R of the crack front edge of the rustless steel strand after the unwinding is finished can be obtained firstly by replacing R in the formula (5.11a) and being equal to the formula (5.2)_cThe calculation expression of (1):

for the additional displacement u of the contact surface caused by the corrosion of the steel strand_r|_r＝RjIt can be expressed as:

when the formula (5.10) and the formula (5.13) are equal to each other, the depth x of corrosion intrusion can be obtained>x_cDepth of corrosion intrusion x and radius of crack front after corrosion R_c2The relationship of (c) is as follows:

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

4. The method for predicting the prestress transmission length of a pretensioned PC member under concrete rust cracking according to claim 1, wherein:

in the steps S9-S12, concrete cracking caused by stranded wire shrinkage develops to the surface of the member, namely the stress condition after the concrete protective layer is completely cracked, by adding R₀R in alternative formula (5.11)_cThe displacement u (r) and the corresponding shear strain epsilon of the concrete at any radius r can be obtained_θ(r) is as follows:

wherein epsilon_θcIs the tangential strain of the edge of the protective layer after the concrete is completely cracked, R in the formula (5.22a) is R_jThe replacement is radial displacement of the contact surface after the concrete is completely cracked, and the radial displacement at the moment is displacement u caused by tension release_jAnd additional displacement u caused by corrosion of steel strand_rTaken together, the following expression can be obtained:

thus, the tangential stress ε of the edge of the concrete protective layer_θcThe calculation can be made by equation (5.23):

and u at this time_rThe crack front radius R can be determined by the formula (5.10)_c2By the peripheral radius R of the cylinder₀Instead, the following can be obtained:

the tangential strain distribution of the concrete protective layer along the tangential direction of the protective layer after the concrete protective layer is completely cracked and the residual tensile stress sigma of the cracked concrete cylinder can be obtained through the formula (5.22b)_θCan be obtained by the formula (5.19) where the critical strain ε₁And ε_uCorresponding radius R of₁And R_uAnd epsilon_θcCan be determined by relating epsilon to₁0.0003 and ε_uThe formula (5.22b) can be determined as 0.002:

the concrete is not provided with the constraint stress p after being completely cracked_cTherefore, equation (5.18) can be simplified as:

the formula (5.27) can be used for calculating the expansion stress p of the contact surface in the complete cracking state of the concrete,

and the acting force of the stirrup at the moment is as follows:

the method comprises the following steps:

after the prestress releasing and tensioning of the prestress pre-tensioning method member is finished, the influence of corrosion of the steel strand is considered, the obtained concrete is not cracked, is partially cracked and is in a complete cracking state, the radial pressure of the contact surface of the steel strand and the concrete is calculated, the relation between the corrosion invasion depth x and the radial stress p at different positions in the transmission length range can be deduced based on the process, and then the transmission length under the influence of corrosion is calculated based on the deduced relation.

5. The method for predicting the prestress transmission length of a pretensioned PC member under concrete rust cracking according to claim 1, wherein:

in step S13 — yes S18, in the case that the constraint stress p of the pretensioned steel strand of the pretensioned member is known under different corrosion degrees, 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)

wherein mu is a friction coefficient and can be regarded as a constant under a specific corrosion rate; p is the radial pressure acting on the surface of the steel strand, 5.2 sections are used for calculation, and the bonding stress distribution can be obtained on the premise that the friction coefficients mu and p are known,

mu is taken as the friction coefficient of the prestressed steel strand and the concrete, is determined by the surface condition of the contact surface of the steel strand and the concrete, and changes along with the corrosion degree of the steel strand, and is consistent with the calculation of the friction coefficient mu in the previous section of the bond strength model, namely mu (rho)_p)＝0.343-0.26(x-x_cr) Wherein x is_crIs the corresponding corrosion invasion depth x when the concrete protective layer is completely cracked_cr＝0.031；

Assuming that the bonding stress within each range of the length of Δ z is uniformly distributed, the prestress increment Δ f of the prestressed steel strand accumulated by the bonding stress on each micro-segment_pzCan be expressed as:

considering that the prestress at the end position of the prestressed member is 0, and assuming the strain change Δ ∈ of the steel strand_pzWith initial pre-strain_piTo be uniform, the stress f of the prestressed steel strand is thus at any nth Δ z length_pz,nAnd strain variable Δ ε_pz,nCan be respectively calculated as:

the stress of the steel strand calculated by the formula (5.31) is substituted into the formula (5.3), and the nth delta z length position can be calculatedStrain epsilon of concrete_cz,n，

When the strain variable Delta epsilon of the steel strand is calculated by the formula (5.32)_pz,nAnd the concrete strain ε calculated by the formula (5.33)_cz,nWhen the positions are equal, the steel strand and the concrete do not move any more, and the distance from the position to the end part of the member is the transmission length l of the prestressed steel strand_tr。

Images