CN102375925B - Method for evaluating resistance deterioration of stay cable of steel strand of cable-stayed bridge taking fretting fatigue - Google Patents

Abstract

The invention provides a method for evaluating the resistance deterioration of a stay cable of a steel strand of a cable-stayed bridge taking fretting fatigue. The method comprises the following steps of: sampling practical parameters of steel wires of the stay cable of the cable-stayed bridge on site; obtaining a fretting fatigue depth fitting formula among steel wires of the stay cable of the steel strand of the cable-stayed bridge; dividing resistance deterioration phases of the stay cable; and substituting the actual fretting fatigue depth among the steel wires into the deterioration process of the stay cable of the steel strand of the cable-stayed bridge according to division of different phases to obtain a resistance deterioration equation of the stay cable of the steel strand of the cable-stayed bridge taking the fretting fatigue, and evaluating the situation of the resistance deterioration of the stay cable. The method describes the deterioration process of the stay cable of the steel strand more comprehensively and more accurately, and the evaluation process is simple and practical.

Description

Consider the cable-stayed bridge steel strand stay cable Resistance deterioration appraisal procedure of fretting fatigue

Technical field

The present invention relates to transportation bridge engineeting field, specifically relate to a kind of cable-stayed bridge steel strand stay cable Resistance deterioration appraisal procedure considering fretting damage.

Background technology

The assessment of cable-stayed bridge steel strand stay cable Resistance deterioration is the important research problem in Design of Cable-Stayed Bridge, construction and operation process always.For steel strand stay cable, owing to combining closely between its steel wire, be subject to weather, vehicular load effect time, the vibration that suspension cable occurs makes the anchored end of rope be subject to stretch bending effect repeatedly, interaction force between steel strand stay cable steel wire and fine motion slippage make fretting damage produce, so the Resistance deterioration analysis of steel strand stay cable needs the fretting fatigue considering steel wire.And the Resistance deterioration research of suspension cable at present mainly launches based on parallel steel wire suspension cable, state by being corroded parallel steel wire suspension cable, after fatigue is studied, obtain the Resistance deterioration equation of parallel steel wire suspension cable, do not consider the interaction between parallel steel wire suspension cable steel wire.

Summary of the invention

Technical matters to be solved by this invention is: provide a kind of cable-stayed bridge steel strand stay cable Resistance deterioration appraisal procedure considering fretting fatigue, this method describes the degenerative process of steel strand stay cable more comprehensively, accurately, and evaluation process is simple, easy.

The technical solution adopted in the present invention is: the cable-stayed bridge steel strand stay cable Resistance deterioration appraisal procedure considering fretting fatigue, first carries out spot sampling to the actual parameter of cable-stayed bridge steel strand stay cable steel wire; Then utilize Finite Element Method and bitter end geometric relationship, derive the relational expression between cable-stayed bridge steel strand stay cable steel wire interaction force and fretting amplitude, then draw the fretting wear degree of depth matching formula between cable-stayed bridge steel strand stay cable steel wire; Afterwards according to the matching formula of the described fretting wear degree of depth, obtain the actual fretting wear degree of depth between described steel wire, and the described fretting wear degree of depth and steel wire crack Propagation critical depth are compared, judge that in steel strand stay cable Resistance deterioration process, whether the local pitting corrosion stage exists, to carry out the division in suspension cable Resistance deterioration stage; Finally according to the division of described different phase, and the actual fretting wear degree of depth between described steel wire is substituted into cable-stayed bridge steel strand stay cable degenerative process, obtain the cable-stayed bridge steel strand stay cable Resistance deterioration equation considering fretting fatigue, the situation of described suspension cable Resistance deterioration is assessed.

Described method, the fretting wear degree of depth matching formula between derivation cable-stayed bridge steel strand stay cable steel wire comprises the following steps:

S1) set up cable-stayed bridge steel strand stay cable steel wire finite element model, solve steel wire interaction force P:

P＝∫∫f(s)ds (1)

In formula (1), f (s) is Steel Wire Surface stress distribution function;

S2) steel wire fretting amplitude A is solved according to Inclined Cable Vibration characteristic and bitter end geometric relationship:

$A=\frac{64f_{s}R}{l^2}---\left(2\right)$

In formula (2), R is steel strand wires steel wire radius, f _sfor the amount of deflection increment that suspension cable peak swing causes, l is that suspension cable calculating rope is long;

S3) relational expression between steel wire interaction force P and fretting amplitude A is set up:

$\mathrm{\π}[a_f\left(t\right)-{\mathrm{\α}}_0]\sqrt{[{2\mathrm{Ra}}_f\left(t\right)-a_f^2\left(t\right)][{4\mathrm{Ra}}_f\left(t\right)-{4a}_f^2\left(t\right)]}-2\mathrm{PAkft}=0---\left(3\right)$

In formula (3), α ₀be steel wire wear fitting parameter with k, a _ft () is steel wire fretting wearing depth, f is Inclined Cable Vibration frequency is also steel wire fretting frequency, and t is steel wire service time;

S4) (1) formula, (2) formula are substituted into (3) formula, obtain the fretting wear degree of depth a between cable-stayed bridge steel strand stay cable steel wire _fthe matching formula of (t):

$\mathrm{\π}[a_f\left(t\right)-{\mathrm{\α}}_0]\sqrt{[{2\mathrm{Ra}}_f\left(t\right)-a_f^2\left(t\right)][{4\mathrm{Ra}}_f\left(t\right)-{4a}_f^2\left(t\right)]}-\frac{128f_s\mathrm{Rkft}}{l^2}\∫\∫f\left(s\right)\mathrm{ds}=0---\left(4\right)$

Described method, to the method for the division in suspension cable Resistance deterioration stage is:

S1) steel wire t service time is taken as whole sheath time to rupture point t ' ₁, (4) formula of substitution calculates the actual fretting wear degree of depth a obtained between described steel wire _f(t ' ₁);

S2) actual fretting wear degree of depth a is obtained _f(t ' ₁) and wire galvanization layer local maximum gauge b _localdifference, and by this difference and steel wire crack Propagation critical depth a _cicompare; If (a _f(t ' ₁)-b _local) < a _ci, enter step S3); If (a _f(t ' ₁)-b _local)>=a _ci, enter step S4);

S3) comprise pitting corrosion process, suspension cable Resistance deterioration process is divided into 0≤t < t ' ₁, t ' ₁≤ t < t ' ₂, t ' ₂≤ t three phases, wherein t ' ₂for fatigue crack starts expansion time point;

S4) do not comprise pitting corrosion process, suspension cable Resistance deterioration process is divided into 0≤t < t ' ₁, t ' ₁two stages of≤t.

Described method, when comprising pitting corrosion process, consider that cable-stayed bridge steel strand stay cable Resistance deterioration function R ' (t) of fretting damage is:

$\left\{\begin{array}{c}{a}^{\&prime;}\left(t\right)=a_f\left(t\right)-\left(b_{\mathrm{local}}\right),0\&le;t\&le;{t_1}^{\&prime;}\\ a^{\&prime;}\left(t\right)=C_{1I}{\left({{t-t}_1}^{\&prime;}\right)}^{\mathrm{ni}},{t_1}^{\&prime;}\&le;t\&le;{t_2}^{\&prime;}\\ a^{\&prime;}\left(t\right)=a_{\mathrm{ci}}+\frac{k_0{(\mathrm{\&Delta;K}-{\mathrm{\&Delta;K}}_{\mathrm{th}})}^2\left({{t-t}_2}^{\&prime;}\right)f_z}{2\mathrm{\&pi;}{\mathrm{\&sigma;}}_{\mathrm{FF}}^2},{t_2}^{\&prime;}\&le;t\end{array}\right.---\left(5\right)$

$\left\{\begin{array}{c}{R}^{\&prime;}\left(t\right)=R_0{g}^{\&prime;}\left(t\right)=N{\mathrm{\&tau;}}_N{f}_{\mathrm{td}}{a}_0\&times;g^{\&prime;}\left(t\right)\\ g^{\&prime;}\left(t\right)=1-5.3\&times;{10}^{-5}{\mathrm{\&rho;}}_0{a}^{\&prime;}\left(t\right)-5.9\&times;{10}^{-8}{\mathrm{\&rho;}}_0{a}^{\&prime;2}\left(t\right)\end{array}\right.---\left(6\right)$

In formula (5), (6), a ' (t) is for considering fretting fatigue bridge steel strand stay cable degeneration depth function, C _1Ifor the corrosion penetration of steel First Year, ni is steel corrosion environmental parameter, environment constant k ₀=1.585, Δ K is stress intensive factor range, Δ K _thfor CORROSION FATIGUE THRESHOLD LIMIT, σ _fFfor material critical faulting stress, R ₀for structure initial resistive force, N is steel strand stay cable steel wire sum, f _tdfor steel wire tension design load, a ₀for steel wire initial area, g ' (t) is steel wire Resistance deterioration function, τ _nfor Denier coefficient, ρ ₀for suspension cable cross section steel wire corrosion number ratio, f _zfor Equivalent Fatigue mobile load loading frequency.

Described method, when not comprising pitting corrosion process, consider that cable-stayed bridge steel strand stay cable Resistance deterioration function R ' (t) of fretting damage is:

$\left\{\begin{array}{c}{a}^{\&prime;}\left(t\right)=a_f\left(t\right)-\left(b_{\mathrm{local}}\right),0\&le;t\&le;{t_1}^{\&prime;}\\ a^{\&prime;}\left(t\right)=\frac{k_0{(\mathrm{\&Delta;K}-{\mathrm{\&Delta;K}}_{\mathrm{th}})}^2\left({{t-t}_2}^{\&prime;}\right)f_z}{2\mathrm{\&pi;}{\mathrm{\&sigma;}}_{\mathrm{FF}}^2}+\\ a_f\left({t_1}^{\&prime;}\right)-\left(b_{\mathrm{local}}\right),{t_1}^{\&prime;}\&le;t\end{array}\right.---\left(7\right)$

$\left\{\begin{array}{c}{R}^{\&prime;}\left(t\right)=R_0{g}^{\&prime;}\left(t\right)=N{\mathrm{\&tau;}}_N{f}_{\mathrm{td}}{a}_0\&times;g^{\&prime;}\left(t\right)\\ g^{\&prime;}\left(t\right)=1-5.3\&times;{10}^{-5}{\mathrm{\&rho;}}_0{a}^{\&prime;}\left(t\right)-5.9\&times;{10}^{-8}{\mathrm{\&rho;}}_0{a}^{\&prime;2}\left(t\right)\end{array}\right.---\left(8\right)$

In formula, t ' ₁for whole sheath time to rupture point, also for fatigue crack starts expansion time point.

The present invention has following major advantage:

One. consider steel strand stay cable steel wire fretting fatigue attrition and attack fatigue effect each other, compared with the suspension cable degradation assessment method only considering corrosion fatigue more comprehensively, accurately assess the life-cycle Resistance deterioration process of cable-stayed bridge steel strand stay cable.

They are two years old. and evaluation process is simple, easy, has larger practical engineering application and is worth.

Accompanying drawing explanation

Fig. 1 is steel strand anchoring end steel wire contact FEA illustraton of model.

Fig. 2 is steel strand anchoring end steel wire contact FEA the model calculation.

Fig. 3 is 110m steel strand wires finite element contact model result of calculation.

Fig. 4 is the Cable Stayed Steel twisted wire degeneration equation deterministic process considering fretting damage.

Fig. 5 is embodiment 1 Chinese style (7), formula (10) maps the result compared.

Embodiment

The invention provides a kind of cable-stayed bridge steel strand stay cable Resistance deterioration appraisal procedure considering fretting fatigue, the time dependent expression formula of steel wire fretting wearing depth that the method proposes based on steel wire fretting wear test, by to the geometric analysis of steel strand stay cable anchored end and the transactional analysis of Wire Vibration, solve steel strand stay cable steel wire interaction force and mutually unknown between amplitude problem in the time dependent expression formula of steel wire fretting wearing depth by experiment, set up steel strand stay cable steel wire fretting wearing depth and change expression formula in time.The present invention combines the suspension cable Resistance deterioration function not considering fretting fatigue, by whether reaching the judgement of the even crack propagation critical degree of depth to the fretting wear degree of depth, forms the cable-stayed bridge steel strand stay cable Resistance deterioration function considering fretting fatigue.Contemplated by the invention the fretting fatigue factor in steel strand stay cable Resistance deterioration process, result of calculation and engineering reality more close.

A kind of cable-stayed bridge steel strand stay cable Resistance deterioration appraisal procedure considering fretting fatigue, the method utilizes Finite Element Method and bitter end geometric relationship, solve the problem of cable-stayed bridge steel strand stay cable steel wire interaction force and fretting amplitude the unknown, the fretting wear degree of depth a derived between cable-stayed bridge steel strand stay cable steel wire _ft () matching formula, by considering into cable-stayed bridge steel strand stay cable degenerative process by this matching formula, to wearing depth a _f(t) and steel wire crack Propagation critical depth a _cicompare, judge that in steel strand stay cable degenerative process, whether the local pitting corrosion stage exists, obtain the cable-stayed bridge steel strand stay cable Resistance deterioration equation considering fretting fatigue, assessment cable-stayed bridge steel strand stay cable time-varying reliability.The method comprises following four steps:

(1) set up cable-stayed bridge steel strand stay cable steel wire finite element model and solve steel wire interaction force P

Simplify contact FEA model (accompanying drawing 1) solve equivalent contact force P (accompanying drawing 2) between steel wire, butt contact stress area integration by setting up steel wire, that is:

P＝∫∫f(s)ds

In formula, f (s) is Steel Wire Surface stress distribution function.

(2) steel wire fretting amplitude A is solved according to Inclined Cable Vibration characteristic and bitter end geometric relationship

Steel strand stay cable steel wire fretting amplitude is derived by following method:

1) suspension cable Approximate Equivalent is become a thin beam, the amount of deflection increment f of thin beam under equivalent uniform load q effect _scan be expressed as:

$f_s=\frac{{\mathrm{ql}}^4}{384\mathrm{EI}}$

In formula, EI is beam bendind rigidity.

2) the then amount of deflection increment f that causes of suspension cable peak swing _sbe expressed as with the relation of the radius-of-curvature increment Delta ρ of bitter end:

$\mathrm{\&Delta;\&rho;}=\frac{\mathrm{EI}}{M}=\frac{12\mathrm{EI}}{{\mathrm{ql}}^2}=\frac{l^2}{32f_s}$

In formula, M is the beam-ends moment of flexure under equivalent uniform load q effect.

3) according to steel strand wires bitter end geometric relationship, then the fretting amplitude A between steel wire and radius-of-curvature increment Delta ρ relation can be expressed as:

$A=\frac{2R}{\mathrm{\&Delta;\&rho;}}=\frac{64{\mathrm{Rf}}_s}{l^2}$

In formula, R is steel strand wires steel wire radius, f _sfor the amount of deflection increment that suspension cable peak swing causes, l is that suspension cable calculating rope is long.

(3) steel strand stay cable steel wire fretting lesion depths expression formula is set up

Become 90 degree of angles to launch fretting wear for two steel wires, changing expression formula in time according to its fretting wear degree of depth of right cylinder geometric relationship can be expressed as:

$\mathrm{\&pi;}[a_f\left(t\right)-{\mathrm{\&alpha;}}_0]\sqrt{[{2\mathrm{Ra}}_f\left(t\right)-a_f^2\left(t\right)][{4\mathrm{Ra}}_f\left(t\right)-{4a}_f^2\left(t\right)]}-2\mathrm{PAkft}=0$

In formula: α ₀be parameter with k, a _ft () is steel wire lesion depths, R is steel wire radius, f _sfor suspension cable amplitude, f is Inclined Cable Vibration frequency (being also steel wire fretting frequency), and t is steel wire service time, and l is that suspension cable calculating rope is long, and the Interaction Force of steel wire when P is fine motion vibration, A is steel wire fretting amplitude.

Consider that the angle between steel strand stay cable steel wire is less than 90 degree, the rate of wear is slightly less than steel wire and becomes 90 degree to carry out fretting wear speed, is still taken as above formula in calculating; Steel wire interaction force P, fretting amplitude A are substituted into above formula, then Cable Stayed Steel twisted wire steel wire fretting wearing depth changes expression formula in time and can be expressed as:

$\mathrm{\&pi;}[a_f\left(t\right)-{\mathrm{\&alpha;}}_0]\sqrt{[{2\mathrm{Ra}}_f\left(t\right)-a_f^2\left(t\right)][{4\mathrm{Ra}}_f\left(t\right)-{4a}_f^2\left(t\right)]}-\frac{128f_s\mathrm{Rkft}}{l^2}\&Integral;\&Integral;f\left(s\right)\mathrm{ds}=0$

(4) set up steel strand stay cable degenrate function R (t) not considering fretting fatigue, can be expressed as:

$\left\{\begin{array}{c}a\left(t\right)0,\begin{array}{cccccccccc}L& L& L& L& L& L& L& L& L& L\end{array}0<t<t_1\\ a\left(t\right)=C_{1Z}{\left({t-t}_1\right)}^{\mathrm{nz}},\begin{array}{ccccc}L& L& L& L& L\end{array}{t}_1<t<t_2\\ a\left(t\right){=C}_{1I}{\left({t-t}_2\right)}^{\mathrm{ni}},\begin{array}{ccccc}L& L& L& L& L\end{array}{t}_2<t<t_3\\ a\left(t\right)=a_{\mathrm{ci}}+\frac{k{(\mathrm{\&Delta;K}-\mathrm{\&Delta;}{K}_{\mathrm{th}})}^2\left({t-t}_3\right)f_z}{2\mathrm{\&pi;}{\mathrm{\&sigma;}}_{\mathrm{FF}}^2},t_3<t\end{array}\right.$

$\left\{\begin{array}{c}R\left(t\right)=R_{0}g\left(t\right)=N{\mathrm{\&tau;}}_N{f}_{\mathrm{td}}{a}_0\&times;g\left(t\right)\\ g\left(t\right)=1-5.3\&times;{10}^{-5}{\mathrm{\&rho;}}_{0}a\left(t\right)-5.9\&times;{10}^{-8}{\mathrm{\&rho;}}_0{a}^2\left(t\right)\end{array}\right.$

In formula: a (t) is collapse dept under the multifactor effect of steel strand stay cable steel wire corrosion-wear; G (t) is not for considering the steel strand stay cable steel wire Resistance deterioration function of fretting fatigue; C _1Zfor the corrosion penetration of zinc coat First Year; Nz is zinc coat corrosion environmental parameter; t ₁for whole HDPE sheath time to rupture point (year); t ₂for zinc coat corrodes the time point that complete steel wire matrix starts uniform corrosion and pitting corrosion, t ₃for fatigue crack starts expansion time point.

(5) by steel strand stay cable steel wire fretting lesion depths expression formula in time, steel wire fretting lesion depths when HDPE sheath destroys is obtained, if steel wire fretting lesion depths is greater than zinc coat average thickness b _local, then steel strand stay cable Resistance deterioration function R ' (t) can be expressed as:

$\left\{\begin{array}{c}{a}^{\&prime;}\left(t\right)=a_f\left(t\right)-\left(b_{\mathrm{local}}\right)\begin{array}{cccccc}L& L& L& L& L& L\end{array}0<t<{t_1}^{\&prime;}\\ a^{\&prime;}\left(t\right)=C_{1I}{\left({{t-t}_1}^{\&prime;}\right)}^{\mathrm{ni}},\begin{array}{cccccc}L& L& L& L& L& L\end{array}{t_1}^{\&prime;}<t<{t_2}^{\&prime;}\\ a^{\&prime;}\left(t\right)=a_{\mathrm{ci}}+\frac{k_0{(\mathrm{\&Delta;K}-{\mathrm{\&Delta;K}}_{\mathrm{th}})}^2\left({{t-t}_2}^{\&prime;}\right)f_z}{2\mathrm{\&pi;}{\mathrm{\&sigma;}}_{\mathrm{FF}}^2},{t_2}^{\&prime;}<t\end{array}\right.$

$\left\{\begin{array}{c}{R}^{\&prime;}\left(t\right)=R_0{g}^{\&prime;}\left(t\right)=N{\mathrm{\&tau;}}_N{f}_{\mathrm{td}}{a}_0\&times;g^{\&prime;}\left(t\right)\\ g^{\&prime;}\left(t\right)=1-5.3\&times;{10}^{-5}{\mathrm{\&rho;}}_0{a}^{\&prime;}\left(t\right)-5.9\&times;{10}^{-8}{\mathrm{\&rho;}}_0{a}^{\&prime;2}\left(t\right)\end{array}\right.$

In formula, b _localfor wire galvanization layer local maximum gauge, C _1Ifor the corrosion penetration of steel First Year; Ni is steel corrosion environmental parameter, a _cifor critical depth during crack Propagation, k ₀, (Δ K-Δ K _th) ², for environment constant, t ' ₁for whole HDPE (high density polyethylene) sheath time to rupture point (year); T ' ₂for steel wire uniform corrosion is complete, fatigue crack starts expansion time point, R ₀for structure initial resistive force, N is steel strand stay cable steel wire sum, f _tdfor steel wire tension design load, a ₀for steel wire initial area, g ' (t) is steel wire Resistance deterioration function, τ _nfor Denier coefficient, ρ ₀for suspension cable cross section steel wire corrosion number ratio.F _zfor Equivalent Fatigue mobile load loading frequency, f _z=1 time/min.

(6) if the fretting damage degree of depth is greater than steel wire crack Propagation degree of depth threshold value a _ci, then steel strand stay cable Resistance deterioration function R ' (t) can be expressed as:

$\left\{\begin{array}{c}{a}^{\&prime;}\left(t\right)=a_f\left(t\right)-\left(b_{\mathrm{local}}\right)\begin{array}{cccccc}L& L& L& L& L& L\end{array}0<t<{t_1}^{\&prime;}\\ a^{\&prime;}\left(t\right)=\frac{k_0{(\mathrm{\&Delta;K}-\mathrm{\&Delta;}{K}_{\mathrm{th}})}^2\left({{t-t}_2}^{\&prime;}\right)f_z}{2\mathrm{\&pi;}{\mathrm{\&sigma;}}_{\mathrm{FF}}^2}+\\ a_f\left({t_1}^{\&prime;}\right)-\left(b_{\mathrm{local}}\right),\begin{array}{cccccc}L& L& L& L& L& L\end{array}{t_1}^{\&prime;}<t\end{array}\right.$

$\left\{\begin{array}{c}{R}^{\&prime;}\left(t\right)=R_0{g}^{\&prime;}\left(t\right)=N{\mathrm{\&tau;}}_N{f}_{\mathrm{td}}{a}_0\&times;g^{\&prime;}\left(t\right)\\ g^{\&prime;}\left(t\right)=1-5.3\&times;{10}^{-5}{\mathrm{\&rho;}}_0{a}^{\&prime;}\left(t\right)-5.9\&times;{10}^{-8}{\mathrm{\&rho;}}_0{a}^{\&prime;2}\left(t\right)\end{array}\right.$

In formula, t ' ₁for whole HDPE sheath time to rupture point (year), also for fatigue crack starts expansion time point.

Below in conjunction with accompanying drawing and example, the present invention is described in further details.

Embodiment 1, certain Yangtze Bridge steel strand stay cable Resistance deterioration is assessed

Certain Yangtze Bridge 110m steel strand stay cable, suspension cable Resistance deterioration correlation parameter is as table one, table two

Table one steel strand wires zinc coat, steel parameter

Table two steel strand wires rope basic design parameters

By suspension cable Analysis of Fundamental Frequencies, it is 1.02Hz that the long steel strand stay cable amplitude of this Yangtze Bridge 110m gets single order fundamental vibration frequency, amplitude f _s=max{l/1700,2.5 times of rope diameters }, be taken as 0.5m.

(1) to steel strand stay cable anchored end set up simplify steel wire contact FEA model, can be calculated finite element model stress point be chainlike distribution in Steel Wire Surface, maximum stress reaches 100Mpa, as accompanying drawing 3.

Counter stress distributed points is carried out integration and is obtained:

P＝∫∫f(s)ds＝42N (1)

In formula, f (s) is Steel Wire Surface stress distribution function.

(2) steel wire fretting amplitude A is solved according to Inclined Cable Vibration characteristic and bitter end geometric relationship

$A=\frac{64f_{s}R}{l^2}=13.75\mathrm{\&mu;m}---\left(2\right)$

In formula, R is steel strand wires steel wire radius; f _sfor suspension cable amplitude, l is that suspension cable calculating rope is long.

(3) steel wire fretting amplitude A is brought into and interaction force P obtains steel strand stay cable steel wire fretting wearing depth expression formula:

$\mathrm{\&pi;}[a_f\left(t\right)-{\mathrm{\&alpha;}}_0]\sqrt{[{2\mathrm{Ra}}_f\left(t\right)-a_f^2\left(t\right)][{4\mathrm{Ra}}_f\left(t\right)-{4a}_f^2\left(t\right)]}-\frac{128f_s\mathrm{Rkft}}{l^2}\&Integral;\&Integral;f\left(s\right)\mathrm{ds}=0---\left(3\right)$

$\mathrm{\&pi;}[a_f\left(t\right)-12]\sqrt{[{2\mathrm{Ra}}_f\left(t\right)-a_f^2\left(t\right)][{4\mathrm{Ra}}_f\left(t\right)-{4a}_f^2\left(t\right)]}-58.9t=0---\left(4\right)$

Carry out numerical fitting to above formula to obtain:

a _f(t)＝18.72+16.62t-0.564t ²(5)

In formula, α ₀, k is parameter, is taken as α respectively ₀=12, k=0.05, a _ft () is steel wire lesion depths, R is steel wire radius, f _sfor suspension cable amplitude, f is Inclined Cable Vibration frequency is also steel wire fretting frequency, and t is steel wire service time, and l is that suspension cable calculating rope is long.

(4) the steel strand stay cable Resistance deterioration function considering fatigue is solved:

$\left\{\begin{array}{c}{a}^{\&prime;}\left(t\right)=a_f\left(t\right)-\left(b_{\mathrm{local}}\right)\begin{array}{cccccc}L& L& L& L& L& L\end{array}0<t<{t_1}^{\&prime;}\\ a^{\&prime;}\left(t\right)=C_{1I}{\left({{t-t}_1}^{\&prime;}\right)}^{\mathrm{ni}},\begin{array}{cccccc}L& L& L& L& L& L\end{array}{t_1}^{\&prime;}<t<{t_2}^{\&prime;}\\ a^{\&prime;}\left(t\right)=a_{\mathrm{ci}}+\frac{k_0{(\mathrm{\&Delta;K}-{\mathrm{\&Delta;K}}_{\mathrm{th}})}^2\left({{t-t}_2}^{\&prime;}\right)f_z}{2\mathrm{\&pi;}{\mathrm{\&sigma;}}_{\mathrm{FF}}^2},{t_2}^{\&prime;}<t\end{array}\right.---\left(6\right)$

$\left\{\begin{array}{c}{R}^{\&prime;}\left(t\right)=R_0{g}^{\&prime;}\left(t\right)=N{\mathrm{\&tau;}}_N{f}_{\mathrm{td}}{a}_0\&times;g^{\&prime;}\left(t\right)\\ g^{\&prime;}\left(t\right)=1-5.3\&times;{10}^{-5}{\mathrm{\&rho;}}_0{a}^{\&prime;}\left(t\right)-5.9\&times;{10}^{-8}{\mathrm{\&rho;}}_0{a}^{\&prime;2}\left(t\right)\end{array}\right.$

In formula, b _localfor wire galvanization layer local maximum gauge, C _1Ifor the corrosion penetration of steel First Year; Ni is steel corrosion environmental parameter, a _cifor critical depth during crack Propagation, k ₀, (Δ K-Δ K _th) ², for environment constant, wherein Δ K is stress intensive factor range, Δ K _thfor CORROSION FATIGUE THRESHOLD LIMIT, σ _fFfor material critical faulting stress, t ' ₁for whole HDPE sheath time to rupture point (year); T ' ₂for steel wire uniform corrosion is complete, fatigue crack starts expansion time point, R ₀for structure initial resistive force, N is steel strand stay cable steel wire sum, f _tdfor steel wire tension design load, a ₀for steel wire initial area, g ' (t) is steel wire Resistance deterioration function, τ _nfor Denier coefficient, ρ ₀for suspension cable cross section steel wire corrosion number ratio.F _zfor Equivalent Fatigue mobile load loading frequency f _z=1 time/min.

4.1 steel strand stay cable catagen phase time points calculate:

(1) HDPE sheath corrosion failure time t ₁(t ' ₁)=13.8 year

(2) the time t that the zinc coat that atmospheric corrosion causes destroys ₂=10.9 years

(3) cable wire steel from uniform corrosion to corrosion fatigue crack expansion time t ₃(t ' ₂)=0.45 year

4.2 do not consider that steel strand stay cable Resistance deterioration function R (t) of fretting fatigue is:

$\left\{\begin{array}{c}R\left(t\right)=8037\&times;g\left(t\right)\\ \left\{\begin{array}{c}g\left(t\right)=\mathrm{1,0}<t<24.7\\ g\left(t\right)=1-3\&times;{10}^{-3}{(t-24.7)}^{0.8},24.7<t<25.2\\ g\left(t\right)=0.631+0.0307t-0.00064t^2,t>25.2\end{array}\right.\end{array}\right.---\left(7\right)$

4.3 carry out the analysis of fretting fatigue wearing depth:

When HDPE sheath corrosion failure, t ₁(t ' ₁)=13.8 year, steel strand stay cable steel wire fretting fatigue wear degree of depth a _f(t) be:

a _f(t)＝18.72+16.62t-0.564t ²＝145μm (8)

By (a _f(t)=145 μm) > (b _local=39 μm), and (a _f(t)-b _local=106 μm) > (a _ci=53 μm), (see accompanying drawing 4) is so steel strand stay cable Resistance deterioration function R ' (t) can be expressed as:

$\left\{\begin{array}{c}{a}^{\&prime;}\left(t\right)=a_f\left(t\right)-\left(b_{\mathrm{local}}\right)\begin{array}{cccccc}L& L& L& L& L& L\end{array}0<t<{t_1}^{\&prime;}\\ a^{\&prime;}\left(t\right)=\frac{k_0{(\mathrm{\&Delta;K}-\mathrm{\&Delta;}{K}_{\mathrm{th}})}^2\left({{t-t}_2}^{\&prime;}\right)f_z}{2\mathrm{\&pi;}{\mathrm{\&sigma;}}_{\mathrm{FF}}^2}+\\ a_f\left({t_1}^{\&prime;}\right)-\left(b_{\mathrm{local}}\right),\begin{array}{cccccc}L& L& L& L& L& L\end{array}{t_1}^{\&prime;}<t\end{array}\right.---\left(9\right)$

$\left\{\begin{array}{c}{R}^{\&prime;}\left(t\right)=R_0{g}^{\&prime;}\left(t\right)=N{\mathrm{\&tau;}}_N{f}_{\mathrm{td}}{a}_0\&times;g^{\&prime;}\left(t\right)\\ g^{\&prime;}\left(t\right)=1-5.3\&times;{10}^{-5}{\mathrm{\&rho;}}_0{a}^{\&prime;}\left(t\right)-5.9\&times;{10}^{-8}{\mathrm{\&rho;}}_0{a}^{\&prime;2}\left(t\right)\end{array}\right.$

In formula, t ' ₁for whole HDPE sheath time to rupture point (year) is also for fatigue crack starts expansion time point.

Namely have:

$\left\{\begin{array}{c}{R}^{\&prime;}\left(t\right)=8037\mathrm{KN}\&times;g^{\&prime;}\left(t\right)\\ \left\{\begin{array}{c}{g}^{\&prime;}\left(t\right)=1-3.2\&times;{10}^{-4}t,0<t<13.8\\ g^{\&prime;}\left(t\right)=0.950+0.0121t-6.4\&times;{10}^{-4}{t}^2,t>13.8\end{array}\right.\end{array}\right.---\left(10\right)$

According to (10) formula, namely can obtain considering the cable-stayed bridge steel strand stay cable drag size of fretting fatigue and the relational expression between the time, and can obtain according to this expression formula the situation that drag decays in time, thus make the assessment result of Resistance deterioration.Compare (as accompanying drawing 5) formula (7), formula (10) mapping, the steel strand stay cable Resistance deterioration speed of known consideration fretting fatigue is accelerated.

In a word, the method is by analyzing Inclined Cable Vibration characteristic, set up steel strand wires contact FEA model, by the contact stress between this model analysis steel wire, by the analysis and solution steel strand wires steel wire fretting amplitude to Inclined Cable Vibration characteristic and bitter end steel strand wires steel wire geometric relationship, to solve the problem of cable-stayed bridge steel strand stay cable steel wire interaction force and fretting amplitude the unknown, obtain cable-stayed bridge steel strand stay cable steel wire fretting wearing depth matching formula.Steel strand wires fretting fatigue abrasive action is considered into cable-stayed bridge steel strand stay cable degradation analysis system, to wearing depth a _f(t) and steel wire crack Propagation critical depth a _cicompare, judge that in steel strand stay cable degenerative process, whether the local pitting corrosion stage exists, obtain the cable-stayed bridge steel strand stay cable Resistance deterioration appraisal procedure considering fretting damage.

Claims (1)

1. consider the cable-stayed bridge steel strand stay cable Resistance deterioration appraisal procedure of fretting fatigue, it is characterized in that: first spot sampling is carried out to the actual parameter of cable-stayed bridge steel strand stay cable steel wire; Then utilize Finite Element Method and bitter end geometric relationship, derive the relational expression between cable-stayed bridge steel strand stay cable steel wire interaction force and fretting amplitude, then draw the fretting wear degree of depth matching formula between cable-stayed bridge steel strand stay cable steel wire; Afterwards according to the matching formula of the described fretting wear degree of depth, obtain the actual fretting wear degree of depth between described steel wire, and the described fretting wear degree of depth and steel wire crack Propagation critical depth are compared, judge that in steel strand stay cable Resistance deterioration process, whether the local pitting corrosion stage exists, to carry out the division in suspension cable Resistance deterioration stage; Finally according to the division of described different phase, and the actual fretting wear degree of depth between described steel wire is substituted into cable-stayed bridge steel strand stay cable degenerative process, obtain the cable-stayed bridge steel strand stay cable Resistance deterioration equation considering fretting fatigue, the situation of described suspension cable Resistance deterioration is assessed;

Fretting wear degree of depth matching formula between derivation cable-stayed bridge steel strand stay cable steel wire comprises the following steps:

S1) set up cable-stayed bridge steel strand stay cable steel wire finite element model, solve steel wire interaction force P:

P＝∫∫f(s)ds (1)

In formula (1), f (s) is Steel Wire Surface stress distribution function;

S2) steel wire fretting amplitude A is solved according to Inclined Cable Vibration characteristic and bitter end geometric relationship:

$A=\frac{64f_{s}R}{l^2}---\left(2\right)$

In formula (2), R is steel strand wires steel wire radius, f _sfor the amount of deflection increment that suspension cable peak swing causes, l is that suspension cable calculating rope is long;

S3) relational expression between steel wire interaction force P and fretting amplitude A is set up:

$\mathrm{\&pi;}[a_f\left(t\right)-{\mathrm{\&alpha;}}_0]\sqrt{[(2R{a}_f\left(t\right)-a_f^2\left(t\right)][4{\mathrm{Ra}}_f\left(t\right)-4a_f^2\left(t\right)]}-2\mathrm{PAkft}=0---\left(3\right)$

In formula (3), α ₀be steel wire wear fitting parameter with k, a _ft () is steel wire fretting wearing depth, f is Inclined Cable Vibration frequency is also steel wire fretting frequency, and t is steel wire service time;

S4) (1) formula, (2) formula are substituted into (3) formula, obtain the fretting wear degree of depth a between cable-stayed bridge steel strand stay cable steel wire _fthe matching formula of (t):

$\mathrm{\&pi;}[a_f\left(t\right)-{\mathrm{\&alpha;}}_0]\sqrt{[(2R{a}_f\left(t\right)-a_f^2\left(t\right)][4{\mathrm{Ra}}_f\left(t\right)-4a_f^2\left(t\right)]}-\frac{128f_s\mathrm{Rkft}}{l^2}\&Integral;\&Integral;f\left(s\right)\mathrm{ds}=0---\left(4\right);$

To the method for the division in suspension cable Resistance deterioration stage be:

S1) steel wire t service time is taken as whole sheath time to rupture point t ₁', (4) formula of substitution calculates the actual fretting wear degree of depth a obtained between described steel wire _f(t ₁');

S2) actual fretting wear degree of depth a is obtained _f(t ₁') and wire galvanization layer local maximum gauge b _localdifference, and by this difference and steel wire crack Propagation critical depth a _cicompare; If (a _f(t ₁')-b _local) <a _ci, enter step S3); If (a _f(t ₁')-b _local)>=a _ci, enter step S4);

S3) comprise pitting corrosion process, suspension cable Resistance deterioration process is divided into 0≤t < t ₁', t ₁'≤t < t' ₂, t' ₂≤ t three phases, wherein t' ₂for fatigue crack starts expansion time point;

S4) do not comprise pitting corrosion process, suspension cable Resistance deterioration process is divided into 0≤t < t ₁', t ₁two stages of '≤t;

When comprising pitting corrosion process, consider the cable-stayed bridge steel strand stay cable Resistance deterioration function R'(t of fretting damage) be:

$\left\{\begin{array}{c}{a}^{\&prime;}\left(t\right)=a_f\left(t\right)-\left(b_{\mathrm{local}}\right),0\&le;t<{t_1}^{\&prime;}\\ a^{\&prime;}\left(t\right)=C_{1I}{(t-{t_1}^{\&prime;})}^{\mathrm{ni}},{t_1}^{\&prime;}\&le;t<{t_2}^{\&prime;}\\ a^{\&prime;}\left(t\right)=a_{\mathrm{ci}}+\frac{k_0{(\mathrm{\&Delta;K}-\mathrm{\&Delta;}{K}_{\mathrm{th}})}^2(t-{t_2}^{\&prime;})f_z}{2\mathrm{\&pi;}{\mathrm{\&sigma;}}_{\mathrm{FF}}^2},{t_2}^{\&prime;}\&le;t\end{array}\right.----\left(5\right)$

$\left\{\begin{array}{c}{R}^{\&prime;}\left(t\right)=R_0{g}^{\&prime;}\left(t\right)=N{\mathrm{\&tau;}}_N{f}_{\mathrm{td}}{a}_0\&times;g^{\&prime;}\left(t\right)\\ g^{\&prime;}\left(t\right)=1-5.3\&times;{10}^{-5}{\mathrm{\&rho;}}_0{a}^{\&prime;}\left(t\right)-5.9\&times;{10}^{-8}{\mathrm{\&rho;}}_0{a}^{{\&prime;}^2}\left(t\right)\end{array}\right.---\left(6\right)$

In formula (5), (6), a'(t) for considering fretting fatigue bridge steel strand stay cable degeneration depth function, C _1Ifor the corrosion penetration of steel First Year, ni is steel corrosion environmental parameter, environment constant k ₀=1.585, Δ K is stress intensive factor range, Δ K _thfor CORROSION FATIGUE THRESHOLD LIMIT, σ _fFfor material critical faulting stress, R ₀for structure initial resistive force, N is steel strand stay cable steel wire sum, f _tdfor steel wire tension design load, a ₀for steel wire initial area, g'(t) be steel wire Resistance deterioration function, τ _nfor Denier coefficient, ρ ₀for suspension cable cross section steel wire corrosion number ratio, f _zfor Equivalent Fatigue mobile load loading frequency;

When not comprising pitting corrosion process, consider the cable-stayed bridge steel strand stay cable Resistance deterioration function R'(t of fretting damage) be:

$\left\{\begin{array}{c}{a}^{\&prime;}\left(t\right)=a_f\left(t\right)-\left(b_{\mathrm{local}}\right),0\&le;t<{t_1}^{\&prime;}\\ a^{\&prime;}\left(t\right)=\frac{k_0{(\mathrm{\&Delta;K}-\mathrm{\&Delta;}{K}_{\mathrm{th}})}^2(t-{t_2}^{\&prime;})f_z}{2\mathrm{\&pi;}{\mathrm{\&sigma;}}_{\mathrm{FF}}^2}+\\ a_f\left({t_1}^{\&prime;}\right)-\left(b_{\mathrm{local}}\right),{t_1}^{\&prime;}\&le;t\end{array}\right.----\left(7\right)$

$\left\{\begin{array}{c}{R}^{\&prime;}\left(t\right)=R_0{g}^{\&prime;}\left(t\right)=N{\mathrm{\&tau;}}_N{f}_{\mathrm{td}}{a}_0\&times;g^{\&prime;}\left(t\right)\\ g^{\&prime;}\left(t\right)=1-5.3\&times;{10}^{-5}{\mathrm{\&rho;}}_0{a}^{\&prime;}\left(t\right)-5.9\&times;{10}^{-8}{\mathrm{\&rho;}}_0{a}^{{\&prime;}^2}\left(t\right)\end{array}\right.---\left(8\right)$

In formula, t ₁' be whole sheath time to rupture point, also for fatigue crack starts expansion time point.