**5. Working life prediction method and evaluation technology for cable based on breakage safety**

#### **5.1 Working life prediction method of service cable**

Based on the experimental analysis, compared with non-corroded steel wire, the pitting corrosion, fracture stress, and elongation after fracture of corroded steel wire in the nucleation stage are basically unchanged, the remaining life prediction of this stage is not considered. The process of corrosion pits growing and expanding until they turn into small cracks is a common stage of the steel wire in service cable. At this stage, the life of the steel wire is reduced. After the appearance of the crack, the corrosive environment and the coupling of alternating stresses made the expansion rate of the crack rapidly increase to the maximum, breaking stress and elongation after fracture are sharply reduced, the steel wire toughness is reduced, and brittleness enhanced, there is a turn from tough to brittle at this stage [19]. In order to prevent the appearance of brittle fracture, the steel wire of the cable needs to be avoided in this stage of work, the critical crack appears, and the steel wire brittle fracture will occur at any time, the study of its life is also not considered. Therefore, the reliability of the steel wire of the service cable is affected by the development of corrosion pits until the appearance of small cracks and the appearance of small cracks until the appearance of the critical crack size *ac* [15].

The law of fatigue crack expansion can be used to analyze the reliability of inservice ties as well as risk of safety. The fatigue crack expansion rate function commonly used in engineering is the Paris-Erdogan model, that is, the Paris formula, which establishes the relationship between the stress intensity factor and the crack expansion rate and is the theoretical basis for predicting the fatigue crack expansion life in the form shown in Eq. (1):

$${}^{da}\!\!/\_{dN} = \mathbf{C} (\Delta \mathbf{K})^n \tag{1}$$

where *a* is the crack length; *N* is the number of stress cycles; *da*/*dN* is the crack expansion rate; *C*, *n* is the material constant, environmental factors such as temperature, humidity, media, loading frequency, etc. are obtained by fitting the experimental data.

Δ*K* is the stress intensity factor amplitude and is described by Eq. (2):

$$
\Delta K = K\_{\text{max}} - K\_{\text{min}} = f \Delta \sigma \sqrt{\pi a} \tag{2}
$$

*Corrosion Fatigue Behavior and Damage Mechanism of the Bridge Cable Structures DOI: http://dx.doi.org/10.5772/intechopen.109105*

where *f* is a function of member geometry and crack size; *Kmax* and *Kmin* are the maximum and minimum values of the stress intensity factor at the crack; and Δ*σ* is the stress amplitude at the crack.

At the stage in which corrosion pits develop into small cracks, each growth of initial defect is independent, the growth of corrosion pit is slow and also affected by accumulation of corrosion product and other factors. while in the small crack until the critical crack size *ac* stage, the growth of crack depth is faster, the growth rate increased rapidly to the maximum, the expansion rate is significantly greater than the previous stage.

On the basis of the Paris formula, the corrosion fatigue calculation is simplified into two linear stages, namely, the life of the corrosion pit development to the crack stage *N1* and the life of the tough and brittle expansion stage *N2*, the total life of the wire of the cable is equal to the sum of life of these two stages, expansion of corrosion fatigue crack generally meets the deformation form of Paris formula, as shown in Eq. (3):

$$d\omega\_{dN}^{\prime} = D(t)(\Delta K)^m \tag{3}$$

where *m* is the material constant, generally close to the parameter *m* in simple fatigue; *D*(*t*) is related to the material medium system, loading frequency and load form and other factors, and is a function about time *t*, instead of the original constant *c*.

In a comprehensive analysis, the key to applying this method is to determine the range of different stress intensity factors ΔK, the initial corrosion pit size *a0*, the initial crack size *af*, and the critical crack size *ac* for these two segments. based on this, the life of each stage and the total life *N* are described by Eqs. (4) to (6).

$$N\_1 = \int\_{at\_0}^{t\_f} da \left[ \begin{matrix} \\ \overline{\mu} \end{matrix} \right]\_1 \tag{4}$$

$$N\_2 = \int\_{a\_f}^{a\_c} d\omega \left[\_{\overline{\text{dN}}}^{a}\right]\_2\tag{5}$$

$$N = N\_1 + N\_2 \tag{6}$$

#### **5.2 Safety assessment of damage cable**

Cable may fail in two ways after experiencing long-term accumulated damage: One way is that the damage makes the cable reach the critical state that is unsuitable for use and even makes the cable fail under the action of conventional load; another way is that the accumulated damage causes the cable resistance to decay. By now, although the damage does not reach the critical value, part of the steel wire or even the whole cable may suddenly fail when the lasso encounters the action of accidental extreme load. Therefore, it is necessary to effectively detect, monitor, and predict the process of damage accumulation of the cable, grasp the law that its resistance decay with the accumulation of damage, and timely repair and reinforce cable to ensure the safe operation of the cable structure and avoid the occurrence of accidents to the maximum extent possible [19–22].

(1) Evaluation of steel wire corrosion based on appearance: Relying on a unified hard and fast objective standard, so that each tester is bound by this standard and does not rely entirely on subjective guesses to draw conclusions.

(2) Assessment of effective strength reduction factor based on appearance quality: Usually based on the experimental analysis of some representative bridge cables, the empirical formula for regression analysis of cable's strength is used to assess the safety of cables or investigate the influence of defects and diseases of a large number of bridge cables on the bearing capacity of the bridge, determine a more reasonable strength reduction factor according to the investigation results, and then discount the strength of the cable. In this method, the effective resistance of the damaged cable is reduced, and the actual resistance of the damaged cable is equal to the standard value of the steel wire tensile strength of the cable multiplied by the strength reduction factor. The calculation formula is shown in Eq. (7):

$$R^{\pi} = \nu R^b \tag{7}$$

where *R<sup>τ</sup>* is the estimated actual tensile strength of the damaged cable, *Rb* is the design tensile strength value of the cable, and *v* is the strength reduction factor of the damaged cable.

(3) Safety assessment of cable strength based on the overall level of load effect: A detailed appearance survey is carried out for the cable, and the cable is classified according to the results of appearance survey, and its resistance calculation coefficient is determined according to the classification. The corresponding calculation coefficient *Z1* value is based on a large number of engineering accumulation and experimental statistics. Based on the detailed appearance investigation of the cable, the value of *Z1* can be referred to **Table 5**. The safety assessment of the cable based on the overall level of load effect is to judge the safety condition of the cable by comparing the relationship between the magnitude of the load effect and the resistance of the cable. *Z1* is used as the reduction factor of the resistance of the cable when calculating the resistance of the cable, and the actually allowable resistance of the cable can be calculated by Eq. (8):

$$\left[\boldsymbol{R}\right]^{\mathbf{r}} = \boldsymbol{Z}\_{\mathbf{k}} \mathbf{g} \left[\boldsymbol{R}\right]^{d} \tag{8}$$


**Table 5.** *Values of* Z*<sup>1</sup> discount factor for cable resistance.* *Corrosion Fatigue Behavior and Damage Mechanism of the Bridge Cable Structures DOI: http://dx.doi.org/10.5772/intechopen.109105*

where [*R*] *<sup>d</sup>* is the designed value of the cables' resistance and *Z1* is the reduction factor of the cables' resistance.

(4) Method that evaluates identification factor based on safety factor: For the safety evaluation of cable, the identification factor method from abroad can be used. In the design stage of cable-stayed bridges, a certain safety factor is usually taken for the design resistance of the cable, considering various factors such as material safety, load impact effect, and cable fatigue. The safety factor is defined as the ratio of the load effect to the resistance effect of the cable. Comparing the actual safety factor with the design safety factor, according to the design requirements, if *η*>[*η*], it means that the cable is in a safe state; if *η* = [*η*], it means that the cable is in a safe critical state; if *η*< [*η*], the smaller the measured safety factor *η* the less safe the cable is under stress. The specific calculation method is shown in Eq. (9):

$$
\eta = \mathbb{R}\zeta\_{\mathbb{S}} \tag{9}
$$

where *η* is the safety factor calculated by the actual measurement of the cable, *R* is the force of cable generated by the load effect, and *S* is the actual ultimate resistance of the cable.

(5) Assessment methods based on reliability theory: Safety after damage is an important indicator to evaluate the actual working performance of cable-stayed bridges in service. However, due to the differences in the working environment of the structure and the materials themselves, such as geometric characteristics, mechanical properties of the materials, load rating and distribution, and other parameters cannot be unified in the safety assessment of cableways, making the safety assessment of damaged cables uncertain. The assessment method of probabilistic reliability defines this uncertainty as a random variable to achieve the assessment of cable safety. According to the current unified standards for the design and definition of structural reliability of buildings, structural reliability is expressed as the probability that the structure will complete its intended function within a scheduled time and under specified conditions. For the cable, the safety function of the strength is first established as shown in Eq. (10):

$$Z = r - s \tag{10}$$

where *s* is the ultimate or design strength of the cables and *r* is the measured or estimated strength of the cables corresponding to *s*.

Since both *r* and *S* are random variables, *Z* is also a random variable. When *Z* > 0, the structure is in a reliable state; when *Z* = 0, the structure reaches the limit state; and when *Z* < 0, the structure is in a failure state. The static model of reliability analysis is applicable to the resistance of the structure, which hardly changes with time in the process of use, so the resistance can be regarded as a constant value. For real structures, on the other hand, the structural resistance is constantly changing over time, so the resistance of the existing structure must be simulated using a stochastic process. Assuming that the probability density function of *Z* is *f*(*Z*), the general representation of the component safety probability calculation formula is Eq. (11).

$$R = P(Z > 0) = \bigcap\_{0}^{\infty} f(Z)dZ \tag{11}$$

The key to the safety assessment of cable strength using the reliability method is to establish the functional function and determine the distribution parameter.

The general functional functions are divided into two types, one is based on the functional function of cable resistance and another is based on the safety factor. The structural resistance and load effects in the functional functions generally obey normal distribution, lognormal distribution, and Weibull distribution. The relevant research results suggest that for generally engineering design, it is reasonable and biased toward safety to use normal distribution for member resistance and load effects. For members with high safety requirements, lognormal or two-parameter Weibull distributions are recommended.

(6) Fatigue-based safety assessment of cable: For the safety analysis of fatigue strength of cable, the Palmgren-Miner linear cumulative damage theory is commonly used in engineering, and the corresponding stress-fatigue life curve is statistically analyzed and summarized through the steel wire fatigue experiment, form a standard for the service life of cables under different load cycles, which is used to predict and evaluate the fatigue safety performance of cable.
