Research on Equivalent Wind Load Spectrum Acquisition and Remaining Fatigue Strength Method of Double Active Arm Type Holding Pole

Abstract

In order to study the wind load fatigue residual strength in in-service holding poles for preventing disasters, and improving load-bearing capacity reliability. Firstly, using the Davenport wind speed spectrum to simulate the wind speed, it will be the input load of the finite element dynamics model of the 2 × 40 kN double active arm type holding pole (DAATHP). The stress time history curve at the welding section of the main chord at the bottom of the tower shaft is obtained at the working condition with the wind angle of 0°, 45°, 90°, and 135°. The rain flow counting method derives the statistical characteristics of the mean, amplitude, and frequency of stress at four wind angles. Secondly, the frequency of wind load under four wind angles is extended, and the synthesized fatigue load spectrum is used as the equivalent wind load spectrum. Finally, the Paris-Eadogan equation is used to derive the calculation formula of fatigue residual strength of the main chord at the bottom of the tower shaft, and the functional function of the main chord is established and then the failure probability formula is derived by combining the equivalent wind load spectrum. It is found that, compared with the ANSYS calculation results, the error of the predicted stress spectrum of the proposed method is minor, only 7.95%. It is verified that the proposed method can efficiently obtain the equivalent load spectrum of the DAATHP. The results indicate that it is feasible and practical to apply the proposed method to calculate the remaining fatigue life of the DAATHP.

1 Introduction

A holding pole is an everyday lifting and hoisting of mechanical equipment when lifting large components for ultra-high voltage (UHV) system transmission towers. Using a holding pole can shorten the construction period and reduce costs, so its reliability is directly related to the safety and quality of tower construction and is an essential guarantee for the smooth progress of the transmission project. Chi et al. [1] used finite element software to analyzed the overall stress state of the inner suspension holding pole used in the 1000 kV project. Lin et al. [2] established a finite element model of the 40 m all-steel internal suspended holding pole, and obtained the stress distribution of each part of the holding pole. Huang [3] developed a detection system to prevent the internal suspension holding pole from tilting due to inclination angle and tensile force exceeding the limit. Ding et al. [4] considered the influence of self-weight, lifting load, wind load, and lifting deflection, and find the weak parts in the structure. Zhang et al. [5] conducted finite element analysis on the inner suspension and external tension holding poles at different inclination angles, and the results showed that the tensile force, displacement, and stress increased with the increase of the inclination angle. Tan et al. [6] used the finite element method to analyze the mechanical properties of the DAATHP. The results showed that to ensure construction safety, the height of the free section should be reduced as much as possible. Wang et al. [7] used ANSYS software to analyze the static force of the holding pole under various typical working conditions to study its bearing law. The results showed that the axial force of the main chord of the tower structure was large, and the axial force of the oblique bar was relatively small.

Most existing studies focus on the holding pole’s stability, nonlinearity, and strength. In contrast, research on wind-induced fatigue damage and fatigue residual strength of the holding pole is rarely reported, which increases the unsafety of the holding pole structure. DAATHP is a common tool used in the lifting and assembly process of UHV transmission towers, so studying its working reliability under wind load has important practical and theoretical significance to reducing construction risks. From the above analysis, the main chord of the pole holding tower shaft, under the action of alternating stress, the initial crack of the welding section, easily expands and makes the pole fatigue failure. Hence, this paper takes the welding section of the main chord at the bottom of the tower shaft of 2 × 40 kN DAATHP as the research object, and the residual strength of its fatigue is studied.

Load spectrum is a key factor in calculating the remaining fatigue strength of structures. Due to the high randomness of wind load during the operation, the uncertainty of stress at the weld section of the main chord at the bottom of the tower is increased. This makes it impossible to directly use the measured wind load for theoretical analysis and engineering practice. Furthermore, due to the limitations of field test conditions, it is difficult to conduct a large number of actual load spectrum tests. There needs to be more research on the load spectrum of tower structures domestically and internationally, with most existing research focused on other structural fields, such as cranes. Joo et al. [8] conducted specimen tests and statistical analysis on carbon fiber epoxy resin composite materials, and established standards for the optimal load spectrum’s reliable truncation level and load coefficient. Wang et al. [9, 10] found a load spectrum based on damage consistency, aiming to reproduce the damage of the frame in working conditions by considering the difference between measured fatigue damage and calculated damage as the objective function. Li et al. [11, 12] obtained the load’s average frequency and amplitude-frequency matrix through the rain flow counting method, established load spectra for the main shaft and other structures, and applied them to fatigue life prediction. Through experiments, Pokorný et al. [13] determined the load spectrum under cyclic amplitude load for railway bearings. Zhu et al. [14] proposed a Bayesian method for correcting fatigue load spectrum for ship life estimation. Xu et al. [15] established a nonlinear mapping relationship between crane working cycles and different rated weights and lifting loads using a support vector machine model, which was the equivalent load spectrum of the crane.

The methods mentioned above to obtain load spectra are targeted at single or simple structures. At the same time, the holding pole belongs to a complex truss structure composed of many beams and rod components. The formation mechanism of the dangerous section load spectrum under wind load is still unknown. Therefore, it is urgently necessary to seek a quick and efficient method to obtain the load spectrum of the tower brace to evaluate its fatigue remaining strength.

In this study, the Davenport wind speed spectrum is used to simulate the wind speed, and the wind load during a working period is collected as the external load of the finite element dynamic simulation model. The axial stress time history at the weld section of the main chord at the bottom of the tower shaft is obtained considering the geometric nonlinearity effect at 0°, 45°, 90°, and 135° wind directions. The statistical characteristics of stress mean, amplitude, and frequency in the four wind directions are obtained using the rain flow counting method. The mean frequency of stress amplitude in the four wind directions is expanded, and the synthesized fatigue load spectrum is used as the wind load equivalent load spectrum. Based on linear elastic fracture mechanics theory and Miner fatigue damage accumulation theory, the theoretical formula for fatigue remaining strength of main chord components is derived using the Paris-Eadogan equation. The effectiveness of this method is verified by comparing the failure probability of the main chord obtained through the ANSYS stochastic finite element module.

2 DAATHP Tower Shaft Wind Load Equivalent Load Spectrum Acquisition

2.1 Equivalent Load Statistical Analysis

The 2 × 40 kN DAATHP is used as the research object, as shown in Fig. 1. The height of the holding pole H is 24.35 m, the self-weight is 16.883 kN, the component material is Q345B, the active arms on both sides of the holding pole are lifted at the same time, the lifting angle is maintained at 10°, the lifting load on both sides is 40 kN, the main specifications of the holding pole are as follows: (1) tower shaft: height 12 m, section 656 mm × 656 mm; (2) Tower cap: height 11.571 mm, bottom section size 970 mm × 970 mm, top section 650 mm × 650 mm; (3) Active arms: the length of the single arm is 11.37 m, the section of the active section is 420 mm × 520 mm, and the section of the middle arm section is 465 mm × 565 mm.

Fig. 1

2 × 40 kN DAATHP wind load diagram

Based on the Davenport wind speed spectrum Eq. (1), the harmonic synthesis method simulates the pulsating wind speed. The average wind speed at the height of 10 m is 26.8 m/s on the B-level ground and a working cycle is 300 s.

$$S_{v} (n) = 4K\overline{v}_{10}^{2} \frac{{x^{2} }}{{n(1 + x^{2} )^{(4/3)} }}$$

(1)

where, $\overline{v}_{10}$ is the average wind speed vector representation of pulsating wind force at an altitude of 10 m above the ground; x is the coefficient, $x = \frac{1200n}{{\overline{v}_{10} }}$; n is the pulsating wind frequency, HZ; K is the roughness factor of the ground.

In Fig. 1, F is the lifting load of the holding pole, θ is the angle between the active arm and the horizontal direction, v is the wind speed, and α is the angle between the wind speed and the DAATHP.

Wind load P(t) to which the pole is exposed:

$$P(t) = \frac{\gamma }{2g}\mu_{s} (z)A(z)[\overline{v}(z) + v_{d} (z,t)]^{2}$$

(2)

where, γ is the air bulk density, kg/m³; g is the acceleration due to gravity, kg/m³; ${\mu }_{s}\left(z\right)$ is the body shape coefficient at structural height z; $\overline{v}(z)$ is the average wind speed at height, m/s; ${v}_{d}(z,t)$ is the pulsating wind speed at height z, m/s; $A(z)$ is the effective windward area at structural height z, m².

The wind load is applied to the finite element model of the holding pole according to different heights. The axial stress time history curve at the welding section of the main chord at the bottom of the tower shaft is shown in Fig. 2 when α is 0°, 45°, 90° and 135°.

Fig. 2

Axial stress time history curve of the main chord at the bottom of the tower shaft

The axial stress time history curve corresponding to the wind direction angle of 0°, 45°, 90°, and 135° is calculated by using the rain flow counting algorithm, and the amplitude, average value, and frequency of axial stress at the welding section of the main chord at the bottom of the tower shaft are obtained as shown in Fig. 3.

Fig. 3

Rain flow counting results of axial stress

The lognormal, normal, and Weibull distributions test the axial stress relationship between amplitude-frequency and the mean-frequency at different wind angles. The results show that the relationship between the mean-frequency of axial stress under the four wind angles follows the normal distribution, and the relationship between the amplitude-frequency of axial stress obeys the Weibull distribution.

When the amplitude and mean distributions are independent, the amplitude-mean combined probability density function can determine the frequency of the load spectrum at all levels. According to Fischer’s theorem, two random variables approximately obey the chi-square χ² distribution with $(r - 1) \cdot (s - 1)$ degrees of freedom:

$${\chi }^{2}=\frac{1}{\lambda }{\sum }_{i=1}^{r}{\sum }_{j=1}^{s}\frac{(\lambda {\lambda }_{ij}-{\lambda }_{i}{\lambda }_{j}{)}^{2}}{{\lambda }_{i}{\lambda }_{j}}$$

(3)

where, λ is the sample size; r, s are the mean, magnitude grading; λ_i is the frequency at which the mean is at level i; λ_j is the frequency of the amplitude at the j-th level; λ_ij is the frequency with the mean at level i and the amplitude at class j.

According to the rain flow calculation results and the chi-square test results of the mean amplitude at the four wind angles, it can be seen that the amplitude and mean of the axial stress at the four wind angles are independent of each other when the significance level is 0.05, and the mean amplitude is combined with the probability density function $f(X,Y)$

$$f(X,Y)=\frac{1}{\sigma \sqrt{2\pi }}\frac{\alpha }{\beta }(\frac{Y-\varepsilon }{\beta }{)}^{\alpha -1}{e}^{-(\frac{Y-\varepsilon }{\beta }{)}^{\alpha }-(\frac{X-\mu }{\sigma \sqrt{2}})}$$

(4)

where, X, Y are the stress mean and amplitude; μ, σ are the normal distribution mean and standard deviation; α, β, ε are the shape, scale parameters, and threshold parameters of the Weibull distribution.

According to the normal distribution and Weibull distribution, the mean-frequency and amplitude-frequency histograms of the axial stress time history of the main chord at the bottom of the 2 × 40 kN DAATHP under four wind directions were curved fitted, as shown in Figs. 4 and 5, respectively.

Fig. 4

The mean of stress-frequency normal distribution fitting

Fig. 5

The mean of stress-frequency Weibull distribution fitting

The average and amplitude distribution fitting parameters of the axial stress of the main chord at the bottom of the tower shaft at four wind angles of 2 × 40 kN double rocker holding poles are shown in Table 1.

Table 1 Fitting parameters for mean axial stress and amplitude distribution

2.2 DAATHP Wind Load Synthesis and Extrapolation

To include the wind load condition with a small probability of occurrence, it is necessary to extend the frequency of the wind load and extrapolate the extreme value. The cumulative frequency is extended to 1.0 × 106 cycles, and the wind load expansion frequency under the four wind direction angles is calculated as follows:

$$N_{k} = N^{\prime } \left( {\frac{{m_{k} \lambda_{k} }}{{s_{k} }}} \right)/\sum\limits_{k = 1}^{4} {\left( {\frac{{m_{k} \lambda_{k} }}{{s_{k} }}} \right)}$$

(5)

where, N_k is the frequency of the spread of the wind angle k; N′ is the cumulative number of cycles in which extreme loads occur once; ${m}_{k}$ is the number of cycles obtained by counting the wind direction angle k rain flow; λ_k is the proportion of time occupied by the wind direction angle k; s_k is the actual time occupied by the wind angle k.

After the wind load is synthesized, the axial stress extremum and amplitude extremum of the main chord at the bottom of the tower shaft is the maximum values in each wind direction angle. The Ref. [16] of this paper divides the load spectrum into eight levels, the mean interval is divided by equidistant intervals based on the mean extremes, and the amplitude interval is divided by the ratio coefficient method based on the extreme amplitude of the non-equidistant interval. The ratio coefficients of each level are 1, 0.95, 0.85, 0.725, 0.575, 0.425, 0.275, and 0.125.

The axial stress cycle number N_kab corresponding to the amplitude interval of class a and class b at four wind angles is obtained by the mean amplitude combined probability density function $f(X,Y)$ by the following equation:

$${N}_{kab}={N}_{k}{\int }_{{\tau }_{a}}^{{\tau }_{a+1}}{\int }_{{\varphi }_{b}}^{{\varphi }_{b+1}}f(X,Y)dXdY$$

(6)

where, τ_a, τ_a+1 are the lower and upper limits of the mean stress of class a; φ_b, φ_b+1 are the lower and upper limits of the amplitude stress of class b.

The cyclic number N_ab of the mean of class a and the amplitude of class b in the eight-stage two-dimensional wind load equivalent load spectrum synthesized from multiple wind angles is obtained by the following equation:

$${N}_{ab}=\sum\limits_{k=1}^{4}{N}_{kab}$$

(7)

3 Theoretical Derivation of Fatigue Residual Strength of the Main Chord of the Tower Shaft

Aiming at the fatigue fracture, which is the primary failure form of the metal structure of the DAATHP, this paper takes the welding section of the main chord at the bottom of the tower shaft as the research object and uses the previous section method to obtain the two-dimensional wind load equivalent load spectrum, and uses the Paris-Eadogan equation to derive the fatigue residual strength calculation formula based on the linear elastic fracture mechanics theory and the Miner fatigue damage accumulation theory.

3.1 Miner Linear Damage Accumulation Theory and Equivalent Stress Amplitude

The DAATHP lifts the work piece while being subjected to random wind load, and the welding section of the main chord at the bottom of the tower shaft is subjected to cyclic stress during this process. Miner’s linear damage accumulation theory holds that under a given stress or strain level, the damage is linearly accumulated in the number of stress or strain cycles, and failure will occur when the damage accumulation reaches a particular critical value [17].

This paper uses the Miner linear damage accumulation theory to evaluate fatigue damage of the welding section of the main chord at the bottom of the tower shaft. Under the action of stress σ_i, the damage of the main chord of the holding pole after each stress cycle is 1/N_i, and the damage value after the n_i cycle is:

$${D}_{i}=\frac{{n}_{i}}{{N}_{i}}$$

(8)

The total damage accumulation value of the welding section of the main chord at the bottom of the tower shaft during fatigue failure is:

$$D=\sum\limits_{i=1}^{M}\frac{{n}_{i}}{{N}_{i}}$$

(9)

where, ${n}_{i}$ is the actual number of Class i stress cycles; ${N}_{i}$ is the number of failure cycles under Class i stress cycles.

When fatigue failure occurs in the main chord at the bottom of the tower shaft, D = 1. In classical fatigue theory, the S–N curve at a specific stress ratio R can be described by the Basquin equation:

$${\sigma }_{1}^{m}{N}_{1}={\sigma }_{2}^{m}{N}_{2}=C$$

(10)

where, ${\sigma }_{i}$ is the stress amplitude of Class $i$, $i=1,\hspace{0.33em}2$; ${N}_{i}$ is the number of failure cycles under Class $i$ stress cycles; m, C are the constants related to material, stress ratio, and loading mode.

When the welding section of the main chord of the holding tower shaft fails, the simultaneous (9) and (10) can be obtained:

$$\sum {\sigma }_{i}^{m}{n}_{i}={\sigma }_{j}^{m}{N}_{j}$$

(11)

To simplify the complex calculation, this paper adopts the Miner stress amplitude equivalence method, that is, replacing the stress amplitude of each stage in the variable amplitude stress with equivalent force amplitude, and the equivalent force amplitude formula is:

$${\sigma }_{eq}=\sqrt[m]{\sum {\sigma }_{i}^{m}\frac{{n}_{i}}{{N}_{f}}}$$

(12)

where, ${\sigma }_{eq}$ is the equivalent force amplitude; ${\sigma }_{i}$ is the stress amplitude of Class i; ${N}_{f}$ is the total number of cycles at failure; ${n}_{i}$ is the actual number of class i stress cycles; $m$ is the material constant, due to the lack of holding pole experimental data, this paper draws on the crane data in the literature [18], that is, Q235, m = 3.214, Q345, m = 7.806.

3.2 Fatigue Residual Strength Calculation Formula Based on Linear Elastic Fracture Mechanics Theory

The DAATHP is a welded structure, and the full fatigue strength of most welded structural parts is determined by the crack expansion stage [19, 20]. Therefore, according to the literature [21], the fatigue crack growth strength at the welding section of the main chord of the tower shaft is calculated by the fracture mechanics method, which is used as the fatigue residual strength of the DAATHP. The correlation between crack propagation velocity and stress intensity factor amplitude is obtained by the linear elastic fracture mechanics Paris formula:

$$\frac{dl}{dN}=C(\Delta K)^{n}$$

(13)

where, l is the crack length; N is the number of stress cycles; $\frac{dl}{dN}$ is the crack propagation velocity; n, C is the material constant; $\Delta K$ is the Stress intensity factor amplitude.

$$\Delta K={K}_{\max}-{K}_{\min}=Y \sigma \sqrt{\pi l}$$

(14)

where, Y is the stress intensity factor correction coefficient; σ is the amplitude of stress at the crack.

By substituting Eq. (14) into Eq. (13) and integrating to obtain the number of stress cycles experienced when the welding section of the main chord at the bottom of the pole holding tower shaft is extended from the initial crack ${l}_{0}$ to the failure crack ${l}_{f}$, that is, the fatigue residual strength of the holding pole, ${N}_{f}$ the formula is as follows:

$${N}_{f}=\frac{2({l}_{f}^{1-0.5n}-{l}_{0}^{1-0.5n})}{(2-n)C{\pi }^{n/2}(Y\sigma {)}^{n}}$$

(15)

Under the action of the wind load equivalent load spectrum, the functional function of the main material at the bottom of the tower shaft can be obtained from Eq. (16):

$$Z={N}_{f}-{N}_{ab}$$

(16)

According to the stress intensity interference theory, the failure probability of the main material at the bottom of the tower shaft is P_f:

$${P}_{f}=P(Z<0)=\int\limits_{-\infty }^{0}\frac{1}{{\sigma }_{z}\sqrt{2\pi }}\mathit{exp}\left[-\frac{1}{2}{\left(\frac{z-{\mu }_{z}}{{\sigma }_{z}}\right)}^{2}\right]dz$$

(17)

where, ${\mu }_{z}$ is the mean of the functional function; ${\sigma }_{z}$ is the standard deviation of the functional function.

4 Example

A finite element model of the main chord at the bottom of the 2 × 40 kN DAATHP is built, as shown in Fig. 6. The length is 1000 mm, one end is fixed, the other is free, and the free end is subjected to the axial force P = 40 kN, the transverse wind load is applied, and the cross-sectional size is 63 mm × 63 mm × 5 mm × 5 mm, considering the uncertainty of cross-section size, elastic modulus, and density, the random conditions are as follows: (a) the cross-sectional edge length and thickness are uniformly distributed by ±0.1; (b) The mean modulus of elasticity is 210 GPa, following the Gaussian distribution and the standard deviation is 0.05 times the mean; (c) The density follows a uniform distribution, with an average value of 7800 kg/m³; (d) Transverse wind velocity v adopts the Davenport wind speed spectrum, which obeys the Weibull distribution.

Fig. 6

The main chord at the bottom of the 2 × 40 kN DAATHP

The working time of the DAATHP is taken as 60 min, of which the time ratio of 0°, 45°, 90°, and 135° wind direction angle is 2:3:3:2. The two-dimensional wind load equivalent load spectrum in Eq. (7) is sorted out into a one-dimensional wind equivalent load spectrum by variable mean method and low–high–low loading sequence, as shown in Table 2.

Table 2 One-dimensional wind load equivalent load spectrum

According to the literature [20], for the main chord material is Q345B, take n = 3.5, C = 2.52 × 10^–13, for the general weld size, the stress concentration correction coefficient Y = 1.12 is used the initial crack length ${l}_{0}$ of the weld is 0.68 mm by measured and statistical analysis, according to the fracture test of the main chord of the holding pole, the standard lifting load of the DAATHP is taken as 40 kN. The literature shows that the weld’s failure crack length $l$ s is 95 mm [21]. The above parameters are combined with the stress amplitude-frequency and cumulative frequency substitution in Table 2 into Eqs. (11) and (12) to obtain the wind load equivalent stress spectrum and the equivalent force value at the welding section of the main chord at the bottom of the rod-holding tower. The residual fatigue strength at the welding section of the main chord at the bottom of the rod-holding tower is obtained by substituting the equivalent force value into Eq. (15). The comparison results of the failure probability of the primary metal and ANSYS stochastic finite element are obtained from Eqs. (16) and (17), shown in Table 3.

Table 3 Comparison of failure probabilities

It can be seen from Table 3 that by comparing the probability of failure of the main chord at the bottom of the tower shaft calculated by the ANSYS stochastic finite element method and the proposed method, it is found that the failure probability obtained by the proposed method is larger, but the error between the two methods is only 7.95%, indicating that the proposed method is feasible and effective.

5 Conclusions

The wind speed spectrum of Davenport is used to simulate the wind speed, the axial stress time history of the welding section of the main chord at the bottom of the tower shaft of the 2 × 40 kN DAATHP at the wind angle of 0°, 45°, 90°, and 135° is obtained by finite element method. The stress amplitude, average value, and frequency are statistically analyzed by the rain-flow counting method, the wind load frequency of the four wind directions is extended, the fatigue strength formula at the welding section of the main chord at the bottom of the tower shaft is derived, and the functional function is established. Based on this, the failure probability formula of the main chord at the bottom of the tower shaft is further derived, and specific conclusions are as follows:

• (1) The dynamic simulation model of the DAATHP can be established by using the finite element method, which can take into account the geometric nonlinearity of the structure and easily calculate the stress value at the welding section of the main chord at the bottom of the tower shaft, which provides a quick method for fatigue calculation of complex structures;

• (2) The average axial stress frequency of the main chord at the bottom of the tower shaft under different wind loads obeys the normal distribution, the axial stress amplitude-frequency follows the Weibull distribution, and the wind direction angle is different, and the distribution parameters are also different;

• (3) By comparing with the ANSYS stochastic finite element method, the main chord’s failure probability at the bottom of the tower shaft is 0.326, and the error is 7.95%, which verifies the feasibility and correctness of the proposed method.