Stochastic stress response and dynamic reliability evaluation for transmission towers with semi-rigid behaviors

Brief review of semi-rigid connected joints in STTTs

In STTTs, tube-gusset K-joints are specially designed to connect main chords and branch tubes, as illustrated in Figure 1. In detail, the K-joint consists of the main chord, branch tubes, C-type inserted plates, annular plates, gusset plates, and bolts for connecting gusset and C-type inserted plates. Therefore, this kind of K-joint is actually a semi-rigid connected joint, and its rotational stiffness $$ k $$ is an important parameter to determine its semi-rigid behaviors. Via determining the scope of common-used dimensions for the semi-rigid connected joints, a scope of rotational stiffness is obtained as [39.71, 333.06] (unit: kN·m/rad)^[21].

Figure 1. K-joint and steel tube member with semi-rigid behaviors.

Fitting formula for stability coefficient of STMs-SRB

Since the joint of STTTs is determined as a semi-rigid connection, the steel tube connected by this joint can be regarded as a member with semi-rigid behaviors. In our previous work^[19], the experimental investigation and the numerical analysis have been conducted to study the stability behavior of STMs-SRB, and their comparison shows that the numerical analysis results fit well with the experimental results, as shown in Figure 2.

2. Comparison of experimental investigation and numerical analysis for stability behaviors of STMs-SRB.

In order to assess the bearing capacity of STMs-SRB by a fitting formula of the stability coefficient, the parametric numerical analysis is conducted in this paper by adopting the verified FE model of STMs-SRB, and the numerical analysis results could be further as a sample set to fit the stability coefficient of STMs-SRB. The varying parameters of the parametric numerical analysis include the rotational stiffness and the slenderness ratio, in which the rotational stiffness is taken as 50.99 kN$$ \cdot $$m/rad kN$$ \cdot $$m/rad, 102.32kN$$ \cdot $$m/rad, 201.34kN$$ \cdot $$m/rad, and 333.06kN$$ \cdot $$m/rad among the certain range of rotational stiffness, respectively, and the slenderness ratio is taken from 50 to 200. Through the parametric numerical analysis, a number of samples for the stability bearing capacity of STMs-SRB are obtained. Then, the stability coefficient of STMs-SRB under axial compression is defined as

where $$ \varphi $$ denotes the stability coefficient of STMs-SRB; $$ A $$ denotes the area of cross-sections of STMs-SRB; $$ f_{y} $$ is the yield strength of steel materials; $$ F_{s} $$ denotes the stability bearing capacity of STMs-SRB. Thereby, a number of samples for the stability coefficient of STMs-SRB are obtained. Finally, the fitting formula can be fitted for the stability coefficient of STMs-SRB based on these samples.

Combined with the least squares fitting method and the quadratic polynomial regression, the fitting formula for the stability coefficient $$ \varphi $$ of STMs-SRB is expressed as

where $$ \lambda $$ denotes the slenderness ratio; $$ E $$ is the elastic modulus, $$ \alpha_{1} $$, $$ \alpha_{2} $$, and $$ \alpha_{3} $$ are the coefficients that are fitted by samples of the stability coefficient of STMs-SRB. The fitting results are shown in Table 1.

The comparison of the stability coefficient between fitting and numerical results is shown in Figure 3A. It can be observed that the fitting results are in good agreement with the numerical results, and the maximum relative error is 2.8%. In addition, within the range of rotational stiffness, the stability coefficient can be obtained by the linear interpolation. The prediction of the stability coefficient by the proposed fitting formula is shown in Figure 3B, and the maximum relative error is 2.2%, indicating that the proposed fitting formula has satisfactory accuracy in predicting the stability coefficient of STMs-SRB.

Figure 3. Accuracy validation of proposed fitting formula. (A) Comparison of fitting and numerical results; (B) Prediction by proposed fitting formula.

In STTTs, there are two typical kinds of steel-tube members, namely, the tension-bending and compression-bending members. Tension-bending members that are subjected to the bending moment and axial tension force always exhibit strength failure. Regarding compression-bending members that are subjected to the bending moment and axial compression force, their failure is controlled by their stability behavior. However, in the dynamic analysis, it is slightly impracticable to concentrate on the time-efficiency and mature method for estimating the buckling point for all tower members at all-time steps. Thus, the critical strength approach suggested by design specifications is adopted to determine the dynamic stress of members^[5]; i.e., the dynamic stress of steel-tube members is calculated via their stability coefficient.

For compression-bending members, their dynamic stress could be calculated by^[22]

where $$ \sigma_{c}(t) $$ denotes the dynamic stress of compression-bending members; $$ N(t) $$ denotes the dynamic axial compression force; $$ \varphi $$ denotes the stability coefficient, which is calculated by the proposed fitting formula for the stability coefficient of STMs-SRB, i.e., Equations (2)-(4); $$ W_{s} $$ denotes the cross-section modulus; $$ \gamma_{\mathrm{m}} $$ denotes the plastic development coefficient of cross-sections; $$ {N_{\mathrm{E}}^{\prime}}=\pi^{2}EA/({1.1\lambda^{2}}) $$; $$ M(t) $$, $$ \xi_{z}(t) $$ and $$ \xi_{y}(t) $$ are expressed as

where $$ M_{yj} $$ and $$ M_{yi} $$ denote the bending moment around the $$ y $$-axis at nodes $$ j $$ and $$ i $$, respectively; $$ M_{zj} $$ and $$ M_{zi} $$ denote the bending moment around the $$ z $$-axis at nodes $$ j $$ and $$ i $$, respectively; $$ \lambda $$ denotes the slenderness ratio; $$ N_\mathrm{E} $$ = $$ \pi^{2}EA/\lambda^{2} $$; $$ M_{1z}(t) $$, $$ M_{2z}(t) $$, $$ M_{1y}(t) $$, and $$ M_{2y}(t) $$ are expressed as

Regarding tension-bending members, due to their failure controlled by their strength, their dynamic stress $$ \sigma_{t}(t) $$ can be determined by Equation (5) as long as let $$ \varphi=1 $$. Meanwhile, the dynamic axial compression force needs to be replaced with the dynamic axial tension force.

Stochastic dynamic FE model of STTTs-SRB under stochastic wind excitations

A transmission tower can be considered as an integrated structural system. Assume that $$ \mathit{\boldsymbol{\Theta}}={[\mathit{\boldsymbol{\Theta}}_{s}, \mathit{\boldsymbol{\Theta}}_{w}]}^\mathrm{T} $$ represents a random vector including stochastic structural parameter vector $$ \mathit{\boldsymbol{\Theta}}_s $$ (namely, materials and dimensions) and stochastic load parameter vector $$ \mathit{\boldsymbol{\Theta}}_w $$ (namely, wind loads). Thus, the randomness of the system is only presented by $$ \mathit{\boldsymbol{\Theta}} $$. Without loss of generality, the stochastic dynamic FE model of STTTs-SRB under stochastic wind excitations could be given as the following matrix forms.

where $$ \ddot{\boldsymbol{\delta}}(t) $$, $$ \dot{\boldsymbol{\delta}}(t) $$, and $$ \boldsymbol{\delta}(t) $$ are the vector of acceleration, velocity, and displacement of nodes, respectively; $$ \mathbf{K}\left(\boldsymbol{\Theta}_{s}\right) $$, $$ \mathbf{M}\left(\boldsymbol{\Theta}_{s}\right) $$, and $$ \mathbf{C}\left(\boldsymbol{\Theta}_{s}\right) $$ are the stochastic stiffness, mass, and damping matrices of STTTs-SRB, respectively, which are determined in the following subsections; $$ \mathbf{F} $$($$ \boldsymbol{\Theta}_{w} $$, $$ t $$) is the external stochastic wind load vector and is calculated as^[23]

where $$ \bar{v}(h) $$ is the mean wind speed at the tower height of $$ h $$; $$ \boldsymbol{v}\left(\boldsymbol{\Theta}_{w}, t\right) $$ is the stochastic vector of fluctuating wind speeds, which is simulated and determined by the dimension-reduced probabilistic simulation approach^[17]; $$ \mu_{s} $$ is the drag coefficient of transmission towers; $$ \rho_{a} $$ is the air density; $$ A_{s} $$ is the projected area of transmission towers.

Stochastic stress response analysis of STTTs considering effects of semi-rigid connected joints and semi-rigid-constrained stability behaviors

According to the principle of probability conservation^[25], the GDEE for the concerned response $$ H(\boldsymbol{\Theta}, t) $$ (e.g., the dynamic stress of STMs-SRB $$ \sigma (\boldsymbol{\Theta}, t) $$ of STTTs-SRB can be expressed in the differential form^[20,26] or the integral form^[27–30] as

where $$ p_{H \boldsymbol{\Theta}}(H, \boldsymbol{\Theta}, t) $$ is the joint PDF of $$ H $$ and $$ \boldsymbol{\Theta} $$; $$ \dot{H}(\boldsymbol{\Theta}, t) $$ is the derivative of $$ H(\boldsymbol{\Theta}, t) $$ with respect to time; $$ p_H(H, t) $$ is the PDF of $$ H(\boldsymbol{\Theta}, t) $$; $$ p_{\boldsymbol{\Theta}}(\boldsymbol{\Theta}) $$ is the PDF of the random vector; $$ \delta(\cdot) $$ denotes the Dirac delta function; $$ g(\cdot) $$ denotes a monotone function of mapping such that $$ \boldsymbol{\Theta}=g^{-1}(H, t) $$ holds true in stochastic dynamic systems.

Actually, the integral form of the GDEE is equivalent to the differential form of the GDEE in the sense of probability conservation, and the former can deduce the latter^[30]. However, it is worth noting that solving the GDEE in the differential form needs the derivative of responses. In this paper, we are concerned about the stress of members, while the derivative of the stress of members may not be easily obtained. Accordingly, due to no need for the derivative of responses, solving the GDEE in the integral form might be a more appropriate way for this paper.

Generally, the analytical solution of the GDEE is hard to obtain owing to the complex behavior of dynamic response and the complexity of mapping. To this end, two techniques are introduced in the GDEE: (1) partition of input probability space; (2) smoothing of Dirac delta function. Based on the Voronoi cell-based partition technique and GF-discrepancy-based point selection strategy^[31], the former can reduce the numerical error by generating representative points and assigned probabilities that represent the probability space. The latter aims to improve the integral accuracy by smoothing the integrand, e.g., the Gaussian function. By employing a family of Dirac's sequences^[27–29] based on the normal distribution, Equation (22) can be further presented as

where $$ \Omega_\Theta $$ is the probability space of $$ \boldsymbol{\Theta} $$; $$ n_{r p t} $$ is the number of representative points; $$ \sigma_d $$ denotes the smoothing parameter of the Dirac delta function; $$ \boldsymbol{\theta}_q $$ denotes the $$ q $$-th representative point in the input probability space; $$ P_q $$ is the assigned probability of the $$ q $$-th representative point, which is given as

where $$ \Omega_q $$ denotes the probability space of the $$ q $$-th representative point and meets requirements of $$ \Omega_q \cap \Omega_p=\emptyset $$, $$ \forall p \ne q $$, and $$ \cup_{q=1}^{n_{r p t}}\Omega_q=\Omega_\Theta $$.

Dynamic reliability evaluation of STTTs considering effects of semi-rigid connected joints and semi-rigid-constrained stability behaviors

Based on the extreme-value distribution method^[20], once the critical dynamic response exceeds the safety limit at a time instant, the structure will be considered to have failed. Therefore, the key stochastic dynamic response at each discrete time instant can compose a series system in the time domain $$ [0, T] $$.

In transmission towers, the stress response of members can be constructed as an extreme-value variable. Further, the performance function of the extreme-value variable can be constructed as

where $$ Z_{e x t} $$ is the performance function of the extreme-value variable. The dynamic reliability of STTTs considering effects of semi-rigid connected joints and semi-rigid-constrained stability behaviors can be evaluated by the failure probability and reliability index below

where $$ P_f $$ denotes the failure probability; Pr($$ \cdot $$) denotes the probability of the random event in the bracket; $$ \beta $$ denotes the reliability index; $$ \Phi^{-1}(\cdot) $$ denotes the inverse cumulative density function (CDF) of standard normal distributions.

Implementation procedure

In general, the flowchart of the stochastic stress response analysis and dynamic reliability evaluation of STTTs considering effects of semi-rigid connected joints and semi-rigid-constrained stability behaviors is shown in Figure 4, and the corresponding implementation procedure can be summarized as follows.

Figure 4. Flowchart of stochastic stress response analysis and dynamic reliability evaluation.

Step 1: Take the random variables of STTTs-SRB (namely, semi-rigid connections, wind loads, dimensions, and materials).

Step 2: Determine the representative point set and obtain its corresponding assigned probability by the GF-discrepancy-based point selection strategy. Step 3: Simulate the stochastic wind loads of STTTs based on the dimension-reduced probabilistic simulation approach.

Step 4: Determine the structural characteristics of STTTs-SRB, including the stiffness, mass, and damping matrices, to construct their stochastic dynamic FE model.

Step 5: Carry out the deterministic dynamic response analysis at each representative point and obtain the sample set of the stochastic stress response.

Step 6: Carry out the stochastic dynamic response analysis by solving the GDEE via a family of Dirac's sequences.

Step 7: Obtain the PDF evolution process of the stochastic stress response of STTTs considering effects of semi-rigid connected joints and semi-rigid-constrained stability behaviors.

Step 8: Evaluate the dynamic reliability of STTTs considering effects of semi-rigid connected joints and semi-rigid-constrained stability behaviors based on the extreme-value distribution method.

Description of the transmission tower

An STTT is adopted to assess its wind-induced dynamic stress. Because of unpredictable computational cost involved in such a complex and huge structural form, it is quite time-consuming to calculate the wind speed for all nodes of transmission towers. Accordingly, for simplification, transmission towers are often properly divided into a number of simulation areas along the tower height, and each simulation area is represented by a point to calculate the wind speed^[23,32]. By adopting this way, the employed transmission tower is simplified as nine appropriate simulation points to calculate the wind speed, as illustrated in Figure 5. Some design parameters of the tower and lines are listed in Table 2. It is assumed that the wind direction is perpendicular to the transverse plane, and the span of the employed tower is taken as 500 m.

Figure 5. Simulation areas and points of wind loads.

By determining the stiffness, mass, and damping matrices of the STTT and the wind load vector, its dynamic FE model is established. For better properties of efficiency and fitting actual engineering practice, the gusset plates and bolts of joints are not established in the FE model^[14,17,32], and the semi-rigid connections are adopted to represent the joints^[23]. It is assumed that the elastic modulus, the Poisson's ratio, and the density of steel material are taken as 206 GPa, 0.3, and 7, 850 kg/$$ \mathrm{m}^{3} $$, respectively. Moreover, the damping ratio of the tower structure is considered to be 0.02^[5]. It is worth pointing out that the nonlinearity barely affects the structural dynamic response of STTTs-SRB at the design wind speed according to the pre-analysis. However, there are iterations in the nonlinear analysis at all-time steps, and thus, the nonlinear analysis is much more time-consuming than the linear analysis. Therefore, the linear analysis is conducted in this paper.

Determination of random variables

Some basic random variables are selected, and their statistical properties are listed in Table 3. Referring to the probabilistic model code^[33], the probability distribution of the elastic modulus, Poisson ratio, and yield strength is taken as the lognormal distribution, and the probability distribution of the thickness and outer diameter of steel tube members is taken as the normal distribution. Meanwhile, the probability distribution of the damping ratio is also taken as the normal distribution^[34]. In addition, the fluctuating wind speed is simulated as a stochastic process by taking two independent random variables ($$ \mathit{\Theta}_{1} $$ and $$ \mathit{\Theta}_{2} $$) that both follow the uniform distribution in the interval of $$ [0, 2\pi] $$^[17]. Moreover, the rotational stiffness of semi-rigid connected joints is taken as the uniform distribution^[23]. The rationality of the aforementioned assumption of probability distributions of the random variables has been proven in the previous studies^[17,19,23] on the stochastic response analysis and reliability evaluation of transmission towers. As for the representative point set, the sample size of the point set is taken as 500.

Results of stochastic stress response and dynamic reliability

Generally, the maximum response of structures is the key and concerned response. Accordingly, in this practical example, the maximum member stress in transmission towers is selected to illustrate the results of stochastic stress responses of the STTT considering effects of semi-rigid connected joints and semi-rigid-constrained stability behaviors. The location of the maximum member stress in the transmission tower is shown in Figure 6. Figure 7 shows the PDF evolution process of the maximum member stress of the STTT considering effects of semi-rigid connected joints and semi-rigid-constrained stability behaviors within the time interval from 100 s to 500 s. It can be clearly seen that the evolution process of the PDF surface and contour with the time-variant peak is noticeably non-smooth, which indicates that the stress of members induced by the random variables presents strongly stochastic and time-variant characteristics. Correspondingly, Figure 8 shows the time histories of the mean and standard deviation (STD) of maximum member stress. Similar to Figure 7, the time histories of the mean and STD of maximum member stress are time-variant, in which its mean fluctuates around 100 MPa and STD fluctuates around 40 MPa.

Figure 6. Location of the maximum member stress in the transmission tower.

Figure 7. PDF evolution process of maximum member stress within the time interval [100 s-500 s]. (A) The PDF surface; (B) The PDF contour.

Figure 8. Time histories of the mean and STD of maximum member stress. (A) The mean; (B) The STD.

In terms of the dynamic reliability, the reliability index of the STTT considering effects of semi-rigid connected joints and semi-rigid-constrained stability behaviors is calculated as 4.1039 according to the established performance function of extreme-value variables.

Comparisons of analysis results of transmission towers with different connections and stability coefficients

In order to study the effects of semi-rigid connected joints and semi-rigid-constrained stability behaviors on the structural dynamic response and dynamic reliability of transmission towers, the stochastic stress response analyses and reliability evaluations of transmission towers considering pinned and rigid connected joints are also conducted, respectively. It is noted that once the pinned or rigid connected joint is adopted for the transmission tower, the corresponding stability coefficient would change with it. Therefore, in the following subsections, when the different (semi-rigid, rigid, and pinned) connections are adopted, the corresponding stability coefficients are calculated by members with distinct (semi-rigid, rigid, and pinned) connections, respectively.

A sample of stochastic stress response with different connections and stability coefficients

To investigate the effects of semi-rigid connected joints and semi-rigid-constrained stability behaviors on the value of structural responses of transmission towers, the comparisons of a sample of maximum member stress with various connections and stability coefficients are shown in Figure 9. Generally, the variation trend of samples of maximum member stress with rigid and pinned connected joints is basically consistent with that with semi-rigid connected joints. The sample of maximum member stress with semi-rigid connected joints is between that with rigid and pinned connected joints. Table 4 shows the statistical parameters of time-history curves for towers with different connections and stability coefficients, including the mean and STD. Specifically, the mean and STD of time-history curves for the transmission tower considering semi-rigid connected joints are both larger than those considering rigid connected joints while are less than those considering pinned connected joints. This can be explained by the fact that the stress of members is composed of the stress caused by axial forces and bending moments [i.e., Equation (5)]. Although pinned connected joints ignore bending moments, the stability coefficient of members with pinned behaviors is the smallest among the three connection types, leading to the relatively large stress caused by axial forces. On the contrary, rigid connected joints cause the largest bending moments, but the largest stability coefficient of members with rigid behaviors leads to the relatively small stress caused by axial forces. Thus, after comprehensively considering the two parts of the stress, the stress calculated by Equation (5) would present the maximum value for members with pinned behaviors, while that would present the minimum value for members with rigid behaviors, and that would present the median value for members with semi-rigid behaviors.

Figure 9. Comparisons of a sample of maximum member stress with different connections and stability coefficients.

Table 4

Comparisons of mean and STD of time-history curves with different connections and stability coefficients (unit: MPa)

	Rigid connection	Semi-rigid connection	Pinned connection
Mean	72.7135	86.3197	101.1282
STD	23.9091	29.3765	35.7644
Maximum value	258.8285	86.3197	297.2877

Stochastic stress response of towers with different stability coefficients and connections

Figure 10 compares the PDF curves of maximum member stress for towers with different stability coefficients and connections at three typical time instants, i.e., $$ t=100 $$ s, $$ t=300 $$ s, and $$ t=500 $$ s. It can be observed that the PDF shows different curves at various time instants; the complexity of PDF evolutions of the maximum member stress is further illustrated. Furthermore, the different connection and stability coefficients obviously affect the PDF curves of the maximum member stress. Figure 11 shows the comparisons of time-history curves of the mean and STD of PDF for transmission towers with different stability coefficients and connections. It can be found that the time-history curves of the mean and STD for the transmission tower considering semi-rigid connected joints are both higher than those considering rigid connected joints while being lower than those considering pinned connected joints.

Figure 10. Comparison of PDF for the maximum member stress with different stability coefficients and connections at three typical time instants.

Figure 11. Comparisons of the mean and STD of PDF for maximum member stress with different stability coefficients and connections. (A) The Mean; (B) The STD.

Dynamic reliability of transmission towers with different stability coefficients and connections

The PDF and CDF of the performance function with different stability coefficients and connections are compared in Figure 12. It can be clearly observed that the PDF and CDF curves of the performance function with pinned connected joints are closest to the failure boundary (i.e., 0 MPa), and those with rigid connected joints are farthest from the failure boundary. The PDF and CDF of the performance function with semi-rigid connected joints are between those with pinned and rigid connected joints. Furthermore, the dynamic reliability of transmission towers with different stability coefficients and connections is compared in Table 5. It can be obtained that the reliability index of the transmission tower considering rigid connected joints is calculated as 5.3967 and that considering pinned connected joints is calculated as 3.0304. The reliability index of the transmission tower considering semi-rigid connected joints is smaller than that considering rigid connected joints and is larger than that considering pinned connected joints.

Figure 12. Comparisons of PDF and CDF of the performance function with different stability coefficients and connections. (A) The PDF; (B) The CDF.

Table 5

Comparisons of reliability indexes for transmission towers with different stability coefficients and connections

	Rigid connection	Semi-rigid connection	Pinned connection
Reliability index	5.3967	4.1039	3.0304

In summary, the semi-rigid connected joints and semi-rigid-constrained stability behaviors would significantly affect the stochastic stress response and dynamic reliability of STTTs. The time-history curves of the mean and STD of member stress of STTTs considering semi-rigid connected joints and semi-rigid-constrained stability behaviors are between those considering rigid and pinned connected joints and corresponding constrained stability behaviors. Similarly, the dynamic reliability of STTTs considering semi-rigid connected joints and semi-rigid-constrained stability behaviors is smaller than that considering rigid connected joints and rigid-constrained stability behaviors and is larger than that considering pinned connected joints and pinned-constrained stability behaviors. Thus, it is essential to take into account the semi-rigid behavior for STTTs, and the adoption of pinned or rigid behavior in the traditional design and analysis may be inappropriate for these structures.

Acknowledgments

The work presented in this paper was fully supported by the Special Support of Chongqing Postdoctoral Research Project (Grant No. 2022CQBSHBT3009) and the Special Postdoctoral Support Project of Chongqing Research Institute of HIT (Grant No. KY506023002). The authors would like to express their gratitude for all the support received

Authors' contributions

Methodology, validation, formal analysis, software, investigation, data curation, visualization, and writing - original draft: Tang ZQ Conceptualization, methodology, software, resources, supervision, and writing - review and editing: Wang T Project administration and writing - review and editing: Li ZL Writing - review and editing: Lu DG, Tan YQ

Availability of data and materials

Some or all data and materials that support the findings of this study are available from the corresponding author upon reasonable request.

Financial support and sponsorship

The research received funding from the Special Support of Chongqing Postdoctoral Research Project (Grant No. 2022CQBSHBT3009) and the Special Postdoctoral Support Project of Chongqing Research Institute of HIT (Grant No. KY506023002).

Conflicts of interest

All authors declared that there are no conflicts of interest.

Ethical approval and consent to participate

Not applicable.

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Copyright

© The Author(s) 2023.