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DOI: 10.11776/j.issn.1000-4939.2025.03.011
Study on Deflection and Additional Bending Moment of Multi-Pyramidal Steel Tubular Poles Considering Flange Bolt Slip
PAN Feng^1, CHEN Saihui^2, CHEN Yuan^3, DING Lei^1
(1. Zhejiang Electric Power Design Institute, China Energy Engineering Group Co., Ltd., 310012 Hangzhou;
2. State Grid Zhejiang Economic Research Institute, 310016 Hangzhou; 3. Department of Civil Engineering, Zhejiang University, 310030 Hangzhou)
Abstract: Multi-pyramidal steel tubular poles are widely used in urban peripheral transmission lines, and flange connections are usually adopted for connecting one tube to another. In view of the relatively limited theoretical calculation methods currently available for the deflection of multi-pyramidal steel tubular poles, this study, based on the differential-equation theory of the deflection curve of beams and incorporating bolt slip at flange joints, derives calculation methods for the deflection and additional bending moment of multi-pyramidal steel tubular poles under common loads (bending moment $M$, horizontal force $P$, and uniformly distributed force $q$). It is recommended that the slip of the flange plate be calculated according to the clearance of the bolt holes, and that the corresponding rotation amplification factor be considered. Finally, typical steel tubular poles are selected for comparative analysis, and several important conclusions are obtained: the displacement calculated without considering flange bolt slip has an error of about 7% compared with the finite-element solution, and the deflection caused by the additional bending moment accounts for about 2% of the total deflection.
Keywords: transmission line; tapered steel tubular pole; bolt slip; moment of inertia of section; deflection; additional bending moment
Chinese Library Classification No.: TU311
Document Code: A
Article No.: 1000-4939(2025)03-0580-10
Study on deflection and additional bending moment of multi-pyramidal steel pole considering flange bolt slip
PAN Feng^1, CHEN Saihui^2, CHEN Yuan^3, DING Lei^1
(1. Zhejiang Electric Power Design Institute, China Energy Engineering Group Co., Ltd., 310012 Hangzhou, China;
2. State Grid Zhejiang Economic Research Institute, 310016 Hangzhou, China;
3. Department of Civil Engineering, Zhejiang University, 310030 Hangzhou, China)
Abstract: Multi-pyramid steel pipe pole is widely used in transmission lines around cities, and flange connection is usually used for pipe to pipe connection. In view of the relatively single theoretical calculation method for the deflection of multi pyramid steel pipe pole, this paper will combine the bolt slip of flange joint according to the deflection curve differential equation theory of beam, and deduce the common load (bending moment $M$, horizontal force $P$ and uniformly distributed force $q$) calculation method of deflection and additional bending moment of multi pyramid steel pipe pole under action. It is suggested to calculate the slip amount of flange plate according to the clearance of bolt hole and consider the corresponding angle amplification factor. Finally, some important conclusions are drawn from the comparative analysis of typical steel pipe pole. The displacement value calculated without considering the slip of flange bolt has about 7%
Date received: 2023-01-16
Fund project: National Natural Science Foundation of China-funded project (No. 51878607)
Corresponding author: PAN Feng, Senior Engineer. E-mail: qiushif@163.com
Citation format: PAN Feng, CHEN Saihui, CHEN Yuan, et al. Study on deflection and additional bending moment of multi-pyramidal steel tubular poles considering flange bolt slip[J]. Chinese Journal of Applied Mechanics, 2025, 42(3): 580-589.
PAN Feng, CHEN Saihui, CHEN Yuan, et al. Study on deflection and additional bending moment of multi-pyramidal steel pole considering flange bolt slip[J]. Chinese journal of applied mechanics, 2025, 42(3): 580-589.
error with the solution of finite element method, and the deflection caused by additional bending moment accounts for about 2% of the total deflection.
Key words: transmission line; taper steel pipe pole; bolt slip; section moment of inertia; deflection; additional bending moment
With the rapid development of urban construction, corridor resources for line routes have become very tight, and overhead steel tubular poles are one of the main means of effectively solving corridor problems. Compared with transmission towers, steel tubular poles have such advantages as a small footprint, small line corridors, and an attractive appearance. However, the design of steel tubular poles and the control of pole-top displacement are key and difficult points in design: structural safety must be ensured, while technical economy must also be achieved. Therefore, design codes for steel tubular poles at home and abroad all contain corresponding displacement-control requirements.[1–2] At present, the connections between steel tubular pole segments mainly include the following three types[3]: circumferential butt-welded connection, flange connection, and sleeve connection, among which flange connection is the mainstream connection type.
At present, many studies have been carried out at home and abroad on displacement calculation for steel tubular poles, whereas relatively few studies have addressed displacement calculation methods for sleeve connections. Zhu Zhengyi et al.[4–5] derived deformation calculation formulas for tapered steel tubes under various loads and, together with these formulas, compiled a set of coefficient tables for use. Huang Jiayin et al.[6] and Nie Guoyi[7] used the virtual unit-load method and Euler’s critical-force method, respectively, to calculate the deflection of tapered poles; by using Mohr integration, the deflection at any section height of a pole can be obtained, and general curves were provided for the convenience of designers. Wang Jianhua et al.[8–11], Wang Yanshan et al.[12], and Lin Shibao[13], starting from the approximate differential equation of the deflection curve of a beam, derived analytical expressions for calculating the deflection of steel tubular poles. Luo Lie et al.[14] investigated the load-bearing performance and force-transfer mechanism of plug-in joints, obtained a calculation method for bending-resistance stiffness reduction, and verified the correctness of using simplified formulas to calculate the overall deformation of plug-in single-tube towers. Xu Jianshe et al.[15] used numerical methods to give calculation formulas for bolt-hole wall deformation and connection slip. Huang Weidong et al.[16] carried out numerical simulation of existing single-bolt slip tests by the finite element method. Yao Kuan et al.[17] and Yang Fengli[18] proposed, through finite element analysis, that bolt slip affects the axial force of the main diagonal member of the tower body and the deformation of the steel tower.
In summary, given that there are relatively few studies on calculation methods for the deflection of polygonal tapered steel tubular poles, this study, based on the differential-equation theory for the deflection curve of beams and combined with a flange-joint bolt-slip model, derives calculation methods for the deflection and additional bending moment of polygonal tapered steel tubular poles under common loads (bending moment \(M\), horizontal force \(P\), and uniformly distributed load \(q\)), so as to provide a reference for the calculation and design of flange-connected steel tubular poles.
1 Sectional Characteristics of Steel Tubular Poles
Among the most commonly used cross-sectional forms of steel tubular poles, 12-sided and 16-sided sections are the most common, as shown in Fig. 1. The integral formula for the section moment of inertia is[13]
\[ I=\omega \cdot \left(D^{3}t+Dt^{3}\right),\quad A_{\mathrm{g}}=\omega_{\mathrm{g}}\cdot D\cdot t \tag{1} \]
where \(D\) is the distance across flats of the steel tube; \(t\) is the steel-tube thickness; \(\omega\) is the coefficient of section moment of inertia; \(A_{\mathrm{g}}\) is the sectional area; and \(\omega_{\mathrm{g}}\) is the area coefficient. According to the ASCE definition[2], the values of \(\omega\) and \(\omega_{\mathrm{g}}\) are shown in Table 1.
Fig. 1 Typical section (16 side) and calculation diagram of steel pipe pole
Table 1 The value of section moment of inertia coefficient
| Parameter | 4-sided | 6-sided | 8-sided | 12-sided | 16-sided | Circular |
|---|---|---|---|---|---|---|
| \(\omega\) | 0.666 | 0.481 | 0.438 | 0.411 | 0.403 | 0.393 |
| \(\omega_{\mathrm{g}}\) | 4.000 | 3.460 | 3.320 | 3.220 | 3.190 | \(\pi\) |
When \(D/t \ge 10\), the \(Dt^{3}\) term in Eq. (1) can be neglected[13]; therefore, the section moment of inertia in Eq. (1) can be expressed as
\[ I=\omega D^{3}t \tag{2} \]
According to Eq. (2), the stiffness expression for the top section of a tapered steel tubular pole is
\[ EI_{0}=E\cdot \omega \left(D_{0}-t\right)^{3}t \tag{3} \]
where \(D_{0}\) is the outside distance across flats of the top steel-tube section, and \(E\) is the elastic modulus of steel.
For any height \(x\), the sectional stiffness can be expressed as
\[ EI_{x}=E\cdot \omega \left(D_{x}-t\right)^{3}t =E\cdot \omega \left[\left(D_{0}-t\right)+2i\cdot x\right]^{3}t \tag{4} \]
where \(i\) is the taper of the steel tube, \(i=\tan \alpha\). Thus Eq. (4) can be further expressed as
\[ \begin{aligned} EI_{x} &=E\cdot \omega \left(D_{0}-t\right)^{3}t \left(1+\frac{2i}{D_{0}-t}\cdot x\right)^{3} \\ &=EI_{0}\left(1+\frac{2i}{D_{0}-t}\cdot x\right)^{3} \end{aligned} \tag{5} \]
To simplify the calculation, the variation of the steel pipe pole stiffness along the pole length can be expressed as
\[ EI_x=EI_0\cdot(1+n\cdot x)^3,\quad n=\frac{2i}{D_0-t} \tag{6} \]
Therefore, when \(x=H\), Eq. (6) can be expressed as
\[ EI_H=EI_0\cdot(1+n\cdot H)^3,\quad 1+n\cdot H=\sqrt[3]{\frac{I_H}{I_0}} \tag{7} \]
Define the following formula:
\[ \mu=\sqrt[3]{\frac{I_H}{I_0}},\quad n=\frac{\mu-1}{H} \tag{8} \]
Combining Eqs. (3) and (8), one obtains
\[ \mu=\sqrt[3]{\frac{\omega(D_A-t)^3t}{\omega(D_0-t)^3t}} =\frac{D_A-t}{D_0-t}\approx\frac{D_A}{D_0} \tag{9} \]
In the equation, \(D_A\) is the across-corner outer diameter of the steel-pipe section at the bottom.
2 Calculation of rotation angle and displacement of a single-segment steel pipe pole
The load, rotation angle, and displacement calculation for the steel pipe pole are shown in Fig. 2.
Fig. 2 Load and displacement calculation diagram of steel pipe pole
According to material mechanics, the differential equation of the deflection curve for any section is
\[ EI_x\cdot\frac{\mathrm{d}^2 f}{\mathrm{d}x^2}=-M_x \tag{10} \]
where \(I_x\) and \(M_x\) are the moment of inertia and bending moment of the section at a distance \(x\) from the top, respectively.
According to Fig. 2, the bending moment at any section can be expressed as
\[ M_x=-M-Px-\frac{q}{2}x^2 \tag{11} \]
where \(M\) is the top bending moment of the steel pipe pole segment; \(P\) is the top horizontal force; and \(q\) is the uniformly distributed load.
Combining Eqs. (6) and (11), it can be expressed as
\[ EI_0\cdot(1+n\cdot x)^3\cdot\frac{\mathrm{d}^2 f}{\mathrm{d}x^2} =M+Px+\frac{q}{2}x^2, \]
\[ \frac{\mathrm{d}^2 f}{\mathrm{d}x^2} = \frac{M+Px+\frac{q}{2}x^2}{EI_0\cdot(1+n\cdot x)^3} \tag{12} \]
The above equation can be further expressed as
\[ \frac{\mathrm{d}^2 f}{\mathrm{d}x^2} = a_M\cdot\frac{1}{(1+n\cdot x)^3} + a_P\cdot\frac{x}{(1+n\cdot x)^3} + a_q\cdot\frac{x^2}{(1+n\cdot x)^3} \]
where
\[ a_M=\frac{M}{EI_0},\quad a_P=\frac{P}{EI_0},\quad a_q=\frac{q}{2EI_0} \tag{13} \]
By integration, the rotation-angle equation at any section is obtained as
\[ \theta_x=\frac{\mathrm{d}f}{\mathrm{d}x} = \int \frac{M_x}{EI_0\cdot(1+n\cdot x)^3}\mathrm{d}x+C \]
\[ = a_M\cdot\int\frac{1}{(1+n\cdot x)^3}\mathrm{d}x + a_P\cdot\int\frac{x}{(1+n\cdot x)^3}\mathrm{d}x + a_q\cdot\int\frac{x^2}{(1+n\cdot x)^3}\mathrm{d}x \tag{14} \]
After integration, the rotation angle at any section is
\[ \theta_x = -a_M\cdot\frac{1}{2n(1+nx)^2} -a_P\cdot\frac{(1+2nx)}{2n^2(1+nx)^2} + a_q\cdot \left[ \frac{\ln(1+nx)}{n^3} + \frac{3+4nx}{2n^3(1+nx)^2} \right] +C \tag{15} \]
According to the boundary condition, when \(x=H\), \(\theta_H=0\), and the value of the constant \(C\) is obtained as
\[ C = a_M\cdot\frac{1}{2n\mu^2} + a_P\cdot\frac{2\mu-1}{2n^2\mu^2} - a_q\cdot \left[ \frac{\ln\mu}{n^3} + \frac{4\mu-1}{2n^3\mu^2} \right] \tag{16} \]
The rotation angle at the top \((x=0)\) is
\[ \theta_0 = -a_M\cdot\frac{1}{2n} -a_P\cdot\frac{1}{2n^2} + a_q\cdot\frac{3}{2n^3} +C \]
\[ = a_M\cdot \left[ -\frac{1}{2n} + \frac{1}{2n\mu^2} \right] - a_P\cdot \left[ \frac{1}{2n^2} - \frac{2\mu-1}{2n^2\mu^2} \right] + a_q\cdot \left[ \frac{3}{2n^3} - \frac{\ln\mu}{n^3} - \frac{4\mu-1}{2n^3\mu^2} \right] \]
\[ = a_M\cdot \left[ -\frac{H\cdot(1+\mu)}{2\mu^2} \right] - a_P\cdot \left( \frac{H^2}{2\mu^2} \right) - a_q\cdot \left[ \frac{H^3}{2(\mu-1)^2} \cdot \left( \frac{1-3\mu}{\mu^2} + \frac{2\ln\mu}{\mu-1} \right) \right] \tag{17} \]
The displacement equation at any section is
\[ f_x=\int \theta_x\,\mathrm{d}x+D \]
\[ = -\frac{a_M}{2n}\cdot\int\frac{1}{(1+nx)^2}\mathrm{d}x - \frac{a_P}{2n^2}\cdot\int\frac{1+2nx}{(1+nx)^2}\mathrm{d}x + \frac{a_q}{n^3}\cdot \int \left[ \ln(1+nx)+\frac{3+4nx}{2(1+nx)^2} \right]\mathrm{d}x + Cx+D \tag{18} \]
After integration, the displacement at an arbitrary section is
$$ \begin{aligned} f_x={}&-\frac{a_M}{2n}\cdot\left[-\frac{1}{n(1+nx)}\right] -\frac{a_P}{2n^2}\left[\frac{1}{n(1+nx)}+\frac{\ln(1+nx)}{n}\right] \\ &+\frac{a_q}{n^3}\left[x\ln(1+nx)-x+\frac{3\ln(1+nx)}{n} +\frac{1}{2n(1+nx)}\right]+Cx+D \end{aligned} \tag{19} $$
According to the boundary condition, when \(x=H\), \(f_H=0\), the value of the constant \(D\) is
$$ \begin{aligned} D&=D_M+D_P+D_q,\\ D_M&=-\frac{a_M}{2n^2}\cdot\frac{\mu+nH}{\mu^2},\\ D_P&=\frac{a_P}{2n^3}\cdot\left[2\ln\mu-\frac{2\mu^2-4\mu+1}{\mu^2}\right],\\ D_q&=-\frac{a_q}{n^4}\cdot\left[3\ln\mu-\frac{2\mu^3+2\mu^2-6\mu+1}{2\mu^2}\right] \end{aligned} \tag{20} $$
The displacement at the top \((x=0)\) is
$$ f_0=\frac{a_M}{2n^2}-\frac{a_P}{2n^3}+\frac{a_q}{2n^4}+D_M+D_P+D_q \tag{21} $$
After simplification, the above equation can be further expressed as
$$ \begin{aligned} f_0={}&a_M\cdot\frac{H^2}{2\mu^2} +a_P\cdot\frac{H^3}{2(\mu-1)^2}\cdot \left(\frac{1-3\mu}{\mu^2}+\frac{2\ln\mu}{\mu-1}\right)\\ &+a_q\cdot\frac{H^4}{(\mu-1)^4}\cdot \left(\frac{2\mu^3+3\mu^2-6\mu+1}{2\mu^2}-3\ln\mu\right) \end{aligned} \tag{22} $$
Define the following deformation parameters:
$$ \begin{aligned} \beta_1&=\frac{1+\mu}{2\mu^2},\quad \beta_2=\frac{1}{2\mu^2},\\ \beta_3&=\frac{1}{2(\mu-1)^2}\cdot \left(\frac{1-3\mu}{\mu^2}+\frac{2\ln\mu}{\mu-1}\right),\\ \beta_4&=\frac{1}{(\mu-1)^4}\cdot \left(\frac{2\mu^3+3\mu^2-6\mu+1}{2\mu^2}-3\ln\mu\right) \end{aligned} \tag{23} $$
The formula for calculating the rotation angle at the top \((x=0)\) is
$$ \begin{aligned} \theta_0 &=-\frac{M}{EI_0}\cdot H\cdot\beta_1 -\frac{P}{EI_0}\cdot H^2\cdot\beta_2 -\frac{q}{2EI_0}\cdot H^3\cdot\beta_3\\ &=-\frac{1}{EI_0}\cdot \left(\beta_1\cdot M\cdot H+\beta_2\cdot P\cdot H^2+\beta_3\cdot q\cdot\frac{H^3}{2}\right) \end{aligned} \tag{24} $$
The formula for calculating the displacement at the top \((x=0)\) is
$$ \begin{aligned} f_0 &=\frac{M}{EI_0}\cdot H^2\cdot\beta_2 +\frac{P}{EI_0}\cdot H^3\cdot\beta_3 +\frac{q}{2EI_0}\cdot H^4\cdot\beta_4\\ &=\frac{1}{EI_0}\cdot \left(\beta_2\cdot M\cdot H^2+\beta_3\cdot P\cdot H^3+\beta_4\cdot q\cdot\frac{H^4}{2}\right) \end{aligned} \tag{25} $$
For different values of \(\mu\), the values of the deformation parameters \(\beta_1\sim\beta_4\) are shown in Table 2.
Table 2 Values of section deformation parameters
Tab. 2 The value of section deformation parameters
| \(\mu\) | \(\beta_1\) | \(\beta_2\) | \(\beta_3\) | \(\beta_4\) | \(\mu\) | \(\beta_1\) | \(\beta_2\) | \(\beta_3\) | \(\beta_4\) |
|---|---|---|---|---|---|---|---|---|---|
| 1.02 | 0.971 | 0.481 | 0.319 | 0.238 | 1.80 | 0.432 | 0.154 | 0.087 | 0.059 |
| 1.05 | 0.930 | 0.454 | 0.299 | 0.222 | 1.85 | 0.416 | 0.146 | 0.082 | 0.055 |
| 1.10 | 0.868 | 0.413 | 0.269 | 0.199 | 1.90 | 0.402 | 0.139 | 0.077 | 0.052 |
| 1.15 | 0.813 | 0.378 | 0.243 | 0.178 | 1.95 | 0.388 | 0.131 | 0.072 | 0.049 |
| 1.20 | 0.764 | 0.347 | 0.221 | 0.161 | 2.00 | 0.375 | 0.125 | 0.068 | 0.046 |
| 1.25 | 0.720 | 0.320 | 0.201 | 0.146 | 2.05 | 0.363 | 0.119 | 0.064 | 0.043 |
| 1.30 | 0.680 | 0.296 | 0.184 | 0.132 | 2.10 | 0.351 | 0.113 | 0.061 | 0.040 |
| 1.35 | 0.645 | 0.274 | 0.169 | 0.121 | 2.20 | 0.331 | 0.103 | 0.055 | 0.036 |
| 1.40 | 0.612 | 0.255 | 0.155 | 0.110 | 2.30 | 0.312 | 0.095 | 0.049 | 0.032 |
| 1.45 | 0.583 | 0.238 | 0.143 | 0.101 | 2.40 | 0.295 | 0.087 | 0.044 | 0.029 |
| 1.50 | 0.556 | 0.222 | 0.133 | 0.093 | 2.50 | 0.280 | 0.080 | 0.040 | 0.026 |
| 1.55 | 0.531 | 0.208 | 0.123 | 0.086 | 2.60 | 0.266 | 0.074 | 0.037 | 0.023 |
| 1.60 | 0.508 | 0.195 | 0.114 | 0.079 | 2.70 | 0.254 | 0.069 | 0.034 | 0.021 |
| 1.65 | 0.487 | 0.184 | 0.106 | 0.074 | 2.80 | 0.242 | 0.064 | 0.031 | 0.019 |
| 1.70 | 0.467 | 0.173 | 0.099 | 0.068 | 2.90 | 0.232 | 0.059 | 0.028 | 0.018 |
| 1.75 | 0.449 | 0.163 | 0.093 | 0.064 | 3.00 | 0.222 | 0.056 | 0.026 | 0.016 |
When an external load \((M, P, q)\) acts at an arbitrary height point \(B\) of the steel pipe pole, as shown in Fig. 3, then according to Eqs. (24) and (25), the rotation angle and displacement at point \(B\) are
\[ \theta_B = -\frac{1}{EI_0}\cdot \left( \beta_1\cdot M\cdot H_B + \beta_2\cdot P\cdot H_B^2 + \beta_3\cdot q\cdot \frac{H_B^3}{2} \right), \]
\[ f_B = \frac{1}{EI_0}\cdot \left( \beta_2\cdot M\cdot H_B^2 + \beta_3\cdot P\cdot H_B^3 + \beta_4\cdot q\cdot \frac{H_B^4}{2} \right), \]
\[ \mu=\frac{D_A-t}{D_B-t}\approx\frac{D_A}{D_B} \tag{26} \]
Fig. 3 Calculation diagram of load and displacement at any height of steel pipe pole
At this time, the formulas for the rotation angle and displacement at the top of the steel pipe pole can be approximately expressed as
\[ \theta_0=\theta_B, \qquad f_0=f_B+\left|\theta_B\right|\cdot (H-H_B) \tag{27} \]
3 Calculation of Rotation Angle and Displacement of Assembled Steel Pipe Pole
The main member of the steel pipe pole is often composed of several segments, as shown in Fig. 4.
Fig. 4 Overall assembly and deformation calculation diagram of steel pole
When calculating the overall deformation, the rotation angles and displacements at the top of each segment can first be calculated separately, and then the overall deformation calculation is carried out. For the top of each segment, without considering stiffness reduction, the rotation-angle calculation formulas are
\[ \theta_1 = -\frac{1}{EI_1}\cdot \left( \beta_{11}\cdot M_1\cdot H_1 + \beta_{21}\cdot P_1\cdot H_1^2 + \beta_{31}\cdot q_1\cdot \frac{H_1^3}{2} \right), \]
\[ \theta_2 = -\frac{1}{EI_2}\cdot \left( \beta_{12}\cdot M_2\cdot H_2 + \beta_{22}\cdot P_2\cdot H_2^2 + \beta_{32}\cdot q_2\cdot \frac{H_2^3}{2} \right), \]
\[ \cdots \]
\[ \theta_n = -\frac{1}{EI_n}\cdot \left( \beta_{1n}\cdot M_n\cdot H_n + \beta_{2n}\cdot P_n\cdot H_n^2 + \beta_{3n}\cdot q_n\cdot \frac{H_n^3}{2} \right) \tag{28} \]
The displacement calculation formulas are
\[ f'_1 = \frac{1}{EI_1}\cdot \left( \beta_{21}\cdot M_1\cdot H_1^2 + \beta_{31}\cdot P_1\cdot H_1^3 + \beta_{41}\cdot q_1\cdot \frac{H_1^4}{2} \right), \]
\[ f'_2 = \frac{1}{EI_2}\cdot \left( \beta_{22}\cdot M_2\cdot H_2^2 + \beta_{32}\cdot P_2\cdot H_2^3 + \beta_{42}\cdot q_2\cdot \frac{H_2^4}{2} \right), \]
\[ \cdots \]
\[ f'_n = \frac{1}{EI_n}\cdot \left( \beta_{2n}\cdot M_n\cdot H_n^2 + \beta_{3n}\cdot P_n\cdot H_n^3 + \beta_{4n}\cdot q_n\cdot \frac{H_n^4}{2} \right) \tag{29} \]
According to the analysis by Xu Jianshe et al. \([15]\) of the deformation and slip process of the bolt-hole wall, the flange bolt slip model is shown in Fig. 5.
(a) Before slip
(b) After slip
Fig. 5 Schematic diagram of flange bolt slip model
Let the bolt diameter be \(d\) and the hole diameter be \(d_0\). Under the action of external force, the slip of the flange plate is
\[ u=\Delta+u_1+u_2,\qquad \Delta=d_0-d \tag{30} \]
where \(\Delta\) is the clearance of the bolt hole; \(u_1\) and \(u_2\) are respectively the deformation of the hole wall of the upper flange plate and that of the lower flange plate.
Because, in ordinary steel pipe pole connection joints, the upper and lower flange plates are relatively thick and have relatively good stiffness, \(u_1\) and \(u_2\) may be neglected. Therefore, the slip of the flange plate is the clearance of the bolt hole, namely
\[ u=\Delta \tag{31} \]
Considering the bolt slip of the above flange, then at the flange connection joint, the
The amplification factor for the rotation angle \(\theta\) is
\[ K_{\theta i}=\frac{f_i+\Delta}{f_i} \tag{32} \]
where \(f_i\) is the overall deflection value of node \(i\).
It should be noted that the above calculations of rotation and displacement are all derived from the differential equation of the deflection curve at an arbitrary section in mechanics of materials. The differential equation of the deflection curve itself is an approximate differential equation for a curve. When additional bending moments are not considered, the displacement values obtained by the method in this study still have certain errors compared with finite-element solutions. Define this amplification factor as
\[ \gamma=\sum_{i=1}^{n}\frac{f_{i,\mathrm{FEA}}}{f_i}/n \tag{33} \]
where \(f_{i,\mathrm{FEA}}\) is the finite-element displacement solution at a node \(i\) at a certain height of the steel tubular pole.
According to the above formulas, considering the error of the deflection-difference method and flange-node bolt slip, the total displacement at the top of the assembled steel tubular pole should take into account the horizontal displacement of each segment and the displacement effect caused by the rotation angle at the top of each segment. The calculation of displacement for each segment is shown in Fig. 6, and the formula for calculating the total displacement at the pole top is
\[ \begin{aligned} f =&\ \gamma \times \bigl\{ f'_1+H_1\cdot\left(|\theta_2K_{\theta 2}|+|\theta_3K_{\theta 3}|+\cdots+|\theta_nK_{\theta n}|\right) +f'_2 \\ &+H_2\cdot\left(|\theta_3K_{\theta 3}|+|\theta_4K_{\theta 4}|+\cdots+|\theta_nK_{\theta n}|\right) +\cdots+f'_i \\ &+H_i\cdot\left(|\theta_{i+1}K_{\theta,i+1}|+|\theta_{i+2}K_{\theta,i+2}|+\cdots+|\theta_nK_{\theta n}|\right) +\cdots+f'_n \bigr\} \end{aligned} \]
Expression 1:
\[ \gamma\times\left[ \sum_{i=1}^{n} f'_i+ \sum_{i=1}^{n-1}\left( H_i\cdot\sum_{j=i+1}^{n}|\theta_jK_{\theta j}| \right) \right] \]
Expression 2:
\[ \gamma\times\left[ \sum_{i=1}^{n} f'_i+ \sum_{i=2}^{n}\left( |\theta_iK_{\theta i}|\cdot\sum_{j=1}^{i-1}H_j \right) \right] \tag{34} \]
Fig. 6 Calculation diagram of top displacement of assembled steel pipe pole
4 Calculation of Additional Bending Moment of Assembled Steel Tubular Pole
Because the displacement of the steel tubular pole itself causes the center of gravity of each segment to shift, a corresponding additional bending moment is generated. The calculation formula for the additional bending moment at any height of the steel tubular pole is
\[ M_{G1}=0, \]
\[ M_{G2}=G_1\cdot(f_1-f_2), \]
\[ M_{G3}=G_1\cdot(f_1-f_3)+G_2\cdot(f_2-f_3), \]
\[ \cdots \]
\[ M_{Gi}=G_1\cdot(f_1-f_i)+G_2\cdot(f_2-f_i)+\cdots+ G_{i-1}\cdot(f_{i-1}-f_i), \]
\[ \cdots \]
\[ M_{Gn}=G_1\cdot(f_1-f_n)+G_2\cdot(f_2-f_n)+\cdots+ G_{n-1}\cdot(f_{n-1}-f_n) \]
\[ \Rightarrow M_{Gi}=\sum_{j=1}^{i-1}G_j\cdot(f_j-f_i) \tag{35} \]
According to the above equation, the corresponding additional displacement \(f_G\) caused by the additional bending moment can be calculated; then, through iterative calculation, the second-order effect caused by the additional bending moment can be calculated. Thus, the formula for calculating the total displacement of node \(i\) is
\[ f_{\text{total},i}=f_i+f_{Gi} \tag{36} \]
5 Example Analysis
A typical 220 kV transmission strain steel tubular pole SGJ-24 is selected as the research object. The design wind speed is \(V=35\ \mathrm{m/s}\), the ice thickness is \(C=5\ \mathrm{mm}\), the conductor is \(4\times\mathrm{AAC}\ 500\ \mathrm{mm^2}\), and the ground wire is OPGW. The section is a 12-sided polygon, and Q420 steel is adopted. The strain pole SGJ3-24 has an overall height of \(46.9\ \mathrm{m}\), a nominal height of \(24\ \mathrm{m}\), a rotation-angle number of \(0\sim90^\circ\), a pole-top steel tube diameter of \(660\ \mathrm{mm}\), a pole-base diameter of \(2\,800\ \mathrm{mm}\), and a taper \(i=1/21.9\). The single-line and sectional-dimension schematic diagrams for the example are shown in Fig. 7, and the full-scale pole test and load arrangement are shown in Fig. 8.
Fig. 7 Schematic diagram of single line and section size of typical steel pipe pole
Fig. 8 Schematic diagram of true steel pole test and load layout
Using ANSYS finite element software, the steel tubular pole was modeled and analyzed. The yield strength of the steel was taken as 420 MPa, the elastic modulus as \(E=2.06\times10^{11}\ \mathrm{N/m^2}\), the density as \(\rho=7\,850\ \mathrm{kg/m^3}\), and Poisson’s ratio as \(\nu=0.3\). Table 3 is a comparison between the displacement values calculated by the method of this study (without considering flange bolt slip) and the finite element solution. It can be seen from Table 3 that the displacement values calculated by the method of this study have a certain error compared with the finite element solution, and the amplification factor \(\gamma\) is approximately 1.07.
Table 4 is the calculation table for the deflection of the steel tubular pole (without considering additional bending moment), where \(f_{Q1}\) is the deflection value obtained by considering the error of the deflection-curve differential equation (\(\gamma=1.07\), without considering flange bolt slip), \(f_{Q2}\) is the deflection value after considering \(\gamma\) and flange bolt slip, and \(f_{Q3}\) is the displacement value from the full-scale test. From Table 4, it can be seen that: ① after considering the error of the deflection-curve differential equation (\(\gamma=1.07\)), the calculated deflection of the steel tubular pole is basically close to the finite element solution; ② according to the method of this study, after considering flange bolt slip, the calculated deflection of the steel tubular pole is basically close to the full-scale test value, the error at the pole top can be ignored, and the error at the pole bottom is slightly larger, approximately 13%; ③ according to this study, when the error \(\gamma\) of the deflection-curve differential equation and the effect of flange bolt slip are considered simultaneously, the calculated deflection of the steel tubular pole can be used as a relatively accurate method for predicting the actual deflection value. The schematic diagram for calculating the deflection of the assembled steel tubular pole is shown in Fig. 9.
Fig. 9 The deflection calculation diagram of assembled steel pipe pole
Table 3 Comparison of deflection without considering bolt slip and finite element solution
| \(H/\mathrm{m}\) | Rotation angle \(\theta\) | Segmental deflection / mm | Calculated deflection \(f\) without considering slip / mm | Finite element calculated displacement / mm | Amplification factor \(\gamma\) |
|---|---|---|---|---|---|
| 46.90 | \(-0.002\,62\) | 13.594 | 1 524.43 | 1 626.50 | 1.07 |
| 38.00 | \(-0.001\,63\) | 1.993 | 1 034.79 | 1 105.50 | 1.07 |
| 35.90 | \(-0.005\,76\) | 15.111 | 923.90 | 987.25 | 1.07 |
| 31.00 | \(-0.008\,91\) | 24.025 | 682.90 | 731.03 | 1.07 |
| 25.90 | \(-0.002\,60\) | 2.485 | 469.22 | 503.26 | 1.07 |
| 24.00 | \(-0.014\,91\) | 72.927 | 401.02 | 430.71 | 1.07 |
| 14.40 | \(-0.011\,39\) | 44.961 | 139.15 | 149.85 | 1.08 |
| 6.40 | \(-0.008\,29\) | 26.073 | 27.90 | 29.53 | 1.06 |
| 0.00 | 0.000 00 | 0.000 | 0.00 | 0.00 | — |
Table 5 is the deflection calculation table for the steel tubular pole considering additional bending moment. From Table 5, it can be seen that: ① the additional deflection caused by the additional bending moment accounts for 0.57%–0.69% of the total deflection, representing a relatively small proportion overall; ② under the actual instantaneous high-wind condition, the deflection effect will further increase. Therefore, according to the design provisions for steel tubular poles \([1]\), it is reasonable to consider second-order effects with a factor of 1.05–1.10.
Table 4 Deflection calculation table for the steel pipe pole (without considering additional bending moment)
Tab. 4 Deflection calculation of steel pipe pole (without considering additional bending moment)
| No. | Segment | $H$/m | Rotation $\theta$ | Segmental deflection $f$/mm | Angular amplification coefficient $K_{\theta i}$ | Total deflection $f_{Q1}$/mm | Total deflection $f_{Q2}$/mm | Total deflection $f_{Q3}$/mm | Deflection ratio $f_{Q2}/f_{Q1}$ | Deflection ratio $f_{Q3}/f_{Q2}$ |
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 46.90 | −0.002 81 | 14.545 | – | 1 629.19 | 1 724.06 | 1 726.00 | 1.06 | 1.00 |
| 2 | 46.00 | −0.002 74 | 12.040 | 1 575.17 | 1 666.43 | 1.06 | ||||
| 3 | 44.00 | −0.002 26 | 6.982 | 1 455.65 | 1 538.89 | 1.06 | ||||
| 4 | 42.00 | −0.001 56 | 3.135 | 1 337.34 | 1 412.57 | 1.06 | ||||
| 5 | 40.00 | −0.000 78 | 0.782 | 1 220.52 | 1 287.73 | 1.06 | ||||
| 6 | 38.00 | −0.001 75 | 2.133 | 1 105.27 | 1 164.47 | 1.05 | ||||
| 7 | 37.00 | −0.001 13 | 0.658 | 1 048.31 | 1 103.50 | 1 107.00 | 1.05 | 1.00 | ||
| 8 | 2 | 35.90 | −0.006 16 | 16.169 | 1.124 | 986.62 | 1 037.40 | 1.05 | ||
| 9 | 35.00 | −0.005 26 | 11.019 | 937.08 | 984.94 | 1.05 | ||||
| 10 | 33.00 | −0.002 82 | 2.860 | 830.27 | 871.64 | 1.05 | ||||
| 11 | 31.00 | −0.009 53 | 25.707 | 728.75 | 763.64 | 1.05 | ||||
| 12 | 29.90 | −0.007 79 | 16.155 | 675.43 | 706.75 | 1.05 | ||||
| 13 | 27.90 | −0.004 11 | 4.161 | 583.85 | 608.68 | 634.00 | 1.04 | 1.04 | ||
| 14 | 3 | 25.90 | −0.002 78 | 2.659 | 1.752 | 500.11 | 518.45 | 1.04 | ||
| 15 | 25.00 | −0.001 48 | 0.740 | 464.88 | 482.18 | 1.04 | ||||
| 16 | 24.00 | −0.015 95 | 78.032 | 427.13 | 443.29 | 1.04 | ||||
| 17 | 22.40 | −0.013 48 | 54.464 | 369.87 | 384.18 | 1.04 | ||||
| 18 | 20.40 | −0.010 22 | 30.738 | 304.03 | 316.04 | 1.04 | ||||
| 19 | 18.40 | −0.006 84 | 13.670 | 244.84 | 254.55 | 1.04 | ||||
| 20 | 16.40 | −0.003 42 | 3.412 | 192.47 | 199.88 | 226.00 | 1.04 | 1.13 | ||
| 21 | 4 | 14.40 | −0.012 19 | 48.108 | 1.042 | 146.94 | 152.05 | 1.03 | ||
| 22 | 12.40 | −0.009 06 | 26.864 | 107.96 | 111.79 | 1.04 | ||||
| 23 | 10.40 | −0.005 97 | 11.844 | 75.21 | 77.76 | 1.03 | ||||
| 24 | 8.40 | −0.002 95 | 2.935 | 48.57 | 49.84 | – | 1.03 | – | ||
| 25 | 5 | 6.40 | −0.008 87 | 27.898 | 1.072 | 27.90 | 27.90 | – | ||
| 26 | 4.00 | −0.005 44 | 10.757 | 10.76 | 10.76 | – | ||||
| 27 | 2.00 | −0.002 67 | 2.660 | 2.66 | 2.66 | – | ||||
| 28 | 0.00 | 0.000 00 | 0.000 | 0.00 | 0.00 | – | – | – |
Table 5 Total deflection calculation table for the steel pipe pole
Tab. 5 The total deflection calculation of steel pipe pole
| No. | Segment | $H$/m | Deflection $f$/mm | Additional deflection $f_G$/mm | Total deflection $f_{\text{总}}$/mm | Proportion of additional deflection/% |
|---|---|---|---|---|---|---|
| 1 | 1 | 46.90 | 1 724.06 | 11.97 | 1 736.03 | 0.69 |
| 2 | 46.00 | 1 666.43 | 11.58 | 1 678.02 | 0.70 | |
| 3 | 44.00 | 1 538.89 | 10.71 | 1 549.61 | 0.70 | |
| 4 | 42.00 | 1 412.57 | 9.84 | 1 422.41 | 0.70 | |
| 5 | 40.00 | 1 287.73 | 8.97 | 1 296.70 | 0.70 | |
| 6 | 38.00 | 1 164.47 | 8.10 | 1 172.57 | 0.70 | |
| 7 | 37.00 | 1 103.50 | 7.67 | 1 111.17 | 0.69 |
Continued Table 5
| No. | Segment No. | $H/\mathrm{m}$ | Deflection $f/\mathrm{mm}$ | Additional deflection $f_c/\mathrm{mm}$ | Total deflection $f_{\text{total}}/\mathrm{mm}$ | Proportion of additional deflection / % |
|---|---|---|---|---|---|---|
| 8 | 2 | 35.90 | 1 037.40 | 7.20 | 1 044.60 | 0.69 |
| 9 | 35.00 | 984.94 | 6.82 | 991.76 | 0.69 | |
| 10 | 33.00 | 871.64 | 6.01 | 877.65 | 0.69 | |
| 11 | 31.00 | 763.64 | 5.23 | 768.87 | 0.69 | |
| 12 | 29.90 | 706.75 | 4.82 | 711.57 | 0.68 | |
| 13 | 27.90 | 608.68 | 4.12 | 612.80 | 0.68 | |
| 14 | 3 | 25.90 | 518.45 | 3.47 | 521.92 | 0.67 |
| 15 | 25.00 | 482.18 | 3.22 | 485.40 | 0.67 | |
| 16 | 24.00 | 443.29 | 2.95 | 446.23 | 0.67 | |
| 17 | 22.40 | 384.18 | 2.55 | 386.73 | 0.66 | |
| 18 | 20.40 | 316.04 | 2.09 | 318.13 | 0.66 | |
| 19 | 18.40 | 254.55 | 1.69 | 256.24 | 0.66 | |
| 20 | 16.40 | 199.88 | 1.32 | 201.20 | 0.66 | |
| 21 | 4 | 14.40 | 152.05 | 1.00 | 153.05 | 0.66 |
| 22 | 12.40 | 111.79 | 0.72 | 112.52 | 0.65 | |
| 23 | 10.40 | 77.76 | 0.49 | 78.26 | 0.64 | |
| 24 | 8.40 | 49.84 | 0.31 | 50.16 | 0.62 | |
| 25 | 5 | 6.40 | 27.90 | 0.17 | 28.06 | 0.60 |
| 26 | 4.00 | 10.76 | 0.06 | 10.82 | 0.58 | |
| 27 | 2.00 | 2.66 | 0.02 | 2.67 | 0.57 |
6 Conclusions
1) Based on the differential equation for beam deflection curves, and in combination with flange bolt slip, the proposed calculation method for the deflection and additional bending moment of multi-pyramid tapered steel tubular poles is correct and reliable, and can be used as a method for calculating the deflection of steel tubular poles.
2) In deflection calculation, when the method of this study is adopted (without considering flange bolt slip), the displacement values obtained have a certain error compared with the finite-element solution, and the amplification factor $\gamma$ is approximately 1.07.
3) The slip of the flange plate is considered according to the clearance of the bolt holes, and the corresponding amplification factor $K_\theta$ for the rotation angle $\theta$ is considered.
4) According to actual calculations, the calculated deflection of the steel tubular pole is basically close to the full-scale test value of the steel tubular pole. The error at the pole top may be neglected, while the error at the pole base is slightly larger.
5) According to actual calculations, the additional deflection caused by the additional bending moment accounts for only a small proportion of the total deflection, generally less than 2%.
6) When the actual strong-wind load case is considered, the deflection effect will further increase; therefore, it is reasonable for the design code to take the second-order effect as 1.05–1.10.
References:
[1] National Energy Administration. Technical code for the design of pole and tower structures of overhead transmission lines: DL/T 5486—2020 [S]. Beijing: China Electric Power Press, 2020.
[2] ASCE. Design of steel transmission pole structures: ASCE 48-11 [S]. Reston, VA, USA: ASCE, 2011.
[3] 王方林, 唐国安. 套接连接在多棱锥型钢管杆中的应用与探讨[J]. 特种结构, 2006, 23(3): 52-56.
WANG Fanglin, TANG Guoan. Sleeve joint multi-pyramidal steel pipe application and discussion[J]. Special structures, 2006, 23(3): 52-56 (in Chinese).
[4] 朱正仪, 上官敦家. 拔梢钢管杆的变形计算[J]. 电力建设, 1989, 10(3): 23-27.
ZHU Zhengyi, SHANGGUAN Dunjia. Deformation calculation of steel pole with tip pulling[J]. Electric power construction, 1989, 10(3): 23-27 (in Chinese).
[5] 上官敦家, 朱正仪. 棱锥形钢管杆的挠度计算[J]. 电力建设, 2007, 28(10): 51-52.
SHANGGUAN Dunjia, ZHU Zhengyi. Deflection calculation of pyramid shaped steel tube pole[J]. Electric power construction, 2007, 28(10): 51-52 (in Chinese).
[6] 黄嘉印, 郭明昌. 拔梢钢管电杆的挠度计算[J]. 电力建设,
1991, 12(3): 30–32.
HUANG Jiayin, GUO Mingchang. Deflection calculation of steel pole with tip pulling [J]. Electric Power Construction, 1991, 12(3): 30–32 (in Chinese).
[7] NIE Guoyi. Calculation of deflection and additional bending moment of steel pole with tip pulling [J]. Electric Power Construction, 1997, 18(7): 32–37.
NIE Guoyi. Calculation on deflection and additional bending moment of steel pole with tip pulling [J]. Electric Power Construction, 1997, 18(7): 32–37 (in Chinese).
[8] WANG Jianhua. Calculation of internal force and deflection of steel tubular poles considering axial pressure [J]. Electric Power Construction, 1995, 16(1): 7–9.
WANG Jianhua. Calculation of internal force and deflection of steelpole considering axial pressure [J]. Electric Power Construction, 1995, 16(1): 7–9 (in Chinese).
[9] WANG Jianhua. Deformation calculation of tapered steel tubular poles [J]. Electric Power, 1995, 28(6): 56–59.
WANG Jianhua. Deformation calculation of tapered steelpole [J]. Electric Power, 1995, 28(6): 56–59 (in Chinese).
[10] WANG Jianhua, LIU Xindong. Analytical expression for calculating deflection of steel pole with tip pulling [J]. Journal of Xi’an Mining Institute, 1995, 15(Suppl. 1): 81–83, 76.
WANG Jianhua, LIU Xindong. Analytical expression for calculating deflection of steel pole with tip pulling [J]. Journal of Xi’an Mining Institute, 1995, 15(S1): 81–83, 76 (in Chinese).
[11] WANG Jianhua, NIU Shaoqin, LIANG Jingru. Deflection and strength calculation of multi-lateral inserted steel towers [J]. Electric Power Construction, 1995, 16(11): 25–27.
WANG Jianhua, NIU Shaoqin, LIANG Jingru. Deflection and strength calculation of multi lateral inserted steel tower [J]. Electric Power Construction, 1995, 16(11): 25–27 (in Chinese).
[12] WANG Xieshan, GU Yujiong. Analytical and simplified methods for deflection of steel tubular poles for transmission lines [J]. Modern Electric Power, 2002, 19(3): 35–40.
WANG Xieshan, GU Yujiong. Analytical and simplified methods for deformation of transmit-electricity steel tube Pole [J]. Modern Electric Power, 2002, 19(3): 35–40 (in Chinese).
[13] LIN Shipao. Deflection calculation method for variable-section steel tubular poles [J]. Steel Construction, 2012, 27(9): 42–44.
LIN Shipao. Deflection calculation of variable cross-section steel tube pole [J]. Steel Construction, 2012, 27(9): 42–44 (in Chinese).
[14] LUO Lie, GAO Shuangxi, HUANG Juzhi. Comparison among finite-element analysis, simplified calculation, and tests for the overall deformation of slip-joint monopoles [J]. Structural Engineers, 2009, 25(3): 34–39.
LUO Lie, GAO Shuangxi, HUANG Juzhi. Comparison among deflection calculation results of slip-joint monopoles with finite element method, simplified method and experiment [J]. Structural Engineers, 2009, 25(3): 34–39 (in Chinese).
[15] XU Jianshe, CHEN Yiyi, HAN Lin, et al. Slip process analysis of regular bolt and bearing-type high-strength bolt shear connections [J]. Journal of Tongji University (Natural Science), 2003, 31(5): 510–514.
XU Jianshe, CHEN Yiyi, HAN Lin, et al. Slip process analysis of regular bolt and bearing type high-tensile bolt shear connections [J]. Journal of Tongji University (Natural Science), 2003, 31(5): 510–514 (in Chinese).
[16] HUANG Weidong, CHEN Haibo, WANG Peng, et al. Numerical simulation of bolt slippage in lap joints of transmission towers [J]. Chinese Journal of Solid Mechanics, 2014, 35(Suppl. 1): 215–220.
HUANG Weidong, CHEN Haibo, WANG Peng, et al. Finite element simulation of bolt slippage in the lap joints of transmission tower [J]. Chinese Journal of Solid Mechanics, 2014, 35(S1): 215–220 (in Chinese).
[17] YAO Kuan, ZHANG Qian, CAI Jianguo, et al. Study on the effect of bolted joint slippage on deformation of transmission-tower structures [J]. Steel Construction, 2019, 34(5): 39–44.
YAO Kuan, ZHANG Qian, CAI Jianguo, et al. Effect of bolted joint slippage on deformation of transmission tower structure [J]. Steel Construction, 2019, 34(5): 39–44 (in Chinese).
[18] YANG Fengli. Structural analysis of a transmission tower body considering the influence of bolt-connection slippage [J]. Engineering Mechanics, 2018, 35(Suppl. 1): 193–199.
YANG Fengli. Structural analysis on a typical transmission tower body section with bolt slippage effects [J]. Engineering Mechanics, 2018, 35(S1): 193–199 (in Chinese).
(Editor: Li Kunrui)