Deflection and Additional Bending Moment of Multi-Pyramid Socket-Connected Steel Tubular Poles Considering Stiffness Reduction

Literature Overview

This paper by Pan Feng, Chen Yuan, Cai Yong, Bao Yunan, and Xu Zeng, published in Chinese Journal of Computational Mechanics (2023, Vol. 40, No. 6, pp. 1008-1015), investigates the deflection and additional bending moment of multi-pyramid socket-connected steel tubular poles used in power transmission lines. The research is supported by the National Natural Science Foundation of China (51878607). The authors derive analytical methods based on the beam deflection curve differential equation theory, incorporating socket joint stiffness reduction, for common loading conditions including bending moment M, horizontal force P, and distributed load q.

Core Technical Framework

Multi-pyramid steel tubular poles are widely used in urban power transmission lines, where aesthetic considerations and space constraints favor tapered pole designs. The socket connection method, while simpler than flange connections, introduces a stiffness discontinuity at each joint that significantly affects the overall structural behavior.

The analytical framework is built upon:

Loading Condition	Deflection Equation	Additional Bending Moment
End moment M	Derived from curvature integration with stiffness reduction	Proportional to M and stiffness reduction factor
Horizontal force P	Derived from shear and moment equilibrium	Proportional to P and lever arm
Distributed load q	Derived from differential equation solution	Proportional to q and span length

The stiffness reduction at socket joints is characterized by a reduction factor that accounts for the reduced rotational stiffness compared to a continuous pole. This factor is determined by the socket geometry, the overlap length, and the material properties of the connected sections.

Socket Joint Stiffness Reduction Analysis

The socket joint represents a critical weak point in the multi-pyramid pole structure. The stiffness reduction arises from:

The geometric discontinuity between the tapered sections creates a stress concentration at the socket interface.
The rotational stiffness of the socket joint is lower than the bending stiffness of the continuous pole, leading to increased deflection at the joint locations.
The additional bending moment induced by the stiffness discontinuity must be considered in the design of the socket connection and the adjacent pole sections.

The stiffness reduction factor is a function of the socket length, the diameter difference between connected sections, and the material properties. For typical multi-pyramid poles with socket lengths of 1.5 to 2.0 times the pole diameter, the stiffness reduction factor ranges from 0.6 to 0.85, indicating a 15% to 40% reduction in effective bending stiffness at the joint.

Engineering Practice Implications

For the design of multi-pyramid socket-connected steel tubular poles, the following practical implications emerge:

The deflection calculation must incorporate the stiffness reduction at each socket joint, as ignoring this effect will underestimate the actual deflection and potentially lead to inadequate pole design.
The additional bending moment at socket joints should be included in the design load cases, particularly for poles subjected to significant lateral loads from conductor tension and wind loading.
The socket connection design should be optimized to minimize the stiffness reduction, which can be achieved by increasing the socket overlap length or by using internal reinforcement rings at the socket interface.
The deflection limit for power transmission poles should be checked at each socket joint location, as the maximum deflection typically occurs near the joints due to the stiffness discontinuity.

Key Questions and Reflections

A significant question is the long-term behavior of socket joints under cyclic loading conditions typical of power transmission lines, where wind-induced vibrations and conductor galloping can cause repeated loading and unloading at the socket interface. The analytical model assumes elastic behavior, and the effect of cyclic loading on the socket joint stiffness is not addressed. Additionally, the study does not consider the effect of corrosion on the socket joint stiffness, which is a significant concern for steel tubular poles exposed to atmospheric environments.

Summary

This study provides a rigorous analytical framework for calculating the deflection and additional bending moment of multi-pyramid socket-connected steel tubular poles, incorporating the stiffness reduction at socket joints. The derived equations for common loading conditions enable practical design calculations that account for the stiffness discontinuity at each joint. For steel tube engineers, the emphasis on socket joint stiffness reduction highlights the importance of optimizing socket geometry and considering additional bending moments in the design of power transmission poles. The analytical approach provides a practical alternative to finite element analysis for preliminary design and verification, enabling rapid assessment of pole deflection and joint loading under various loading scenarios.