Zhuojin Pipe Fitting Co., LtdDeflection and Additional Bending Moment of Multi-Pyramid Socket-Welded Steel Pipe Poles with Stiffness Reduction
Literature Overview
This paper, published in the Chinese Journal of Computational Mechanics (2023, Vol. 40, No. 6, pp. 1008–1015), addresses a critical yet under-researched problem in transmission line engineering: the deflection and additional bending moment of multi-pyramid socket-welded steel pipe poles. The authors—Pan Feng, Chen Yuan, Cai Yong, Bao Yunan, and Xu Zheng—belong to the China Energy Engineering Group Zhejiang Electric Power Design Institute, Dalian Maritime University, and State Grid Zhejiang Electric Power Co., Ltd. The work is supported by the National Natural Science Foundation of China (Grant No. 51878607).
In urban and suburban transmission line projects, multi-pyramid (tapered, multi-segment) steel pipe poles are increasingly used for their aesthetic advantages and ease of installation compared to traditional lattice towers. While flange connections dominate the current practice, socket (lap) connections offer simpler field assembly and lower labor costs. However, the mechanical behavior of socket joints—particularly the stiffness degradation at the overlap region—remains poorly characterized in existing design codes. This gap has practical consequences: if the stiffness reduction is ignored, deflections may be underestimated and secondary bending moments overlooked, potentially leading to serviceability failures or even structural instability under wind or ice loading.
Core Technical Approach
The authors derive closed-form analytical expressions for deflection and additional bending moment under three common load cases: concentrated moment M, horizontal force P, and uniformly distributed load q. The derivation is grounded in the classical Euler-Bernoulli beam deflection curve differential equation, but with a critical modification: the section moment of inertia I is not treated as a simple step function at the socket joint, but is instead reduced by an empirical stiffness degradation factor to reflect the partial continuity loss at the overlap.
Stiffness Reduction Model
The key innovation lies in the stiffness reduction factor applied at the socket joint. In a socket connection, the outer pipe segment slides into the inner pipe segment with a certain overlap length. The joint does not behave as a rigidly continuous beam because:
- The overlap region has a different effective cross-sectional geometry (double-wall versus single-wall).
- Frictional resistance and bolt preload (if any) provide partial moment transfer, but not full continuity.
- Local deformation at the joint under bending leads to a measurable increase in curvature compared to a monolithic section.
The stiffness reduction factor is typically expressed as a ratio of the effective moment of inertia at the joint to the nominal moment of inertia of the larger-diameter segment. Based on experimental and numerical calibration, this factor ranges between 0.6 and 0.85 depending on the overlap length-to-diameter ratio, the fit clearance, and the presence of welding or mechanical fasteners at the joint.
Governing Equations
For a multi-pyramid pole with n segments, each having its own outer diameter D_i, wall thickness t_i, and length L_i, the deflection at any section is obtained by integrating the curvature equation segment by segment, applying compatibility conditions at each joint. At a socket joint between segment i and segment i+1, the compatibility conditions become:
- Displacement continuity: w_i(L_i) = w_{i+1}(0)
- Modified moment continuity: EI_i w''_i(L_i) = k EI_{i+1} * w''_{i+1}(0), where k is the stiffness reduction factor
- Shear force continuity: EI_i w'''_i(L_i) = EI_{i+1} w'''_{i+1}(0)
The additional bending moment arises because the stiffness discontinuity causes a redistribution of internal forces that would not exist in a perfectly uniform beam. This additional moment is superimposed on the primary bending moment and can be significant near the joints, particularly for poles with high taper ratios.
Typical Parameters and Case Study
The paper analyzes a representative multi-pyramid socket-welded pole with the following typical parameters:
| Parameter | Value |
|---|---|
| Total pole height | 40 m |
| Number of segments | 4 |
| Top outer diameter | 400 mm |
| Bottom outer diameter | 800 mm |
| Wall thickness (top to bottom) | 8–14 mm |
| Socket overlap length | 1.5–2.0 m |
| Material grade | Q345B |
| Stiffness reduction factor k | 0.70 (typical) |
| Design wind speed | 35 m/s |
The comparison between the stiffness-reduced model and the conventional model (which assumes full continuity at joints) shows that:
| Metric | Conventional Model | Stiffness-Reduced Model | Difference |
|---|---|---|---|
| Maximum deflection | 186 mm | 224 mm | +20.4% |
| Maximum additional bending moment | 0 kN·m | 48 kN·m | Significant |
| Deflection at 20 m height | 92 mm | 112 mm | +21.7% |
| Stress at joint section | 142 MPa | 178 MPa | +25.4% |
These results demonstrate that ignoring stiffness reduction can lead to underestimation of deflection by approximately 20% and completely miss the additional bending moment, which can increase local stresses by 25% or more.
Engineering Practice Implications
Design Recommendations
Based on the study, the following recommendations emerge for practical engineering design:
- Stiffness reduction must be explicitly accounted for in the analysis of socket-welded multi-pyramid poles. The factor k should be determined from experimental calibration for the specific joint configuration and should not be assumed to be unity.
- The additional bending moment should be included in stress calculations at joint sections. This moment, while not present in the primary analysis, can govern the local design at the socket overlap.
- Overlap length should be optimized: longer overlap lengths improve stiffness continuity but increase material usage and transportation costs. A minimum overlap-to-diameter ratio of approximately 2.0–2.5 is recommended to achieve a stiffness reduction factor above 0.75.
- Welding at the socket joint (post-installation field welding) can significantly improve stiffness continuity, potentially raising k to 0.90 or higher. However, this introduces welding quality control challenges in the field, including weld penetration, HAZ toughness, and residual stress management.
Connection to Welding Quality Control
From a welding engineering perspective, the socket joint in a socket-welded pole typically involves a circumferential fillet weld or a full-penetration butt weld at the lap end. The weld quality directly affects the stiffness reduction factor:
- A fully penetrated, sound butt weld can approach k = 0.90–0.95.
- A fillet weld with incomplete penetration may yield k = 0.60–0.75.
- A mechanically clamped joint without welding may have k = 0.50–0.65.
Quality control measures should include:
| Inspection Method | Purpose | Acceptance Criteria |
|---|---|---|
| Visual Testing (VT) | Surface defects, weld profile | No undercut > 1 mm, no cracks |
| Ultrasonic Testing (UT) | Internal defects, penetration | ISO 17640, Level II |
| Radiographic Testing (RT) | Volumetric defects | ISO 17636, Grade B |
| Hardness Testing | HAZ properties | ≤ 350 HV for Q345B |
Key Questions and Reflections
The paper raises several important questions that warrant further investigation:
- Long-term stiffness degradation: Over decades of service, fatigue, corrosion, and cyclic loading may further degrade the joint stiffness. The initial k value may not remain constant, and a time-dependent stiffness model would be more realistic for lifecycle analysis.
- Interaction with buckling: The additional bending moment near joints reduces the effective length for buckling calculations. The paper focuses on deflection and bending but does not address the stability implications, which is a significant omission for slender pipe poles.
- Dynamic effects: Transmission line poles are subjected to dynamic loads from wind turbulence, galloping, and ice shedding. The stiffness reduction at joints affects the natural frequencies and mode shapes, potentially leading to resonance conditions not captured in static analysis.
- Standardization gap: Current standards such as GB/T 50009 (Load Code for Building Structures) and DL/T 5092 (Design Code for Overhead Transmission Lines) do not provide explicit guidance on socket joint stiffness reduction. The methodology proposed in this paper could serve as a basis for future standard revisions.
Study Insights and Implications
This research fills a meaningful gap in the structural analysis of transmission line pipe poles. The analytical framework is elegant and practical: it requires no complex finite element modeling, yet captures the essential physics of stiffness degradation at socket joints. For design engineers, the key takeaway is that socket connections, while offering construction advantages, introduce mechanical discontinuities that must be quantified and accounted for in both serviceability and strength checks.
The stiffness reduction factor k is not merely a numerical parameter—it represents the physical reality of how well the joint transfers bending moments. In my experience reviewing pipe pole designs, I have seen several cases where socket joints were treated as fully continuous, leading to deflections that exceeded allowable limits under extreme wind loading. This paper provides the analytical tools to avoid such errors.
From a manufacturing and quality control standpoint, the paper implicitly advocates for higher-quality socket joint fabrication. The correlation between weld quality and stiffness continuity is direct: a well-executed full-penetration butt weld with proper HAZ control will yield a higher k value, reducing deflection and additional moments. This reinforces the importance of welder qualification, pre-qualification welding procedures (WPQR), and rigorous non-destructive testing at socket joints.