# Fragility Assessment and Reinforcement Strategies for Transmission Towers Under Extreme Wind Loads
## 5 Concurrent Multi-Scale Reinforcement Modeling  
### Establishment of Concurrent Multi-Scale Model  
As identified in Section 3, the lower halves of the main structural members in the third segment of the tower--specifically, rods No. 1 through No. 4--constitute the most critical region of vulnerability under strong wind loading. To capture the localized structural behavior of this zone with higher fidelity, a detailed sub-model is developed and integrated into a concurrent multi-scale modeling framework. In multi-scale simulations, effective coupling between global and local models requires compatibility in degrees of freedom (DOFs) at the interface. When the DOFs are consistent between different element types, direct nodal connections can be used. However, in cases where discrepancies in DOFs exist, constraint equations must be formulated to enforce kinematic compatibility in both translational and rotational motions across the interface [34]. This relationship is mathematically expressed as  
\[C(I)=\sum_{I=1}^{N}[Co(I)\cdot U(I)] \tag{8}\]  
where \(C(I)\) represents the constant term; \(U(I)\) is the degree of freedom variable; \(Co(I)\) is the coefficient of \(U(I)\); and \(N\) is the total number for the constraint equations. To implement this coupling, the Multi-Point Constraint (MPC) method in Abaqus is adopted. The MPC algorithm automatically generates constraint equations and is capable of accommodating nonlinear phenomena such as large displacements and the birth or death of elements. This makes it especially effective for modeling complex joint behavior, including rigid rods and rigid beams. In this study, beam and solid elements are integrated using the MPC approach, where the node of the beam element is designated as the control node. This node governs the displacement and rotation of the adjacent solid element nodes, ensuring continuity and mechanical consistency across the model interface. The coupling configuration is illustrated in Figure 16.  
To strengthen the identified weak region in the third section of the tower, the traditional perforated plate reinforcement method is implemented. This technique involves the installation of perforated steel connecting plates at both ends of the primary structural members (typically angle steel rods), forming a bolted assembly that redistributes internal forces and improves local stability. The perforated plate reinforcement method is defined as a structural strengthening approach in which plates with pre-drilled bolt holes are used to bridge critical members through frictional and mechanical interlock provided by bolted joints, thereby enhancing both the stiffness and load-bearing capacity of the target region [24; 26]. Mechanically, the reinforcement effect is achieved through dual mechanisms:  
Figure 15: Failure of the third section. (1) direct force transmission via bolt preload and bearing contact between the plate and the member, and (2) confinement effect, where the plate assembly constrains the local deformation and delays the onset of instability. Compared to full-member replacement or welded stiffening techniques, the perforated plate method offers several key advantages, such as non-destructive installation, better stress redistribution, ease of customization, as well as improved fatigue resistance. The reinforced structural configuration is illustrated in Figure 17. Given the complexity of simulating contact interactions between bolts and surrounding structural components, a simplified modeling approach is adopted. The normal contact behavior is defined as hard contact, preventing interpenetration of mating surfaces. Tangential behavior is modeled using a penalty friction formulation, with a friction coefficient set to 0.3 to represent realistic slip resistance during shear load transfer. To replicate the mechanical effect of bolted joints while minimizing computational cost, all nodes on the bolt hole surface are coupled to a single central control node positioned at the geometric center of the bolt hole.