Vertical Bending Research Articles

Consequences of the Improved Wave Statistics on a Hull Girder Reliability of Double Hull Oil Tankers

This paper investigates the change in hull girder failure probabilities and partial safety factors caused by the implementation of the new procedure for direct computation of wave loads recommended by the International Association of Classification Societies (IACS). Differences between new and previous procedures are primarily related to the different associated scatter diagrams, and secondarily due to the assumptions on wave spectrum, wave energy spreading, and ship speed. This study performs a comparative structural reliability analysis of the global longitudinal bending of five oil tankers of different sizes between two procedures for wave load computation. Firstly, failure probabilities are compared, and secondly, modified partial safety factors are proposed, resulting in similar failure probabilities according to two separate procedures. It is found that implementation of the new revision of the IACS procedure for direct computation of wave loads results in a reduction of the minimum required ultimate vertical bending capacity of a ship hull by 10%. In addition to the novel investigation of the safety of oil tankers using a revised wave scatter diagram, this study offers a new rapid method for calculation of extreme vertical wave bending moments based on the regression of the parameters of the Weibull function, used for the long-term probability distribution of wave-induced loads.

Analysis of Influence Characteristics of Site Conditions on Seismic Response of Utility Tunnel

Utility tunnels are crucial infrastructure projects for cities. Earthquakes are a leading cause of damage to these tunnels. The effect of a non-homogeneous site, as a complex site, on the seismic response of utility tunnels cannot be ignored. Therefore, in this study, a non-homogeneous three-dimensional finite element model of a utility tunnel was created using the ABAQUS 2021 software. El Centro seismic waves were inputted while changing the structural depth of burial to investigate the utility tunnel’s seismic response. This research is expected to further clarify the seismic variation in the utility tunnel under complex site conditions. The results show the following: (1) The impact of structure burial depth on the utility tunnel in a non-homogeneous site should not be neglected. The peak acceleration shows a decreasing and then increasing trend with increasing depth of burial within 10 m. Under the same site conditions, the peak accelerations and amplification factors of the soil in the clay are larger than those in the sand. (2) In identical site conditions, the utility tunnel structure experiences larger peak displacements in clay compared to sand. As the structure is buried deeper, the peak displacement of the utility tunnel gradually decreases, while the rate of decrease in the peak displacement of the utility tunnel structure gradually increases. The rate of decrease in peak displacement in clay is greater than that in sand. (3) The stress distribution of the utility tunnel structure changes as the depth of the structure increases. In the clay area, the utility tunnel structure shifts from transverse bending deformation to vertical bending deformation. In the sand area, it maintains transverse bending deformation due to the non-homogeneous site. The findings from this study offer guidance for building utility tunnels on non-homogeneous sites.

Дослідження відносних деформацій міжуламкового регенерату в моделях різних варіантів остеосинтезу кісток гомілки з переломами в середній третині у дітей з недосконалим остогенезом та незавершеним ростом

Introduction. Osteogenesis imperfecta is a genetic pathology that leads to a violation of the processes of formation of collagen fibers, formation of bone matrix and its mineralization and, as a result, to the formation of bone tissue with low strength properties. The most promising means of stabilizing bone fragments under this condition are intramedullary fixators of various designs. Aim. To investigate the relative deformations of the interfragmentary regenerate under the influence of various types of loads in models of the tibia with fractures of both bones in the middle third and their osteosynthesis using intramedullary rods of various designs in children with osteogenesis imperfecta and incomplete growth. Materials and methods. Mathematical modeling of options for osteosynthesis of tibia bones with a fracture in the middle third in children with osteogenesis imperfecta was performed. Two variants for tibial osteosynthesis were modelled: a rod without rotational stability; rod with rotational stability. Osteosynthesis of the fibula was not modelled in all variants. The stress-strain state of the model under the influence of vertical compressive, bending and torsional loads, as well as the magnitude of the relative deformations of the interstitial regenerate, were studied. The results. A “growing” intramedullary shear with rotational stability provides essential advantages in the case of torsional loads. The presence of rotational stability makes it possible to ensure two times lower values of relative deformations of interfragmentary regenerates compared to osteosynthesis with a rotationally unstable rod. Under compressive and bending loads, both rods showed almost identical results of relative deformations of bone regenerates. High rates of deformation of interfragmentary tibial regenerates are due to the lack of longitudinal axial stability of both rods, which is the basis for the possibility of increasing their length during the growth of the patient. Conclusions. The use of osteosynthesis with intramedullary rods, which increase during the treatment of tibial fractures in patients with osteogenesis imperfecta, does not ensure a sufficient level of stability for the fixation of bone fragments under compression and bending loads, which leads to the greatest deformations of bone regenerates. A rod with rotational stability provides advantages in resistance to torsional loads, which is determined by twice lower relative deformations of interfragmentary regenerates. No conflict of interests was declared by the authors.

Study on Abnormal Pattern Detection Method for In-Service Bridge Based on Lasso Regression

The real-time operational safety of in-service bridges has received wide attention in recent years. By fully utilizing the health monitoring data of bridges, a structural abnormal pattern detection method based on data mining can be established to effectively ensure the safety of in-service bridges. This paper takes a large-span arch bridge as the research object, analyzes the time-based variation of the main monitoring data of the structure, establishes Lasso regression models for load characteristic indicators and vertical bending fundamental frequency of the structure under different time scales, and uses the residuals of the Lasso model to indicate the structural state and identify abnormal patterns. Firstly, the monitoring data of bridge structural temperature, girder end displacement, and girder acceleration were analyzed, and the interrelationships were studied to extract characteristic parameters of structural load characteristics and structural frequency. Then, the time-varying patterns of structural response were analyzed, and Lasso regression models and their regression variables were discussed based on monitoring data under two different time scales: daily cycle and annual cycle. The abnormal pattern detection method for bridge structures was developed. Finally, the effectiveness of this method was verified by taking the bridge deck pavement replacement as the abnormal pattern. The research results indicate that the proposed bridge structure abnormal pattern detection method based on Lasso regression can effectively monitor changes in the state of the bridge, and the residual dispersion of the model established on the annual cycle scale is relatively smaller than that on the daily cycle scale, resulting in better abnormal detection performance.

Seismic performance of joints of prefabricated corrugated steel utility tunnels Part(I)–Experimental analysis

This paper presents the seismic performance of a corrugated steel plate utility tunnel, based on an engineering prototype. The research combines quasi-static testing and numerical simulation to investigate the behavior of the structure under cyclic loads in different directions. Specifically, six loading scenarios were considered: the quasi-static test was conducted for transverse tension and compression, transverse bending and shear, vertical shear test (Conditions (1–3)), while numerical simulations were performed for longitudinal tension and compression, transverse shear, and vertical bending shear test (Condition (4–6)) in Part(II). The results reveal comprehensive hysteresis curves for the plate under diverse loading directions, with an extended yield platform section that underscores its significant plastic deformation capacity. However, the “pinch” phenomenon of the hysteresis curve is obvious. The existence of rounded corners not only makes the hysteresis curve of the plate asymmetrical but also makes a great difference to the bearing capacity of the structure in the direction of positive and negative loading. The component exhibits superior bearing capacity and stiffness along the longitudinal direction compared to other orientations. Additionally, the structure demonstrates remarkable energy dissipation capabilities because of the inherent “accordion effect” of the corrugated steel plate. Moreover, the maximum energy dissipation coefficient in three directions is more than 1.0, illustrating its excellent energy dissipation capacity. The results provided valuable insight into the seismic performance of joints and contributed to the safe design of prefabricated corrugated steel utility tunnels.

Train–Bridge Coupled Vibration of a Long-Span Steel Truss Suspension Bridge Under Complex Driving Conditions

To investigate the vibration characteristics of long-span suspension bridges under the impact of high-speed trains, a method for conducting a time–frequency domain analysis of the dynamic response of a coupled system is presented in this study. First, the finite element model of a suspension bridge is established to conduct modal analysis, and simulation results are compared with measured data to validate the correctness of the bridge model. Subsequently, a multi-body dynamics model of a high-speed train is developed. Then, two approaches, the dummy method and flexible-track method, are employed to create the train–bridge coupled system. Finally, the flexible-track method is used to perform time–frequency domain analysis of the system under complex operational conditions. The research revealed that large-span suspension bridges exhibit lower natural frequencies. The vertical bending vibrations of the main beam have the potential to significantly impact train operations. Under the combined impact of cable tension and wheel–rail forces, the vibration patterns of large-span suspension bridges become notably intricate. The number of trains and their loading configurations both exert influence over the dominant frequencies and amplitudes in the PSD of bridges. In the complex traffic of large-span suspension bridges, the rational arrangement of travel lanes can reduce the possibility of resonance in the bridge–train system.

In-plane and out-of-plane one-way vertical bending behavior interaction analysis of unreinforced masonry walls with newly developed load capacity interaction curve

The structural behavior of unreinforced masonry (URM) walls under in-plane (IP) or out-of-plane (OOP) loading in masonry buildings has been extensively investigated in the public literature. However, studies focusing on the URM walls subjected to concurrent IP and OOP loadings are limited. Neglecting IP and OOP interaction effects may lead to unsafe design practices. As such, this study conducts a comprehensive numerical investigation into the behavior of URM walls under combined IP and OOP loading, focusing on the influences of aspect ratio (i.e., height-to-length ratio), slenderness ratio (i.e., height-to-thickness ratio), and pre-compression load levels. To capture the possible failure modes of URM walls under various loading scenarios, the simplified micro modeling approach is employed. The analysis results indicate that the presence of OOP loading leads to a substantial reduction in IP capacities. Longer walls, characterized by smaller AR, exhibit more pronounced IP and OOP interaction effects. Additionally, highly slender walls show significant additional moments due to second-order effects under OOP loading, thereby negatively affecting the IP capacity. Low-level pre-compression loads are beneficial in diminishing capacity interaction effects, while high pre-compression loads exert a negative influence. To facilitate the consideration of IP and OOP capacity interaction effects in an efficient manner, a simplified analytical model is developed through curve fitting, in which the effects of aspect ratio, slenderness ratio, and pre-compression load are incorporated.

ANALYTICAL DEFLECTIONS AND STRESS ANALYSIS IN SLENDER BAR FRAMED STRUCTURES SUBJECTED TO ANTISYMMETRIC LATERAL LOADS

Mechanical analysis of the lateral deflections in slender structures is an interesting task to model the mechanical response to transverse forces like seismic or wind forces. Special interest is focused on buildings, communication towers, or tower cranes, as well as any other kind of framed structures. A pure analytical solution for the lateral deflections and vertical displacements, stresses and bending moments of a framed structure under power-law lateral loads is presented in a non-dimensional form. The internal forces of the structure are stated by the force method which drives to a set of linear equations, which are analytically solved, thus reaching an exact, continuous, and differentiable solution. A parametric analysis is performed on a simplified case to show the influence of the dimensionless parameters on the solution. An explicit solution for deflections and bending moments is included for one and two structural levels. The solution is validated via numerical software, using STR and ABAQUS, upon a population of ten, building-like structures. Keywords: structure, deflections, force method, tower, building, lateral load, non-dimensional, antisymmetric

Study on the effect of tunneling and secondary grouting on the deformation of existing pipelines in bedrock raised strata

Shield crossing the bedrock raised strata easily leads to large deformation of the soil, resulting in deformation and damage of the upper pipelines. To investigate the law of pipeline deformation caused by shield underpassing in bedrock raised strata, the changes in the action range of the additional thrust of the cutter head (p1), the friction resistance of the shield shell (p2), and the additional grouting force (p3) were analysed. A convergence model of the excavation face suitable for bedrock raised strata was proposed. The calculation formula of soil displacement at the axis of the pipeline was derived by using Mindlin’s solution and stochastic medium theory. The solutions of the vertical displacement, bending moment and strain of the pipeline were obtained by the energy variational method and the principle of minimum potential energy. Pipeline deformation analysis and reliability verification were carried out by an engineering case in Hangzhou, China. The influence of bedrock distribution and secondary grouting behind the pipe wall on pipeline deformation was studied. The results show that the calculated results are similar to the measured data with a high degree of coincidence, and the degree of agreement is higher compared to the existing method. When the shield tunneling direction is well controlled, the settlement and bending moment of the pipeline caused by the shield crossing the raised bedrock stratum will be reduced, and the reduction increases with increasing bedrock length and intrusion depth. The secondary grouting behind the pipe wall can reduce the settlement of the pipe line, but it is necessary to reasonably control the three parameters of grouting amount per unit length (Vinj), grouting ring length (Lh), and grouting ring angle (δ). Lh should not be too long and can be controlled below 30 m, δ should be controlled within 60°, and Vinj can be adjusted according to the deformation size of the pipeline to prevent excessive or insufficient correction.

Frequency-Domain 3D Computer Program for Predicting Motions and Loads on a Ship in Regular Waves

The development of an in-house computer program for determining the motions and loads of advancing ships through sea waves in the frequency domain, is described in this paper. The code is based on the potential flow formulation and originates from a double-body code enhanced with the regular part of the velocity potential computed using the pulsing source Green function. The code is fully developed in C++ language with extensive use of the object-oriented paradigm. The code is capable of estimating the excitation and inertial radiation loads or arbitrary incoming wave frequencies and incidence angles. The hydrodynamic responses such as hydrodynamic coefficients, ship motions, the vertical shear force and the vertical bending moment are estimated. A benchmark container ship and an LNG carrier are selected for testing and validating the computer code. The obtained results are compared with the available experimental data which demonstrate the acceptable compliance for the zero speed whereas there are some discrepancies over the range of frequencies for the advancing ship in different heading angles.

Numerical study on the dynamic amplification factors of highway continuous beam bridges under the action of vehicle fleets

The mechanism and reasonable values of the dynamic amplification effect of highway continuous beam bridges under the action of vehicle fleets are quite complex and very unclear at present. This study, using 23 highway continuous beam bridges with the main span lengths from 32 m to 165 m as research objects, investigates the mechanism and influencing factors of the dynamic amplification factors (DAFs) of highway continuous beam bridges under the action of vehicle fleets based on the vehicle-bridge coupled vibration analyses. The results show that the travelling speeds and vehicle intervals (VIs) exert an important influence on the DAFs of continuous beam bridges. The maximum dynamic response at different travelling speeds and VIs is searched in this study, the maximum static response at different VIs is also searched, and their ratio is regarded as the effective DAFs. Based on the calculation results, it is concluded that when the roughness surface condition (RSC) is ‘good’ (or ‘average’), the effective DAFs for the vertical displacement and vertical bending moment at the mid-spans can safely take a value of 1.15 (or 1.30); those for the vertical bending moment at the inner supports and shear force can safely take a value of 1.10 (or 1.20); and when the bridge fundamental frequency is less than 1.0 Hz, all DAF values can take a value of 1.05 (or 1.10). Meanwhile, the relevant provisions in AASHTO LRFD (2017) and CEN (2003) are considered on the safe side when RSC is ‘good’ or ‘average’.

Radiated power and acoustic convergence zone properties of the vibration of the shaft-hull system in an ocean acoustic channel

To compute the vibroacoustic radiation of the shaft-hull system in an ocean acoustic channel, we apply the finite element method/boundary element method. For the convergence zone effect, we use the beam displacement ray normal mode to examine the profile of the speed of sound in seawater. To evaluate the radiation property of the system, we propose the concept of effective radiated acoustic power, where only reversing acoustic rays and seabed totally reflected rays are included. We reveal that the far-field radiation can be approximately inferred from the effective radiated acoustic power. Effective radiated acoustic power can be utilized to extract the main radiating modes, which dominate the far-field radiation. The axial force induces vibration, including breathing and bending modes, resulting in the highest peak of the radiated sound pressure level (SPL). The lateral and vertical excitation results in horizontal and vertical bending modes, respectively, and the SPL in convergence zones induced laterally are higher than those induced vertically.

Research on the hydroelastic response of ice floes and wave scattering field

Abstract The marginal ice zone (MIZ) is the area between sea ice and open water, the structure of which is mainly determined by wave and ice interactions. Thus mastering the characteristics of MIZ is of great significance to the Arctic routes opening and the natural resources development. In this paper, the hydroelastic response of ice floes in waves is studied, a three-dimensional numerical wave tank is established based on the computational fluid dynamics (CFD) technology. The finite volume method (FVM) and finite element method (FEM) are respectively utilized for discrete fluid domain and ice domain. A mapping interface at the junction of the fluid and ice floes domains is created to perform data mapping by the shape function interpolation method and the least square method. This work presents a series of numerical simulations to study the fluid-solid interaction (FSI) of waves and ice floes. Under the given incident wave parameters, the vertical bending deformation of ice floes with different shapes under the excitation of waves, the effect of ice floes' deformation on the wave field are studied, and the effect of wave overwash on the transmitted wave field is emphasized. Results show that the shape of the ice floes significantly affects its elastic deformation and scattered wave field, and the wave overwash phenomenon attenuates the scattering wave.

Adaptive multi-point flexible-bending technology for improving the quality of profile using discrete dies driven by snake-shaped motion mechanism

Based on the application of snake-spine-like structures, this paper proposes a roller-type multi-point flexible 3D profile bending process with snake-spine-like structures (Flexible stretch-bending with roller dies - driven by snake-like mechanism, FSBRD-S), and establishes a finite element model. Taking the L-shaped cross-section profile as the research object, different working modes and contact modes are defined to study their impact on crease defects during the vertical bending step, and the new structure is designed to improve the forming quality of the products' surface. The research results show that changing the contact mode leads to a change in the relative angle between the die and the billet. In the process of larger angle vertical bending, the FSBRD-S process can effectively reduce the crease defects generated by the traditional FSBRD process, especially in the area near the clamp. Furthermore, this article proposes the FSBRD-S shape-adjusting technology, which allows for adjustments to the FSBRD-S process based on actual needs, to obtain products with different surface qualities.

Improved semi-analytical solution for longitudinal mechanical response of tunnels crossing active faults considering nonlinear soil-tunnel interactions and shear effects

Tunnels crossing active faults are susceptible to severe damage. This study establishes an improved semi-analytical methodology for analyzing the mechanical responses of tunnels subjected to active fault dislocation. The methodology incorporates nonlinear axial, transverse, and vertical soil-tunnel interactions, shear effects, and geometric nonlinearity within its governing equations. Compared with existing methods, the proposed method has a significantly extended application range with improved accuracy, and the solution procedure demonstrates exceptional computational efficiency, with each case typically being solved in less than 1 s. The proposed methodology demonstrates outstanding qualitative and quantitative agreement with 3D FEM results across various fault types, as well as with the results obtained from the model test. Additionally, neglecting shear effects results in approximately 1.11 to 1.20 times higher bending moments and 1.13 to 1.19 times higher shear forces of the tunnel. Finally, a parametric analysis was conducted based on the proposed methodology to investigate the influence of critical parameters, such as fault displacement, buried depth, tunnel diameter, soil cohesion, and soil friction angle, on the tunnel response under fault dislocation. The results suggest that the tunnel responses positively correlate with the fault displacement, buried depth, and soil cohesion. Increasing the fault displacement amplifies the vertical shear force and bending moment asymmetry while increasing buried depth reduces this asymmetry. An increase in the tunnel diameter and soil friction angle is associated with a decrease in the peak axial force and an increase in the vertical shear force and bending moment. Additionally, variations in the friction angle do not exert a significant effect on the transverse shear force and bending moment.

A quasi-solid-state self-healing flexible zinc-ion battery using a dual-crosslinked hybrid hydrogel as the electrolyte and Prussian blue analogue as the cathode material.

Due to their excellent safety and lower cost, aqueous Zn-ion batteries (AZIBs) have garnered extensive interest among various energy-storage systems. Here we report a quasi-solid-state self-healing AZIB by using a hybrid hydrogel which consists of dual-crosslinked polyacrylamide and polyvinyl alcohol as a flexible electrolyte and a cobalt hexacyanoferrate (K3.24Co3[Fe(CN)6]2·12.6H2O) Prussian blue analogue as the cathode material. The obtained hybrid hydrogel showed a superhigh fracture strain of up to 1490%, which was almost 15 times higher than that of the original size. Due to the fast formation of hydrogen bonds, the self-healed hydrogel from two pieces still displayed 1165% strain upon failure. As a result, the self-healed battery delivered stable capacities of 119.1, 108.6 and 103.0 mA h g-1 even after being completely cut into 2, 3 and 4 pieces, respectively. The battery capacity recovery rates for each bending cycle exceeded 99.5%, 99.8%, 98.6% and 98.9% during four continuous bending cycles (30 times bending at 90° for each cycle), which indicates outstanding flexibility and self-healing capability. In parallel, the hydrogel electrolyte displayed a broader electrochemically stable window of 3.37 V due to the suppression of water splitting and low overvoltage during the 500 h cycling in a symmetric cell. Zinc dendrites were also suppressed as evidenced in symmetric cell measurements. The assembled AZIB exhibited an initial capacity of 176 mA h g-1 upon vertical bending. The battery showed a reliable capacity of 140.7 mA h g-1 at 0.2 A g-1 after 100 cycles along with a coulombic efficiency of >99%. A reliable capacity of nearly 100 mA h g-1 was retained after 300 cycles at 1.0 A g-1. The highly flexible and self-healing AZIB demonstrates great potential in various wearable electronic devices.

Real-time prediction of wave-induced hull girder loads for a large container ship based on the recurrent neural network model and error correction strategy

Real-time acquisition of wave-induced hull girder loads of a sailing ship will help the captain make reasonable decisions, which is of great significance for improving the safety of the ship's navigation. This paper investigates the real-time prediction method of hull girder loads based on the Recurrent Neural Network (RNN) model and error correction strategy. Firstly, taking the vertical bending moment, horizontal bending moment, and torsional moment at the mid-ship position of a large container ship as examples, corresponding neural network prediction models are established through parameter influence analysis. Secondly, various sea state conditions are used to verify the feasibility of established network prediction models to predict the hull girder loads in real-time. The VBM prediction model performs better than the TM prediction model and HBM prediction model, and the errors of the TM prediction model and HBM prediction model are slightly larger in some cases. Lastly, an improved prediction model based on an error correction strategy is proposed to improve the prediction accuracy of the neural network prediction model, and the adequate performance of the error correction strategy is discussed.

Physical and mechanical properties of foam-type panels manufactured from recycled cardboard

This study investigated different properties of foam-type composites made from recycled cardboard. Two types of panels, type A (237.36 kg/m3) and type B (284.28 kg/m3) were manufactured varying the share of corn starch as binder. Dimensional stability to water, vertical density profile (VDP), thermal conductivity, bending characteristics, and internal bond strength tests were conducted. The results revealed that the higher corn starch content improved the modulus of elasticity (MOE), modulus of rupture (MOR), and internal bond strength (IB) by 56%, 33%, and 30% for panel type B, respectively. However, no significant difference was observed for the internal bonding strength test. Statistical analysis revealed that no substantial difference in the thermal conductivity property of the two types of panels was determined at a 95% confidence level. The water absorption and thickness swelling increased for both samples after 24 h water immersion. Foam-type panels made from waste cardboard would have the potential to be used for certain applications such as insulation purposes within the perspective of sustainability and a green approach.

Linear Longitudinal Strength Analysis of a Multipurpose Cargo Ship under Combined Bending and Torsional Load

Cargo ships with wide hatches usually have thin walls and limited torsional rigidity. Consequently, conducting a comprehensive torsional analysis is important because these loads can exert a significant impact. In this paper, the structural response of a multipurpose cargo ship to combined bending and torsional loads is studied using finite element analysis. The bending and torsional moments are calculated following the rules and standard regulations followed by the classification society. The ship’s 3D finite element model was verified using beam theory and direct calculations. In contrast, the accuracy of torsional stress was confirmed by comparing thin wall girder theory with direct calculation results. This study thoroughly examined the impacts of the still water bending moment, the vertical wave bending moment, and the wave-induced torsional moment on the structural response of ships. Furthermore, it scrutinised the impact of torsion on both open-deck and closed-deck ships. Hull girder normal stresses at midship due to still water and the vertical wave bending moment are shown to contribute to almost 70% of total stress in an inclined condition; stresses resulting from the horizontal wave bending moment contribute nearly 10%, while warping stresses contribute approximately 20% in open-deck ships. It is also shown that torsion has little impact on closed-deck ships. Finally, a buckling analysis was conducted to assess the ship’s buckling criteria, confirming that the linear buckling criteria were satisfied.

A novel multi-modal analytical method focusing on dynamic mechanism of bridge flutter

A novel three-dimensional (3D) multi-modal analytical method is proposed in the present study for investigating the dynamic mechanism of long-span bridge flutter. The multi-modal coupled flutter differential equations are derived in a practical manner utilizing the principle of virtual work in this method, and the explicit expressions of system damping and stiffness are established through the excitation-feedback mechanism. Consequently, this method can not only conveniently identify the critical point of 3D bridge flutter, but also provide a comprehensive understanding of the underlying mechanism involved in 3D bridge flutter. The well-known coupled aerodynamic damping related to flutter derivatives A1∗ and H3∗, which is principally responsible for the coupled flutter, is further subdivided in this study to reveal the intrinsic coupled mechanism of flutter participating modes: The aerodynamic coupled degree between a certain vertical bending mode and the fundamental torsional mode can be ascertained by evaluating the similarity factor between their mode shapes; the phase angle between a certain vertical bending mode and the fundamental torsional mode, which is largely dominated by the numerical relationship between natural and flutter frequency, determines whether the aerodynamic coupled effect associated with the two modes drives or restrains flutter.