Dynamic instability criteria of large-span ground-supported truss arches

Abstract

This research focuses on ground-supported steel truss arches for a gymnasium in Guangzhou, China, addressing potential elastoplastic dynamic instability issues caused by typhoons. An energy conversion criterion was initially proposed based on a discrete Lyapunov functional. The vibration responses, energy conversion patterns, and plastic deformation characteristics of the truss arches under 1.0, 1.2, and 1.3 times the reference wind pressure (RWP) were examined using the dynamic relaxation method in conjunction with finite element analysis (FEA). The results indicated that, under a load release of 1.0 to 1.2 times the RWP, the system exhibited stable periodic energy conversion, with an absolute Lyapunov exponent value of less than 1, demonstrating the dynamic stability of the structure. However, as the load release increased to 1.3 RWP, the energy conversion process displayed abrupt changes, leading to the formation of plastic hinges at the abutments, which resulted in asymmetric degradation of structural stiffness. The Lyapunov exponent diverged, causing the system to enter a state of nonlinear dynamic instability. The release of wind load resulted in an asymmetric distribution of plastic hinges along both sides of the truss arches, complete interruption of energy conversion, and significant vibration amplitudes in the Poincaré section, ultimately leading to a loss of load-bearing capacity. This research highlights the effectiveness of the discrete Lyapunov function in exploring dynamic stability, providing a theoretical basis for safety control in wind resistance engineering.

Keywords

steel truss arch elastoplastic dynamic instability discrete Lyapunov functional energy conversion criterion finite element analysis

Introduction

The study of Lyapunov energy-based method demonstrates rich dynamics, making it ideal for studying nonlinear systems and modeling real-world applications like truss arches and bridges. It’s also valuable for dynamic stability analysis in 2D or 3D nonlinear spatial structures.^1–3 Ground-supported truss arches, recognized for their expansive spans, free-form designs, and high construction efficiency, have emerged as a predominant structural form in large-span public buildings.⁴ The arch-shaped load-bearing system consists of steel trusses that are directly anchored to the ground or foundation at both ends. This configuration enables the realization of a large, column-free space by balancing the axial compressive force of the arch with the bending moment resistance of the truss. The primary engineering applications of these structures are found in sports venues. They are also commonly employed in exhibition centers and industrial buildings, where they fulfill the requirements for supporting heavy suspended equipment and facilitating flexible exhibition arrangements.⁵

As large-span flexible structures, steel ground-supported truss arches are commonly utilized as roof support systems.⁶ These structures are inherently exposed to dynamic loads, such as wind forces. Wind loads consist of lateral wind pressure distributed along the arch span, which can lead to overall lateral displacement or even overturning of truss arch structures. Pulsating wind, characterized by a broad spectrum and significant spatial correlation, may induce vortex-induced vibrations (VIVs) or galloping.⁷ Furthermore, the vertical bending modes of large-span arches can resonate with wind vibration frequencies, potentially resulting in resonance failure. Ground-supported truss arches in the southeastern coastal region of China are particularly susceptible to typhoons. When the lift or lateral forces generated by static wind pressure exceed the critical threshold of structural bending stiffness, the structure experiences overall lateral buckling.⁸ This type of instability is characterized by a relatively low critical wind speed and commonly occurs in flat or truss arches with small height-to-width ratios. Due to the combined effects of the diverse characteristics of wind loads acting on truss arches and the complexity of their spatial structure, developing dynamic stability judgment criteria for ground-supported truss arches is particularly challenging.⁹

The first type of static stability problem in arches arises from the bifurcation of the arch’s equilibrium state, leading to a loss of stability in the original equilibrium state and a transition to a new equilibrium state. This includes both in-plane and out-of-plane buckling.¹⁰ Similar to the Euler critical load for centrally compressed columns, mathematically solving the first type of stability problem for arches involves addressing the eigenvalue problem of the elastic equilibrium differential equation for each micro-element that composes the arch, considering the effects of internal and external forces.¹¹ During the solution process, the geometric boundary conditions of the homogeneous equation must be determined based on the deformation of the arch’s deflection curve and its geometric relationships. Additionally, the relationship between internal forces and deformations should be established using a constitutive equation that reflects the physical relationship between these internal forces and deformations.¹² The second type of static stability problem in arches is a nonlinear plastic analysis issue. Similar to the elastic-plastic buckling associated with the P-Δ second-order effect in eccentrically compressed columns, this problem arises from the combined effects of geometric and physical nonlinearities.¹³ By introducing a local coordinate system for the arch into the elastic equilibrium differential equation, the spatial equilibrium equation for the arch is derived.

The judgment criterion for the dynamic stability of arches is fundamentally rooted in the elastic equilibrium equation governing the plane deflection curve of the arch structure. Due to the large span of steel truss arches, conventional structural laboratories are unable to conduct full-scale structural experiments and can only perform aerodynamic performance test, supplemented by numerical analysis methods to verify the stability theories of various steel truss arches.¹⁴ Partial model-based damage identification was employed to evaluate the stability of a long-span steel truss bridge (New Yellow River Bridge), while the stiffness separation method was utilized to reduce the computational time associated with damage identification in the bridge.¹⁵ To analyze the dynamic behavior, we incorporate D'Alembert’s principle, which introduces inertial forces into the system. This approach allows us to consider the effects of inertia alongside the elastic forces acting on the arch.¹⁶ When a trussed arch in a high potential energy state experiences dynamic instability due to the fracture of stay cables, the dynamic relaxation method can be employed to establish the dynamic equilibrium equations.^17,18 For large-scale, flexible, deployable trussed arches in spacecraft subjected to thermal excitation, various techniques, including L-chain transfer, center manifold, and feedback shape control methods, can be utilized in conjunction with the Routh-Hurwitz criterion to assess dynamic stability.^19–23 A criterion for determining instability cannot be established merely by introducing inertial forces into static stability analyses and deriving vibration analytical solutions under the assumption of simple harmonic vibration. Consequently, existing arch stability theories are inadequate for solving nonlinear problems, let alone those involving non-uniform stiffness.²⁴ The aforementioned differential equations lose their theoretical assumptions of elastic deformation when analyzing the dynamic stability of trussed arches under large deformations. Furthermore, the non-uniform stiffness characteristics of trussed arches disrupt the constant stiffness condition, which is essential for solving the equilibrium equations.²⁵ Due to the inability to develop a criterion analogous to Euler’s critical load theorem for trussed arches, a criterion can be established for elastic-plastic dynamic instability in trussed arches by applying discrete Lyapunov functional theory and considering discrete variables provided by various numerical analysis methods.^26,27

Both theories of elastic equilibrium-based arch stability calculation and physical model testing procedures encounter significant limitations due to the inherent geometric nonlinearity of truss arch spatial structure components, joint material nonlinearities, and nonlinear boundary conditions, such as seismic and wind effects.²⁸ The finite element method (FEM) and finite difference method offer distinct advantages in addressing truss arch dynamic stability problems by discretizing nonlinear spaces into linear subspaces and incorporating nonlinear constitutive equations.^29,30 Researchers have conducted extensive analyses using ANSYS and SAP2000.^31,32 To address fluid-structure coupled vibrations between wind and truss arches, the system is decomposed into wind and arch subsystems, and independent vibration equations are derived for each subsystem, followed by cyclic coupling analysis.³³ Research has demonstrated that wind-induced vibration frequency significantly affects truss arch dynamic responses, while adjustments to structural stiffness and damping can effectively mitigate coupled vibration responses and reduce collapse risks.³⁴

Analyses of instability mechanisms have demonstrated that truss arch failure begins with non-uniform material failure, leading to the degradation of localized stiffness and subsequent global instability. This process exhibits cross-scale evolutionary properties, ranging from heterogeneous materials to component stiffness softening and ultimately to system-level dynamic instability. Throughout this evolution, the conservative conversion between kinetic energy (KE) and strain energy (SE) is associated with dynamic stability. First, a discrete Lyapunov function for evaluating the stability of nonlinear dynamic systems was established based on the conservation of system energy conversion. Then, using Python to drive ABAQUS, ground-supported steel truss arches for a gymnasium at a university in Guangzhou were rapidly constructed.^35,36 Through the dynamic release method, the time history of SE and KE conversion of the ground-supported truss arch was analyzed under standard wind pressures of 1, 1.2, and 1.3 times. Finally, the energy conversion time history was incorporated into the Lyapunov function to quantitatively assess the dynamic stability of the ground-supported truss arch. During this process, the strange attractor at the maximum dynamic response Poincaré sections was outputted to further validate the dynamic stability of the arch.

Overview of a steel truss arch project

The university stadium in Guangzhou, South China, features a football field, a large running track, and a grandstand structure with a canopy. The upper section of the structure includes an arched steel canopy supported by ground-based arches, achieving a cantilevered roof with a maximum span of 101.4 m. The transverse truss system consists of three triangular-section trusses, with a maximum longitudinal height of 21.3 m. The bases of the arches are anchored to the ground, and the transverse truss arches are interconnected by ten longitudinal trusses with triangular cross-sections. The joints within the truss system are welded, while the lower portion of the ground-based arches comprises concrete pedestals integrated with seating installations. Figure 1 illustrates the spatial configuration of the ground-based truss arches. Under normal service conditions, the roof system primarily sustains wind loads. The upper chord arches are connected to steel cables to restrict displacements in the z-direction, while displacements and rotations along the x-, y-, and z-directions are constrained by fixed hinge supports at the bases of the arches.

Figure 1.

Schematic diagram of ground-based truss arch spatial structure.

All truss arch members are composed of Q235-grade circular steel tubes with welded joints. The upper and lower chord members have a diameter of 137 mm and a wall thickness of 12 mm. The lower diagonal and horizontal braces have a diameter of 163 mm and a wall thickness of 13 mm, while the upper diagonal and horizontal braces have a diameter of 110 mm and a wall thickness of 10 mm. Table 1 summarizes the detailed material parameters of the steel tubes.

Table 1.

Specific parameters of steel truss.

Type	Profile mm	Area mm²
1	Φ137 × 12	4710
2	Φ51 × 4	628
3	Φ70 × 6	1205.7
4	Φ110 × 10	3140
5	Φ97 × 8	2235.9
6	Φ57 × 4	665.68
7	Φ163 × 13	6132
8	Φ35 × 3	301.44

Dynamic instability critical state of steel truss arches

Under the influence of wind-induced vibrations, ground-based truss arches experience oscillatory motion characterized by alternating conversions of SE and kinetic energy KE among the structural members. In accordance with the law of conservation of mechanical energy within the system, the minimum free energy (U_r) of each member in Euclidean space is defined as follows^37,38:

U_{r} = K E_{r} - S E_{r} + W_{r}

(1)

where W_r was the work done by external forces of the i-th member.

By aggregating the mechanical energies of all components across both spatial and temporal domains, the free energy ( $U_{t r u s s}$ ) of the truss arch system can be expressed as follows:

U_{t r u s s} = \sum_{r = 1}^{n} U_{r}

(2)

While equation (2) offers conceptual clarity, the geometric complexity of the spatial truss arch structure impedes its direct analytical solution. To address this challenge, Lyapunov functionals are employed to assess the convergence of differential equations instead of explicitly solving them.

The asymptotic solution ( $x (t, x_{0}) = e^{A t} x_{0}$ ) of the linear differential equation ( $\dot{x} = A x$ ) plays a crucial role in algebraic methods for assessing the stability of dynamical systems.³⁹ Linearly independent differential equations are solved using the coefficient matrix (A), which can be directly addressed through linear algebra. Explicit solutions to these differential equations can be derived when the eigenvalues and eigenvectors of (A) are known. Stability analysis primarily depends on the real parts of the Jordan matrix, which govern the exponential growth characteristics of the solutions. These characteristics can be further characterized using Lyapunov exponents ( $L (λ)$ ) and their corresponding Lyapunov subspaces.⁴⁰

The symbol (λ) is commonly used to describe time-evolving systems. Let ( $x (\cdot, x_{0})$ ) denote the solution to the linear differential equation ( $\dot{x} = A x$ ):

λ (x_{0}, A) = \lim_{t \to \infty} \sup \frac{1}{t} \log ‖ x (t, x_{0}) ‖

(3)

Substituting ( $x (t, x_{0}) = e^{A t} x_{0}$ ) into equation (3) gives the following:

\lim_{t \to \pm \infty} \frac{1}{t} \log | e^{λ t} x_{0} | = \lim_{t \to \pm \infty} \frac{1}{t} \log (e^{λ t}) + \lim_{t \to \pm \infty} \frac{1}{t} \log (x_{0}) = λ

(4)

where λ is the Lyapunov exponent of the temporal differential equation (

\dot{x} = - λ x

).^41,42

While the aforementioned formulation facilitates Lyapunov functional analysis in continuous-time systems, it is also applicable to discrete-time sequences as follows:

\lim_{n \to \pm \infty} \frac{1}{t} \log | λ^{n} x_{0} | = \lim_{n \to \pm \infty} \frac{1}{n} l og ({| λ |}^{n}) + \lim_{n \to \pm \infty} \frac{1}{n} \log | x_{0} | = | λ |

(5)

where

{| λ |}^{n}

is a positive-definite matrix and

| λ |

is a Lyapunov functional at discrete time step (n).^43,44

FEA validation

To verify the critical state of dynamic instability in steel ground-based truss arches, the parameters λ related to SE and KE conversions are incorporated into the FEA models of the truss arches. This integration facilitates the development of essential physical equations required for analyzing the dynamic stability of truss arches. The specific analysis encompasses the following aspects.

General condition of FEA

ABAQUS software was utilized to conduct FEA. We developed a Python-driven program, Truss_ABAQUS_Py, to construct the steel ground-based truss arch. Two-node linear beam elements in space (B31) were employed to simulate the trusses. To mitigate the impact of element singularity, the element size was set to 0.1 m. This discretization method ensured a uniform number of models, thereby enhancing computational efficiency. Prior to performing the dynamic stability analysis using the dynamic relaxation method (DRM), it was essential to determine the natural frequencies and mode shapes of the steel truss arch. For this purpose, the Lanczos method was applied to calculate the first six natural frequencies and eigenvalues, as presented in Table 2. Due to the structural configuration, where horizontal stiffness significantly exceeded vertical stiffness, the first three dominant mode shapes predominantly occurred in the vertical direction, as illustrated in Figure 2. Notably, the first and second modes exhibited concentrated frequencies and eigenvalues. In contrast, the fourth to sixth mode shapes were primarily manifested in the horizontal direction (Figure 3), with frequencies and eigenvalues substantially higher than those of the first three modes. Consequently, only vertical wind pressure was considered in the elastoplastic dynamic analysis using DRM.

Table 2.

Frequency and eigenvalues of steel truss arches.

Direction	Vertical			Lateral
Mode	1	2	3	4	5	6
Freq.	1.85	1.92	3.41	4.34	5.46	6.70
Value	135.25	146.01	458.74	742.85	1175.3	1773.3

Figure 2.

Vertical mode: (a) Mode 1; (b) Mode 2; (c) Mode 3.

Figure 3.

Lateral mode: (a) Mode 4; (b) Mode 5; (c) Mode 6.

DRM analysis result

The truss arch of the university stadium in Guangzhou, South China, and the return period for reference wind pressure (RWP) was established at 100 years. The basic wind pressure is obtained from the local meteorological statistical data of Guangzhou, and the RWP can be calculated through height correction and shape coefficient modification. The wind pressure at the arch crown, located at a height of 80 m, was calculated to be 2.99 kPa, which was considered the first release load intensity in the DRM analysis. The DRM process is implemented in two steps: first, the RWP is converted into nodal forces and applied to the FEA models nodes, obtaining the internal forces of the steel truss arch through dynamic analysis. Then, the calculated internal force results are assigned as initial conditions to the DRM analysis model, and the boundary conditions are released, allowing the structure to enter the free vibration stage, thereby achieving the DRM process.

Given that the truss arch is situated in a typhoon-prone region, a second release load intensity of 1.2 times the RWP was applied during the DRM analysis. To assess the dynamic stability-bearing limit of the truss arches, a third release load intensity of 1.3 times the RWP was implemented. The selection of 1.0, 1.2, and 1.3 RWP levels for wind load analysis was a strategic decision aimed at ensuring safety, compliance with standards, and a thorough understanding of structural behavior under varying wind conditions. The 1.2 RWP served as a crucial intermediate case that balanced the need for detailed analysis with practical design considerations.

Figure 4 illustrates the Mises stress distribution within the truss arch under the RWP conditions. The Mises stress was generally uniform across the truss arch members; however, significant non-uniformity was observed in the diagonal braces, particularly in the vertical members. The maximum stress value of 214 MPa was recorded in the vertical members of the diagonal braces, while the corresponding stress in other members was approximately 100 MPa. Stress magnitudes in the main ribs were all below 50 MPa, indicating that, under these conditions, the truss arch remained in an elastic stress state. Following the release of the RWP, the truss arch transitioned into a state of free vibration. According to the principle of mechanical energy conservation, during free vibration, KE and SE undergo mutual conversion. Figure 5 illustrates the time history of this energy interconversion.

Figure 4.

Steel truss arch stress state for RWP.

Figure 5.

Release 1 RWP: truss arch SE and KE conversion time history.

The truss arch SE and KE undergo periodic conversion at a constant frequency, demonstrating the structure’s full capacity for energy transformation. This behavior confirms that the truss arch operates within its elastic working state.

Based on discrete Lyapunov functional equation (5), Lyapunov exponent was defined as the normal of the difference between SE and KE at the cross-section of each time series, where buckling deformation SE and truss arch KE undergo periodic transformations.

λ = \frac{1}{n} \sum_{i = 1}^{n} \log \frac{| K E_{i}^{p e a k} - S E_{i}^{p e a k} |}{| K E_{0}^{p e a k} - S E_{0}^{p e a k} |}

(6)

As shown in Figure 5, the variable “i” in equation (10) represents the i-th half cycle intercept of the energy conversion time curve. $K E_{i}^{p e a k}$ and $S E_{i}^{p e a k}$ represent the peak values of strain energy and kinetic energy at the i-th half cycle intercept, while $K E_{0}^{p e a k}$ and $S E_{0}^{p e a k}$ represent the strain energy and kinetic energy under initial conditions, respectively. The parameter $| λ | ⋖ 1$ , which corresponds to equation (5), converges to a bounded value according to probability 1. The physical significance of this is that the SE and KE of the steel truss arch are considered as a random process during the conversion. As the number of conversions “n” increases, the difference between $K E_{i}^{p e a k}$ and $S E_{i}^{p e a k}$ gradually decreases compared to the initial difference between $K E_{0}^{p e a k}$ and $S E_{0}^{p e a k}$ , indicating that this random process is in a steady state.⁴⁵

Table 3 summarizes the values of λ calculated using equation (6). As shown in the table, the absolute value of λ remains less than 1 and exhibits a steady-state decreasing trend, indicating that the truss arch structure undergoes a convergent vibration process with dynamically stable characteristics.

Table 3.

λ of truss arch energy conversions.

i	1 RWP		1.2 RWP		1.3 RWP
i	λ	Time	λ	Time	λ	Time
1	−0.130	0.010	−0.269	0.011	−0.255	0.014
2	−0.132	0.021	−0.249	0.021	−0.392	0.023
3	−0.159	0.032	−0.326	0.032	−1.192	0.032
4	−0.194	0.042	−0.357	0.043
5	−0.237	0.053	−0.429	0.053
6	−0.284	0.063	−0.467	0.063
7	−0.331	0.073	−0.533	0.073
8	−0.381	0.083	−0.570	0.083
9	−0.420	0.093	−0.627	0.094
10	−0.451	0.103	−0.650	0.104

At (i = 1), the truss arch KE reached its maximum. Figure 6 illustrates the velocity distribution, which exhibits central symmetry among all members, peaking at 0.928 m/s in the X-brace and arched roof, while significantly decreasing in other areas. The physical reason for this phenomenon was that, due to the action of wind pressure, the X-brace, as the main load-bearing component, stored more strain energy compared to other truss members. When this strain energy was released, it would naturally respond more violently.

Figure 6.

Release 1 RWP: i = 1 truss arch velocity.

Due to the spatial geometry and the spatial distribution characteristics of truss stiffness, the dynamic responses at different zones within the truss arch varied. The dynamic responses were most pronounced at the edges of the structure and in areas of weak stiffness. Lateral and vertical amplitudes corresponding to the velocities at the node of the upper ribs on the exterior side of the arch crown were selected for constructing Poincaré sections, as illustrated in Figure 7.⁴⁶ These sections clearly demonstrate the presence of strange attractors resulting from the dissipation inherent in truss arch systems. The lateral amplitude (U1) was 0.002 m, with a vibration velocity of 0.18 m/s, while the vertical vibration amplitude (U3) and its corresponding velocity were 0.004 m and 0.7 m/s, respectively.

Figure 7.

Release 1 RWP: Poincaré sections of arch top node: (a) Lateral amplitude U1; (b) vertical amplitude U3.

As illustrated in Figure 8, when the SE reaches its peak value, the logarithmic strain components (LE) of the truss arch display a relatively uniform distribution. However, localized concentrations are observed near the arch foot and adjacent diagonal braces, with maximum values of 0.00,103, while the corresponding values for other regions remain below 0.0005. This strain pattern confirms that, under these loading conditions, the truss arch operates within the elastic regime. The vertical rods at the arch foot served as components connecting the structure and the foundation. When the structure was in a state of high SE energy storage during contraction, it reached a peak pressure-bearing state, resulting in the aforementioned mechanical state of strain peaks.

Figure 8.

Release 1 RWP: i = 2 LE of truss arch.

The release of 1.2 RWP induced free vibrations in the truss arch. Figure 9 illustrates the time history of the conversion of SE to KE. During this process, KE and SE were effectively and continuously converted into one another, indicating stable structural stiffness. Figure 10 presents the velocity contour plot of the truss arch at i = 3. Notably, the vibration velocities of the steel truss arch members exhibited a centrosymmetric distribution with generally uniform magnitudes. The maximum velocity of 1.114 m/s was observed at the diagonal braces and the arch crown, while velocities at other locations remained below 0.7 m/s. Furthermore, the periodic pattern of KE-SE conversion demonstrated increasing regularity, suggesting the onset of dynamic stability within the truss arch system.

Figure 9.

Release 1.2 RWP: Truss arch SE and KE conversion time history.

Figure 10.

Release 1.2 RWP: i = 3 truss arch velocity.

Figure 11 illustrates the Poincaré sections of the arch crown top node depicted in Figure 7. In this condition, increased wind pressure led to heightened SE accumulation within the truss system. Following the release of SE, localized plastic deformation occurred in the truss members, resulting in a non-recoverable plastic deformation of approximately 10 mm at the arch crown in both vertical and horizontal directions. Concurrently, peak vertical vibration velocities reached 1 m/s, while horizontal vibrations remained at 0.2 m/s. Notably, the emergence of strange attractors in both vertical and horizontal directions at the arch crown demonstrated that, despite minor plastic deformations, the truss arch maintains dynamic stability following vibrational loading.

Figure 11.

Release 1.2 RWP: Poincaré sections of arch top node: (a) Lateral amplitude U1; (b) vertical amplitude U3.

As illustrated in Figure 12, where the SE reached its maximum value, the magnitude and spatial distribution of LE showed no significant deviations from those presented in Figure 8. However, the critical member location exhibited the largest shifts in LE within the arch foot region. A comparison with Figure 11(a) indicated that this transfer of maximum LE may be attributed to a minor lateral displacement in the arch, which redistributed strain concentrations among the members near the arch foot. The release of 1.2 RWP induced free vibrations in the truss arch. The time history of the conversion from SE to KE is depicted in Figure 13.

Figure 12.

Release 1.2 RWP: i = 2 LE of truss arch.

Figure 13.

Release 1.3 RWP: Truss arch SE and KE conversion time history.

Continuous energy conversions between KE and SE were unable to persist in this state when (i = 5). Figure 14 illustrates the development of plastic hinges on both sides of the arch. However, the locations of these plastic hinges displayed geometric asymmetry, characterized by rib members positioned on one side and diagonal braces arranged on the opposite side. This spatial non-uniformity resulted in an asymmetric stiffness distribution within the truss arch system, manifesting as differential load-transfer mechanisms across its cross-section.

Figure 14.

Release 1.3 RWP: i = 5 truss arch velocity.

As illustrated in Figure 14, Poincaré sections at identical locations further revealed asymmetric vibrational characteristics between the vertical and lateral directions. Additionally, a significant amplification in vibration magnitude was observed, indicating pronounced nonlinear dynamic responses within the system. The horizontal amplitude increased to 0.02–0.04 m, accompanied by a rise in vibration velocity to 0.4 m/s, while the vertical amplitude exhibited an irreversible increase to 0.03 m during the first vibration cycle, with a maximum vertical vibration velocity reaching 0.75 m/s. These observations indicate substantial non-recoverable plastic deformations within the truss arch systems. Furthermore, the absence of strange attractors in the dynamic responses confirmed that, under these conditions, the truss arch experienced dynamic instability.

As illustrated in Figure 15, where SE reached its maximum value, the magnitude and spatial distribution of LE were found to be similar to those presented in Figure 12. However, a subset of structural members experienced plastic yielding, resulting in their subsequent exclusion from load-bearing functionality. The location of the critical member hosting the maximum LE shifted within the arch foot region. A comparison with Figure 16 revealed that this transfer of maximum LE was attributed to significant arch displacement, which redistributed strain concentrations among the members near the arch foot.

Figure 15.

Release 1.3 RWP: LE of truss arch at i = 2.

Figure 16.

Release 1.3 RWP: Poincaré sections of arch top node: (a) Lateral amplitude U1; (b) vertical amplitude U3.

Conclusions

In this study, a discrete dynamic stability criterion was developed for steel truss arches based on Lyapunov’s theory of dynamic instability. Assuming that in the critical state of truss arch dynamic instability, FEA presented effective conversions between SE and KE, the system’s vibration energy half-cycle attenuation rate, denoted by λ, should be defined. In addition, this research established a new theoretical framework and quantitative criteria to ensure truss arch structural safety under winds. The main findings of this research were summarized as follows:

(1) Dynamic instability mechanism: The critical state of truss arch dynamic instability was triggered by abnormal SE-KE conversions, exacerbated by asymmetric plastic hinge formation under excessive loads (e.g., 1.3 RWP). This resulted in abrupt stiffness degradation and non-recoverable plastic deformation, particularly in diagonal braces and arch ribs.

(2) Lyapunov exponent application: Discrete Lyapunov exponent (λ) served as a robust quantitative criterion for stability evaluation. Stable systems exhibited $λ < 1$ with convergent energy conversion, while divergent λ values with alternating signs indicated instability. FEA results confirmed this behavior, where λ was stabilized under 1.2 RWP but sharply diverged at 1.3 RWP.

(3) Safety implications: The weakest part of this steel arch frame was located at arch foot. When wind pressure just exceeded the estimated value of typhoon wind pressure, plastic hinges might occur at arch foot. Consequently, local reinforcement could be carried out by enlarging arch foot cross-section or filling with concrete.

(4) Scientific analysis method: Numerical simulations integrating Lyapunov theory provided predictive tools to identify vulnerable zones (e.g., arch foot regions) and optimize reinforcement measures. This approach ensured structural safety under extreme wind pressures.

Footnotes

ORCID iD

Chao Guo

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by the Projects of Talents Recruitment of GDUPT (No. 2024rcyj1056 and No. 2024rcyj1055).

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Appendix

References

Alanazy

Moatimid

Amer

, et al. An innovative methodology in analyzing two-degrees-of-freedom of coupled dynamical system. Ain Shams Eng J 2025; 16(12): 103770.

Amer

Moatimid

Zakria

, et al. Dynamical analysis of a four-degree-of-freedom vibratory structure: bifurcation, stability, and resonance exploration. J Low Freq Noise Vib Act Control 2025; 44(3): 1555–1575.

Abohamer

Amer

Abdelhfeez

, et al. Vibration analysis, stability assessment, and chaotic behavior of a damped oscillator coupled with a spherical pendulum and a piezoelectric transducer. Nonlinear Dyn 2025; 113(18): 24343–24381.

Dong

Kato

. Seismic design for steel trussed arch to multi-support excitations. J Constr Steel Res 2007; 63(6): 725–734.

Cheng

. Optimum design of steel truss arch bridges using a hybrid genetic algorithm. J Constr Steel Res 2010; 66(8–9): 1011–1017.

Guo

Zhao

Dou

, et al. Out-of-plane strength design of spatially trussed arches with a rectangular lattice section. J Constr Steel Res 2013; 88: 321–329.

, et al. Experimental study on wind force coefficient of a truss arch tower with multiple skewbacks. Adv Struct Eng 2020; 23(12): 2614–2625.

Adeli

Park

. Optimization of space structures by neural dynamics. Neural Netw 1995; 8(5): 769–781.

Ario

. Homoclinic bifurcation and chaos attractor in elastic two-bar truss. Int J Non Lin Mech 2004; 39(4): 605–617.

10.

Halpern Allison

Sigrid

. Nonlinear elastic in-plane buckling of shallow truss arches. J Bridge Eng 2015; 20(10): 1–11.

11.

Chen

Huang

, et al. Fatigue performance test and finite-element analysis of concrete-filled steel-tube K-joints. J Bridge Eng 2023; 28(3): 04023003.

12.

Dou

Guo

Jiang

, et al. In-plane buckling and design of steel tubular truss arches. Thin-Walled Struct 2018; 130: 613–621.

13.

Dou

Guo

Zhao

, et al. Elastic out-of-plane buckling load of circular steel tubular truss arches incorporating shearing effects. Eng Struct 2013; 52: 697–706.

14.

Bing

Ning

, et al. (2022) Mingzhuwan bridge: the largest span three-main-truss steel arch bridge. Struct Eng Int 33(1): 89–95.

15.

Xiao

Mao

Tian

, et al. Partial-model-based damage identification of long-span steel truss bridge based on stiffness separation method. Struct Control Health Monit 2024; 2024: 5530300.

16.

Finozzi

Sanfedino

Alazard

. Parametric sub-structuring models of large space truss structures for structure/control co-design. Mech Syst Signal Process 2022; 180: 109427.

17.

Rezaiee-Pajand

Estiri

. Computing the structural buckling limit load by using dynamic relaxation method. Int J Non Lin Mech 2016; 81: 245–260.

18.

Wang

Cai

Yang

, et al. Form-finding of deployable mesh reflectors using dynamic relaxation method. Acta Astronaut 2018; 151: 380–388.

19.

Zhang

, et al. Thermal design optimization method of mesh reflector antennas considering the interaction between cable net and flexible truss. Struct Multidiscip Optim 2023; 66(4): 68.

20.

Dong

, et al. Multi-point suspension design and stability analysis of a scaled hoop truss antenna structure. Int J Struct Stabil Dynam 2021; 21(6): 2150077.

21.

Zhang

Zhong

, et al. Thermal design optimization method of mesh reflector antennas considering deformation compatibility conditions. J Aero Eng 2022; 35(4): 1–10.

22.

Zhang

Zhou

Wang

. Local bifurcation of a circus truss antenna considering the effect of space thermal excitation. Chaos Solitons Fractals 2022; 162(Suppl C): 112437.

23.

Xie

Shi

Alleyne

, et al. Feedback shape control for deployable mesh reflectors using gain scheduling method. Acta Astronaut 2016; 121: 241–255.

24.

Mikhlin

Manucharyan

. Determination of the chaos onset in mechanical systems with several equilibrium positions. Meccanica 2006; 41(3): 253–267.

25.

Bontempi

. A study of the chaotic behaviour of the elastic-plastic euler arch. Nonlinear Dyn 1995; 7(2): 217–229.

26.

Potapov

. Stability of a shallow arch under action of deterministic and stochastic load taking into account nonlocalness of elasticity and damping. Acta Mech 2014; 225(3): 919–927.

27.

Wolf

Swift

Swinney

, et al. Determining lyapunov exponents from a time series. Phys Nonlinear Phenom 1985; 16(3): 285–317.

28.

Haddad

Bonnet

Zahira

IAA

, et al. A new solution for stenting large right ventricular outflow tracts before transcatheter pulmonary valve replacement. Can J Cardiol 2022; 38(1): 31–40.

29.

Guo

Zhao

, et al. A novel modular deployable mechanism for the truss antenna: assembly principle and performance analysis. Aero Sci Technol 2020; 105: 105976.

30.

Guo

Zhao

, et al. Design and analysis of truss deployable antenna mechanism based on a novel symmetric hexagonal profile division method. Chin J Aeronaut 2021; 34(8): 87–100.

31.

Hegyi

. Numerical stability analysis of arch-supported membrane roofs. Structures 2021; 29: 785–795.

32.

Huang

Liu

, et al. The nonlinear bifurcation and chaos of coupled heave and pitch motions of a truss spar platform. J Ocean Univ China 2015; 14(5): 795–802.

33.

. Aeroelastic model design and sensitivity analysis of a complicated steel truss arch tower to skew incident winds based on wind tunnel tests. J Wind Eng Ind Aerod 2021; 214(Suppl C): 104646.

34.

Hao

Zhang

Wang

, et al. Stability and seismic performance evaluation of long-span arch bridge deck system after strengthening of girder. J Appl Sci Eng 2022; 26(1): 49–60.

35.

Guo

. Probabilistic analysis of derrick frame in a formwork support system. Ain Shams Eng J 2022; 14(5): 101977.

36.

Guo

Chen

. Multi-scale mechanism of steel tube restraint effect on concrete of CFST. Mech Adv Mater Struct 2023; 31(14): 1–16.

37.

Peek

Triantafyllidis

. Worst shapes of imperfections for space trusses with many simultaneously buckling members. Int J Solid Struct 1992; 29(19): 2385–2402.

38.

Peek

. Worst shapes of imperfections for space trusses with multiple global and local buckling modes. Int J Solid Struct 1993; 30(16): 2243–2260.

39.

Ageno

Sinopoli

. Lyapunov’s exponents for nonsmooth dynamics with impacts: stability analysis of the rocking block. Int J Bifurcat Chaos Appl Sci Eng 2005; 15(6): 2015–2040.

40.

Mosta

Bucher

Schorling

. Dynamic stability analysis of non-linear structures with geometrical imperfections under random loading. J Sound Vib 2004; 276(1): 381–400.

41.

Xie

Ariaratnam

. Numerical computation of wave localization in large disordered beam-like lattice trusses. American Institute of Aeronautics and Astronautics 1994; 32(8): 1724–1732.

42.

Xie

. Vibration mode localization in disordered large planar lattice trusses. Chaos Solitons Fractals 1997; 8(3): 433–454.

43.

Chen

Gong

Zhang

, et al. Development of a novel muck removal hydraulic manipulator for automated steel arch assembly of tunnel boring machine. J Braz Soc Mech Sci Eng 2024; 46(7): 403.

44.

Chen

Gong

Zhou

, et al. Backstepping based trajectory tracking control of a TBM steel arch splicing manipulator. Int J Control Autom Syst 2024; 22(2): 648–660.

45.

Liang

Guo

Tian

, et al. Dynamic instability of coal rock burst in spatiotemporal multi-crack RVE model. Int J Multiscale Comput Eng 2025; 23(4): 69–94.

46.

Zhang

Zheng

Liu

, et al. Multi-pulse jumping double-parameter chaotic dynamics of eccentric rotating ring truss antenna under combined parametric and external excitations. Nonlinear Dyn 2019; 98(1): 761–800.