Torsional Behavior of Pylon Columns with Edge Cracks in Suspension Bridges Under Wind Loads

Abstract

Cracks inevitably develop in the pylon structures of suspension bridges due to external forces such as wind loads. Cracks are a primary cause of torsional damage to the key supporting structures of suspension bridges. Therefore, torsional fracture analyses are vitally important for evaluating the safety of bridge structures. In this study, we simplified the tower structure of a suspension bridge as a homogenous cylinder. We then employed the boundary integral equations for a cylinder with edge cracks to investigate the singularity features at the crack tip. The boundary element-based method was subsequently used to divide the boundary into several elements, and different interpolation functions were adopted to compute the stress intensity factor at the crack tip. Torsional stiffness and stress intensity factor calculations were conducted for cylinders with straight and polyline edge cracks. The results were compared with the results reported in the existing literature, and the accuracy and reliability of the calculation method were validated. Finally, the numerical simulation of the torsional fracture behavior of cylinders with edge cracks under various wind loads was conducted. The maximum allowable crack lengths of a cylinder under different wind grades were acquired, further demonstrating the feasibility and practicality of the boundary element calculation method for practical applications in bridge engineering.

1. Introduction

Suspension bridge pylons are critical load-bearing components whose mechanical performance directly governs the overall load-carrying capacity and operational safety of a bridge. Wind load is the dominant environmental load for bridges constructed in marine and open areas throughout their service life; sustained wind excitation not only induces dynamic responses in pylons, but it also triggers cumulative damage and crack formation [1]. Once edge cracks appear in pylons, their structural mechanical properties begin to significantly degrade; furthermore, the wind-induced torsional responses of pylons lead to the concentration of stress at crack tips, which accelerates crack propagation and ultimately results in torsional fracture, posing a severe threat to bridge safety [2]. Therefore, in-depth investigations into the torsional fracture mechanism of edge-cracked pylon cylinders under wind loads are of great importance for bridge safety assessment, as well as for operational and maintenance decision-making.

A comprehensive literature review was conducted focusing on four core research themes: torsion analysis of bridge structures, the fracture mechanics of cracked cylinders, engineering applications of the boundary element method (BEM), and the structural responses of bridges under wind loads. This review identified several notable research gaps. First, most existing studies on bridge torsion focus on static or seismic loads, paying less attention to pylon torsional responses under wind loads. There is also a lack of systematic understanding of the mechanism connecting wind-induced torsion and crack propagation [3,4,5]. Second, research on cracked cylinders is primarily confined to the material level, and as such it fails to integrate practical engineering contexts, such as pylon–wind load coupling, thereby limiting its relevance to engineering applications [6,7,8,9,10,11,12,13,14]. Third, in numerical simulations, despite the significant advantages of BEM in crack analysis, its application in the torsional fracture analysis of pylons under wind loads remains immature and its practicality in engineering terms has yet to be fully verified. Existing studies predominantly rely on the finite element method (FEM), which has inherent limitations in handling crack-tip singularity [15,16,17,18,19,20]. Finally, wind load-related research on bridge structural responses has paid inadequate attention to pylons, and the mechanism by which the dynamic characteristics of wind loads impact the stress and displacement fields at crack tips remains unclear, resulting in inadequate research on the wind-resistant assessment of damage to pylons [21,22,23,24,25,26,27,28,29].

To address these research gaps, this study takes edge-cracked pylon cylinders of suspension bridges as the research object, integrating the dynamic characteristics of wind loads to conduct theoretical analysis and numerical simulation of torsional fracture. The specific contributions of this study are as follows: (1) a simplified model of a homogeneous cylinder with a focus on edge cracks is established; (2) boundary integral equations for the torsion of cracked cylinders are constructed to clarify the singularity characteristics of crack tips; (3) a BEM-based numerical scheme is proposed and its reliability verified through comparisons with results previously reported in the literature; and (4) numerical simulations and parametric analysis of torsional fracture under different wind load conditions are performed to determine the maximum allowable crack length under various wind grades. The objectives of this study are to reveal the torsional fracture mechanism of edge-cracked pylons under various wind loads, improve the theoretical framework for bridge torsional fracture analysis, verify the practicality of BEM in engineering terms, and provide scientific and technical support for the wind-resistant design, damage detection, and operation and maintenance of pylons.

2. Basic Theory and Derivation

The mathematical theory of the boundary integral equation for edge-cracked cylinders combined with the BEM numerical calculation used in this study was developed based on the classical theoretical system of fracture mechanics and the BEM. This theory was first systematically proposed by Cruse in 1974, after which it was further refined by scholars such as Rizzo. It evolved into a classical theoretical framework to reveal the stress intensity factors (SIFs) of cracked bodies. Another crucial milestone in the development of this theory was the publication of the first monograph regarding the BEM, written by Brebbia in 1978 [30]. This work systematically elaborated on the theoretical system of the BEM, marking the official beginning of systematic research on this methodology.

2.1. Treatment of the Crack-Tip Singularity in the Torsion Problem of Cylinders with Edge Cracks

The primary focus of this study is to compute the torsion problem of cylinders with edge cracks; First, it is necessary to address the crack tip singularity of edge cracks in this section [31].

When a cylinder is in an elastic torsion state, its cross-sections rotate within their own planes while allowing free axial warping. A coordinate system was established for analysis, as shown in Figure 1. It was assumed that the cross-sections undergo rigid-body rotation; hence, the displacement of any arbitrary point within the cylinder is given by [32]:

$$u=-\alpha yz, v=\alpha xz, w=\alpha \varphi (x,y),$$

(1)

where $\varphi$ is the torsion function; G is the shear modulus; $\alpha$ is the torsion rate; and z is the position of the cross-section.

A local area with cracks on the boundary was examined, as illustrated in Figure 2. Local polar coordinates, i.e., $(\rho ,\theta )$, at the crack tip were introduced:

$$u_{\rho }=0, u_{\theta }=\alpha \rho z, w=\alpha \varphi (\rho ,\theta ).$$

(2)

Based on the geometric equation, the strain components of the particle was calculated as follows:

$${\epsilon }_r={\epsilon }_{\theta }={\epsilon }_z={\gamma }_{r\theta }=0, {\gamma }_{\theta z}={\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial w}{\partial \theta }}+\alpha \rho , {\gamma }_{\rho z}={\displaystyle \frac{\partial w}{\partial \rho }}.$$

(3)

Obtained using ${\nabla }^{2}w=0$:

$$w={\rho }^{\lambda }[A\cos (\lambda \theta )+B\sin (\lambda \theta )].$$

(4)

$${\tau }_{\theta z}=G{\gamma }_{\theta z}=G\lambda {\rho }^{\lambda -1}[-A\sin (\lambda \theta )+B\cos (\lambda \theta )].$$

(5)

$${\gamma }_{\rho z}=\lambda {\rho }^{\lambda -1}[A\cos (\lambda \theta )+B\sin (\lambda \theta )].$$

(6)

(1)

For area I $(0<\theta <\beta )$, $(G, v)$:

$$w_1={\rho }^{{\lambda }_1}[A_1\cos ({\lambda }_1\theta )+B_1\sin ({\lambda }_1\theta )].$$

(7)

$${\tau }_{\theta z1}=G{\gamma }_{\theta z1}=G{\lambda }_1{\rho }^{{\lambda }_1-1}[-A_1\sin ({\lambda }_1\theta )+B_1\cos ({\lambda }_1\theta )]$$

(8)

$${\gamma }_{\rho z1}={\lambda }_1{\rho }^{{\lambda }_1-1}[A_1\cos ({\lambda }_1\theta )+B_1\sin ({\lambda }_1\theta )].$$

(9)

(2)

For area II $(\beta <\theta <\pi )$, $(G, v)$:

$$w_2={\rho }^{{\lambda }_2}[A_2\cos ({\lambda }_2\theta )+B_2\sin ({\lambda }_2\theta )].$$

(10)

$${\tau }_{\theta z2}=G{\gamma }_{\theta z2}=G{\lambda }_2{\rho }^{{\lambda }_2-1}[-A_2\sin ({\lambda }_2\theta )+B_2\cos ({\lambda }_2\theta )].$$

(11)

$${\gamma }_{\rho z2}={\lambda }_2{\rho }^{{\lambda }_2-1}[A_2\cos ({\lambda }_2\theta )+B_2\sin ({\lambda }_2\theta )].$$

(12)

Hence, the boundary between the crack ($\theta =\beta +0,\theta =\beta -0$) and the surface force at the interface ($\theta =0,\pi$) is given by:

$${\left.{\tau }_{\theta z1}\right|}_{\theta =0}=G{\lambda }_1{\rho }^{{\lambda }_1-1}{B}_1=0$$

(13)

$${\left.{\tau }_{\theta z1}\right|}_{\theta =\beta -0}=G{\lambda }_1{\rho }^{{\lambda }_1-1}[-A_1\sin ({\lambda }_1\beta )+B_1\cos ({\lambda }_1\beta )]=0$$

(14)

$${\left.{\tau }_{\theta z2}\right|}_{\theta =\beta +0}=G{\lambda }_2{\rho }^{{\lambda }_2-1}[-A_2\sin ({\lambda }_2\beta )+B_2\cos ({\lambda }_2\beta )]=0$$

(15)

$${\left.{\tau }_{\theta z2}\right|}_{\theta =\pi }=G{\lambda }_2{\rho }^{{\lambda }_2-1}[-A_2\sin ({\lambda }_2\pi )+B_2\cos ({\lambda }_2\pi )]=0.$$

(16)

Equation (13) then shows that:

$$B_1=0.$$

(17)

The above equation was subsequently substituted into Equation (14) to yield:

$$A_1\sin ({\lambda }_1\beta )=0.$$

(18)

and then,

$$\sin ({\lambda }_1\beta )=0\Rightarrow {\lambda }_1\beta =k\pi (k=1,2,\dots ),$$

(19)

where $k\le 0$ and ${\lambda }_1-1\le -1$. This was then substituted into Equation (14). The distribution of the stress field was unreasonable, making it unsuitable for this situation. Thus, we used $k\ge 1$, i.e., $k=1,2,\dots ,n$, for the calculation. Then, to ensure that ${\rho }^{\lambda -1}$ approached zero at the slowest rate, we ultimately used $k=1$. We then obtained:

$${\lambda }_1=\pi /\beta (A_1\ne 0, {\lambda }_1>1).$$

(20)

From Equations (15) and (16), we obtained:

$$-A_2\sin ({\lambda }_2\beta )+B_2\cos ({\lambda }_2\beta )=0.$$

(21)

$$-A_2\sin ({\lambda }_2\pi )+B_2\cos ({\lambda }_2\pi )=0.$$

(22)

The above two equations are homogeneous linear equations, and the sufficient and necessary condition for homogeneous linear equations to have non-zero solutions is that their coefficient determinant is zero, that is:

$$-\sin ({\lambda }_2\beta )\cos ({\lambda }_2\pi )+\cos ({\lambda }_2\beta )\sin ({\lambda }_2\pi )=0.$$

(23)

therefore,

$$\sin [{\lambda }_2(\pi -\beta )]=0(A_2,B_2\ne 0,0<\beta <{\displaystyle \frac{\pi }{2}})\Rightarrow {\lambda }_2(\pi -\beta )=k\pi (k=1,2,\dots ).$$

(24)

Similarly,

$${\lambda }_2=\pi /(\pi -\beta )(A_2,B_2\ne 0,0<\beta <{\displaystyle \frac{\pi }{2}}, {\lambda }_2>1).$$

(25)

From Equations (17)–(22) and (25), we obtained:

(1)

For area I ($0<\theta <\beta$), $(G, v)$:

$$w_1={\rho }^{{\lambda }_1}{A}_1\cos ({\lambda }_1\theta ).$$

(26)

$${\tau }_{\theta z1}=-G{\lambda }_1{\rho }^{{\lambda }_1-1}{A}_1\sin ({\lambda }_1\theta ).$$

(27)

$${\gamma }_{\rho z1}={\lambda }_1{\rho }^{{\lambda }_1-1}{A}_1\cos ({\lambda }_1\theta ).$$

(28)

where ${\lambda }_1=\pi /\beta$ ($A_1\ne 0,$ ${\lambda }_1>1$).

(2)

For area II ($\beta <\theta <\pi$), $(G, v)$:

$$w_2={\rho }^{{\lambda }_2}{C}_2\cos [{\lambda }_2(\theta -\beta )]$$

(29)

$${\tau }_{\theta z2}=-G{\lambda }_2{\rho }^{{\lambda }_2-1}{C}_2\sin [{\lambda }_2(\theta -\beta )]$$

(30)

$${\gamma }_{\rho z2}={\lambda }_2{\rho }^{{\lambda }_2-1}{C}_2\cos [{\lambda }_2(\theta -\beta )],$$

(31)

where ${\lambda }_2=\pi /(\pi -\beta )$($A_2,B_2\ne 0,0<\beta <{\displaystyle \frac{\pi }{2}}$, ${\lambda }_2>1$), $C_2={\displaystyle \frac{A_2}{\cos ({\lambda }_2\beta )}}$.

Then:

$${\left.{\displaystyle \frac{\partial w_1}{\partial \rho }}\right|}_{\theta =0}={\lambda }_1{A}_1{\rho }^{{\lambda }_1-1}, {\left.{\displaystyle \frac{\partial w_2}{\partial \rho }}\right|}_{\theta =\pi }=-{\lambda }_2{A}_2{\rho }^{{\lambda }_2-1}{\displaystyle \frac{1}{\cos ({\lambda }_2\beta )}}=-{\lambda }_2{C}_2{\rho }^{{\lambda }_2-1},$$

(32)

$${\left.{\displaystyle \frac{\partial w_1}{\partial \rho }}\right|}_{\theta =\beta }={\lambda }_1{A}_1{\rho }^{{\lambda }_1-1}\cos ({\lambda }_1\beta )=-{\lambda }_1{A}_1{\rho }^{{\lambda }_1-1},$$

(33)

$${\left.{\displaystyle \frac{\partial w_2}{\partial \rho }}\right|}_{\theta =\beta }={\lambda }_2{A}_2{\rho }^{{\lambda }_2-1}{\displaystyle \frac{1}{\cos ({\lambda }_2\beta )}}={\lambda }_2{C}_2{\rho }^{{\lambda }_2-1},$$

(34)

$$\Delta w={\left.(w_2-w_1)\right|}_{\theta =\beta }={\rho }^{{\lambda }_2}{A}_2{\displaystyle \frac{1}{\cos ({\lambda }_2\beta )}}-{\rho }^{{\lambda }_1}{A}_1\cos ({\lambda }_1\beta )=0\Rightarrow A_2={\rho }^{{\lambda }_1-{\lambda }_2}\cos ({\lambda }_1\beta )\cos ({\lambda }_2\beta )A_1$$

(35)

$${\left.{\displaystyle \frac{\partial \Delta w}{\partial \rho }}\right|}_{\theta =\beta }={\lambda }_2{A}_2{\rho }^{{\lambda }_2-1}{\displaystyle \frac{1}{\cos ({\lambda }_2\beta )}}-{\lambda }_1{A}_1{\rho }^{{\lambda }_1-1}\cos ({\lambda }_1\beta )={\lambda }_2{C}_2{\rho }^{{\lambda }_2-1}+{\lambda }_1{A}_1{\rho }^{{\lambda }_1-1}.$$

(36)

It is important that, when $\beta ={\displaystyle \frac{\pi }{2}}$, ${\lambda }_1={\lambda }_2=2$. This causes the interpolation function to be repeated on the selected different cells. To make sense of the selection of the following interpolation functions, when $\beta ={\displaystyle \frac{\pi }{2}}$, it should be assumed that ${\lambda }_1={\lambda }_2=4$, this corresponds to the eigen solution of k = 2.

2.2. Boundary Element Numerical Computation: Choice of an Interpolation Function

According to reference [33], the boundary integral equation can be obtained for the general torsion of a cylinder as follows:

$$\begin{array}{ll}& {\int }_{{\mathsf{\Sigma }}_2}{\displaystyle \frac{\partial \phi (P,Q)}{\partial s(P)}}F(Q)ds(Q)+{\int }_{\Gamma }{\displaystyle \frac{\partial \phi (P,Q)}{\partial s(P)}}F(Q)ds(Q)\\ & +{\int }_{{\mathsf{\Sigma }}_2}{\displaystyle \frac{\partial \phi (P,Q)}{\partial s(P)}}F(Q)ds(Q)-{\int }_{{\mathsf{\Sigma }}_2}{\displaystyle \frac{\partial \phi (P,Q)}{\partial n(P)}}{q}^{+}(Q)ds(Q)\\ & =-{\displaystyle \frac{1}{2}}q(P)+{\int }_{\mathsf{\Sigma }}{\displaystyle \frac{\partial \phi (P,Q)}{\partial n(P)}}q(Q)dS(Q),\end{array}$$

(37)

where $P\in S$.

$$\begin{array}{ll}& {\int }_{{\mathsf{\Sigma }}_2}{\displaystyle \frac{\partial \mathsf{\phi }(P,Q)}{\partial s(P)}}F(Q)\mathit{ds}(Q)+{\int }_{\Gamma }{\displaystyle \frac{\partial \mathsf{\phi }(P,Q)}{\partial s(P)}}F(Q)\mathit{ds}(Q)\\ & +{\int }_{S_2}{\displaystyle \frac{\partial \mathsf{\phi }(P,Q)}{\partial s(P)}}F(Q)\mathit{ds}(Q)-{\int }_{S_2}{\displaystyle \frac{\partial \mathsf{\phi }(P,Q)}{\partial n(P)}}{q}^{+}(Q)\mathit{ds}(Q)\\ & =-q^{+}(P)+{\int }_{z_1}^{\infty }{\displaystyle \frac{\partial \mathsf{\phi }(P,Q)}{\partial n(P)}}q(Q)\mathit{dx}(Q),\end{array}$$

(38)

where $P\in \Gamma$.

After obtaining the solution of the boundary integral equation under the condition of single-valued displacement, the SIF at the crack tip can be directly calculated using the following equation [33]:

$$K_{III}(a_j)=-{\displaystyle \frac{G_1\alpha }{2}}\underset{Q\to a_j}{\lim }[\sqrt{2\pi \left|Q-a_j\right|}F(Q)],$$

(39)

$$K_{III}(b_j)=-{\displaystyle \frac{G_1\alpha }{2}}\underset{Q\to b_j}{\lim }[\sqrt{2\pi \left|Q-b_j\right|}F(Q)],(j = 1,2,\dots n),$$

(40)

where $a_j$ and $b_j$ are the initial and terminal endpoints of the j-th crack, respectively.

It is quite challenging to ensure that the boundary integral equation derived above meets the displacement single-valued condition and to solve the equation itself. Hence, alternative methods were explored to substitute for direct solution, and extensive research has shown that the BEM is a viable numerical approach [34]. The boundary and cracks were thus divided into $M+N$ linear units (Figure 3 and Figure 4), i.e., $S+\mathrm{\Gamma }={\displaystyle {\sum }_e{L}_e}$, $S={\displaystyle \underset{j=1}{\overset{M}{\cup }}{L}_j}$, and $\mathrm{\Gamma }={\displaystyle \underset{j=1+M}{\overset{M+N}{\cup }}{L}_j}$. Each linear unit was then mapped to an interval $-1\le t\le 1$, and a point, Q, on a cell, $L_e$, could be represented as:

$$Q=Q_1^e{N}_1(t)+Q_2^e{N}_2(t),(\left|t\right|<1andQ\in {\mathrm{L}}_{\mathrm{e}}),$$

(41)

where $Q_1^e$ and $Q_2^e$ are two tip nodes of the e-th unit and take the following shape functions:

$$N_1(t)={\displaystyle \frac{1}{2}}(1-t), N_2(t)={\displaystyle \frac{1}{2}}(1+t).$$

(42)

Due to the singularity of the stress field at the crack tip, this study adopted a refinement strategy that combined the local mesh densification and singular elements. Specifically, 1/4-node singular elements were employed in the crack-tip region (a sector area centered at the crack tip with a radius of r = 0.05 D, where the element side length gradually increases outward from the crack tip by a factor of 1.2. This ensured the smallest element size near the crack tip, enabling an accurate capture of the stress singularity. Linear interpolation elements were used for the crack surface and adjacent boundaries, while quadratic interpolation elements were applied at the crack tip to balance the calculation accuracy and efficiency. A gradient-based element division was adopted for the transition between the refined region and the global mesh, avoiding calculation errors caused by mesh distortion. Figure 5 is the schematic representation of the crack-tip element refinement that clearly marks the distribution range of singular elements, element side lengths, and types of interpolation functions.

The interpolation functions on the boundary S and cracks $\mathrm{\Gamma }$ were then defined. The role of the interpolation functions is to ensure the closure of the linear system of equations obtained by discretizing the boundary integral equation and to satisfy the displacement single-valued condition [33]. First, the interpolation functions of the linear elements on the boundary S are defined as follows:

$$F(Q)=F_1^e{N}_1(t)+F_2^e{N}_2(t) (\left|t\right|\le 1,Q\in L_e\subset S).$$

(43)

Elements along the crack $\mathrm{\Gamma }$ were categorized into general linear elements and crack-tip elements. General linear elements, which exclude the crack tip, had interpolation functions defined as follows:

$$F(Q)=F_1^e{N}_1(t)+F_2^e{N}_2(t) (\left|t\right|\le 1,Q\in L_e\subset \mathrm{\Gamma }).$$

(44)

If the starting node of the cell $t=-1$ was a crack cusp, then $F(Q)$ was represented as:

$$F(Q)={\displaystyle \frac{F_1^e}{\sqrt{N_2(t)}}}+(F_2^e-F_1^e)N_2(t) (\left|t\right|\le 1,Q\in L_e\subset \mathrm{\Gamma }).$$

(45)

Unit $e_1$ on the boundary S adjacent to the unit $e_2$, as shown in Figure 4, follows the interpolation function as:

$$F(Q)=(F_1^{e_1}-F_2^{e_1})N_1(t)+{\displaystyle \frac{F_2^{e_1}}{{[N_1(t)]}^{1-{\lambda }_1}}} (\left|t\right|\le 1,Q\in L_{e_1}\subset S).$$

(46)

For unit $e_3$ on the boundary S adjacent to the unit $e_2$, the unknown function $F(Q)$ can be expressed as:

$$F(Q)={\displaystyle \frac{F_1^{e_3}}{{[N_2(t)]}^{1-{\lambda }_2}}}+(F_2^{e_3}-F_1^{e_3})N_2(t) (\left|t\right|\le 1,Q\in L_{e_3}\subset S)$$

(47)

For the unit $e_2$ on the crack $\mathrm{\Gamma }$ where the boundary intersects, the following interpolation function was introduced:

$$F(Q)={\displaystyle \frac{F_{2c}^{e_2}}{{[N_1(t)]}^{1-{\lambda }_2}}}+(F_{1c}^{e_2}-F_{2c}^{e_2}-Q_{2c}^{e_2})[N_1(t)]+{\displaystyle \frac{Q_{2c}^{e_2}}{{[N_1(t)]}^{1-{\lambda }_1}}} (\left|t\right|\le 1,Q\in L_{e_2}\subset \mathrm{\Gamma })$$

(48)

On the unit $e_1$:

$$F(Q)={\displaystyle \frac{\partial \varphi (Q)}{\partial s(Q)}}={\left.-{\displaystyle \frac{\partial \varphi (Q)}{\partial \rho }}\right|}_{\theta =0}=-{\displaystyle \frac{1}{\alpha }}{\displaystyle \frac{\partial w_1(\rho ,0)}{\partial \rho }}=-{\displaystyle \frac{1}{\alpha }}{\lambda }_1{A}_1{\rho }^{{\lambda }_1-1}$$

(49)

$$F(Q)=(F_1^{e_1}-F_2^{e_1})N_1(t)+{\displaystyle \frac{F_2^{e_1}}{{[N_1(t)]}^{1-{\lambda }_1}}}, \rho =l_{e_1}{N}_1(t).$$

(50)

hence,

$$\underset{\rho \to 0}{\lim }[{(\rho )}^{1-{\lambda }_1}F(Q)]=-{\displaystyle \frac{1}{\alpha }}{\lambda }_1{A}_1={(l_{e_1})}^{1-{\lambda }_1}{F}_2^{e_1}.$$

(51)

thus,

$$F_2^{e_1}=-{\displaystyle \frac{1}{\alpha }}{\lambda }_1{A}_1{(l_{e_1})}^{{\lambda }_1-1}.$$

(52)

On the unit $e_2$:

$$F(Q)={\displaystyle \frac{\partial \Delta \varphi (Q)}{\partial s(Q)}}={\left.-{\displaystyle \frac{\partial \Delta \varphi (Q)}{\partial \rho }}\right|}_{\theta =\beta }=-{\displaystyle \frac{1}{\alpha }}{\displaystyle \frac{\partial [w_2(\rho ,\beta )-w_1(\rho ,\beta )]}{\partial \rho }}=-{\displaystyle \frac{1}{\alpha }}({\lambda }_2{C}_2{\rho }^{{\lambda }_2-1}+{\lambda }_1{A}_1{\rho }^{{\lambda }_1-1})$$

(53)

$$F(Q)={\displaystyle \frac{F_{2c}^{e_2}}{{[N_1(t)]}^{1-{\lambda }_2}}}+(F_{1c}^{e_2}-F_{2c}^{e_2}-Q_{2c}^{e_2})[N_1(t)]+{\displaystyle \frac{Q_{2c}^{e_2}}{{[N_1(t)]}^{1-{\lambda }_1}}}, \rho =l_{e_2}{N}_1(t).$$

(54)

Hence,

$$\underset{\rho \to 0}{\lim }[{(\rho )}^{1-{\lambda }_2}F(Q)]=-{\displaystyle \frac{1}{\alpha }}{\lambda }_2{C}_2-{\displaystyle \frac{1}{\alpha }}{\lambda }_1{A}_1{\rho }^{{\lambda }_1-1}{\rho }^{1-{\lambda }_2}=F_{2c}^{e_2}{(l_{e_2})}^{1-{\lambda }_2}+Q_{2c}^{e_2}{[N_1(t)]}^{{\lambda }_1-1}{\rho }^{1-{\lambda }_2}.$$

(55)

Then,

$$F_{2c}^{e_2}=-{\displaystyle \frac{1}{\alpha }}{\lambda }_2{C}_2{(l_{e_2})}^{{\lambda }_2-1}$$

(56)

$$Q_{2c}^{e_2}=-{\displaystyle \frac{1}{\alpha }}{\lambda }_1{A}_1{(l_{e_2})}^{{\lambda }_1-1}.$$

(57)

On the unit $e_3$:

$$F(Q)={\displaystyle \frac{\partial \varphi (Q)}{\partial s(Q)}}={\left.{\displaystyle \frac{\partial \varphi (Q)}{\partial \rho }}\right|}_{\theta =\pi }=-{\displaystyle \frac{1}{\alpha }}{\displaystyle \frac{\partial w_2(\rho ,\pi )}{\partial \rho }}=-{\displaystyle \frac{1}{\alpha }}{\lambda }_2{C}_2{\rho }^{{\lambda }_2-1}$$

(58)

$$F(Q)={\displaystyle \frac{F_1^{e_3}}{{[N_2(t)]}^{1-{\lambda }_2}}}+(F_2^{e_3}-F_1^{e_3})N_2(t), \rho =l_{e_3}{N}_2(t).$$

(59)

Hence,

$$\underset{\rho \to 0}{\lim }[{(\rho )}^{1-{\lambda }_2}F(Q)]=-{\displaystyle \frac{1}{\alpha }}{\lambda }_2{C}_2={(l_{e_3})}^{1-{\lambda }_2}{F}_1^{e_3}.$$

(60)

Then,

$$F_1^{e_3}=-{\displaystyle \frac{1}{\alpha }}{\lambda }_2{C}_2{(l_{e_3})}^{{\lambda }_2-1}.$$

(61)

Similarly,

$$F_{2c}^{e_2}={(l_{e_2})}^{{\lambda }_2-1}{(l_{e_3})}^{1-{\lambda }_2}{F}_1^{e_3}$$

(62)

$$Q_{2c}^{e_2}={(l_{e_1})}^{1-{\lambda }_1}{(l_{e_2})}^{{\lambda }_1-1}{F}_2^{e_1}.$$

(63)

The boundary integral equation for the torsional problem is [33]:

$${\displaystyle \underset{S+\mathrm{\Gamma }}{\int }\frac{\partial \varphi (P,Q)}{\partial s(P)}}F(Q)ds(Q)={\displaystyle \underset{S}{\int }\frac{\partial \varphi (P,Q)}{\partial n(P)}}q(Q)ds(Q)-{\displaystyle \frac{1}{2}}f(P), (p(x,y)\in S+\mathrm{\Gamma })$$

(64)

The integrals contained within the boundary integral of Equation (64) and the displacement single-valued condition were rewritten as the sum of the integrals of the boundary. The interpolation Equations (43)–(48) were then substituted separately based on the element type. Furthermore, by selecting the source points P in the integral Equation (64) as the mid-nodes of each element and adding the already derived Equations (62) and (63), a closed algebraic system of equations for the undetermined values $\left\{F_j\right\}$ of the interpolation functions $F(Q)$ was obtained, and its coefficient matrix was composed of regular integrals and singular integrals. For non-singular integrals, the Gauss–Legendre quadrature equation was directly applied for computation. In cases where the source point $P(x_0,y_0)$ was the mid-node of the integral element $L_e$, the boundary integral Equation (64) included two singular integrals as follows:

$${\displaystyle \underset{L_e}{\int }\frac{\partial \varphi (P,Q)}{\partial n(P)}}q(Q)ds(Q)={\displaystyle \underset{L_e}{\int }\frac{1}{2\pi r^2}}[\overrightarrow{n}(P)·\overrightarrow{Q}(P)]q(Q)ds(Q)=0, \mathrm{for} P(x_0,y_0)\in L_e$$

(65)

$${\displaystyle \underset{L_e}{\int }\frac{\partial \varphi (P,Q)}{\partial s(P)}}F(Q)ds(Q)={\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-1}^{1}F(Q(t))}{\displaystyle \frac{dt}{t}}, \mathrm{for} P={\displaystyle \frac{1}{2}}(Q_1^e+Q_2^e)\in L_e.$$

(66)

By substituting the interpolation Equations (43)–(45) into the above equation, the following singular integral equation was obtained:

$${\displaystyle {\int }_{-1}^{1}F(Q(t))}{\displaystyle \frac{dt}{t}}=F_2^e-F_1^e, (\mathrm{for} \mathrm{a} \mathrm{linear} \mathrm{unit} \mathrm{on} \mathrm{a} \mathrm{boundary} \mathrm{S}, L_e\subset S);$$

(67)

$${\displaystyle {\int }_{-1}^{1}F(Q(t))}{\displaystyle \frac{dt}{t}}=F_2^e-F_1^e, (\mathrm{for} \mathrm{a} \mathrm{general} \mathrm{unit} \mathrm{\Gamma }\mathrm{on} \mathrm{a} \mathrm{crack} \mathrm{that} \mathrm{does} \mathrm{not} \mathrm{contain} \mathrm{a} \mathrm{crack} \mathrm{tip}, L_e\subset \mathrm{\Gamma });$$

(68)

$${\displaystyle {\int }_{-1}^{1}F(Q(t))}{\displaystyle \frac{dt}{t}}=F_1^e\sqrt{2}\ln (3-2\sqrt{2})+(F_2^e-F_1^e), (\mathrm{for} \mathrm{the} \mathrm{start} \mathrm{node} \mathrm{element},L_e\subset \mathrm{\Gamma })$$

(69)

For the units e₁, e₂, e₃,when any value was used for $\beta$, ${\lambda }_1$, and ${\lambda }_2$ were not natural numbers, hence, Newton’s binomial theorem was no longer valid. Thus, we only calculated when $\beta =\pi /2$ at ${\lambda }_1={\lambda }_2=4$. Then:

$${\displaystyle {\int }_{-1}^{1}F(Q(t))}{\displaystyle \frac{dt}{t}}=-F_1^{e_1}+{\displaystyle \frac{1}{6}}{F}_2^{e_1}, (\mathrm{for} \mathrm{the} \mathrm{unit} e_1\mathrm{on} \mathrm{the} \mathrm{boundary} S, L_e\subset S);$$

(70)

$${\displaystyle {\int }_{-1}^{1}F(Q(t))}{\displaystyle \frac{dt}{t}}=-F_{1c}^{e_2}+{\displaystyle \frac{1}{6}}{F}_{2c}^{e_2}+{\displaystyle \frac{1}{6}}{Q}_{2c}^{e_2}, (\mathrm{for} \mathrm{the} \mathrm{unit} e_2\mathrm{at} \mathrm{the} \mathrm{intersection} \mathrm{of} \mathrm{the} \mathrm{crack} \mathrm{and} \mathrm{the} \mathrm{boundary}, L_e\subset \mathrm{\Gamma });$$

(71)

$${\displaystyle {\int }_{-1}^{1}F(Q(t))}{\displaystyle \frac{dt}{t}}=-{\displaystyle \frac{1}{6}}{F}_1^{e_3}+F_2^{e_3}, (\mathrm{for} \mathrm{the} \mathrm{unit} e_3\mathrm{on} \mathrm{the} \mathrm{boundary} S, L_e\subset S).$$

(72)

The above algebraic equations can solve $\left\{F_j\right\}$, $Q_{2c}^{e_2}$ and ${\lambda }_i$ in the unknown function $F(Q)$. Once these unknowns are solved, the SIF and torsional stiffness, D, can be calculated according to the method of Lu, Z.Z. [33], yielding the following result:

$$K_{\mathrm{I}\mathrm{I}\mathrm{I}}(a_j)=-{\displaystyle \frac{G\alpha }{2}}\sqrt{2\pi l_e}{F}_1^e$$

(73)

$$K_{\mathrm{I}\mathrm{I}\mathrm{I}}(b_j)=0,$$

(74)

where $a_j$ and $b_j$ represent the initial and terminal endpoints of the j-th crack, respectively.

After discretization, the closed integral equation system involves the calculation of integrals over singular elements and non-singular elements. The integrals over singular elements can be computed analytically [35], while the integrals over non-singular elements can be numerically calculated using the Gauss-Legendre quadrature formula [36].

3. Numerical Example of the Torsional Problem of a Cylinder with an Edge Crack

To demonstrate the correctness and effectiveness of the proposed method, two examples of torsional problems of a cylinder with a straight and a broken edge crack are given in this section.

3.1. Cylinders with Straight Edge Cracks

Figure 6 illustrates the torsion of a cylinder containing a straight edge crack. The crack length is 2c, the distance from the center of the circle to the midpoint of the straight crack is d, the outer radius of the annular structure is R, and the inner radius is r. The crack was divided into 200 elements, while the outer boundary was divided into 320 elements. To ensure the accuracy of the calculations, a local mesh refinement strategy was adopted near the crack tip, where the element size decreased geometrically. Moreover, the minimum element size was approximately 0.5% of the crack length. By systematically varying the number of global elements, convergence analysis of the dimensionless SIF was performed, as shown in Figure 7. When the number of crack-tip elements increased to approximately 200, the calculated SIF tended to stabilize, with a relative error of less than 0.3% observed between adjacent meshes. This result indicated that the numerical solution achieved good convergence.

Figure 8 shows the changes in the dimensionless SIF $K_{\mathrm{III}}^{*}=K_{\mathrm{III}}/(G\alpha R\sqrt{\pi c})$ and the dimensionless torsional stiffness $D^{*}=D/({\displaystyle \frac{\pi }{2}}G{R}^4)$ with the ratio of the edge crack length to the outer cylinder radius (2c/R), where $K_{\mathrm{I}\mathrm{I}\mathrm{I}}^{*}$(B) = 0. These parameters were computed via the Gauss–Chebyshev quadrature method reported in the literature [37], FEM, and the BEM of this study. The computed results were relatively consistent with the results reported in the literature [37], and demonstrated higher accuracy than the FEM results. This demonstrated the effectiveness of the BEM for simulating the singular behavior at crack tips and its suitability for torsion calculations related to cylinders with straight edge cracks.

To quantitatively evaluate the influence of the mesh density on the calculation results, the mesh density sensitivity analysis plot is presented below. The plot shows the sensitivity coefficients and error ranges of each calculation index under different crack lengths, thereby verifying the robustness of the numerical scheme.

Figure 9 shows that when the number of elements varied within the range of 640–960, the dimensionless SIF and torsional stiffness exhibited only slight fluctuations, tending to stabilize at N = 800. This indicated that the mesh scheme adopted in this study had good stability. The calculation results were insensitive to variations in the mesh density, and the robustness of the numerical method met the requirements of academic research and engineering analyses.

The computation of the dimensionless SIF and torsional stiffness for cylinders with straight edge cracks offers critical quantitative support for the anti-crack design of engineering structures. The SIF acts as a quantitative measure of the stress field strength at the crack tip. It directly determines whether the crack will undergo unstable propagation under loading and functions as an early warning sign for structural fracture damage—the higher its value, the greater the critical risk posed by the crack. The torsional stiffness represents the component’s capacity to resist torsional deformation. A reduction in stiffness implies that the component’s torsional deformation will increase under the same torque, allowing for the extent to which the crack impairs the structural torsional load-bearing capacity to be quantified and thereby providing a quantitative criterion for residual stiffness assessment. When these two parameters are applied to the tower of a cross-sea suspension bridge, the tower serves as the core component bearing the torsional load of the bridge deck. In the case of construction defects or fatigue cracks during the operational period, the dimensionless SIF proposed in this study, together with the actual material parameters and load conditions of the tower, can be used to rapidly assess whether the crack is likely to enter the propagation phase. In addition, the impact of cracks on the torsional performance of the tower can be assessed using the dimensionless torsional stiffness. This provides guidance for determining the timing of the tower crack repair and formulating reinforcement plans, thus ensuring the long-term operational safety of the cross-sea bridge.

3.2. Cylinders with Cracks at the Broken Edge

Figure 10 illustrates the torsion of a cylinder that contains a broken-line edge crack, as computed in this section. The crack length is a + c; the distance from the center of the circle to the crack’s inflection point is d; the angle between the broken line and the x-axis is $\beta$; the outer radius of the annular structure is R; and the inner radius is r. The crack was divided into 300 elements, while the outer boundary was divided into 400 elements. Figure 11 presents the changes in the dimensionless torsional stiffness D* and dimensionless SIF $K_{\mathrm{I}\mathrm{I}\mathrm{I}}^{*}=K_{\mathrm{I}\mathrm{I}\mathrm{I}}/(G\alpha R\sqrt{\pi (a+c)})$ (computed via the method of this study) with the broken-line angle $\beta$, where $K_{\mathrm{I}\mathrm{I}\mathrm{I}}^{*}$(B) = 0. In the specific calculation process, the values r/R = 0.1, a/R = 0.3, c/R = 0.1, and d/R = 0.7 were used. The results showed that when the values of $\beta$ were centered at 0, the distribution on either side was generally symmetric. At $\beta =0$, the torsional stiffness reached its minimum value, while the SIF reached its maximum value. The computed results further demonstrated that the BEM was equally applicable for the computation of torsion in cylinders with broken-line edge cracks.

In contrast to simple straight cracks, broken-line cracks tend to generate mixed-mode SIFs more easily due to their geometric discontinuity. This indicates that the stress field at the crack tip is more complex. Instead of propagating in the original direction, the crack may deflect or branch, leading to a less predictable propagation path and potentially higher destructiveness.

The model constructed in this research was subjected to a rigorous mesh independence validation to ensure the accuracy and reliability of the boundary element numerical computation results. The purpose of this analysis is to demonstrate that the key solutions do not rely on the mesh density, thereby ensuring the robustness of the conclusions. We chose a representative calculation working condition as the test case. On this basis, a series of mesh schemes ranging from coarse to fine were generated systematically, with the total number of mesh elements was gradually increased from approximately 50 to 500. Special attention was paid to the mesh quality of the crack area during the mesh division process. The computation results indicated that as the number of meshes increased, the value of the monitored quantity significantly changed significantly before gradually stabilizing. Once the total number of mesh elements reached 200, further increasing the number of mesh elements to 500 led to a change of merely 0.25% in the monitored quantity, which was well below the preset convergence threshold of 0.5%. Hence, it was concluded that the solution had good independence at this mesh density.

The flowchart of the FORTRAN 2018 program is presented in Figure 12.

4. An Example of a Torsional Analysis of Suspension Bridge Towers Under Wind Load

4.1. Torsional Effect of Wind Load on the Tower Column Body

The research object of this study was the reinforced concrete pylon of a cross-sea suspension bridge in China, with a main span of 1288 m and a rhombus-frame structure. The key geometric and performance parameters of the pylon were as follows:

(1)

Cross-sectional dimensions of the pylon: The cross-section of the lower pylon column is a circular ring with an outer radius R₀ = 6.5 m, an inner radius R_i = 5.0 m, and a wall thickness of 1.5 m. The cross-section of the upper pylon column tapers gradually to an outer radius R₀ = 4.8 m and an inner radius R_i = 3.8 m.

(2)

Height parameters of the pylon: The total height is 215 m, including the lower pylon column of 80 m, a middle pylon column of 105 m, and a upper pylon column of 30 m.

(3)

Geometric settings of the cracks: Based on the statistics of the actual pylon construction defects and operational damage, the cracks are set at height of 40 m on the lower pylon column (a torque-concentrated region). The length range of the straight cracks is 0.5–3.0 m, and the turning angles of the zigzag cracks are 15°, 30°, and 45°, which are consistent with the typical distribution characteristics of cracks in practical engineering.

(4)

Material properties of the pylon: The main structure of the pylon is made of C50 reinforced concrete, with a modulus of elasticity of E = 3.45 × 104 MPa, a Poisson’s ratio v = 0.2, a compressive strength of f_c = 23.1 MPa, and a tensile strength of f_t = 2.64 MPa. For the fracture mechanics parameters, the fracture toughness was K_IIIc = 1.5 MPa·m^1/2 (as measured via the three-point bending test of C50 concrete).

Based on references [38,39,40], the wind load acting on the tower was calculated using the following equation:

$$F=k\cdot k_z\cdot {\beta }_z\cdot P_0\cdot A,$$

(75)

where k denotes the shape coefficient of the wind load, i.e., the pressure generated when the wind acts on the building surface, whose magnitude is associated with the building dimensions; and k_z stands for the height-dependent coefficient of the offshore wind pressure, which increases with height above the sea level; ${\beta }_z$ denotes the wind oscillation coefficient; P₀ denotes the fundamental wind pressure; and A refers to the outline projection area of the component that is perpendicular to the wind direction.

As illustrated in Figure 13, the pylons of the cross-sea suspension bridges were subjected to vertical pressure from the main cables throughout the year, along with long-term fluctuating torsional loads induced by wind, seismic activity, and vehicle loads. In particular, during typhoon events, the interaction between unbalanced live loads on either side of the main cables and strong wind loads results in considerable torque in the lower section of the pylon, particularly in the pylon columns.

4.2. Example of the Numerical Calculation for the Pylon Column

In this section, the torsional fracture of the suspension bridge pylon column with a straight edge crack is computed under different wind loads conditions. The pylon column segment was treated as a simplified cylinder made of high-strength concrete and containing a straight edge crack. To achieve an accurate analysis of the torsion problem of suspension bridge pylons with edge cracks under wind loads while balancing the operability of mechanical modeling and computational efficiency, we simplified the primary force-bearing section of the pylon into a uniform-section cylinder made of high-strength concrete with straight edge cracks. This simplification was based on three key assumptions and a corresponding rationality verification: (1) in terms of the geometric shape, the local structural details, such as construction joints, were neglected; (2) in terms of the material properties, the pylon was regarded as a homogeneous isotropic material; and (3) in terms of the crack morphology, the cracks were simplified as straight through cracks. For the rationality verification, a comparison with the prototype of a C50 high-strength concrete pylon of a suspension bridge with a main span of 1200 m showed that the deviations of the section moment of inertia and torsional moment of inertia between the simplified model and the prototype were 3.2% and 4.1%, respectively. The errors of the torsional stiffness and SIF were both less than 4% for cases without cracks and for cases with 50 mm cracks. Moreover, the consistency with mainstream domestic and foreign studies, as well as engineering practice of a cross-sea bridges, confirmed the effectiveness of this simplification [41,42,43,44]. It should be noted that this model is applicable to a linear elastic torsion analysis of the main force-bearing section of the pylon, and secondary corrections can be introduced through relevant correction coefficients for extreme working conditions. With the wind loads applied, a numerical computation program was developed using the aforementioned data in accordance with the BEM recommended in this study. This resulted in the dimensionless torsional stiffness and SIF being as follows:

$$D*=0.996652, K_{\mathrm{III}}^{*}(A)=-0.597362, K_{\mathrm{I}\mathrm{I}\mathrm{I}}^{*}(B)=0.$$

(76)

The strain energy density factor criterion for pure type III cracks is given in [45,46,47]:

$$K_{\mathrm{III}c}=\sqrt{1-2\nu }{K}_{\mathrm{I}c}=0.816K_{\mathrm{I}c}=0.90168(MN/m^{3/2}), \left(\nu =0.167\right).$$

(77)

Wind speeds corresponding to various wind grades were acquired through reviewing an extensive range of literature. In accordance with the fracture criterion, $K_{\mathrm{III}}\le K_{\mathrm{III}C}$, the maximum permissible crack length of the cylindrical portion of the suspension bridge pylon with straight edge cracks under different wind load actions was computed. These results are illustrated in Figure 14.

It can be observed from the figure that, as the offshore wind load increases, the maximum permissible crack length of the cracked cylinder pylon column steadily decreases. This critical value not only directly governs the operational safety of the bridge, but also has a far-reaching influence on the long-term design as well as operation and maintenance strategies.

It is worth noting that treating the maximum allowable crack length as a deterministic threshold has inherent limitations. Therefore, we conducted an uncertainty analysis based on the Monte Carlo simulation. The key uncertain factors were selected as follows:

(1)

Core material parameters of the pylon: The tensile strength (assumed to follow a normal distribution with a mean of 2.64 MPa and a coefficient of variation of 0.05); and the elastic modulus (assumed to follow a normal distribution with a mean of 3.45×10⁴ MPa and a coefficient of variation of 0.03);

(2)

Peak wind load under different wind levels (assumed to follow a lognormal distribution, with parameters determined based on Specification GB 50009-2012 [48] and field measurement data);

(3)

Crack size measurement error (assumed to follow a uniform distribution with an error range of ±0.02 mm).

Through 1000 random samplings, the maximum allowable crack length corresponding to each sampling group was calculated using the BEM, and the corresponding probability distribution histograms and statistical parameter tables were obtained. Wind level 8 was used as the representative wind condition (as shown in Figure 15). Under this condition, the maximum allowable crack length had a mean of 130 mm, a standard deviation of 0.63 mm, and a 95% confidence interval of [129.88, 130.12] mm. These results indicated that the maximum allowable crack length was not a fixed threshold but followed a well-defined probability distribution, which effectively compensated for the limitations of a purely deterministic analysis.

In terms of safety, the critical crack length serves as a key threshold at which the structure shifts from a damaged condition to failure. As the primary load-bearing component of a suspension bridge, the pylon sustains a substantial axial force and, at the same time, is subjected to bending moments induced by wind loads and seismic activities, which tend to result in significant stress concentration at the crack tip. The computed critical crack length represents the precise turning point for unstable crack propagation under the most adverse load combination. If the actual crack length approaches or exceeds this threshold, a brittle fracture may occur in the pylon body. This leads to sudden changes in structural stiffness, imbalance in the internal force redistribution, and potentially a chain reaction involving pylon failure and main cable malfunction, which ultimately jeopardize the safety of the entire bridge. Hence, this result provides a critical basis for assessing the remaining service life of the structure and for implementing early risk warnings. To comply with the safety design requirements of practical bridge engineering while considering the pylon as a Class-I critical component, safety factor calibration and threshold correction were supplemented. We referred to the safety reserve requirements for brittle fracture components specified in the General Code for Design of Highway Bridges and Culverts (JTG D60-2015 [49]). We then used the maximum allowable crack length obtained in this study as the baseline, and a safety factor of 1.2 was applied by accounting for the brittle characteristics of the crack propagation and the core load-bearing role of the pylon. This value ensures adequate safety reserves while avoiding excessive conservatism that would lead to unnecessary engineering costs. The corrected threshold values corresponding to different wind grades are presented in Figure 16.

The corrected maximum allowable crack length provides an explicit quantitative criterion for bridge health monitoring and periodic inspection at the operational and maintenance (O&M) management level. The core value of this lies in scientifically defining the thresholds for crack alarm and repair, thereby equipping the formulation of inspection cycles and maintenance decision-making with a reliable quantitative foundation. This ensures that cracks are identified in a timely manner and addressed prior to propagating to a critical state. Based on this engineering threshold, we further developed a three-level inspection early-warning standard as follows:

(1)

Low-risk level: When the measured crack length is less than 0.5 times the corrected threshold, a routine inspection cycle (once every two years) is adopted.

(2)

Medium-risk level: When the measured crack length ranges from 0.5 times the corrected threshold to the corrected threshold, the inspection cycle is shortened to once per year, with simultaneous enhancement of real-time monitoring.

(3)

High-risk level: When the measured crack length reaches or exceeds the corrected threshold, it is imperative to immediately suspend operation, conduct a comprehensive inspection, and formulate a targeted reinforcement scheme.

This hierarchical standard transforms bridge operation and maintenance from a passive response mode to an active prevention mode, facilitating the upgrade of O&M strategies toward refined and predictive management.

5. Conclusions

This study focuses on the torsional behavior of edge-cracked pylon cylinders of suspension bridges under wind loads. Through a combination of theoretical analysis and numerical simulation, the key mechanical characteristics of pylon torsional fracture and the engineering application potential of the BEM are systematically investigated. The main conclusions are drawn as follows:

(1)

Based on the torsion theory of elastic mechanics, boundary integral equations for edge-cracked cylinders were established, which clarified the singularity characteristics of the stress field at crack tips. This research addresses the deficiency of insufficient theoretical analysis of wind-induced torsional fracture in suspension bridge pylons in existing studies, providing a solid theoretical basis for the accurate calculation of SIFs for such problems.

(2)

A BEM-based numerical algorithm tailored for the torsional fracture analysis of pylons was proposed. Through rational boundary element division and the selection of multiple forms of interpolation functions, high-precision calculations of torsional stiffness and SIFs for cylinders with straight or polygonal edge cracks were achieved. Verification against theoretical solutions and experimental data reported in the literature confirms that the proposed algorithm exhibits high accuracy and reliability. The method fully leverages the advantages of BEM in handling crack-related discontinuities and accurately captures crack-tip singularities, thereby filling the gap in the application of BEM in the field of torsional fracture analysis of bridge pylons.

(3)

Numerical simulations under different wind load conditions were performed, which quantified the effects of crack length, crack form (straight or polygonal), and wind load grade on the attenuation of torsional stiffness and the evolution of SIFs at crack tips in pylon cylinders. The maximum allowable crack length thresholds under various wind grades were determined, which were directly correlated with the local crack damage in pylons under actual wind load conditions in engineering practice, thereby providing a quantitative basis for damage assessment and safety early warning of suspension bridge pylons.

In summary, this study systematically reveals the torsional fracture mechanism of edge-cracked pylon columns of suspension bridges under wind loads, improves the theoretical framework for torsional fracture analysis of bridge structures, and verifies the engineering applicability of BEM in torsional fracture analysis of pylon engineering. The research findings not only provide direct technical support for the wind-resistant design and damage detection of suspension bridge pylons, but they can also be extended to the overall bridge safety evaluation system. By quantifying the influence of local cracks in pylons on main cable displacement and deck stability, a multilevel safety evaluation logic from the local to global levels is established, which ultimately provides systematic and comprehensive theoretical and technical support for the full life-cycle safety management of cross-sea suspension bridges.

It should be noted that the simplification of pylon cracks as single edge cracks in this study is intended to eliminate complex interfering factors, focus on the core mechanical mechanism by which cracks affect the torsional bearing capacity of pylons, and prioritize verification of the validity of fundamental theories and numerical calculation methods (e.g., the reliability of the BEM is verified by comparing with results reported in the existing literature), thereby laying a solid foundation for subsequent research on complex crack problems. In future work, the calculation method for the maximum allowable crack length proposed in this study can be further expanded and optimized by introducing multicrack interference theory and adjusting the calculation parameters based on the actual crack morphology obtained from field inspection. These improvements will enhance the applicability of the method to complex crack scenarios in practical engineering, thus providing more accurate technical references for evaluating the safety of actual pylons.