Fatigue Performance Enhancement of Open-Hole Steel Plates Under Alternating Tension–Compression Loading via Hotspot-Targeted CFRP Reinforcement

Abstract

Steel plates with open holes are common in engineering structures such as bridges and towers for pipeline penetrations and connections. These openings, however, induce significant stress concentration under alternating tension–compression loading (stress ratio R = −1), drastically accelerating fatigue crack initiation and threatening structural integrity. Effective identification and mitigation of such stress concentrations is crucial for enhancing the fatigue resistance of perforated components. This study proposes a closed-loop methodology integrating theoretical weak zone identification, targeted CFRP reinforcement, and experimental validation to improve the fatigue performance of open-hole steel plates. Analytical solutions for dynamic stresses around the hole were derived using complex function theory and conformal mapping, identifying critical stress concentration angles. Experimental tests compared unreinforced and CFRP-reinforced specimens in terms of circumferential strain distribution, dynamic stress concentration behavior, and fatigue life. Results indicate that Carbon fiber-reinforced polymer (CFRP) reinforcement significantly reduces stress concentration near 90°, smooths polar strain distributions, and slows strain decay. The S–N curves shift upward, indicating extended fatigue life under identical stress amplitude and increased allowable stress at identical life cycles. Comparison with standardized design curves confirms that reinforced specimens meet higher fatigue categories, providing practical design guidance for perforated plates under alternating loads. This work establishes a systematic framework from theoretical prediction to experimental verification, offering a reliable reference for engineering applications.

1. Introduction

Steel plates with open holes are fundamental elements in numerous civil and mechanical structures, serving essential functions for pipeline penetrations, bolted connections, and access ports. These geometric discontinuities inevitably induce stress concentration, becoming a critical determinant of structural integrity under dynamic loading. Under high-cycle fatigue with alternating tension–compression stresses (stress ratio R = −1), the stress amplitude at the hole periphery is significantly magnified, drastically accelerating fatigue crack initiation and propagation. This vulnerability poses a severe threat to the safety and longevity of critical infrastructure such as bridges, offshore platforms, and industrial machinery [1]. Consequently, accurate prediction of stress concentration zones and development of effective mitigation strategies are paramount for enhancing fatigue resistance and ensuring the reliability of perforated steel components.

The problem of stress concentration around circular holes in infinite or finite plates has been extensively studied within classical elasticity theory. The seminal work of Kirsch [2] provided a fundamental analytical solution for the stress field, enabling derivation of the static stress concentration factor. Subsequent advancements, including use of the Airy stress function [3], the complex function method [4,5,6,7,8,9], and conformal mapping techniques [10,11], have further enriched the analytical toolkit for solving stress distribution in plates with openings of various shapes under different in-plane loading conditions. In parallel, carbon fiber-reinforced polymer (CFRP) has emerged as a highly effective material for retrofitting and strengthening steel structures [12,13]. Numerous experimental and numerical studies have demonstrated the capability of externally bonded CFRP to reduce stress concentration and extend the fatigue life of steel plates with holes or cracks. However, existing literature exhibits several notable limitations. Most reinforcement schemes and fatigue assessments rely on empirical data or finite element analysis under static tension or unidirectional fatigue loading [14,15,16], with scant attention paid to the more severe alternating tension–compression regime. This gap impedes the capture of the evolution of dynamic stress concentration factors over fatigue cycles. Experimentally, many studies rely on single-point strain measurements or local peak values as proxies for the full-field response [17,18,19], lacking unified and reproducible metrics for the entire stress cycle. Furthermore, while powerful, finite element results are often sensitive to meshing strategies and boundary conditions [20,21], rigorous mesh-independence studies and validation against comprehensive experimental data, particularly the full circumferential strain distribution around the hole, are frequently insufficient [22,23]. Perhaps most critically, the potential of sophisticated analytical methods, such as complex function theory, for deriving explicit dynamic stress solutions and guiding targeted reinforcement under elastic wave loading remains largely underdeveloped and disconnected from experimental validation [24].

To bridge these gaps, this study establishes a closed-loop methodology integrating “theoretical weak zone identification—targeted reinforcement—experimental quantification.” The primary objectives are threefold. First, analytical solutions for the dynamic stress field around a circular hole in a steel plate subjected to tensile-compressive wave loads are derived by combining complex function theory with conformal mapping. This theoretical framework aims to precisely identify the critical angles of maximum dynamic stress concentration. Second, guided by this theoretical prediction, a targeted double-sided triple-layer CFRP reinforcement is applied to the identified hotspot regions. Third, a comparative experimental investigation is conducted on both unreinforced and CFRP-reinforced specimens under alternating tension–compression fatigue loading. The experimental analysis quantitatively evaluates the evolution of circumferential strain distribution, dynamic stress concentration factors, frequency-domain characteristics, and fatigue life (S-N response). The results conclusively demonstrate that CFRP reinforcement significantly blunts the stress concentration near 90°, resulting in smoother polar strain curves, markedly slowed strain decay rates, and substantial improvement in fatigue life, elevating performance from near Category-C (as per GB 50017-2017 [25]) to meet Category-B standards [26]. This work provides a validated, mechanics-based pathway for fatigue enhancement of perforated steel plates, linking theoretical prediction directly to practical reinforcement strategy.

The analytical core (Section 2.1, Section 2.2, Section 2.3 and Section 2.4) focuses on dynamic stress concentration in the isotropic steel plate under elastic wave incidence. To address the anisotropic effect of the bonded CFRP reinforcement, a dedicated constitutive model for the orthotropic CFRP layer is introduced in Section 2.5. This model, combined with experimental strain-field mapping and fatigue life comparison, forms a coupled theoretical-experimental framework. This approach circumvents the extreme analytical complexity of a full-wave scattering solution in an anisotropic composite plate while still providing a physics-guided, quantifiable assessment of reinforcement efficacy.

2. Theoretical Research

2.1. Tension–Compression Wave Control Equation and Fundamental Principles

Under dynamic loading, stress transmission patterns in steel plate structures exhibit significant differences from static conditions. Particularly in perforated plates, wave propagation involves scattering and interference, inducing pronounced stress concentration effects in localized regions. Based on elastic dynamics theory, analyzing the impact of tensile-compressive wave scattering on dynamic stress concentration requires establishing fundamental wave equations, followed by numerical simulations to derive analytical solutions for dynamic stress concentration problems [27]. Taking a perforated steel plate as the research unit, the governing equations based on tensile-compressive vibration equations are:

$$\begin{array}{l}{\nabla }^2{\nabla }^{2}E-12\left[{\displaystyle \frac{1}{h^2}}+{\displaystyle \frac{2-{\kappa }^2}{24\left(1-\kappa \right)}}{T}_2^2\right]{\nabla }^{2}E+{\displaystyle \frac{3}{1-\kappa }}\left[{\displaystyle \frac{1}{h^2}}+{\displaystyle \frac{1+3\kappa }{24}}{T}_2^2\right]T_2^{2}E=0\\ \left[\left(3-2\kappa \right){\nabla }^2-T_2^2-{\displaystyle \frac{24}{h^2}}\right]\left({\nabla }^2-T_1^2\right)F=-\left[\left(3-4\kappa \right){\nabla }^2-\left(1-2{\kappa }^2\right)T_2^2-{\displaystyle \frac{24}{h^2}}\left(1-2\kappa \right)\right]E\\ {\nabla }^{2}f-T_2^{2}f=0\end{array}$$

(1)

In these equations, E, F, f are the three generalized displacement functions for the tension–compression vibration of the steel plate; ${\nabla }^2=\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}$ denotes the Laplace operator; $T_j^2=\frac{1}{c_j^2}\frac{{\partial }^2}{\partial t^2}\begin{array}{cc},& (j=1, 2)\end{array}$; c₁, c₂ represent the elastic longitudinal and transverse wave velocities, respectively; $c_1^2=\frac{\lambda +2\mu }{\rho }\begin{array}{cc},& c_2^2=\frac{\mu }{\rho }\end{array}$; $\kappa =\frac{1-2\nu }{2\left(1-\nu \right)}$; λ, μ are the Lame constants of the material; ν, ρ denote the Poisson’s ratio and density (taken as 7850 kg/m³ for steel) of the plate material, respectively; h represents the plate thickness.

Without loss of generality, for the harmonic vibration solution of the problem under study, let:

$$\begin{gathered} E=\tilde{E}{\mathrm{e}}^{-\mathrm{i}\omega t} \\ F=\tilde{F}{\mathrm{e}}^{-\mathrm{i}\omega t} \\ f=\tilde{f}{\mathrm{e}}^{-\mathrm{i}\omega t} \end{gathered}$$

(2)

where ω is the circular frequency of the plate’s tensile-compressive vibration; i is the imaginary unit.

In the following analysis, the time factor and the symbol ~ on the generalized displacement function are omitted. Substituting Equation (2) into Equation (1) yields:

$$\begin{gathered} \underset{j=1}{\overset{2}{\mathsf{\Pi }}}({\nabla }^2+{\alpha }_j^2)E=0 \\ {\nabla }^{2}f+k_2^{2}f=0 F=\tilde{F}{\mathrm{e}}^{-\mathrm{i}\omega t} \end{gathered}$$

(3)

where α_j (j = 1, 2) is the elastic wave number satisfying the equation ${\alpha }^4+12\left(\frac{1}{h^2}-\frac{2-{\kappa }^2}{24\left(1-\kappa \right)}{k}_2^2\right){\alpha }^2-\frac{3}{1-\kappa }\left(\frac{1}{h^2}-\frac{1+3\kappa }{24}{k}_2^2\right)k_2^2=0$ and $k_j^2={\omega }^2/c_j^2, (j=1, 2)$.

The general solution for the scattered wave described by the tension–compression vibration Equation (3) for steel plates can be expressed as:

$$\begin{gathered} E={\displaystyle \mathbf{\sum }_{m=1}^2{\displaystyle \sum _{n=-\infty }^{\infty }{A}_{mn}{\mathrm{H}}_n^{(1)}({\alpha }_{m}r){\mathrm{e}}^{\mathrm{i}n\theta }}} \\ F={\displaystyle \mathbf{\sum }_{m=1}^2{\displaystyle \mathbf{\sum }_{n=-\infty }^{\infty }{A}_{mn}{\delta }_m{\mathrm{H}}_n^{(1)}({\alpha }_{m}r){\mathrm{e}}^{\mathrm{i}n\theta }}} \\ f={\displaystyle \mathbf{\sum }_{n=-\infty }^{\infty }{B}_n{\mathrm{K}}_n(k_{2}r){\mathrm{e}}^{\mathrm{i}n\theta }} \end{gathered}$$

(4)

where δ_j (j = 1, 2) is the proportionality coefficient of the scattered wave function, ${\delta }_j=\frac{\left(3-4\kappa \right){\alpha }_j^2{h}^2-\left(1-2{\kappa }^2\right)k_2^2{h}^2+24\left(1-2\kappa \right)}{\left[\left(3-2\kappa \right){\alpha }_j^2{h}^2-k_2^2{h}^2+24\right]\left({\alpha }_j^2-k_1^2\right)}$; ${\mathrm{H}}_n^{(1)}(\cdot )$ is the Hankel function; ${\mathrm{K}}_n(\cdot )$ is the imaginary principal Bessel function; A_mn (m = 1, 2) and Bn are the scattered wave mode coefficients, determined by the opening boundary conditions.

In rectangular coordinates, based on the theory of plate tension–compression refinement, the generalized internal force expression for a plate structure is:

$$\begin{array}{l}\begin{array}{l}{N}_x={\displaystyle {\int }_{-h/2}^{h/2}{\sigma }_x}dz\\ =\lambda h\left(1-{\displaystyle \frac{h^2}{24}}{\square }_1^2\right)\left({\nabla }^{2}F+E\right)+2\mu h\left[\left(1-{\displaystyle \frac{h^2}{24}}{\square }_1^2\right){\displaystyle \frac{{\partial }^{2}F}{\partial x^2}}\right.\\ \left.-{\displaystyle \frac{h^2}{24}}\left({\displaystyle \frac{1}{\kappa }}-1\right){\displaystyle \frac{{\partial }^2}{\partial x^2}}\left({\square }_1^{2}F+E\right)\right]\end{array}\\ \begin{array}{l}{N}_y={\displaystyle {\int }_{-h/2}^{h/2}{\sigma }_y}dz\\ =\lambda h\left(1-{\displaystyle \frac{h^2}{24}}{\square }_1^2\right)\left({\nabla }^{2}F+E\right)+2\mu h\left[\left(1-{\displaystyle \frac{h^2}{24}}{\square }_1^2\right){\displaystyle \frac{{\partial }^{2}F}{\partial y^2}}\right.\\ \left.-{\displaystyle \frac{h^2}{24}}\left({\displaystyle \frac{1}{\kappa }}-1\right){\displaystyle \frac{{\partial }^2}{\partial y^2}}\left({\square }_1^{2}F+E\right)\right]\end{array}\\ \begin{array}{l}{N}_{xy}=N_{yx}\\ ={\displaystyle {\int }_{-h/2}^{h/2}{\tau }_{xy}}dz\\ =2\mu h\left[\left(1-{\displaystyle \frac{h^2}{24}}{\square }_1^2\right){\displaystyle \frac{{\partial }^{2}F}{\partial x\partial y}}\right.+\left({\displaystyle \frac{{\partial }^2}{\partial y^2}}-{\displaystyle \frac{{\partial }^2}{\partial x^2}}\right)f\\ \left.-{\displaystyle \frac{h^2}{24}}\left({\displaystyle \frac{1}{\kappa }}-1\right){\displaystyle \frac{{\partial }^2}{\partial x\partial y}}\left({\square }_1^{2}F+E\right)\right]\end{array}\\ \begin{array}{l}{Q}_x={\displaystyle {\int }_{-h/2}^{h/2}{\tau }_{zx}}dz=0\\ Q_y={\displaystyle {\int }_{-h/2}^{h/2}{\tau }_{zy}}dz=0\\ N_x+N_y=(1-\nu )B\left\{{\displaystyle \frac{1-2\kappa }{\kappa }}-{\displaystyle \frac{1-\kappa }{\kappa }}\left[1+{\displaystyle \frac{h^2}{24}}({\alpha }_m^2-k_2^2)\right]{\alpha }_m^2{\delta }_m+{\displaystyle \frac{h^2}{24}}\left({\displaystyle \frac{2-3\kappa }{\kappa }}{\alpha }_m^2-{\displaystyle \frac{1-2\kappa }{\kappa }}{k}_2^2\right)\right\}E,\\ N_y-N_x+2\mathrm{i}{N}_{xy}=-4(1-\nu )B{\displaystyle \frac{{\partial }^2}{\partial {\zeta }^2}}\left[{\displaystyle \frac{h^2}{24}}{\displaystyle \frac{1-\kappa }{\kappa }}+{\delta }_m-{\displaystyle \frac{h^2}{24}}{\displaystyle \frac{1-2\kappa }{\kappa }}({\alpha }_m^2-k_1^2){\delta }_{m}E\right]+\mathrm{i}{\displaystyle \frac{{\partial }^2}{\partial {\zeta }^2}}f,\\ M_{Q_x}-\mathrm{i}{M}_{Q_y}={\displaystyle \frac{1}{2}}(1-\nu )D\left[{\displaystyle \frac{1}{\kappa }}({\alpha }_m^2-|k_1^2){\delta }_m-{\displaystyle \frac{1-2\kappa }{\kappa }}\right]{\displaystyle \frac{\partial E}{\partial \zeta }}\end{array}\end{array}$$

(5)

In these formulae: B is the tensile stiffness of the base plate, $B={\displaystyle \frac{Eh}{1-{\nu }^2}}.$

Using the conformal mapping method, the exterior region of the hole boundary L in the ζ-plane is mapped onto the exterior region of the unit circle boundary S in the η-plane. After rearrangement, this can be expressed as:

$$\zeta =\mathsf{\Omega }(\eta )=c\eta +\mathsf{\Phi }(\eta )$$

(6)

In the formula: Φ(η) is a holomorphic function.

Equation (6) can be expressed in polar coordinates (r, β) and rearranged as follows:

$$\begin{array}{c}{N}_r+N_{\beta }=N_x+N_y,\\ N_{\beta }-N_r+2i{N}_{\eta \beta }=(N_y-N_x+2i{N}_{xy})\exp (2i\beta ),\\ M_{Q_r}-i{M}_{Q_{\beta }}=(M_{Q_x}-i{M}_{Q_y})\exp (i\beta ).\end{array}$$

(7)

Thus, in the plane η = ρexp(iθ), Equation (7) can be written as:

$$\begin{array}{l}{N}_{\rho }+N_{\theta }=(1-\nu )B\left\{{\displaystyle \frac{1-2\kappa }{\kappa }}-{\displaystyle \frac{1-\kappa }{\kappa }}\left[1+{\displaystyle \frac{h^2}{24}}({\alpha }_m^2-k_2^2)\right]{\alpha }_m^2{\delta }_m+{\displaystyle \frac{h^2}{24}}\left({\displaystyle \frac{2-3\kappa }{\kappa }}{\alpha }_m^2-{\displaystyle \frac{1-2\kappa }{\kappa }}{k}_2^2\right)\right\}E,\\ N_{\theta }-N_{\rho }+2\mathrm{i}{N}_{\rho \theta }=-4(1-\nu )B{\displaystyle \frac{{\eta }^2}{\mid \eta {\mid }^2}}{\displaystyle \frac{1}{\overline{\mathsf{\Omega }(\eta )}}}{\displaystyle \frac{\partial }{\partial \eta }}\left({\displaystyle \frac{1}{\mathsf{\Omega }(\eta )}}{\displaystyle \frac{\partial }{\partial \eta }}\right)\times \\ \left\{\left[{\displaystyle \frac{h^2}{24}}{\displaystyle \frac{1-\kappa }{\kappa }}+{\delta }_m-{\displaystyle \frac{h^2}{24}}{\displaystyle \frac{1-2\kappa }{\kappa }}({\alpha }_m^2-k_1^2){\delta }_m\right]E+\mathrm{i}f\right\}\\ M_{Q_{\rho }}-i{M}_{Q_{\theta }}={\displaystyle \frac{1}{2}}(1-\nu )D\left[{\displaystyle \frac{1}{\kappa }}({\alpha }_m^2-k_1^2){\delta }_m-{\displaystyle \frac{1-2\kappa }{\kappa }}\right]{\displaystyle \frac{\eta }{\rho |\mathsf{\Omega }(\eta )|}}{\displaystyle \frac{\partial }{\partial \eta }}E\end{array}$$

(8)

According to Equation (8), the expression for the generalized internal forces in a flat plate on the η plane can be derived as:

$$\begin{array}{l}\begin{array}{l}{N}_r={\displaystyle {\int }_{-h/2}^{h/2}{\sigma }_r}dz\\ =2\lambda {\displaystyle \frac{\sin \left(\frac{h}{2}{\square }_1\right)}{{\square }_1}}\left({\nabla }^{2}F+E\right)+4\mu {\displaystyle \frac{\partial }{\partial r}}\left[{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial \theta }}{\displaystyle \frac{\sin \left(\frac{h}{2}{\square }_2\right)}{{\square }_2}}f\right]\\ +4\mu \left({\nabla }^2-{\displaystyle \frac{{\partial }^2}{\partial r^2}}\right)\left[{\displaystyle \sum _{j=1}^2{\left(-1\right)}^{j-1}\frac{\sin \left(\frac{h}{2}{\square }_j\right)}{{\square }_j}}{T}_1^{-2}\left({\square }_1^{2}F+E\right)+{\displaystyle \frac{\sin \left(\frac{h}{2}{\square }_1\right)}{{\square }_1}}F\right]\end{array}\\ \begin{array}{l}{N}_{r\theta }={\displaystyle {\int }_{-h/2}^{h/2}{\tau }_{r\theta }}dz\\ =2\mu \left({\nabla }^2-2{\displaystyle \frac{{\partial }^2}{\partial r^2}}\right){\displaystyle \frac{\sin \left(\frac{h}{2}{\square }_2\right)}{{\square }_2}}f+4\mu \left({\displaystyle \frac{1}{r}}{\displaystyle \frac{{\partial }^2}{\partial \theta \partial r}}-{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{\partial }{\partial \theta }}\right)\\ \times \left[{\displaystyle \sum _{j=1}^2{\left(-1\right)}^{j-1}}{\displaystyle \frac{\sin \left(\frac{h}{2}{\square }_j\right)}{{\square }_j}}{T}_1^{-2}\left({\square }_1^{2}F+E\right)+{\displaystyle \frac{\sin \left(\frac{h}{2}{\square }_1\right)}{{\square }_1}}F\right]\end{array}\\ \begin{array}{l}{N}_{\theta }={\displaystyle {\int }_{-h/2}^{h/2}{\sigma }_{\theta }}dz\\ =2\lambda {\displaystyle \frac{\sin \left(\frac{h}{2}{\square }_1\right)}{{\square }_1}}\left({\nabla }^{2}F+E\right)+4\mu {\displaystyle \frac{1}{r^2}}{\displaystyle \frac{\partial }{\partial \theta }}{\displaystyle \frac{\sin \left(\frac{h}{2}{\square }_2\right)}{{\square }_2}}f+4\mu {\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial r}}\\ \times \left[{\displaystyle \sum _{j=1}^2{\left(-1\right)}^{j-1}\frac{\sin \left(\frac{h}{2}{\square }_j\right)}{{\square }_j}}{T}_1^{-2}\left({\square }_1^{2}F+E\right)+{\displaystyle \frac{\sin \left(\frac{h}{2}{\square }_1\right)}{{\square }_1}}F\right]\end{array}\\ \begin{array}{l}{Q}_r={\displaystyle {\int }_{-h/2}^{h/2}{\tau }_{zr}}dz=0\end{array}\\ \begin{array}{l}{Q}_{\theta }={\displaystyle {\int }_{-h/2}^{h/2}{\tau }_{z\theta }}dz=0\end{array}\end{array}$$

(9)

where ${\square }_j^2={\nabla }^2-T_j^2$ is the Lorentz operator; ${\nabla }^2=\frac{{\partial }^2}{\partial r^2}+\frac{1}{r}\frac{\partial }{\partial r}+\frac{1}{r^2}\frac{{\partial }^2}{\partial {\theta }^2}$.

2.2. Total Wavefield for Tension-Compression Scattering

Assume a tensile-compressive elastic wave incident along the positive x-axis on the tension side of the steel plate, expressed as:

$$E^{(i)}=E_0{\mathrm{e}}^{\mathrm{i}{\alpha }_{1}x}{F}^{(i)}={\delta }_1{E}^{(i)}{f}^{(i)}=0$$

(10)

where E₀ is the amplitude of the generalized displacement of the incident wave along the x-axis.

2.3. Boundary Conditions and Mode Coefficients

Assume that during tensile-compressive vibration on the tension side of the steel plate, the hole on the η plane is a free boundary. The plate theory satisfies the following six boundary conditions (three force and three moment conditions):

$$\begin{gathered} N_{\rho }^m{\mid }_{\rho =a} = 0 \\ {N}_{\rho \theta }^m{\mid }_{\rho =a} = 0 \\ {M}_{Q_{\rho }}^m{\mid }_{\rho =a} = 0 \end{gathered}$$

(11)

where m = 1, 2; a is the hole radius; L represents the characteristic length related to the plate geometry and boundary conditions in the conformal mapping function.

Substituting Equation (10) into Equation (11) yields an infinite system of algebraic equations determining $A_{1n}^{\left(1\right)}$, $A_{2n}^{\left(1\right)}$, $B_n^{\left(1\right)}$, $A_{1n}^{\left(2\right)}$, $A_{2n}^{\left(2\right)}$, and $B_n^{\left(2\right)}$:

$$\sum _{j=1}^6\sum _{n=-\infty }^{\infty }{E}_n{X}_n^j=E$$

(12)

Multiplying both sides of Equation (12) by e^−isθ yields the infinite system of algebraic equations:

$$\sum _{j=1}^6\sum _{n=-\infty }^{\infty }{E}_{ns}{X}_n^j=E_i$$

(13)

where $E_{ns}={\displaystyle \frac{1}{2\pi }}{\int }_{-\pi }^{\pi }{E}_n{\mathrm{e}}^{-\mathrm{is}{\theta }_j}\mathrm{d}{\theta }_j, E_s={\displaystyle \frac{1}{2\pi }}{\int }_{-\pi }^{\pi }{E}_i{\mathrm{e}}^{-\mathrm{is}{\theta }_j}\mathrm{d}{\theta }_j.$

$$\begin{array}{l}\begin{array}{l}{E}_n=\left[\begin{array}{cccccc}{E}_{11}^n& E_{12}^n& E_{13}^n& E_{14}^n& E_{15}^n& E_{16}^n\\ E_{21}^n& E_{22}^n& E_{23}^n& E_{24}^n& E_{25}^n& E_{26}^n\\ E_{31}^n& E_{32}^n& 0& E_{34}^n& E_{35}^n& E_{36}^n\\ E_{41}^n& E_{42}^n& E_{43}^n& E_{44}^n& E_{45}^n& E_{46}^n\\ E_{51}^n& E_{52}^n& E_{53}^n& E_{54}^n& E_{55}^n& E_{56}^n\\ E_{61}^n& E_{62}^n& E_{63}^n& E_{64}^n& E_{65}^n& 0\end{array}\right],\hspace{1em}{X}_n^j=\left[\begin{array}{c}{A}_{n1}^{(1)}\\ A_{n2}^{(1)}\\ B_n^{(1)}\\ A_{n1}^{(2)}\\ A_{n2}^{(2)}\\ B_n^{(2)}\end{array}\right],\hspace{1em}E=\left[\begin{array}{c}{E}_1\\ E_2\\ E_3\\ E_4\\ E_5\\ E_6\end{array}\right]\end{array}\\ \begin{array}{l}{E}_1=-\{{\displaystyle \frac{1-2\kappa }{\kappa }}-{\displaystyle \frac{1-\kappa }{\kappa }}[1+{\displaystyle \frac{h^2}{24}}({\alpha }_1^2-k_2^2)]{\alpha }_1^2{\delta }_1+{\displaystyle \frac{h^2}{24}}({\displaystyle \frac{2-3\kappa }{\kappa }}{\alpha }_1^2-{\displaystyle \frac{1-2\kappa }{\kappa }}{k}_2^2)-\\ \left.{\alpha }_1^2\left[{\displaystyle \frac{h^2}{24}}\left({\displaystyle \frac{1}{\kappa }}-1\right)+{\delta }_1-{\displaystyle \frac{h^2}{24}}{\displaystyle \frac{1-2\kappa }{\kappa }}\left({\alpha }_1^2-k_1^2\right){\delta }_1\right]\times \mathrm{Re}\left({\displaystyle \frac{{\eta }_1}{\overline{{\eta }_1}}}{\displaystyle \frac{ \mathsf{\Omega }\prime ({\eta }_1)}{\overline{ \mathsf{\Omega }\prime ({\eta }_1)}}}\right)\right\}{E}_0\exp (\mathrm{i}{\alpha }_1\mathrm{Re}(\mathsf{\Omega }({\eta }_1)))\end{array}\\ \begin{array}{l}{E}_2=\mathrm{i}{\alpha }_1^2[{\displaystyle \frac{h^2}{24}}({\displaystyle \frac{1}{\kappa }}-1)+{\delta }_1-{\displaystyle \frac{h^2}{24}}{\displaystyle \frac{1-2\kappa }{\kappa }}({\alpha }_1^2-k_1^2){\delta }_1]\times \mathrm{Im}({\displaystyle \frac{\eta }{\overline{\eta }}}{\displaystyle \frac{{\Omega }^{\prime }({\eta }_1)}{\overline{{\Omega }^{\prime }({\eta }_1)}}})E_0\exp (\mathrm{i}{\alpha }_1\mathrm{Re}(\mathsf{\Omega }({\eta }_1)))\end{array}\\ \begin{array}{l}{E}_3=-2\mathrm{i}{\alpha }_1\left[{\displaystyle \frac{1}{\kappa }}({\alpha }_m^2-k_1^2){\delta }_m-{\displaystyle \frac{1-2\kappa }{\kappa }}\right]\times \mathrm{Re}\left({\displaystyle \frac{{\eta }_1}{\rho }}{\displaystyle \frac{ \mathsf{\Omega }\prime ({\eta }_1)}{| \mathsf{\Omega }\prime ({\eta }_1)|}}\right)E_0\exp (\mathrm{i}{\alpha }_1\mathrm{Re}(\mathsf{\Omega }({\eta }_1)))\end{array}\\ \begin{array}{l}{E}_4=-\left\{{\displaystyle \frac{1-2\kappa }{\kappa }}-{\displaystyle \frac{1-\kappa }{\kappa }}\left[1+{\displaystyle \frac{h^2}{24}}({\alpha }_1^2-k_2^2)\right]{\alpha }_1^2{\delta }_1+{\displaystyle \frac{h^2}{24}}\left({\displaystyle \frac{2-3\kappa }{\kappa }}{\alpha }_1^2-{\displaystyle \frac{1-2\kappa }{\kappa }}{k}_2^2\right)-\right.\\ \left.{\alpha }_1^2\left[{\displaystyle \frac{h^2}{24}}\left({\displaystyle \frac{1}{\kappa }}-1\right)+{\delta }_1-{\displaystyle \frac{h^2}{24}}{\displaystyle \frac{1-2\kappa }{\kappa }}\left({\alpha }_1^2-k_1^2\right){\delta }_1\right]\times \mathrm{Re}\left({\displaystyle \frac{{\eta }_2}{\overline{{\eta }_2}}}{\displaystyle \frac{ \mathsf{\Omega }\prime ({\eta }_2)}{\overline{ \mathsf{\Omega }\prime ({\eta }_2)}}}\right)\right\}{E}_0\exp (\mathrm{i}{\alpha }_1\mathrm{Re}(\mathsf{\Omega }({\eta }_2)))\end{array}\\ \begin{array}{l}{E}_5=\mathrm{i}{\alpha }_1^2\left[{\displaystyle \frac{h^2}{24}}\left({\displaystyle \frac{1}{\kappa }}-1\right)+{\delta }_1-{\displaystyle \frac{h^2}{24}}{\displaystyle \frac{1-2\kappa }{\kappa }}({\alpha }_1^2-k_1^2){\delta }_1\right]\times \mathrm{Im}\left({\displaystyle \frac{{\eta }_2}{\overline{{\eta }_2}}}{\displaystyle \frac{ \mathsf{\Omega }\prime ({\eta }_2)}{\overline{ \mathsf{\Omega }\prime ({\eta }_2)}}}\right)E_0\exp (\mathrm{i}{\alpha }_1\mathrm{Re}(\mathsf{\Omega }({\eta }_2)))\end{array}\\ \begin{array}{l}{E}_6=-2\mathrm{i}{\alpha }_1\left[{\displaystyle \frac{1}{\kappa }}({\alpha }_m^2-k_1^2){\delta }_m-{\displaystyle \frac{1-2\kappa }{\kappa }}\right]\times \mathrm{Re}\left({\displaystyle \frac{{\eta }_2}{\rho }}{\displaystyle \frac{ \mathsf{\Omega }\prime ({\eta }_2)}{| \mathsf{\Omega }\prime ({\eta }_2)|}}\right)E_0\exp (\mathrm{i}{\alpha }_1\mathrm{Re}(\mathsf{\Omega }({\eta }_2)))\end{array}\\ \begin{array}{l}{\eta }_1=\exp (\mathrm{i}a{\theta }_1),\hspace{1em}{r}_2=\sqrt{a^2+d^2+2ad\sin {\theta }_1},\hspace{1em}{\theta }_2=\arccos \left({\displaystyle \frac{a\cos {\theta }_1}{r_2}}\right)\end{array}\\ \begin{array}{l}{\eta }_2=\exp (\mathrm{i}a{\theta }_2),\hspace{1em}{r}_1=\sqrt{a^2+d^2-2ad\sin {\theta }_2},\hspace{1em}{\theta }_1=-\arccos \left({\displaystyle \frac{a\cos {\theta }_2}{r_1}}\right)\end{array}\end{array}$$

(14)

The expression for the incident wave at the boundary is:

$${\sigma }_{\theta }^{(i)}={\mu }_M\left\{\left({\displaystyle \frac{1-2\kappa }{\kappa }}-{\displaystyle \frac{1-\kappa }{\kappa }}{\alpha }_1^2{\delta }_1\right)+{\alpha }_1^2{\delta }_1\mathrm{Re}\left({\displaystyle \frac{{\eta }_i^2}{{\rho }_i^2}}{\displaystyle \frac{ \mathsf{\Omega }\prime ({\eta }_i)}{\overline{ \mathsf{\Omega }\prime ({\eta }_i)}}}\right)\right\}\exp \left[\mathrm{i}{\alpha }_1\mathrm{Re}(\mathsf{\Omega }({\eta }_i))\right]$$

(15)

The expression for the scattered wave at the j-th aperture boundary is:

$$\begin{array}{l}{\sigma }_{\theta }^{(s)}={\mu }_M{\displaystyle \sum _{j=1}^2\sum _{m=1}^2\sum _{n=-\infty }^{\infty }}\left({\displaystyle \frac{1-2\kappa }{\kappa }}-{\displaystyle \frac{1-\kappa }{\kappa }}{\alpha }_m^2{\delta }_m\right)A_{mn}^{(j)}{H}_n^{(1)}\left({\alpha }_m{r}_j\right){\mathrm{e}}^{\mathrm{i}n{\theta }_j}-\\ 2{\mu }_M\left\{{\displaystyle \sum _{j=1}^2\sum _{m=1}^2\sum _{n=-\infty }^{\infty }}{A}_{mn}^{(j)}{\displaystyle \frac{{\alpha }_m^2{\delta }_m}{4}}\right.\left[{\displaystyle \frac{{\eta }_i^2}{{\rho }_i^2}}{\displaystyle \frac{\mathsf{\Omega }({\eta }_i)}{\overline{\mathsf{\Omega }({\eta }_i)}}}{H}_{n-2}^{(1)}({\alpha }_m{r}_j){\mathrm{e}}^{\mathrm{i}(n-2){\theta }_j}+{\displaystyle \frac{{\overline{{\eta }_i}}^2}{{\rho }_i^2}}{\displaystyle \frac{\overline{\mathsf{\Omega }({\eta }_i)}}{\mathsf{\Omega }({\eta }_i)}}{H}_{n+2}^{(1)}({\alpha }_m{r}_j){\mathrm{e}}^{\mathrm{i}(n+2){\theta }_j}\right]\\ +\left.{\displaystyle \frac{k_2^2}{4}}\mathrm{i}{\displaystyle \sum _{j=1}^2\sum _{n=-\infty }^{\infty }}{B}_n^{(j)}\left[{\displaystyle \frac{{\eta }_i^2}{{\rho }_i^2}}{\displaystyle \frac{ \mathsf{\Omega }\prime ({\eta }_i)}{\overline{ \mathsf{\Omega }\prime ({\eta }_i)}}}{K}_{n-2}(k_2{r}_j){\mathrm{e}}^{\mathrm{i}(n-2){\theta }_j}-{\displaystyle \frac{{\overline{\eta }}_i^2}{{\rho }_i^2}}{\displaystyle \frac{\overline{ \mathsf{\Omega }\prime ({\eta }_i)}}{ \mathsf{\Omega }\prime ({\eta }_i)}}{K}_{n+2}(k_2{r}_j){\mathrm{e}}^{\mathrm{i}(n+2){\theta }_j}\right]\right\}\end{array}$$

(16)

The expression for the dynamic stress concentration factor is:

$${\sigma }_{\theta }^{\ast }={\displaystyle \frac{{\sigma }_{\theta }}{{\sigma }_0}}={\displaystyle \frac{{\sigma }_{\theta }^{(i)}+{\sigma }_{\theta }^{(s)}}{{\sigma }_0}}$$

(17)

where σ₀ is the stress amplitude of the incident wave along the positive x-direction, ${\sigma }_0=\left[{\lambda }_M(1-{\delta }_1{\alpha }_1^2)-2{\mu }_M{\delta }_1{\alpha }_1^2\right]E_0.$

Unlike previous studies that only considered short-term impact loads from 0 to t and numerical simulations using finite element methods, this study presents a complete solution derived explicitly under dynamic elastic conditions.

2.4. Numerical Example

With n = 10, Poisson’s ratio ν = 0.3, and hole-to-plate thickness ratio a/h = 0.1 to 5.0, and dimensionless wave number α₁a = 0.1 to 5.0, the distribution of the dynamic stress concentration factor along the hole edge is shown in Figure 1. The upper half depicts the numerical distribution of the dynamic stress concentration factor along the hole edge at t = 0, while the lower half shows the distribution at t = T/4. Figure 2 illustrates the variation curve of the dynamic stress concentration factor with respect to the incident wave number.

Comparing Figure 1a,b shows that at α₁a = 0.1, the dynamic stress concentration factor remains relatively stable, exhibiting a maximum value at θ = π/2, with the numerical distribution being roughly symmetric about θ = π/2. Comparing Figure 1a,c indicates that at a/h = 0.1 and L/a = 2.1, an increase in α₁a causes gradual change in the distribution of the dynamic stress concentration factor. The maximum value remains at θ = π/2, but negative stresses appear at θ = −2π/3. Comparing Figure 1b,e shows that at a/h = 5.0 and L/a = 4.0, as α₁a increases, the distribution of the dynamic stress concentration factor gradually changes, the maximum value remains at θ = π/2, but negative stresses appear at both θ = −2π/3 and θ = π. Comparing Figure 1c,d reveals that with a/h and L/a unchanged, an increase in α₁a causes irregular changes in the distribution of dynamic stress concentration factors, with negative stresses appearing in multiple directions, significantly affecting component stability. Comparison of Figure 1d,f shows that at relatively large α₁a but small a/h, fluctuations in the dynamic stress concentration factors remain intense.

The theoretical distributions in Figure 1 represent dynamic stress concentration under ideal elastic wave incidence in an infinite plate, serving to identify the critical angles (θ = 90°) where stress concentration is most severe. While the experimental setup involves finite plates and alternating tension–compression loading (rather than harmonic wave incidence), the predicted hotspot angle (≈90°) aligns consistently with the experimental strain concentration location (Section 4.3), thereby validating the theoretical identification of weak zones. The anisotropic effect of bonded CFRP is not incorporated in the current analytical model; its efficacy is evaluated through comparative experimental quantification of strain redistribution and fatigue life enhancement.

As shown in Figure 2, when the incident wave number and the ratio of aperture diameter to plate thickness are small, the maximum dynamic stress concentration factor reaches 3.3. As the incident wave number and the aperture-to-plate-thickness ratio gradually increase, the fluctuation of the dynamic stress concentration factor intensifies, eventually approaching 1.

The theoretical analysis above provides the dynamic stress concentration in the isotropic steel plate without reinforcement. The following section introduces the constitutive model for the anisotropic CFRP reinforcement layer, crucial for understanding the load-sharing mechanism in the composite system.

2.5. Constitutive Modeling of Anisotropic CFRP Reinforcement Layer

Externally bonded CFRP is a typical orthotropic material. Its constitutive relationship in the principal material coordinates (1-direction along fibers, 2-direction transverse) is given by:

$$\left[\begin{array}{c}{\sigma }_{11}\\ {\sigma }_{22}\\ {\tau }_{12}\end{array}\right]=\left[\begin{array}{ccc}{Q}_{11}& Q_{12}& 0\\ Q_{12}& Q_{22}& 0\\ 0& 0& Q_{66}\end{array}\right]\left[\begin{array}{c}{\epsilon }_{11}\\ {\epsilon }_{22}\\ {\gamma }_{12}\end{array}\right]$$

(18)

where Q_ij are the reduced stiffness components, expressible in terms of the engineering constants: E₁, E₂ (Young’s moduli), ν₁₂ (major Poisson’s ratio), and G₁₂ (in-plane shear modulus). The explicit expressions are [28]:

$$Q_{11}={\displaystyle \frac{E_1}{1-{\nu }_{12}{\nu }_{21}}}, Q_{22}={\displaystyle \frac{E_2}{1-{\nu }_{12}{\nu }_{21}}}, Q_{12}={\displaystyle \frac{{\nu }_{12}{E}_2}{1-{\nu }_{12}{\nu }_{21}}}, Q_{66}=G_{12}$$

(19)

where ν₂₁ = ν₁₂(E₂/E₁). For unidirectional CFRP with fibers aligned circumferentially (hoop direction) around the hole, the 1-direction coincides with the hoop direction (θ), and the 2-direction with the radial direction (r).

Theoretical Treatment in the Composite System:

In the proposed “steel–adhesive–CFRP” composite plate under alternating tension–compression, the dynamic response can be conceptually decomposed into two coupled problems:

(1)

Stress concentration in the isotropic steel plate (solved via Section 2.1, Section 2.2, Section 2.3 and Section 2.4), providing the driving stress field (${\sigma }_{\theta \theta }^{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{e}\mathrm{l}}$) at the potential hotspot (θ ≈ 90°).

(2)

Load sharing and constraint provided by the bonded anisotropic CFRP layer. The CFRP, due to its high stiffness in the fiber (hoop) direction, directly reduces the hoop strain in the steel at the hotspot. This effect can be represented in a simplified analytical form by considering force equilibrium and strain compatibility at the steel–adhesive interface:

$${\sigma }_{\theta \theta }^{steel}\cdot t_{steel}+{\sigma }_{\theta \theta }^{CFRP}\cdot t_{CFRP}={\sigma }_{\theta \theta }^{effective}\cdot (t_{steel}+t_{CFRP})$$

(20)

where ${\sigma }_{\theta \theta }^{\mathrm{C}\mathrm{F}\mathrm{R}\mathrm{P}}=Q_{11}{\epsilon }_{\theta \theta }^{\mathrm{C}\mathrm{F}\mathrm{R}\mathrm{P}}$ (assuming fibers dominate hoop stiffness), and ${\epsilon }_{\theta \theta }^{\mathrm{C}\mathrm{F}\mathrm{R}\mathrm{P}}\approx {\epsilon }_{\theta \theta }^{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{e}\mathrm{l}}$ at the interface (perfect bond assumption for elastic stage). This leads to an effective reduction in the dynamic stress concentration factor (DSCF) at the steel surface. It is acknowledged that CFRP strengthening is typically tension-dominant [29]. Under compression cycles, the CFRP may experience fiber micro-buckling or reduced effective stiffness, and the adhesive interface faces different shear stress states. The current model assumes linear elastic behavior for both materials under alternating loads, focusing on the overall load-sharing effect observed in the elastic range of the steel. A more detailed model incorporating compression-specific degradation is a subject for future study.

While a complete closed-form solution for wave scattering in a finite, anisotropic composite plate is prohibitively complex, the presented theoretical framework identifies the critical hotspot based on base plate mechanics and provides a clear stiffness-based mechanism for CFRP efficacy. The potential for interface degradation, stress redistribution away from the hole, and the evolution of load transfer mechanisms during fatigue are assessed experimentally. Quantitative validation of this mechanism—i.e., the reduction in strain concentration at the predicted angle and subsequent fatigue life extension—is rigorously provided by the experimental results in Section 4.3, Section 4.4 and Section 4.5.

To validate the theoretical load-sharing mechanism, the predicted hoop strain reduction at θ ≈ 90° from Equation (20) was compared with experimental normalized strain data at N = 10³ cycles. For Δσ = 320 MPa, the theoretical model predicted a 34% reduction in ε_θθ at the hotspot, while the experimental reduction was 29% ± 4%, showing good agreement within the elastic range. This consistency supports use of the orthotropic constitutive model for guiding reinforcement design under dynamic loading.

3. Experimental Setup

3.1. Material Properties

Test specimens were fabricated from hot-rolled Q355 steel plates using unidirectional pultruded carbon fiber-reinforced polymer (CFRP). A two-component epoxy structural reinforcement adhesive, Sikadur-30CN (S30) [30], was employed for CFRP bonding. Bonded test specimens underwent curing at room temperature for 7 days. Tensile tests were conducted on the materials according to testing standards [31,32,33]. Primary mechanical properties of the materials are shown in

Table 1. Mechanical Properties of Tested Materials.

Material	Thickness (mm)	Young’s Modulus (GPa)	Tensile Strength (MPa)	Yield Stress (MPa)	Poisson’s Ratio
steel	6	210	414	360	0.3
S30	1	2.97	40	-	0.35
CFRP	0.15	164	2610	-	0.3

. Figure 3 shows close-up photos of the CFRP sheet and the S30 epoxy adhesive used in this study.

CFRP sheets were bonded using the two-component epoxy adhesive (Sikadur-30CN) under controlled laboratory conditions (temperature: 23 ± 2 °C, relative humidity: 50 ± 5%). The adhesive was applied uniformly with a notched trowel to ensure a consistent bond line thickness of 1.0 ± 0.1 mm. Curing was carried out at room temperature for 7 days, followed by post-curing at 40 °C for 24 h to achieve full mechanical properties.

3.2. Sample Preparation

To obtain test specimens with precise aperture dimensions and smooth edges, holes were cut in the steel plate using a fiber laser cutting machine (FSCUT4000, Huagong Laser Technology Co., Ltd., Wuhan, China). Laser power was set to 2 kW, with a cutting speed of approximately 600 mm/min. The area surrounding the cut holes on the steel plate was ground, and both CFRP and steel plate surfaces were cleaned with acetone to achieve a clean, rough surface. Steel plate dimensions were 360 × 120 × 6 mm. A circular hole (diameter D = 40 mm) was machined at the plate center, as shown in Figure 3. Two comparison groups were used: unreinforced bare steel and externally bonded CFRP reinforcement, with three specimens per group.

Specimen dimensions and hole diameter were selected based on typical geometric ratios in bridge gusset plates and tubular joint connections [34,35], where the hole-diameter-to-plate-width ratio (D/W ≈ 0.333) and hole-diameter-to-plate-thickness ratio (D/t ≈ 6.67) fall within common engineering ranges. Selected stress levels (Δσ = 240 MPa and 320 MPa) correspond to approximately 0.58 and 0.77 of the yield strength (σ_y = 414 MPa), representing high-cycle fatigue regimes relevant to practical alternating load scenarios such as wind, traffic, or wave-induced vibrations in structures [36,37].

CFRP was applied using unidirectional fabric with symmetrical bonding on both sides. To monitor strain around the opening, uniaxial strain gages (A_θ) were radially bonded along θ = 0°, 30° … 360° (with tangential orientation to the circumference) on one side of the steel plate, centered at the hole with a radius r = 20 mm (10 mm from the hole edge). θ represents the angular position of the gage, as shown in Figure 4. One axial strain gauge was placed at the plate’s neutral axis, away from the hole edge and fixture (≥2 mm from hole edge), to measure nominal strain ε_nom. For the CFRP group, the “strain gauges first, then fabric” method was applied, with strain gauges positioned at the steel-adhesive interface and wires routed radially outward.

Terminal-free, solderless strain gauges (with pre-attached lead wires) (120-3AA, Shenzhen Luojia Technology Co., Ltd., Shenzhen, China) were bonded using a cyanoacrylate adhesive after surface preparation, including grinding, acetone cleaning, and neutralization. Gauge alignment was verified using a microscope to ensure tangential orientation within ±1°. For CFRP-reinforced specimens, gauges were installed before CFRP application, with wires routed through small grooves in the adhesive layer to avoid interfacial disturbance (as shown in Figure 5). Post-test inspection confirmed no debonding initiation at gauge locations, indicating negligible interference with the steel–adhesive–CFRP load transfer mechanism.

Specimen types are denoted as “H1-B0” for bare steel plates with holes and “H1-D3L” for CFRP-reinforced steel plates with holes. H indicates holes, B0 denotes no CFRP reinforcement, and D3L signifies reinforcement with three layers of CFRP.

3.3. Loading Procedure

The electro-hydraulic servo testing machine press applied σ(t) = σasin (2πft) under force control, where σ_a = Δσ/2, with a stress ratio of −1 and a frequency of 10 Hz. The frequency of 10 Hz was selected as a compromise between test efficiency and minimizing potential frequency-dependent effects while still representing a realistic range for many structural fatigue scenarios induced by mechanical vibrations or cyclic traffic loads [38]. Two design amplitude levels were employed: ±120 MPa (Δσ = 240 MPa) and ±160 MPa (Δσ = 320 MPa). Observation windows were set at N = 10³, 10⁴, and 10⁵ to calculate the mean and standard deviation of peak-to-peak stress amplitude over 10s intervals. Incoming strain (closed-loop channel), actuator force and displacement, and circumferential strain at the hole edge were recorded.

The sinusoidal waveform was verified prior to testing using an oscilloscope connected to the load cell output. The recorded waveform showed a harmonic distortion of less than 2%, confirming the purity of the alternating tension–compression loading.

Testing terminated upon meeting any of the following conditions, which are standard criteria in fatigue testing to ensure consistent failure definition and prevent overloading the testing machine:

①

Visible main crack length at the hole edge ≥ 1 mm;

②

Stiffness reduction ≥ 5% (change in displacement amplitude/force amplitude ratio);

③

Amplitude tracking error exceeding ±2 MPa for 10 consecutive seconds.

3.4. Test Setup and Monitoring

An SDZ-0100 electro-hydraulic servo fatigue testing machine (Changchun Zhongji Testing Co., Ltd., Changchun, China) was used. The machine has a maximum dynamic load capacity of ±100 kN, an operating frequency range of 0.1–100 Hz, and a stroke of ±50 mm. It is capable of performing tensile and compressive tests, high-cycle and low-cycle fatigue tests, as well as conventional mechanical property tests, as shown in Figure 6. Crack initiation and propagation were monitored using a digital microscope (50× magnification) at each observation interval (N = 10³, 10⁴, 10⁵ cycles). Acoustic emission (AE) sensors were placed near the hole to detect early damage in the CFRP layer (as shown in Figure 7). Strain data from both steel and CFRP surfaces were synchronously collected via a dynamic strain analyzer.

Crack observation followed a standardized procedure: at each observation window, loading was paused, and the hole edge was examined microscopically. A crack was recorded as initiated when its length exceeded 1 mm. AE activity was continuously monitored to identify early damage events such as fiber breakage or adhesive microcracking. This multi-method approach ensured reliable tracking of both steel crack growth and CFRP system degradation throughout the fatigue process.

4. Test Results

4.1. Failure Mode

Testing employed Δσ = 240 MPa and 320 MPa equivalent tensile-compressive wave loading. Specimen H1-B0 exhibited a consistent overall failure path: crack initiation at the hole edge → propagation along the minimum net cross-section → net cross-section fracture (Figure 8a). Figure 8b shows a close-up of the final fracture surface of specimen H1-B0. The crack originates approximately 90° from the hole edge and propagates along the net cross-section until fracture.

Specimen H1-D3L exhibited a composite failure mode involving steel–adhesive–fiber interaction. Due to the strong local bond strength between CFRP and both the steel plate and adhesive, CFRP fiber rupture (FR), longitudinal cracking along fiber orientation (LS), and random bundle-level rupture (RR) occurred at the steel plate failure site. This damage redistributed loads between steel and reinforcement layers, significantly delaying crack initiation and reducing propagation rates. Steel plate cracking was substantially postponed, ultimately evolving into composite failure characterized by “damage within reinforcement + steel plate penetration.” (Figure 8c,d). Figure 8e provides a detailed close-up view of the final fracture zone in specimen H1-D3L, clearly illustrating the synergistic failure interaction between the CFRP reinforcement and the underlying steel substrate. The term “retarded crack” is used here to describe the significantly delayed crack initiation and slowed propagation in the reinforced specimen.

Acoustic emission data indicated early fiber–matrix micro-damage in CFRP starting at approximately 60% of total fatigue life, followed by localized fiber rupture near the hole edge. Strain readings from gauges beneath CFRP remained consistent until near-final failure, suggesting that interface degradation did not significantly affect strain measurement reliability until late stages.

4.2. Strain Response Around the Hole

To eliminate the influence of minor fluctuations in loading amplitude and temperature drift on the results, this study employs the “normalized circumferential strain amplitude” as a comparative metric. The specific procedure is as follows: near a given number of cycles N, data is segmented into discrete complete cycles using the zero crossings of the loading signal. For each of 10 consecutive cycles, the “half peak-to-peak” amplitude (i.e., half the difference between the maximum and minimum values of each cycle) is recorded. For both unreinforced and reinforced groups, the amplitude at the same angle and the same N0 of their respective specimens is used as the denominator.

$$A_n={\displaystyle \frac{\underset{t}{\max }{\epsilon }_{\theta }(t)-\underset{t}{\min }{\epsilon }_{\theta }(t)}{2}}, \hspace{1em}n=1, \dots , 10$$

(21)

The average is then calculated to obtain the representative amplitude at the specified angle θ around the borehole for that N.

$$A_{\theta }(N)={\displaystyle \frac{1}{10}}\sum _{n=1}^{10}{A}_n$$

(22)

Baseline definition: Select the baseline moment N₀ = 10³ (when the equipment is stable and no cracks have initiated). Place baseline strain gauges on the same specimen far from the hole and calculate the average amplitude over 10 cycles using the same method.

$$A_{\theta }(N_0)={\displaystyle \frac{1}{10}}\sum _{n=1}^{10}{A}_n^{(N_0)}$$

(23)

This serves as the baseline amplitude for that angle. Thus, the normalized circumferential strain amplitude is defined as:

$$R(\theta ,N)={\displaystyle \frac{A_{\theta }(N)}{A_{\theta }(N_0)}}$$

(24)

Figure 9 shows the evolution of normalized circumferential strain versus N for different cycle nodes. All data are normalized to the 10-cycle average amplitude at baseline time N₀ = 10³ for the same angle, with consistent aperture for both groups. At Δσ = 240 MPa, the most pronounced decrease in R(90, N) occurs at 90°. Performing first-order linear regression on ${\log }_{10}\mathrm{N}$ yields corresponding logarithmic domain slopes of −0.132/dec for H1-B0 and −0.031/dec for H1-D3L. Reinforcement reduces the degradation rate by approximately 3.7-fold, with maximum baseline deviation reduction reaching 62.8%. At Δσ = 320 MPa, corresponding logarithmic domain slopes for H1-B0 and H1-D3L were −0.191/dec and −0.035/dec, respectively. Reinforcement reduced the degradation rate by approximately 5.5-fold, with maximum baseline deviation reduction reaching 74.9%. As load amplitude increases, the degradation rate of the reinforced specimen decreases more significantly, leading to hysteresis phenomena.

The 0° curve closely follows the baseline and remains nearly constant with N, indicating weak stress concentration in this direction. The 90° curve declines most rapidly with N, revealing the highest localized stress concentration at this angle along the hole edge.

4.3. Dynamic Stress Concentration Angle

Figure 10 and Figure 11 present the normalized circumferential strain polar distributions at various angles for H1-B0 and H1-D3L specimens. To quantify the degree of “curve roundness/uniformity,” this study introduces the roundness index C(N) and the stability criterion for the hotspot angle θ*(N):

$$R_{\max }(N)=\underset{\theta }{\max }R(\theta ,N),\hspace{1em}{R}_{\min }(N)=\underset{\theta }{\min }R(\theta ,N),\hspace{1em}{\theta }^{\ast }(N)=\arg \underset{\theta }{\max }R(\theta ,N)$$

(25)

$$\mathrm{Roundness}:C(N)=1-{\displaystyle \frac{R_{\max }(N)-R_{\min }(N)}{R_{\max }(N)}}\in (0, 1]$$

(26)

Results indicate that the unreinforced H1-B0 specimen exhibits significant inward contraction and elliptical deformation during cycling, demonstrating enhanced anisotropy. In contrast, the reinforced H1-D3L specimen shows outward expansion and greater circularity, with peak values in the 90°/270° (±5°) neighborhood blunted and angular differences significantly reduced. The roundness coefficient C(N) of the reinforced specimen increased by only about 0.03 compared to the unreinforced specimen in early stages, but the gap widened rapidly with cycling. By N ≈ 2.1 × 10⁵, improvement reached approximately 0.26, indicating significant suppression of anisotropy. As stress amplitude Δσ increases, stress concentration and anisotropy in unreinforced specimens are amplified; whereas the “blunting-rounding-hysteresis” effect of CFRP becomes more pronounced. Notably, the hot spot angle consistently remains near the theoretically weak angle (≈90°), indicating that CFRP does not alter the hot spot direction but effectively reduces its severity.

Linkage to Theoretical CFRP Constitutive Model:

The observed “blunting” of the strain polar plot and increase in roundness index C(N) for the CFRP-reinforced specimen provide direct experimental validation of the load-sharing mechanism described by the anisotropic constitutive model in Section 2.5. The CFRP’s high stiffness in the hoop direction (Q₁₁) directly counteracts peak tensile and compressive strains at the critical angle (θ ≈ 90°), leading to a more uniform strain distribution around the hole. This experimental evidence confirms that the CFRP layer functions as an effective orthotropic constraint, reducing the dynamic stress concentration factor (DSCF) in the underlying steel plate as conceptualized in Equation (20). The stability of the hotspot angle further confirms that reinforcement mitigates the effect without shifting the fundamental weak zone identified by base plate theory.

4.4. Frequency-Domain Characteristics of Dynamic Stress Concentration

Figure 12 shows the variation in the dynamic stress concentration factor (Δσ) with wave number at Δσ = 320 MPa for the specimen, compared with the theoretical curve. Compared to theory, the unreinforced specimen exhibits higher peak-to-trough oscillations in the low-to-mid wave number range. This occurs primarily because theoretical models often assume ideal boundaries or infinite plates, whereas actual specimens are finite plates influenced by fixtures, making them more prone to reflections and standing waves in the low-to-mid wave number range [15]. Near the first minimum, the valley depth of H1-D3L is markedly shallower than that of H1-B0, indicating significant suppression of low-order dynamic amplification. Subsequently, across all peak-to-valley oscillations, the H1-D3L curve exhibits lower peaks and faster amplitude decay. This demonstrates that CFRP wrapping reduces dynamic concentration and enhances response stability across a broad wave number range.

4.5. Fatigue Life

Investigating the fatigue life of structural components is crucial for ensuring engineering safety and reliability [39]. Since fatigue life is influenced by multiple parameters under cyclic loading, its response mechanism is complex and typically requires empirical methods for description and simulation. The most widely used method for determining fatigue life is the stress-life curve (S-N curve) [5]. This curve reflects the relationship between stress amplitude in the far field and number of cycles to failure. To quantify fatigue performance of specimens under cyclic loading, this study employs the Basquin relationship to perform regression analysis on scatter data in the logarithmic domain. The stress range Δσ_e (MPa) and number of cycles N are expressed as:

$${\log }_{10}N={\log }_{10}C-m{\log }_{10}\mathsf{\Delta }{\sigma }_e$$

(27)

where m is the equivalent slope (Basquin index) and C is the material’s scale parameter.

Figure 13 presents the S-N curve for the specimens. Under both stress amplitude levels, the S–N data for the H1-D3L group generally exceed those of the H1-B0 group. While slopes m of both groups are approximately consistent, intercepts exhibit positive shifts, indicating that reinforcement primarily manifests as an increase in allowable stress amplitude at equal life and extension of life at equal amplitude. Taking N = 10⁶ as an example, the allowable stress amplitude of H1-D3L increased by approximately 45% compared to H1-B0. Under stress amplitudes of Δσ = 320 MPa and 240 MPa, fatigue life of the steel plate increased by approximately 3.8 times and 5.2 times, respectively.

Fatigue life data are presented as mean ± standard deviation (n = 3). The coefficient of variation (COV) for the H1-B0 and H1-D3L groups was 12.3% and 9.7%, respectively. Sample size was determined based on a pre-test power analysis (α = 0.05, β = 0.2) to detect a minimum life improvement of 50% with 80% statistical power.

Overall, three-layer CFRP wrapping elevates fatigue performance of the perforated plate from near Category-C to meet Category-B standards, approaching compliance with design line requirements of GB 50017-2017.

5. Conclusions

This study establishes a comprehensive “theoretical identification—targeted reinforcement—experimental quantification” framework to enhance fatigue performance of open-hole steel plates under alternating tension–compression loading. The main findings are summarized as follows:

A coupled theoretical framework was developed, combining dynamic stress analysis for isotropic perforated plates (using complex function theory and conformal mapping) with an orthotropic constitutive model for the CFRP reinforcement layer. This framework successfully identifies the critical weak zone at θ* ≈ 90° ± 5°, which remained stable throughout fatigue cycles and was consistently corroborated by the experimental hotspot location.

Targeted double-sided triple-layer CFRP reinforcement significantly mitigated dynamic stress concentration. At Δσ = 320 MPa, CFRP reduced the logarithmic degradation rate of normalized strain at 90° by approximately 5.5 times (from −0.191/dec to −0.035/dec) and reduced baseline deviation by up to 74.9%. A similar strong effect was observed at Δσ = 240 MPa.

CFRP reinforcement effectively suppressed anisotropy of strain distribution around the hole. The roundness index C(N) of the polar strain plot for the reinforced specimen improved by 0.26 over the fatigue life, demonstrating a pronounced “blunting” effect that smoothed circumferential strain distribution while keeping the hotspot angle stationary.

The anisotropic CFRP layer provided substantial fatigue life extension. At an equivalent life of N = 10⁶ cycles, allowable stress amplitude increased by approximately 45%. Under constant stress amplitudes of Δσ = 240 MPa and 320 MPa, fatigue life improved by 5.2 times and 3.8 times, respectively, elevating performance from near Category-C to meet Category-B standards.

The study bridges the gap between isotropic base-plate theory and anisotropic reinforcement action. While a full-wave scattering solution in the composite plate is analytically complex [40], the proposed approach—using theory to identify the critical zone and explain the load-sharing mechanism, followed by rigorous experimental validation—provides a practical and reliable pathway for engineering applications.

This work demonstrates that a mechanics-guided, hotspot-targeted CFRP reinforcement strategy is highly effective in enhancing fatigue resistance of perforated steel components under severe alternating loads, offering valuable design insights for bridges, towers, and other critical infrastructure.

It is acknowledged that the theoretical model assumes an infinite isotropic plate under harmonic wave incidence, while the experiments used finite plates under sinusoidal mechanical loading. This discrepancy may lead to differences in wave reflection and boundary effects, particularly at low wave numbers. However, consistent identification of the hotspot angle (θ ≈ 90°) and quantitative strain reduction validate the practical utility of the theoretical framework for guiding targeted reinforcement in engineering applications.