An Artificial Neural Network-Based Strategy for Predicting Multiaxial Fatigue Damage to Welded Steel Structures

Abstract

Fatigue failure constitutes an issue that cannot be ignored when designing welded steel structures due to the initiation of cracks at weld toes and defects under cyclic loading conditions. Traditional methods, such as the notch stress approach, estimate fatigue life by modeling local stress distributions using idealized weld geometries. While these methods are widely accepted in design codes, they can be limited by complexity and reduced accuracy in real-world applications. This study explores the use of artificial neural networks (ANNs) to enhance fatigue life prediction through data-driven modeling. The proposed method involves training an ANN using synthetic data generated through finite element simulations of S355 steel weldments under various loading histories, rates, and frequencies. The objective is to capture the influence of local geometric and stress features without relying solely on assumptions used in conventional approaches. The FEM-based training data incorporate both classical experimental findings and validated modeling practices. While performance evaluation of the ANN model is reserved for future work, this study lays the groundwork for replacing or supplementing the notch stress approach with a more adaptable and efficient predictive tool. The integration of machine learning into fatigue assessment has the potential to improve reliability, reduce computational burden, and support more informed maintenance and design decisions.

1. Introduction

Material fatigue is the damage or failure of materials and components under time-varying, frequently repeated stress loading. Material failure tends to occur around defects, notches, and cross-sectional transitions after a small or large number of oscillation cycles. The cracks increase with further oscillation cycles, and finally, the remaining fracture occurs. Notably, this occurs at a load level well below the static strength [1]. Thus, fatigue represents one of the most formidable challenges faced by designers and structural engineers when dealing with welded steel structures.

To address this issue, extensive research has explored methodologies to predict the anticipated service life of critical welded components. These methodologies fall under the category of “local methods”, as they primarily impact the area surrounding the weld joint [2]. They are designed to counteract the adverse effects of welding, such as residual stress and the local stress concentration, which weaken the welded region. Local methods can be classified into two prominent families based on their primary influence. The first family comprises methods that alter the stress distribution near the weld toe, thereby creating a favorable compressive field. An example is the high-frequency mechanical impact treatment (HFMI treatment) [3]. The second family focuses on improving the topography of the weld toe to reduce the stress concentration factor, as seen in methods such as Tungsten Gas remelting (TIG) [4].

The literature review of experiments on the fatigue lifetime of S355 steel can be classified into experimental methods and experimental specimens. The experimental data were obtained from two relevant sources. Olivier and Ritter [5] provide a broad overview of SN curves of weldment joints. In the past decades, several advanced local stress-based approaches have been developed and validated to improve the accuracy of estimating the fatigue lifetime of welded connections. Among these approaches, the notch stress approach is gaining increasing attention in industrial applications and finding access to fatigue design recommendations. The basic idea of this approach is to model the weld toe or root with a reference radius, thereby evaluating the local principal or von Mises stress and comparing it against allowable values. This method is state-of-the-art according to Eurocode 3 [6] and also IIW recommendations [7]. This method assumes that fatigue strength can be estimated accurately using linear-elastic notch stresses determined by rounding either the weld toes or the weld roots. By taking advantage of the micro-support theory proposed by Neuber to model sharp cracks, Radaj proposed to use a fictitious weld toe/root radius, r, of 1 mm to assess the fatigue strength of welded connections with a thickness larger than (or equal to) 5 mm. Contrarily, for thin-welded details with thicknesses lower than 5 mm, a fictitious radius, r, of $0.05$ mm is recommended by Morgenstern et al. [8]. It is worth noting that this recommendation is not supported by the IIW recommendations [7].

In the finite lifetime area, the fatigue behaviour of welded high-strength steel joints is most beneficial due to the high yield strength of the base material. In the high-cycle fatigue zone, the notch topography, microstructure in the heat-affected zone, and residual stress state significantly influence the fatigue lifetime. According to the International Institute of Welding (IIW) recommendations [7], the fatigue life of welded steel components is independent of their yield strength. However, additional high-frequency mechanical impact (HFMI) treatment offers the possibility to increase fatigue life, especially for high-strength steel.

Fatigue analysis is typically conducted using empirical or semi-empirical approaches. One of the most classical methods is the Wöhler curve, also known as the S–N curve [9], which relates the stress amplitude of cyclic loading to the number of cycles to failure. This approach is based on extensive experimental data and is widely used for estimating fatigue initiation life. To account for the effects of mean stress and cyclic plasticity on fatigue life, models such as the Morrow model [10] and the Smith–Watson–Topper (SWT) model [11] have been introduced. These models incorporate contributions from plastic strain and maximum stress to more accurately predict fatigue initiation. For welded structures, four main methodologies are commonly used to assess fatigue performance [7]: the nominal stress approach, the hot spot stress approach, the effective notch stress approach, and the fracture mechanics-based approach. Except for the fracture mechanics method, these approaches typically rely on S–N curves derived from material-specific fatigue data. Beyond empirical and semi-empirical methods, fatigue life can also be estimated using physics-based approaches. Assuming isotropic and ductile damage, Lemaitre [12] proposed a phenomenological continuum damage mechanics (CDM) model for fatigue prediction. Building upon this, Dhar et al. [13] integrated CDM with large deformation theory to simulate crack initiation in ductile materials. Fatigue crack propagation can further be simulated using numerical methods. However, traditional computational approaches often rely on predefined crack paths, leading to significant mesh dependency. To address this, various numerical techniques have been developed, including the boundary element method (BEM) [14,15], meshfree methods [16,17], and the finite element method (FEM).

Classical statistical methods and finite element analysis often struggle to accurately and efficiently predict the service life of structures. This issue persists due to the complexity of the problems, the nonlinear behavior of structures, data uncertainty, and limited data availability. Recent advances in artificial intelligence and machine learning have improved its applications in various fields ranging from the molecular dynamic simulation models [18] to FEM models [19,20]. Artificial Intelligence offers a complementary paradigm for fatigue prediction by enabling direct learning of damage evolution from time-resolved data. In contrast to traditional physics-based models, AI-based approaches do not require explicit cycle decomposition or predefined equivalent stress criteria. Instead, they can learn complex nonlinear mappings between multi-component stress–strain histories and fatigue response directly from simulation or experimental data, making them particularly attractive for multi-axial fatigue problems where load paths and phase relationships play a dominant role.

Various studies have been performed in the recent decade to apply machine learning as well as advanced neural network models to solve the fatigue life problem. Surrogates have been developed for various remaining life prediction models. One such model was developed by Feng et al. [21], who used FNN to construct a real-time remaining fatigue life prediction model. Similarly, other studies in this domain have been performed by either varying the material under study or by editing the geometry of the component under study, by joining various components together, and changing the loading conditions [22,23,24,25,26,27,28,29,30,31,32].

The objective of this study is to develop an artificial neural network tool for fatigue analysis to predict the likelihood of multi-axial failure in mechanical components, accounting for loading history, rates, and frequencies. We aim to provide a more accurate and efficient method for fatigue analysis than traditional approaches, such as cumulative damage design and finite element-based strategies. In particular, the artificial intelligence-based model replaces the notch stress approach, while data for training this model will be generated by the structural analyses using finite element analyses that entail elastoplastic material models.

2. Materials and Methods

2.1. Joints

2.1.1. Cruciform Joints with HY Seam

Welded cruciform joints are integral components in numerous engineering applications. Thus, the fatigue behavior of these structures plays a critical role in determining their service life [33]. Through a combination of experimental testing and finite element analysis, this paper aims to provide a comprehensive understanding of the fatigue performance of such joints.

The cruciform joints with HY seam were also taken from the SN curve in Olivier and Ritter [34]. The first joint has a throat thickness of $13.8$ mm, a root face of 13 mm, and a plate thickness of 25 mm, as shown in

Table 1. Fatigue lifetime for cruciform joints [34].

Joints	Stress Ratio	Stress Amplitude (N/mm2)	Fatigue Lifetime (Cycles)
Cruciform joint 1	0	78	31.2 × 103
		70	49.3 × 103
		60	170.5 × 103
		53	210.4 × 103
		45	378 × 103
		35	538.1 × 103
Cruciform joint 2	0	78	103.7 × 103
		70	308.1 × 103
		60	3118.4 × 103
		55	280.9 × 103
		50	680.9 × 103
		40	2267.7 × 103

; the second one has a different geometry, with a throat thickness of 16.6 mm, a root face of 13 mm and a plate thickness of 25 mm, as shown in Table 1.

To visualize the stress distribution in the weld seams of the end plate connection, experimental investigations were carried out to determine fatigue strength and the location of the crack initiation. The specific geometry is shown in Figure 1.

2.1.2. T Joints with Double Fillet

Further experimental data were obtained from the literature, specifically one uniaxial tension-compression test [36] and one three-point bending test [4]. We used the specific geometries from these tests to build our finite element models. The specific geometry is given in Figure 2. Furthermore, the uniaxial loading condition with constant loading ratio of $R=-0.29$ is given in Figure 3, and the fatigue lifetime is shown in

Table 2. Fatigue lifetime for T-joint 2 [36].

Joints	Stress Ratio	Stress Amplitude (N/mm2)	Fatigue Lifetime (Cycles)
T Joint 2	−0.29	145	8.84 × 105
		139	3.1 × 106
		145	1 × 106
		175	7.57 × 105
		170	6.85 × 105
		165	8.59 × 106
		160	9.87 × 106
		150	1.13 × 106
		150	1.6 × 106
		150	2.42 × 106
		150	3.25 × 106

.

The other two T-joints are from the SN curve literature [5], considering two types of loading conditions. Fatigue tests were conducted under constant-amplitude tension–compression loading at stress ratios R of $-0.5$ and $-1$. The corresponding fatigue lifetime data are presented in

Table 3. Fatigue lifetime for T-joints [5].

Joints	Stress Ratio	Stress Amplitude (N/mm2)	Fatigue Lifetime (Cycles)
T Joint 3	−1	177	420 × 103
		177	1500 × 103
		167	550 × 103
		167	670 × 103
		157	1050 × 103
		147	4660 × 103
T Joint 4	−0.5	140	750 × 103
		140	1600 × 103
		133	1070 × 103
		129	1420 × 103
		125	1200 × 103
		125	3300 × 103

.

2.2. Constitutive Model and Material Properties

The material model used in this study is an elastoplastic model with kinematic hardening. Considering small deformations in our formulation, the strain tensor is defined by:

$$\begin{array}{c}\hfill \mathit{\epsilon }={\nabla }_s\mathit{u}={\displaystyle \frac{1}{2}}\left(\nabla \mathit{u}+{\nabla }^T\mathit{u}\right).\end{array}$$

(1)

The total strain, $\mathit{\epsilon }$, is decomposed into an elastic contribution, ${\mathit{\epsilon }}^e$, and a plastic contribution, ${\mathit{\epsilon }}^p$, written as:

$$\begin{array}{c}\hfill \mathit{\epsilon }={\mathit{\epsilon }}^e+{\mathit{\epsilon }}^p.\end{array}$$

(2)

Introducing the elastic deviatoric strain contribution, ${\mathit{\epsilon }}_D^e={\mathit{\epsilon }}_D-{\mathit{\epsilon }}_D^p$, where ${\mathit{\epsilon }}_D^p:={\mathit{\epsilon }}^p-{\displaystyle \frac{1}{3}}\mathrm{tr}\left({\mathit{\epsilon }}^p\right)\mathit{I}$ refers to the deviatoric part of plastic strain; the free energy density is defined as

$$\begin{array}{c}\hfill \psi ={\displaystyle \frac{1}{2}}K\mathrm{tr}{(\mathit{\epsilon }-{\mathit{\epsilon }}^p)}^2+G\left({\mathit{\epsilon }}_D-{\mathit{\epsilon }}_D^p\right):\left({\mathit{\epsilon }}_D-{\mathit{\epsilon }}_D^p\right)+\sum _{i=1}^N{\displaystyle \frac{1}{2}}{C}_i{\mathit{\beta }}_i:{\mathit{\beta }}_i,\end{array}$$

(3)

where K and G are the bulk and shear modulus, respectively, and N is the number of backstresses. We chose $N=3$, as proposed by Chaboche [37]. The value $C_i$ is the initial kinematic hardening module, and ${\mathit{\beta }}_i$ is a strain-like internal variable associated with the backstress. The dissipation potential is then written as

$$\begin{array}{c}\hfill \varphi =f(\mathit{\sigma }-\mathit{\alpha })+\sum _{i=1}^N{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\gamma }_i}{C_i}}{\mathit{\alpha }}_i:{\mathit{\alpha }}_i,\end{array}$$

(4)

where ${\gamma }_i$ determines the rate at which the kinematic hardening module $C_i$ decreases with increasing plastic deformation. The yield surface $f(\mathit{\sigma }-\mathit{\alpha })$ is written as

$$\begin{array}{c}\hfill f(\mathit{\sigma }-\mathit{\alpha })=\sqrt{{\displaystyle \frac{3}{2}}\left(\mathbf{S}-{\mathit{\alpha }}^{dev}\right):\left(\mathbf{S}-{\mathit{\alpha }}^{dev}\right)}-{\sigma }^0=0,\end{array}$$

(5)

where ${\mathit{\alpha }}^{dev}$ is the deviatoric part of the backstress $\mathit{\alpha }:={\sum }_{i=1}^N{\mathit{\alpha }}_i$, $\mathbf{S}$ is the deviatoric stress tensor and ${\sigma }^0$ is the initial yield stress. The stress tensor $\mathit{\sigma }$ is then written as

$$\begin{array}{c}\hfill \mathit{\sigma }={\displaystyle \frac{\mathsf{\partial }\psi }{\mathsf{\partial }\epsilon }}=\mathit{E}:(\mathit{\epsilon }-{\mathit{\epsilon }}^p),\end{array}$$

(6)

where $\mathit{E}$ is the fourth-order elasticity tensor. The plastic evolution laws are given by

$$\begin{array}{cc}\hfill {\dot{\mathit{\epsilon }}}^p& ={\displaystyle \frac{\mathsf{\partial }f(\mathit{\sigma }-\mathit{\alpha })}{\mathsf{\partial }\mathit{\sigma }}}{\dot{\overline{\epsilon }}}^p,\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill -{\dot{\mathit{\beta }}}_i& ={\dot{\overline{\epsilon }}}^p{\displaystyle \frac{\mathsf{\partial }\varphi }{\mathsf{\partial }{\mathit{\alpha }}_i}}=-{\dot{\overline{\epsilon }}}^p{\displaystyle \frac{\mathrm{dev}[\mathit{\sigma }-\mathit{\alpha }]}{\left|\right|\mathrm{dev}[\mathit{\sigma }-\mathit{\alpha }]\left|\right|}}+{\dot{\overline{\epsilon }}}^p{\displaystyle \frac{\gamma }{C_i}}{\mathit{\alpha }}_i,\hfill \end{array}$$

(8)

where ${\dot{\mathit{\epsilon }}}^p$ is the plastic flow rate and ${\dot{\overline{\epsilon }}}^p$ represents the equivalent plastic strain rate, ${\dot{\overline{\epsilon }}}^p=\sqrt{{\displaystyle \frac{2}{3}}{\dot{\mathit{\epsilon }}}^p:{\dot{\mathit{\epsilon }}}^p}$. Using ${\mathit{\alpha }}_i=C_i{\mathit{\beta }}_i$, one obtains

$$\begin{array}{c}\hfill {\dot{\mathit{\alpha }}}_i=C_i{\dot{\mathit{\epsilon }}}^p-{\gamma }_i{\mathit{\alpha }}_i{\dot{\overline{\epsilon }}}^p.\end{array}$$

(9)

The loading and unloading conditions are written as

$$\begin{array}{c}\hfill {\dot{\overline{\epsilon }}}^p\ge 0,f\le 0,{\dot{\overline{\epsilon }}}^{p}f=0.\end{array}$$

(10)

The material properties of S355 steel used in this paper are listed in

Table 4. Material properties of S355 [38,39].

Elastic properties
Elastic modulus	204,437 MPa
Poisson ratio	0.3
plastic properties
${\sigma }^0$	386 MPa
$C_1$	5327 MPa
${\gamma }_1$	75
$C_2$	1725 MPa
${\gamma }_2$	16
$C_3$	1120 MPa
${\gamma }_3$	10

. The good agreement between the stress–strain curves of the experiment and simulation, as shown in Figure 4, indicates the feasibility of the current elasto-plastic constitutive model.

2.3. FEM Model Setup for Data Generation

According to the IIW Recommendation [7], an effective notch radius of $r=1$ mm is applied for as-welded toes and roots. The maximum mesh size in these regions is ${\displaystyle \frac{r}{6}}$ for linear elements and ${\displaystyle \frac{r}{4}}$ for quadratic elements. To ensure smooth computational results, the mesh at the notches is generated to approximately follow a cubic shape [40]. The dataset is generated by sampling SN curves with 50% probability. The sampling strategy is as follows: one data point is collected every $1\times {10}^4$ cycles in the range of $5\times {10}^4$ to $1\times {10}^5$ cycles, one data point is collected every $1\times {10}^5$ cycles in the range of $1\times {10}^5$ to $4\times {10}^6$ cycles, and one data point is collected every $2\times {10}^6$ cycles in the range of $4\times {10}^6$ to $2\times {10}^7$ cycles. The functions describing the SN curves for the specified specimens are written as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathrm{Cruciform}\mathrm{joint}1:& \log \Delta \sigma =-0.26967\log N+3.14634\hfill \\ \hfill \mathrm{Cruciform}\mathrm{joint}2:& \log \Delta \sigma =-0.27173\log N+3.28603\hfill \\ \hfill \mathrm{Cruciform}\mathrm{joint}3:& \log \Delta \sigma =-0.33322\log N+4.14685\hfill \\ \hfill \mathrm{Cruciform}\mathrm{joint}4:& \log \Delta \sigma =-0.33080\log N+4.10714\hfill \\ \hfill \mathrm{T}\text{-}\mathrm{joint}1:& \log \Delta \sigma =-0.10756\log N+2.83925\hfill \\ \hfill \mathrm{T}\text{-}\mathrm{joint}2:& \log \Delta \sigma =-0.20904\log N+3.67739\hfill \\ \hfill \mathrm{T}\text{-}\mathrm{joint}3(\mathrm{R}=-1):& \log \Delta \sigma =-0.27148\log N+3.77845\hfill \\ \hfill \mathrm{T}\text{-}\mathrm{joint}4(\mathrm{R}=-0.5):& \log \Delta \sigma =-0.27072\log N+3.81953.\hfill \end{array}\end{array}$$

(11)

For ultra-high cycle fatigue, that is, fatigue problems with more than 2 million cycles, the SN curves are intentionally set with a slight slope to represent the kink. The current approach is set so that the stress range after $2\times {10}^7$ cycles takes the value of 95 % of the stress range for $2\times {10}^6$ cycles. The chosen nominal stress ranges are used to determine the loads on the specimens, and the corresponding fatigue life is used as the input data for AI model training.

The FEM simulations were performed in the commercial software Abaqus using fully integrated 8-node linear elements. The meshing strategy follows the IIW recommendations for welded joints. Considering that all fatigue tests are based on cyclic loading, two static analytical steps are defined and named $Ste{p}_{preload}$ and $Ste{p}_{cyclic}$. The Abaqus equation solver is direct, and the solution technique is the full Newton method.

2.4. Dataset

Initially, the operating conditions, loading conditions, and material properties of the structures, as well as the geometry of the weldment, were specified to define different types of connection specimens. These generalized weldment joints were validated to be in correlation with in-service loads and relevant structural geometrical conditions. After that, the relevant information from existing literature was collected for fatigue evaluation. This included fatigue data obtained from experiments, simulations, or case studies, as well as details on the methods used to measure and analyse fatigue. From the literature, experimental stress-fatigue lifetime (SN) curves for different geometries and loading conditions were reported. For the geometry, the T-joint with a base plate and a pipe, the cruciform joint, and the T-joint were considered for the final study. All mentioned joints were studied under various loading conditions, including tension, bending, Torsion, and multiaxial loading.

Using cases reported in the literature, FEM simulations were reconstructed to understand the stress–strain behavior of a particular joint and to gather data related to it. The data available from the experiment, for example, included the maximum nominal stress and the fatigue life for the T-joint experiments with a base plate and a circular pipe. Using these data, a nonlinear FE model was developed to analyze complex stress and strain states, with the input parameter being the loading conditions and the output being the fatigue life. The accuracy of the models was evaluated using cross-validation, mesh convergence studies, and/or testing on the global displacement dataset. Based on the validated FE models, stress and strain under multiple load conditions were evaluated to determine the strain state. The resulting nonlinear section stress and strain states were multi-axial and complex and depended on geometrical and material characteristics and the boundary conditions specified.

The final six generalized stress and strain components ${\sigma }_{xx},{\sigma }_{yy},\dots {\sigma }_{xz}$, generated by the FE model over 21 timesteps, will be used as input to the AI model. These data are stored in a separate file that contains different numbers of life cycles.

2.5. Recurrent Neural Network Model

A Long Short-Term Memory (LSTM) architecture is a distinctive variant of a Recurrent Neural Network (RNN), also called a gated RNN. They are explicitly designed to use their memory for generating complex sequences containing long-range dependencies based on the RNN architecture. LSTM can selectively retain or forget long-term information while effectively storing short-term information within its memory. This section provides an overview of the memory storage mechanism, the internal structure of the LSTM cell, and the computational expressions utilized for output prediction within the LSTM neural network. The cyclic structure of an LSTM neural network is represented in a time-unrolled format, as illustrated in Figure 5.

This diagram provides a visual mapping of the network’s circuitry, with the right side showing the unfolded computational graph corresponding to the circuit on the left. Each square in the diagram represents a specific time instance. The network defines a circuit that operates in multiple steps, so that the current state can influence its future state [42]. By unrolling the network over time, we observe that the LSTM model maintains a consistent input size regardless of the input sequence length. This implies that the dimensionality of the input for an LSTM neural network can be expressed as (batch size, sequence length, input size). Consequently, the LSTM network can efficiently process variable-length sequences without requiring adjustments to the input size. Furthermore, the mapping function employed within the LSTM architecture remains consistent across different time steps. This means that the same set of parameters is applied to each time step of the network, ensuring that the structure of the LSTM layer remains unchanged throughout the processing of the sequence. In other words, every cell within the LSTM layer utilizes an identical set of weights and biases, contributing to the consistent behavior and information flow within the network. In contrast to traditional RNNs, which only incorporate the output as part of the input, LSTMs introduce a more intricate memory mechanism that involves the cell state and a set of gating mechanisms. The key distinction between LSTMs and RNNs lies in the architecture of their memory units. While RNNs have a simple recurrent structure, LSTMs incorporate memory cells with dedicated input, forget, and output gates. These gates enable LSTMs to control the flow of information, retain or forget memorized information, and prevent the vanishing gradient problem from impeding learning. Each gate within an LSTM cell operates autonomously to regulate the flow of information. The input gate determines which information is allowed to enter the cell, enabling the LSTM to selectively update its memory with new input. The forget gate determines which information is removed from the cell memory, allowing the network to discard irrelevant or outdated information. Lastly, the output gate determines which information is output from the cell, influencing the hidden state. What is also crucial in the LSTM cell is the cell state $C_t$ and the hidden state $h_t$, which are the key to storing information from new inputs and loading memory into the current task in the LSTM network. The internal cell state $C_t$ acts as the internal memory and is updated after each calculation, while the hidden state $h_t$ is propagated to the next time step. The hidden state $h_t$, also known as the output state, carries information directly from previous events and is overwritten at every step. The hidden state is influenced by the cell state and computed from the input at the current time step and the previous hidden state. It serves as a summary or compressed representation of the relevant information extracted from the input sequence up to that point. On the other hand, the cell state $C_t$ serves as a long-term memory mechanism that stores information from events that may not immediately precede, allowing the network to remember important information over long sequences.

AI Model

The model used in this paper was trained by the Keras library. The network consists of an LSTM layer with 64 neurons with tanh activation with an input shape of $21\times 2340$, where 21 is the number of timesteps, and 2340 is the number of stresses (${\sigma }_{xx},{\sigma }_{yy},\dots {\sigma }_{xz}$) for all the elements on the radius. The LSTM layer is followed by a dense layer with one neuron and a linear activation function, which serves as the output layer. Before feeding the input and output to the network, both must be within the same range. Therefore, input is normalized using mean normalization, and output is normalized using standard scalar normalization. Based on the above-mentioned network information, the trainable parameters are shown in

Table 5. Trainable parameters in the neural network model.

Model: "sequential"
Layer (type)	Output Shape	Parameters
lstm (LSTM)	(None, 64)	615,680
dense (Dense)	(None, 1)	65
Total Params: 615,745
Trainable Params: 615,745
Non-trainable params: 0

.

The mean squared error (MSE) is used as the evaluation criterion during training. To update the weights and biases of the model, the Adam optimization algorithm is employed with a learning rate of 0.001. The total number of samples available for the current study was 53, split between low-cycle fatigue and high-cycle fatigue. The network was trained and validated on 43 samples. These 43 samples were split into training and validation datasets (90–20%) using the validation split method, keeping the shuffle value as "True". The batch size used during training was set to two. The training duration for the neural network, with the specified architecture, was approximately 11 ms per epoch and 500 epochs in total. The training was conducted on a system equipped with an M1 Pro chip and 16 GB of memory storage.

3. Results and Discussion

3.1. Fatigue Lifetime of the Joints

The experimental fatigue lifetime of the cruciform joints is given in Figure 6. Figure 6 illustrates the experimental data for the S–N curves taken from a series of tests conducted on AW (as-welded) specimens. The fatigue test results correspond to a stress ratio of $R=0$. Furthermore, a comparison is drawn between two test results with distinct plate depths, namely, one at 30 mm and the other at 25 mm. Evidently, under identical stress ratios and magnitudes, the specimen with a depth of 25 mm exhibits a prolonged fatigue lifespan. The experimental fatigue lifetime of the T-joints is shown in Figure 7.

3.2. FEM Model of the Welded Joints and Post-Processing of Simulation Results

Cruciform joints 1 and 2 differ primarily in plate thickness, but for illustration purposes, let us focus on the cruciform joint 1 model. Due to the geometric symmetry and load conditions of the specimen, an eighth of the model is utilized for finite element analysis, as depicted in Figure 8. The number of elements for cruciform joints 1 and 2 is 300,669 and 330,003, respectively. The boundary conditions are illustrated in Figure 8. The displacement of symmetry planes A, B, and C is fixed in the x-, z, and y-directions, respectively, in all steps based on the symmetry restriction. A distributed surface traction $loa{d}_1$ is applied to the surface (D), the magnitude of which reaches the average of the periodic load at the end of $Ste{p}_{preload}$ and remains constant at $Ste{p}_{cyclic}$. A sinusoidally distributed traction $loa{d}_2$ is loaded to the surface (D) at $Ste{p}_{cyclic}$ with a period of the total duration of $Ste{p}_{cyclic}$. The amplitude of $loa{d}_2$ is identical to the nominal stress range.

The model setup for cruciform joints 3 and 4 follows the boundary conditions shown in Figure 9. A reference point is coupled to the inner surface of the hole. The reference point is fixed in the x-, y-, and z-directions. The symmetry planes, A and B, of the specimen are fixed in the z- and y-direction, respectively. The number of elements for cruciform joints 3 and 4 is 341,771 and 372,648, respectively. A distributed surface traction $loa{d}_1$ is applied to the surface C, the magnitude of which reaches the average of the periodic load at the end of $Ste{p}_{preload}$ and remains constant at $Ste{p}_{cyclic}$. A sinusoidally distributed traction $loa{d}_2$ is loaded to the surface C at $Ste{p}_{cyclic}$ with a total duration of $Ste{p}_{cyclic}$. The amplitude of $loa{d}_2$ is identical to the nominal stress range.

The model setup for T-joint 1 is simplified, as shown in Figure 10. The displacement of symmetry planes A and B is fixed in the x- and z-directions, respectively, in all steps based on the symmetry restriction. The displacement of edge E is fixed in the y-direction. The number of elements is 225,450. A distributed surface traction $loa{d}_1$ is applied to surface C, the magnitude of which reaches the average of the periodic load at the end of $Ste{p}_{preload}$ and remains constant at $Ste{p}_{cyclic}$. A sinusoidally distributed traction $loa{d}_2$ is loaded to the surface C at $Ste{p}_{cyclic}$ with a total duration of $Ste{p}_{cyclic}$. The amplitude of $loa{d}_2$ is identical to the nominal stress range.

Considering the symmetry of the specimen geometry and loads of T-joint 2, a fourth model was built for FEM analysis. The number of elements in the specimen is 209,920. The boundary conditions are illustrated in Figure 11. Two pins are modeled as rigid bodies to represent the loading method in the three-point bending test. Pin A is fixed in all directions, while pin B can move along the y-direction. The contact between the specimen and pins is modeled in the "interaction" module of Abaqus. The contact property is defined as "hard normal contact" with a friction coefficient of 0.3. The displacement of symmetry planes C and D is fixed in the x- and z-direction, respectively, in all steps based on the symmetry restriction. A concentrated load $loa{d}_1$ is applied to pin A, the magnitude of which reaches the average of the periodic load at the end of $Ste{p}_{preload}$ and remains constant at $Ste{p}_{cyclic}$. A sinusoidal concentrated load $loa{d}_2$ is loaded to surface E at $Ste{p}_{cyclic}$ with a total duration of $Ste{p}_{cyclic}$. The relation between the concentrated force and the nominal force is formulated as

$$\begin{array}{c}\hfill \Delta loa{d}_2={\displaystyle \frac{\Delta \sigma I}{124.5}},\end{array}$$

(12)

where I is the section modulus.

The model setup for T-joints 3 and 4 is illustrated in Figure 12. The number of elements is 321,540. A distributed surface traction $loa{d}_1$ is applied to surface C, the magnitude of which reaches the average of the periodic load at the end of $Ste{p}_{preload}$ and remains constant at $Ste{p}_{cyclic}$. A sinusoidally distributed traction $loa{d}_2$ is loaded to the surface C at $Ste{p}_{cyclic}$ with a total duration of $Ste{p}_{cyclic}$. The amplitude of $loa{d}_2$ is identical to the nominal stress range.

Since the stress concentration in all specimens is localized at the notches, nodal stress and strain histories along the entire critical notch radius are extracted for post-processing. Figure 13 presents the stress distributions along both the weld toe and the weld gap following one loading cycle.

To ensure data smoothness and dimensional consistency across the welding area, linear interpolation is applied. Specifically, 36 and 180 interpolation points are selected for the weld toe (small radius) and the weld gap (large radius), respectively. For each interpolation point, the corresponding strain and stress histories are illustrated in Figure 14. The output dimension is $36\times 20\times 13$ for the weld toe and $180\times 20\times 13$ for the weld gap. Here, the second output dimension represents 21 historical data points per cycle, while the third dimension comprises six strain components, six stress components, and one accumulated plastic strain. The historical data points refer to uniformly spaced sampling points over one full loading cycle in the FEM simulations. These data points are not arbitrary subdivisions of a single cycle, but rather discrete time steps that capture the evolution of the local stress–strain field throughout the cyclic loading history.

3.3. AI Model Results

The model developed and outlined above was specifically designed to predict the fatigue life of a range of joint configurations. To achieve this, the model underwent extensive training using a comprehensive dataset comprising various geometric shapes and loading conditions. This process ensured that the model could accurately learn the characteristics of each geometry. Once trained with an adequate number of samples for each joint type, the model was used to predict the fatigue life of unseen joint samples across the range of loading scenarios and geometries. In this section, we present the results from both the training and testing phases of our analysis. We focus on the model’s performance across various joint types, analyzing outcomes for both low-cycle fatigue, which typically involves fewer load cycles at higher stress levels, and high-cycle fatigue, characterized by more cycles at lower stress levels.

3.3.1. Cruciform Joints

The finite-life fatigue region of the SN curve was used to train the RNN model mentioned in the Methods section. The model is trained until each joint reaches a minimum loss value, which is the mean squared error in our case, over the epochs.

Table 6. Training results for prediction using the RNN and the radial element of critical stress location.

Joints	Parameter	Value
Cruciform	Training loss	9.9162 × 10−4
Joint 1	Validation loss	8.2632 × 10−4
Cruciform	Training loss	8.8828 × 10−4
Joint 2	Validation loss	6.5718 × 10−4
Cruciform	Training loss	0.0029
Joint 3	Validation loss	0.0044
Cruciform	Training loss	4.4592 × 10−4
Joint 4	Validation loss	0.0011

provides the training information about the final training loss and validation loss. The trend of these losses over training can be plotted against training epochs, as shown in Figure 15. It can be seen in Figure 15A,B that the training is completed after 250 epochs for cruciform joints 1 and joint 2, where it takes 500 epochs to finish the training process for cruciform joints 3 and joint 4 Figure 15C,D.

Having finalized the training model and monitored the final loss, we then predicted the number of cycles from the input stress for the entire dataset (training and validation sets). This prediction was then compared with the actual number of cycles for the respective stress states reported in the literature or in simulations. In general, an ideal network should predict the exact number of cycles for a given stress state. Therefore, when plotted together, it was expected to follow a straight line dividing the two axes. However, in practice, this is not possible, and some points are not exactly predicted and do not lie on the stress line, denoting 100% accurate prediction. The distance between the point and the straight line refers to the prediction error. The relation between the predicted and actual value of the number of cycles is presented in Figure 16. It can be inferred from this figure that the prediction was almost 100% accurate except for cruciform joint 3, where the lower and middle datapoints were not predicted with 100% accuracy, as seen in Figure 16C. These datapoints include the entire dataset of 53 samples, which consists of the training dataset and validation dataset, as well as the test dataset, which was never seen by the network while training. The prediction accuracy of the proposed regression model is quantitatively evaluated using the root mean square error (RMSE) and relative $R^2$ score. These metrics provide a scale-independent assessment of fatigue life prediction accuracy. Over the entire dataset, the model achieves an average RMSE of 0.013 and a relative $R^2$ score of 0.99, indicating excellent agreement with the reference fatigue life.

Once the prediction was accurate, it was plotted against the nominal stress of the respective case to generate SN curves. These plots help us assess the network’s accuracy in terms of SN curves. Figure 17 shows the SN curve for all cruciform joints for the finite life fatigue region. These predicted SN curves were comparable to those from the experimental or theoretical values. The numerical difference between them was negligible.

Following the finite life fatigue region of the SN curve, we also explored the high-cycle fatigue region of the SN curve, where the low-stress value corresponds to a higher number of cycles. The trained model for the finite-life fatigue region was used to assess the robustness of the proposed RNN model.

Next, the network was analyzed in the low-stress state with a higher number of cycles. The stress states for this case were re-simulated, and the corresponding stress values were calculated for the fatigue region beyond two million cycles. The model predicts high cycle numbers from low-stress states and was compared with the target high number of cycles. This comparison can be seen in the following results.

In all the results, we observe good agreement between the predicted and actual numbers of cycles. In Figure 18A,B, a slight curvature in the plots can be observed. Even though the curvature of the points tends to move away from the exact one-to-one relation, the distance between the straight line and the predicted line is negligible, which means the prediction error is negligible. Therefore, the prediction is considered acceptable, and the model is well-trained and can predict results on an unseen dataset.

3.3.2. T-Joints

A similar model is trained for T-joints with different loading conditions, as described in the Methods section. After hyperparameter tuning, the model remained exactly the same as the earlier model, with 3000 epochs. The results for the T-joint were summarized similarly to those of the earlier joints, but with different types of loading. The relation between predicted and actual unnormalized data can be seen in Figure 19. Figure 19A gives us the relation between predicted and actual number of cycles for T-joint type 1 with loading type 1, as mentioned in Section 2.1.2, whereas Figure 19B, C gives us the relation between predicted and actual number of cycles for T-joint type 2 and loading type 1 and loading type 2 repespectively Section 2.1.2. The last Figure 19D represents the same results for T-joint 3 and loading type 1.

After achieving accurate predictions, the results were plotted against the corresponding nominal stress levels to construct S–N curves. Figure 20 shows the S–N curves for all T-joints in the high-cycle fatigue regime. The predicted curves closely match the experimental and theoretical references, with deviations remaining within a small margin, indicating a high level of predictive accuracy of the proposed network.

Similar to the cruciform joint, the t-joint was also tested for high-cycle fatigue, which is the region beyond two million cycles. The model was trained to predict high cycle numbers from low-stress values, and the same architecture also learned the high-cycle fatigue values. The comparison between the actual and predicted high-cycle fatigue can be seen in Figure 21 in a sequence for all the t-joint cases, as mentioned above.

Based on the above results, the developed model can predict the number of cycles across various regions, geometries, and joint types. A combined model can be developed with all the above-mentioned factors as input to predict the number of cycles for any joint under any loading.

4. Conclusions

This study presents a detailed data-driven surrogate model for predicting the fatigue life of welded joints under multiaxial cyclic loading. A Long Short-Term Memory (LSTM) neural network was developed to learn the relationship between local stress–strain histories and the corresponding number of cycles to failure. The data used to train the neural network model were generated via finite element simulations of a nonlinear constitutive model for S355 steel. These simulations were validated against experiments available in the literature for the same geometries. The current model achieves high accuracy for various cruciform and T-joint specimens under various loading conditions, as mentioned in the Methods section. The finite element models were constructed in accordance with IIW recommendations, and local stress and strain histories were extracted along the critical notch regions. These histories, sampled over one full loading cycle, served as sequential inputs to the LSTM model, enabling it to naturally capture load-path dependence and cyclic memory effects inherent to fatigue damage accumulation. Compared with conventional fatigue assessment methods, the proposed framework offers significant advantages in computational efficiency and adaptability, even though the current model is trained for limited welded joint loads and geometries. These models can accommodate every type of geometry for further training. In the future, the dataset will be extended further in size and diversity. In addition, the development of a unified model capable of handling arbitrary joint geometries and loading scenarios will be pursued.