1. Introduction
The monitoring of precast construction is gaining attention nowadays [
1]. Among precast members, prestressed structures are of the utmost importance. Prestressing is a technique in which high-strength steel tendons are stretched or tensioned to induce pre-compression in concrete. These tendons are commonly stretched between two abutments and jacked at the end [
2]. Using prestressed tendons helps reduce cracking in structures that require long timespans, such as bridges, parking lots, balconies, water tanks, concrete pipes, floor slabs, and driven piles. The provision of long spans allows for uninterrupted space with fewer joints in prestressed structures. [
3]. Implementing prestressing techniques in construction enhances the project’s quality by minimizing the utilization of materials and streamlining the integration of large component assembly, leading to more efficient and effective construction. The fatigue and seismic resistance of a structure can be significantly improved using prestressed concrete, as it increases the overall stiffness of the structure [
4].
Prestressed concrete structures have had a significant impact on infrastructure networks. However, the prestressing process can be challenging to implement due to the need for specialized equipment, skilled labor, and strict quality control measures [
5]. The lack of proper equipment and labor can also significantly affect the quality of the structure. Researchers have attempted to improve the quality of prefabrication through field tests and numerical simulations [
6]. Quality control during prestressing can be achieved by carefully ensuring that the tensioning capacity and specifications meet relevant standards for use. The corrosion of steel tendons is a common cause of failure in prestressed concrete. The proper inspection of tendons can prevent this by identifying broken wires and rusting, which can reduce stress in tendons [
7]. In addition, deviations in applied tensioning stress can necessitate repeating the entire prestressing process, and excessive prestressing force can cause significant cracks and blow-outs [
8]. Therefore, quality control through stress monitoring during the manufacturing of precast prestressed structures is critical, as the long-term behavior of these structures can be unpredictable due to issues such as corrosion [
9].
Monitoring the tendon force is essential to assessing the health and quality of prestressed concrete (PSC). Over time, the prestressing force in tendons may be reduced due to the long-term effects on prestressing steel. Variations in prestressing stress compared to design strength can result in cracks, creep, or shrinkage in the structure [
10]. If this damage goes unnoticed for an extended period, it can lead to structural failure. It is particularly important to monitor tendon force in prestressed structures, as excessive creep may occur if the tendons are tensioned with a higher force than intended [
11]. The behavioral effect of effective prestress was investigated in several studies [
12,
13]. Tanchan et al. [
14] conducted a numerical analysis on unbonded tendons. They found that effective prestress has a more substantial impact on the stress increment in the tendons than on their ultimate moment capacity. These studies highlight the importance of accurately estimating stresses in tendons and their impact on various parameters.
In addition to quality control, the prestressing technique is often used to achieve the desired camber for girders by applying the required tendon force [
15]. Camber is the upward deflection of a member, which tends to return to a neutral state upon applying load. The camber can be controlled by adjusting the tendon stresses [
16]. Tadros et al. [
17] noted that errors in estimating prestressing force and girder self-weight could lead to variability in the initial camber, causing construction challenges or the rejection of the girder. The authors suggested using modern methods for calculating the modulus of elasticity, creep, shrinkage, and other parameters in order to reduce errors in estimating the camber.
The accurate estimation of deflection and strain values is essential for evaluating the serviceability of prestressed bridges. Several methods have been used to estimate prestressing force and observe its effects on various parameters [
18,
19]. J. Hsiao et al. [
20] studied the effect of prestressing force on midspan deflection using hand calculation. They found that neglecting the P-delta effect resulted in relatively small deflections. In contrast, by considering the moment of inertia of the gross concrete section, the calculations resulted in more considerable deflections. Ghallab et al. [
21] reviewed equations for calculating the tendon stress increase at the ultimate stage, but the study was limited to externally tensioned tendons and did not examine internally stressed tendons. However, hand calculations are prone to human errors; therefore, software is preferred for fast and accurate calculations.
In general, two approaches are commonly used for monitoring prestressed structures: the direct onsite monitoring of prestressing tendons and using FEM. The onsite monitoring of prestressed girders can be carried out using strain-based methods [
22,
23,
24,
25], some of which depend on electrical impedance, vibration, and wave velocity [
19]. It is challenging to apply these methods in actual construction sites and structures since the prestressed tendons are not visible from the outside and are filled with mortar. In other words, onsite monitoring necessitates direct access to prestressing tendons to be effective, limiting its applicability. The use of FEM is rapidly gaining fame and is replacing other estimation methods. Turmo et al. [
26] examined the shear behavior of a prestressed concrete beam by modifying the jacking prestress using a 2D flat-joint numerical model. Other studies [
27,
28] examined the impact of changes in the prestressing and aspect ratios on the nonlinear structural behavior of prestressed concrete segments using FEM. Halder et al. [
27] found that the stress in unbonded tendons before failure varies from 0.66 to 0.94 fpy for longer tendons using FEM, which are significantly underestimated compared to ACI318-19 code predictions [
29,
30]. Michael et al. [
31] developed a verified 3D finite element model for a parametric study and observed the behavior of concrete bridges under different loading conditions and tendon stress increments. Other researchers have investigated corrosion, creep, and shrinkage in prestressing structures using FEM analysis [
10,
32]. However, the numerical method can be time-consuming and cost ineffective; consequently, more sophisticated and advanced technology is required for real-time tendon stress estimation.
A more advanced approach to estimating structural stresses in real-time is to use machine learning (ML) models, also known as surrogate models, to replace computationally expensive nonlinear numerical simulations [
33]. A surrogate model that utilizes ML models is a simplified and efficient substitute for a complex and expensive model that is used in engineering and scientific simulations to make predictions and optimize designs with fewer computational resources. Creating a surrogate model is also called metamodeling in research fields other than civil engineering [
34]. ML models, such as random forest regression (RFR), XGBoost, support vector machine for regression (SVR), multi-layer perceptron (MLP), and K-nearest neighbor (KNN), have been used in the past as substitutes for FEM in different fields of research [
35,
36,
37,
38]. Other approaches, such as the Kriging model, genetic algorithms [
39,
40], Gaussian process regression [
28], and Bayesian interference framework [
29], have also been used as surrogates for FE models. However, this research was limited to predicting the long-term deflections of structures, and further research is needed to explore the use of ML models for detecting tendon stresses [
41]. Junkyeong et al. [
42] used embedded elastic–magnetic sensors and machine learning methods to estimate tendon stresses in prestressed bridges. Giulio et al. [
43] detected tendon malfunction using machine learning on monitored stress data. However, neither of these studies used a FEM-based surrogate model. Torgeir et al. [
44] recently used an artificial neural network to estimate the tendon stress in unbonded members utilizing a database of experimental results from the literature. Currently, for real-time prediction, there are few or no applications of strain-based tendon stress estimation in bonded prestressed members using machine learning and FEM analysis.
The current study presents a novel framework that employs machine learning techniques to the enable immediate estimation of tendon stress based on strain data. The novelty of the research is the use of strain data from girders with bonded tendons to predict tendon stress in real time. The bridge girder’s finite FE model was created based on field observations. Three ML models, namely random forest (RF), support vector regression (SVR), and artificial neural network (ANN), were trained using the dataset obtained from the FE model, which was validated using analytical calculations. The proposed model demonstrated the most exceptional performance, and hence it was utilized to make accurate predictions of tendon stresses, affirming the effectiveness of the proposed tendon stress estimator. The ANN-based calculator enabled real-time stress estimation on new data and provided insights into the significance of various sensor permutations for optimizing the sensor number and location.
2. Surrogate Model Framework
The proposed tendon stress estimation model comprises two main components: FEM and machine learning (ML). The proposed framework is illustrated in
Figure 1. The FEM analysis of the bridge girder was conducted by the varying tendon stress and generating data. Using the reverse engineering approach, which includes breaking down the individual components of more oversized products to extract design information, the individual strain response of the girder was used to estimate the applied stress on the structure. The FEM input was treated as the ML model output to create a surrogate model that can predict the tendon stress in real time for inputs in the form of girder strain values and an elastic modulus, as explained further in section II-D. The concept of this framework includes training the ML model on a validated FE dataset, and the trained model utilizes strain data as an input to predict tendon stresses. The input strain values were obtained for field application from strain gauges that were installed onsite at various girder locations.
In some cases, the most critical areas of a structure or material may experience high deformation or stress. It is essential that the sensors are placed in these locations to capture accurate measurements. The high-stress regions of symmetrical structures can be assumed; however, the stress distribution differs for unsymmetrical members, and the assumption cannot be made easily. Therefore, the surrogate model framework also provides information regarding the best location and number of strain gauges to be installed, based on identifying the essential features in the ML model.
2.1. Girder Specifications
The selection of the girder was based on two factors. Firstly, the research required using a bonded tendon girder, which was a prestressed concrete beam with bonded tendons running along its length. Secondly, the girder was selected from available girders on the field to validate the accuracy of the FEM model by comparing the results with the analytical solutions that the field experts performed.
The specific girder chosen for the study was a 45 m long I-beam girder, as illustrated in
Figure 2, along with its various components.
Table 1 and
Table A1 in
Appendix A provide detailed specifications of the girder, including its dimensions, material properties, cross-sectional area, and section modulus.
The I-beam girder featured four parabolic bonded tendons anchored at both ends. These tendons allowed the prestressed force to create a camber at the midspan of the girder. This means that the girder had a slight upward curve in the center, which helps to counteract the sag that would otherwise occur due to the weight of the girder itself and any additional loads it may be supporting.
2.2. FEM Modeling
The provided CAD drawings were used to develop a three-dimensional FEM model (
Figure 3) of an I-beam girder using Abaqus software [
45]. The software utilizes the implicit scheme to solve the problem. The model aims to simulate the actual onsite conditions and validate its accuracy by comparing the results with existing analytical solutions. The girder specifications used to develop the FEM model are provided in
Table 1.
The main parts of the model consist of the concrete I-beam, steel rebars and tendons. The girder was modeled as C3D8R, an eight-node linear brick element that has six faces and four integration points per face. In terms of deformation capabilities, the C3D8R element can undergo plastic deformation, exhibit large strains, and can handle material nonlinearity. It can also model crack propagation, contact behavior, and damage evolution. Meanwhile, a T3D2, a three-dimensional, two-node truss element, was used to model the reinforcement of steel bars and pre-tensioned tendons. The interaction between the steel and concrete was modeled as surface-to-surface contact, with embedded constraint properties. The girder was modeled as a simply supported beam with one end hinged and the other end set as roller support. The diameter of the steel bars was kept at around 16 mm. The four bonded tendons were designed using a truss element following a parabolic curve and represented by embedded constraints. Prestressing was applied to each tendon using a predefined field, with the tendon stress involved in the x-direction specified as sigma11 in Abaqus FEM, along the length of the girder. A fine meshing of 50 mm in size was assigned to the structure. An implicit scheme was utilized by performing a static linear analysis under gravity load, allowing the model to deform in all directions. The details for process of girder analysis are presented in the
Supplementary Materials.
2.3. FEM Validation
The FEM model of the bridge girder was validated using analytical solutions obtained from the field. The formulas used to compute the tendon force (
), midspan stress at top fibers (
), and bottom fiber (
) of the I-beam girder are mentioned below:
The parameters used in the above equations and their values are listed in
Table 2.
The limit of initial prestressing, as per the PCI bridge design manual section 17.8.7 or ACI 318-19 article 20.3.2.5, is given as follows [
30]:
where the
is the initial tensile stress in the tendon after prestressing or re-stressing in MPa, and
is the ultimate tensile strength of the prestressing steel in MPa. The ACI 318 code requires that the initial stress in the tendon is limited to a value that ensures that the ultimate tensile strength of the prestressing steel is not exceeded.
Based on the analytical solutions, a tendon stress of approximately 1109 MPa was applied as initial prestressing, less than the tendon stress limit of 1488 MPa provided by Equation (4). The stress in concrete immediately after the introduction of prestress to the girder and the resulting deflected shape is shown in
Figure 3. The prestressing creates a camber at the midspan of the girder. This means that the girder had a slight upward curve in the center, which helps to counteract the sag that would otherwise occur due to the weight of the girder itself and any additional loads it may be supporting. The girder had a camber of approximately 35.4 mm, which falls within the limit of 50 mm, as specified by the ACI 318 standards based on the span length, specific design requirements, and loading conditions of the girder.
The analytical solutions were compared with those obtained from the finite element model, and the differences were negligible. This demonstrates the validity of the finite element model. The comparison results are summarized in
Table 2. After validation, as mentioned in the following sections, the Python script from this basic FEM model served as a tool to generate data for training ML models.
2.4. Statistical Background of Parameters
An investigation of the statistical properties of the material strength of the concrete, rebar, and tendon strand commonly used in domestic construction sites was performed by Paik et al. [
46] in order to provide a foundation of reliability-based design codes for concrete structures. The authors conducted tensile strength tests on concrete, rebar, and strand samples. They analyzed the test results using statistical methods, including the standard deviation, mean, and coefficient of variation in the material strength for each type of material. The authors found that the material strength of concrete had the highest coefficient of variation, followed by rebar and strand. The study results showed that the tensile strength of rebar and strand had a relatively low coefficient of variation, indicating that these materials have fairly consistent mechanical properties. On the other hand, a higher coefficient of variation was observed in the compressive strength of concrete, indicating that the mechanical characteristics of concrete may have more variability. Based on the literature, the parameters selected in the present research for data generation were the strain values as its strain-based stress estimation, along with concrete elastic modulus and tendon stress values.
2.5. Data Generation and Preprocessing
The Abaqus script was used to generate a total of 5000 data points, with the elastic modulus (EM) and tendon stress (TS) serving as input parameters, resulting in strain values at the locations S1, S2, S3, S4, S5, S6, and S7, as shown in
Figure 2a. The strain locations were set based on the regions in which possible damage could occur, such as the flexural failure region at the mid and the shear failure region on supports, during overstressed conditions. Simulations were performed on a personal computer using an AMD Ryzen™ 7-5700G processor and 16 GB of operational memory.
The data generation process involved varying the tendon stress within a range of 100 to 5000 MPa and the elastic modulus within a range of 20,000 to 50,000 MPa. Using the concept of reverse engineering, the generated data were split into input and output features for the ML models. The eight input features were the strains at locations S1, S2, S3, S4, S5, S6, and S7, as well as the elastic modulus, while the tendon stress was taken as the output label.
2.6. Machine Learning (ML) Models
Past research has shown that the accuracy of ML models that consist of decision trees is better than those that use artificial neural networks [
47]. Therefore, in the present study, the analysis was performed by increasing the number of hidden layers of an artificial neural network and using state-of-the-art ML models for comparison [
48].
Supervised ML models that rely on decision tree ensembles, such as RFR, are commonly employed for regression analysis. In contrast, SVR is a supervised ML model that utilizes a hyperplane instead of a line to make predictions. In contrast, RFR does not require feature scaling [
49].
This study used a grid search method to determine the optimal parameters for the ML models, including RFR and SVR. The grid search was conducted using the sci-kit-learn Python library, and the models were evaluated based on their mean square error or accuracy for each combination of hyperparameters. In addition to these models, an artificial neural network (ANN) model was also trained to compare the performance of each model in terms of percentage errors. The flow chart of the framework is shown in
Figure 4, depicting the training of ML models and the selection of the model with the least MSE for the stress calculator. The stress calculator is capable of taking inputs from field sensors to estimate the tendon stress, along with providing suggestions regarding the number of strain gauges and their locations.
In this study, the ANN model’s hyperparameters were tuned to obtain a minimum mean square error score. The ANN architecture comprises a single input layer, two hidden layers, and one output layer. The number of hidden layers was determined based on the model performance and overall comparison of the model with other ML models. A total of 128 neurons for the first hidden layer and 512 neurons for the second hidden layer were finalized; refer to
Figure 5 for the network architecture.
The architecture of the ANN utilizes Relu and linear activation functions. The optimization process utilized the Adam optimizer with a learning rate of 0.001, a batch size of 256, and early stopping at the lowest validation score at epoch 606. The advantage of using the ML model is its ability to make predictions very quickly compared to the FEM model, making it ideal for applications in which real-time predictions are required, such as structural health monitoring.
In order to train the ML models, the dataset was divided into training and testing subsets, where 80% of the data was allocated for training the model, and the rest was reserved for assessing its performance in the current study. The present study utilized the programming language Python and its commonly used libraries for code execution. By utilizing CPU and GPU resources, the conducted research effectively trained and evaluated the performance of the proposed ML models.
4. Practical Considerations
One of the promising outcomes of the developed framework is its ability to monitor the initial tendon stress so that it is within the limits, as per ACI 318-19 [
30]. Ensuring limit controls the excessive creep, cracks or shrinkage, and ultimately protects the structure from failure. In addition, properly monitoring the tendon stress during prestressing provides the assurance of achieving the desired camber in the girder. Apart from advantages such as computational efficiency and cost effectiveness, the surrogate model has made it convenient to utilize strains or deflection measurements at different girder locations for the serviceability evaluation of the prestressed bridges.
To provide a more accessible method for estimating the stress in tendons during application, a more user-friendly approach is required. Therefore, for real-time tendon stress estimation, the stress estimator is proposed. The stress estimator takes, as input, the elastic modulus and strain values that are monitored in the field from the sensor and returns the tendon stress applied on the girder in real time. For safety purposes, the stress calculator considers the initial tendon stress limit provided in Equation (9). The predicted tendon stress is then compared to the stress limit to determine whether it is within the limit or not. If stress exceeds the provided limit, the user can issue a warning to check the design tendon stress during the prestressing process, ensuring quality control.
In addition, the ANN tendon stress calculator displays the sequence of strain variations from high to low for strain gauges at different locations of the girder, and an optimum number of strain gauges is suggested. The strain at S7, located at the midspan of the girder, showed the highest strain fluctuation when the tendon stress was applied. Therefore, it is convenient to place the strain gauges at locations in which the strain varies the most. The user-friendly calculator can be used prior to sensor placement to best optimize the sensor location and number.
This exploratory study presents some limitations based on simplifying a numerical model. The impact of environmental factors, such as humidity and temperature fluctuations, can influence stress and strains, which is one of the limitations of this study. The present study suggests the use of more real-world data in order to create accurate models that represent actual prestress losses.
Further, the type of girder used can differ; therefore, for each girder type, there is a requirement for an updated FE model to provide a training dataset. Once the model is trained, the tendon stress is easily estimated using the proposed framework.
The ANN-based tendon stress calculator can be modified for future use as an application that receives sensor data from the cloud, predicts the stress values, and suggests critical strain values to give a warning for safety assessment, and can propose optimal gauge number and locations.
5. Conclusions
The present research demonstrates the feasibility of using an ANN-based surrogate model framework to predict the stress in the tendons of a bridge girder. The research implemented a novel approach involving machine learning-based tendon stress estimation and strain gauge optimization in real time using the strain data from the FE model as inputs to a ML model.
Among other models, the ANN has the most accurately generated R2 values of 0.998 and 0.997 for the training and testing datasets, respectively. The average prediction error or MAE was less than 10% for the ANN and SVR depicting the sound prediction of the models. However, the ANN model with the lowest RMSE values demonstrated the best performance; therefore, it was selected for the surrogate model framework.
The proposed surrogate model framework using ANN estimated the tendon stress in real time, with only a 0.4% difference between it and the FEM model, where mostly strains at different girder locations were specified as inputs. The ANN-based tendon stress calculator showed a practical benefit of using the proposed framework, via the addition of the computed prestress loss (11.4%) to the predicted results. The added loss compensated for the losses expected on the field, and the estimated stress was compared with the standard limits for quality assurance and structure safety. The present framework can be utilized in situations where only strain data are available to monitor the tendon stress for the purposes of structural health monitoring.
Further, the proposed model utilizes permutation importance to select appropriate locations and the number of strain gauge to be installed. The sensor with the most strain variation was found to be located at the midspan (S7), as expected from the behavior of a simply supported girder bridge. Therefore, the sensors can be placed at locations where there is more strain variation. In the present research, three strain gauges were suggested at locations S7, S5 of the midspan, and S2 at the support, thereby providing the advantage of improved structural health monitoring and reducing unnecessary gauge installment.
The proposed surrogate model is a feasible and quick means of tendon stress estimation. The framework can be utilized for issuing warnings during the prestressing process to keep tendon stress within specified limits. In the present scenario, during the prestressing process, it is crucial to monitor the tendon stress in real time to check whether it is within the standard limits or not to avoid any consequences due to over or under-prestressing. Apart from the adaptability, the ML models are easy to use, making them accessible to a wide range of users. The research focuses on the efficient use of the ML model to replace FEM models for the tendon stress estimation of bridge girders. Tendon stress estimation benefits the industry by enhancing the safety and reliability of prestressed structures.
An application that utilizes the Internet of Things (IoT) to monitor girders during prestressing in real time is recommended as future work.