Monitoring–Modeling Integrated Assessment of Temperature-Induced Prestress Variations in Prestressed Concrete Beams During Construction

Abstract

Prestressed concrete (PSC) beams are widely used in bridges and large structures due to their high load-bearing capacity and crack resistance. However, owing to their complex construction process, they are highly sensitive to temperature variations. Implementing temperature monitoring at this stage helps assess the actual mechanical behavior and effective prestress of the beam, providing a reliable basis for construction control and prestress adjustment. This study aims to investigate the mechanical performance of a bidirectionally stiffened composite tensioning and anchoring system developed earlier by the research team during the construction phase and to elucidate the effect of temperature on the mechanical behavior of pretensioned prestressed concrete beams. By deploying a monitoring system integrated with high-precision sensors, synchronized temperature and displacement data during tensioning, pouring, and curing stages were obtained. Field-measured data were used to validate finite element models under different thermal load conditions. The results indicate that the heat of hydration of concrete causes a temperature difference of 12.0 °C at the end form, leading to a maximum displacement of 0.2 mm at the top of the anchor plate. Notably, a temperature change of 22 °C induces a prestress fluctuation of 0.12% to 0.3%. The numerical model demonstrates strong accuracy, with the highest agreement with experimental data and an error of less than 7.5%. These findings provide a scientific basis for compensating prestress losses and controlling the deformation of prestressed concrete beam structures, playing a critical role in ensuring the long-term safety and performance of structures under complex working conditions.

1. Introduction

1.1. Engineering Background and Motivation

Prestressed concrete (PSC) beams [1] are widely used in bridges and large-scale structures due to their high load capacity and crack resistance. During construction, PSC beams undergo complex processes such as tensioning, dimensioning, and demolding, where the structure is statically indeterminate and highly sensitive to environmental and operational variations [2]. Temperature effects [3] significantly impact PSC beam behavior during construction. Temperature fluctuations cause thermal expansion and contraction, leading to additional deflections and profile deviations that mask true load-induced displacements. Furthermore, temperature changes create strain incompatibility between concrete and prestressing tendons, altering prestress loss [4]. Temperature-induced [5] stress redistribution may also reduce effective prestress [6,7,8], affecting stiffness and crack resistance. In practice, neglecting accurate monitoring and assessment of thermal effects can result in erroneous prestress loss estimates, excessive deflections, and structural degradation, ultimately compromising construction safety. Therefore, simultaneous monitoring [9,10,11,12] of temperature and structural response is essential.

1.2. State-of-the-Art Reviews

1.2.1. Construction-Stage Structural Monitoring

The construction stage is widely recognized as a critical phase with a relatively high incidence of safety-related incidents in engineering projects [13]. During this stage, structural systems are often subjected to incomplete boundary conditions, evolving load paths, and temporary environmental actions, which may lead to unfavorable stress concentrations in localized regions. For instance, Aksoylu et al. [14] reported that excessive snow accumulation could induce brittle shear failure at dapped-end regions of prefabricated purlins due to severe stress concentration effects, potentially resulting in progressive roof collapse. This example illustrates that localized structural weaknesses under temporary loading conditions may significantly threaten safety during construction. Existing studies on construction-stage structural monitoring have primarily focused on the observation of structural responses and environmental factors [15]. Regarding structural response monitoring, Nicoletti et al. [16] integrated non-destructive testing with ambient vibration measurements to track variations in modal properties during key construction stages, thereby enabling the assessment of construction scheme rationality and the timely identification of abnormal structural behavior. Zheng et al. [17] combined in situ deflection and stress monitoring with a MIDAS finite element model to monitor the construction process of a 13-span prestressed concrete continuous girder bridge, allowing for the prediction and correction of construction-stage deflection deviations and ensuring compliance with design requirements throughout construction. Uva et al. [18] conducted real-time monitoring over the entire construction period of a prestressed concrete viaduct and, through comparisons between measured strain data and theoretical predictions at different construction stages, proposed a monitoring-data-based approach for safety limit assessment.

Among various environmental factors, temperature effects play a pivotal role in governing structural behavior. Traditional temperature monitoring approaches mainly rely on manual inspections or wired structural health monitoring systems [19,20,21]. While manual inspection is straightforward and cost-effective, it lacks sufficient accuracy and continuity for large-scale structures. Wired monitoring systems [22,23] generally provide higher stability and reliability; however, they are often associated with high installation costs and difficulties in construction-stage deployment and maintenance. In recent years, wireless structural monitoring systems have been increasingly adopted for temperature monitoring during construction [24,25,26], enhancing monitoring flexibility to some extent. Overall, although substantial progress has been achieved in construction-stage monitoring of structural responses and temperature, existing studies remain constrained by limitations in sensor deployment imposed by construction conditions, as well as by insufficient correspondence between selected monitoring indicators and actual construction-stage risks. As a result, these approaches are often inadequate for fully capturing the true stress state and safety condition of structures subjected to temperature effects during construction.

1.2.2. Temperature Effects in Prestressed Concrete Structures

Temperature effects arising from hydration heat and environmental temperature variations during construction are important factors influencing stress distribution and deformation in prestressed concrete structures [27,28,29]. Extensive research has been conducted on the influence of construction-stage temperature effects on the mechanical behavior of prestressed concrete structures. In the context of hydration heat [30,31], existing studies have predominantly employed combined experimental monitoring and finite element analysis to investigate the distribution and evolution of temperature fields in early-age concrete. Zhang et al. [32] integrated heat transfer theoretical models, field temperature measurements, and numerical simulations to examine the temperature evolution associated with hydration heat in mass concrete, demonstrating that cooling water temperature is the dominant factor governing internal temperature variations. Han et al. [33] investigated hydration heat-induced temperature evolution and early-age cracking risk in cold-region concrete box girders using field monitoring and FEA. Their findings reveal three temperature stages and that optimized cement parameters mitigate thermal stress and cracking. In contrast, studies addressing the influence of environmental temperature variations [34] on prestressed concrete structures during the construction stage remain relatively limited. Existing investigations are often restricted to specific engineering cases or simplified loading conditions, and the coupled relationship between temperature variations and structural responses under realistic construction conditions has yet to be fully elucidated.

To evaluate the effects of temperature actions on the mechanical performance of prestressed concrete structures during construction, several studies [35,36] have examined prestress loss, stress redistribution, and deformation characteristics under thermal loading. However, in most cases, temperature effects are treated independently from time-dependent phenomena such as creep and shrinkage [37,38,39], making it difficult to accurately represent the actual stress state of structures during construction. In summary, Progress in analyzing hydration heat and temperature stress in construction-stage concrete is notable, yet gaps remain in understanding coupled effects of multiple temperature sources, evolving temperature effects throughout construction, and synergistic mechanisms between temperature and prestress responses. Further systematic research under realistic construction conditions is thus essential.

1.2.3. Monitoring Techniques and Numerical Modeling Approaches for Prestress Assessment

The performance of prestressed structures is highly dependent on construction quality and its time-dependent evolution. Owing to construction deviations, material nonlinearity, and time-dependent effects, prestress behavior is often characterized by considerable complexity and uncertainty [40]. Accordingly, the evaluation and tracking of prestress behavior have become critical issues in the assessment of structural safety and service performance. In recent years, increasing attention has been directed toward the integration of monitoring technologies with numerical modeling approaches for the assessment and prediction of prestress behavior [41]. Monitoring techniques provide direct information on actual structural responses, while numerical models offer theoretical insights into underlying mechanisms and evolutionary trends. Consequently, the combined use of monitoring data and numerical modeling, through model calibration and validation based on measured data, has emerged as an effective approach for prestress evaluation [42]. From a monitoring perspective, various sensing technologies have been employed to directly or indirectly measure prestress states [43,44]. Meanwhile, numerical modeling methods have been widely adopted to simulate the mechanical behavior of prestressed structures. Aksoylu et al. [45] conducted an integrated experimental and finite element investigation on prestressed concrete purlins with longitudinal web openings, where the numerical models were calibrated and validated against experimental results, demonstrating the capability of finite element approaches to accurately predict stiffness, load-carrying capacity, and prestress-related behavior. Kulprapha et al. [46] investigated the feasibility of structural health monitoring for continuous prestressed concrete bridges by integrating environmental temperature monitoring with thermo-mechanical coupled models. Their findings indicate that environmental temperature variations can effectively reflect the severity and spatial distribution of distributed bending damage in bridges. Jiao et al. [47] proposed a novel buckling-based measurement device to evaluate and reliably monitor the thermal response of prestressed concrete bridges subjected to continuous temperature variations. Moreover, semantic segmentation networks like DeepLab [48] use atrous convolution for multi-scale feature extraction, enabling precise pixel-level recognition of small deformations in complex environments. EfficientNet [49] employs a compound scaling strategy that reduces computational cost while maintaining accuracy, making it suitable for real-time on-site monitoring. Integrating such vision-based models with finite element analysis could lead to a more refined monitoring-simulation framework, improving predictions of effective prestress evolution under thermal effects. Despite these advances, further research is still required to achieve deeper integration between monitoring data and numerical models, as well as more refined characterization of the evolution mechanisms of prestress behavior.

1.3. Existing Gaps

Based on the foregoing reviews, the below gaps can be identified: (i) During construction of prestressed concrete structures, synchronizations are required between temperature monitoring and structural analysis; (ii) It is necessary to investigate the impact of temperature on prestressed concrete by real-world monitoring; (iii) A numerical model is demanding to establish for prestressed concrete structures, coupling construction-phase temperature effects into prestressing evaluation.

1.4. Aim and Methodology

The primary objective of this study is to systematically investigate how temperature variations during construction affect the structural response and prestressing behavior of prestressed concrete (PSC) beams. By integrating field monitoring data with numerical simulations, this work elucidates the evolution of beam displacement, stress, and prestress under thermal effects, offering critical insights for construction control and safety evaluation. The paper is structured as follows: Section 2 outlines the engineering background, monitoring scheme, and data acquisition methods. Section 3 describes the collection and processing of monitoring data. Section 4 establishes and validates a finite element model against monitoring results and numerical simulations, which is then used to analyze thermal effects and prestressing behavior. Finally, conclusions are presented, along with a discussion of limitations and suggestions for future research.

2. Design and Implementation of Monitoring Systems

The principal stages in fabricating pretensioned prestressed concrete beams are tendon tensioning, concrete casting, curing, and tendon release. During construction, the thermal behaviors [50] of the main girder is mainly influenced by ambient temperature and concrete hydration heat. The experiment involved continuous monitoring of the relative displacement between the concrete beam and the reaction device, along with simultaneous recording of ambient temperature and internal concrete hydration heat temperature, to analyze the correlation between temperature variations and relative displacement.

2.1. Test Specimen

The bidirectional stiffening superimposed prestressed tensioning and anchoring system comprises three primary components: the tensioning base, the reaction tensioning anchor plate, and the steel pipe truss. The reaction tensioning anchor plate is connected to the concrete beam through prestressed steel strands, forming a complete force-transferring system. The reinforced concrete tensioning platform measures 3 m in length, 60 cm in width, and 70 cm in height, with a burial depth of 40 cm. The reaction force tensioning anchor plate consists of a front anchor plate and a rear anchor plate, each measuring 4 m in height and 1.24 m in width. The front anchor plate has a thickness of 6 cm and connects to the tensioning base via a 50 cm long transition section. The rear anchor plate is constructed from two steel plates, each 4 cm thick, with a 22 cm-thick orthogonal special-shaped plate positioned between them. Stiffening ribs connect the front and rear anchor plates. The steel pipe truss includes upper chord members, two lower chord members, and multiple K-shaped node web members, extending a total length of 31 m with a K-shaped node spacing of 1 m. Bolts secure the connection between the steel pipe truss and the tensioning anchor plate. The structure of this bidirectional stiffening superimposed prestressed tensioning and anchoring system is illustrated in Figure 1.

The experimental specimen is a full-scale I-shaped beam, 29.96 m in length and 2.0 m in section height. Cross-sections at the mid-span and support points are illustrated in Figure 2. The boundary condition arrangement is consistent with the actual construction site. The upper and lower flanges each have a width of 600 mm, while the web thickness is 200 mm. The height of the lower flange varies from 300 mm at the mid-span to 600 mm at the fulcrum. Symmetrically arranged along the lower flange are 20 straight steel strands, designated N1 to N20 [51,52]. These strands consist of 1860 grade 1 × 715.2 low relaxation steel, with a tensioning control stress of 1395 MPa, as shown in Figure 3.

2.2. Instrumentation Layout

Temperature measurements [53] were conducted at three locations: the upper, middle, and lower sections of the reaction tensioning anchor plate, as well as the corresponding sections of the main beam end formwork. Additionally, one measurement point was established for each row of steel strands. Following the concrete pour, temperature assessments were expanded to include the reaction force tensioning anchor plate, steel strands, and end formwork, along with an additional measurement of the concrete transformation temperature. Measurement points were selected at intervals of 1/4 along the beam length to evaluate the upper, middle, and lower sections of the formwork. After the removal of the formwork from the prefabricated concrete beam, surface temperature measurements of the concrete were conducted at corresponding positions along the beam length. For monitoring the relative displacement of the tensioning system, one displacement gauge was positioned at the anchor plate of the tensioning platform seat. Three additional displacement gauges were placed above, in the middle, and below the concrete beam formwork and prestressed anchor plate, while two gauges were positioned between the anchor plate and the truss. The layout of the measurement points is illustrated in Figure 4.

The parameters of the temperature sensor and displacement sensor selected for this experiment are shown in

Table 1. Technical Parameters of Monitoring Instruments.

Types	Selected Sensors	Sensitivities	Ranges
Temperature sensors	FeelElec FR03D Infrared Surface Thermometer	0.1 °C	−20 to 550 °C
Displacement sensors	Fiaye FER20 Series LVDT Differential Displacement Extensometer	0.1 μm	0 to 5 mm

. Prior to the experiment, the temperature sensor and displacement sensor were strictly calibrated. For the temperature sensor, first, based on the material properties of the measured object, the corresponding emissivity was set in the device menu before each measurement to correct the influence of surface emissivity on temperature measurement results. Second, a standard blackbody furnace was used as a reference source to perform single-point temperature compensation calibration on the thermal imager. By inputting the temperature difference correction value, the device reading was aligned with the reference source temperature. For the displacement sensor, first, the sensor probe was fully retracted, and the zero potentiometer was adjusted to zero the output. Subsequently, a high-precision displacement stage was used to push the probe to the full-scale position, and the gain potentiometer was adjusted to align the output with the theoretical full-scale value. To ensure linearity, five equally spaced displacement points within the 0 to 5 mm range were selected for multi-point calibration. A linear equation was fitted using the least squares method, and the correction parameters were written into the data acquisition system.

The LVDT displacement sensors with a sensitivity of 0.1 μm and a measurement range of 0–5 mm were used in this study. Given that the maximum displacement observed was 0.2 mm, which is three orders of magnitude larger than the sensor resolution, we ensured that the measurement precision was adequate for detecting small-scale structural deformations. The LVDT sensors are equipped with self-compensation mechanisms to mitigate thermal drift. Zero calibration was performed prior to each monitoring phase to minimize drift during multi-day measurements. The displacement sensors were mounted on rigid steel brackets directly fixed to the reaction frame to minimize installation compliance, ensuring that the sensors’ movement is largely restricted to the deformation of the concrete structure. To further ensure measurement reliability, the time-series data from the sensors exhibit smooth and continuous trends without stepwise fluctuations, indicating that the sensors performed consistently throughout the testing period.

2.3. Testing Protocol

This test comprises several primary construction stages, including steel strand tensioning, concrete pouring, the concrete’s attainment of its prestressed load-holding age, and the removal of formwork following steel strand tensioning. Prior to concrete pouring, after the steel strands have been tensioned, the displacement and surface temperature of the reaction tensioning anchor plates are measured every 30 min for a duration of 3 h, resulting in a total of 7 measurements. A thermometer is installed on the prefabricated beam formwork in advance to monitor temperature changes in the concrete. The interval from concrete pouring to formwork removal spans a total of 16 h, during which tests are conducted every 2 h, amounting to 8 measurements. Following the removal of the formwork, tests are scheduled every 4 h for the first 12 h, and subsequently every 12 h for a total duration of 4 days. Throughout the temperature testing process, the displacement of the measurement points is assessed concurrently.

During the monitoring program, readings were taken seven times within the first 3 h after tendon tensioning at 30 min intervals and eight times within the first 16 h after casting at 2 h intervals. Potential aliasing would occur only if significant high-frequency components existed in the displacement or temperature responses. However, the observed thermo-mechanical response is governed by heat diffusion and elastic deformation, which evolve continuously. The recorded displacement time histories exhibit smooth and gradual variations without visible stepwise fluctuations, indicating that high-frequency components are unlikely to be present within the adopted sampling interval. Therefore, aliasing effects are considered negligible for the reported displacement trends.

Although environmental parameters such as humidity and wind speed were not separately monitored, their effects on the structural response are considered secondary. The mechanical analysis in this study is directly based on measured temperature data, which inherently reflect the combined environmental influences on the thermal field. Furthermore, the monitoring was conducted under relatively stable winter conditions, during which variations in humidity and wind speed were limited.

3. Acquisition and Processing of Monitoring Data

3.1. Temperature Monitoring in the Construction Phase

The average ambient temperature during the experimental test ranged from 16 to 18 degrees Celsius. Temperature measurement points were strategically positioned on the surface of the reaction tensioning anchor plate, bottom formwork, and concrete formwork. By conducting on-site temperature measurements, the temperature variations in these components at various construction stages were recorded. The findings are detailed in

Table 2. Temperature monitoring variation under various phases (unit: °C).

Phases	Tensioning Anchor Plates			Bottom Formwork			Concrete Formwork
	Peak	Valley	Range	Peak	Valley	Range	Peak	Valley	Range
Post-tensioning	18.7	16.8	1.9	18.0	16.6	1.4	-	-	-
Post-casting	20.5	15.5	4.0	28.9	16.9	12.0	42.1	20.1	20.0
Post-stripping	31.4	14.8	16.6	39.9	21.8	18.1	45.6	25.2	20.4
Post-release	-	-	-	-	-	-	-	-	-

.

Table 2 indicates that the temperature differences for the reaction force tensioning anchor plate and the end template after tensioning are 1.9 °C and 1.4 °C, respectively. These temperature differences are relatively minor, with the actual temperatures remaining close to the ambient temperature. Following the pouring process, the temperature difference for the reaction force tensioning anchor plate increased to 2.8 °C, while the actual temperature continued to approximate the ambient temperature. In contrast, the temperature difference for the end formwork reached 12.0 °C, reflecting a significant variation. This pronounced temperature change can be attributed to the heat generated by the hydration of the concrete after pouring, which is transferred to the end formwork. After removing the formwork, the temperature difference for the tensioning anchor plate was 6.8 °C, while the end formwork showed a temperature difference of 18.1 °C, influenced by the concrete’s hydration heat. The temperature difference change in the end formwork exceeded that of the tensioned anchor plate. The temperature gap between the concrete formwork surface post-pouring and post-formwork removal was notably large, peaking at 20.4 °C. Hydration heat release was evident, with the lowest temperature surpassing the ambient temperature. Meanwhile, the heat generated by hydration is transferred to the reaction anchor plate through the steel strands. The cross-section of the steel strands is relatively small, limiting their heat transfer capacity, while the concrete formwork and end plates, being directly connected to the concrete, are more significantly affected by the hydration heat. Consequently, the temperature difference between the end plates and the concrete formwork becomes greater than that of the reaction anchor plate.

It should be noted that internal temperature measurements within the concrete core were not conducted in this study. This is because the focus of the present research is on the temperature-induced mechanical response of the tensioning system during construction rather than on the detailed characterization of hydration heat development within the concrete mass. Future work will incorporate high-precision distributed fiber optic sensors to obtain supplementary temperature data, particularly at the beam mid-span and within the concrete interior, in order to provide a more comprehensive understanding of temperature field distribution.

3.2. Monitoring of Concrete Shrinkage

Synchronous tests were carried out on concrete shrinkage changes during the construction of prefabricated tensioned beams. Three specimens were produced on-site for a 28-day test to determine concrete shrinkage values.

Table 3. Variation in Mean shrinkage with testing time.

Test Time	Average Shrinkage/mm	Percentage to 7-Day Value (%)	Percentage to 28-Day Value (%)
Day 1	0.042	0.27	0.19
Day 3	0.116	0.74	0.54
Day 5	0.146	0.74	0.68
Day 7	0.156	-	0.73
Day 28	0.214	-	-

displays the time-dependent variations in the average shrinkage values of the three specimens.

Table 3 illustrates a progressive rise in concrete shrinkage as the number of days post-pouring increases. The total shrinkage of concrete reaches 0.214 mm after 28 days, with a value of 0.156 mm at 7 days, representing 73% of the total shrinkage. Hence, concrete shrinkage exhibits a notable impact by the 7th day post-pouring.

3.3. Influence of Temperature on Reaction System Displacement

The physical properties of steel dictate that the tensioning and anchoring system will experience deformation in response to temperature variations. When multiple steel strands are tensioned, temperature fluctuations induce changes in the tensioning levels of these strands, leading to a prestressed tensioning effect. Consequently, conducting displacement tests on the tensioning reaction force system at varying temperatures is essential.

3.3.1. Tensioning Stage

Within three hours following the tensioning of the steel strands, temperature and displacement measurements of the reaction tensioning anchor plate were recorded seven times, allowing for the assessment of temperature and displacement distribution along the height of the reaction tensioning anchor plate frame. The results are illustrated in Figure 5.

Figure 5a illustrates that the reaction force tensioning anchor plate’s overall temperature ranges from 18 °C to 19 °C during tensioning. Temperature rises at 30 and 150 min but decreases at other time points compared to the initial stage due to temperature fluctuations. Vertically, the top exhibits slightly higher temperatures than the bottom, with a small temperature gradient along the height. However, this gradient is not substantial.

Figure 5b shows minimal displacement of the reaction tensioning anchor plate during tensioning, mainly concentrated at the top and upper-middle sections. The lower-middle parts and the bottom experience negligible displacement. At 120 min, the maximum displacement at the top of the reaction support is −0.03 mm, indicating minimal temperature influence on anchor plate displacement within this temperature range.

3.3.2. Pouring to Solidification Stage

The temperature and displacement of the reaction tensioning anchor plate were measured eight times over a 16 h period during the concrete pouring stage. The temperature and displacement distribution along the height of the reaction tensioning anchor plate during the pouring to solidification stage was subsequently extracted. The results are presented in Figure 6.

Figure 6a illustrates that the temperature range at each measurement point on the anchor plate during the pouring to solidification stage is between 16 °C and 20 °C, closely aligning with the ambient temperature. Between 2 and 10 h post-pouring, the temperature distribution along the height of the anchor plate exhibits a pattern of lower temperatures at the bottom and higher temperatures at the top, primarily influenced by the hydration heat generated during the pouring process. However, the significant temperature change observed in the end formwork during this stage suggests that the reaction tensioning anchor plate is not substantially affected by the hydration heat of the concrete. After 10 h, the temperature distribution along the height indicates a consistent trend of higher temperatures at the top, reflecting a gradual decline in the hydration heat reaction of the concrete.

As illustrated in Figure 6b, during the transition from pouring to solidification, the displacement of the reaction tensioning anchor plate remains stable and relatively minor. In contrast to the displacement distribution along the height of the anchor plate, the displacements observed at the top and upper middle sections are significantly larger, while those at the middle lower and bottom sections are comparatively small. The predominant displacements are concentrated at the top and upper middle regions. In the middle lower and bottom sections of the support, displacement is virtually negligible. After 16 h of pouring, the maximum displacement recorded at measurement point 5 was 0.05 mm, suggesting that the temperature during the pouring phase exerted a minimal influence on the reaction anchor plate.

3.3.3. Formwork Removal and Curing Stage

Tests are conducted every 4 h for the first 12 h post-removal, followed by measurements every 12 h for a total duration of 4 days. The displacements recorded at each measurement point across varying temperatures during this stage are illustrated in Figure 7.

Figure 7a indicates that the temperature range at each measurement point on the anchor plate during the formwork removal and curing stage spans from 16 °C to 32 °C. This relatively broad temperature range primarily results from the extended testing period. Additionally, during the tests conducted at 12, 36, 60, 84, and 108 h, the temperatures were notably elevated. Furthermore, as the temperature increases, the gradient effect becomes more pronounced.

Figure 7b shows that, during the formwork removal and curing stage, the displacement of the anchor plate is predominantly observed at the top, with minimal displacement at the bottom. As the temperature rises, the displacement at the top gradually increases, reaching a maximum of 0.2 mm.

4. Simulation Analysis of Thermal Effects on Pretensioned Prestressing

4.1. Finite Element (FE) Modeling

The simulation analysis utilized the finite element software ABAQUS 2022 [54] to construct a three-dimensional finite element model of the bidirectional stiffening superimposed prestressed tensioning and anchoring system. The tensioning anchor plates and triangular anchor plates were represented by shell unit S4R, the base by solid unit C3D8R, and the prestressed steel strands by truss unit T3D2 [55]. The material properties [56] for each component are detailed in

Table 4. Material properties for FE models.

Parts	Density (kg/m3)	Elastic Modulus (GPa)	Poisson’s Ratio	Linear Expansion Coefficient (1/°C)
Prestressing anchor plate	7850	206	0.3	1.2 × 10−5
Strand	7850	195	0.3	1.2 × 10−5
Reaction frame	2420	34.5	0.2	1.0 × 10−5

.

The tensioning anchoring system components were assembled using Boolean operations. Tie constraints were applied to connect the tensioning anchor plate to the prestressing steel strands and to the pedestal. The boundary conditions of the pedestal were defined as follows: one end of its bottom surface was permitted to rotate only about the X-axis, while the other end was restrained from displacing in the X- and Y-directions; the remaining bottom surfaces were constrained only against vertical displacement. Displacement in the X- and Z-directions was restricted at the bottom of the reaction frame. A schematic of the three-dimensional finite element model is shown in Figure 8.

The tensile force of prestressed steel bundles was simulated using the cooling method [57]. The ambient temperature was modeled through a temperature field. Initially, the ambient temperature was defined as 18 °C before tensioning. Subsequently, each following analysis step introduced a temperature increment of 2 °C, raising the temperature progressively to 40 °C, and then decreasing it incrementally from 40 °C to 0 °C in decrements of −2 °C.

Field measurements indicate that nonuniform temperature gradients may exist along the beam height and near the formwork due to hydration effects. However, monitoring results show that the global structural response of the reaction bed–strand system is primarily governed by overall temperature variation rather than localized thermal gradients. Considering the high stiffness of the pre-tensioning system and the relatively small measured displacements, a spatially uniform temperature field was adopted in the numerical model to represent the dominant first-order thermal effect. This simplification allows for clear identification of the thermo-mechanical coupling mechanism between temperature variation and prestress evolution, while maintaining model transparency and interpretability.

4.2. Validation of the FE Model

The data obtained from each measurement point post-tensioning were juxtaposed with the results of the finite element analysis, depicted in Figure 9. It is evident from Figure 9 that the longitudinal displacement diminishes progressively from measurement points w1 to w7 during the tensioning process. A satisfactory concordance is observed between the measured values and the outcomes of the finite element analysis. Specifically, at measurement point w1, the measured tensioning stage value and the longitudinal displacement in the finite element analysis are 0.040 mm and 0.043 mm, respectively, exhibiting a minor disparity of 7.5%.

4.3. Analysis of the Influence of Temperature on Post-Tensioning

4.3.1. Effect of Post-Tensioning Temperature on Anchor-Plate Displacement

After tendon stressing, the displacement distribution along the height of the reaction anchorage plate and the temperature-dependent displacement responses at the monitoring points are illustrated in Figure 10 and Figure 11.

As illustrated in Figure 10 and Figure 11, the displacement of the anchor plate, subjected to tension from the reaction force, varies along its height. Specifically, the upper section displaces outward as a unit, while the middle and lower sections shift inward collectively. The displacement phenomenon observed in the reaction force tension anchor plate can be explained from both mechanical and thermal perspectives. From the mechanical perspective, concentrated reaction forces are transferred to the anchor plate, generating a non-uniform stress distribution along its height after tendon tensioning. This results in a bending deformation pattern in which the middle and lower regions mainly experience compressive inward movement, while the upper region undergoes outward displacement due to bending effects. Moreover, the bottom of the anchor plate is rigidly connected to the pedestal and reaction frame, providing strong boundary constraints, whereas the upper region is comparatively less restrained. Under combined axial force and bending effects, structural displacement tends to amplify at locations with weaker constraints. Consequently, the upper region becomes the primary deformation concentration zone.

In addition, thermal expansion further enhances this response. When temperature increases, the steel plate expands. Because the base is strongly constrained, free thermal deformation is restricted, producing additional bending deformation and amplifying displacement at the top. This deformation is reversible, and displacement decreases correspondingly with temperature reduction. It should be noted that although the upper displacement is relatively larger, the overall magnitude remains small, indicating high global stiffness of the anchorage system and no adverse impact on structural safety. However, such displacement variations may induce minor prestress fluctuations, highlighting the necessity of considering temperature effects during construction-stage prestress evaluation.

After the steel bundle was tensioned, the temperature decreased from 22 °C to 0 °C. Figure 12 and Figure 13 illustrates the distribution of the displacement of the reaction tensioning anchor plate along its height, as well as the variation in displacement at each measurement point in relation to temperature.

As illustrated in Figure 12 and Figure 13, the tensioning anchor plate shifts inward as the temperature decreases, with the displacement at the top exceeding that at the bottom. The temperature decreased from 22 °C to 0 °C, resulting in a reduction in the displacement at measurement point w1 from 2.51 mm to 2.08 mm, a change of −0.43 mm, or 17.13%. At measurement point w2, the displacement decreased from 0.77 mm to 0.48 mm, reflecting a reduction of −0.29 mm, or 37.7%. For measurement point w3, the displacement changed from −0.31 mm to −0.51 mm, indicating a reduction of 0.2 mm, or 64.5%. The displacements of the other measurement points exhibited no significant variation with temperature.

As indicated in the preceding text, following the tensioning of the steel bundle, variations in temperature will cause the reaction tensioning anchor plate to experience longitudinal displacement, thereby impacting the effectiveness of the actual prestress. The prestress loss caused by the deformation of the anchor is determined by Equation (1):

$${\sigma }_1={(E}_p·\sum \mathsf{\Delta }L)/L$$

(1)

where $\mathsf{\Delta }L$ denotes the deformation of the anchor at the tensioning end and the deformation of the steel bundle (mm); $L$ represents the distance from the tensioning end to the anchoring end (mm); $E_p$ represents the elastic modulus of the steel bundle.

It is important to clarify that the temperature increase from 18 °C to 40 °C considered in this section represents a parametric temperature sweep analysis, rather than the short-term temperature fluctuations observed during construction. The imposed temperature increment is significantly larger than that recorded during the casting stage and is intended to evaluate structural sensitivity under broader environmental temperature variations. Since thermal deformation is approximately proportional to the temperature increment under linear thermal expansion assumptions, the larger displacement obtained in the numerical analysis is consistent with the substantially greater imposed temperature variation compared to field measurements.

4.3.2. Effect of Post-Tensioning Temperature on Prestress

After tensioning, the results of displacement and prestress increment for steel strands N1 to N20 as temperature increases are presented in

Table 5. Strand displacement and prestress increment (temperature increases).

ID	Displacement (mm)							$∆\mathit{w}$ (mm)	${\mathit{\sigma }}_1$ (MPa)
	18 °C	20 °C	24 °C	28 °C	32 °C	36 °C	40 °C
N1	−1.51	−1.50	−1.47	−1.44	−1.41	−1.38	−1.35	0.16	1.93
N2	−1.47	−1.46	−1.43	−1.40	−1.37	−1.35	−1.32	0.15	1.86
N3	−1.25	−1.24	−1.21	−1.19	−1.16	−1.14	−1.11	0.14	1.69
N4	0.10	0.13	0.19	0.25	0.32	0.38	0.44	0.34	4.21
N5	−1.13	−1.11	−1.09	−1.07	−1.04	−1.02	−0.99	0.14	1.65
N8	−1.45	−1.43	−1.40	−1.36	−1.33	−1.29	−1.26	0.19	2.30
N9	−1.51	−1.49	−1.46	−1.42	−1.39	−1.36	−1.33	0.18	2.19
N10	−1.53	−1.51	−1.48	−1.45	−1.42	−1.39	−1.36	0.17	2.10
N11	−1.53	−1.52	−1.49	−1.46	−1.43	−1.40	−1.37	0.17	2.01
N12	−1.42	−1.40	−1.38	−1.35	−1.32	−1.30	−1.27	0.15	1.79
N13	−1.34	−1.33	−1.30	−1.28	−1.25	−1.23	−1.20	0.14	1.73
N14	−0.96	−0.95	−0.93	−0.90	−0.88	−0.85	−0.83	0.13	1.62

. The displacement increment ($∆w$) is defined as the displacement of the steel strand at 40 °C minus that at 18 °C.

As indicated in Table 5, the prestressed steel strands gradually expand outward with rising temperatures, resulting in an increase in displacement. Specifically, when the temperature increased from 18 °C to 40 °C, the displacement increment of the steel strands varied from 0.13 mm to 0.34 mm. It is noteworthy that strand N4 exhibited a significantly larger displacement increment compared to the other strands, exhibited a displacement increase from 0.1 mm to 0.44 mm. This phenomenon can be attributed to its geometric position at the upper part of the anchor plate, which is farther from the centroid of the anchor plate. Under temperature variations, the anchor plate undergoes thermal expansion, and the displacement magnitude increases proportionally with the distance from the centroid. Therefore, strands located farther from the centroid, such as N4, experience greater displacement increments. This behavior reflects the rigid-body thermal expansion characteristics of the anchor plate rather than any abnormal mechanical response of the strand itself. This observation is consistent with the theoretical thermal expansion relationship, where displacement is proportional to the distance from the expansion center.

The displacement growth of the other steel strands ranged from 10.37% to 13.82%. Concurrently, as the temperature rose, the prestress in the steel strands consistently increased, with increments between 1.62 MPa and 4.21 MPa. When the tensioning control stress was set at 1395 MPa, the growth ranged from 0.12% to 0.3%.

After tensioning, the results pertaining to the displacement and prestress increment of N1 to N20 steel strands, as influenced by temperature changes, are presented in

Table 6. Strand displacement and prestress increment (temperature drop).

ID	Displacement (mm)								$∆\mathit{w}$ (mm)	${\mathit{\sigma }}_1$ (MPa)
	22 °C	20 °C	18 °C	16 °C	12 °C	8 °C	4 °C	0 °C
N1	−1.48	−1.50	−1.51	−1.53	−1.55	−1.58	−1.61	−1.64	−0.16	−1.93
N2	−1.44	−1.46	−1.47	−1.49	−1.51	−1.54	−1.57	−1.60	−0.15	−1.86
N3	−1.23	−1.24	−1.25	−1.26	−1.29	−1.31	−1.34	−1.37	−0.14	−1.69
N4	0.16	0.13	0.10	0.07	0.00	−0.06	−0.12	−0.19	−0.35	−4.21
N5	−1.10	−1.11	−1.13	−1.14	−1.16	−1.19	−1.21	−1.24	−0.14	−1.65
N8	−1.41	−1.43	−1.45	−1.47	−1.50	−1.53	−1.57	−1.60	−0.19	−2.30
N9	−1.47	−1.49	−1.51	−1.52	−1.56	−1.59	−1.62	−1.65	−0.18	−2.19
N10	−1.50	−1.51	−1.53	−1.54	−1.58	−1.61	−1.64	−1.67	−0.17	−2.10
N11	−1.50	−1.52	−1.53	−1.55	−1.58	−1.61	−1.64	−1.67	−0.17	−2.01
N12	−1.39	−1.40	−1.42	−1.43	−1.46	−1.48	−1.51	−1.54	−0.15	−1.79
N13	−1.32	−1.33	−1.34	−1.35	−1.38	−1.41	−1.43	−1.46	−0.14	−1.73
N14	−0.94	−0.95	−0.96	−0.98	−1.00	−1.02	−1.05	−1.07	−0.13	−1.62

.

The data indicate that as the temperature decreases, the prestressed steel strands progressively contract, resulting in a gradual reduction in displacement. Specifically, when the temperature fell from 22 °C to 0 °C, the displacement increments for the steel strands varied between −0.35 mm and −0.13 mm. Notably, the displacement of the N4 steel strands diminished from 0.16 mm to −0.19 mm, reflecting a reduction of 2.17 times, while the displacement of the other steel strands decreased by −10.57% to −14.18%. Concurrently, the stress experienced by the steel strands also decreased with the drop in temperature, with increments ranging from −4.21 MPa to −1.62 MPa. When the tensioning control stress is set at 1395 MPa, the observed decrease falls between 0.12% and 0.3%.

In conclusion, the prestressed tensioning process is influenced by temperature fluctuations, which affect the longitudinal displacement of steel strands and consequently alter the prestress. An increase in temperature causes the tensioning anchor plate to expand outward, resulting in the stretching of the prestressed steel strands and a gradual increase in prestress. Specifically, when the temperature rises from 18 °C to 40 °C, the prestress increases by 0.12% to 0.3%. Conversely, a decrease in temperature leads to the contraction of the tensioned anchor plates, which in turn causes the prestressed steel strands to contract and the prestress to diminish gradually. For instance, when the temperature drops from 22 °C to 0 °C, the prestress decreases by 0.12% to 0.3%. Furthermore, as temperature variations occur, both the longitudinal displacement of the anchor plate and the prestress along the height of the tensioned anchor plate gradually decrease.

4.4. Engineering Implications

The monitored results demonstrate that temperature rise causes expansion of the anchorage plate and elongation of the prestressing strands, leading to an increase in prestress. Conversely, temperature reduction results in contraction and prestress loss. This reversible thermal effect should be considered during construction-stage tensioning operations. According to the experimental results, a temperature variation of 22 °C resulted in a prestress fluctuation of approximately 1.62 to 4.21 MPa, corresponding to 0.12% to 0.30% of the control stress. This indicates that temperature variation, although relatively small, may lead to measurable stress deviations during construction. Therefore, a temperature correction coefficient $k$ is introduced for construction-stage application. The recommended correction expression is shown in Equation (2):

$${\sigma }_{\mathrm{actu}\mathrm{a}\mathrm{l}}={\sigma }_{\mathrm{design}}+k\times (T_{\mathrm{now}}-T_{\mathrm{ref}})$$

(2)

where ${\sigma }_{\mathrm{actual}}$ is actual control stress (MPa); ${\sigma }_{\mathrm{design}}$ is design control stress (MPa); $T_{\mathrm{now}}$ represents current temperature (°C) while $T_{\mathrm{ref}}$ represents reference temperature (°C); $k$ represents the temperature correction coefficient derived from the experimental data. Based on the test results, $k$ is recommended to be taken as 0.10 to 0.15 MPa/°C for practical field application.

To facilitate engineering implementation, temperature correction thresholds are proposed:

No correction is required when $\Delta T$ ≤ 5 °C;

Correction is recommended when 5 °C < $\Delta T$ < 10 °C;

Correction is mandatory when $\Delta T$ ≥ 10 °C;

where $\Delta T$ represents temperature variation.

Additionally, abrupt temperature changes—such as after formwork removal, sudden solar radiation variations, or rapid cooling by cold air should be avoided during tensioning.

Although the temperature-induced prestress variation observed in this study appears relatively small compared with the total control stress level, its engineering significance should be interpreted in the context of construction-stage control accuracy. In practical pre-tensioning construction, prestress control is typically verified through the applied jacking stress, the effective stress after release, and the structural response of the member. The observed thermal variation is lower than common construction control tolerances and would not independently require re-tensioning or modification of standard construction procedures. Therefore, the primary contribution of this study is not to alter established construction practices, but to quantify the magnitude of environmental thermal effects and improve the interpretation of field monitoring data. By identifying the thermal component of short-term prestress fluctuation, the study enhances the reliability of construction-stage monitoring assessment.

5. Conclusions

This study systematically analyzed the mechanical response characteristics of the reaction tensioning anchor plate under thermal loading through an integrated approach combining experimental investigation and finite element simulation. The main findings are summarized as follows:

• Due to diurnal ambient variations, a persistent vertical temperature gradient formed across the reaction tensioning anchor plate during monitoring, characterized by consistently elevated temperatures in the upper region. In contrast, the effect of concrete hydration heat was limited, as heat transfer from the end formwork to the plate exhibited clear attenuation.
• Under combined thermal and prestressing actions, displacements in the reaction prestressing anchor plates were minimal (≤0.20 mm), occurring mainly in the crown and upper-middle regions, while the base remained essentially stable. This confirms the high global stiffness of the composite system, which ensures deformation control.
• Comparison between experimental data and finite element results shows good agreement in the longitudinal displacement pattern of the reaction tensioning anchor plate, with a maximum deviation of 7.5%, confirming the validity of the numerical model.
• The reaction tensioning anchor plate exhibits reversible thermal expansion and contraction. An outward expansion of 0.43 mm (17.7%) was recorded at point w1 as temperature increased from 18 °C to 40 °C, while a corresponding inward contraction of 0.43 mm (17.13%) occurred as temperature decreased from 22 °C to 0 °C.
• Prestress variation correlates directly with temperature changes. A temperature increases from 18 °C to 40 °C resulted in a prestress gain of 1.62–4.21 MPa, corresponding to 0.12–0.3% of the control stress (1395 MPa). Conversely, a temperature drops from 22 °C to 0 °C led to a prestress loss of 1.62–4.21 MPa, representing a reduction of 0.12–0.3%.

While this study provides insights into the mechanical behavior and temperature effects on prestressed concrete beams, several limitations should be noted. The analysis considered only extreme thermal conditions, excluding long-term temperature cycles. The tensioning system was tested solely on variable-section I-beams, limiting its generalizability to other beam types. Additionally, the influence of wind and humidity was not accounted for. Future research could adopt computer vision-based non-contact monitoring, converting pixel-level displacements into physical values via video calibration to continuously track anchor plate and formwork deformation.