State-of-the-Art Review on Determining Prestress Losses in Prestressed Concrete Girders

: Prestressing methods were used to realize long-span bridges in the last few decades. For their predictive maintenance, devices and dynamic nondestructive procedures for identifying prestress losses were mainly developed since serviceability and safety of Prestressed Concrete (PC) girders depend on the e ﬀ ective state of prestressing. In fact, substantial long term prestress losses can induce excessive deﬂections and cracking in large span PC bridge girders. However, old unsolved problematics as well as new challenges exist since a variation in prestress force does not signiﬁcantly a ﬀ ect the vibration responses of such PC girders. As a result, this makes uncertain the use of natural frequencies as appropriate parameters for prestress loss determinations. Thus, amongst emerging techniques, static identiﬁcation based on vertical deﬂections has preliminary proved to be a reliable method with the goal to become a dominant approach in the near future. In fact, measured vertical deﬂections take accurately and instantaneously into account the changes of structural geometry of PC girders due to prestressing losses on the equilibrium conditions, in turn caused by the combined e ﬀ ects of tendon relaxation, concrete creep and shrinkage, and parameters of real environment as, e.g., temperature and relative humidity. Given the current state of quantitative and principled methodologies, this paper represents a state-of-the-art review of some important research works on determining prestress losses conducted worldwide. The attention is principally focused on a static nondestructive method, and a comparison with dynamic ones is elaborated. Comments and recommendations are made at proper places, while concluding remarks including future studies and ﬁeld developments are mentioned at the end of the paper.


Introduction
The first applications of prestressing methods to concrete structures go back to the first half of the 20th century. Today prestressing is widely used for many applications, ranging from small members, such as railway sleepers, to more important structures such as bridges, long and light precast flooring, and roofing elements for constructions. Serviceability and safety of Prestressed Concrete (PC) structures rely on the effective state of prestress force [1]. In fact, prestressing methods are principally utilized to reduce deflections and to partially counterbalance the effect of dead and live loads in the case of bridges [2]. As a result, an extreme loss of prestressing may cause excessive deflections or jeopardize the performance of large span PC girders by indicating cracking phenomena [3]. For this reason, devices and dynamic approaches capable of determining prestress losses largely developed. In particular, dynamic Structural Health Monitoring (SHM) techniques also generated for

Works Conducted by Researchers Worldwide
Research was devoted to analyzing the influence of prestress force on the dynamic behavior of PC girders. A state-of-the-art review of the primary contributions on this topic is presented herein, including laboratory, numerical investigations, and testing methods.

Laboratory Investigations
Hop [31] monitored the vibration responses of a number of PC beams. Investigation focused on the influence of prestress forces on frequency and damping. The author found that applying an increase in prestress force, acting unevenly on the member, would increase the vibration frequency. In many cases, it was measured that the application of further levels of prestressing increase would result in drop of vibration frequency.
Similar experimental results were obtained by Saiidi et al. [32] on a PC girder with concentric tendon. The research proved an increase of the first eigenfrequency from 11.41 Hz for the case of null prestressing, to 15.07 Hz for the maximum magnitude of prestressing. Besides, the authors observed that an increase in prestressing seems to influence microcrack closure and, consequently, increment stiffness and natural frequencies of PC beams.
Miyamoto et al. [33] tested the dynamic response of PC girders, strengthened with external tendons. According to their results, prestress forces applied to external tendons influence the frequency vibrations of PC girders.
Lu and Law [34] tested a PC beam with a straight concentric tendon. Two conditions were studied, id est, with and without the prestress force of 66.7 kN. The authors noted that the prestressing induced an increment in the first three eigenfrequencies within a range of 0.4-2.1%.
Xiong and Zhang [35] tested three simply supported PC girders with different configurations of external tendons. The authors observed that the natural frequency initially increased with the increase in prestress force. Vice versa, the natural frequency decreased after cracks induced by prestressing.
Kim et al. [36] tested a PC girder with many damage scenarios of prestress losses. Starting from a state of absence of prestress losses, the prestress force was then gradually reduced to zero. During this unloading process, vibration measurements allowed to estimate reductions of the first four eigenfrequencies up to values of 4.0-4.4% from the initial stage to the final one.
Jang et al. [37] tested a number of six PC beams with a bonded tendon. By applying continuously an increase of prestressing from 0 to 523 kN, the authors observed a progressive increase of the first eigenfrequency from 7.6 to 8.7 Hz.
Noh et al. [38] performed tests on three PC girders with different configurations of tendons. The researchers detected that the natural frequency generally increases as tension force in prestressing steel increases. Additionally, they observed that natural frequencies of PC girders are affected by other parameters, such as beam camber, geometric stiffness of tendon, and stiffness effect of beam-tendon system.
The results of the aforementioned relevant works, id est, of Hop [31], Saiidi et al. [32], Kim et al. [36], and Jang et al. [37], declared that for lower values of prestress force, an increase in prestressing generates an increase in eigenfrequencies, especially for the fundamental frequency. Conversely, for higher levels of prestressing, the rate of increase of the eigenfrequencies tends to decrease. Moreover, changes in vibration frequencies were observed to be higher for smaller eccentricities of prestressing tendon. By considering such investigations, it can generally be claimed that prestress force slightly affects the dynamics of PC girders (Table 1). Nevertheless, relevance of these modifications depends on many characteristics (e.g., cracking and nonlinearity of concrete, bonding and eccentricity of tendon) which generate counterbalancing effects making it difficult to determine an evident relation between dynamic properties and prestress force.

Numerical Investigations
Numerical studies were conducted by addressing the effect of prestress force on the dynamics of PC girders. During simulation of a moving force identification method which assumed the effects of prestressing, Chan and Yung [39] found that the natural frequencies of a PC bridge decrease with an increase in prestress force. This is well-known as the "compression-softening" effect and, in general, occurs in Euler-Bernoulli beams and PC members preserved against crack formation [40][41][42].
Kim et al. [43] investigated prestress loss identifications in PC girders based on the measurement variations of natural frequencies. Comparison between the experimental results obtained by Saiidi et al. [32] and the previsions of their model validated their method.
Law and Lu [44] analyzed the time-domain response of a PC girder under dynamic excitation. By comparing their results of numerical simulations with theoretical findings, the authors identified the prestress force in the time domain by displacement and strain measurements. According to their findings, the natural frequencies decrease as the prestressing increases.
Hamed and Frostig [45] developed a nonlinear solution for modeling the behavior of PC girders with a bonded or an unbonded tendon. Based on the derived governing equations, the authors proved that prestressing does not influence the natural frequencies of PC girders.
Jaiswal [46] underlined that the increase of a PC girder's stiffness (and frequency) depends on the tendon's eccentricity, thus inducing greater moment and stiffening effect along the member.
Limongelli et al. [47] investigated the prediction of early warning signs of deterioration in a PC beam caused by prestress losses. The investigators pointed out that vibration frequencies of PC girders significantly change only under the effects of crack initiation or crack re-opening.
Gan et al. [48] validated the experiments done by Jang et al. [37] and Noble et al. [49] by a Finite Element (FE) model, where the influence of prestressing on the natural frequencies was simulated by the existence of early-age shrinkage cracks inside the concrete.
Bonopera et al. [50,51] found that the fundamental frequency of uncracked PC girders with a parabolic tendon is unaffected by the prestress force because the course of the "compression-softening" theory being cancelled out by the increase of elastic modulus due to the concrete's consolidation/hardening with time [52].
Luna Vera et al. [53] investigated the flexural performance of two PC girders under uncracked and cracked conditions for further applications of SHM. In relation to the uncracked state, the authors affirmed that changes in natural frequency due to prestress losses are negligible.
Looking at the relevant works above mentioned, id est, at Hamed and Frostig [45], Jaiswal [46], Limongelli et al. [47], and Luna Vera et al. [53], no significant agreement between the effect of prestress force and dynamics of uncracked PC girders was observed ( Table 2). As a result, natural frequencies are generally considered inappropriate parameters for determining prestress losses, as also indicated by Saiidi et al. [32] and Bonopera et al. [50,51].

Testing Methods
Testing methods are in general required for estimating prestress losses, and include five typologies (Table 3): (1) static load testing to determine crack initiation or crack re-opening loads to obtain the available compressive stress in the bottom flange of a PC girder [23,28,54,55]. (2) Severing the prestressing tendon by cutting it into a representative exposed length after placing strain gauges on the tendon [23,56]. (3) Relating the tension in the tendon to a vertical deflection recorded when known weights are suspended from it on a representative exposed length [54,57]. (4) Determining the side pressure to close the induced crack in a small cylindrical hole drilled adjacent to the tendon in the bottom flange of a PC girder [55,58]. (5) Vibration testing to determine the natural frequencies and/or dynamic responses of a PC girder [59][60][61][62][63][64][65]. Table 3. Testing methods for determining prestress losses in PC girders.

Reference Testing on a PC Girder Result and Finding
(1) Destructive [23,28,54,55] Static test to determine crack initiation or crack re-opening loads to obtain the compressive stress in the bottom flange It causes damages (2) Destructive [23,56] Severing the prestressing tendon by cutting it into an exposed length after placing strain gauges It causes damages (3) Semidestructive [54,57] Relating the tension in the tendon to a vertical deflection recorded when weights are suspended from it on an exposed length It causes partial damages

Reference Testing on a PC Girder Result and Finding
(4) Nondestructive [55,58] Determining the side pressure to close the induced crack in a small cylindrical hole drilled adjacent to the tendon in the bottom flange The embedded tendon, if is too close to the concrete surface, could be damaged by the deep hole-cut (5) Nondestructive [59][60][61][62][63][64][65][66][67][68] Vibration test to determine natural frequencies and/or dynamic responses It requires an accurate selection of the mode shape Methods 1 and 2 are destructive approaches which cause damages. In Method 1, a large amount of vertical loads must be applied to involve crack initiation, or to force a main crack to form in the same location as a monitored crack.
Method 3 is a semidestructive test for exposed tendons, and involves accurately determining the exposed length for calculations.
Method 4 is a NDT approach for embedded tendons which involves a negligible local damage along the PC span girder. This is not possible when the tendon is too close to the concrete surface because it could be damaged by the deep hole-cut. Notably, further studies in a controlled environment with monitoring of actual prestress forces were suggested in order to validate its applicability for PC girders with a parabolic tendon [55].
Method 5 represents a series of NDT techniques which require local vibration measurements along the PC girder. With regards, Law et al. [66], Li et al. [67], and Xiang et al. [68] performed numerical simulations using the dynamics of PC girders to moving vehicular loads. In their methods, prestress force identifications were obtained using vibration measures if the prestress force in the tendon is assumed as equivalent to an external compressive force applied to the beam ends [69][70][71][72][73][74][75][76][77][78][79][80][81][82]. Consequently, the natural frequency of the PC girder tends to increase with a decrease in prestress force according to the "compression-softening" theory. Notably, whether prestressed beams are subjected to the "compression-softening" effect was extensively discussed as, e.g., in the literature review by Noble et al. [49,83]. Most of the dynamic methods, cited therein, are based on the identified modal characteristics of a PC girder, id est, natural frequencies and/or mode shapes. The modal characteristics depend on the PC girder's stiffness and, accordingly, are affected by the prestress force. Particularly, such vibration-based identifications require an accurate selection of the mode shape to be utilized in the procedures. In fact, selecting the optimal frequency a priori is challenging, and different frequencies yield varying degrees of accuracy in prestress force determinations.

Static NDT Methods
Amongst vibration-based techniques, static identifications using vertical deflections have proved to be reliable methods for axial force identification in beam elements. Indeed, measured vertical deflections take accurately into account the changes of structural geometry of the members due to axial force variations on the equilibrium conditions [40,41,71,74,77,84] (Figure 1a,b). Experimental simulations were as well conducted on members belonging to space frames and trusses [78,79,85]. Likewise, Bonopera et al. [80] verified the feasibility of estimating prestress force in a PC girder using vertical deflections measured by three-point bending tests. It is also worth noting that this approach only adopts static parameters, thus, in contrast to vibration-based techniques, does not require selecting experimental data for use in the algorithms.

Brief on Works Conducted by Bonopera et al. (2018)
The static approach was originally developed for detecting axial force in compressed steel beams [77]. Subsequently, it was employed for identifying prestress forces in PC girders [80]. In this last case, the reference model comprised a simply supported Euler-Bernoulli beam of 250 mm in width, 400 mm in height, and length L of 6.62 m, made with a high strength concrete, and prestressed by a straight unbonded tendon, where the prestress force N was assumed as an external compressive force eccentrically applied to the end constraints N e (Figure 2a). The cross sectional second moment of the area of the PC beam's section I was equal to 1.3333 × 10 9 mm 4 . Besides, a bending deflection v (1) along the aforementioned beam's model, of 0.01 mm in accuracy, was properly approximated by multiplying the corresponding first-order deflection by the "magnification factor" of the second-order effects, id est, according to the "compression-softening" theory [40][41][42] (Figure 2b). Experiments on a PC beam specimen, having the test configuration above mentioned (Figure 2a,b), were arranged in the laboratory of the National Center for Research on Earthquake Engineering (NCREE) of Taipei, Taiwan [80], where a research program, based on testing uncracked PC bridge member prototypes, began in 2015 ( Figure 3). All geometrical dimensions were checked by laser rangefinder and caliper, of 0.01 mm in tolerance, after the PC beam was fixed on the simple supports. First, deflected-shape measurements v (1) along its span, obtained from 27 three-point bending tests with different applied prestress forces N, measured by a load cell placed at both end constraints [86], were examined to verify the accuracy of the assumptions of the beam's model (Figure 2b). The span of the PC beam specimen was specifically instrumented to short term measure such static deflection shapes, with an accuracy of 0.01 mm, by a set of displacement transducers. Temperature and relative humidity of real environment of the PC beam were not continuously recorded during testing. Second, based on the "magnification factor" formula of the "compression-softening" theory [40][41][42], prestress force determinations were obtained using two series of vertical deflections v (1) , id est, those recorded at the quarter v 2 (1) and at the midspan v 4 (1) of the PC beam, respectively. Information regarding the flexural rigidity of the PC beam were in addition required. In detail, average values of the chord elastic modulus E aver of the high strength concrete, with an accuracy of 1 MPa, were estimated by compression tests on a series of cylinders cast at the same time of the PC beam [87] to determine its increment caused by the concrete's consolidation/hardening with time [52].  Table 4 lists the prestress force determinations N a obtained using the vertical deflections v 2 (1) and corresponding experimental values Ψ = F L 3 /E aver I in Equation (8a) (Test 1), as well as the vertical deflections v 4 (1) and corresponding parameters Ψ in Equation (8b) (Test 2), respectively.

Prestress Force Determinations Obtained by Bonopera et al. (2018)
Equations (8a) and (8b) are illustrated in Bonopera et al. [80]. In specific, the nine test combinations represent the best prestress force determinations among the three test repetitions performed (Section 3.1). The chord elastic modulus E aver of the high strength concrete, for each day of execution of the experiments (Section 3.1), was utilized as parameter in the identification process. The corresponding first-order vertical deflections were instead estimated by Equations (4a) and (4b), as similarly reported in Bonopera et al. [80]. Table 4 additionally shows the related percentage errors ∆ = (N a − N) / N. In general, poor estimates N a were obtained when prestressing N equal to 617 and 620 kN were assigned (Figure 2a,b). Vice versa, the test combinations with prestressing that induced second-order effects greater than 6.5%, id est, when N 721 kN, furnished excellent identifications of prestress forces N a . In fact, in this last case, estimation errors were lower, in absolute value, than 6.1%. Sensitivity analyses were elaborated for the prestress force determinations based on Equations (8a) and (8b). The vertical deflections v 2 (1) and v 4 (1) , calculated with 0.01 mm in accuracy by Equations (3a) and (3b) reported in Bonopera et al. [80], and parameter Ψ were modified to generate possible experimental errors. In detail, deflections v 2 (1) , v 4 (1) , and Ψ were alternatively multiplied by 0.99 and 1.01 to reproduce 14 combinations of simulated values for nine different assumed prestress forces N. The average value of the applied vertical loads, F aver = 22.6 kN (Figure 2b), was taken into account in the manipulations. Figure 4 depicts a comparison between the worst determined N a and assumed values N conducted using vertical deflections v 2 (1) and v 4 (1) , both of which yielded a constant error of about ±107 kN. Based on all the results obtained, a favorable correspondence between analytical N and experimental determinations of prestress force N a was found when midspan deflections v 4 (1) were taken into account.

Concluding Remarks
A state-of-the-art review of some important research works conducted worldwide on determining prestress losses in PC girders allowed to analyze various information and trace future developments. Some references affirm that a variation in prestress force does not significantly affect the vibration responses of PC girders. Accordingly, this makes uncertain the use of natural frequencies as appropriate parameters for prestress loss determinations. Vice versa, most of laboratory works show a slight increment in the eigenfrequencies under the increase in prestress force. This behavior is related to the concrete mechanics, and is a main consequence of the effect of crack and microcrack closure along PC girders. However, vibration-based identification methods require an accurate selection of the mode shape because different natural frequencies provide varying degrees of accuracy in prestress force evaluations.
By considering these characteristics, the manuscript focused then on the static NDT method preliminary proposed by Bonopera et al. [80] through laboratory simulations on an uncracked PC beam specimen. The procedure can accurately and instantaneously determine the effective prestress force using vertical deflection measurements, of 0.01 mm in accuracy, under actual ambient conditions [51,81]. The precision of estimation improved when the PC beam was subjected to a high prestress force and, moreover, when midspan deflections were taken into account. Information regarding the flexural rigidity of a PC girder under investigation are also necessary. With regards, an average value of the chord elastic modulus, of 1 MPa in accuracy, must be estimated by compression tests on a set of concrete cores, drilled along its span, at the time of deflection measurements. Besides, the static NDT method does not require any direct measure of the tension force in the tendon and, mostly, in contrast to dynamic NDT ones, does not request selecting experimental data for use in the algorithms (Table 5). Table 5. Characteristics of the static NDT method preliminary proposed by Bonopera et al. [80] for determining prestress losses in PC girders.

Advantages Disadvantages
(1) Precise determinations by vertical deflection measurements of 0.01 mm in accuracy In conclusion, to make the NDT method applicable in situ, further studies should focus on the measurement of vertical deflections induced by bending tests with vehicle loading along PC bridge girders [88][89][90], in which their constraint stiffness should be evaluated with unknown boundary conditions. In fact, static vertical deflections take accurately and instantaneously into account the changes of structural geometry due to prestressing losses on the equilibrium conditions [51,80,81], in turn caused by the combined effects of tendon relaxation, concrete creep and shrinkage, and parameters of real environment of the PC girder as, e.g., temperature and relative humidity [49] (Figure 3a). The FBG-DSM liquid-level system [77,91] is instead an effective measurement device because it can provide bridge deflections over long distances up to 0.01 mm in accuracy (Figure 3b), referred to an absolute point, without any external physical reference and requirements of good environmental conditions, accessibility and visibility in situ.