METHOD FOR IDENTIFYING PRESTRESS FORCE IN SINGLE-SPAN OR MULTI-SPAN PCI GIRDER-BRIDGES

Abstract

A method for identifying prestress force in single-span or multi-span PCI girder-bridges is provided. The method includes non-destructive steps for obtaining a set of parameters of the PCI girder-bridge under investigation, and combines various analyses to identify the change of prestress force. Therefore, the losses of prestress force are tracked and predicted. The method does not cause structural damages along the PCI girder-bridge, and the cost of the identification is significantly decreased.

Claims

1. A method for identifying prestress force in single-span or multi-span PCI girder-bridges, comprising the steps of: (A) obtaining a total length (L) and a first-order fundamental frequency (f.sub.1,I) of a PCI girder-bridge, and calculating or measuring an initial tangent Young's modulus (E.sub.exp,c,t) and a cross-sectional second moment of area (I.sub.1,I) of the PCI girder-bridge; (B) performing a three-point bending test through a vertical load (F) for measuring a static vertical deflection at the PCI girder-bridge's midspan (v.sub.tot,mid) and a loading parameter (ψ); (C) calculating a non-dimensional prestress force (n.sub.a) by an equation (I): $\begin{array}{cc}{n}_a={\pi }^2\left(1-\frac{\psi }{𝒳v_{\mathrm{tot},\mathrm{mid}}}\right)x^2;& \left(I\right)\end{array}$ wherein x is 1 and χ is 48 when the PCI girder-bridge is a single span of length L; x is 2 and χ is 534.26 when the PCI-girder-bridge is an equidistant two-span of length L; x is 3 and χ is 2356.35 when the PCI-girder-bridge is an equidistant three-span of length L; and (D) determining the prestress force (N.sub.a) by an equation (II): $\begin{array}{cc}{N}_a=\frac{E_{\exp ,c,t}I}{L^2}{n}_a.& \left(\mathrm{II}\right)\end{array}$

2. The method of claim 1, wherein step (A), when the initial tangent Young's modulus (E.sub.exp,c, t) and the PCI girder-bridge's total self-mass per unit length (m.sub.PCI+d) are known, the first-order fundamental frequency (f.sub.1,I) is evaluated.

3. The method of claim 2, wherein step (A), when the PCI girder-bridge is the single-span of length L, the first-order fundamental frequency (f.sub.1,I) is calculated by an analytical solution, the cross-sectional second moment of area (I.sub.1,I) is calculated by an equation (III-1) based on Euler-Bernoulli theory: $\begin{array}{cc}{I}_{1,I}=\frac{4f_{1,I}^2{m}_{\mathrm{PCI}+d}{L}^4}{{\pi }^2{E}_{\exp ,c,t}g};& \left(\mathrm{III}-1\right)\end{array}$ wherein g=9.81 m/s.sup.2.

4. The method of claim 3, wherein the first-order fundamental frequency (f.sub.1,I) is calculated by the analytical solution shown in equation (2): $\begin{array}{cc}{f}_{1,I}=\sqrt{\frac{E_{\exp ,c,t}{I}_{\mathrm{tot},\mathrm{mid}}{\pi }^5+32\lambda f_t{L}^2}{2m_{\mathrm{PCI}+d}\pi L^4}};& \left(2\right)\end{array}$ wherein I.sub.tot,mid is the cross-sectional second moment of area of the PCI girder-bridge's midspan; λ is a first-order coefficient, f.sub.t is a deflected shape of a parabolic tendon.

5. The method of claim 4, wherein the first-order coefficient λ is calculated by equation (3): $\begin{array}{cc}\lambda =\frac{E_t{A}_t}{L_t}\left[\frac{16f_t}{\pi L}-\frac{2L^3}{E_{\exp ,c,t}{I}_{\mathrm{tot},\mathrm{mid}}{\pi }^3}\left(-m_{\mathrm{PCI}+d}\right)\right];& \left(3\right)\end{array}$ wherein E.sub.t is a Young's modulus of the parabolic tendon; A.sub.t is a cross-sectional area of the parabolic tendon; L.sub.t is an effective length of the parabolic tendon.

6. The method of claim 2, wherein step (A), when the PCI girder-bridge is single or multi-span, the first-order fundamental frequency (f.sub.1,I,FE) is calculated by a Finite Element (FE) model, the cross-sectional second moment of area (I.sub.1,I,FE) is consequently determined based on Euler-Bernoulli theory; when the PCI girder-bridge is the single-span of length L, the cross-sectional second moment of area (I.sub.1,I,FE) is calculated by equation (III-2-1): $\begin{array}{cc}{I}_{1,I,\mathrm{FE}}=\frac{4f_{1,I,\mathrm{FE}}^2{m}_{tot}{L}^4}{{\pi }^2{E}_{\exp ,c,t}g};& \left(\mathrm{III}-2-1\right)\end{array}$ when the PCI girder-bridge is the equidistant two-span of length L, the cross-sectional second moment of area (I.sub.1,I,FE) is calculated by equation (III-2-2): $\begin{array}{cc}{I}_{1,I,\mathrm{FE}}=\frac{f_{1,I,\mathrm{FE}}^2{m}_{tot}{L}^4}{4{\pi }^2{E}_{\exp ,c,t}g};& (\mathrm{III}-2\text{-2)}\end{array}$ and when the PCI girder-bridge is the equidistant three-span of length L, the cross-sectional second moment of area (I.sub.1,I,FE) is calculated by equation (III-2-3): $\begin{array}{cc}{I}_{1,I,\mathrm{FE}}=\frac{f_{1,I,\mathrm{FE}}^2{m}_{tot}{L}^4}{20.25{\pi }^2{E}_{\exp ,c,t}g};& \left(\mathrm{III}-2-3\right);\end{array}$ wherein equations (III-2-1) to (III-2-3), g=9.81 m/s.sup.2, m.sub.tot is the PCI girder-bridge's total self-mass per unit length, whereas I.sub.1,I,FE is regarded to the cross-sectional second moment of area (I.sub.1,I) for subsequent steps; wherein eccentricities of parabolic tendon e.sub.1 and e.sub.2, or e.sub.1, e.sub.2 and e.sub.3 are considered in the FE models.

7. The method of claim 1, wherein step (A), when the cross-sectional second moment of area (I.sub.1,I) of the PCI girder-bridge is unknown, and when the PCI girder-bridge is multi-span or single-span, the first-order fundamental frequency (f.sub.1,exp) is measured through free bending vibration tests, whereas the cross-sectional second moment of area (I.sub.1,I,exp) is calculated based on the Euler-Bernoulli theory: when the PCI girder-bridge is the single-span of length L, the cross-sectional second moment of area (I.sub.1,I,exp) is calculated by equation (III-3-1): $\begin{array}{cc}{I}_{1,I,\exp }=\frac{4f_{1,\exp }^2{m}_{tot}{L}^4}{{\pi }^2{E}_{\exp ,c,t}g};& (\mathrm{III}-3\text{-1)}\end{array}$ when the PCI girder-bridge is the equidistant two-span of length L, the cross-sectional second moment of area (I.sub.1,I,exp) is calculated by equation (III-3-2): $\begin{array}{cc}{I}_{1,I,\exp }=\frac{f_{1,\exp }^2{m}_{tot}{L}^4}{4{\pi }^2{E}_{\exp ,c,t}g};& \left(\mathrm{III}-3\text{-2}\right)\end{array}$ and when the PCI girder-bridge is the equidistant three-span of length L, the cross-sectional second moment of area (I.sub.1,I,exp) is calculated by equation (III-3-3): $\begin{array}{cc}{I}_{1,I,\exp }=\frac{f_{1,\exp }^2{m}_{tot}{L}^4}{20.25{\pi }^2{E}_{\exp ,c,t}g};& \left(\mathrm{III}-3-3\right)\end{array}$ wherein equations (III-3-1) to (III-3-3), g=9.81 m/s.sup.2; m.sub.tot is the PCI girder-bridge's total self-mass per unit length; a calibrated cross-sectional second moment of area (I.sub.1,I,cal) is consequently calculated by an equation (IV):
I.sub.1,I,cal=0.93×I.sub.1,I,exp  (IV); wherein the calibrated cross-sectional second moment of area (I.sub.1,I,cal) is regarded as the cross-sectional second moment of area (I.sub.1,I) for subsequent steps.

8. The method of claim 1, wherein step (B), the loading parameter (y) is measured by an equation (V): $\begin{array}{cc}\psi =\frac{{\mathrm{FL}}^3}{E_{\exp ,c,t}I}.& \left(V\right)\end{array}$

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0039] FIG. 1 is a schematic view of a single-span PCI girder-bridge;

[0040] FIG. 2 is a schematic view of a two-span PCI girder-bridge;

[0041] FIG. 3 is a schematic view of a three-span PCI girder-bridge;

[0042] FIG. 7 is the first flow chart of evaluation of the cross-sectional second moment of area of one embodiment of the present invention;

[0043] FIG. 8 is the second flow chart of evaluation of the cross-sectional second moment of area of one embodiment of the present invention;

[0044] FIG. 9 is a schematic view of a single-span PCI girder-bridge of one embodiment of the present invention;

[0045] FIG. 10 is a schematic view of a single-span PCI girder-bridge of one embodiment of the present invention;

[0046] FIG. 11 is a schematic view of a two-span PCI girder-bridge of one embodiment of the present invention;

[0047] FIG. 12 is a schematic view of a three-span PCI girder-bridge of one embodiment of the present invention;

[0048] FIG. 13 is a schematic view of a single-span PCI girder-bridge of one embodiment of the present invention;

[0049] FIG. 14 is a schematic view of a two-span PCI girder-bridge of one embodiment of the present invention;

[0050] FIG. 15 is a schematic view of a three-span PCI girder-bridge of one embodiment of the present invention;

[0051] FIG. 16 is a schematic view of a single-span PCI girder-bridge for determining the vertical load F of three-point bending test of one embodiment of the present invention;

[0052] FIG. 17 is a schematic view of a two-span PCI girder-bridge for determining the vertical load F of three-point bending test of one embodiment of the present invention;

[0053] FIG. 18 is a schematic view of a three-span PCI girder-bridge for determining the vertical load F of three-point bending test of one embodiment of the present invention;

[0054] FIG. 19 is a schematic view which shows the test layout of the single-span PC girder-bridge of the laboratory simulation of the present invention;

[0055] FIG. 20 shows the accelerations (A3, m/s.sup.2) measured at cross-section i=3 at 291 days of prestressing of one embodiment of the present invention;

[0056] FIG. 21 shows the fast Fourier transform of A3 of one embodiment of the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

[0057] [Single-Span PC Girder-Bridge Prototype]

[0058] The PC girder-bridge prototype was composed of a high-strength concrete made in Taiwan, and reinforced with steel rebars and stirrups with a unit weight (ρ.sub.s) of ≈1.23 kN/m.sup.3. The concrete's unit weight was 22.90 kN/m.sup.3. As illustrated in FIG. 19, two pinned-end supports were arranged at its ends for a clear span (L) of 6,870 mm. The ultimate yield strength (σ.sub.uy), Young's modulus (E.sub.t), and unit weight of the strands (ρ.sub.t) were 1,860 MPa, 200 GPa, and 76.65 kN/m.sup.3, respectively. The cross-sectional second moment of area of the PC girder-bridge's concrete-only section (I) was 1.2775×10.sup.9 mm.sup.4. The corresponding cross-sectional area (A) was 9.727×10.sup.4 mm.sup.2. Furthermore, the cross-sectional area of the parabolic tendon (A.sub.t) was 973 mm.sup.2, whereas its effective length was L.sub.t=[1+8/3×(f.sub.t/L).sup.2]×L=6,886 mm.

[0059] [Measurement of Prestress Losses]

[0060] The PC girder-bridge prototype was positioned in a test rig (FIG. 19). At one of its ends, a hydraulic oil jack was used to create a prestress force (N.sub.0×,aver) of ≈600 kN at an age of concrete of 127 days by pulling the parabolic tendon outwardly. A sensor was arranged at both ends to measure the prestress forces N.sub.0×1 and N.sub.0×2 caused by elastic shortening phenomena (Table 1). A mean prestress force (N.sub.0×,aver) of 557 kN was then measured after 7 days of curing of cement mortar, with which the parabolic tendon was injected, that was at a concrete age of 134 days. At this time, the shortening prestress losses were at last of 7.2%. Notably, the end of the 7 days of curing of cement mortar was assumed as the initial time of prestressing. Refer to Table 1, the prestress forces (N.sub.0×,aver) were subsequently measured at durations of 3, 8, 10, 15, 17, 24, 29, 31, 43, 45, 57, and 66 days.

TABLE-US-00001 TABLE 1 Age of Age of Prestress concrete prestressing N.sub.0×1 N.sub.0×2 N.sub.0x, aver losses N.sub.x1 N.sub.x2 F v.sub.1 v.sub.2 v.sub.3 v.sub.4 v.sub.5 v.sub.6 v.sub.7 (days) (days) (kN) (kN) (kN) (%) (kN) (kN) N.sub.x, aver (kN) (mm) (mm) (mm) (mm) (mm) (mm) (mm) 127 — ~629 ~571 ~600 — — — — — — — — — — — — 134 1 586 527 557 −7.2 — — — — — — — — — — — 136 3 586 527 557 −7.2 — — — — — — — — — — — 141 8 586 527 557 −7.2 — — — — — — — — — — — 143 10 585 527 556 −7.3 — — — — — — — — — — — 148 15 585 526 555 −7.5 — — — — — — — — — — — 150 17 585 526 556 −7.3 — — — — — — — — — — — 157 24 584 525 555 −7.5 — — — — — — — — — — — 162 29 583 524 554 −7.7 — — — — — — — — — — — 164 31 583 525 554 −7.7 — — — — — — — — — — — 176 43 582 524 553 −7.8 — — — — — — — — — — — 178 45 582 524 553 −7.8 — — — — — — — — — — — 190 57 581 523 552 −8.0 — — — — — — — — — — — 199 66 579 521 550 −8.3 — — — — — — — — — — — 421 288 533 499 516 −14.0 533 499 516 23.2 1.18 2.13 2.79 3.15 2.86 2.17 1.26 34.2 1.76 3.16 4.15 4.65 4.25 3.23 1.87 42.2 2.18 3.93 5.16 5.75 5.27 4.01 2.33 423 290 533 499 516 −14.0 533 499 516 24.2 1.22 2.21 2.89 3.26 2.97 2.25 1.29 33.2 1.68 3.06 4.01 4.50 4.11 3.10 1.78 42.8 2.22 4.00 5.24 5.85 5.36 4.07 2.35 51.4 2.63 4.80 6.33 7.04 6.45 4.89 2.81 424 291 532 498 515 −14.2 532 498 515 26.5 1.36 2.44 3.21 3.55 3.27 2.48 1.43 34.1 1.76 3.16 4.16 4.59 4.22 3.20 1.82 42.1 2.18 3.93 5.20 5.72 5.26 4.00 2.25 51.9 2.69 4.88 6.45 7.12 6.53 4.96 2.78

[0061] [Free Bending Vibration Tests]

[0062] Free vibrations were generated by breaking a series of steel rebars of a diameter of 8 mm which were installed near the PC girder-bridge's midspan. Its self-mass per unit length (m.sub.PCI) was 2.392 kN/m (concrete+rebars). When the rebars ruptured, the PC girder-bridge was vertically excited by small unbalanced forces. Therefore, its vibrational response was measured along the strong axis. Vibration measurements were repeated thrice at prestressing durations of 288, 290, and 291 days, respectively. The average measurements of the applied prestress forces N.sub.0×1 and N.sub.0×2 for every test day were listed in Table 1.

[0063] [Three-Point Bending Tests]

[0064] A vertical load (F) of different values was applied by a transverse steel beam at the PC girder-bridge's midspan at prestressing durations of 288, 290, and 291 days. Displacement transducers were used to measure the static vertical deflections v.sub.i, for i=0, . . . , 8 (FIG. 19). The results were reported in Table 1.

[0065] [Estimation of Young's Modulus]

[0066] The Young's modulus of the PC girder-bridge prototype was measured by compression tests, according to ASTM C 469/C 469M-14 (Annual Book of ASTM Standards 2016). The results were reported in Table 2. The mean compressive cylinder strength (f.sub.c,aver,28) and the average chord Young's modulus (E.sub.exp,28) at 28 days were 88 and 35,060 MPa, respectively (Table 2). The mean compressive strength (f.sub.c,aver,431) and the average chord Young's modulus (E.sub.exp,431) of the drilled cores at 431 days of concrete curing were instead 92 and 37,889 MPa, respectively, i.e., 4.5 and 8.1% higher than the corresponding values at 28 days.

[0067] In addition, the initial tangent Young's modulus of the high-strength concrete (E.sub.exp,c,431) at 431 days was evaluated by equation (1), according to Model B4-TW, wherein the Young's modulus (E.sub.exp,c,431) is expressed in kg/cm.sup.2:

E.sub.exp,c,431=12,000√{square root over (f.sub.ck,aver,431)}  (1)

TABLE-US-00002 TABLE 2 Age of concrete t N.sub.0x, aver f.sub.ck, aver, t E.sub.exp, c, t (days) (kN) (MPa) (MPa) 28 — 88 35,198 431 516 92 36,054

[0068] [Evaluation of the Cross-Sectional Second Moment of Area—Analytical Solution]

[0069] When the PCI girder-bridge is a single-span, its cross-sectional second moment of area (I.sub.1,I) is determined by substituting the first-order fundamental frequency (f.sub.1,I) into equation (III-1) based on the Euler-Bernoulli theory.

[0070] In equation (III-1), g=9.81 m/s.sup.2. The fundamental frequency (f.sub.1,I) is calculated by the analytical solution, which includes the following equations (2) and (3). I.sub.tot,mid is the cross-sectional moment of area of the PC girder-bridge's midspan (concrete and tendon), which was assumed to be equal to 1.3261×10.sup.9 mm.sup.4, according to the design. λ is a first-order coefficient which is calculated by equation (3).

[0071] The effective cross-sectional second moment of area (I.sub.tot,mid) and that obtained from the aforementioned procedure (I.sub.1,I) were reported in Table 3. According to the results, the value of the cross-sectional second moment of area (I.sub.1,I) evaluated by the analytical solution, and based on the Euler-Bernoulli theory, was reliable. Consequently, its use as parameter within the present invention was implemented in the subsequent calculations.

[0072] [Evaluation of the Cross-Sectional Second Moment of Area—Finite Element Model]

[0073] In the present embodiment, when the PCI girder-bridge is a multi-span, its fundamental frequency (f.sub.1,I,FE) is determined according to the Finite Element (FE) model (Jaiswal O R, 2008, Effect of prestressing on the first flexural natural frequency of beams, Structural Engineering and Mechanics, 28(5):515-524). Its cross-sectional second moment of area (I.sub.1,I,FE) is then determined by substituting the FE fundamental frequency (f.sub.1,I,FE) into equation (III-2), which represents the first-order fundamental frequency of a single-span Euler-Bernoulli beam, wherein m.sub.tot=(m.sub.PCI+m.sub.t)=[m.sub.PCI+(ρ.sub.t×A.sub.1)]=2.4666 kN/m.

[0074] The FE fundamental frequency (f.sub.1,I,FE), cross-sectional second moment of area (I.sub.1,I,FE), and effective cross-sectional second moment of area (I.sub.1,I) obtained from the aforementioned procedure were also reported in Table 3. According to the results, the value of the cross-sectional second moment of area (I.sub.1,I) evaluated by the analytical solution, and based on the Euler-Bernoulli theory, was reliable. Consequently, its use as a parameter within the present invention was implemented in the subsequent calculations.

[0075] [Evaluation of the Cross-Sectional Second Moment of Area—Experimental Method]

[0076] In the present embodiment, when the PCI girder-bridge is single-span or multi-span, and its design parameters are unknown, the flexural rigidity is estimated through free bending vibrations. In short, its first-order fundamental frequency (f.sub.1,exp) is obtained using free bending vibration tests. The test results were shown in FIG. 20 and FIG. 21, wherein acceleration versus time for instrumented section A3 (m/s.sup.2) at 291 days of prestressing was shown in FIG. 20. The Fast Fourier transform for a block size of 65,536 samples, and using Peak Picking method, was instead shown in FIG. 21. The experimental fundamental frequencies (f.sub.1,exp) provided by the seismometer A3 were 15.60, 15.60 and 15.62 Hz at 288, 290 and 291 days of prestressing, respectively. Next, the fundamental frequencies (f.sub.1,exp) were substituted into equation (III-3), based on the Euler-Bernoulli theory, for the cross-sectional second moment of area (I.sub.1,I,exp) of a single-span PCI girder-bridge.

[0077] In equation (III-3), g=9.81 m/s.sup.2; m.sub.tot=(m.sub.PCI+m.sub.t) is the total self-mass per unit length given by the sum of self-mass per unit length of PCI girder-bridge and that of parabolic tendon.

According to the results (Table 3), and based on the aforementioned calculations, when E.sub.exp,c,431=36,054 MPa is brining into the equations, I.sub.1,I,exp is 1.54285×10.sup.9 mm.sup.4 at 288 and 290 days, whereas is equal to 1.54680×10.sup.9 mm.sup.4 at 291 days of prestressing. Next, a calibrated cross-sectional second moment of area (I.sub.1,I,cal) is obtained by the calibration equation (IV). The results of calibration were also reported in Table 3.

TABLE-US-00003 TABLE 3 Age of prestressing f.sub.1, I I.sub.1, I I.sub.1, I f.sub.1, I, FE I.sub.1, I, FE f.sub.1, exp I.sub.1, I, exp I.sub.1, I, cal (days) (Hz) (mm.sup.4) (mm.sup.4) (Hz) (mm.sup.4) (Hz) (mm.sup.4) (mm.sup.4) 288, 290 15.32 1.43209 × 10.sup.9 1.44167 × 10.sup.9 15.17 1.44879 × 10.sup.9 15.60 1.54285 × 10.sup.9 1.43485 × 10.sup.9 291 1.43211 × 10.sup.9 1.44167 × 10.sup.9 1.44879 × 10.sup.9 15.62 1.54680 × 10.sup.9 1.43853 × 10.sup.9

[0078] [Identification of Prestress Forces]

[0079] Firstly, equation (4) is the formula of the magnification factor as follows:

[00019]$\begin{array}{cc}{v}_{tot,\mathrm{mid}}d=\frac{v_{I,\mathrm{mid}}}{1-N_x/N_{crE}},& \left(4\right)\end{array}$

[0080] wherein v.sub.tot,mid is the static vertical deflection at the PCI girder-bridge's midspan; v.sub.I,mid is the corresponding first-order static vertical deflection; N.sub.x is the existing prestress force; whereas N.sub.crE is the PC girder-bridge's Euler buckling load. Equation (4) is then transformed into equation (5) with simple manipulations:

[00020]$\begin{array}{cc}{N}_x=N_{\mathrm{crE}}\left(1-\frac{v_{I,\mathrm{mid}}}{v_{\mathrm{tot},\mathrm{mid}}}\right).& \left(5\right)\end{array}$

[0081] A first-order static vertical deflection v.sub.I(x) along a single-span PCI girder-bridge can be determined by equation (6):

v.sub.I(x)=(ψ/12)×(x/L)[¾−(x/L).sup.2]  (6).

[0082] v.sub.I,mid=ψ/48 is gained by substituting x=L/2 into equation (6), wherein the loading parameter ψ is expressed by equation (V):

[00021]$\begin{array}{cc}\psi =\frac{{\mathrm{FL}}^3}{E_{\exp ,c,t}I}.& \left(V\right)\end{array}$

[0083] The Euler buckling load of a single-span PCI girder-bridge is calculated by equation (7):

[00022]$\begin{array}{cc}{N}_{crE}={\pi }^2\frac{E_{\exp ,c,t}I}{L^2}.& \left(7\right)\end{array}$

[0084] The non-dimensional prestress force (n.sub.x) is instead calculated by equation (8):

[00023]$\begin{array}{cc}{n}_x=\frac{N_x{L}^2}{E_{\exp ,c,t}I}.& \left(8\right)\end{array}$

[0085] Equation (I) for the non-dimensional prestress force (n.sub.a) was obtained by substituting equation (5), v.sub.I=ψ/48, equation (V), and equation (7) into equation (8).

[0086] The prestress force (N.sub.a) can consequently be identified by substituting n.sub.a into equation (II), which is transformed from equation (8).

[0087] At last, the prestress force (N.sub.a) is identified by substituting the initial tangent Young's modulus (E.sub.exp,c, t); the cross-sectional second moment of area obtained from different procedures, including I.sub.1,I from the analytical solution, I.sub.1,I,FE from the FE model, and I.sub.1,I,cal from free bending vibration tests and subsequent calibration; and the static vertical deflection (v.sub.tot,mid) measured with the three-point bending test into equation (I) and equation (II). The results were shown in Table 4, wherein the identifications were obtained assuming the initial tangent Young's modulus (E.sub.exp,c, t) and the vertical deflections (v.sub.4) measured at the PC girder-bridge's midspan (FIG. 19).

TABLE-US-00004 TABLE 4 Age of prestressing N.sub.x, aver F N.sub.a Δ N.sub.a, FE Δ.sub.FE N.sub.a, cal Δ.sub.cal (days) (kN) (kN) n.sub.a (kN) (%) n.sub.a, FE (kN) (%) n.sub.a, cal (kN) (%) 288 516 23.2 0.42 466 −9.7 0.47 519 0.6 0.38 414 −19.8 34.2 0.44 480 −7.0 0.48 534 3.5 0.39 429 −16.9 42.2 0.46 502 −2.7 0.50 556 7.8 0.41 451 −12.6 290 516 24.2 0.35 383 −25.8 0.39 437 −15.3 0.30 332 −35.7 33.2 0.41 446 −13.2 0.45 501 −2.9 0.36 396 −23.3 42.8 0.49 535 3.7 0.53 588 14.0 0.44 483 −6.4 51.4 0.50 556 7.8 0.55 610 18.2 0.46 504 −2.3 291 515 26.5 0.29 325 −36.9 0.34 378 −26.6 0.27 301 −41.6 34.1 0.34 375 −27.2 0.39 429 −16.7 0.32 351 −31.8 42.1 0.43 473 −8.2 0.48 526 2.1 0.41 449 −12.8 51.9 0.52 573 11.3 0.57 626 21.6 0.50 549 6.6

[0088] In summary, the method for identifying prestress force in single or multi-span PCI girder-bridges, provided by the present invention, can be performed without causing any structural damage along the PCI bridge. Notably, the structural damage of drilling cores for measuring the initial tangent Young's modulus (E.sub.exp,c, t), when it is necessary, is not serious. The prestress losses can then precisely be predicted through free bending vibration and three-point bending tests. Thus, the cost of identifying prestress force is significantly decreased.

[0089] The aforementioned laboratory simulations were intended to illustrate the embodiments of the subject invention and the technical features thereof, but not for restricting the scope of protection of the subject invention. Other possible modifications and/or variations can be made without departing from the spirit and scope of the invention as hereinafter claimed. Particularly, this is referred to the analytical and experimental evaluation of the initial tangent Young's modulus of concrete at the time of testing, and to the assumption of different geometrical properties and boundary conditions along the PCI girder-bridges. The scope of the subject invention is based on the claims as appended.