PSI - Issue 78

Marco Bonopera et al. / Procedia Structural Integrity 78 (2026) 1143–1150

1146

( ) I, shear a v x

( )

.

(5)

( ) tot, shear x

( ) x

v

=

1

N N

−

crE,shear,1

Such a factor coincides with the ratio v I,shear tot,shear ( x ) ( x ). A static bending test with an assigned post – tensioning ( N ) and an additional vertical load ( F ) located at a cross-section, can be executed to measure the total deflection, v tot,shear ( x ) ( x ) [Fig. 1(c) in Bonopera et al. 2018]. No stiffening effect is induced by the straight tendon. Consequently, the ratio v I,shear ( a ) ( x ) / v tot,shear ( x ) ( x ) and the definition of the magnification factor including shear deformation can be utilized to estimate the post – tensioning force ( N a ,shear ) using the following equation, and according to the nondestructive method proposed by Bonopera and Chang (2021) based on the Euler – Bernoulli theory: ( a ) ( x ) / v

  

   

( ) I, shear a x v x ( ) tot, shear v

( )

.

(6)

1  −

,shear N N = a

crE,shear,1

( ) x

In general, the prestressing force identification must be conducted through the following phases: (1) Measure a total deflection along the simply supported PC girder-bridge [ v tot,shear ( x ) ( x )] by following the application of a vertical load ( F ); (2) Determine the first-order critical buckling load N crE,shear,1 [Eq. (4)]; (3) Solve Eq. (6) to identify the prestressing force ( N a ,shear ) by computing the first-order deflection [ v I,shear ( a ) ( x )] [Eq. (1)]. Notably, both deflection measurement [ v tot,shear ( x ) ( x )] and analytical first-order one [ v I,shear ( a ) ( x )] must be assumed at the same cross-section along the PC girder-bridge. 3. Simply supported prestressed concrete girder-bridge specimen The PC girder-bridge specimen used by Bonopera et al. (2018), composed of a high-strength concrete, was taken into consideration (Fig. 3 in Bonopera et al. 2018). Two pinned-end restraints were positioned at its ends for a span L = 6.62 m. Such supports did not allow significant friction. The specimen was post – tensioned by a straight tendon with a small eccentricity e =50 mm . The tendon included seven steel “7 - wire” strands (diameter=15.2 mm) inserted into a metallic pipe which was embedded along the specimen. The pipe was not injected after post-tensioning. The specimen’s rectangular cross-sectional area A =1.0×10 5 mm 2 (b=250mm×h=400mm), whilst its cross-sectional second moment of the area I = 1.3333 × 10 9 mm 4 . Also, its slenderness ratio was 57, whereas its length/height ( L /h) ratio was 17. At one of the specimen ’s ends, a hydraulic oil jack was utilized to create a post – tensioning force ( N 0 ) by pulling the tendon outward. A 1,000 kN load cell was arranged at the other specimen ’s end to measure the effective post – tensioning force ( N 0 ) caused by elastic shortening losses (Table 1). Specifically, average post – tensioning forces ( N 0,aver ) equal to ≈ 618, 722 and 820 kN were applied and measured at a concrete exposure age of 426, 427 and 433 days respectively (Table 1) At each post – tensioning force ( N 0 ), a vertical load ( F ) was assigned by a hydraulic actuator at the midspan with an initial magnitude of ≈ 20.0 kN, then gradually incremented to ≈ 22.5 and ≈ 25.0 kN (Table 1). Short-term transverse deformations were induced along the specimen which, in turn, was characterized by a second order initial curvature due to the post – tensioning by causing two equal bending moments at the supports, and an axial end constraint due to the stiffness of the composite section formed of concrete and tendon (Ilanko 1990; Carpinteri et al. 2014). The tendon was always in contact with the surrounding cross-section. Such a test condition was repeated thrice resulting at last in 27 experiments. The elastic shortening post – tensioning losses were due to the probable effect of the concrete early-age creep and tendon relaxation (Table 1). Vice versa, the early and long-age shrinkage occurred before post – tensioning ( N 0 ). Consequently, the post – tensioning ( N 0 ) closed the shrinkage-type cracks and/or microcracks inside the concrete (Bonopera et al. 2021). Nine Linear Variable Differential Transformer (LVDT) sensors were located in correspondence of the specimen axis ’ s cross-sections, i.e., at i = 0, …, 8 (Fig. 5 in Bonopera et al. 2018). The average measurements of post – tensioning forces ( N 0 and N x ), vertical load ( F ) and deflections v i , at i = 1, …., 7, for a repetition of the test combinations were listed in Table 1. No cracks exhibited during and after experiments. More information on the experiments and test layout are illustrated in Bonopera et al. (2018).