PSI - Issue 78

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

1148

the average elastic shear modulus G aver = E aver / [2 (1+  )] exhibited the same increments compared to the value obtained at 28 days (12,733 MPa). Pois son’s ratio  =0.2. Instead, the average compressive cylinder strength ( f ck,aver ) was 66 MPa for all compression tests (Table 2). These results confirmed that the long-term consolidation/hardening is the main factor influencing the increment in the time-dependent elastic moduli of high-strength concretes of PC girder-bridges (Bonopera and Chang 2021). More information on the strain measurement and loading system of the compression tests are described in Bonopera et al. (2018).

Table 2. Measured concrete compressive strength and elastic moduli obtained through compression tests.

Age of concrete (days)

Var. (%)

Var. (%)

f ck (MPa)

f ck,aver (MPa)

E (MPa) 28734 32124 30823 34732 35008 39602 35634 39407 38174

E aver (MPa)

G aver (MPa)

N 0,aver (kN)

Cyl.

Date

A B C

15-Oct-2015

28

–

–

–

30560

–

12733

–

1 2 3 4 5 6

72 72 68 65 59 57

16-Nov-2016 426

618

34870 +14.1 14529 +14.1

17-Nov-2016 427

722

66

37618 +23.1 15674 +23.1

23-Nov-2016 433

820

38791 +26.9 16163 +26.9

5. Analysis of the effective shear deformation Table 1 compares the deflection measurements v i , at i = 1, …., 7 (Fig. 5 in Bonopera et al. 2018), after applying each vertical load ( F ) and regarding a repetition of the three-point bending test’s combinations . The reference initial deflected-shape curve [ v (0) ] was assumed to be the one with the post – tensioning force N 0 . Specifically, the deflection measurements ( v i ) were compared with the corresponding total deflections including shear deformation [ v tot,shear ( a ) ( x )] and obtained by Eq. (2). The related applied post – tensioning ( N 0 and N x ) and average chord elastic modulus E aver (Table 2) at each test execution day were respectively assumed for computation (Table 1). Excluding the deflections v 1 , which exhibited a systematic error due to the LVDT sensor malfunction (at i = 1), the absolute average error between analytical total deflections [ v tot,shear ( a ) ( x )] and deflection measurements ( v i ) was 0.04 mm, corresponding to a average percentage error of − 1.5%. Particularly, considering the cross-sections at i = 2 and 3, a maximum percentage error of − 3.3% was recorded for the 9 test combinations. In Table 1, the same values of total deflections [ v tot,shear ( a ) ( x )] were achieved by using the magnification factor formula including shear deformation [Eq. (5)] and a FE model assuming geometric nonlinearities. In fact, the maximum absolute error between such analytical and numerical deflections was 0.02 mm. In detail, the second-order FE analyses were conducted in STRAND7 (2010) by discretizing the PC girder-bridge specimen into 8 Timoshenko beam elements. The post – tensioning forces ( N 0 and N x ) were externally assigned. Notably, in Table 1, the total deflections denoted “ Analytical – E. – B. ” are the displacements gained by the Euler – Bernoulli theory, and likewise reported in Table 3 in Bonopera et al. (2018). Since the maximum post – tensioning force ( N x = 820 kN; Table 1) was only 7.4% of the average critical buckling load N crE,shear,1,aver = 11,138 kN, the first-order deflections v I,shear ( a ) ( x ) [Eq. (1)] were magnified by a factor equal to [1/(1 – 820 kN/11,138 kN)] = 1.079. The aforementioned outcomes match with the findings regarding (axially unloaded) simply supported beams of rectangular cross-section, where the additional effect of the shearing force on the first-order deflections [ v I ( a ) ( x )] is ≈ 4% for a ratio L /h = 10 (Timoshenko 1946). Accordingly, the compression-softening theory is valid even when the shear deformation, in terms of second-order effects, is considered and the crack formation is precluded. More information on the small-deflection measurements are illustrated in Bonopera et al. (2018). 6. Identification of the existing post – tensioning forces The post – tensioning forces ( N x ) at 426, 427 and 433 days (Table 1) were properly identified through the nondestructive method proposed by Bonopera and Chang (2021) and according to the considerations regarding the use of the magnification factor formula including shear deformation [Eq. (5)] (Section 2). Table 3 depicts the analytical