Page 10 - Open-Access-September-2020

TECHNICAL PAPER

APPENDIX

Analytical Procedure for the Flexural Study on the BFRP-PSC Beams in the Current Work

Prestress losses For the lowest prestressing tendon placed at the farthest
distance from the neutral axis,
The loss of prestress in the concrete beams pre-tensioned with 2 48080 2 48080 65 583200

.
the BFRP tendons in real-life conditions occurs due to elastic f cp1 610 4 42910 6 42910 6 292 MPa,

.
.
shortening, creep, shrinkage, and relaxation of the tendons. It is
notable that, for the sake of completeness of design approach, and hence Δf es1 = 5.56 MPa, and for the second prestressing
these losses have been discussed here, whereas the presented tendon in the next layer closer to the neutral axis,

experimental work may not necessarily simulate it. Therefore, f 2 48080 2 48080 65 583200 192 MPa,
.

the beam was assumed to have a higher initial prestressing cp2 610 4 18 10 6 18 10 6
ratio so that it maintains the same effective prestressing forces and hence Δf es2 = 3.64 MPa.
as applied in the experimental setup after accounting for the
prestress losses. The assumption was made for the sake of 2. Loss of stress due to shrinkage of concrete can be
[21]
validation of the analytical results of the present approach with calculated as Δf s = ε s E f = 0.0006 × 50840 = 30.5 MPa.

the experimental findings. Therefore, the initial prestressing
ratio was assumed as 85.7% of the ultimate tensile strength 3. Loss of stress due to creep of the BFRP tendons can be
of the BFRP tendons, which results in a total initial prestress calculated as Δf cr = 0.1785 f pi = 170.72 MPa, where the initial
of 956.4 MPa and initial prestressing force of 48.08 kN in each prestress f pi = 0.857 × 1116 = 956.4 MPa. Here, the factor
tendon. (0.1785) was extrapolated based on the data given in the
literature [11] where the creep rate of BFRP tendons was
1. Loss due to elastic shortening of concrete can be provided up to 70% initial prestress.
obtained as [21] Δf es = mf cp where 'm' is the modular ratio,
m = E f / E ci = 1.9. Here 'E f ' is the elastic modulus of the 4. Loss of stress due to relaxation of the BFRP tendons
BFRP tendon, presented in Table 1, and 'E ci ' is the can be estimated based on the ratio of the initial stress
elastic modulus of the concrete at the prestress transfer to the ultimate strength of the tendon material as
stage, which can be calculated from the compressive Δf r = 0.085 f pi = 81.29 MPa. Here, the factor (0.085) was
strength of concrete at the prestress transfer stage, extrapolated based on the data given in the literature [23]
on the 14 day of the concrete age, as per ACI 318-19 where the stress relaxation rate of BFRP tendons was
th
2019 , as 'E ci ' = 57000 f' ci . Here 'f' ci ' is expressed in psi provided for initial prestress up to 55%.
[22]

unit, which results in SI unit as E ci = 26.77 GPa, where the
compressive strength of the concrete at the prestress The total loss of prestress becomes 288.07 MPa for the first
transfer was f' ci = 32 MPa; and 'f cp ' is the compressive stress prestressing tendon placed farthest from the neutral axis and

in the concrete at the tendon level, f cp = F pi / A + F pi × e / 286.16 MPa for the second prestressing tendon in the next layer
S b – M d / S b . Here 'S b ' is the section modulus, 'F pi ' is the closer to the neutral axis relatively.
total initial prestressing force, 'e' is the eccentricity of the
total prestressing force, 'A' is the cross-sectional area, and Checking for stresses
'M d ' is the moment due to dead loads. The prestressing
forces in both the tendons were mentioned earlier in this The checking for stresses in the prestressed concrete beams is
section as F p1 = F p2 = 48.08 kN; the cross-sectional area carried out under the permanent loads as follows.

A = 200 × 300 = 6 × 10 mm ; the eccentricities of both Top fiber stress,
4
2
the prestressing forces were e 1 = 105 mm and e 2 = 25 mm,
and hence the eccentricity of the total prestressing force f t = F pi / A – F pi × e / S t + M d / S t = –0.29 MPa < (3 f' ci = 1.41 MPa) ,
[22]

was e = (F p1 × e 1 + F p2 × e 2 ) / (F p1 + F p2 ) = 65 mm; the

moment of inertia of the cross-section was I = 200 × 300 / where the unit of 'f ' ci ' in the formula of the permissible stress
3

[22]
7
12 = 45 × 10 mm ; the section moduli corresponding at top fiber (3 f' ci ) is expressed in psi unit , however, the
4
to both the prestressing tendons levels were S b1 = I / results are converted to the SI unit, 'S t ' is the section modulus
6
e 1 = 4.29 ×10 mm and S b2 = I / e 2 = 18 × 10 mm ; and corresponding to the extreme top fiber, S t = 3×10 mm ,
3
6
3
3
6

2
–6
M d = 200 × 300 × 24 × 10 × 1800 / 8 = 583200 N⋅mm, where f ' ci = 32 MPa as defied earlier, and the remaining terms are also

the density of concrete was 24 × 10 N/mm . defined earlier.
–6
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28 THE INDIAN CONCRETE JOURNAL | JANUARY 2021