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TECHNICAL PAPER

200
0.003 0.85 f c ' = 34
40 c = 59 mm β 1 c = 44.9 mm (4)
(4) (4) 49
175 0.00097
300 (2) 255 (2)

ε pmi = 0.013 0.0059 961
(3) (1) (3) (3) (1)
45 ε pmi = 0.013 0.00996 506 1116

Cross-section Strain diagram Stress diagram (MPa)
Prestressing tendon Non-prestressing tendon Dimensions in mm
Figure 11: Strain and stress diagrams of the beam section at the balanced condition

Bottom fiber stress, neutral axis, after accounting for prestress losses, which can be
obtained as ε pmi = (956.4 – 288.07) / E f = 0.013.

f b = F pi / A + F pi × e / S b – M d / S b = 3.49 MPa < (0.6 f ' ci = 19.20 MPa) ,
[22]
β 1 = [0.85 – 0.05 × ( f ' c × 145.038 – 4000) / 1000] = 0.7599 (4)
where 'S b ' is the section modulus corresponding to the extreme
6
bottom fiber, S b = 3 × 10 mm . Substituting in Equation (3), the balanced ratio is obtained as
3
The above stress calculations show a camber in the BFRP-PSC 40 0 003
.
0850 7599 0 0053.
.
.
.
b
beam under the permanent loads, in which tensile stresses were 1116 0 003 0 0231 0 013
.
.
.
experienced by the top fibers and compressive stresses were
experienced by the bottom fibers; however, the tensile and Reinforcement ratio
compressive stresses at the top and bottom fibers, respectively, The reinforcement ratio (ρ) can be obtained as
under the permanent loads were found within the specified p
permissible limits. ∑ A α
fi i
ρ = i= 1 (5)
Balanced ratio bd m
where 'A fi ' is the cross-sectional area of the tendon, which is
The balanced ratio (ρ b ) is obtained based on the strain
considered as positive in the tension zone and negative in
compatibility in the cross-section and represents the
the compression zone; α i = f bi / f fu , where 'f bi ' is the stress in the
reinforcement ratio at which the concrete failure in compression
and rupturing of reinforcements in tension occur simultaneously. tendon at the balanced condition; 'b' is the width of the beam;
'd m ' is the distance of the bottommost prestressing tendon from
The balanced ratio is based on several assumptions such as:
[21]
the extreme compression fiber; and 'p' is the total number of
(a) the ultimate compression strain in the extreme concrete fiber
reinforcing materials/ tendons. Labeling the reinforcement
is ε cu = 0.003, (b) the nonlinear behavior of concrete is considered materials/ tendons as: (1) for the lowest prestressing tendon
as equivalent rectangular Whitney’s stress block, and (c) the farthest from the neutral axis, (2) for the second prestressing
ultimate tensile strain in the tendons is 'ε fu '. The balanced ratio tendon of the next layer relatively closer to the neutral axis,
can be obtained as [21] (3) for the bottom non-prestressing tendons, and (4) for the top
f c cu non-prestressing tendons, as shown in Figure 11, where 'c' is
085. 1 f (3) the neutral axis depth at the balanced condition. The strains
b
fu
cu
fu
pmi
in the different reinforcement materials can be obtained as:
where 'β 1 ' is the ratio of the depth of the equivalent rectangular ε 1 = 0.0231, ε 2 = 0.0189, ε 3 = 0.00996, and ε 4 = 0.00097, where the
stress block to the depth of the neutral axis, which is obtained strain in the bottom non-prestressing tendons (ε 3 ) is obtained
from Equation (4) [22] ; 'f ' c ' is the specified compressive strength of from the difference between the ultimate rupture strain (0.0231)
concrete (40 MPa); 'f fu ' is the tensile strength of the prestressing and the initial prestressing strain in the prestressing tendon
tendons (1116 MPa); ε fu = 0.0231; and 'ε pmi ' is the initial prestressing of the same layer after accounting for the prestress losses
strain in the lowest prestressing tendons farthest from the (ε pmi = 0.013). The stresses in the different reinforcement materials

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