Reinforced concrete beams encountered in practice can be singly or doubly reinforced.
Five failure modes of FRP-ER strengthened beams were discussed earlier in Section 5.9. Analyses of these five different failure modes in terms of stresses and strains in concrete, steel, and FRP-ER are presented in the following sections.
5.12.1 TENSION-CONTROLLED FAILURE WITH FRP-ER RUPTURE
As shown in Figure 5.21, steel is assumed to yield (εs≥ 0.005) and the FRP-ER is at the point of incipient rupture (εfrp = εfrpu). The strain in concrete corresponding to the nominal strength of the strengthened beam is obtained from Equation 5.20b:
(5.32a)
or
(5.32b)
Note that εfrpu has been substituted for εfrp in Equation 5.20b, where εfrpu is the strain in the fiber reinforcement at incipient rupture.
ff(εbl)Ef; ff(εb−εbi)Ef; ff=εbEf−εbiEf
Substitution of c = a/β1 in Equation (5.32b) yields
(5.32c)
where β1 is defined by ACI 318-02, Section 10.2.7.3.
Likewise, the strain in steel is also obtained from Equation 5.20b and Equation 5.32a:
(5.33a)
so that
(5.33b)
The maximum strain in FRP-ER just before rupture:
(5.33c) The tension and compression force equilibrium:
(5.34) Solving for the depth of the equivalent rectangular stress block a,
FIGURE 5.21 Force distributton in a tension-controlled failure with FRP rupture.
b
(5.35)
The nominal strength is obtained by taking moments of Ts and Tfrp about the com-pression resultant C:
(5.36a)
The nominal flexural strength of a FRP-ER-strengthened beam is determined as the sum of flexural strength provided by steel and the FRP (Equation 5.36a). The strength reduction factor for tension-controlled failure with steel yield and FRP rupture is φ = 0.9. In addition to the use of the strength reduction factor (φ) required by ACI 318, an additional strength reduction factor (ψf = 0.85) is applied to the flexural strength provided by the FRP-ER only. From Equation 5.4, the design strength φMn ≥ Mu; accordingly, the design strength of the FRP-ER strengthened beam is obtained by modifying Equation 5.36a with the appropriate strength reduc-tion factors as follows:
(5.36b)
5.12.2 TENSION-CONTROLLED FAILURE WITHOUT FRP RUPTURE
Tension-controlled failure without FRP rupture is characterized by an “adequate”
amount of steel yielding (εs≥ 0.005) without FRP-ER rupture (εfrp < εfrpu, Figure 5.22). This failure mode eventually leads to secondary compression (crushing of the concrete beam in the compression zone) or shear compression failure. Due to
FIGURE 5.22 Force distribution in a tension-controlled failure without FRP rupture.
a A f A E
secondary compression failure, assuming that the full usable strain in concrete has occurred is reasonable (i.e., concrete has crushed) so that εc = εcu = 0.003.
Strains in steel reinforcement and FRP-ER corresponding to the nominal strength of the strengthened beam are obtained from Equation 5.20b by substituting εcu for εc. Thus, the strain in steel is obtained from similar triangles shown in Figure 5.18:
(5.37a)
The depth of the neutral axis from the extreme compression fiber, c, and the depth of compression block a are related by a = β1c:
(5.37b)
The strain in FRP is
(5.38a)
which, upon substituting c = a/β1, yields
(5.38b)
Force Equilibrium. In the tension-controlled failure mode with steel yield but without FRP rupture, concrete is eventually crushed (εc = εcu = 0.003). When tension steel has yielded with εs≥ 0.005, the stress in the steel is given by Equation 5.13b.
Equating tensile and compressive forces:
(5.39a) or, expressing ffrp in terms of strain in FRP-ER,
(5.39b) Substituting the value of εfrp from Equation 5.38b into Equation 5.39b yields,
εs εcu
(5.40) Equation 5.40 is a quadratic, which can be solved for a.
Nominal Strength. The nominal strength of the beam can be expressed in terms of a:
(5.41a)
where
The strength reduction factor for this failure remains the same as the tension-controlled failure with FRP rupture, i.e., φ = 0.9. In addition to the use of the strength reduction factor φ required by ACI 318, an additional strength reduction factor, ψf
= 0.85, is applied to the flexural strength provided by the FRP-ER only (ACI 440.2R-02). Noting from Equation 5.4 that φMn≥ Mu, Equation 5.41a can be expressed as:
(5.41b)
where
5.12.3 TENSION- AND COMPRESSION-CONTROLLED FAILUREWITH
STEEL YIELDING AND WITHOUT FRP RUPTURE
The tension- and compression failure mode is characterized by steel yielding (εsy≤ εs < 0.005) without FRP rupture (εfrp < εfrpu, Figure 5.23) that eventually leads to secondary compression or shear compression failure (εc = εcu = 0.003). Analysis of beams with tension- and compression-controlled failure with tension steel yielding and without FRP rupture remains the same as explained in Section 5.12.2. However, the strength reduction factor for this failure mode is (from Figure 5.17 and Section 5.8.10).
0 85. f bac′ 2+[A Efrp frp(εcu+εbi)−A f as y] −β1AfrpEEfrp cuε h= 0
5.12.4 COMPRESSION-CONTROLLED FAILURE WITHOUT STEEL
YIELDINGAND WITHOUT FRP RUPTURE
This mode of failure is characterized by no tension steel yielding (εsy≤ εs), no FRP rupture (εfrp < εfrpu), and the crushing of concrete (εc = εcu = 0.003, Figure 5.24).
The strength reduction factor for the compression-controlled failure mode is 0.7.
However, designing beams for compression-controlled failure without steel yielding and without FRP rupture is not recommended by ACI 318-02. In addition, such compression-controlled failures may not be economical.
5.12.5 BALANCED FAILURE
Balanced failure mode is a hypothetical failure mode that is assumed to occur when strains in extreme tension and compression fibers have reached their limit values simultaneously as follows (also see Figure 5.25):
FIGURE 5.23 Force distribution in a tension- and-compression-controlled failure with steel yielding and without FRP rupture.
FIGURE 5.24 Force distribution in a compression-controlled failure without steel yielding and without FRP rupture.
Strain in concrete: εc = εcu = 0.003
Strain in steel: εs ≥ εsy (this is as a consequence of strain in extreme FRP tension fiber reaching its ultimate value)
Strain in FRP-ER: εfrp = εfrpu
The strain conditions at balanced failure can be expressed as follows from the similar triangles principle:
= 0.015 to 0.025 for glass FRP
Force Equilibrium. Forces in concrete, steel, and the FRP-ER are assumed given, respectively, by Equation 5.22, Equation 5.23a and b, and Equation 5.24a. Substi-tution of these values in force equilibrium Equation 5.34 gives:
(5.43) The above equation can be solved for ab, where ab is the depth of the rectangular stress block corresponding to the balanced condition.
By knowing all other parameters in the force equilibrium equation, the area of FRP reinforcement for a balanced failure (Afrp,b) is:
FIGURE 5.25 Force distribution in a balanced failure with concrete crushing, steel yielding, and FRP rupture.
(5.44)
The nominal flexural strength of the beam is obtained by taking the moments of Ts and Tfrp forces about Cc:
(5.45a)
The strength reduction factor for this failure mode remains the same as tension-controlled failure with FRP rupture, i.e., φ = 0.9 (εs≥ 0.005). In addition to the use of the strength reduction factor (φ) required by ACI 318, an additional strength reduction factor (ψf = 0.85) is applied to the flexural strength provided by the FRP-ER. Again, noting that φMn≥ Mu, Equation 5.45a can be expressed as:
(5.45b)
5.13 TRIAL-AND-ERROR PROCEDURE FOR ANALYSIS