The fundamental assumptions made in deriving the nominal strength of FRP-ER strengthened beams are the same as those for conventional concrete beams [MacGre-gor 1998; Nawy 2002; Park and Paulay 1975; Wang 1992] and specified in ACI 318, Section 10.2. The strain distribution is assumed linear and the stress distribution is assumed uniform over the compression zone of concrete (see Figure 5.18). How-ever, due to the addition of FRP-ER, the following assumptions are made:
1. No relative slip or deformation exists between the FRP-ER and concrete substrate to which it is bonded.
2. FRP-ER follows a linear stress-strain relation (Hooke’s law) up to failure (rupture).
The derivation of moment capacities of FRP-ER beams is based on satisfying two basic conditions: the static equilibrium of forces in concrete, steel, and FRP-ER; and the compatibility of strains in concrete, steel, and FRP-ER. Referring to Figure 5.18, the maximum usable strain in concrete is assumed to be 0.003 (εc = εcu
= 0.003). The strain in steel reinforcement at yield is based on Hooke’s law, i.e., the yield stress fy of the reinforcing steel is equal to Es times the steel yield strain, where Es is the modulus of elasticity of steel. Assuming Es = 29,000 ksi, the following values of yield strains are obtained for Grade 40 and Grade 60 reinforcement:
Grade 40 reinforcement:
(5.15a)
Grade 60 reinforcement:
εy y
s
f
= E = 40 000 = ≅
29 000 000, 0 00138 0 0014
, , . .
(5.15b)
The failure strain of FRP composites varies depending on their type and com-position. For design purposes, note that the maximum (ultimate) tensile stress in a conventional reinforced beam corresponds to the steel yield strain value, whereas for a FRP-ER strengthened beam, the maximum tensile stress corresponds to FRP rupture strain. A discussion on the mechanical properties of FRP-ER composites (e.g., carbon and glass fibers) is provided in Chapter 2. The following strain values will be used to establish strain compatibility between various elements of the beam:
Concrete (all strengths) (εc = εcu) 0.003 Steel reinforcement (yield) (εs = εsy) Grade 40 0.0014
Grade 60 0.002 Carbon fiber fabric (rupture) (εfrp = εfrpu) 0.01–0.015 Glass fiber fabric (rupture) (εfrp = εfrpu) 0.015–0.025
To establish the strain compatibility relationships, refer to Figure 5.18b, which shows strains in extreme compression fibers, steel reinforcement, and extreme tension fibers prior to and after strengthening of the beam with the fiber FRP-ER. The strain in the FRP-ER is assumed to be the same as in the extreme tension fiber of the beam, an assumption justified in view of the small thicknesses of the FRP-ER laminate and the interfacial adhesive layer. Figure 5.18b shows various strains as follows:
Before strengthening of beam:
εci= initial strain in extreme compression fibers εsi= initial strain in steel reinforcement
εbi= initial strain (i.e., due to dead loads) in extreme bottom tension fibers prior to FRP-ER strengthening
where the second subscript i refers to the initial conditions corresponding to the nominal strength of the beam without external reinforcement. In parameter εbi, the subscript b denotes the “bonded side” of the beam. In a simply supported beam, the bonded side is its tension side. In a cantilever beam, the top side (not the bottom side) of the beam will be in tension, which will be bonded with FRP reinforcement for strengthening purposes.
Upon installing the external FRP reinforcement (strengthening) with fibers ori-ented in the longitudinal direction, the beam develops additional flexural strength.
An FRP-ER strengthened concrete beam under live loads, sustained loads, temper-ature, and other loads will be subjected to additional strains. These additional strains (Figure 5.18b) are as follows:
εy y
s
f
=E = 60 000 = ≅
29 000 000, 0 00207 0 002
, , . .
εcl = additional strain in extreme compression fibers after strengthening and loading
εsl = additional strain in steel reinforcement after strengthening and loading εci = εfrp = εfrpl = additional strain in extreme tension fibers after strengthening
and loading (note that due to the very small thickness of FRP-ER, the strain in the extreme bottom tension fiber in concrete and the bonded FRP is assumed to be the same)
The subscript l refers to values resulting after beam strengthening and subsequent loading. The total strain values in concrete (εc), steel (εs), and at the extreme tension fibers where FRP-ER is bonded (εb) are defined as follows:
(5.16) (5.17)
(5.18) Strain Compatibility. Referring to similar triangles in Figure 5.18b, the following relations are established. Before installing the FRP-ER (initial condition, typically under the effect of dead loads):
(5.19)
After installing the FRP-ER:
(5.20a)
Noting that the additional strain near the extreme tension fiber (i.e., at the bonded location εbl, which develops after installing the FRP reinforcement and subsequent loading) is equal to the strain in the FRP-ER, εfrp (i.e., εbl = εfrp), Equation 5.20a can be expressed as:
(5.20b)
From Equation 5.19, εbi can be expressed as
(5.21)
Force Equilibrium: The compressive force in concrete (Cc) is
(5.22) The tensile force in steel (Ts) is
(5.23a)
(5.23b) The tensile force in FRP-ER (Tfrp) before or at rupture (εfrp < εfrpu) is
(5.24a) The tensile force in FRP-ER (Tfrp) after rupture is
(5.24b) 5.11.1 DEPTH OF NEUTRAL AXIS (c = kd) WITH AND
WITHOUT FRP-ER
The depth of neutral axis in a beam, kd (Figure 5.19), under service loads with and without FRP-ER is given by the following equations based on elastic analysis similar to conventional steel-reinforced concrete beams. The factor k is sometimes referred to as the neutral axis factor.
FIGURE 5.19 Stress and strain distribution in FRP-ER reinforced concrete beam under service loads.
Cc=0 85. f abc′
Ts=A fs s=A Es s sε (before steel yielding, εs<εyy)
Ts=A fs y=A Es s yε (at and after steel yielding,, εs≥ )εy
Tfrp=Afrpffrp=Afrp(Efrpεfrp)
Tfrp= 0
Neutral axis b
d − c h − c εcl εci
εsi εsl
εbi
fcbc
Af
Ts Tf p fc
d h
c = kd C =
Ast
1 2
εfrpl
5.11.2 VALUEOF NEUTRAL AXIS FACTOR (k) WITH FRP-ER Based on the linear stress-strain relationship (Hookes’ Law),
(5.25) Considering the force equilibrium in a cracked beam with FRP-ER and substituting the above stress values,
(5.26a)
(5.26b)
From Equation 5.20a, we obtain,
Substituting the above values into Equation 5.26b,
(5.26c)
Dividing Equation 5.26c throughout by εc and simplifying,
(5.26d)
(5.26e)
(5.26f)
Quantities (Es/Ec = ns) and (Ef/Ec = nf) can be expressed as modular ratios for steel and FRP reinforcement, respectively, in Equation 5.26f, which is quadratic in c.
With the above substitutions, Equation 5.26f can be rewritten as:
fs=Es sε, ff =Efεf, fc=Ec cε
(5.26g)
Dividing Equation 5.26g throughout by bd yields
(5.26h)
Equation 5.26h can be expressed in terms of the reinforcement ratios, defined as (As/bd) = ρs and (Af/bf) = ρf:
(5.26i)
Rearranging various terms in Equation 5.26i and writing it as a quadratic in c, we obtain
(5.26j)
Solving Equation 5.26j for c,
(5.27a)
which, when simplified, yields
(5.27b)
Noting that c = kd, Equation 5.27b gives the value of the neutral axis factor k for a concrete beam with internal steel reinforcement and external FRP reinforcement under service load conditions:
5.11.3 VALUEOF k WITHOUT FRP-ER
The value of neutral axis factor k for a beam without the external FRP reinforcement can be obtained from Equation 5.28 by setting the terms related to FRP-ER, i.e., ρf
to zero. Thus,
(5.29)
5.11.4 SERVICE LOAD STRESSES (POST-CRACKING) IN STEEL
Strain and stress conditions in a FRP-ER strengthened beam during the post-cracking stage due to service loads are shown in Figure 5.20.
Taking moments about the centroid of compression force resultant (Figure 5.20), and by noting that fs = εsEs and ff = εfEf, we obtain:
(5.30a)
Substituting εf = εb – εbi into Equation 5.30a,
(5.30b)
(5.30c)
Substituting εsEs = fs,s and from Equation 5.20a into Equation 5.30c gives, FIGURE 5.20 Tensile and compressive forces and lever arms at service loads.
Neutral axis
(5.30d)
Rearranging various terms of Equation 5.30d, we obtain:
(5.30e)
Multiply both sides of Equation 5.30e by Es (note that εsEs = fs,s):
(5.30f)
Rearranging and solving Equation 5.30f for fs,s,
(5.30g)
Substituting c = kd into Equation 5.30g yields the following equation, which is same as Equation 9-12 of ACI440.2R-02:
(5.30h)
5.11.5 SERVICE LOAD STRESSES IN FRP-ER
The stresses in the FRP-ER due to loads applied subsequent to its installation can be determined from Hooke’s law. The strain in the FRP-ER at any given loading equals the difference between the final strain in the bonded surface of the beam (εb) that occurs after the load is applied and at the same surface before the loads were applied (εbi). Thus, εbl = εb – εbi (see Equation 5.18) so that
(5.31a) Substitution for εb from Equation 5.20a and c = kd yields
(5.31b)
or, expressing εs as Equation 5.31b can be expressed as
(5.31c)
Under the full service load condition (fs = fs,s) in Equation 5.31c and denoting the stress in FRP-ER ( ff) at service load as ff,s:
(5.31d)
5.12 NOMINAL FLEXURAL STRENGTH OF A SINGLY