- Open Access
Balanced Ratio of Concrete Beams Internally Prestressed with Unbonded CFRP Tendons
- C. Lee^{1}Email author,
- S. Shin^{2} and
- H. Lee^{2}
https://doi.org/10.1007/s40069-016-0171-6
© The Author(s) 2016
- Received: 9 October 2015
- Accepted: 28 September 2016
- Published: 27 December 2016
Abstract
The compression or tension-controlled failure mode of concrete beams prestressed with unbonded FRP tendons is governed by the relative amount of prestressing tendon to the balanced one. Explicit assessment to determine the balanced reinforcement ratio of a beam with unbonded tendons (\( \rho_{pfb}^{U} \)) is difficult because it requires a priori knowledge of the deformed beam geometry in order to evaluate the unbonded tendon strain. In this study, a theoretical evaluation of \( \rho_{pfb}^{U} \) is presented based on a concept of three equivalent rectangular curvature blocks for simply supported concrete beams internally prestressed with unbonded carbon-fiber-reinforced polymer (CFRP) tendons. The equivalent curvature blocks were iteratively refined to closely simulate beam rotations at the supports, mid-span beam deflection, and member-dependent strain of the unbonded tendon at the ultimate state. The model was verified by comparing its predictions with the test results. Parametric studies were performed to examine the effects of various parameters on \( \rho_{pfb}^{U} \).
Keywords
- balanced ratio
- carbon-fiber-reinforced polymer
- modeling
- prestressed concrete
- unbonded
1 Introduction
Carbon Fiber Reinforced Polymers (CFRPs) have a number of valuable advantages: corrosion-free; high strength in tension; lower unit weight than steel; and low linear expansion coefficient. As a substitute material for steels, different types of CFRP have been suggested in various applications of concrete structures, mainly for flexure as internal or external CFRP tendons and for CFRP stirrups (Elrefai et al. 2012; Lee et al. 2013; Han et al. 2015; Lee et al. 2015a, b; Girgle and Petr 2016).
Concrete beams prestressed with an insufficient amount of FRP tendons are subject to a catastrophic brittle failure of the beam resulting from a sudden release of elastic energy at the moment of tensile rupture of the FRP tendons. A section is regarded as the tension-controlled section if the FRP tendon rupture governs the beam failure with a prestressing ratio (ρ _{ pf }) less than the balanced ratio of the prestressing tendon (ρ _{ pfb }). On the other hand, if concrete crushing governs beam failure where ρ _{ pf } is greater than ρ _{ pfb }, the section is regarded as the compression-controlled section (ACI 440.1R-03 2003; ACI 440.4R-04 2011).
The balanced reinforcement ratio of bonded FRP tendons (\( \rho_{pfb}^{B} \)) is presented in ACI 440.4R-04 (2011), and was developed based on the compatibility at a section. Linearly varying strength reduction factors were suggested from 0.65 for compression-controlled sections to 0.85 for tension-controlled sections with CFRP (aramid fiber-reinforced polymer) tendons. Determining \( \rho_{pfb}^{U} \) becomes more challenging for prestressed beams with unbonded FRP tendons, as the tensile stress of the unbonded tendons depends on the averaged elongation of concrete at the level of unbonded tendons along the beam span (Naaman and Alkhairi 1991a, b; Kato and Hayashida 1993; Maissen and De Semet 1995; Grace et al. 2006, 2008; Du et al. 2008; Heo et al. 2013). Consequently, it is expected that \( \rho_{pfb}^{U} \) is always smaller than \( \rho_{{_{pfb} }}^{B} \), and prestressing the beam with a reinforcement ratio of unbodned CFRP tendon (\( \rho_{pf}^{U} \)) greater than \( \rho_{pfb}^{B} \) would preserve the compression-controlled section with a higher degree of plasticity, avoiding an abrupt brittle failure. However, this may result in overdesign and underutilize maximum tensile capacity of unbonded tendon. In order to realize a more structurally reliable and yet cost-effective design of prestressed concrete beams with unbonded CFRP tendons, it seems necessary to have a rational evaluation of \( \rho_{pfb}^{U} \) available.
An estimation of \( \rho_{pfb}^{U} \) is not provided in the ACI 440.4R-04 (2011) or any other literature to authors’ best knowledge. In this study, modeling of tensile strain of unbonded FRP tendon at ultimate state is presented first and compared with test results of the beam prestressed with unbonded CFRP tendons. Based on the developed iterative algorithms, a practical formula of Δf _{ pfu } is presented. The model is then modified and the iterative algorithm determining \( \rho_{pfb}^{U} \) is presented.
2 Equivalent Curvature Blocks at Ultimate State
2.1 Assumptions
2.2 Equivalent Rectangular Curvature Blocks and Unbonded Tendon Strain
At the ultimate loading state, one center rectangular curvature block between the loading points with ϕ _{ p } and L _{ p } and two symmetrically-located adjacent rectangular blocks, each with ϕ _{ eq } and L _{ eq }, were assumed. The use of three equivalent rectangular curvature blocks enables the approximation of beam deflection between the loading points and rotations at the beam end supports, as a result of which the overall beam deflection could closely approximate that of linearly-varying curvature distributions ϕ(x) in Fig. 3b (Fig. 3d). In addition, adopting three rectangular curvature blocks facilitates the integration of curvatures, as this concept assumes the distribution of constant strain at beam sections within a block along the beam span.
For the 4-point loading case, the magnitude (ϕ _{ eq }) and length (L _{ eq }) of the adjacent equivalent rectangular curvature blocks between the loading points and supports in Fig. 3c were determined according to Eq. (1).
where L _{ sp } is the distance between the support and loading point (mm).
For the 3-point loading case, L _{ p } = L/10 could be used (Lee et al. 1999). It is worth mentioning that the model adjusts the size of the equivalent blocks for a given magnitude of L _{ p } so that the beam end rotation and mid-span deflection become the same as those obtained from the linearly varying curvature distributions between characteristic sections.
Expressions of multipliers for Λ and Ω in \( \phi_{eq} = \frac{3}{4} \cdot \frac{{(\Lambda )^{2} }}{\Omega } \) and \( L_{eq} = - \frac{2}{3} \cdot \frac{\Omega }{\Lambda } \) (Eq. (2)).
Sections | L _{ s } | ω _{ s } | κ _{ s } | |
---|---|---|---|---|
Locations | Symbols (s) | |||
Beam end | es | \( - \frac{{M_{es} }}{{M_{us} }}L_{sp} \) | L _{ es } | ω _{ es }(3L _{ sp } - L _{ es }) |
Cracked section | cs | \( \frac{{M_{es} + M_{cs} }}{{M_{us} }}L_{sp} \) | − (L _{ cs } + L _{ us }) | ω _{ cr }(L _{ cs } + 2L _{ ys } + 3L _{ us }) |
Yield section | ys | \( \frac{{M_{ys} - M_{cs} }}{{M_{us} }}L_{sp} \) | − (L _{ ys } + L _{ us }) | ω _{ ys }(L _{ ys } + 2L _{ us }) |
Ultimate section | us | \( \left( {1 - \frac{{M_{ys} }}{{M_{us} }}} \right) \cdot L_{sp} \) | − L _{ us } | ω _{ us } · L _{ us } |
2.3 Moment and Curvature at Characteristic Sections
3 Iterative Algorithm for the Prediction of ɛ _{ pfu } at Ultimate State
3.1 Modified Algorithm to Include Frictional Losses
- (1)At the beginning of iteration, assume Δɛ _{ pfu } as a fraction of the difference between the strain of the maximum stress of the CFRP tendon (ɛ _{ pfm }) and the sum of the initial prestressing strain (ɛ _{ pfi }) and decompression strain (ɛ _{ dc }):$$ \Delta \varepsilon_{pfu,s} = \alpha \cdot \left\{ {\varepsilon_{pfm} - (\varepsilon_{pfi} + \varepsilon_{dc} )} \right\},\,0.0 \le \alpha \le 1.0 $$(12a)$$ \varepsilon_{pfu,s,o} = \varepsilon_{pfi} + \varepsilon_{dc} + \Delta \varepsilon_{pfu,s} $$(12b)
- (2)
Using \( \varepsilon_{pfu,s,o}^{{}} \), evaluate (M _{ es }, ϕ _{ es }), (M _{ cs }, ϕ _{ cs }), (M _{ ys }, ϕ _{ ys }), and (M _{ us }, ϕ _{ us }) based on Eqs. (6), (7), (9), and (11), respectively. Then obtain the characteristic lengths from Table 1.
- (3)
From Eq. (2), obtain ϕ _{ eq } and L _{ eq }. From Eq. (5), find \( \Delta \varepsilon_{pfu} = \frac{{\Delta_{eq} + \Delta_{p} }}{L} \).
- (4)Obtain the revised Δɛ _{ pfu,s } resulting from the losses by wobble friction (κ) and curvature friction (η).$$ \theta_{s} = \left\{ {\begin{array}{*{20}c} {\frac{1}{2} \cdot \phi_{u} \cdot L_{p} ( = \theta_{us} )} & {\text{for\,ultimate\,section}} \\ {\phi_{u} + \int_{{L_{p} /2}}^{{L_{s} }} {\phi_{eq} \cdot \left\langle {\frac{{L_{s} - L_{p} /2}}{{L_{eq} }} - 1} \right\rangle \cdot dx} } & {\text{for\,other\,sections}} \\ \end{array} } \right. $$(13a)$$ \Delta \varepsilon_{pfu,s} = e^{{ - (\kappa \cdot L + \eta \cdot \theta_{s} )}} \cdot \Delta \varepsilon_{pfu} $$(13b)
where, \( < E > = \left \{ {\begin{array}{*{20}c} {L_{eq} } & {if\,\,E \ge 0} \\ {L_{s} } & {if\,\,E < 0} \\ \end{array} } \right. \)
- (5)Find the updated value of \( \varepsilon_{pfu,s,n} \) for each critical section by:$$ \varepsilon_{pfu,s,n} = \varepsilon_{pfi} + \varepsilon_{dc} + \Delta \varepsilon_{pfu,s} $$
- (6)
If \( \hbox{max} \left| {\frac{{\varepsilon_{pfu,s,n} - \varepsilon_{pfu,s,o} }}{{\varepsilon_{pfu,s,o} }}} \right| \le \) tolerance, then the strain of the unbonded tendon is converged with \( \varepsilon_{pfu,s} = \varepsilon_{pfu,s,n} \). Otherwise, let \( \varepsilon_{pfu,s,o} = \varepsilon_{pfu,s,n} \) and repeat from step 2).
3.2 Validity of the Model
Mechanical properties of tendons, auxiliary bonded reinforcement and stirrup.
Types | Diameter (mm) | Effective area (mm^{2}) | CFRP | Steel | |||
---|---|---|---|---|---|---|---|
f _{ pfm } or f _{ bfm } (MPa) | E _{ pf } or E _{ bf } (MPa) | ɛ _{ pfm } or ɛ _{ bfm } (%) | f _{ y } (MPa) | ||||
Tendons | CFCC | 7.5 | 30.4 | 1880 | 144,000 | 1.30 | – |
10.5 | 55.7 | 1880 | 144,000 | 1.30 | – | ||
15.2 | 113.6 | 1750 | 135,000 | 1.30 | – | ||
DWC | 9.5 | 70.9 | 2500 | 135,000 | 1.85 | – | |
Auxiliary bars | DWC | 6.0 | 28.3 | 2200 | 139,000 | 1.58 | – |
Steel | 6.0 | 31.6 | – | – | – | 420 | |
Stirrups | DWC | 6.0 | 28.3 | 2200 | 139,000 | 1.58 | – |
Steel | 6.0 | 31.6 | – | – | – | 420 |
Details of tested beam specimens.
Specimens | Prestressing reinfocement ratio \( \rho_{pf}^{U} /\rho_{pfb}^{B} \) | Initial prestrssing f _{ pfi }/f _{ pfm } | Loading type, L _{ sp } (mm) | Reinforcement | ||||||
---|---|---|---|---|---|---|---|---|---|---|
Concrete compressive strength, \( f_{c}^{'} \) (MPa) | Unbonded FRP tendons | Bonded reinforcing bar | ||||||||
Types | Diameter (mm) | A _{ pf } (mm^{2}) | Types | Diameter (mm) | A _{ bf } (mm^{2}) | |||||
RU50 | 0.53 | 0.5 | 4-point (900) | 35 | CFCC | 10.5 | 55.7 | Steel | 6.0 | 63.2 |
RU70 | 0.37 | 0.7 | 4-point (900) | 35 | CFCC | 10.5 | 55.7 | Steel | 6.0 | 63.2 |
RO50 | 1.51 | 0.5 | 4-point (900) | 35 | CFCC | 7.5 | 30.4 | Steel | 6.0 | 63.2 |
CFCC | 15.2 | 113.6 | ||||||||
RO55 | 1.85 | 0.55 | 4-point (900) | 40 | DWC | 9.5 | 70.9 | CFRP | 6.0 | 84.9 |
TB45 | 0.84 | 0.45 | 4-point (1200) | 40 | DWC | 9.5 | 141.8 | CFRP | 6.0 | 56.6 |
TO45 | 1.36 | 0.45 | 4-point (1200) | 40 | DWC | 9.5 | 70.9 | CFRP | 6.0 | 56.6 |
DWC | 9.5 | 141.8 |
Comparisons of model predictions with test results.
Specimens | Prestressing reinforcement ratios | Max. load | Tendon strains | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
\( \rho_{pf}^{U} \) | \( \rho_{pfb}^{B} \) | \( \rho_{pfb}^{U} \) | \( \rho_{pf}^{U} /\rho_{pfb}^{U} \) | Test (kN) | Thy/Test | ɛ _{ pfu } | Δɛ _{ pfu } | |||||
Test (mm/mm) | Model/Test | Test (mm/mm) | Model/Test | Equation (14)/Test | Equation (15)/Test | |||||||
RU50 | 0.0019 | 0.0036 | 0.0014 | 1.36 | 47.2 | 1.03 | 0.0088 | 1.10 | 0.0032 | 1.45 | 1.16 | 1.62 |
RU70 | 0.0019 | 0.0052 | 0.0025 | 0.76 | 58.4 | 0.94 | N.A | N.A | N.A | N.A | N.A | N.A |
RO50 | 0.0053 | 0.0035 | 0.0013 | 4.08 | 80.2 | 0.95 | 0.0082 | 1.06 | 0.0020 | 1.34 | 1.14 | 0.94 |
RO55 | 0.0024 | 0.0030 | 0 | – | 116.1 | 0.81 | 0.0142 | 0.99 | 0.0040 | 0.92 | 0.74 | 0.11 |
TB45 | 0.0021 | 0.0025 | 0.0013 | 1.62 | 118.5 | 1.06 | N.A | N.A | N.A | N.A | N.A | N.A |
TO45 | 0.0034 | 0.0025 | 0.0012 | 2.83 | 150.3 | 0.94 | 0.0112 | 1.13 | 0.0043 | 1.39 | 1.11 | 0.64 |
Average Standard deviation | 0.96 0.09 | 1.07 0.06 | 1.27 0.24 | 1.04 0.20 | 0.83 0.63 |
The model was able to predict an ultimate load capacity for the tested specimens with reasonable accuracy regardless of the sectional shape, prestressing reinforcement ratio, amount of initial prestressing, and type of auxiliary bar. The average (μ) and standard deviation (σ) of the ratios of beam strength at ultimate state predicted by the model to that from the test were 0.96 and 0.09, respectively. Those for the ratios of the predicted values of ɛ _{ pfu }(Δɛ _{ pfu }) to the corresponding experimental values at ultimate state are 1.07 and 0.06 (1.27, 0.24), respectively.
3.3 Practical Equation of Δf _{ ps }
As can be seen in Table 4, Eq. (14) was able to predict test results with a reasonable accuracy. The μ and σ for the ratios of prediction made by Eq. (14) to test result are 1.04 and 0.20, respectively. However, rather scattered predictions were made by Eq. (15) with 0.83 and 0.63 for the μ and σ, respectively. Figure 8 exhibits that better predictions were made by Eq. (14) than Eq. (15) for both values of Δf _{ ps } from test result and model prediction. Similar trends in predicting Δf _{ ps } are shown between Eqs. (14) and (15): overestimation for RU50 and underestimation of RO55. It is worth mentioning that Eq. (15) was empirically suggested by Alkhairi (1991) for concrete beams with unbonded steel tendons in such a way that most of the predicted values are smaller than the experimental results. As a result, Eq. (15) estimated rather conservative values of Δf _{ ps } for the beams with auxiliary bonded steel bars, compared with those from the test results and three equivalent curvature block model (Fig. 8c). In addition, Eq. (15) for 4-point loading case was empirically developed mostly based on the third point test results for the beams with unbonded steel tendons and auxiliary bonded steel bars. Consequently, Eq. (15) resulted in a significant deviation from model predictions and test results when it predicted the values of Δf _{ ps } for the beams with auxiliary bonded CFRP bars (Fig. 8d). Equation (15) was also shown to disregard the effect of L _{ p }/L on Δf _{ ps }.
4 Iterative Algorithm for the Determination of ρ _{ pfb } ^{ U }
4.1 Development of a Balanced Ratio
- (1)
Assume \( \rho_{pfb}^{U} = 0.5 \cdot \rho_{pfb}^{B} \)
- (2)
Let \( \Delta \varepsilon_{pfu,o} = \varepsilon_{pfm} - \varepsilon_{pfi} - \varepsilon_{dc} \) and \( \varepsilon_{pfu,o} = \varepsilon_{pfm} \)
- (3)
Obtain \( A_{pf} = \rho_{pfb}^{U} \cdot b \cdot d_{pf} \)
- (4)
Estimate the tensile force of the tendon by \( T_{pf} = A_{pf} \cdot E_{pf} \cdot \varepsilon_{pfu,o} \). Obtain the L _{ eq } and ϕ _{ eq } from Eq. (2).
- (5)
Obtain the updated value of Δɛ _{ pfu,n } from Eq. (5). Find the renewed unbonded tendon strain by \( \varepsilon_{pfu,n} = \varepsilon_{pfi} + \varepsilon_{dc} + \Delta \varepsilon_{pfu,n} \).
- (6)
If \( \left| {\frac{{\varepsilon_{pfu,n} - \varepsilon_{pfu,o} }}{{\varepsilon_{pfu,o} }}} \right| \le \) tolerance, then let \( \varepsilon_{pfu} = \varepsilon_{pfu,n} \) and go to step 7). Otherwise, let \( \varepsilon_{pfu,o} = \varepsilon_{pfu,n} \) and repeat from step 4).
- (7)
If \( \left| {\frac{{\varepsilon_{pfu} - \varepsilon_{pfm} }}{{\varepsilon_{pfm} }}} \right| \le \) tolerance, then the \( \rho_{pfb}^{U} \) is converged and stop. Otherwise, update \( \rho_{pfb}^{U} \) in Eq. (18) using the revised values of ϕ _{ eq }, L _{ eq } andc _{ eq }. Repeat from step 3).
4.2 Estimation of \( \rho_{pfb}^{U} \) and Failure Modes for Tested Beams
For each tested beam, evaluated values of \( \rho_{pfb}^{B} \) using Eq. (16) and \( \rho_{pfb}^{U} \) via the above algorithms are presented in Table 4. All tested beams presented in Table 4 except for beam RU70 are shown to be compression-controlled sections with a \( \rho_{pf}^{U} \) greater than the corresponding \( \rho_{pfb}^{U} \). Note that the \( \rho_{pf}^{U} \) s of beams RU50, RU70 and TB45 are less than the corresponding \( \rho_{pfb}^{B} \) s. The beam RO55 was found to reach its ultimate state without tensile fracture regardless of the sectional area of unbonded tendon due to a relatively large sectional area of the auxiliary bars.
From Table 4, it can be seen that the average of the ratios of theoretically estimated \( \varepsilon_{pfu} \) values to experimentally obtained ones is 1.07. A more conservative \( \rho_{pfb}^{U} \) is, therefore, implied by the model with its greater prediction on the tensile strain of unbonded tendon. From experiments, it was observed that all of the tested beams, including the under-reinforced RU70 beam with a ratio of \( \rho_{pf}^{U} /\rho_{pfb}^{U} \) equal to 0.76, failed in compression-controlled mode by the concrete crushing without tensile fracture of the unbonded tendons (Heo et al. 2013).
For the under-reinforced beam RU70, if the strain of the unbonded tendon increases over its ɛ _{ pfm } without fracture at the ultimate state, the developed model predicts the strain value at the ultimate state corresponding to the 1.05 times ɛ _{ pfm }. This increase in tendon strain seems to be relatively marginal compared with the provided sectional area of the tendon, which was 24 % less than the sectional area for the balanced condition. The compression-controlled failure in beam RU70 can be explained by the marginally increasing tendon strain over ɛ _{ pfm } even for relatively small \( \rho_{pf}^{U} /\rho_{pfb}^{U} \) in conjunction with the conservative overestimation of the model for the tendon strain.
5 Parametric Studies for \( \rho_{pfb}^{U} \)
Balanced ratio with each parameter.
No. | Effects | Initial prestress | Compressive concrete strength | Bonded FRP tendon | Effective depth | Distance between loading points | Balanced ratio | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\( f_{pfi} \) (MPa) | \( f_{pfi} /f_{pfm} \) | \( f_{c}^{'} \) (MPa) | \( f_{c}^{'} /f_{c,std}^{'} \) | A _{ bf } (mm^{2}) | A _{ bf }/A _{ bf,min} | d _{ pf } (mm) | L/d _{ pf } | L _{ p } (mm) | L _{ p }/L | \( \rho_{pfb}^{U} \) | \( \rho_{pfb}^{B} \) | \( \rho_{pfb}^{U} /\rho_{pfb}^{B} \) | ||
1 | Standard | 845 | 0.5 | 40 | 1.0 | 745 | 1.0 | 560 | 26.8 | 4667 | 0.33 | 0.0027 | 0.0039 | 0.68 |
2 | \( f_{pfi} \) | 676 | 0.4 | 40 | 1.0 | 745 | 1.0 | 560 | 26.8 | 4667 | 0.33 | 0.0021 | 0.0033 | 0.63 |
3 | 1014 | 0.6 | 0.0037 | 0.0051 | 0.73 | |||||||||
4 | f _{ c } ^{’} | 845 | 0.5 | 30 | 0.75 | 745 | 1.0 | 560 | 26.8 | 4667 | 0.33 | 0.0021 | 0.0032 | 0.65 |
5 | 50 | 1.25 | 0.0032 | 0.0046 | 0.70 | |||||||||
8 | d _{ pf } | 845 | 0.5 | 40 | 1.0 | 745 | 1.0 | 600 | 25 | 4667 | 0.33 | 0.0029 | 0.0042 | 0.71 |
9 | 520 | 28.8 | 0.0025 | 0.0039 | 0.65 | |||||||||
10 | A _{ bf } | 845 | 0.5 | 40 | 1.0 | 372.5 | 0.5 | 560 | 26.8 | 4667 | 0.33 | 0.0030 | 0.0043 | 0.70 |
11 | 1118 | 1.4 | 0.0024 | 0.0037 | 0.65 | |||||||||
12 | L _{ p } | 845 | 0.5 | 40 | 1.0 | 745 | 1.0 | 560 | 26.8 | 0 | 0 | 0.0015 | 0.0035 | 0.43 |
13 | 2333 | 0.17 | 0.0020 | 0.0036 | 0.55 | |||||||||
14 | 7000 | 0.50 | 0.0032 | 0.0039 | 0.83 |
5.1 Effect of \( f_{c}^{'} \)
5.2 Effect of f _{ pfi }
As shown in Fig. 9b, the increase in f _{ pfi } resulted in an increase in both \( \rho_{pfb}^{B} \) and \( \rho_{pfb}^{U} \). With a larger value of \( f_{pfi} \), less additional tensile strain on the tendon is available at the ultimate state for the balanced beam. This increases the depth of the neutral axis, increasing the area of concrete in compression; hence, the sectional area of tendon increases with the increase in \( f_{pfi} \) for the sectional equilibrium.
5.3 Effect of d _{ pf }
In Fig. 9c, increasing tendencies for both \( \rho_{pfb}^{B} \) and \( \rho_{pfb}^{U} \) and deceasing tendencies for the strains of auxiliary bonded rebars were observed with an increase in d _{ pf }. Since the amount of available tensile strain in the tendon in addition to its initial strain is the same regardless of the d _{ pf }, an increase in d _{ pf } increases the sectional area of concrete in compression due to the increased depth of neutral axis and decreases the strain of the auxiliary bonded re-bars. Equilibrium at a section, therefore, requires a larger sectional area of tendon with a larger d _{ pf }.
5.4 Effect of A _{ bf }
Figure 9d illustrates that with the increase in A _{ bf }, both \( \rho_{pfb}^{B} \) and \( \rho_{pfb}^{U} \) decrease while the strain of auxiliary bars remains almost constant. As A _{ bf } increases, the tensile resistance of bonded auxiliary bars increases. For the beam prestressed with bonded tendon, as the area of A _{ bf } increases, the area of bonded tendon reduces to satisfy sectional equilibrium without change in strain configuration corresponding to the balanced condition. A similar explanation can also be applicable to an unbonded case even though there would be some changes in strain configuration at balanced condition with the increase of A _{ bf } due to the member-dependency of ɛ _{ pfu }.
5.5 Effect of L _{ p }/L
The effect of L _{ p }/L is presented in Fig. 9e. When the ratio of L _{ p }/L increases, the value of \( \rho_{pfb}^{U} \)(\( \varepsilon_{bf}^{U} \)) tends to increase (decrease) in an almost linear fashion. Decrease in the value of \( \varepsilon_{bf}^{U} \) with the increase in L _{ p }/L implies less curvature at critical sections, beam deflection, and ɛ _{ pfu } at ultimate state with the increase in L _{ p }/L. Consequently, a greater area of A _{ pfu } is needed with a smaller value of ɛ _{ pfu } to balance a compressive resultant in a section resulting from a smaller curvature as L _{ p }/L increases.
6 Conclusions
- (1)
The model based on three equivalent curvature blocks was validated by comparing its predictions with test results: the average and standard deviation of the ratios of the predicted to the measured beam strengths (total strain of unbonded tendon) at the ultimate state were 0.96 and 0.09 (1.07 and 0.06), respectively.
- (2)
Performing regression analysis on the values of Δf _{ ps } generated by the model, a practical formula based on the concept of bond reduction factor was suggested for the estimation of Δf _{ ps }. The formula predicted test results and model simulated values with reasonable accuracy, regardless of type of auxiliary bonded bars and different values of L _{ p }/L.
- (3)
Predictions made by the equation provided by ACI 440.4R-04 showed significant deviations from the test results and model predictions, particularly for the beams reinforced with unbonded CFRP tendons and auxiliary bonded CFRP bars.
- (4)
By extending the concept of three equivalent curvature blocks, a stable iterative algorithm in two-loop was developed to find fixed points for the member-dependent quantities of the \( \rho_{pfb}^{U} \).
- (5)
It was found that the values of \( \rho_{pfb}^{U} /\rho_{pfb}^{B} \) remained in a range between 0.43 and 0.83 for the beams considered in this parametric study. Relatively small values of \( \rho_{pfb}^{U} /\rho_{pfb}^{B} \) imply that a beam prestressed with unbonded tendons would fail in a compression-controlled mode with sufficient margin of safety if prestressed with \( \rho_{pf}^{U} \ge \rho_{pfb}^{B} \).
- (6)
From the parametric studies, an increase in \( \rho_{{_{pfb} }}^{U} \) was more heavily influenced by increases in L _{ p }, d _{ pf }, f _{ pfi }, and \( f_{c}^{'} \) in order. However, the increase in \( A_{bf} \) resulted in the decrease in \( \rho_{{_{pfb} }}^{U} \).
- (7)
The suggested model for \( \rho_{pfb}^{U} \) can be of useful guide for a cost-effective design with minimum amount of unbonded CFRP tendons for compression-controlled beam section, avoiding a catastrophic failure resulting from undesired tensile fracture of tendons.
Declarations
Acknowledgments
This research was supported by the Chung-Ang University Research Scholarship Grants in 2016.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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