Analytical solution for the pull-out response of FRP rods embedded in steel tubes filled with cement grout
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Abstract
Fiber-reinforced plastic (FRP) tendons have been widely used for ground anchors in civil engineering. Although various pull-out tests of FRP rods from grout-filled steel tubes have been conducted to simulate ground anchors in rock, there are relatively few theoretical studies reported in the literature for this type of bonded anchorages. The intention of this paper is to present an analytical solution for predicting the maximum pull-out load of FRP rods embedded in steel tubes filled with cement grout. First, the expression of the shear stress along the thickness direction of the grout layer is obtained analytically. The tensile stress in the rod and the interfacial shear stress at the rod–grout interface are formulated at different loading stages. By modeling interfacial debonding as an interfacial shear crack, the pull-out load is then expressed as a function of the interfacial crack length. Finally, based on the Lagrange multiplier method, the maximum pull-out load and the critical crack length are determined. The validity of the proposed analytical solution is verified with the experimental results obtained from literature. It can be concluded that the proposed analytical solution can predict the maximum pull-out load of spiral wound and indented rods embedded in steel tubes filled with cement grout with reasonable accuracy. The proposed solution can be also applied in predicting the pull-out capacity of steel bars from concrete.
Keywords
FRP rod Cement grout Steel tube Anchorage Maximum pull-out load1 Introduction
Fiber-reinforced plastic (FRP) rods are regarded as a better alternative to steel bars due to their high strength-to-weight ratio, resistance to corrosion, and ease of transportation and handling. Integrating fiber optic sensors into the FRP rods allows us to monitor the behavior of anchors satisfactorily [33]. As the FRP rod is widely used in ground anchorages such as grouted anchors embedded in rock, the bond properties between the FRP rod and cement grout have gradually attracted more attention of researchers and engineers.
Erki and Rizkalla [9, 10] introduced detailed anchorages for glass FRP (GFRP) and carbon FRP (CFRP) tendons, aramid fiber ropes and aramid FRP (AFRP) rods giving due considerations of their characteristics. These types of anchorages deserve more attention in practical applications due to the high axial-to-lateral strength of FRP materials [22]. The bonded anchorage is one of the currently-used methods for FRP rods, in which a FRP rod is bonded into a steel pipe or tube filled with cement or resin grout [32]. The steel pipe or tube acts as an infinite rock mass [30]. Pull-out tests of FRP rods from grout-filled steel tubes are generally carried out to simulate the ground anchors in rock [5, 7, 22, 27, 30, 32, 33]. Mckay and Erki [22] found that the performance of cement grouted anchors depends on the confinement, moist curing and stiffness properties of the grout. Budelmann et al. [7] investigated the fatigue behavior of FRP bars anchored in a cylindrical steel tube filled with a quartz sand mortar. The introduction of sand and swelling agent into grout can create pressure on the rod and therefore increase the shear bonding resistance [5, 22, 23]. On the other hand, the shrinkage of cement grout decreases the shear strength [5, 32]. The load-bearing capacity of the anchorage increases as the bonding length and compressive strength of cement grout, and the elastic modulus of the steel sleeve increase [3, 5, 8]. If the steel sleeve too large, however, the effect of its stiffness becomes less significant [2]. Different surface geometries and mechanical properties of FRP rods can yield different bonding resistances and pull-out properties [6, 30]. Moreover, the multi-grouted anchor has been recommended for practical engineering applications due to its higher stiffness and load-bearing capacity than the single grouted anchor [33]. Zhang et al. [34] carried out a test on a full-scale ground anchor with fiber-reinforced polymeric 9-bar tendons and found that the tendons perform satisfactorily in post-tensioning applications. Benmokrane et al. [4] replaced steel tubes with concrete in laboratory tests and with rock in field tests as host media for pull-out tests. Results showed that the bond strength from the laboratory tests is higher than that from the field tests.
Although much experimental work has been carried out on grouted FRP bars anchored in mortar-filled steel tubes, theoretical studies are relatively few in the literature. Zhang et al. [32] presented an analytical model for predicting the tensile capacity of bonded anchorages. In their model, four parameters related to the FRP rod–grout interface, as well as the distribution of the bond stress along the embedment length in the ultimate state, are required to write equilibrium equations. The theoretical results are in good agreement with the test results.
Since FRP rods are a possible replacement of steel bars, current studies for pull-out of metallic bars from cementitious matrix can provide some references on the bonding characteristics between the FRP rod and cement grout. The interaction between the bar and concrete is generally characterized by four different stages [11]. At the first stage, bond efficiency is assured by chemical adhesion without any slip. Then the chemical adhesion breaks down, and interfacial slip occurs. The bond strength and stiffness are assured mostly by the interlocking among the reinforcements at the following stage. Finally, splitting-induced pull-out failure or single pull-out failure occurs depending on the transverse confinement or the thickness of the concrete cover. Li et al. [20] introduced two methods to improve the bond properties by modifying the matrix and the rebar surface. Results showed that both methods can (a) improve the bond performance, (b) increase the interfacial microhardness and (c) reduce the porous region of the interface. Bamonte et al. [1] investigated the size effect of bonding of anchorages to ordinary and high performance concrete, and found that the bond is less size-dependent in high performance concrete due to the greater tensile strength, homogeneity and chemical adhesion with the bar.
Moreover, the models for pull-out of fibers from matrix provide some theoretical guidance. A shear-lag model was frequently applied in theoretical studies, in which the tensile stresses in the matrix are negligible compared with those in the fiber [13]. Based on this model, the behavior of the composites before and after debonding was extensively analyzed [14, 15, 16, 17, 18, 19, 29]. The fracture energy-based criterion [12, 17, 19, 24, 28, 29] was used to describe the interfacial debonding behavior. The advantages of this criterion were detailedly discussed by Stang et al. [29]. Zhang et al. [31] presented an improved model to obtain the stress fields in both bonded and debonded regions at the fiber–matrix interface by considering a pull-out rate-dependent frictional coefficient. The whole process of pulling out a single fiber was then modeled numerically [21]. Naaman and Namur et al. [25, 26] accurately obtained the entire pull-out load-end slip relationship of fibers using a dynamic mechanism, in which the Poisson’s effect, shrink-fit and fiber–matrix misfit theory were incorporated.
Based on the pioneer work by Zhang et al. [32], an analytical solution is proposed in this paper for the maximum pull-out load of FRP rods embedded in steel tubes filled with cement grout. One advantage of the proposed solution is that no assumption is made on the bond stress distribution along the embedment length. Finally, the validity of the proposed solution is verified against some experimental results obtained from the literature.
2 The maximum pull-out load of FRP rods
2.1 Basic assumptions
- a.
The cement grout and steel tube are both linear elastic materials with Young’s moduli E _{c} and E _{s}, respectively.
- b.
The bond between the tube and grout is perfect, i.e., there is no shear slip, elastic slip or debonding at the interface.
- c.
The normal stress is uniform on the cross-section of the steel tube.
- d.
The grout with shear modulus G is in a state of pure shear.
- e.Based on the experimental results from Zhang and Benmokrane [30], the relationship between the shear stress τ and slip δ at the rod–grout interface is multi-linear as shown in Fig. 2. The grout sleeve is modeled as a shear-lag member, whose shear stiffness k is equal to the slope of the ascending portion in the τ–δ curve. Thus, the relationship is given by$$ \tau = k\delta \;\;\;\;\left( {0 \le \delta \le \delta_{1} } \right) $$(1a)$$ \tau = \frac{{\tau_{u} \delta_{2} - \tau_{s} \delta_{1} }}{{\delta_{2} - \delta_{1} }} - \frac{{\tau_{u} - \tau_{s} }}{{\delta_{2} - \delta_{1} }}\delta \;\;\;\;\left( {\delta_{1} < \delta \le \delta_{2} } \right) $$(1b)where τ _{u} and τ _{s} are the shear strength and residual frictional stress at the rod–grout interface, respectively.$$ \tau = \tau_{s} \;\;\;\;\left( {\delta > \delta_{2} } \right), $$(1c)
- f.
All radial effects of the rod, grout and steel tube are neglected.
- g.
Since the radial stiffness of the confining medium (i.e., the steel tube) is relatively large for this type of anchored FRP rods and the shear dilation of the grout is neglected, interfacial debonding at the rod–grout interface is the most likely failure mode.
2.2 The numerical model
3 Experimental verification and discussion
Basic geometrical and mechanical parameters of anchorage specimens from Zhang et al. [32]
No. | Type | D (mm) | t (mm) | b (mm) | L (mm) | E_{c} (GPa) | Poisson’s ratio υ of grout | E_{p} (GPa) | E_{s} (GPa) |
---|---|---|---|---|---|---|---|---|---|
1 | Round sanded + CG1 | 7.5 | 21.75 | 3.0 | 100 | 17.4 | 0.11 | 60.83 | 195 |
2 | Round sanded + CG2 | 7.5 | 21.75 | 3.0 | 100 | 18.6 | 0.11 | 60.83 | 195 |
3 | Round sanded + CG3 | 7.5 | 21.75 | 3.0 | 100 | 22.9 | 0.10 | 60.83 | 195 |
4 | Round sanded + CG4 | 7.5 | 21.75 | 3.0 | 100 | 16.7 | 0.12 | 60.83 | 195 |
5 | Round sanded + CG4 | 7.5 | 21.75 | 3.0 | 200 | 16.7 | 0.12 | 60.83 | 195 |
6 | Round sanded + CG4 | 7.5 | 21.75 | 3.0 | 350 | 16.7 | 0.12 | 60.83 | 195 |
7 | Spiral wound + CG1 | 8.0 | 21.50 | 3.0 | 100 | 17.4 | 0.11 | 43.50 | 195 |
8 | Spiral wound + CG2 | 8.0 | 21.50 | 3.0 | 100 | 18.6 | 0.11 | 43.50 | 195 |
9 | Spiral wound + CG3 | 8.0 | 21.50 | 3.0 | 100 | 22.9 | 0.10 | 43.50 | 195 |
10 | Spiral wound + CG4 | 8.0 | 21.50 | 3.0 | 100 | 16.7 | 0.12 | 43.50 | 195 |
11 | Spiral wound + CG4 | 8.0 | 21.50 | 3.0 | 200 | 16.7 | 0.12 | 43.50 | 195 |
12 | Spiral wound + CG4 | 8.0 | 21.50 | 3.0 | 350 | 16.7 | 0.12 | 43.50 | 195 |
13 | Indented + CG1 | 7.9 | 21.55 | 3.0 | 100 | 17.4 | 0.11 | 163.33 | 195 |
14 | Indented + CG2 | 7.9 | 21.55 | 3.0 | 100 | 18.6 | 0.11 | 163.33 | 195 |
15 | Indented + CG3 | 7.9 | 21.55 | 3.0 | 100 | 22.9 | 0.10 | 163.33 | 195 |
16 | Indented + CG4 | 7.9 | 21.55 | 3.0 | 100 | 16.7 | 0.12 | 163.33 | 195 |
17 | Indented + CG4 | 7.9 | 21.55 | 3.0 | 200 | 16.7 | 0.12 | 163.33 | 195 |
Mixture proportions of four grouts from Zhang et al. [32]
No. | Mixture proportions |
---|---|
CG1 | Type 10 portland cement (ASTM 1) |
CG2 | Type 30 portland cement (ASTM II) + superplasticizer solids (1% by weight of cement) |
CG3 | Type 10 portland cement (ASTM 1) + sand (40% by weight of cement) |
CG4 | Type SF cement (blended Type I cement containing 8% silica fume) + swelling agent (0.004% by weight of cement and other additives) |
Interfacial parameters obtained from Zhang et al. [32]
Type | No. | τ_{u} (MPa) | τ_{s} (MPa) | δ_{1} (mm) | δ_{2} (mm) |
---|---|---|---|---|---|
Round sanded + CG1 | Specimen 1 | 8.2 | 2.8 | 1.31 | 3.86 |
Round sanded + CG2 | Specimen 2 | 7.9 | 2.5 | 1.05 | 6.10 |
Round sanded + CG3 | Specimen 3 | 8.4 | 3.1 | 0.72 | 5.60 |
Round sanded + CG4 | Specimens 4–6 | 8.7 | 2.6 | 0.66 | 4.18 |
Spiral wound + CG1 | Specimen 7 | 12.3 | 3.3 | 2.34 | 7.66 |
Spiral wound + CG2 | Specimen 8 | 7.9 | 2.4 | 2.30 | 6.48 |
Spiral wound + CG3 | Specimen 9 | 12.3 | 3.3 | 1.78 | 7.80 |
Spiral wound + CG4 | Specimens 10–12 | 13.2 | 3.8 | 2.50 | 6.50 |
Indented + CG1 | Specimen 13 | 13.1 | 4.1 | 3.32 | 9.60 |
Indented + CG2 | Specimen 14 | 10.6 | 3.1 | 2.97 | 9.95 |
Indented + CG3 | Specimen 15 | 12.4 | 4.4 | 2.61 | 8.70 |
Indented + CG4 | Specimens 16–17 | 14.4 | 5.6 | 2.90 | 6.40 |
Comparison between analytical results and experimental results from Zhang et al. [32]
Numbers of specimens | Types of Specimens | P _{max} ^{a} (kN) | P _{max} ^{e} (kN) | (P _{max} ^{e} − P _{max} ^{a} )/P _{max} ^{e} × 100 (%) |
---|---|---|---|---|
1 | Round sanded + CG1 | 18.8 | 19.4 | 3.1 |
2 | Round sanded + CG2 | 18.3 | 18.6 | 1.6 |
3 | Round sanded + CG3 | 19.5 | 19.9 | 2.0 |
4 | Round sanded + CG4 | 20.0 | 20.6 | 2.9 |
5 | Round sanded + CG4 | 36.9 | 26.9 | −37.2 |
6 | Round sanded + CG4 | 51.9 | 37.1 | −39.9 |
7 | Spiral wound + CG1 | 30.0 | 30.9 | 2.9 |
8 | Spiral wound + CG2 | 19.4 | 20.0 | 3.0 |
9 | Spiral wound + CG3 | 30.0 | 31.0 | 3.2 |
10 | Spiral wound + CG4 | 31.8 | 33.3 | 4.5 |
11 | Spiral wound + CG4 | 56.2 | 55.6 | −1.1 |
12 | Spiral wound + CG4 | 74.5 | 67.9 | −9.7 |
13 | Indented + CG1 | 32.3 | 32.6 | 0.9 |
14 | Indented + CG2 | 26.2 | 26.7 | 1.9 |
15 | Indented + CG3 | 30.6 | 30.8 | 0.6 |
16 | Indented + CG4 | 35.3 | 35.8 | 1.4 |
17 | Indented + CG4 | 68.0 | 67.6 | −0.6 |
It can be seen from Table 4 that, except for Nos. 5 and 6 specimens, the calculated results are in good agreement with the experimental results. For Nos. 5 and 6 specimens, the surface of FRP rods is round sanded and their embedment lengths are relatively long. The bond between the round sanded rod and grout mainly depends on chemical adhesion and friction once the interfacial slip occurs [30]. In addition, since the elastic modulus of the round sanded rod is relatively low (60.83 GPa), the effect of the radial shrinkage in the rod becomes more significant due to the larger maximum pull-out strains. As a result, the interfacial shear strength τ _{u} and residual frictional stress τ _{s} decrease, which has not been taken into account in the proposed analytical solution. For other specimens, however, the bond between the spiral wound or indented rod and grout is mainly due to the interlocking interaction. The compressive interaction between the spiral or rib and grout provides the resistance for the rod. Thus, the radial shrinkage in these rods has a minor effect on the interfacial shear strength τ _{u} and residual frictional stress τ _{s}. Therefore, the effect of the radial shrinkage in FRP rods should further be taken into account in future studies to predict the maximum pull-out load more accurately.
Basic geometrical and mechanical parameters of anchorage specimens from Zhang and Benmokrane [30]
Numbers of specimens | Types of specimens | D (mm) | t (mm) | b (mm) | L (mm) | E_{c} (GPa) | Poisson’s ratio υ of grout | E_{p} (GPa) | E_{s} (GPa) |
---|---|---|---|---|---|---|---|---|---|
SP1 | Round sanded + EM | 7.5 | 8.95 | 4.8 | 40 | 22.6 | 0.24 | 60.8 | 195 |
SP2 | Round sanded + EM | 7.5 | 8.95 | 4.8 | 80 | 22.6 | 0.24 | 60.8 | 195 |
SP3 | Ribbed + CM | 7.9 | 8.75 | 4.8 | 40 | 26.6 | 0.22 | 163.3 | 195 |
SP4 | Ribbed + CM | 7.9 | 8.75 | 4.8 | 80 | 26.6 | 0.22 | 163.3 | 195 |
SP5 | Ribbed + EM | 7.9 | 8.75 | 4.8 | 40 | 22.6 | 0.24 | 163.3 | 195 |
SP6 | Ribbed + EM | 7.9 | 8.75 | 4.8 | 80 | 22.6 | 0.24 | 163.3 | 195 |
Mixture proportions of grout CM and EM from Zhang and Benmokrane [30]
Type | Mixture proportions |
---|---|
CM | Type 10 SF cement (Blended Type I cement containing 8% silica fume) + sand (50% by weight of cement) + superplasticizer solids (1.0% by weight of cement) |
EM | Type 10 SF cement (Blended Type I cement containing 8% silica fume) + sand (50% by weight of cement) + superplasticizer solids (1.0% by weight of cement) + swelling agent (0.005% by weight of cement) |
Interfacial parameters obtained from Zhang and Benmokrane [30]
Types of specimens | Corresponding specimens | τ_{u} (MPa) | τ_{s} (MPa) | δ_{1} (mm) | δ_{2} (mm) |
---|---|---|---|---|---|
Round sanded + EM | Specimens SP1 and SP2 | 14.85 | 3.93 | 1.22 | 3.25 |
Ribbed + CM | Specimens SP3 and SP4 | 23.75 | 6.78 | 4.49 | 8.99 |
Ribbed + EM | Specimens SP5 and SP6 | 21.54 | 8.06 | 4.22 | 7.89 |
Comparison between analytical results and experimental results from Zhang and Benmokrane [30]
Numbers of specimens | Types of specimens | P _{max} ^{a} (kN) | P _{max} ^{e} (kN) | (P _{max} ^{e} − P _{max} ^{a} )/P _{max} ^{e} × 100 (%) |
---|---|---|---|---|
SP1 | Round sanded + EM | 13.8 | 14.0 | 1.4 |
SP2 | Round sanded + EM | 26.7 | 20.3 | −31.5 |
SP3 | Ribbed + CM | 23.5 | 23.6 | 0.4 |
SP4 | Ribbed + CM | 46.6 | 45.5 | −2.4 |
SP5 | Ribbed + EM | 21.3 | 21.4 | 0.5 |
SP6 | Ribbed + EM | 42.3 | 37.0 | −14.3 |
4 Conclusions
An analytical solution has been presented for predicting the maximum pull-out load of FRP rods embedded in steel tubes filled with cement grout. In the proposed solution, four parameters concerning the rod–grout interface, i.e., the interfacial shear strength, the slip corresponding to the shear strength, the residual frictional stress and the slip when the residual frictional stress first occurs, are needed. The shear stress along the thickness direction of the grout layer, the tensile stress in the rod and the interfacial shear stress at the rod–grout interface have been derived in an analytical manner. By modeling interfacial debonding as an interfacial shear crack, the pull-out load has been expressed as a function of the interfacial crack length. With the help of the Lagrange multiplier method, the maximum pull-out load has been determined. By comparing the analytical solution with the experimental results obtained from literature, it can be concluded that it can predict the maximum pull-out load of spiral wound and indented rods embedded in steel tubes filled with cement grout with reasonable accuracy. But for the rod with round sanded surface and low elastic modulus, the proposed solution seems inapplicable at the present stage. Besides, the proposed model can be in principle extended to reinforced concrete, to predict the pull-out capacity of a steel bar. It is mainly because that the concrete Poisson’s ratio is much smaller than that of the polymeric rods and the shear-lag model is also applicable.
Notes
Acknowledgment
The financial support from the National Natural Science Foundation with Grant No. 50578025, of the People’s Republic of China, is greatly acknowledged.
Open Access
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