Abstract
With the corrosion resistance of glass fiber reinforced polymer bars, the durability of concrete structures can be improved. The tensile strength of a glass fiber reinforced polymer bar is primarily dependent on the tensile strength of the fibers and the total cross sectional area of the fibers, which are determined by the nominal diameter of the bar and the volume fraction of the fibers. Furthermore, the uneven distribution of fibers due to the manufacturing process may have a degrading effect. However, the shear lag effect also influences the strength of the bar, as it causes an uneven normal stress distribution among the individual fibers of the glass fiber reinforced polymer bars. Numerical modeling of a standard tensile test setup of a glass fiber reinforced polymer bar was performed to investigate the intensity of the shear lag effect at varying fiber volume fractions. Fibers and matrix were modeled separately assuming the matrix as an embedding continuum around the individual, non-contacting, evenly arranged, parallel fibers. The results were in good agreement with the manufacturer's data. The shear lag effect was shown to be more prominent at higher fiber volume fractions.
1 Introduction
The durability of today's reinforced concrete structures is reduced due to the corrosion of the steel reinforcement [1]. A possible solution to this problem is the use of non-metallic reinforcement, in particular composite bars instead of steel bars. This usually means the use of Fiber Reinforced Polymer (FRP) bars. FRP bars are increasingly used in constructions, where durability is important, or where an aggressive environment is present [2].
FRP bars are composite materials consisting of fibers and an embedding material (matrix). The fibers in the FRP bar provide the strength and stiffness of the material. Fibers are the main load bearing components. In contrast, the matrix holds the fibers together, transfers the loads, and protects them from external influences.
As a result of the composite nature of the FRP bars, their mechanical behavior is greatly different from that of traditional civil engineering materials. They are anisotropic, and they can withstand significantly more loading in the fiber direction than in the transverse direction [1]. However, the force transfer between the fibers is not perfect. It is mainly influenced by the stiffness of the matrix and causes a shear lag effect inside the bar [2].
1.1 Shear lag effect
Since the mechanical properties of the matrix material are low, the tensile force causes a significant shear deformation within the FRP bar, as can be seen in Fig. 1. At the critical cross section, the tensile strains in the fibers closer to the constrained surface of the bar are higher than those closer to the center of the bar.
Shear lag effect in FRP bars (Source: Authors')
Citation: Pollack Periodica 2025; 10.1556/606.2024.01049
The normal strain is highest at the edge of the bar and lowest in the middle of the bar. Therefore, the normal stress is the lowest in the center of the bar and the highest at the edge of the bar. Failure occurs when the maximum normal stress reaches the tensile strength of the fiber. After fiber rupture, a progressive failure occurs, causing the entire cross section to rupture suddenly, which is unfavorable in terms of safety [3].
Due to the uneven normal stress distribution, the tensile strength of the FRP bar is lower than the idealistic composite strength of the bar. The tensile strength is the integral of the normal stress in the critical cross section with the function of the cross sectional area. When tensile tests are performed, the maximum force at the point of failure can be attained. From this, the average stress can be calculated in the critical cross section, substituting the uneven stress distribution with an even one.
1.2 Tensile test
The most exact way to acquire the tensile strength and modulus of elasticity of a material is to perform a laboratory test. In case of traditional steel bars, it is simple to do, as conventional gripping mechanisms can grab the bare steel bar directly, without causing premature failure in the specimen.
In the case of FRP bars, however, the use of a special anchorage is necessary, since the material cannot withstand the transverse pressure arising during the experiment. For the geometry of these anchors, the ASTM D7205/D7205M-06:2016 [4] and the Canadian CSA S806:2012 (R2017) [5] codes give recommendations. The recommended anchors usually consist of steel tubes and expansive cement grout, to provide rigidity and eliminate any slippage during testing.
The typical tensile test setup can be seen in Fig. 2. During tensile tests, failure of the FRP bars occurs near the end of the anchorages, where the shear lag effect is the most dominant. The fibers will rupture at one end or the other of the free length. In addition to the shear lag effect, imperfections during manufacture will also decrease the tensile strength of the bar.
Tensile test setup (Source: Authors')
Citation: Pollack Periodica 2025; 10.1556/606.2024.01049
Due to the anchorages, the specimen is much longer than for steel bars. The anchorage length is diameter dependent, for large diameters (25–30 mm), the recommended length of the specimen can be more than 2 m. This makes testing FRP bars less productive, compelling researchers to develop reliable analytical solutions and numerical models. The same problem arises also in the case of prestressing [6] FRP tendons, as adequate grip is necessary to tension the tendons.
2 Solutions in the literature
There are many factors that influence the shear lag effect, including the material properties of the constituent materials, the manufacturing process, and the diameter of the bar and the volume fraction of the fiber. This article focuses on the influence of fiber content on the tensile strength of FRP bars.
2.1 Analytical solutions
The Rule of Mixtures (RoM) is commonly used to calculate the mechanical properties of composite plies [1]. It assumes that the fiber and the matrix are homogeneous and linear elastic. It does not consider voids and other imperfections.
Equation (2) predicts the modulus of elasticity of the FRP bar with great precision, but Eq. (1) greatly overestimates the tensile strength of the FRP bar. This is because this formula assumes a uniform stress distribution and perfect geometry.
You et al. [8] proposed a different analytical model that assumes a quadratic stress distribution in the cross section. They performed experiments on Glass Fiber Reinforced Polymer (GFRP) rebars, with different diameter tubes inserted in the middle, making them hollow. In this way, they could measure the strains not just outside the FRP bar but also inside, at the point of the tube. They also used a reduction factor denoted with
Also, with their quadratic model, they could calculate the tensile strength of solid and hollow bars with similar precision using the same reduction factor, showing that quadratic stress distribution is a good estimate for the stress distribution inside FRP bars.
2.2 Numerical solutions
The use of numerical models can help analyze theoretical models and reduce the number of experiments needed to perform. Vo and Yoshitake [9] aimed to develop a numerical model that can predict the tensile strength of FRP bars. They used volume elements with the Reference Volume Element (RVE) method. This way the finite elements are either assigned as fiber or matrix properties adequately, without manually modifying the geometry.
They performed tensile tests on aramid FRP bars with four different diameters: 3, 4, 6 and 8 mm. By comparing the experimental and numerical results, they found a 5–8% deviation. Bars with smaller diameters, tend to have fewer imperfections than bars with larger diameters. This also showed in their data that the deviation increased with the diameter. This is to be expected, as they only modeled the shear lag effect and did not take into account imperfections.
3 Numerical modeling
The downside of numerical modeling is that the number of imperfections is needed from the experiments to calibrate the model. However, this can also be used as an advantage. The shear lag effect is difficult to measure in an experiment since the tensile strength is affected by the imperfections and the inside of the bar is difficult to measure. In a numerical model, however, perfect geometry can be modeled, thus the intensity of the shear lag effect can be grasped.
3.1 Geometry
Instead of modeling with the RVE method as Vo and Yoshitake [9], a simplified approach was used in the ATENA software. The matrix material was modeled with volume elements, while the fibers were embedded as 1D reinforcement. Figure 3 shows how the geometry of the created model looks. The examined bar has grooves, however they are made by grinding into the bar, thus they do not contain continuous fibers. Thus, they have minor influence on the tensile strength. Because of this, only the core of the bar was modeled.
The matrix material modeled with volume elements (left) and the fibers modeled with 1D reinforcement (right) (Source: Authors')
Citation: Pollack Periodica 2025; 10.1556/606.2024.01049
The matrix and the fibers are working together perfectly, and no slip was modeled. Fibers have the same tensile strain as the matrix at the point where they are connected. The key factor is the distance from the center of the bar. Therefore, the fibers had to be placed around a circle, and the farthest circle had to be 6 mm from the center of the bar.
In Fig. 4, it can be seen, how the fibers were placed on four concentric circles initially, making the distance between the fibers 1.5 mm. During the verification step, the number of modeled fibers increased (section 3.4). In the final model the fibers are placed on 10 circles, making the distance between them 0.6 mm.
Cross section with 63 modeled fibers (left) and with 345 modeled fibers (right) (Source: Authors')
Citation: Pollack Periodica 2025; 10.1556/606.2024.01049
The matrix was modeled with an 8 edge polygon cross section, which has a 6 mm inner radius. This geometric simplification made the models run significantly faster, without affecting the results, because the matrix material has low mechanical properties.
3.2 Material properties
The matrix and the fibers were modeled as linear isotropic. The material parameters of the Schöck Combar [11] were used. The failure of the matrix was not taken into account, as it has a higher deformation capacity than the fibers. In the model, the entire cross section is filled with the matrix material. To model the real normal stiffness of the matrix, the modulus of elasticity was multiplied by the volume fraction of the matrix. In this way, the amount of matrix material can be considered by changing the modulus of elasticity. The material properties used in the numerical model are summarized in Table 1.
Material properties of the constituent materials
Component | E | V | Modeled | Modeled | |
(GPa) | (MPa) | (%) | V (%) | E (GPa) | |
Matrix | 3 | – | 25 | 100 | 0.75 |
Fibers | 80 | 3,500 | 75 | 75 | 80.00 |
3.3 Boundary conditions
Half of the setup was modeled, with an anchorage length of 40 cm (according to ASTM D7205/D7205M-06:2016 [4]) and half the free length of 30 cm. At the other end, a rigid loading element was installed and a displacement was defined for the point condition. This represents reality, as in the middle of the setup, the stress distribution can be considered even, because it is far enough from the anchorage, where normal stresses are disturbed.
For the non-linear solution, the Newton-Raphson method was used with a loading increment of 0.1 mm.
3.4 Verification
The thorough verification and validation of the model was necessary to attain sensible results. First, the number of modeled fibers was tested. The number of modeled circles as it can be seen in Fig. 4 was increased until the failure load became constant. In Fig. 5 with 10 circles (345 fibers), the failure load does not change significantly with increasing number of fibers.
Change in failure load with increase in the number of modeled fibers (Sources: Authors')
Citation: Pollack Periodica 2025; 10.1556/606.2024.01049
Second, the mesh was changed until the failure load became constant. In Fig. 6, it can be seen that the mesh was changed considering two parameters: the number of finite elements within the cross section; and the number of finite elements along the length of the bar. In the end, an even 1 mm finite element mesh size was used. Figure 7 (right) shows that the failure load converges as the mesh size decreases.
Course mesh around 6–10 mm (left) fine mesh around 1 mm (right) (Source: Authors')
Citation: Pollack Periodica 2025; 10.1556/606.2024.01049
Change in failure load with decrease in mesh size (Source: Authors')
Citation: Pollack Periodica 2025; 10.1556/606.2024.01049
3.5 Validation
For validation, the stress distribution and failure mode from the literature were used. Even in the simplest model, the stress distribution was already quadratic, as it can be seen in Fig. 8. However, the mesh size needs to be properly adjusted to get back the desired failure mode. There is a stress concentration and the end of the anchorage, near the surface of the bar. Here is where failure happens first. To model this accurately, the finite element mesh must be small enough.
Change in stress distribution with decrease in mesh size near failure load (the mesh becomes finer from top to bottom) (Source: Authors')
Citation: Pollack Periodica 2025; 10.1556/606.2024.01049
4 Results
The results are summarized in Table 2. Since fibers are the main load carrying phase, the failure load increases with the increase in the fiber volume fraction. Figure 9 shows that the increase of the failure load follows a linear trend in function of the fiber volume fraction if there is no imperfection present.
Result of the numerical models
Vf | F | σav | E | Vfsl | Psl | ||
FEM | FEM | ROM | FEM | ROM | FEM | FEM | |
(−) | (kN) | (MPa) | (MPa) | (GPa) | (GPa) | (−) | (−) |
0.20 | 76 | 672 | 805 | 19 | 18 | 0.16 | 0.19 |
0.30 | 101 | 892 | 1,142 | 27 | 26 | 0.23 | 0.24 |
0.40 | 126 | 1,111 | 1,479 | 35 | 34 | 0.30 | 0.26 |
0.50 | 139 | 1,228 | 1,816 | 43 | 42 | 0.33 | 0.34 |
0.60 | 159 | 1,409 | 2,153 | 50 | 49 | 0.39 | 0.35 |
0.70 | 178 | 1,571 | 2,489 | 58 | 57 | 0.44 | 0.37 |
0.75 | 186 | 1,640 | 2,658 | 62 | 61 | 0.46 | 0.39 |
0.80 | 191 | 1,689 | 2,826 | 66 | 65 | 0.48 | 0.41 |
Failure load to fiber volume fraction from numerical data (Source: Authors')
Citation: Pollack Periodica 2025; 10.1556/606.2024.01049
The longitudinal strains in the matrix and the longitudinal stresses in the fibers it can be seen in Fig. 10, near the end of the anchorage. In the matrix, a stress concentration is present at the edges. This represents reality, as with the presence of expansive cement grout, slippage can be eliminated as it is described in ASTM D7205/D7205M-06:2016 [2]. This stress concentration in the matrix also causes a stress concentration in the fibers, resulting in rupture.
Tensile strain distribution in the matrix near the failure load (left) and the stress distribution in the fibers at the same time (right) (Source: Authors')
Citation: Pollack Periodica 2025; 10.1556/606.2024.01049
Schöck manufactures Combar with a 75% fiber volume fraction [8]. According to the tests performed by Weber et al. [9], a 12 mm bar has a failure load of 135 kN. In this case
Degradation parameter from the numerical models to fiber volume fraction (Source: Authors')
Citation: Pollack Periodica 2025; 10.1556/606.2024.01049
Lee and Hwang [5] found a V-shaped connection with the degradation parameter and the volume fraction of the fiber. From the numerical results, this can only be true if the degradation caused by imperfections increases with decreasing fiber volume fraction.
Thus, degradation at low fiber volume fractions is due to nonhomogeneous fiber spread, the distribution of fiber orientation, and the shear lag effect, and as the fiber volume fraction increases, the effects from imperfection decrease, while the degradation from the shear lag effect increases, making the diagram for the degradation parameter V-shaped.
5 Conclusions
The numerical modeling of the tensile test setup of a 12 mm diameter GFRP Schöck Combar was carried out. An idealistic model was created without imperfections. The matrix was modeled with volume elements and the fibers were embedded as 1D elements. The core of the bar was modeled without grooves.
The verification and validation of the model was carried out by considering the number of modeled fibers, the size of the mesh, and the failure mode. The modulus of elasticity of the bars was the same as expected. The tensile strength was higher than in laboratory tests because imperfections were not considered.
Laboratory tests from the literature show a V-shaped connection between the degradation parameter and the fiber volume fraction. The results in this paper showed that the strength degradation of the bar due to the shear lag effect increases linearly with increasing fiber volume fraction. The difference can be explained by the imperfections. At low fiber volume fraction, the degradation from imperfections is dominant, while it decreases with increasing fiber volume fraction, and the shear lag effect becomes more dominant.
The numerical model needs to be validated with laboratory experiments in the future, by measuring the strains both in the middle of the bar and near the anchorages. In addition, the shear lag effect is also needed to investigate at different diameters. This will allow for an understanding of the three main factors that influence the shear lag effect: fiber content, manufacturing imperfections, and bar diameter.
Acknowledgments
This research has been implemented with the support provided from the National Research, Development and Innovation Fund of Hungary, financed under the 2019-1.3.1-KK funding scheme of Project no. 2019-1.3.1-KK-2019-00004.
Supported by the ÚNKP-23-I-BME-57 New National Excellence Program of the Ministry for Culture and Innovation from the source of the National Research, Development and Innovation fund.
The project supported by the Doctoral Excellence Fellowship Programme (DCEP) is funded by the National Research Development and Innovation Fund of the Ministry of Culture and Innovation and the Budapest University of Technology and Economics, under a grant agreement with the National Research, Development and Innovation Office.
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