The bearing capacity of tensile steel rods can be improved by fixing additional elements made of carbon composite materials. The easiest way to fix strengthening elements is to use adhesive bonding. Strengthening elements must overlap the weakened steel rod and extend beyond the weakened area. Reinforcement is performed symmetrically to eliminate eccentricities and prevent bending moments in the cross-section. To assess the bearing capacity of a strengthened rod, it is necessary to experimentally determine the strength of the carbon-fiber-reinforced composite material, and the length of adhesive bonding that ensures the structural performance of CFRP. It is necessary to theoretically substantiate the resulting strength of the element and conduct tests to identify its bearing capacity. Numerical computations should be performed using the finite element method and simulation of a steel rod, CFRP, and the adhesive layer.
2.1. Experimental Studies
Let us focus on the rod made of steel with the yield strength of 312.5 MPa and the ultimate strength of 445.8 MPa. The rod was strengthened by CFRP lamellas FibArm Lamella-12/5 that were 1.2 mm thick. According to the manufacturer, the strength of lamellas was 2800 MPa; their modulus of elasticity was 1.65 × 105 MPa. A lamella was fixed to the steel rod by the two-component epoxy adhesive FibArm Resin Laminate+ with a shear strength of 15 MPa.
Before studying the strengthening of steel rods, the strength and rigidity of the lamellas were evaluated. For this purpose, six 25 mm wide specimens of lamellas were tensile tested (
Figure 1).
Two steel plates were bonded to the ends of the specimens to prevent damage or slippage in the clamps of the testing machine. The velocity of clamps under tension varied from 0.5 to 1 mm/s, and all specimens were brought to failure in the process of testing.
As a result of the experiment, the authors identified the length of the adhesive joint, that ensured the consolidated strength performance of lamellas and the steel rod without debonding. Special specimens (
Figure 2) were applied to test the strength of the adhesive bond.
The specimens were two 12 mm thick, 55 mm wide steel plates made of S245 steel. There was a 2 mm gap between the plates, and CFRP lamellas were bonded to them on both sides. Each specimen had a unique length of adhesive bond
l. In total, eight adhesive bond strength tests were conducted for four adhesive bond length values of 170, 200, 280, 310 mm. If the length of the adhesive bond area was minimal, adhesive bond strength was to withstand normal stresses in the lamellas reaching 2125 MPa. As for other lengths of the adhesive bond, lamellas, rather than adhesive layers, were to fail. Strength tests of the adhesive bond were used to identify normal stresses
arising in a lamella to trigger the failure of the adhesive layer. Notations and parameters, used in the course of adhesive strength testing, are shown in
Table 1.
In
Table 1, the sample type is indicated by two numbers—the first number is the length of the CFRP, the second number is the sample number.
The next stage of experimental studies was the tensile testing of parameters, used in the course of adhesive strength testing of specimens, that were made of sheet steel and had an intricate shape (
Figure 3).
The shape of the specimen was intricate in order to make its midsection fail. CFRP was bonded to the steel element along the entire length of a CFRP strip that extended beyond the less strong area, so that the strengths of these areas could ensure the strength of the adhesive joint when ultimate stresses arose in the lamella. It was assumed that the adhesive bond of lamellas along the entire length would make it possible to take maximal advantage of the CFRP strength. The cross-sectional area of the less strong part of the steel specimen was 4.8 cm2; the total area of the CFRP lamellas was 0.6 cm2; the length of the CFRP lamella was 605 mm. CFRP lamellas were bonded using two-component adhesive FibArm Resin Laminate+. Then, four specimens were tensile tested.
Specimen clamping areas were free of lamellas to prevent the CFPR damage when the specimen was subjected to loading. The specimens were clamped by the jaws of the testing machine so that the axes of the specimen and the testing machine coincided. In the course of testing, clamps of the testing machine moved at a speed of 2 mm/s, and the tensile force was registered by the sensors. In addition, strain gauges were attached to the two specimens to measure strain more accurately. Two sensors were mounted on the lamellas of one specimen along its axis. The same sensors were mounted on the lamellas of the other specimen, and two additional sensors were mounted on the side faces of the steel specimen.
To clarify the parameters before the testing of strengthened specimens, non-strengthened specimens were tested to determine the mechanical characteristics of steel specimens and identify the effect of strengthening on the bearing capacity. The general view of tested specimens is shown in
Figure 4.
2.2. Theoretical Evaluation of Strength of a Strengthened Tensile Element
The following assumptions precede the theoretical evaluation of strength of a CFRP-strengthened tensile steel element:
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Strain values of steel and CFRP coincide due to the small thickness of the adhesive layer before the failure of the adhesive joint;
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The modulus of elasticity of CFRP before failure is close to the modulus of elasticity of steel ;
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The value of normal stresses in CFRP does not exceed experimental values of in CFRP, determined by the strength of the adhesive bond;
- −
To take into account the thicknesses of the CFRP, the coefficient of operating conditions is used, where is the thickness of lamellas in mm, the coefficient of working conditions takes into account that the limiting stresses in carbon fiber of a different thickness will differ from those experimentally determined for a 1.2 mm thick lamella, this coefficient allows you to convert experimental limiting stresses into limiting stresses in carbon fiber of a different thickness than in experiments.
The final two conditions mean that no strengthening is effective for CFRP, when stresses reach the value of
. In case of such stresses, the adhesive bonding, used to fix the strengthening element to the steel profile, is destroyed and strengthening becomes ineffective. The longitudinal relative strain, corresponding to the failure of the adhesive layer, is
The strength of the adhesive joint is affected by both longitudinal and transverse deformations in carbon fiber and steel. This is important when analyzing the mechanism of the destruction of glue. When studying the load-bearing capacity of a CFRP-reinforced stretched steel element, the mechanism of glue destruction was not studied due to the recommendations of the carbon fiber manufacturer on the use of an adhesive joint performed using the glue and bonding technology recommended by the carbon fiber manufacturer. This made it possible to ensure the standard quality and strength of the adhesive joint. Tests of the adhesive joint showed its high strength, sufficient for practical use when reinforcing steel elements with carbon fiber and confirmed the validity of the manufacturer’s recommendations on the choice of glue and the technology of the connection. It has been experimentally established that, in this case, the normal stresses in carbon fiber cannot exceed a certain value, the achievement of a longitudinal force corresponding to these stresses in CFPR leads to the destruction of the adhesive and the exhaustion of the strength of the reinforced element. Thus, it is the longitudinal deformations and forces that are important for assessing the bearing capacity of a stretched element.
When a strengthened steel element stretches, several stages of its behaviour can be observed.
At the first stage, characterized by the elastic steel behaviour, normal stresses in steel and CFRP are approximately equal due to the close values of the modulus of elasticity of both materials. At the first stage, stresses , arising in steel, do not exceed the yield strength equal to .
At the second stage, stresses, arising in the steel rod, reach the yield strength of steel . As the load, applied to the rod, increases, and same about the strain, stresses change little in steel. Normal stresses in CFRP continue to grow in proportion to the modulus of elasticity up to the value equal to . When stresses reach , the adhesive layer fails and the same about the entire rod.
When the strengthened rod stretches, the stress-strain state of CFRP changes as follows:
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At the first stage, strains and stresses are distributed along CFRP same as in the steel specimen;
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At the second stage, stresses rise sharply in CFRP in the area of plastic strains, arising in the steel specimen;
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The brittle failure of the adhesive strengthening layer is observed when stresses are reached in CFRP.
Let us make a theoretical analysis of the ultimate load that a tensile steel rod, symmetrically strengthened by CFRP, can resist. One end of the rod was rigidly fixed; the other free end was subjected to the longitudinal force (
Figure 5).
The following notations are used: is the initial length of the rod, is the elongation of the rod in tension.
Let us assume that strains in the adhesive layer are negligibly small; in this case strains in steel and CFRP were the same before the failure of the adhesive. We also disregarded a change in the cross-sectional area of the steel specimen after it reached the yield strength point.
At the loaded end, the total force in steel and carbon fiber was equal to the load applied to the rod. The movements of the steel rod and carbon fiber at the loaded end of the rod were equal.
The total longitudinal force is the sum of longitudinal forces in the steel rod and CFRP:
where
is the total longitudinal force resisted by the rod,
is the longitudinal force resisted by the steel part of the rod,
is the longitudinal force resisted by CFRP.
It follows from the equality of displacements of the steel part of the rod and CFRP that:
where
is the length of the rod,
is the cross-sectional area of steel,
is the area of CFRP.
Based on (3), longitudinal forces
and
can be expressed as follows:
Having made several transformations, we obtained:
where
,
Then formula (2) could be written as follows:
When making calculations using formula (8), we assumed that strains in the rod were determined by strains in CFRP. The presence of the yield plateau of steel was not taken into account here. It was assumed that stresses rose in steel up to its failure at the initial modulus of elasticity of steel. This behaviour pattern generated an excessively high value of the bearing capacity.
Formula (9) assumes that the strength of the strengthened rod is exhausted when stresses, arising in steel, reach the yield point. In this case, stresses in CFRP did not reach the limit value, and the longitudinal force value turned out to be underestimated.
The development of plastic strains was acceptable in steels whose yield point did not exceed 440 MPa; therefore, the bearing capacity of CFPR-reinforced steel rods was determined taking into account the behaviour of steel in the plastic range. Only when plastic strains developed in steel could limit stresses arise in CFPR and the adhesive layer could fail.
To clarify the value of the bearing capacity taking into account the development of plastic strains in steel, we used the following diagram of the steel behaviour (see
Figure 6).
In the diagram, is the proportional limit, is the yield point, is the strength limit, is stress at the moment of the specimen rupture, is the relative strain of the proportional limit, is the relative strain of the yield point, is the relative strain of the beginning of the yield plateau, is the relative strain of the end of the yield plateau, is the relative strain of the strength limit, is the relative strain of the specimen rupture. The letters A-F indicate the characteristic fracture points on the graph.
Let us identify the bearing capacity of a steel rod strengthened by CFRP taking into account the above diagram.
Strains corresponding to the yield point are equal to:
Parameters of the diagram depend on the yield point and are as follows:
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for steels with the yield point below 355 MPa:
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for steels with the yield point from 355 MPa to 440 MPa:
Let us assume that after reaching the proportional limit, the modulus of elasticity of steel
is constant and it is determined by the diagram of the steel behaviour in the AC section of
Figure 6:
Hence,
- −
for steels with the yield point below 355 MPa:
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for steels with the yield point from 355 MPa to 440 MPa:
Therefore, formula (1) can be represented as follows:
where
is the portion of longitudinal force, arising in the steel rod and resisted before the proportional limit is reached;
is the portion of longitudinal force resisted if the proportional limit is exceeded,
is the longitudinal force resisted by CFRP.
Let us suppose that the ultimate elongation of the rod is determined by the ultimate stresses in CFRP. In this case, strains will be calculated as follows:
Then the bearing capacity of the strengthened rod will be calculated as follows:
- −
for steels with the yield point below 355 MPa:
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for steels with the yield point from 355 MPa to 440 MPa:
Formulas (19) and (20) obtained using the normative diagram of steel work (
Figure 6) could be applied to make a theoretical evaluation of the strength of a tensile steel rod, strengthened by CFRP.
2.3. Numerical Studies
The analysis of CFRP-strengthened tensile steel rods was performed using the finite element method and the Nasnran software.
The computational model of a strengthened tensile steel element was developed using 3D elements of the solid type, which had the shape of an 8-node hexahedron. Nodes of finite elements were united at the interface between different materials (steel-adhesive, adhesive-lamella).
Interaction between a steel rod and strengthening elements was simulated by adding finite elements of the adhesive layer to the computational model. The characteristics of this layer were clarified as a result of successive computations of models, having parameters of the adhesive, and the outcomes of computations were compared with the experimental data obtained in the course of studying the strength of the adhesive bond between CFRP and the steel rod. This approach enabled the authors to set the parameters of the material that simulated the adhesive layer, so that the computation results corresponded to the test data. In the second case, the adhesive was not included in the computational model, and it was taken into account due to the fact that CFRP had a strength of 686 MPa in the finite–element model. This value corresponded to stresses in the lamella that caused the adhesive layer to fail during the experiment. When these stress values were attained, CFRP stopped resisting any loads, which were thereafter resisted by steel only.
The material was assumed to be isotropic in the finite–element model. Its properties were the same in all directions. Characteristics of the isotropic material were set using the following basic parameters: Young modulus, Poisson’s ratio, and the limit stress of the material. The value of the shear modulus was calculated using the following formula:
where
is the modulus of elasticity, MPa;
is the Poisson’s ratio.
To take into account the physical nonlinearity of the system, functional dependences between stresses () and strains () were set. They described the relationship between strains and stresses. A general model of an elastic–plastic material, called Plastic, was used. The properties of this nonlinear material include the stress–strain curve, the Yield Criterion, the Yield Stress, and the Hardening Rule.
Initially, the modulus of elasticity of steel was constant, and the stress–strain relationship was linear. After the yield point was reached, two options of the steel behaviour diagram were used. For the steel, used in the experiments (SteelE), a bilinear diagram was used. For S245 and S440 steels, the generalized steel diagram was used (
Figure 6). Parameters of the generalized diagram, made for steels S245 and S440, are listed in
Table 2.
The CFRP lamella had an elastic modulus of 190,000 MPa at normal stresses not exceeding 1480 MPa, a shear modulus of 79,166 MPa. The steel had an elastic modulus of 206,000 MPa, a shear modulus of 79,230 MPa. The adhesive had a modulus of elasticity of 200,000 MPa, a shear modulus of 76,923 MPa.
Figure 7 shows the dependence between the stresses and strains for the materials of a strengthened steel element.
Numerical studies were made for two finite-element models. The first one simulated the experimental specimens, used to evaluate the strength of the adhesive joint (
Figure 2). The second one was used to analyze the CFPR-strengthened steel element (
Figure 3). To reduce the calculation time, 1/8th of the experimental specimen was used in the computational model, given the symmetry of the specimen.
Figure 8 shows parts of experimental specimens, used in the calculation, as well as their FEM models.
The longitudinal force in the specimens was set by displacing one of the ends of the computational model, while the other end remained still. The calculation was performed using the extended nonlinear transient solver of the Nastran software, that was titled “23...Advanced Nonlinear Transient”. It takes into account geometrical and physical nonlinearities.
When the adhesive was added to the finite-element models, its strain and strength characteristics were selected so that the numerical results corresponded to the data obtained during the experimental studies of the adhesive layer and its strength.
Two options of adding the adhesive to the finite-element models were considered:
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The first option: the adhesive layer consisted of one row of finite elements with a thickness of 0.1 mm;
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The second option: the adhesive layer had two rows of finite elements with a total thickness of 0.4 mm.
The cross-sections of elements in the computational model were as follows: steel plate—27.5 × 6 mm; two CFPR lamellas—12.5 × 1.2 mm. The length and width of all finite elements were 2 to 5 mm. In terms of thickness, the steel element was divided into six rows of elements; the CFPR lamella was simulated by one row. The mesh, selected to split the specimen into finite elements, was checked during the tensile test calculations of steel rods, which confirmed the reliability of numerical results and acceptable computing time.