Test Results and Nonlinear Analysis of RC T-beams Strengthened by Bonded Steel Plates

The test specimen design was based on a 16 m long RC T-beam that was designed in accordance with the Chinese Highway Bridge Design Code (1974). The test specimen was designed based on similarity theory (Zhao et al. 2011). Similar constants are the geometric similarity constant \( C_{l} = 1/8 \), the similarity constant of the elastic modulus \( C_{E} = 1 \), and the similarity constant of Poisson’s ratio \( C_{n} = 1 \).

According to the reinforcement ratio similarity criterion π ₁, \( \rho_{m} = \rho_{p} \) is obtained, (i.e., the reinforcement ratio of the specimen is the same as that of the original beam). According to the flexural reinforcement ratio similarity criterion π ₂, \( \rho_{fm} = \rho_{fp} \) is obtained (i.e., the flexural reinforcement ratio obtained from the specimen is the same as that of the original beam). According to the loading ratio similarity criterion π ₃, \( P_{m} = P_{p} /64 \) (i.e., the load ratio obtained from the original beam is 64 times more than that of the test specimen). According to the unit weight similarity criterion π ₄, \( \gamma_{m} = 8\gamma_{p} \) (i.e., the load ratio obtained from the original beam is eight times more than that of the specimen).

Based on the above similarity analysis, the similarity parameters are given as: the similarity constant of the dead load \( C_{w} = 8 \), and the similarity constant of the load \( C_{F} = 1/64 \).

This test program consisted of testing five 2,100 mm (with a clear span of 2,000 mm) long RC T-beams. The geometric dimensions of the test specimen are shown in Table 1 and Fig. 1, which are determined according to the above-mentioned similarity constants.

The measured material properties of the steel reinforcing bars, the steel plate, and the concrete are shown in Table 2.

A total of five beam specimens were tested. The beam dimensions are shown in Fig. 2. The beams were reinforced with steel plates bonded to the tension surface of the beam. All the beams were designed to fail in flexure, even after strengthening with steel plates, which made it possible to estimate the increase in flexural strength. Beam B-0 was kept as the reference (unstrengthened) beam, and other beams were bonded with steel plates. Each beam was reinforced with (2)–10 mm diameter bars in the flexural tension region, and 6 mm diameter stirrups spaced at 65 mm in the shear span and 100 mm in the region of pure flexure. Concrete was cast in two different batches and was designated as Type I and Type II (Table 2). One 1,800 mm long steel plate with a thickness of 2.0 mm was bonded to the bottom of each of the strengthened beams using epoxy adhesive. Two different plate widths, 21 and 42 mm, were investigated. The properties of the steel reinforcing bars and the steel plates are summarized in Table 2. Before bonding the steel plates, the bottom surface of the beam was roughened by mechanical grinding and cleaned thoroughly with acetone. The bonding faces of the steel plates were also sand blasted and cleaned thoroughly with acetone. In order to attach the steel plates tightly to the beams, M8 bolts (diameter: 8 mm, length: 40 mm) were used, and within the region of pure flexure, two anchor spacings, 250 mm (Fig. 2a) and 1,000 mm (Fig. 2b), were investigated. Test specimen information is summarized in Table 3. The test results are presented in Sect. 4.

All beam specimens were tested under four-point loading over a span of 2,000 mm. Strains in the tensile reinforcing bars, the steel plates at different locations, and the concrete were recorded. Strain gauges were attached to the bottom edge and the mid-thickness of the steel plates at the middle of each shear span, as well as at the midspan in the longitudinal direction. Midspan displacement was measured by a linear variable displacement transducer (LVDT). The specimens were loaded under displacement control (Fig. 3). Crack initiation and propagation were monitored by visual inspection during testing, and crack patterns were marked.

In numerical analysis of RC structures, the concrete constitutive relation has a significant impact on the results. The commercial program used in this study, ABAQUS, provides different types of concrete constitutive models including (1) a smeared crack model; (2) a discrete crack model; and (3) a damage plasticity model (ABAQUS 2010). The concrete damaged plasticity model, which can be used for modeling concrete and other quasi-brittle materials, was used in this study. This model combines the concepts of isotropic damaged elasticity with isotropic tensile and compressive plasticity to model the inelastic behavior of concrete. The model assumes scalar (isotropic) damage and can be used for both monotonic and cyclic loading conditions, and has better convergence. Elastic stiffness degradation from plastic straining in tension and compression were accounted in this study (Lubliner et al. 1989, Lee and Fenves 1998). Cicekli et al. (2007) and Qin et al. (2007) proved that damaged plasticity model could provide an effective method for modeling the concrete behavior in tension and compression.

The main parameters required in the concrete damage plasticity model, including the constitutive relationship of concrete, are defined by the user. The study described in this paper used the constitutive relation of concrete from the Chinese code GB 50010-2002 (China 2002), Guo (2001) and Xue et al. (2010).

The stress–strain relation for concrete under uniaxial tension equation is described using Fig. 4 and Eq. (1). Damage is assumed to occur only after the peak stress is reached.

where α _t is a decline curve parameter of concrete under uniaxial tension. α _t  = 0 means the concrete constitutive relation becomes a horizontal line (i.e., fully plastic) after peak load, while α _t  = ∞ means the concrete constitutive relation becomes a vertical line (i.e., fully brittle) after the peak load. f _t is the tensile strength of concrete. Similarly the stress–strain relation for concrete under uniaxial compression can be described using Fig. 5 and Eq. (2).

where α _d is a decline curve parameter of concrete under uniaxial compression and f _c is the compression strength of concrete. According to the conservation of energy, the complementary energy of damaged material is equal to that of the elastic material, as long as the stress is converted into an equivalent stress or the elasticity modulus is equal to an equivalent elastic modulus when the material is damaged.

In Eq. (3), \( W_{0}^{e} \) is the complementary energy of undamaged material given in Eq. (4):

And \( W_{d}^{e} \) is the complementary energy of the damaged material given in Eq. (5):

where E ₀ and E _d are the undamaged and damaged elasticity moduli, respectively.

The equivalent stress is described using Cauchy’s effective stress tensor in Eq. (6), in which D is the damage variable:

where A and \( A^{'} \) are the effective bearing area of undamaged and damaged section, respectively; and σ and \( \sigma^{'} \) are the effective stress of undamaged and damaged section. Combining Eqs. (3), (4), (5) and (1) results in Eqs. (7a) and (7b) below:

Adopting the principles above, the uniaxial tensile damage equation can be described in Eqs. (8a) and (8b):

Similarly, the uniaxial compressive damage equation is described in Eqs. (9a) and (9b):

Material models exhibiting softening behavior and stiffness degradation often lead to severe convergence difficulties in implicit analysis programs. A common technique to overcome some of these convergence difficulties is the use of a viscoplastic regularization of the constitutive equations, which would lead to the consistent tangent stiffness of the softening material to become positive for sufficiently small time increments.

Using the viscoplastic regularization with a small value for the viscosity parameter (this paper takes a value of 0.0005) usually helps to improve the rate of convergence of the model in the softening regime and without compromising results. The lower the coefficient, the higher the calculation accuracy, but the calculation is more time consuming.

In order to accurately simulate the actual behavior of the RC T-beams investigated in this study, a description of the material, model configuration, boundary conditions, and loading are required. For the linear elastic behavior simulations, at least two material constants are required: Young’s modulus (E) and Poisson’s ratio (\( \nu \)). For nonlinear analysis, the steel and concrete uniaxial behaviors beyond the elastic range must be defined to simulate their behavior at higher stresses. The minimum input parameters required to define the concrete material are the uniaxial compression curve, the ratio of biaxial and uniaxial compressive strength, and the uniaxial tensile strength. The bond between the steel plate and concrete surface was model by cohesive behavior. The anchor was modeled by the node coupled method, where the nodes of the steel reinforcing bar and the steel plate were coupled.

Figure 10a–e shows analytical and experimental load–deflection curves of the five beam specimens. In these figures, indices I–III indicate the loading states. I indicates initiation of flexural cracking, II indicates the yielding of reinforcing bar and/or steel plate, and III indicates the maximum load.

The analytical results for beam B-0 (Fig. 10a) agree well with the experimental results. By comparing the reinforced RC beams shown in Fig. 10b–e, it is confirmed that the stiffness of the RC beam decreases sharply at points I and II due to opening of the flexural cracks and yielding of reinforcing bar and/or steel plate, respectively. After reinforcing bar yielding, the load–deflection curves are almost linear up to the maximum load point. At point III, the applied load decreases suddenly due to concrete crushing.

It can be seen from beam B-5-2 (Fig. 10c) that there is significant difference between analytical and experimental curves after point II. Experimental values of beam B-5-2 are obviously too small after point II by comparison with the other strengthened beams (Fig. 10b, d and e). The reason is that the steel plate was damaged in advance, which happened before the reinforcing bar yielding, because the anchoring holes at the loading point weakened the effective area of the steel plate. The numerical analysis did not consider the debilitating effects of the anchor hole on the steel plate and therefore predicts that beam B-5-2 fails in the sequence of yielding of reinforcing bar, yielding of steel plate, and then crushing of concrete.

These discussions suggest that the load-carrying capacity and failure behavior of RC beams reinforced in flexure with externally bonded steel plate can be simulated with good accuracy using the proposed numerical analysis method.

Figure 11 compares the analytical and experimental strain distributions of the steel plate at the following loading stages: 30, 35, 40, 45, and 50 kN. In this figure, analytical results and experimental results are illustrated using the strains measured over the 1/2 length of the span.

Comparisons for beams B-2-2, B-2-4, B-5-2, and B-5-4 show that analytical results agree roughly with the experimental results at each of the loading states.

Based on the observed analytical results, the strain of steel plate was found to increase suddenly at the section of concrete cracking (plastic strain section). This shows that the cohesive behavior can be used to simulate the stress transfer between concrete and steel effectively. It also can be seen that the strain in the steel plate and local concrete increases rapidly at the end of steel plate. This is consistent with the experimental observation.

In this research, the steel plate was bonded to the concrete by an epoxy adhesive and with mechanical anchors (i.e., bolts). Due to the existence of the bolts, the slip phenomenon between the concrete and steel plate was been minimized, thus there was a relatively small slip phenomenon between the bolts, which can be observed in Fig. 11.

The failure load for the numerical analysis was defined as peak load in the \( P - \delta \) curves. Table 4 compares the cracking load, yield load, and failure loads from the experiments with those from the numerical analysis, \( P_{Test} \) and \( P_{FE} \), respectively. With the exception of beam B-5-2, it can be observed that the correlation is quite good between the numerical results and the experimental data for the cracking load, yield load, and failure loads. The error is within 8.9 % in cracking load, 7.7 % in the yield load, and 9.7 % in failure load.

It can be seen from Table 4 that a relatively large deviation between the experimental and numerical results was found for beam B-5-2. By comparing with the experimental observations, the reason could be attributed to the steel plate damage in advance which influenced the response before the yield load, because the anchor hole at the loading point weakened the effective area of the steel plate (as discussed previously).

Figure 12 shows the destruction of the five beams. The failure modes of beams B-0, B-2-2, B-5-2, and B-5-4 were steel plate (rebar) yield/damage and concrete crushing, whereas the failure mode of beam B-2-4 was bolt failure as seen in Fig. 12d. Figures 13, 14 and 15 show the failure modes of the beams predicted by the proposed numerical model. Comparing with the experiment results, it is clearly seen that the proposed FEM model could accurately predict the failure modes and failure loads of the tested beams.