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A numerical simulation of the hysteresis performance of corroded reinforced concrete (RC) frame columns was conducted. Moreover, the results obtained were compared with experimental data. On this basis, a degenerated three-linearity (D-TRI) restoring force model was established which could reflect the hysteresis performance of corroded RC frame columns through theoretical analysis and data fitting. Results indicated that the hysteretic bearing capacity of frame columns decreased significantly due to corrosion of the rebar. In view of the characteristics of the hysteresis curve, the plumpness of the hysteresis loop for frame columns decreased and shrinkage increased with increasing rebar corrosion. All these illustrated that the seismic energy dissipation performance of frame columns reduced but their brittleness increased. As for the features of the skeleton curve, the trends for corroded and noncorroded members were basically consistent and roughly corresponded to the features of a trilinear equivalent model. Thereby, the existing Clough hysteresis rule can be used to establish the restoring force model applicable to corroded RC frame columns based on that of the noncorroded RC members. The calculated skeleton curve and hysteresis curve of corroded RC frame columns using the D-TRI model are closer to the experimental results.

The RC frame column is currently one of the most widely used structural forms. With the increase in its service life and the direct or indirect effects of external corrosion media, structural materials will degrade and undergo surface cracking, carbonisation, desquamation, corrosion-induced expansion of the rebar, and so forth, as shown in Figure

Cracks, desquamation of concrete cover, and corrosion of rebar in the frame beam and column.

Beam

Column

A frame column, which supports structures such as beams and slabs, is considered the foremost load-bearing member in an RC framed structure. Once a frame column is broken, it exerts a more severe influence on the damage suffered by beams, slabs, and filler walls. As for the frame structures constructed in areas where earthquake frequently occurs, the frame column bears not only the vertical load, but also the brunt of any seismic action. The frame column, as a kind of eccentric compression member, has a lower stiffness than that of the beam and bears mainly vertical load. Therefore, even if only a small number of columns are damaged in a frame structure, the whole structure is likely to collapse (Figure

Earthquake damage to the bottom columns in a frame structure.

The hysteresis curve and restoring force model are two important indicators used when analysing the seismic behaviour of RC frame columns [

The hysteresis curve reflects the characteristics of structures or members during repetitive stress, including deformation, stiffness degradation, and energy consumption [

So far, existing research on the seismic behaviour of engineering structures is mainly conducted on those to be built, while ignoring the relationship between the seismic behaviour and service time of the structure. As a matter of fact, with the extension of the service of RC frame structures, the corrosion and degradation accumulate continuously. Under such circumstances, the original seismic design of structures fails to reflect the seismic safety situation of corroded structures which have been used for a certain time. Therefore, the seismic safety and durability of corroded and degraded RC frame structures need to be reevaluated and tested, so as to clarify the current performance of such structures. Based on the evaluated results, different measures can be applied to reinforce the structures according to the importance of, and extent of the damage to, the structures. After reinforcement, structures are expected to resist possible seismic actions and avoid, or reduce, causalities and economic loss throughout their service lives.

At present, the hysteresis performance and restoring force models for noncorroded RC frame structures or members have been explored widely in engineering. However, there are relatively few research reports on degraded RC structures or members (particularly frame columns) affected by corrosion. Existing studies mainly focus on the evaluation of the macroscopic seismic behaviour and reinforcement effect of reinforced corroded columns [

With the rapid development of nonlinear finite element techniques and computer technology, numerical simulation has been widely applied in civil engineering. Moreover, numerical simulation presents numerous advantages such as short computation time and low cost, and it considers the influences of various parameters. Thereby, numerical simulation is used based on the experimental research to study the hysteresis performance of corroded RC members under different working conditions and obtain sufficient relationship data between the restoring force and deformation. In this way, a simple, practical, mathematical model can be established: this is an easy way to construct the restoring force model of corroded members.

According to the above analysis, an RC frame column was studied in terms of the influences of different corrosion rates on the hysteresis performance of the column using nonlinear FE software ABAQUS. Thereafter, the simulation results were compared with the experimental results to verify the reliability of the FE numerical simulation [

A concrete damaged plasticity (CDP) model is a native constitutive model for concrete materials implicit to the algorithm used in the ABAQUS software. The model was first proposed by Lubliner et al. [

Concrete structures, or members, will produce plastic deformation and cracking under low intensity cyclic loading, and both the accumulation of plastic deformation and the expansion of cracks can induce stiffness degradation or softening. So, a damage factor _{0} is the initial elastic modulus of the concrete; _{c} is the scale factor between plastic strain and inelastic strain

Empirical values of parameters in the CDP model.

Parameter | | | | | |
---|---|---|---|---|---|

Value | 30 | 0.1 | 1.16 | 0.6667 | 0.0005 |

Under cyclic loading, the influence of the Bauschinger effect produced by loading and unloading on the stiffness degradation of rebar must be taken into account. Owing to the actual factors affecting the Bauschinger effect being complicated, some researchers have simplified the constitutive model of rebar based on experimental data to facilitate analysis. Typically, the USTEEL02 subprogram for rebar modelling in the uniaxial hysteresis constitutive model set of PQ-Fibre [

Loading and unloading: the Clough model [

As shown in Figure

USTEEL02 rebar model [

In the atmosphere, the corrosion of rebar in RC structures is mainly caused by carbonisation, cracking, and spalling of the concrete cover. It can be assumed that the corrosion is uniform. As there are many corrosion products, such as rust, on the surface of the rebar during corrosion, the geometric parameters of the rebar, including its linear mass and effective cross-sectional area and its mechanical properties, such as yield strength, can be reduced to some extent. Although the corrosion of rebar is discrete, to a certain extent, the related model for the decline in yield strength and elastic modulus of rebar can be obtained through statistical analysis of multiple experiments [

As for actual RC structures and members, if the concentration of external harmful media, such as chlorides and sulphates, is high, besides, the mass of concrete being inhomogeneous and variable, the invasion time of the harmful media to the rebar will vary. Inevitably, the corrosion time for rebar is different; that is to say, nonuniform corrosion occurs. In addition, the nonuniformity is more significant with increasing corrosion which induces local pitting corrosion on the rebar which can lead to stress concentration and a further reduction in the bearing capacity of structures and members. The yield strength reduction of rebar under the effects of pitting corrosion can be computed by the following formula [

With respect to the influence of corrosion on the elastic modulus of the rebar, some researchers find that the elastic modulus of rebar changed little with increasing corrosion rate [

For analysis, the rebar model can be reasonably simplified. (1) Assuming that the corrosion rates of stirrups and the main longitudinal reinforcement are the same and that the constraining effects of the stirrups on the core concrete are ignored, the beneficial effect of any stirrups can be neglected owing to the stirrup ratio of general RC members being low and the constraint of stirrups on the core concrete being slight. (2) As for the local nonuniform pitting corrosion of longitudinal reinforcement, the shape and distribution of pitted zones are random. If the finite element model is established according to the actual shape and distribution of these pitted zones, the modelling computational effort becomes large, while the computational efficiency decreases. Some researchers have studied the influence characteristics of pitting corrosion on RC members under uniform corrosion. Besides, they provided a conversion formula between local pitting corrosion and uniform corrosion [

In (

There is relative slippage between concrete and rebar, when RC members are subjected to low frequency cyclic loading. To some extent, slippage can absorb the energy in members produced by external loads. One of the macro-behaviours is a pinch effect caused by hysteresis. The bond-slip stress between concrete and rebar mainly originates from friction, cohesive forces from the cement material on the rebar, and the mechanical interaction between the surface of the deformed rebar and the concrete before corrosion. The surface roughness of the rebar decreased and some threads on the surface were lost: this affected the stick-slip behaviour after corrosion of the rebar.

The relative slippage between concrete and rebar could be simulated by defining a nonlinear spring element in ABAQUS. Owing to load being added to the top side of the frame column during simulation, slippage of the rebar mainly appeared along the longitudinal column axis (the ^{11} to 2 ^{13} N/mm) were set on the overlapped nodes of the concrete and rebar in the

According to the experience of the authors, the calculation mode of slippage-induced shear stress-displacement, as defined by CEB-FIB MODEL CODE 1990 [

To compare with existing results [

Dimensions and reinforcement layout.

To consider the influence of corrosion on the bond-slip between the rebar and the concrete, discrete modelling was used, including C3D8R for concrete elements with a unit size of 100 ^{12} N/mm was applied to the section orientation, that is to say,

Finite element models for concrete and rebar.

Concrete model

Framework model of reinforcement

Owing to the bottom of frame column being fixed, a fixed constraint was applied at the nodes of grid at its base. A concentrated force of 325.08 kN was applied to simulate the axial compression. Reference point RP1 was established at 100 mm below the top surface of column and it was coupled with the top surface. The displacement loading method was adopted to simulate the effect of lateral repeated force. Cyclic loading was preformed according to displacement amplitudes

To understand the influence of different amounts of rebar corrosion on the hysteresis performance of a frame column, four sets of operating conditions, including noncorroded, slight corrosion, moderate corrosion, and severe corrosion, along with the influence of local pitting corrosion, were selected. According to the literature [

Calculation parameters for different degrees of corrosion.

Corrosion rate (pitting corrosion rate) | 0% (0%) | 5% (6.3%) | 10% (25.0%) | 15% (56.3%) |
---|---|---|---|---|

Mass loss rate | 0% | 5.1% | 10.5% | 15.7% |

Yield strength (MPa) | 415.6 | 400.1 | 363.3 | 296.4 |

Elastic modulus (GPa) | 200 | 197.7 | 176.3 | 164.5 |

Degradation coefficient of bond-slip | 1.0 | 0.75 | 0.56 | 0.47 |

To obtain the yield displacements of each frame column under various corrosion conditions during loading, firstly a uniaxial pushover analysis was conducted on each frame column under different cases. The target displacement was set to 45 mm according to Chinese Standard GB50011-2010 (Code for the Seismic Design of Buildings) [

Uniaxial pushover analysis of frame columns with different corrosion rates.

As shown in Figure

Based on the aforementioned parameters, low frequency cyclic loading and unloading were performed on frame columns under various load regimes. The comparison of simulated hysteresis curves with the experimental results [

Hysteresis curves for RC columns under different corrosion rates.

Noncorrosion

Slight corrosion

Moderate corrosion

Serious corrosion

The comparison of the FE simulated results and the experimental results shows that the simulated hysteresis curves are similar in shape to those obtained experimentally [

Skeleton curves are obtained by connecting the ultimate loading points for each tension or compression load cycle, in the same direction, on the hysteresis curve successively. Skeleton curves describe the trajectory of the maximum peak horizontal force at each cyclic loading stage and reflect the changes in the stress and deformation of members at different stages. Therefore, skeleton curves are an important index for assessing the seismic performance of members or structures, as well as a significant basis for determining the feature points in the restoring force model for such members.

The skeleton curves of frame columns with different corrosion rates were extracted (see Figure

Skeleton curves for RC columns under different corrosion rates.

FE simulation results

Experimental results

It needs to be pointed out that the FE simulation results and experimental results in Figures

Based on the above analysis, the following suggestions can be applied to the seismic reinforcement of corroded RC frame columns: for mildly corroded members (i.e., those with a rebar corrosion rate within 5%) whose hysteretic bearing capacity reduces slightly to within 10%, then such structures, or members, can be slightly, or non-, reinforced according to their importance in engineering practice. Regarding the slightly corroded members with corrosion rates of 5% to 10%, their hysteretic bearing capacity was reduced by 10% to 20%, indicating that members were in a poor state. Therefore, such members are expected to be reasonably reinforced. Owing to the hysteretic bearing capacity of moderately corroded members (with corrosion rates of between 10% and 15%) decreased by 20% to 30%, the members need to be given more attention when designing/assessing their reinforcement as they are in poorer condition. In view of the more seriously corroded members, the corrosion rate of which is above 15%, their hysteretic bearing capacity decreases significantly (by more than 30% generally), indicating there is an increased risk from such members. Thereby, these kinds of members should be replaced.

To understand the final deformation and failure characteristics of corroded frame columns under cyclic loading, the stress distributions and deformed shapes of each column were extracted. Figure

The stress, and final deformation, of corroded frame columns under cyclic loading.

Overall deformation

Deformation of the rebar framework

Figure

Seismic damage to RC columns.

In the seismic response analysis of structures, the actual hysteresis performance curves are commonly modelled (using the restoring force model). For RC structures, the restoring force model is generally divided into two levels. The first level is the restoring force model of the materials used: this is mainly designed to reveal the stress-strain relationship between rebar and concrete and is the basis for modelling the restoring forces in RC members. The second level is the restoring force model of the members, which is used to describe the hysteretic relationship between bending moment and curvature

A suitable restoring force model for RC members needs to meet the following two requirements simultaneously: firstly, the model is expected to exhibit a certain precision, reflect the hysteretic performance of actual structures or members, and replicate experimental results within an acceptable tolerance through numerical simulation. Secondly, the model should be simple and practical, so that it does not present unnecessary complexity which hinders the effective performance of static elastoplastic, or dynamic nonlinear time-history, analysis.

In earthquakes, when RC structures are subjected to elastoplastic stress stages, the plastic deformation of structures can absorb large amounts of the input energy, which endows the relationship between the restoring force and displacement of members with apparent hysteretic nonlinearity. Considering this characteristic, the restoring force model for describing actual RC structures and members is divided into two types with curves and polygonal lines. The curvilinear restoring force model describes the actual stress characteristics of structures more accurately. However, its computations are complex, and the model is thereby rarely applied. As for that using polygonal lines, although it is comprised of several line segments and demonstrates a discontinuous stiffness distribution and has inflection points, it can be used without undue difficulty. Therefore, the restoring force model with polygonal lines has become widely used in practice.

The simplest nonlinear hysteretic model is bilinear elastoplastic model. Its positive loaded skeleton curve is composed of two lines and its shape is determined by yield strength, elastic stiffness, and postyield stiffness of members. As to the negative loaded skeleton curve, its loading and unloading stiffness are constant and equal to the elastic stiffness, which is similar to the positive loaded one. However, one of the shortcomings of this model is the difficulty of accounting for the stiffness deterioration of RC elements during cyclic load reversals. To overcome the problem of stiffness deterioration, Clough [

To verify the effect of different hysteretic models on the dynamic response of RC frames, Anderson and Townsend [

Based on the above analysis, we use the D-TRI model to investigate the hysteresis performance of RC frame columns in this paper. The skeleton curve and basic hysteresis rule of D-TRI model are shown in Figure

D-TRI model.

Skeleton curve: D-TRI model

Basic hysteresis rule: D-TRI model

In Figure

Once the key points of the bearing capacity and displacement were determined on the skeleton curve in the D-TRI model, the values of

After determining the above parameters, the formula for the restoring force model of members can be established. Furthermore, owing to symmetrical reinforcement being generally applied to RC frame columns in practice, the hysteresis curve is basically central-symmetric under cyclic loading. Therefore, to simplify the computation, it was assumed that the restoring force model was central-symmetric, that is,

For corroded RC frame columns, it can be seen from Section

Based on the above analysis, it was assumed that corroded and noncorroded members had similar restoring force models: therefore D-TRI model was applicable to both kinds of members, as displayed in Figure

Skeleton curves for D-TRI model of corroded, and noncorroded, columns.

The determination of the feature point parameters of the skeleton curve for the restoring force model of a noncorroded RC column played a fundamental role in establishing the restoring force model for corroded ones.

The relative height of the compression zone

As the stiffness degradation is not obvious from concrete cracking to rebar yielding in actual frame columns, the member is assumed to be perfectly elastic before rebar yielding. Therefore, the yield displacement

Considering the influence on the mechanical properties of the members of factors such as the properties of the concrete, axial compression ratio, and shear span ratio, during the determination of peak displacement, an empirical method was used to fit the calculated data to obtain an expression for the peak displacement [

The corresponding displacement to failure load

Rebar corrosion can induce a decrease in the cross-sectional area thereof, as well as its yield strength, bond behaviour, ultimate elongation, and so forth. Besides, there are many coupled factors which could interact: generally, corrosion degradation parameters are usually related to the axial compression ratio

Yield load

Ultimate load

Failure load

For noncorroded and corroded members, the corresponding stiffness parameters can be acquired by substituting the bearing capacity and key displacement parameters on the skeleton curve, respectively.

The parameters for each member in Section

Calculated results of yield load (

Corrosion rate (%) | Formula | Numerical ABAQUS | Experiment [ | Relative error (%) |
---|---|---|---|---|

Yield load | ||||

0 | 47.6 | 49.0 | 41.78 | 13.9 |

5 | 43.8 | 45.0 | 41.15 | 6.4 |

10 | 39.3 | 40.5 | 42.05 | −6.5 |

15 | 34.0 | 35.0 | 37.11 | −8.4 |

| ||||

Yield displacement | ||||

0 | 9.8 | 10.0 | 9.32 | 5.2 |

5 | 9.7 | 9.7 | 9.36 | 3.6 |

10 | 9.2 | 8.8 | 10.84 | −15.1 |

15 | 8.2 | 7.6 | 7.85 | 4.5 |

Computed results of ultimate load

Corrosion rate (%) | Formula | Numerical ABAQUS | Experiment [ | Relative error (%) |
---|---|---|---|---|

Ultimate load | ||||

0 | 52.2 | 52.3 | 50.15 | 4.1% |

5 | 49.0 | 47.9 | 49.65 | −1.3% |

10 | 46.5 | 44.8 | 49.5 | −6.1% |

15 | 44.5 | 39.3 | 43.54 | 2.2% |

| ||||

Peak displacement | ||||

0 | 23.6 | 24.5 | 22.54 | 4.7 |

5 | 22.9 | 18.3 | 22.37 | 2.4 |

10 | 20.8 | 25.9 | 19.75 | 5.3 |

15 | 17.9 | 22.4 | 14.32 | 25.0 |

Calculated results of failure load

Corrosion rate (%) | Formula | Numerical ABAQUS | Experiment [ | Relative error (%) |
---|---|---|---|---|

Failure load | ||||

0 | 44.4 | 42.5 | 42.63 | 4.2 |

5 | 41.7 | 43.6 | 42.2 | −1.2 |

10 | 39.5 | 40.2 | 42.07 | −6.1 |

15 | 37.8 | 35.3 | 37.01 | 2.1 |

| ||||

Failure displacement | ||||

0 | 40.9 | 39.5 | 41.54 | 3.4 |

5 | 35.9 | 34.0 | 39.51 | −9.1 |

10 | 31.3 | 30.0 | 32.52 | −3.8 |

15 | 26.8 | 26.3 | 24.1 | 11.2 |

The calculated key-point parameters of skeleton curve for the restoring force model in Section

Comparison of skeleton curves and hysteresis curves obtained by the restoring force model and experiment.

From Figure

ABAQUS finite element software was used for the numerical simulation analysis of the hysteresis performance of RC frame columns with four different amounts of corrosion: noncorroded, slight corrosion, moderate corrosion, and severe corrosion. In addition, the analytical results were compared with published experimental results and actual earthquake damage characteristics of framed columns. On this basis, a D-TRI model was established to reflect the hysteresis performance of corroded RC frame columns. Moreover, the influence of factors including the rebar corrosion rate and axial compression ratio was taken into consideration. The main conclusions were as follows:

The seismic bearing capacity of a frame column would be significantly decreased after corrosion of its rebar. In addition, with increasing rebar corrosion rate, the diminution of the bearing capacity gradually increased, including the corrosion-induced degradation of the bond-slip behaviour between rebar and concrete, which played an important role in weakening the hysteretic bearing capacity.

Along with the increased amount of rebar corrosion, the plumpness of the hysteresis loop of a frame column decreased, while the shrinkage increased. Meanwhile, the hysteresis curve became a reverse S-shape, having originally been arcuate. Furthermore, the area within the hysteresis loop became smaller and generated severe “pinching”: this explained why the seismic energy dissipation capacity of such frame columns decreased, while their brittleness increased.

The FE simulated frame column showed failure and deformations similar to those experiencing actual earthquakes under cyclic loading. This suggested that it was feasible to simulate the hysteresis performance of corroded RC frame columns using the FE method. Furthermore, the simulated results favourably described the failure characteristics of the frame column in earthquake conditions.

The hysteresis, and skeleton, curves of the corroded RC frame column presented basically consistent shapes with those of a noncorroded member under low-cyclic loading, which broadly conformed to the characteristics of a trilinear distribution. Therefore, a restoring force model applicable to corroded RC frame columns, the D-TRI restoring force model, for instance, could be established based on that of noncorroded RC members using the existing Clough hysteresis rule.

The skeleton, and hysteresis, curves of corroded RC frame columns, predicted using the D-TRI model, were similar to those from experimental results. Except for a few data points which showed large discrepancies (25%) with the experimental results, most of the calculated key-point parameters of the skeleton curve presented discrepancies of less than 15%. Moreover, the hysteresis curve showed similar characteristics to the experimental results. This proved that the restoring force model established for the corroded RC members in this research using numerical simulation was simple and applicable, as well as accurate and rational.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research has been funded by the National Natural Science Foundation of China (Grant nos. 51520105012 and 51278303) and Outstanding Young Talent Research Fund of Zhengzhou University (Grant no. 1521322004) and Guangdong Provincial Key Laboratory of Durability for Civil Engineering, Shenzhen University (Grant no. GDDCE 12-06). In addition, the authors would like to thank Dr. Y. W. Zhou and all study participants for their helpful discussions.