FINITE ELEMENT ANALYSIS OF THE BEHAVIOUR OF REINFORCED CONCRETE COLUMNS CONFINED BY OVERLAPPING HOOPS SUBJECTED TO RAPID CONCENTRIC LOADING

The strain rate sensitivity of concrete material was discovered approximately one hundred years ago, and it has a marked effect on the behaviour of concrete members subjected to dynamic loadings such as strong earthquake and impact loading. Because of the great importance of the confined reinforced concrete (RC) columns in RC structures, the dynamic behaviour of the columns induced by the strain rate effect has been studied, but only few experiments and analyses have been conducted. To investigate the behaviour of overlapping hoop-confined square reinforced normal-strength concrete columns, considering the strain rate effect at a strain rate of 10/sec to 10/sec induced by earthquake excitation, an explicit dynamic finite element analysis (FEA) model was developed in ABAQUS to predict the behaviour of confined RC columns subjected to the rapid concentric loading. A locally modified stress-strain relation of confined concrete with the strain rate sensitivity of the concrete material and the confining effect of overlapping hoops were proposed to complete the simulation of the dynamic behaviour of concrete with the concrete plastic-constitutive model in ABAQUS. The finite element predictions are consistent with the existing test results. Based on the FEA model, a parametric investigation was conducted to capture more information about the behaviour of confined RC columns under varying loading rates.


INTRODUCTION
With the increasing strain rate of concrete and reinforcing steel, the properties of these materials change.This refers to the strain rate sensitivity of concrete and reinforcing steel, which has been documented in literature [1][2][3][4][5].The strain rate sensitivity of the materials significantly affects the behaviour of reinforced concrete (RC) members under dynamic loading.As a typical type of dynamic load, earthquake loading induces strain rates on the structures.The quasi-static stain rate is commonly approximately 10 -6 /sec to 10 -5 /sec.Bertero predicted that for a notably rigid RC structure with a fundamental natural period of approximately 0.1 s under earthquake action, the strain rate at some critical regions can be as high as 0.025/sec [6].Generally, the strain rate of materials in RC structures under earthquake loading is approximately from 10 -4 /sec to 10 -1 /sec [7][8][9].Asprone et al. conducted an earthquake evaluation analysis of RC structures to appreciate the effect of the strain rate sensitivity of concrete and reinforcing steel on the global seismic response of RC structures [10].The result shows that considering the updated material properties, to THE CIVIL ENGINEERING JOURNAL 4-2017 ---------------------------------------------------------------------------------------------------------------- account for the earthquake-induced strain rate, a strength reserve of the structural system is experienced when only ductile failure mechanisms are considered; on the other hand, the structural capacity decreases when brittle failure mechanisms are included.The evaluation method with consideration of the strain rate effect of the behaviour of RC structures under earthquake loading appears to be more proper.
The RC column is an elementary member of RC structures.The strain rate effect on the dynamic behaviour of RC columns by directly imparting rapid loading protocols has been partially studied.Reinschmidt et al. conducted several static and dynamic tests on plain and RC columns with a slenderness ratio (L/D) of 3 to 25 under eccentric or concentric axial loads [11].They observed a 30-40 percent increase in the strength of columns under dynamic loads compared with the contrast columns under quasi-static loads, except the notably slender columns with a slenderness ratio of 25 were 70 to 100 percent stronger under dynamic loads than those tested under quasi-static loads.Xu and Zeng [12] found that the lateral inertial effect played an important role in increasing the axial bearing capacity for slender columns under axially rapid loading.Iwai et al. investigated the effect of the axial loading rate on the behaviour of RC columns [13].Four short columns (L/D=6) and eight long columns (l/D=16 and 26) were tested under concentric and eccentric loads.The experimental results show that the maximum load of dynamically loaded columns is 11-14% larger than that statically loaded for concentric compression and 17-37% larger under dynamic load for eccentric compression.
The aforementioned studies did not fully account for the confined effect of hoops under high strain rates.Under a static load, various studies show that the confined concrete behaves better than unconfined concrete.Then, the behaviour of confined concrete under dynamic loading becomes an interesting topic.To investigate the behaviour of concrete confined by hoops under rapid loading, Scott et al. conducted a test programme on twenty-five square short RC columns with different longitudinal steel bars and arrangements of overlapping hoops, which were subjected to concentric and eccentric loads at different strain rates [14].The compressive strain rate was either 0.0000033/sec or 0.0167/sec.The test results show that a high longitudinal strain rate (0.0167/sec) increases the peak stress and descending branch of the core concrete by approximately 25%.A stress-strain curve for confined concrete loaded at a high strain rate was proposed by modifying Kent and Park's [15] stress-strain relation for confined concrete with a dynamic multiplying factor of 1.25 for the peak stress, strain at the peak stress and slope of the falling branch.Li et al. experimentally investigated the behaviour of short reinforced high-strength concrete columns [16].Thirty specimens with different confining reinforcement configurations, yield strengths of transverse reinforcement and concrete compressive strengths, were tested under concentric loads at different strain rates (0.000011/sec and 0.0167/sec).The test results confirmed that high-strength concrete had lower strain rate sensitivity than low-and moderate-strength concrete.Under high strain rate loading, the modulus of elasticity and slope of the descending branch of the stress-strain curve increase, but the effect of the high strain rate on the compressive strength depends on the strength of the transverse reinforcement.Zeng and Xu [17] developed an FEA model in general-purpose finite element computer program ABAQUS to predict the behaviour of laterally confined short RC columns under rapid concentric loading, but the model had a shortcoming: the cover concrete and core concrete were simulated with the same identical constitutive models.
Despite two experimental programmes and limited numerical analyses that investigated the effects of the loading rates on RC columns confined by transverse reinforcement, the state of the art on the subject remains in its infancy.Until now, notably limited experimental data can be acquired to investigate the effect of different parameters at high strain rate, and there is notably little finite element modelling on the dynamic behaviour of RC columns.Because of the high requirement on loading instrument for the rapid loading test, the FEA modelling becomes a good choice for parametric studies.In this paper, the objective is to investigate the dynamic behaviour of square reinforced normal-strength concrete columns confined by overlapping hoops under rapid concentric loading through finite element analysis.To model the behaviour of concrete confined by overlapping hoops (different from the hoops in the study by Zeng and Xu [17]) at high strain rates using the concrete damaged plasticity model in ABAQUS, a locally modified compressive stressstrain relation of concrete was developed based on the work by Zeng and Xu [17], which considers the confining effect of the overlapping hoops and strain rate effect of concrete.Then, an explicit dynamic FEA model was established in ABAQUS to evaluate and predict the behaviour of the overlapping-hoop-confined square reinforced normal-strength concrete columns with two different reinforcement arrangements (type A and type B, as shown in Figure 1).Finally, the effect of the parameters including the loading strain rate (3×10 −5 -3×10 −1 /sec), reinforcement arrangement (type A and type B), longitudinal reinforcement ratios (s: 2.65% and 1.27%) and volumetric ratios of transverse reinforcement (sv: 3% and 1.5%) on the dynamic behaviour of confined RC columns was investigated.

BRIEF DESCRIPTION OF THE RAPID AXIAL LOADING EXPERIMENT ON RC COLUMNS
Until now, the experimental data of square reinforced normal-strength concrete columns confined by overlapping hoops under axial rapid loading have been notably limited and can only be acquired from literature [14].In the paper, eight RC columns confined by overlapping hoops from the rapid concentric loading experiment of Scott et al. [14] were used to verify the following FEA model.Figure 1 shows that all test specimens have identical cross sections of 450 mm × 450 mm and are 1200 mm high.The longitudinal concrete strains in the test units were measured over the central 400-mm gage length of the units.Two reinforcement arrangements (type A and type B, as shown in Figure 1) were used.These arrangements are typical for 8-bar and 12-bar columns.The centre-to-centre spacing of longitudinal bars across the section for type-A and type-B arrangements was 183 mm and 123 mm, respectively.More details of the cross-sections in the test regions of different columns are shown in Table 1.The volumetric ratio of transverse reinforcement sv is 0.0134 to 0.0309.The spacing of transverse hoops was reduced by one-half for 200 mm at each end of the test columns to provide extra confinement and ensure that failure occurred in the 800-mm-long central region.For columns under rapid concentric loadings, the strain rate was 0.0167/sec, which is representative of that expected during seismic loading.The quasi-static test columns with a strain rate of 0.0000033/sec were used for contrast.

EXPLICIT DYNAMIC FEA MODELLING
To avoid numerical difficulty in convergence, the analysis module ABAQUS/Explicit was used to solve the quasi-static and dynamic analysis of RC columns under concentric loading.
The steel rebar was modelled using a 2-node linear 3D truss element (T3D2) in the explicit element library.The concrete was modelled using 8-node brick elements (C3D8R) in the explicit element library with three translational degrees of freedom at each node.The approximate global mesh size of 50 mm was used to discretize the concrete body, which can provide a precise simulation result.The load plate was modelled with an analytical rigid part, which is reasonable to save the computing cost because the load plate of the test machine is sufficiently stiff.
An embedded region constraint was used to simulate the interaction between the steel bars and concrete, which embedded the steel bars (embedded elements) into the concrete (host elements).In other words, the translational degrees of freedom of the embedded node were constrained to the interpolated values of the corresponding degrees of freedom of the host element, but these rotations were not constrained by the embedding [18].General contact in the explicit module was used to simulate the interaction between the surfaces of the rigid load plate and the end of the column, which combines a rough friction formulation for the tangential behaviour and a contact pressure model for the normal direction behaviour.
In Figure 2, a quarter model with symmetric boundaries on the X-Y plane and Y-Z plane was used based on the symmetry, which reduced the computation cost.In ABAQUS, the motion of the rigid load plate at the end of the column was constrained to the motion of a reference point, so the axial load was applied to the top reference point with an allowable translational motion in the direction Z and an allowable rotational motion around the X-axis.The boundary conditions in the FEA model were selected according to the actual experimental boundary conditions.

Concrete material model
The concrete damaged plasticity model [18] was used to simulate the behaviour of concrete.The constitutive model uses concepts of isotropic damaged elasticity in combination with isotropic tensile and compressive plasticity to represent the inelastic behaviour of concrete.The damaged Fig. 2 -Illustration of the finite element model model describes the irreversible damage that occurs during the fracturing process, which enables the user to control the stiffness recovery effects during cyclic load reversals.In this study, the test columns were under a monotonic load, there was no need to define the damaged model, and only the plasticity model was defined in the FEA model.
The plastic constitutive theory in the model is suitable to describe concrete and other quasibrittle materials under fairly low confining pressures (less than four or five times the ultimate compressive stress in uniaxial compression loading) and can consider the strength improvement at the state of triaxial loading.It can also be defined as sensitive to the straining rate.The key parameters of the plasticity model are as follows: the dilation angle, eccentricity, ratio of the biaxial compression strength to the uniaxial compression strength of concrete, ratio of the second stress invariant on the tensile meridian to that on the compressive meridian and viscosity parameter are 30°, 0.1, 1.16, 0.667, and 0.0001, respectively [19].Regarding the dynamic modulus of elasticity, Bischoff and Perry noted the confusion about whether the initial tangent modulus should change with the strain rate because of the confliction from different experimental results [2].In this study, the modulus of elasticity is assumed to be constant for an effective numerical implementation in ABAQUS and is equal to c 4730 f by ACI 318 [20], where fc (N/mm 2 ) is the compressive cylinder strength of concrete under a quasi-static load.Poisson's ratio of concrete is assumed to be constant under dynamic loading, as recommended by CEB [21], and is equal to 0.2.Moreover, an equivalent uniaxial compressive stress-strain relationship and tension stiffening are required in the plasticity model of concrete, which will be described as follows.

Uniaxial compressive stress-strain relation of confined concrete considering the strain rate effect
As previously mentioned, the range of strain rate of the materials in RC structures under earthquake loading is approximately 10 -4 /sec to 10 -1 /sec [7][8][9].To consider the strain rate sensitivity of concrete, the dynamic compressive cylinder strength fcd of concrete can be estimated from Equations 1 and 2 provided by CEB-FIP Model Code 1990 [1].
where fcs is the quasi-static compressive strength of unconfined concrete; fcs=0.85fcconsidering a strength-reduction factor related to the column shape, size and the difference between the strength of in situ concrete and the strength determined from standard cylinder tests [22][23];  is the strain rate (s -1 ); fc0=10 Mpa; and cs  =30×10 -6 s -1 .fcd increases with increasing strain rate; the other parameters in the following uniaxial compressive stress-strain curve correspondingly change, and the dynamic uniaxial compressive stress-strain curve is formed.
The uniaxial compressive stress-strain relation of confined concrete considering the strain rate effect is shown in Figure 3, which is suitable for the dynamic finite element analysis of concrete confined by overlapping hoops using ABAQUS.The basic equations (Equation 3 to 6) of the stress-strain relation, which were proposed by the author [17,19], are used.
where parameters a and d control the slope of the ascending and descending branches of the stress-strain curve, respectively;  and are the strain and stress of the confined concrete under dynamic loading, respectively; and co is the peak strain of the confined concrete (as shown in Figure 3), which is expressed as [17] where sv fh/fcd,sv is the volume ratio of the transverse hoop to the confined concrete core, and fh is the stress in confinement reinforcement at the peak strength of confined concrete.fh was proposed by Le´geron and Paultre [23] as follows:  (8) where fhy is the yield strength of stirrups; se is the effective sectional ratio of confinement reinforcement in the x or y direction (Figure 4); and is a parameter to determine whether the transverse reinforcement yields at the peak strength of confined concrete.se and  are expressed as follows: where s and c (as shown in Figure 4, c=cx=cy) are the spacing of transverse reinforcement and diameter of the core measured centre-to-centre of the hoops, respectively; Ash is the total area of transverse bars in the x or y direction and is defined as 3.41 and 3.61 times the cross-section area of a single tie leg for the type-A and type-B configurations in Table 1, respectively [24]; Es and c are the modulus of elasticity of transverse reinforcement and the axial strain, which corresponds to the concrete cylinder strength, respectively; and ke is the geometrical effectiveness coefficient of confinement, which represents the ratio of the smallest effectively confined concrete area at midway between two layers of stirrups to the nominal concrete core area.ke was proposed by Mander et al. [25] as follows: (11) where wi is the ith clear distance between adjacent longitudinal bars (as shown in Figure 4); s is the clear spacing of transverse reinforcement; c is the ratio of the area of longitudinal steel to the area of the core of the section; and cx and cy are the core dimensions to centrelines of the perimeter hoop in the x and y directions, respectively.
In Equation 4,cc50 is the post-peak axial strain in confined concrete when the capacity decreases to 50% of the confined strength.Based on the expression of cc50 proposed by Le´geron and Paultre [23], the modified expression of cc50 is where c50 is the post-peak axial strain in unconfined concrete under quasi-static load when the capacity decreases to 50% of unconfined strength, and c50=0.004according to the proposal of Le´geron and Paultre [23];

Uniaxial compression stress-strain relations of cover concrete considering the strain rate effect
When the volume ratio of transverse steel is v  =0 and the effective confinement index is e50 I =0 in Equations 7 and 10, respectively, Equation 1 to 13 describe the uniaxial compressive stress-strain relation of cover concrete with the strain rate effect, which can be used to analyse the behaviour of the cover concrete of RC columns.

Uniaxial tensile behaviour
In ABAQUS, the uniaxial tensile behaviour of concrete is taken as a linear elastic relation before reaching the tensile strength [18].The post-failure behaviour for direct straining is modelled with tension stiffening, which enables the user to define the strain-softening behaviour for cracked concrete.This behaviour also enables the effects of the reinforcement interaction with concrete to be simulated in a simple manner.One method to specify tension stiffening is applying a fracture energy cracking criterion.The fracture energy is directly specified as a material property in the model, and a linear loss of strength after cracking is assumed.The fracture energy GF in N/m is determined by the expression proposed by the Fib Model Code for Concrete Structures 2010 [27], where fc is the compressive strength in MPa.
As suggested by available studies [28][29][30] on the dynamic fracture energy of concrete, the dynamic fracture energy of concrete can be assumed to be identical to the static fracture energy for the strain rate range from 10 -4 to 10 -1 under earthquake action.

Steel material model
An isotropic elastic-plastic model was used to describe the constitutive behaviour of the steel bar.The perfect elastic-plastic relation was used as the uniaxial stress-strain relation of steel.Based on the studies on the dynamic behaviour of steel, the steel modulus of elasticity has no relation to the strain rate.The steel modulus of elasticity is constant under dynamic loading.The dynamic yield strength of steel yd  is obtained using the Cowper-Symond equation, where ys  is the static yield strength of steel,  is the strain rate of steel, C=1300 s -1 and q=5 according to Munoz-Garcia et al. [31].

VERIFICATION OF FEA MODEL
Based on the FEA model, the quasi-static and dynamic behaviours of the columns in Table 1 were simulated.Figures 5 and 6 show that the predicted axial load (N) versus the axial strain () curves are consistent with the tested N-curves under the conditions of quasi-static loading and rapid loading.Because the available tested N-curves under rapid loading from the literature are notably limited, the comparison between the predicted N- curves and the tested axial bearing capacity are shown in Figures 5 and 6.Table 2 shows the comparison between experimental and predicted axial bearing capacities (Pe and Pc) and the corresponding average axial strain (c ande).The largest error of the predicted axial bearing capacity is 7% for Unit19.The mean value and standard deviation of Pc/Pe are 0.97 and 0.03, respectively.The largest error of the predicted axial strain at the bearing capacity is 36% for Unit12, but the mean value and standard deviation of c/e are 1.11 and 0.16, respectively.The comparison shows that the FEA model is effective to predict the quasi-static and dynamic behaviour of square reinforced normal-strength concrete columns with different volume ratios of transverse reinforcement.

Fig. 8 -Effect of the transverse reinforcement ratio on DIF
Figure 9 shows that an increase in longitudinal reinforcement ratio from 1.27% to 2.65% makes the DIF slightly decrease.Figure 10 shows the effect of the reinforcement arrangement on the DIF.The two specimens in each group in Figure 10 have identical longitudinal reinforcement ratio and transverse reinforcement volumetric ratio but different reinforcement arrangements.The DIF of specimens with type-B reinforcement arrangement is slightly larger than the DIF of specimens with type-A reinforcement arrangement.In sum, the effect of the longitudinal reinforcement ratio and reinforcement arrangement on DIF is smaller than the effect of the transverse reinforcement ratio, and the effect is also commonly negligible.

Fig. 1 -
Fig. 1 -Typical details of the test units

Fig. 5 -
Fig. 5 -Comparison between experimental and predicted N- curves of columns with type-A reinforcement arrangement