BEHAVIOR EVALUATION FOR REINFORCED CONCRETE COLUMNS WITH RECTANGULAR HOLLOW SECTION SUBJECTED TO AXIAL COMPRESSION AND BIAXIAL BENDING

In order to evaluate the behavior of reinforced concrete columns with rectangular hollow section subjected to axial compression and biaxial bending, the calculation formula of load capacity and moment-curvature relationship are derived according to the distribution type of neutral axis in this paper. The load capacity and rotation ductility of the bottom control section of three reinforced concrete specimen bridge columns with rectangular hollow section under different axial compression ratio, reinforcement ratio and stirrup ratio are analyzed based on these calculation formulae. The Mx–My interaction curves and moment-curvature curves of bridge column specimens derived from the theoretical calculation show good agreement with the experimental data obtained by cyclic testing of three specimens under axial compression and biaxial bending. The results show that the P–Mx–My interaction has considerable effects on the behavior of the reinforced concrete bridge columns with rectangular hollow section. If these interaction effects are ignored, then the load capacity and deformation are overestimated and this fact can be crucial from the viewpoint of design.


Introduction
Reinforced concrete (RC) bridge columns with rectangular hollow section (RHS) have been widely used in engineering practice, especially for bridge columns with high elevations. Hollow section columns offer an optimal strength/ mass ratio and stiffness/mass ratio for bridges in seismic regions and reduce the mass contribution of the column to seismic response (Dong et al. 2014). And the other side, columns with RHS also reduce the tendency for thermallyinduced cracking at an early age resulting from heat-of-hydration temperature variations (Han et al. 2013). Among Chinese bridge engineering, these RC columns with RHS are usually designed as in Fig. 1a what is recommended by Chinese code JTJ004-89: Seismic Design Code of Highway Engineering. However, some hollow columns were damaged seriously during 2008 Wenchuan earthquake in China (Han et al. 2009). Priestley et al. (1996) suggested that large amounts of transverse links and hoops for effective reinforcement of hollow cross section are required to enhance lateral resistance capacity what was adopted in the current Chinese Code JTG/T B02-01(2008): Guideline for Seismic Design of Highway Bridges, as shown in Fig. 1b.
Although this configuration provides excellent confinement to the concrete, its construction practice is relatively complex and expensive compared to the conventional RC columns. Bridge columns are always designed in longitudinal and transverse following the current Chinese seismic design code for highway bridges (Wang et al. 2014). However, the section forms and reinforcement configurations of hollow columns are significantly different from RC solid columns (Hong 2001;Pinto et al. 2003). RC columns will be subjected to vertical loads and bidirectional horizontal loads under earthquake excitation. The solid RC column behavior subjected to axial compression and biaxial bending are different from those subjected to axial compression and uniaxial bending due to the bidirectional coupling effect (Chang 2010;Fafitis 2001;Qiu et al. 2002). The main objectives in this paper are: (1) to evaluate the behavior of RHS of RC columns subjected to axial compression and biaxial bending, (2) to offer a simple and effective method for safety verification on the hollow cross section considering the P-M x -M y interaction.
The load capacity equilibrium equations of RC columns under axial compression and biaxial bending were calculated based on the theory of "simplified rectangular

BEHAVIOR EVALUATION FOR REINFORCED CONCRETE COLUMNS WITH RECTANGULAR HOLLOW SECTION SUBJECTED TO AXIAL COMPRESSION AND BIAXIAL BENDING
block", and load capacity of axial force and moment interaction relationship for different section forms have been carried out by Cengiz (1990). Di Ludovico et al. (2012) researched the experimental behavior of nonconforming RC columns with plain bars under constant axial load and biaxial bending. Stefan, Léger (2010) investigated multi-criteria capacity envelopes for biaxial bending of concrete hydraulic structures. However, there is few research referred to the achievements in load capacity and ductility evaluation of RC columns with RHS. According to the distribution of the neutral axis, the calculation formula of load capacity and curvature is derived in this paper based on the section feature, details of seismic design and stress conditions of existing RC bridge columns with RHS in China. The numerical calculation of bottom section of three RC rectangular hollow columns with different axial compression ratio, reinforcement ratio and stirrup ratio are analyzed based on these calculation formulas and then compared with the experiment results. The load capacity, ductility and its influence parameters of a rectangular hollow section of an existing bridge column have been analyzed, and the P-M x -M y interaction yield surface has also been developed in this paper.

Functional equation of P, M x and M y
The stress distribution of the rectangular hollow section under an axial force (P) and biaxial bending moments (M x and M y ) are shown in Fig. 2. The theoretical model will be developed based on several assumptions, including: a) plane cross section assumption, b) concrete tension is ignore, c) the bond-slip between steel and concrete is not taken into account, d) shear deformation and torsional deformation are not considered. The section size of the hollow rectangular is b×h with wall-thickness of d and concrete cover thickness of c. The depth of compression is R and the angle between neutral axis and x axis is φ. Concrete compressive zone consists of external concrete cover, confined concrete core and internal concrete cover with the area of A 1 , A 2 and A 3 respectively. The area of four triangles with the hypotenuse of l 1 , l 2 , l 3 and l 4 is S 1 , S 2 , S 3 and S 4 respectively.
Equilibrium equations can now be written as follows: , , where -compressive strength of concrete cover, Pa;compressive strength of confined concrete core, Pa; f si -tensile stress of No. i steel, Pa; A st -the total area of the tensile reinforcement, m 2 ; A sc -the total area of the compression reinforcement, m 2 . Those mentioned equilibrium equations illustrate that the load capacity is related to only two geometric parameters, R and φ. Therefore they can also be written as follows: ; ; . (10) Interaction between the axial force (P), and biaxial bending moments (M x and M y ) is represented by a 3D surface based on the theories presented by Bresler (1960). If the load of a cross section is inside the 3D surface, the design will be acceptable. Otherwise the design of the cross section is not safe. There are three methods available to determine the biaxial strength of a column with RHS. These methods are listed as follows: (1) interaction curves for an assumed bending moment ratio; (2) load contours for an assumed axial load; (3) isogonic or three-dimensional curves as proposed by Marin (1979). The interaction curves (Lai et al. 1984;Rodriguez et al. 1999) obtained by these three methods are presented in Fig. 3.

Deformation of the rectangular hollow section
There is a corresponding sectional curvature for an assumed axial load (P) to the rectangular hollow bridge column. And the moment of this section which is corresponding with the curvature that is obtained based on the strain contribution of this cross section. By varying the strain values, a series of moments and curvatures will be calculated and the curve between moment and curvature can also be obtained. Similar to the calculation of load capacity, both unconfined concrete cover and confined concrete core should be considered in the analysis of momentcurvature, shown in Fig. 4.
Equilibrium equations can be written as follows: , , where .
Then, the curvature can be written as that: .
The deformation of bridge column is shown in Fig. 5. Column height = L, m; length of plastic hinge = L p , m; yield curvature of column section = φ y , m -1 ; ultimate curvature of column section = φ u , m -1 ; plastic curvature of column section φ p = φ u -φ y , m -1 ; plastic rotation θ p = L p φ p , nondimensional. Then, yield displacement of column top section, plastic displacement of column top section and total displacement of column top section in horizontal can be calculated by the follow formulae. , , (15) . (16) The length of plastic hinge recommended by Priestley (1996) can be calculated as following: , where L -Column height, m; f y -yield strength of longitudinal reinforcement, MPa; d s -diameter of longitudinal reinforcement, m.

Experiment program
The cross section of the specimen is 0.5×0.36 m, the wall thickness is 120 mm and the height of the column (distance from the top of the RC footing to the loading point) is 1.44 m. Fig. 6 shows the configuration and reinforcement of this rectangular hollow section. The properties of these specimens are listed in Table 1. S1 and S3 were tested under a constant axial load of , and S2 was tested under a constant axial load of , which is the estimated weight of the bridge deck. is the 28-day concrete compressive strength, and A g is the gross section area of RHS. The experiment program was conducted in the Key Laboratory of Urban Security and Disaster Engineering (KLUSDE) of Beijing University of Technology, China. In this research, the prescribed bidirectional displacement cyclic loading protocols were defined based on a predetermined displacement history in terms of the column drift ratio. The lateral load correction scheme and the target lateral displacement history in two perpendicular horizontal directions are shown in Fig. 7. In this figure, two perpendicular axes (the weak axis and strong axis) at the top of column are defined as x axis and y axis, respectively and the corresponding displacements in x axis and y axis are denoted as u and v, respectively.
Compared with the prototype RC column, this research applied the same steel and concrete materials for the specimen, resulting in a stress scaling factor of 1.0. There were two different longitudinal reinforcing bars with diameter of 8 mm or 10 mm using in this experiment and the transverse reinforcing bars were 6 mm in diameter. A design yielding strength of all the reinforcing bars was 300 MPa. The design 28-day compressive strength for concrete was 19.1 MPa. the strength test of the steel and concrete coupons were conducted before which showed that the average measured values of the yielding strength, ultimate stress and ultimate strain of reinforcing steel coupons from standard tensile tests were 385 MPa, 498 MPa and 0.16, respectively. The actual average compressive strength of concrete was = 42.6 MPa, which was determined by a typical 150×150×150 mm cubic compression test after 28-day curing process. Both the measured strengths of reinforcing steel and concrete were higher than the design strengths.
The displacements corresponding to the yield of a longitudinal steel bar are defined as the yield displacements of a column in x and y axes. Strain gauge measurements indicate that the longitudinal steel bars at the corners yielded first and the yield displacements are u y = 8.8 mm and v y = 8.8 mm for specimen S2, and the yield moments in two directions are found to be M yx = 222 kN·m and M yy = 124 kN·m. The flexural moment was calculated by the following equation: , where F -a lateral load, that equals to the developed restoring force in the opposite direction, N; Δ -the corresponding   Table 2 showed the experimental results of all these specimens.

Numerical simulation
The concrete is represented by a Mander et al. (1988) model and the steel reinforcement is represented by a bilinear strain hardening model as shown in Fig. 8.
When the unconfined concrete cover reaches its ultimate compression strain (0.004) or the reinforcement reaches the strain hardening stage (the corresponding tensile strain is 0.01), this section is defined as reaching the yield point.
The complete curve and ideal bilinear model between moment and curvature of these three specimens under a constant axial compression and biaxial bending are shown in Fig. 9. As shown in Fig. 9 and Table 2, skeleton curves  8%, but the difference between calculated and experimental values of yield bearing capacity was less than 7% except for the specimen S3 the error of which was about 30%. The difference between calculated and experimental values of the ultimate horizontal bearing capacity was up to about 9%, while that of ultimate moment was less than 10%. Therefore, the theoretical model calculated by the method presented in this study agreed well with the experimental results, which will offer a simple and accurate method for practical engineering. At the same time, when the axial compression ratio of the three specimens were 0.1 or 0.2, the curves between M x and M y at ultimate state were obtained and shown in Fig. 10.

Comparative analysis of experimental and calculated results
The comparative of analysis of experimental and calculated results are also shown in Fig. 10. It is obvious that experimental results of M x and M y at the ultimate state are outside of the calculated curves but not far from the curves. As shown in Fig. 11, the rupture of tensile longitudinal reinforcing bars at the bottom of the RC columns caused the ultimate failure mode for the specimens dominated by the flexural capacity of the RC columns. In addition, the value of M y at the curves were larger than M x in uniaxial compression bending which illustrated that y axis was strong axis of this rectangular hollow cross section and x axis was weak axis. The experimental and results of these three specimens, such as yield displacement, yield side force, ultimate displacement, ultimate side force and ultimate moment, are listed in Table 2.

Engineering situation
As shown in Fig. 12, the span arrangement of a continuous rigid frame bridge located in Beijing City in China, is 45+75+45+35 m. The concrete type of bridge column is C40 (i.e., the 28-day compression strength is 47.4 MPa). HRB335 steel was applied to the reinforcement with diameter over 12 mm while R235 steel is applied to the reinforcement with diameter less than 12 mm. The highest column of this bridge is the second bent with 7 m × 5 m cross section dimension as shown in Fig. 13.

Analysis of load capacity
In order to compare the difference between uniaxial compression bending and biaxial compression bending, P-M x and P-M y curves at uniaxial compression bending and M x -M y curve at constant axial compression were calculated in this section. As shown in Fig. 14, these three cases may be obtained from three load combination cases in the practical project: I. P = 4 · 10 5 kN, M x = 7.5 · 10 5 kN·m; II. P = 4 · 10 5 kN, M y = 6.5 · 10 5 kN·m; III. P = 4 · 10 5 kN, M x = 7.5 · 10 5 kN·m, M y = 6.5 · 10 5 kN·m. As shown in Figs 14a and 14b, the load points of cases I and II were both inside the curves, which illustrated that the cross section was safe to be used in these two cases. While in Fig. 13c, axial compression, moment in x direction and moment in y direction of case III were the same as cases I and II, but the load point was outside of the curve, which illustrated that the cross section would fail under the combination of the axial load and biaxial bending. The result shows that the coupling effect of M x versus M y will decrease the load capacity of bridge columns and the design method that only considers uniaxial bending is unsafe to estimate the load capacity.

3D yield surface
3D interaction surface for the axial force P and the two bending moments M x and M y can be obtained by the method (2) load contours for a given axial load P, as shown in Fig. 15. This analytical method of 3D interaction surface was adapted by Lai et al. (1984) and Rodriguez et al. (1999). It is seen that there are two limit stages in P-M x -M y interaction yield surface, one is axial load absence (P = 0), the yielding of the column under the interaction effects between M x and M y , but this stage is impossible in practical bridge engineering because of the weight of bridge superstructure P is never zero. The other is when the values of both M x and M y is equal to zero, meaning that the yielding of the column under maximum axial compression.
3D yield surface can be used to verify the safety of cross section in practical engineering. 2D curves of M x -M y will be obtained at each axial compression based on the corresponding axial compression ratio. When the load point stays inside the 3D surface, the design will be considered as safe design. Otherwise the design is unsafe and it is necessary to re-design the cross section. Furthermore, it is obvious in Fig. 14 that when the axial compression ratio ranges from 0.2 to 0.4 (i.e., 160 000 kN to 320 000 kN), the area surrounded by the curve of M x -M y is much larger than others. That is to say, at this case, the material properties are used in the optimal stage to resist the bending moment.

Analysis of moment and curvature
When the angle between neutral axis and x axis range varies from 0° to 90°, the stress states of the section are different. As shown in Fig. 16, variation relationship between moment and curvature of the two principle axis are different.
As for the strong axis, x direction, the ultimate moment and curvature increase with the decreasing angle between neutral axis and x axis; as for the weak axis, y direction, the ultimate moment and curvature increase with the increasing angle. The amplitude of the increasing depends on the sectional properties.

Conclusion
The calculation formula of load capacity and curvature is derived according to the distribution of the neutral axis. The P-M x -M y interaction yield surfaces developed in this paper can be used in evaluating the load capacity and deformation of columns with rectangular hollow section. The following conclusions can be drawn: 1. The load capacity and the relationship between moment and curvature of rectangular hollow section bridge column presented are accurate and reasonable. The moment-curvature curves of bridge column specimens derived from both the theoretical calculation and ideal bilinear model show good agreement with the experimental result. It is convenient and accurate to evaluate the behavior of rectangular hollow section bridge columns applying the bilinear model in practical bridge engineering.
2. There is significant overestimation of the load capacity and deformation if the P-M x -M y interaction effects are ignored and columns with hollow cross section are idealized independently in the transverse and longitudinal directions. The decrease in the load capacity and deformation due to bidirectional interaction is crucial, and must be included in the effective design of the columns with rectangular hollow section for the performance-based seismic design of bridges.
3. The M x -M y interaction curves in different axial compression ratio and three-dimensional yield surface calculation, developed in thi paper, offer a simple and effective method for safety verification of the cross section.