Biaxial bending of slender HSC columns and tubes filled with concrete under short- and long-term loads: II) Verification

ABSTRACT An analytical method that calculates both the shortand long-term response of slender columns made of high-strength concrete (HSC) and of tubes filled with concrete with generalized end conditions that are subjected to transverse loads along the span and to axial loads at the ends (causing singleor double-curvature under uniaxial or biaxial bending) is presented in a companion paper. The columns that can be analyzed with this method include those with solid and hollow (rectangular, circular, oval, C-, T-, L-, or any arbitrary shape) cross sections and columns made of circular and rectangular steel tubes filled with HSC. In this paper, the validity of the proposed method is tested against experimental results from the technical literature that examined over seventy column specimens.

Introduction 123   An analytical method that calculates both the short-and long-term response of slender columns made of high-strength concrete (HSC) and of tubes filled with concrete with generalized end conditions that are subjected to transverse loads along the span and to axial loads at the ends (causing single-or double-curvature under uniaxial or biaxial bending) is presented in a companion paper published by the authors in 2014.
The main objective of this paper is to verify the iterative analytical procedure and corresponding equations that were presented in the companion paper.The proposed model, which is capable of predicting not only the complete load-rotation and load-deflection curves (both the ascending and descending parts) but also the polygon with 16 sides, and the cross-sectional area of the tube is estimated as a total of 20 rebars uniformly distributed around the perimeter of the circle.
Columns after Cederwall et al. 1990-.A series of 22 columns made of 4.72 in.(120-mm) square, steel tubes filled with concrete and with a height of 118.11 in.(3 m) were tested under shortterm loads.Column specimens made of concrete with a strength greater than or equal to 11,603 psi (80 MPa) were selected from this series to study their behavior using the proposed method.These eight specimens were subjected to end loads applied simultaneously to the steel tube and to the concrete core causing a single-curvature up to failure.Table 1 lists the thicknesses of the steel tubes, the applied end eccentricities, the yield strength of the steel tube, the compressive strength of the concrete, and the maximum experimental and theoretical axial load.Excellent agreement between the calculated and the experimental maximum values of the axial load are shown when comparing the last two columns of Table 1.Fig. 1 also shows excellent agreement between the calculated and the experimental curves (loading and unloading load-deflection responses).It is important to note that the effects of the confinement of the concrete provided by the square steel tubes in the experimental results of all specimens subjected to an axial load with low eccentricity are rather insignificant.Columns tested by Hsu et al 1995-.A series of 9 columns identified as L-columns made of high-strength concrete were tested under a short-term axial load and biaxial bending.The effects of concrete strength, axial load eccentricity, steel ratio and ratio eccentricities (Θ= Tan -1 (ey/ex)) were studied.All column specimens had a span length of 48 in.(1.22 m).Table 2 presents the properties of all L-columns and their corresponding experimental and calculated axial load at failure.The proposed model predicts with good accuracy both the axial load and the maximum lateral deflection at failure, as well as the load-deflection response as shown by Figs. 2 and 3. Fig. 3 shows good agreement between the calculated and experimental curves (for both loading and unloading) for the load-deflection responses of specimens L3 and L4.Reinforcements: The longitudinal reinforcement consisted of 12mm steel rebars with a yield strength fy = 62 ksi (430 MPa).The transverse reinforcement consisted of closed steel stirrups that were 4 mm in diameter with a yield strength fy = 65 ksi (450 MPa).
Details of the cross-sectional properties of the columns are shown in Fig. 4. End eccentricities, experimental and theoretical axial load and mid-span deflection at failure are all listed in Table 3. Figure 5 shows the full load-deflection responses.Figure 6 shows the correlations of the ratios between the experimental and the theoretical values for both the axial load and the mid-span deflection.Good agreements between the calculated and experimental results were obtained.In the theoretical analyses, the concrete core was approximated by a regular polygon with 16 sides, and the steel tube was assumed to be equivalent to 20 rebars around the concrete core.The test results from 25 specimens out of the 41 circular tubular steel columns were used in this study.Figure 7 shows correlations between the calculated results and the experimental load-deflection responses for several specimens.Figures 8(a)-(b) show the phenomenon that was observed by other researchers in the columns subjected initially to double-curvature (or axial compressive load with opposite eccentricities at the ends) of an instability or abrupt change in the deflected shape to a more stable single-curvature shape.The anti-symmetric deformed shape (double-curvature) of the column is maintained only up to a certain value of the applied eccentric axial load.However, the proposed model does not capture the phenomenon of instability in the deflected shape, because it assumes a perfectly anti-symmetric moment diagram and consequently a double-curvature deformed shape at all load levels.To capture this phenomenon, the numerical process must be capable of predicting any change in the deflected shape of the column (i.e., it must be controlled by deflection rather than by load).Table 4 shows the applied end eccentricities ea and eb, and the experimental and theoretical axial load values at failure, and Fig. 9 shows the correlation between these axial load values.Columns tested by Claeson andGylltoft 1998, 2000-.Slender columns made of HSC and normal strength concrete (NSC) subjected to long-term loadings (i.e., sustained loads) were tested.
All column specimens had a span length of 4 m.To take into account the long-term effects of creep and shrinkage in the concrete, the creep coefficient proposed by Han [6], χ = 1 and the expression: εsh(t) = 0.004t/(t + 35) (ACI 209 Committee) for shrinkage strain in the concrete were utilized in the proposed method.Because all columns were tested horizontally, the analysis also included the transverse deflections caused by the weight of the columns.Figure 10 shows the calculated (theoretical) M-P-ϕ curves used in the analysis.Fig. 11 shows good agreement between the calculated and experimental curves.

Figure 1 .
Figure 1.Load-Deflection Curves for HSC Filled Columns after Cederwall et al 1990

Figure 4 .Figure 5 .Deflection
Figure 4. Cross Sections tested by Lloyd and Rangan 1996 Columns tested by Kilpatrick and Rangan 1999-.Forty-one circular tubular steel columns filled with concrete were tested under short-term loads.Eleven test specimens were subjected to double curvature, the rest (thirty specimens) to single-curvature.The test specimens were made of 0.094 in.(2.4 mm) thick steel tubes of 4.05 in.(0.1015 m) in diameter and 85.63 in.(2.175 m)

Figure 9 .
Figure 9. Experimental-versus-Calculated Results of Columns tested by Kilpatrick and Rangan 1999

Table 2 .
Hsu et al. 1995byHsu et al. 1995 Theoretical Load at Failure (kN) Lloyd and Rangan 1996-.andRangan 1996-.A series of 36 columns with an effective height of 66.14 in.(1.68 m) were subjected to short-term axial load up to failure at the University of Curtin, Australia.The columns were simply supported and subjected to an eccentric axial load P causing equal moments (Pe) at both ends.The properties of the materials are as follows: Concrete: Series I-IV: 8,410 psi (58 MPa); Series V-VIII: 13,340 psi (92 MPa); Series IX-XIII: 14,065 psi (97 MPa).