As the column capacity axis varies from zero to π/2, the moment capacity also varies accordingly. Figure 3.14 shows the variation of moment capacity at key point U of the column capacity curve. Figure 3.15 is the variation of moment capacity at key point S as θ is varied from zero to π/2. It is obvious from this demonstration of moment capacity that the standard column inter-action formula for biaxial bending, which gives only one set of column capacity values, is no longer warranted in structural analysis.
Figure 3.14 Moment variation at key point U.
z
Moment Variation at Key Point “U”
0
0.000 0.500 1.000 1.500 2.000
Theta in radians
Moment capacity in kN-m
M
It is also clear that for biaxial bending conditions, development of the uniaxial capacity at θ = 0 and θ = π/2 is not possible. This alone makes the application of the standard interaction formula for biaxial bending not only incorrect, but also a violation of the basic principle of structural mechanics that requires consideration of the geometric properties of the section.
3.5.2 Limitations of the standard interaction formula
The significant limitations of the standard interaction formula can be shown from the results of the analytical method. The sum of the ratios of the external load to the internal capacity of a section in orthogonal axes 2-2 and 1-1 can be added up to test for unity as a maximum value. From the tabulated values, it can be easily concluded that application of the standard interaction formula will result in the underutilization of a structural member.
Figure 3.16 shows the capacity curve for Example 3.2 (584 × 432 mm with 14 to 22 mm diameter reinforcing steel bars). Concrete stress fc′ = 34.5 MPa and the useable concrete strain = 0.003. The yield stress of the reinforc-ing steel bar = 414 MPa. The column capacity axis is at the diagonal of the rectangular column. The test is applied at key point U of the capacity curve.
The capacity curve of the column is taken from the Excel worksheet in which the uniaxial capacities of this column are M1 = 946 kN-m, M2 = 625 kN-m, M = 730 kN-m, and Mz = 235 kN-m. These values will be used in the interaction formula to demonstrate the limitations of its effectiveness in predicting the capacity of a column section.
Table 3.5 shows the application of the interaction formula using the results of the analytical method for Example 3.2. Assume that the resultant Figure 3.15 Moment variation at key point S.
Key point “S”
Moment Variation at Key Point “S”
0
0.000 0.500 1.000 1.500 2.000
Theta in radians
Moment capacity in kN-m
M
internal moment capacity is equal to the external load. The resultant bending moment capacity is obtained as MR = {(730)2 + (235)2}1/2 = 767 kN-m.
The load axis and location of the resultant bending moment from the horizontal is given by the expression
θu = α − β (3.81)
in which β = arctan (Mz/M) = arctan (235/730) = 0.3114 radians and α = arctan (b/d) = arctan (17/23) = 0.6365 radians.
Substitute the values in Equation 3.81 to obtain θu = 0.6365 − 0.3114 = 0.3251 radians. The result is resolved along the 1 and 2 axes as
M1′ = MR′ cos θu and M2′ = MR′ sin θu (3.82) Figure 3.16 Column capacity curve.
Column capacity axis
432 mm 10–22mm bars
4–22 mm bars
2
1 1 64 mm typical
584 mm
θ = Arctan (432/584) 2
Moment axis
fc’ = 34.5 MPa fy = 414 MPa Column Capacity Curve
8000 10000 12000
6000
200 600 800 1000
0 2000 4000
0 400 Moment capacity in kN-m.
Axial capacity in kN
θ
The ratios of external to internal moments are R1 = M1′/M1 and R2 = M2′/M2. R = MR′/MR, which is the fraction of the resultant bending moment utilized in the interaction formula. These relationships and above numerical values are used with the standard interaction formula for biaxial bending to construct Table 3.5, in which the usefulness of the standard interaction formula is below 59% of its potential ultimate strength capacity. From these results, we now know that the accuracy of the standard interaction formula is sub-standard and therefore is no longer warranted and should be discarded.
Notations
b: width of a rectangular concrete section c: depth of a concrete section in compression d: depth of a rectangular concrete section
d′: concrete cover from the edge to center of any steel bar d1: distance from the concrete edge to the first steel bar e: eccentricity = M/P
ec: useable concrete strain es: steel yield strain
en: compressive (or tensile) steel strain at the nth bar location eu: eccentricity of the external load (Mu/Pu)
fc: compressive stress in concrete
fc′: specified ultimate compressive strength of concrete fy: specified yield stress of a steel bar
fyn: steel stress ≤ fy
h: overall thickness of a member
Table 3.5 Application of the Interaction Formula to Example Rectangular Column
P = 3105 MR= 767 Beta = 0.3114 PMAX= 10382 M = 730 M1 = 946 Alpha = 0.6365
Mz= 235 M2= 625 Theta U = 0.3251
M1' M2' MR' R1 R2 R1+ R2 R3 R1+ R2 + R3 R %
727 245 767 0.768 0.392 1.160 0.299 1.459 1.000 100
711 240 750 0.751 0.383 1.135 0.299 1.434 0.978 98
687 232 725 0.726 0.370 1.097 0.299 1.396 0.945 95
663 224 700 0.701 0.358 1.059 0.299 1.358 0.913 91
640 216 675 0.676 0.345 1.021 0.299 1.320 0.880 88
616 208 650 0.651 0.332 0.983 0.299 1.282 0.848 85
592 200 625 0.626 0.319 0.945 0.299 1.245 0.815 81
569 192 600 0.601 0.307 0.908 0.299 1.207 0.782 78
545 184 575 0.576 0.294 0.870 0.299 1.169 0.750 75
521 176 550 0.551 0.281 0.832 0.299 1.131 0.717 72
498 168 525 0.526 0.268 0.794 0.299 1.093 0.685 68
474 160 500 0.501 0.255 0.756 0.299 1.055 0.652 65
450 152 475 0.476 0.243 0.719 0.299 1.018 0.619 62
426 144 450 0.451 0.230 0.681 0.299 0.980 0.587 59
403 136 425 0.426 0.217 0.643 0.299 0.942 0.554 55
379 128 400 0.401 0.204 0.605 0.299 0.904 0.522 52
355 120 375 0.376 0.192 0.567 0.299 0.866 0.489 49
332 112 350 0.351 0.179 0.529 0.299 0.829 0.456 46
xm: location of a steel bar from a reference axis
xn: location of a steel bar from the concrete compressive edge : distance from the X-axis of the centroid of internal forces M1: external moment around axis 1-1
M2: external moment around axis 2-2 M: internal ultimate moment capacity
Ml: number of bars along the depth of a section Mu: external resultant moment
N: number of bars along the width of a section P: internal ultimate axial capacity
Pu: external axial load
R: radius of a circular column
θ: inclination of the column capacity axis with the horizontal axis θu: arctan M2/M1 (inclination of the resultant external forces about the
horizontal axis)
α1: central angle subtended by one bar spacing in a circular column α: arctan (b/d) = the inclination of the diagonal of a rectangular
col-umn with the horizontal axis ACI: American Concrete Institute
CRSI: Concrete Reinforcing Steel Institute
Note: All other alphabets and symbols used in mathematical deriva-tions are defined in the context of their use.
z
177
chapter four
Concrete-filled tube columns
4.1 Introduction
The current method of calculating the ultimate strength of concrete-filled tube (CFT) columns employs the column interaction formula for steel and reinforced concrete sections subjected to biaxial bending. In contrast, the analytical method illustrated in this book will eliminate the need to use the column interaction formula by using the column capacity axis not only as the reference for equilibrium of internal and external forces, but also to determine the capacities of the column section at every position of this axis.
This analysis involves calculating the concrete and steel forces separately and then combining these to determine the ultimate strength of CFT col-umns. The equations for the steel forces were presented at the ISEC-02 conference in Rome. The assumptions used in the ultimate strength of rein-forced concrete columns are also applicable in analyzing the ultimate strength of CFT column sections. Equations for determining the centroid of internal forces and factors for external loads are also applicable in this case.
At ultimate conditions of stress and strain, the concrete strain, usually assumed equal to 0.003 by the American Concrete Institute (ACI) method and 0.0035 by Canadian practice, is the pivot point for determining the steel stress and strain. The concrete and steel elements in a CFT column section undergo common deformation when resisting external loads.
This deformation is assumed linear with respect to the neutral axis, whose location is the concrete compressive depth c. As c is varied from the concrete edge it will generate a steel stress–strain diagram consisting of a triangular and rectangular shape. This steel stress–strain diagram will define the steel forces to be added to concrete forces to obtain the ultimate strength of a CFT column section. The compressive and tensile steel stress is limited to the yield strength fy of the material.
The derived equations for circular and rectangular CFT columns are programmed using Microsoft Excel 97 to generate capacity curves for these columns. From the numerical data of a rectangular CFT column, the accuracy of the current standard interaction formula for biaxial bending can be shown to be substandard (see Table 4.1 for details).
178 Structural analysis: The analytical method
4.2 Derivation
The governing equations for determining the ultimate strength of a column section are
P=Pc+Ps (4.1)
M=Mc+Ms (4.2)
The subscripts c and s for P and M denote concrete and steel capacities, respectively. The concrete forces were derived in Chapter 3. Hence, only the equations for the steel forces need to be derived.