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The unified formula, as verified by data, can be used as unified failure criteria of RC components in ultimate state to determine the safety of members under various loading states. If the value of the left side is lesser than that of the right side, the section can be considered sufficiently safe. Otherwise, if the value of the left side is larger than that of the right side, the section is unsafe. This study provides an example of torsion-shear-bending-axial compression inter- action to illustrate this point.
For the planar frame given in Fig. 8, assume that all the columns have the same section details, as do the beams; the sections are all symmetrically reinforced. The section details of the columns and the beams are summarized in Table 9. The value of the concentrated loads at Nodes 3 and 2 in the x-direc- tion are 250 and 125 kN (56.25 and 28.13 kips), respectively. The value of the concentrated loads at Nodes 2 and 8 in the y-direction are 125 and –125 kN (28.13 and –28.13 kips), respectively. The value of the distributed loads at both the first and second floors is 20 kN/m (0.104 kip/in.).
From the planar frame, the maximum value of axial load, shear force, and bending moment of all the components can be calculated. The ultimate pure axial compression capacity, ultimate pure shear capacity, and ultimate bending moment capacity of columns and beams can also be calculated based on the section details. The calculation results of N, M, V,
T, N0, M0, V0, and T0 are substituted into Eq. (35) and are
summarized in Table 10.
As shown in Table 10, the failure factors of Components 7 and 9 are larger than 1, indicating that these components are Table 8—Values of three parameters of unified formula
2 2 2 1 2 3 0 0 0 0 1 T V M N p p q h T V M N a + a + a + − = Interactions a1 a2 a3 Equation Pure force 1 1 1 Torsion-shear 1 1 1 (3) Torsion-bending 1 1 1 (7) Bending-axial compression 1 1 1 (6) Shear-bending — 0.25 1 (24) Torsion-axial compression 1/(4N/N0 + 2)2 — — (26) Shear-bending-axial compression — 0.25 1 (29) Torsion-bending-shear 0.25 1 0.5 (31) Torsion-bending-axial compression 1 1 1 (33) Torsion-bending-shear- axial compression 0.25 1 0.5 (35)
Table 7—Comparison of test data and torsion- bending-shear-axial compression interaction formula
Specimen Nexp/N0 Texp/T0 Vexp/V0 Mexp/M0 Value of Eq. (35)
Zhao et al.36 average = 1.321; COV = 0.245
WV4-3-2 0.357 1.901 0.52 0.432 1.042 WV5-3-3 0.446 2.583 0.65 0.896 1.603 WV6-3-1 0.536 2.273 0.78 1.075 1.640 WV3-3-2b 0.303 2.118 0.508 0.433 1.186 WV3-2-2 0.303 1.76 0.339 0.289 0.775 WV4-2-2 0.401 2.292 0.443 0.377 1.235 WV3-5-1 0.26 1.559 0.945 0.534 1.567 WV3-2-2b 0.303 2.012 0.339 0.289 0.929 WV4-2-3 0.401 3.199 0.443 0.668 1.515 WV3-2-3 0.303 2.805 0.339 0.289 1.547 WV4-2-ul 0.401 2.847 0 0.377 1.572 WV4-3-2m 0.371 1.642 0.426 0.467 0.889 WV5-4-2m 0.464 2.484 0.65 1.326 1.671
Fig. 8—Planar frame. (Note: 1 m = 3.3 ft; 1 kN = 0.225 kips; 1 kN.m = 8.87 kip-in.)
Table 9—Section details of columns and beams
Component bw, mm d, mm rx, % rz, % fy, MPa fc, MPa
Column 500 500 1.5 0.25 400 30 Beam 250 500 1.5 0.3 400 30
not sufficiently safe. Given that the failure factors of Compo- nents 3, 4, 6, 8, and 10 are less than 1, these components are safe. Components 1 and 2 are subjected to axial tension, biaxial shear, biaxial bending, and torsion, so the unified formula is not applicable to these two components. There- fore, the safety of Components 1 and 2 cannot be determined by Eq. (35), which is a flaw of the unified formula. Similarly, Component 5 also cannot be determined.
Components whose failure factors are less than 1 can still sustain additional force. The unified formula can determine how much residual force these components can sustain because it can be used to calculate residual force capacity of components under various kinds of combined loading. Given the planar frame as an example, the residual capacity is calculated by the following steps. First, given the shear force as an example, the overall shear force capacity is calculated by 2 2 0 0 0 0 1 q N h 0.5p M 0.25p T N M T V V p − − − − = ′ (36)
Then, the residual shear force is calculated by
Vs = V′ – V (37)
The results are also shown in Table 10. For torsion, shear force, and bending moment, the residual force is nega- tive when the failure factor is larger than 1, whereas the residual force is positive when the failure factor is less than 1. The negative values mean that the force should be reduced to ensure that the failure factor is less than 1. Given Component 7 as an example, if 194 kN.m (1720.78 kip-in.) of bending moment is reduced and the axial compression, torsion moment, and shear force are unchanged, the failure factor is 1 and the component is under ultimate state. The positive values mean that the components are able to with- stand additional force capacity. Given Component 3 in Table 10 as an example, the component is able to bear an additional 67 kN.m (593.95 kip-in.) of torsion moment if the axial compression, bending moment, and shear force are unchanged, or an additional 200 kN (45.01 kips) of shear force if the axial compression, torsion moment, and bending moment are unchanged, or an additional 244 kN.m (2163.04 kip-in.) of bending moment if the axial compres- sion, torsion moment, and shear force are unchanged. From the table, however, it can be observed that the torsional and
shear residential capacities of Component 7 cannot be calcu- lated. No matter how the values of torsion moment and shear are modified, these components will not be safe. The compo- nents have been already been destroyed even if the torsion and shear are reduced to 0.
CONCLUSIONS
A unified formula capable of predicting the ultimate state of RC sections subjected to combined loading, including bending, shear, torsion, and axial compression, has been developed. The formula, which is based on bending-torsion interaction as a link between shear-torsion and bending-axial compression interactions, directly considers the overall inter- actions of bending, shear, torsion, and axial compression.
Various kinds of interaction equations can be obtained from the unified formula. Interaction formulas have been calibrated by comparisons of experimental results from members loaded in various kinds of load combinations.
Some of the features of the formula are as follows: 1. It directly considers the interaction between various kinds of loading.
2. It can accurately predict test data under various loading combinations.
3. It can be used as unified failure criteria of RC compo- nents in ultimate state to determine the safety of members under various loading states.
4. It can be used to calculate the residual stress capacity of components under various kinds of combined loading.
NOTATION
Ag = cross-sectional area of beam
Ao = gross area enclosed by centerline of shear flow path
As = area of tension steel
As′ = area of compression steel
Ast = total area of longitudinal steel
At = cross-sectional area of one leg of torsional stirrup
Av = cross-sectional area of one leg of shear stirrup
bw = width of beam
c = depth of concrete compression zone
d = effective depth of beam
d′ = effective depth to compression reinforcement
Es = modulus of elasticity of steel
fc = concrete cubic compression strength (150 x 150 x 150 mm [6 x
6 x 6 in.])
fc′ = compressive cylinder strength of concrete
fs′ = stress of compression steel
fy = yield stress of tension steel
fy′ = yield stress of compression steel
fyv = yield stress of stirrups
M0 = ultimate pure bending capacity
Mb = ultimate bending capacity in balanced condition when subjected
to eccentric compression
Mexp = experimental bending capacity Table 10—Failure factor and residual capacity
No.
Maximum force of components Ultimate pure force capacity Failure factor
Safe or not Residual capacity N, kN T, kN·m M, kN·m V, kN N0, kN T0, kN·m M0, kN·m V0, kN Value of Eq. (35) Tr, kN Mr, kN·m Vr, kN 3 398 90 448 145 6524 96 271 467 0.87 Yes 67 244 200 4 139 68 341 123 6524 96 271 467 0.88 Yes 70 214 196 6 134 47 235 78 6524 96 271 467 0.80 Yes 119 374 318 7 64 24 332 183 3262 27 136 258 1.21 No — -194 — 8 200 14 232 143 3262 27 136 258 0.89 Yes 22 100 69 9 42 24 347 178 3262 27 136 258 1.24 No — -221 — 10 78 14 235 134 3262 27 136 258 0.99 Yes 4 11 9
Mr = residual bending capacity
N0 = ultimate pure axial compression capacity
Nb = ultimate axial compression capacity in balanced condition when
subjected to eccentric compression
Nexp = experimental axial compression capacity
Nr = residual axial compression capacity
px = longitude reinforcement ratio
pz = stirrup reinforcement ratio
s = spacing of stirrups
T0 = ultimate pure torsion capacity
Texp = experimental torsion capacity
V0 = ultimate pure shear capacity
V′ = overall shear capacity according to unified formula
Vexp = experimental shear capacity
Vr = residual shear capacity
b1 = equivalent factor of rectangular compressive stress distribution
es′ = strain of compression steel
ey = yield strain of steel
ACKNOWLEDGMENTS
This research described in this paper has been sponsored by the National Natural Science Foundation of China (No. 51078132), the Chang Jiang Scholars Program and Innovative Research Team Project by the Ministry of Education of China (Project No. IRT0917). The authors gratefully acknowl- edge the assistance provided by Z. Xu, H. Yi, S. Chen, J. Li, and B. Yan.
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MS No. S-2011-088.R1 received April 13, 2011, and reviewed under Institute publication policies. Copyright © 2013, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion including author’s closure, if any, will be published in the November-December 2013 ACI Structural Journal if the discussion is received by July 1, 2013.