Study On Second - Order Elastic - Plastic Calculation Method Of Reinforced Concrete Columns

The second-order elastic-plastic performance of reinforced concrete (RC) columns relates to the coupling of material and geometric nonlinearity. The finite element methods are inefficient because modeling requires a lot of time. By analyzing current numerical and simplification methods, here, the inverse methods are theoretically derived to determine the sectional bearing capacity and the moment-curvature relationship through calculating the internal force from strain. These methods are accurate for using the whole constitutive relationships of reinforcement and concrete without simplification as well as no iterative and convergence error for the analytical algorithms. Furthermore, three different precision methods are derived to calculate the second-order elastic-plastic performance of RC columns and three levels of design tools are then provided correspondingly. The validity of these methods are verified by:the reasonability of change regulations of curves that were drawn in large quantities from calculation results; the logical relationships between these curves; the comparison of the existing methods, the current codes, the existing test results, etc.(a) By referring to Eurocode2, all the possible strain distributions at the ultimate limit state of the RC cross-section are obtained for Chinese condition. The relationship between these strains and the ultimate axial force-moment is one-to-one correspondence, and accordingly the sectional bearing capacity can be calculated directly from these strains. Procedures for rectangular and circular sections are derived and sectional axial force-moment interaction curves are obtained, which can be used in section strength checking and reinforcement design of short columns.(b) Through introducing intermediate parameters, one new widely applicable analysis method is derived for determining the moment-curvature relationship of the steel section, the analytical expressions of rectangular and I-shape section are given, and curves are then obtained including the moment-curvature curves with constant axial force, the axial force-moment interaction curves and the axial force-moment curves with constant curvature. These curves reflect the sectional ductility, the relationship and whole change process among moment-curvature-axial force. The results show that:the influence of axial force on the moment-curvature relationship is great, the existence or increase of axial force makes the cross section into the plastic stage earlier and faster, which makes the decreases of the moment carrying capacity. These influences are different with different sectional shape and the I-shape is more unfavorable than the rectangular.(c) Based on the same idea as steel section, the analytical algorithm on the moment-curvature relationship of the RC rectangular section is derived, which can be used to calculate yield curvature and ductility factor at ductility analysis and obtain the relationship curves among the three variables. The results show that:similar with steel section, the existence of axial force on the RC section is generally unfavorable because it decreases the moment bearing capacity, while under small compression it is favorable because the ultimate moment increases slightly with the increasing axial force. The sectional ductility becomes poor dramatically under large compression. The greater the curvature, the closer is axial force-moment to the ultimate value.(d) Based on the analytical algorithm on the sectional moment-curvature relationship, three methods are derived in section strength checking and reinforcement design for slender columns. The first is the improved Newmark method that belongs to numerical methods with high accuracy because of little simplification. The second is the graphic analysis and calculation method on the sinusoid deformation assumption, which can express directly the coupling of material and geometric nonlinearity, and it is relatively high accurate and efficient because of fewer simplification and analytical process. The third is the graph algorithm based on the quadratic parabola curvature assumption and the ultimate curvature simplification model. Although with more simplification, the third method can be used to derive the nomograph as a manual design tool. Especially, the reinforcement design and strength checking for slender columns can be conduct by only calculating three basic variables and doing four auxiliary lines. This achieves a simple calculation of the second-order elastic-plastic performance of RC columns.(e) The results from the column second-order elastic-plastic analysis show that: the ultimate bearing capacity of columns with a double reinforcement is approximately equal to the cross-section. This is false only in rare cases that the slenderness ratio is very large as well as the first-order eccentricity is very little. Under large compression, strength damage may occur for columns over a wide range of the slenderness ratio because the tangent stiffness is far more than0at the ultimate state with the stain restrictions of reinforcement and concrete.