The calculation of ring-section reinforcement considering the second-order effect has triple nonlinearity, that is, the nonlinearity of material, geometry and section-width variation, it’s very difficult to calculate. In order to provide a fast calculation tool for the design work, a simplified model of the ultimate curvature of the ring-section column is proposed through calculation and analysis, then calculated the second-order eccentricity by the quadratic parabola hypothesis of the second-order deflection shape of the member, and finally the simplified n-mu second-order equilibrium equation has been obtained. The three-coordinate nomogram for calculation was drawn to further facilitate the design in the paper. The hand-calculated reinforcement of the ring-section column could be done only by making 4 auxiliary lines in the diagram, avoiding iterative solution of the transcendental equation. Because of the dimensionless derivation, this nomogram is multipurpose, it could solve the calculation problems of second-order reinforcement in the case of arbitrary section size and concrete grade of C15 ~ C50.

Based on Rusch’s creep constitutive relation, differential equations for the redistribution of shrinkage internal force and creep of the composite beam are derived and solved. The closed solution is cumbersome and is inconvenient to be applied practically. It is hard to solve the accurate solution for coupled differential equations. Therefore, a simplified approach is given. However, it ignores the influence of the redistribution of bending moment of the concrete slab on the axial strain and removes the coupling relationship of differential equations so that it makes the solution become convenient. The comparison of the results calculated by the two approaches shows that their calculated errors are small, within 3%, when the stiffness ratio of the concrete slab and the steel beam are less than 0.185. It also shows that the greater the stiffness of the steel beam, the greater the constraint on the creep of the concrete slab, so is the redistribution of internal force.

Calculating the second-order effect and reinforcement of reinforced concrete box section columns has geometric nonlinearity and material nonlinearity. It requires integration and iterative solutions and is inconvenient in practical applications; moreover, China’s “Code for Design of Concrete Structures” (GB 50010-2010) uses the same formula as that for rectangular sections when calculating geometric nonlinearity. To find out a calculation method by hand that is specific to box-shaped sections and does not require iterative procedures, the theoretical derivation is adopted and divided into two gradations: (1) in terms of cross-section: using strain as the known variable to solve the internal force, thus solving the calculation problem of the bearing capacity of the cross-section; (2) in terms of members, the model column method can be used to solve the calculation problem of second-order effects of members. Finally, nomograms that can calculate the second-order effect and reinforcement of columns without iterative calculation are drawn, which contain five parameters, namely first-order bending moment, axial force, curvature, slenderness ratio, and the mechanical ratio of reinforcement. One of the nomograms corresponds to the cross-section resistance, and the other corresponds to the balance of internal resistance and external effect. Compared with the GB 50010-2010, the differences in the total bending moment and reinforcement ratio are within 10% and 20%, respectively. Compared with the numerical calculation results, the remaining examples are within 10% under normal load conditions.