Abstract
Based on modified methods for the results of first-order analysis of RC columns, different codes approximate the second-order effects by using equations focusing on the maximum additional moment through the column height. These equations did not refer to the additional moments between the column and the connected beam, only the effect of the connected beams is taken into consideration by dealing with the effective length of the column, not the total length. Moreover, these equations did not take into account the second-order effect, which is caused by axial force and the inverse moments due to beam restriction for the column ends. This paper presents a new moment magnifiers matrix for the additional moments at the connection between braced columns and the connected beams as a simplified computation that can be used in the design procedure. That is through an equation based on transforming the original long column in second-order analysis to an equivalent isolated column. The equivalent column was represented as an element restricted with rotational spring support at its ends, and it is subjected to lateral distributed loads that have the same influence of the second-order effect on the induced additional moments in the long column. The suggested equivalent column can be used to form the additional bending moment diagram, also to compute the additional deformations as well. Numerous factors were analyzed linearly by using the presented new moment magnifiers matrix and finite element method, and the results proved the efficiency of the proposed model. Although the presented suggested model is based on the isolated analysis of the long column, the effect of the additional moments in the adjacent long column can be considered by presented two suggestions to improve the model. Also, development was proceeded on the model by modifying the flexural rigidity (EI) which is recommended in ACI to appropriate the time of failure. The additional moment values of the developed model were close to the values calculated by the ACI equation.
Highlights
A column is subjected to axial load and equal or unequal end moments, which are caused by the connected beam loads, and deformed laterally due to the existence of end moments. e axial load and occurred lateral deflection cause additional bending moments along the column height which is called second order. e additional bending moments cause additional lateral displacements and rotations of the column and additional rotation of the members connecting into the column
Based on the equivalent column concept, a new moment magnifiers matrix was presented in this paper for computing the additional end moments in the braced long column. e equivalent column was an element restricted at its ends by two spring rotational supports and is subjected to lateral distributed loads, which have the same influence of the secondorder effect in a long column. e additional moments’ diagram and additional deformations can be computed by using the suggested equivalent column taking into consideration the second-order effect, which is caused by the axial load and the inverse moments due to beams restriction for the column ends, this effect is important it is neglected in design codes
E long column in the suggested model was analyzed as an isolated element, but by two presented suggestions, the effect of the additional moments of other adjacent long columns, if any, can be considered. e first suggestion took into account the effect of adjacent additional moments by computing the transmitting additional moments among columns through transfer coefficients depended on the rigidity of the connected beams; the equation of the moment magnifiers matrix was applied more than once for the transmitted moments
Summary
A column is subjected to axial load and equal or unequal end moments, which are caused by the connected beam loads, and deformed laterally due to the existence of end moments. e axial load and occurred lateral deflection cause additional bending moments along the column height which is called second order. e additional bending moments cause additional lateral displacements and rotations of the column and additional rotation of the members connecting into the column. E axial load and occurred lateral deflection cause additional bending moments along the column height which is called second order. Advances in Civil Engineering analysis of the second-order effect can be performed in the finite element method by adding the geometric stiffness matrix to the elastic linear matrix “mechanical stiffness” for the beam column element. In order to account for second-order effects due to geometric and material nonlinearities, the theoretical model (computer software) uses classical stiffness analysis of linear elastic two-dimensional structural frames, the iterative technique combined with an incremental method for computing load-deflection behavior and failure load of the frame, frame discretization to account for column chord (P-Δ) effects and axial load-bending moment-curvature (P-M-φ) relationships to account for effects of nonlinear material behavior [3]
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