Critical Length of Long Hinged and Restrained Concrete Columns


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Title: Critical Length of Long Hinged and Restrained Concrete Columns

Author(s): Wen F. Chang

Publication: Special Publication

Volume: 12


Appears on pages(s): 521-533


Date: 1/1/1965

The behavior of long restrained concrete columns as part of a building frame is much more complicated than that of long hinged concrete columns under eccentric load. A theoretical analysis for determining the critical column length for long hinged concrete columns has been derived previously by the writer. A method for determining the critical column length for long concrete column as part of a box frame is presented here. A long concrete column may buckle laterally as the critical section of the column reaches material failure; but the material failure of a column cannot be used as the criterion to determine the criticalcolumn length. Plastic hinges may be developed in a frame, but a long column may become unstable without developing plastic hinges. An analog computer was used as a tool to determine the critical column lengthfor the following reasons: (1) The problems involve differential equations which are particularly suitable for analog computer solutions (involving typically about 30 sec of computer time for a solution of adequate design accuracy); (2) the plotter, which is a standard unit of the computer, will plot the column or beam deflection curves on graph paper for visual reference; (3) the programmer can more readily make designdecisions by selection of proper constants for each preliminary trail of the problem. Concrete columns, subjected to eccentric loads at the ends will deflect laterally. As the columndeflects laterally the column moment along the column length will be increased by an amount equal to the product of column load and lateral displacement. This increment of moment becomes very important for the analysis of long columns. As the column deflects laterally, cracks will usually appear at the convex side of the column near the region of maximum moment. The error in using a constant EI (modulus of elasticity x moment of inertia) approximation to determine critical column length may be of substance. In considering variable E and I along the deflected column, moment versus edge-strain relationships was derived for a given column with a given column load. A nonlinear second order differential equation can then be obtained from each moment versus edge-strain curve. An analog computer was used to solve the differential equation and the column deflection curves and angle of rotation curves were plotted on graph paper by the computer plotter for a given column with given column load P. For any given values of end moment ME and the column load P, the critical column length for eccentrically loaded hinged column can be easily determined from the column deflection curves. The long column as part of a symmetrical box frame was further studied. It is assumed that all joints are rigid and that the joints do not move laterally. The end rotation 0E of the column must be equal to the end rotation of the beam, and the end moment ME of the column must equal to the end moment of the beam. For a given box frame with given column and beam loads, the critical column height can be determined. It is found that the co-tangency criterion for determining the critical column length for eccentrically loaded hinged column is not always applicable for determining the critical column length for restrained column.