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Unformatted text preview: BIAXIAL BENDING IN COLUMNS By Prof. Abdelhamid Charif 1 INTRODUCTION Columns are usually subjected to two bending moments about two perpendicular axes ( X and Y ) as well as an axial force in the vertical Z direction (see Figure 1). (a) Figure 1: Biaxial bending of columns (a): 3D view (b): Bending about Xaxis (c): Bending about Yaxis (d): Inclined neutral axis in biaxial bending With the shown sign convention, bending about Xaxis causes compression in the top part and tension in the bottom region, whereas bending about Yaxis causes compression in the left hand part and tension in the right part. For symmetric sections subjected to uniaxial bending, the neutral axis is parallel to the moment axis. In biaxial bending (d), the topleft part is subjected to double compression and the bottom right part is subjected to double tension. The remaining parts are subjected to combined compression and tension. This means that the two moments are not X Y M x M y P Y M y X M x (b) (c) (d) X Y M x M y independent but coupled. The resulting neutral axis is inclined with an angle depending on the moment values as well as the section properties. Interaction between the axial force P and the two bending moments M x and M y is represented by a 3D surface. The design surface is inside the nominal surface. The 3D surface is constructed by combining several interaction curves PM at various neutral axis angles. 3D interaction surface P  M x M y Various 2D scans can be extracted from the 3D surface: • Horizontal scan giving interaction curve M x – M y for a given value of the axial force P , also called load contour. • Vertical scan giving interaction curve P – M x for a given value of M y moment. • Vertical scan giving interaction curve P – M y for a given value of M x moment. General section in biaxial bending (a): Inclined neutral axis in global axes (b): Rotated section and use of variables in local axes The figure shows a general section subjected to biaxial bending. With respect to the sign convention shown in the figure, the nominal force and moments in biaxial bending are given by: & + = i si c n F B f P ' 85 . si i si b c nx Y F BY f M & + = ' 85 . si i si b c ny X F BX f M & + = ' 85 . B is the area of the concrete compression block. X b and Y b are coordinates of the centroid of the compression block with respect to X and Y axes having the origin as the centroid of the gross section. Steel bars are described by their coordinates X si and Y si . The compression block may have more than one part as and may contain parts of the possible voids present in the section. For each neutral axis angle, an interaction curve (meridian) PM xM y is constructed by varying the neutral axis depth from pure compression to pure tension. Calculations are complex and are usually carried out in local axes (b)....
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 Winter '09
 parra
 Force, Second moment of area, PNY, biaxial bending

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