Slender member is subjected to axial load and biaxial bending moment and fails due to buckling. This buckling is caused due to slenderness effect also known as ‘P?’ effect. This buckling gives rise to excessive bending moment occurring at a point of maximum deflection. This additional bending moment is considered in second order analysis. The objective of the research reported in this paper is to formulate bending moment equation by using beam column theory and to study the behaviour of solid circular section and hollow circular section of bridge pier. The optimization in area of cross section is done by providing a combination of solid and hollow circular section in place of a solid circular section of pier within permissible limits. A comparative study on behaviour for all three conditions is been carried out.
Piers are not only subjected to axial load but also forces in longitudinal direction as well as in transverse direction. These forces cause moment in longitudinal direction and transverse direction at base of pier. Thus, pier is idealized as a column subjected to axial load and biaxial moment. These forces cause the pier to buckle along its height. The moment due to buckling is not considered in first order analysis.
In order to get accurate forces one has to go for second order analysis where in the buckling effect is considered. Beam column theory is one of the methods to calculate the bending moment by second order analysis.
Iterative neutral axis method is used to design the cross section of pier. In a section subjected to axial load combined with two orthogonal moments, by assuming the neutral axis at certain depth and stress at that point is to be calculated. This stress at neutral axis should be zero or else the procedure is revised for another trail.
II. SECOND ORDER ANALYSIS USING BEAM-COLUMN THEORY
Beams subjected to axial compression with lateral loads act as beam-column. The basic equation for analysis of beam-column can be derived by considering a beam as shown in Figure1.
The beam is subjected to an axial compressive force P and lateral load of intensity ‘q’ which varies with the distance ‘x’ along the beam.
Consider an element of length ‘dx’ between two cross sections taken normal to the original axis of beam as shown in Figure 2.
The lateral load has a constant intensity ‘q’ over a distance ‘dx’ and will be assumed positive when in direction of positive y axis which is downward in this case.
The shearing force V and bending moment M acting on either side of the elements are assumed positive in the downward direction.
The relation between load, shear force and bending moment are obtained from the equilibrium of the element in Figure 2. On summing forces in the y direction it gives.
Taking the moment about point on beam and assuming that angle between the axis of beam and horizontal axis is small, we obtain,
If terms of second-degree are neglected, this equation becomes
If the effects of shearing deformations and shortening of the beam axis are neglected the expression for the curvature of the axis of the beam is,
The quantity EI represents the flexural rigidity of beam in a plane of bending, i.e. XY plane, which is assumed to be plane of symmetry. Combining equation (3) with equation (1) and equation (2) we can express the differential equations of the axis of the beam in the following alternate forms:
Equations (1) to (5) are the basic differential equations for bending of beam-column. If the axial force's P equals zero, these equations reduces to the usual equations for bending by lateral loads only. The nature of the axial forces have significant effect on the deflections and ultimately on the secondary moments.
III. ITERATIVE NEUTRAL AXIS METHOD
Iterative neutral axis method is used for design of slender member which are subjected to axial load and biaxial moment. In this method, some percentage of steel is assumed and the moment of inertia of full section is calculated. Then inclination of neutral axis is calculated. Then, moment of inertia and eccentricity of cracked section is computed. Compute stress at neutral axis, if it is zero, and if stresses at extreme fibers are within permissible limit, the assumed percentage of steel is acceptable otherwise the neutral axis has to be shifted and same procedure has to be carried out.
IV. THEOROTICAL FORMULATION
A. Trapezoidal Load Throughout the Height of Pier
V. PARAMETRIC STUDY
Forces on pier are calculated as specified in IRC and the maximum moment is calculated in Table 1 shown below. Using combined stress equation and keeping the stress constant, behavior of a solid circular and hollow circular section with combination of both is studied. The percentage reduction in volume for combination with solid and hollow pier is plotted for different heights of pier. The variation in area of cross section for different bending moments is studied.
A. As the height of the bridge pier increases the base B.M. value increases and critical B.M develops at the base of the pier.
B. Volume of concrete required increases with increase in base moment. However the rate of increase of volume of concrete required is milder for combination pier in comparison with solid and hollow circular pier.
C. The rate of increase for % reduction in volume of concrete varies from 49% to 83% for solid pier and 37% to 76% for hollow pier for height varying from 15m to 40m respectively.
D. Hence it can be concluded that as the height of pier increases the solid circular section and hollow circular section proves to be uneconomical as compared to combination of solid and hollow circular pier in section.
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 V. K. Raina (1975), “R.C. section subjected to axial load and any axis bending”, Bridge and Structure Journal V-5, 126-140.
 Standard specification and code of practice for Road Bridges, Section-VII, IRC-78, Foundation and substructure, The Indian Road Congress, New Delhi.
 Standard specification and code of practice for Road Bridges, Section-III, IRC-21, Cement concrete (Plain and Reinforced), The Indian Road Congress, New Delhi.
 Analysis and Design, Tata McGraw-Hill Publishing Company Ltd., New Delhi.
 Indian Standard Code of Practice for Plain and Reinforced Concrete IS: 456-2000, Indian Standard Institution, New Delhi.