Stability Analysis of Beam-Columns
The primary objective of this chapter is to develop methods for predicting the deformation response of individual slender members or simple frames composed of such members subjected simultaneously to axial force and bending moment. Such structural members
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5.1 Introduction The primary objective of this chapter is to develop methods for predicting the deformation response of individual slender members or simple frames composed of such members subjected simultaneously to axial force and bending moment. Such structural members are termed beam-columns. In this chapter we are mainly concerned with lateral deformations i.e. deformations perpendicular to the longitudinal axis of the member. The analysis procedures are based upon the solution of appropriate differential equations. It is recognised that the influence of axial force on bending deformations is one of the most important aspects of the structural analysis and design. The lateral loads and/or end moments cause deflections which are further amplified by axial compression causing moment, Py along the member. These additional deflections add significantly to the moments, which may result still further deflections. Finally, a stable situation is reached where deflections correspond to the bending moments due to both lateral load and Py. Because of this interaction between the axial force and the moments, the general superposition procedures are inappropriate. However, as the bending moment approaches zero, the member tends to become axially loaded strut, a problem that has been treated in details in Chap. 4. On the other hand, if the axial force vanishes, the problem reduces to that of a beam.
5.2 Derivation of Basic Equations The iterative process described above actually need not be carried out to obtain a solution. The influence of axial force on the bending moment can be incorporated directly into the governing differential equation: d2y) El ( dx 2
= -Mx = -[Mo(x) + Py(x)]
M. L. Gambhir, Stability Analysis and Design of Structures © Springer-Verlag Berlin Heidelberg 2004
(5.1)
172
5 Stability Analysis of Beam-Columns
where M 0 (x) is the moment due to lateral forces, end moments, or from a known eccentricity of axial force at one or both ends and Py(x) takes into account the added influence of the axial force and deflection. The moment Mo(x) may vary along the length of the member. The moment-equilibrium equation (5.1) can be expressed in the standard form for the case when E I is constant. (
d2
y) +
dx 2
Py __ Mo(x) EI EI
or
y) +
2 (d ctx2
a 2 y = _ Mo(x) EI
(5.2)
J
where a 2 = 1 . As described in Chap. 2, the shear force equilibrium expression of beam-column elements can be obtained by differentiating the moment-equilibrium relation given by (5.1) with respect to x, i.e. 2
d ( EI ctx2 d y) Q(x) = dx
+ p (dy) dx .
If E I is constant (5.3)
Similarly, a second differentiation of (5.1) yields the equilibrium equation for lateral loads, i.e.
For the case when EI is constant. (5.4)
where w(x) is the intensity ofload at a point on the element.
5.3 Analysis of Beam-Columns Beam-column being the basic component of a rigid frame will be treated first, and then the analysis will be extended to the rigid frame. If EI is constant, the general solution of (5.4) has the form y(x) =A sin ax+ B cos ax+
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