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Unformatted text preview: Section 6.1 Solid Mechanics Part I Kelly 215 6.1 Elastic Buckling The initial theory of the buckling of columns was worked out by Euler in 1757, a nice example of a theory preceding the application, the application mainly being for the later invented metal columns in modern structures. 6.1.1 Columns and Buckling A column is a long slender bar under axial compression, Fig. 6.1.1. A column can be horizontal, vertical or inclined; in the latter cases it is termed a strut . The column under axial compression responds elastically in exactly the same way as the axial bar of 4.3. For example, it decreases in length under a compressive force P by an amount given by Eqn. 4.3.5, EA PL / = . However, when the compressive force is large enough, the column will buckle with lateral deflection. This possibility is the subject of this section. Eulers Theory of Buckling Consider an elastic column of length L , pin-ended so free to rotate at its ends, subjected to an axial load P , Fig. 6.1.1. Assume that it undergoes a lateral deflection denoted by v . Moment equilibrium of a section of the deflected column cut at a typical point x , and using the moment-curvature Eqn. 4.6.36, results in 2 2 ) ( ) ( dx v d EI x M x Pv = = (6.1.1) Hence the deflection v satisfies the differential equation ) ( 2 2 2 = + x v k dx v d (6.1.2) where EI P k = 2 (6.1.3) Fig. 6.1.1: a column with deflection v P P x y ) ( x v L Section 6.1 Solid Mechanics Part I Kelly 216 The ordinary differential equation 6.1.2 is linear, homogeneous and with constant coefficients. Its solution can be found in any standard text on differential equations and is given by ( ) ( ) kx B kx A x v sin cos ) ( + = (6.1.4) where A and B are as yet unknown constants. The boundary conditions for pinned- ends are ) ( , ) ( = = L v v (6.1.5) The first condition requires A to be zero and the second leads to ( ) sin = kL B (6.1.6) It follows that either: (a) = B , in which case ) ( = x v for all x and the column is not deflected or (b) ( ) sin = kL , which holds when kL is an integer number of s, i.e. K , 3 , 2 , 1 , = = n L n k , (6.1.7) As mentioned, the solution (a) is governed by the axial deformation theory discussed in 4.3. Concentrating on (b), the corresponding solution for the deflection is K , 3 , 2 , 1 , sin ) ( = = n L x n B x v n (6.1.8) The parameter...
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This note was uploaded on 01/20/2012 for the course ENGINEERIN 1 taught by Professor Staff during the Fall '11 term at Auckland.
- Fall '11