Advanced Methods of Structural Analysis

13.3 Stability of Columns with Rigid andElastic Supports 461

13.3 Stability of Columns withRigid and Elastic Supports

Elastic bar presents a structure with infinite number of degrees of freedom. Such
structures are called the structures with distributed parameters. Their stability anal-
ysis may be effectively performed on the basis of differential equation of the elastic
curve and initial parameter method. Both methods are presented below.

13.3.1 The Double Integration Method.............................

Stability analysis of the uniform compressed columns is based on the moment-
curvature equation

EI

d^2 y

dx^2

DM.x/; (13.4)

wherexandyare the coordinate and lateral displacement of any point of the beam;
EIis the flexural rigidity of the beam;M.x/is the bending moment at the sectionx
of the beam caused by given loads.
This equation allows us to find exact value of the critical load for columns with
rigid and/or elastic supports. To apply this equation, we need to show a column in
deflected state, then it is necessary to construct the expression for bending moment
in terms of lateral displacementyof any point of the column, and to write the
differential equation (13.4). As a result we get the ordinary differential equation,
which could be homogeneous or nonhomogeneous. Then for each specific problem,
we need to integrate this equation and find the constant of integration, using the
boundary conditions. For typical supports they are the following:

Pinned support:yD 0 andy^00 D 0
Clamped support:yD 0 andy^0 D 0
Sliding support:y^0 D 0 andy^000 D 0
Free end:y^00 D 0 andy^000 D 0
For computation of unknown parameters, we get a system of the homogeneous
linear algebraic equations.
Nontrivial solution of this system leads to the equation of stability. Solution of
this equation leads to the expression for critical load.

13.3.1.1 Uniform Clamped-Free Column

Let the column is subjected to axial compressed forceP(Fig.13.7). Elastic curve
of the column is shown by dotted line.
If the lateral displacement of the free end isf, then the bending moment is
M.x/DP.fy/. Corresponding differential equation of the compressed column
becomes

EI

d^2 y

dx^2

DP.fy/ or EI

d^2 y

dx^2

CPyDPf: (13.5)