NON LINEAR BUCKLING OF COLUMNS

The geometric non-linear total potential energy equation is developed and extended to study the behavior of buckling and deflection beyond the bifurcation point and showing columns resistance beyond the Euler load. Three types of boundary conditions are studied (pin ended, fixed ended and cantilever). The equation of non-linear total potential energy is solved by exact method (closed form solution) and compared with other approximated methods (RayleighRitz, Koiter’s theory and non-linear finite difference method). The agreement is found quite enough and satisfactory for most situations of practical cases.

Koiter's theory and non-linear finite difference method ( .

Introduction:
Stability of a body is that condition [1][2][3][4][5][6][7], if after some slight disturbance in the configuration the body returns to the original configuration, this condition is satisfied when there is no change in the total potential energy as the position is varied. In other word, the stability is obtained when the total potential energy in minimum condition and un-stability occurred at maximum total potential energy. For conservative systems, the equilibrium configuration is corresponding to the minimum total potential energy. Mathematically the stability problem is called an Eigen value problem, the critical load (buckling load) is an Eigen value load of the problem, and the deflection w(x) corresponding to the load is an Eigen function. Equilibrium method provides an infinite set of Eigen values (critical loads) where non-trivial configurations could satisfy the requirements of equilibrium and called modes of buckling. The lowest critical value is called the buckling load (Euler load). The equation of minimum total potential energy (functional) or general governing differential equation of the problem can be solved by various techniques such as (closed form or exact solution, finite element method) or approximated methods such as (Rayleigh-Ritz method, Galerkin's method and finite difference method). The bulk of buckling analysis that considered in most references is limited to the linearized Eigen value problems that define buckling load. In this study the geometrical non-linear total potential energy equation is developed for fixed ended and cantilever in addition to pin ended columns and solved by different methods.

Procedures of solution: -
1-For known value of (P/P E ), the value of [ = sin( /2)] is determined from equation (11), by trail and error method to satisfy the value of the integration to be equal to root of the ratio (P/P E ), using numeric integration method. 2-Use equation (14) or equation (15) to determine the value of (w max /L) for the known values of (P/P E ) and ( ) which determined from the previous step. 3-Repeat the procedures for other values of (P/P E ). 4-Tabulate the results and plot the relationship of (P/P E ) versus (w max /L).

II-An Intermediate theory (Rayleigh-Ritz method): -
The total non-linear potential energy equation (3) and the non-linear curvature given in equation (2)

1-Pin ended column:-
The Eigen function is taken as the following.

2-Fixed ended column:-
The suitable Eigen function which satisfy the boundary conditions of the two

IV-Non-linear finite difference method.
The non-linear equation of bending of a member subjected to axial force can be written as the following: EI ( 2 w/ x 2 ) / [1-( w/ x) 2 ] = -P w Re-arrange the above equation to obtain:

Procedures of solution:
1-Assume a specified value of (w max /L). ii-Determinate of [A] = 0; this gives the Eigen value of the problem (K) 5-Find P = EI/h 2 K 6-Find the ratio (P/P E ) corresponding to the assumed value of (w max /L). 7-Repeat the above steps for other values of (w max /L). 8-Plot the relation of (w max /L) versus (P/P E ). 9-For fixed ended and cantilever columns, apply the same procedures using the suitable Eigen function and same changes in the formulation of the non-linear finite difference equation corresponding to the boundary conditions of the problem.

Discussions and conclusions:
The analysis of buckling that considered in most references is limited to the linearized Eigen value problems that define the buckling load. This research is a trail to study the geometric non-linear behavior of buckling beyond the buckling load for different types of columns (pin ended, fixed ended and cantilever). Derivations and results of exact theory are compared with different methods (Rayleigh-Ritz method, Koiter's theory and non-linear finite difference method). Fig(1) shows the comparison of exact solution for all column types and show that the curves are tangent to the horizontal axis at the load ratio (P/P E =1), and then the value of the deflection increased with increasing of (P/P E ). This point is called as the bifurcation point. This behavior means that the column can withstand even higher loads beyond the bifurcation point (buckling load). This situation is called post-buckling stability and the deformation regime beyond this point is called post-buckling regime. Figs (2,3 and 4) show the comparison of the result obtained from the different methods (Rayleigh-Ritz method, Koiter's theory and non-linear finite difference method) for pin ended, fixed ended and cantilever respectively. The results show that all solutions give the same behavior beyond the bifurcation point and agreement between the exact solution and these methods are quite good for (w max /L < 0.3) for pin ended and fixed ended columns and (w max /L < 0.4) for cantilever column, these limits are quite enough and more than satisfactory for most situations of practical cases. The variation of results of all methods in comparison with the exact method is about (±4%) for pin ended columns and (±5%) for fixed ended columns for (w max /L < 0.3) while (±4%) for cantilever columns for (w max /L < 0.4). Fig(1) shows that for constant value of (P/P E ), the cantilever column gives deflection value approximately twice than pin and fixed ended cloumns as shown below: at P/P E = 1. And for constant deflection ratio (w max /L=0.4), the load ratio (P/P E =1.5) for pin and fixed ended columns while (P/P E =1.06) for cantilever column, this mean that cantilever column resist small extra load beyond the bifurcation point while pin and fixed ended resist much more up to (50%) beyond the bifurcation point. at w max /L = 0.4 Type P/P E Pin ended column 1.5 Fixed ended column 1.5 Cantilever column 1.06 Increasing of the deflection (w max /L) up to (0.25) cause increasing the load ratio (P/P E ) to (1.1), only (10%) increasing. But when (w max /L) increase to (0.4) the non-linear load ratio (P/P E ) jumped to the value (1.5) this reflect and explain the effect of large deflection (geometric non-linearity) on the resistance of the columns. The response and behavior of pin and fixed ended columns beyond the bifurcation point are similar while the cantilever columns showed lesser effect.