New analytical approach for geometrically nonlinear buckling analysis of a inclined rod using the arc length method

This paper is concerned with nonlinear buckling problem of inclined rod subjected to concentrated loads and moments at the ends. Rigorous analysis of geometrically nonlinear structures demands creating mathematical models that accurately include loading and support conditions. This work introduces a new analytical approach to construct the governing equations considering large displacement. The mathematical formulation based on geometrical compatibility, equilibrium of forces and moments and constitutive relations considering large displacements. The geometrical compatibility relationship is getting from integrating along the elastic curve of the deformed rod. A system of nonlinear and integral equations with boundary conditions prescribed at both end is constructed. Using the arc length technique the paper developed incremental-iterative algorithm for solving the system of nonlinear equation. Based on the proposed algorithm, the paper established the calculation procedure and the programs for determining the equilibrium path for generally supported inclined rod subjected to concentrated loads and moments at the end.


Introduction
The stability of slender rods is a phenomenon associated with buckling and post bucking. Buckling and post buckling behavior for slender rods is very important since post buckling means loss the stability of structure associated with large displacement and may be lead to destruction the structure. Since the early classical contributions from Bernoulli, Euler and Lagrange in the 18th century, the subject of buckling, post buckling and large deflection analyses of slender rods has experienced significant evolution. In recent years, a large number of research work addressed the buckling and post-buckling behavior of rods [1][2][3][4][5][6]. Most of preceding work deals with numerical approach of problem formulation. This paper proposes a new analytical approach to formulating nonlinear buckling problem of inclined rods subjected to concentrated loads and moments at the ends. The mathematical model is built from geometrical compatibility, equilibrium of forces and moments and constitutive relations considering large displacement. The geometrical relationship is implemented by integrating along the elastic curve of the deformed rod. For solving the system of integral equations with boundary conditions prescribed at both end, the paper presents a new algorithm based on arc length technique [7][8][9]. The calculation programs for solving the nonlinear buckling problem of inclined rods sub which generally supported at two ends. For investigating the nonlinear equilibrium path and critical point presented numerical test for commonly type of inclined rods. Let us consider the geometrical compatibility, equilibrium of forces and moments and constitutive relations for the flexural bar. In formulating nonlinear problem, assuming that mechanical behaviour of materials is ideally elastic. Axial displacement due to stretching is negligible in comparison with the normal displacement. Based on the Euler Bernoulli beam theory [10], from moment-curvature relationship and elementary calculus, we get ''

Problem formulation
Where:  is radius of curvature; d is differential angle; ds is differential curve length; y is displacement function; M is bending moment and EI is flexural stiffness.
The equation (2) is differential equation of elastic curve considering large displacement. It can be compactly written as follows The bending moment is defined by the expression From (2) and (3), we get Taking a derivative of both sides of equation (4) The constants of integration can be determined from the prescribed constraints (the boundary conditions) at the end point A', having Incorporating constants of integration C to (7), we get It can be written in compact form as follows From (8) and (2), getting the function of bending moment From (8) The sign in (10) depends on convexo-concave type of elastic curve. The axial displacement due to stretching is negligible in comparison with the normal displacement and the length of the bar remains unchanged. Integrating along the elastic curve of the deformed rod and from (8) The equilibrium equations are as follows: The system of equations from (11) to (16) is governing system for solving the nonlinear buckling problem.  Figure 2. Incremental-iterative procedure for solving nonlinear buckling problem based on arc length technique

Numerical analysis
Based on proposed above incremental-iterative algorithm, the calculation program for solving nonlinear buckling problem is written in MathCAD software.

Yes
Step: i

Numerical results
The calculating results are equilibrium path, limit points, relationship load-displacement at any point within the rod (shown in Fig. 4-6)

Summary
The proposed method for solving the geometrically nonlinear buckling problem of inclined rod is effective way in analytical approach. The solution of the geometrically nonlinear buckling problem of inclined rod using the proposed analytical formulation has important advantage in possibility of getting the load-displacement relations at any point within element.