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The IUP Journal of Structural Engineering :
Prediction of Postbuckling Behavior of Compound Cantilever Columns Subjected to Different Independent Compressive Loads: A Numerical Experiment
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Several studies on the postbuckling behavior of compound columns subjected to different independent compressive loads are available in literature. The postbuckling problem is popularly known as the ‘Elastica’ problem. The compound columns contain two segments of equal length. The segments are made of two different materials with different Young’s moduli with constant area moment of inertia. Alternately, this configuration can also be treated as a stepped column with constant Young’s modulus, having different area moments of inertia of the segments and having the same Young’s modulus. The postbuckling behavior of these compound cantilever columns, for a specific compressive load, is represented as the ratio of axial load carrying capacity, for a given free end angle in the postbuckling region to the buckling load. This problem is usually solved using either complex continuum or numerical methods. This paper contains a method of evaluation of the postbuckling behavior of the said columns through a numerical experiment that gives reliable results of the postbuckling behavior. However, the accuracy decreases, which is tolerable, as the free end rotation increases, with the limiting value of 80 degrees.

 
 

The large deflections of buckled two-segmented compound cantilever (cc) columns (Elastica) subjected to independently different types of compressive loads are generally called the postbuckling behavior. The two segments are of equal length of L/2, where L is the length that are made up of two different materials, having the Young’s moduli E1 and E2, from the clamped to mid-length and from mid-length to the free ends of the cc columns, respectively. The cc columns are subjected to three types of independent axial compressive loads, which are specified in the initial (un-buckled) axial x-direction of the Cartesian coordinate system (x, y). The cc column is free at one end, where the axial coordinate x = 0, and clamped at the other end, where the axial coordinate x = L, in the Cartesian coordinate system, at the onset of buckling. As has been already mentioned, the two segments contain two different materials, for which the relation of the Young’s moduli of the materials is E1 > E2 and the ratio of E1/E2 = n. For the uniform column E1 = E2, and as such the value of n = 1.0. The area moment of inertia of the cross-section I for both the segments of the cc columns is constant. The ‘Elastica’ problem is well discussed by Timoshenko and Gere (1961). The different cc columns considered in this paper are the steel-aluminum and titanium-aluminum, where the values of n = 1.5556 and 2.9495, respectively. Many practical uses of the compound columns and the development of production technologies (mainly welding) are given hereunder as information.

Compound columns are used in many areas of modern structural engineering applications like bimetallic engine valves, universal joints, gear hubs, diesel injectors, turbine blade assemblies, nuclear reactors, twisted drills, etc., which are used in the fields of automobile, aerospace, defense and power generation. These columns are also used in making consumer products like hand tools and sports equipment. Other use of the compound columns is to achieve near optimum configurations that satisfy stiffness constraints like buckling and postbuckling (Venkayya et al., 1973; Berke and Khot, 1974; Khot et al., 1976; and Rao and Swami, 1980). The production technology is an important aspect in realizing these columns to join the segments made up of different materials. This is accomplished using the modern welding technologies (Dawes, 1977; Aritoshi et al., 1991; and Lee et al., 2006) which assure higher weld efficiency without any undesirable effects.

The aim of the present work is to propose a numerical experiment to obtain the postbuckling behavior of cc column, with two segments of different materials, to the aforementioned loads. Though the numerical experiment does not have a firm mathematical background, it is much simpler than the complex theoretical and numerical formulations and gives accurate postbuckling results. Its effectiveness is demonstrated from the existing solutions that are obtained by using the well-known elliptical, Runge-Kutta and Galerkin Finite Element Method (GFEM). The numerical results, in terms of the ratio of the postbuckling to the buckling load Ppb/Pb, are accurate for the engineering purpose, when the free end angle is of 80 degrees, which is the maximum angle in the present study. However, for smaller free end angles, the present method gives accurate results. The free end angles beyond 80 degrees are not considered in this study to avoid consideration of nonlinear elastic effects, which will arise when the deformations are very large. The configuration of the deformed cantilever column, the three types of independent axial compressive loads and the details of the numerical experiment are discussed in the following sections.

 
 

Postbuckling, Compound cantilever columns, Two segments, Numerical experiment, Accuracy of numerical results, Effect of free end angle