A Master of Science thesis in Mechanical Engineering by Mohammed Ghassan Aldadah entitled, "Buckling of Inhomogeneous and Functionally Graded Columns," submitted in July 2014. Thesis advisor is Dr. Shivakumar Ranganathan and thesis co-advisor is Dr. Farid Abed. Available are both soft and hard copies of the thesis.
Buckling is an instability encountered in a wide variety of problems, both in engineering and biology. Almost all engineering structures are designed with adequate safety factors to prevent failure due to buckling, yielding, or dynamic loads. In a classical sense, design for buckling is done by carefully controlling the modulus of elasticity, moment of inertia, and the length of the structure. Further, such an approach assumes the material to be homogeneous and does not generally account for the microstructural details of the column. In the first part of this thesis, we study the buckling of inhomogeneous columns with a two-phase checkerboard microstructure. Monte Carlo simulations are used to generate microstructures with arbitrary volume fractions and phase contrasts (ratio of the modulus of individual phases). An analytical form is obtained for the ensemble averaged critical buckling load based on the results of over 18,000 eigenvalue problems at arbitrary volume fractions, phase contrasts, and distributions. Further, microstructural realizations that correspond to the highest buckling load (best design) and the lowest buckling load (worst design) are identified and the corresponding distribution of individual phases is determined. The statistical nature of the critical buckling load is discussed by computing the statistical moments that include the mean, coefficient of variation, skewness, and kurtosis. Next, we consider the buckling of long and slender columns with functionally graded microstructure. In such columns, the modulus of elasticity and/or the moment of inertia is varied in a controlled manner along the length of the column. The primary objective is to identify functionally graded microstructures that maximize (and minimize) the critical buckling load when compared to a reference homogeneous column. Several columns with a variety of microstructures are examined and a constraint is imposed on each of the microstructures so that the volume averaged elastic modulus remains the same in all the columns. The buckling load capacity of these microstructures is determined using linear perturbation analysis, as well as the Rayleigh-Ritz method. Finally, microstructures that maximize the critical buckling load are identified and a relationship between the material distribution and the corresponding buckling mode shape is established.