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sigma = stress, sigma-sub-c = classical buckling stress, mu = imperfection amplitude/shell wall thickness
The author's figure caption is:
"Figure 4.3 Knock down curve for
in[90ot/0ot]out antisymmetric cross-ply cylindrical shell under axial compression calculated by different methods"
The author writes about Fig. 4.3:
"Figure 4.3 shows the Knock down factors of in[90ot/0ot]out antisymmetric cross-ply cylindrical shell under axial compression calculated by ABAQUS, Hui's postbuckling method, Koiter's general theory and curve fit of the improved b coefficient, respectively. The result shows that the curve for the Hui's postbuckling method fits the ABAQUS result when the imperfection is up to 18% of the shell thickness. Same phenomenon happens for the result calculated by the improved b coefficient from curve fitting. Koiter's general theory is only valid when the imperfection is less than 5% of the shell thickness. Then it starts to diverge from the ABAQUS result. Again, more than 300% percent improvement of valid region was found for the Hui's postbuckling method compared with the original one. But we can also see that, compared with the previous case; the valid region for the Hui's postbuckling method decreases significantly. So the validity region of the Hui's postbuckling method is varying from case by case. This conclusion is also valid for Koiter's general theory, and the improvement of the valid region is still significant. We can also see the general b coefficient is -0.4528, compared with the improved b coefficient which is -0.2290. There is an about 50% positive shift of the postbuckling b coefficient. This also demonstrates the Koiter's general [theory] significantly overestimates the imperfection sensitivity."
From:
Xu, Hailan (Ph.D Dissertation, Professor David Hui advisor, Engineering and Applied Science, Dept. of Mechanical Engineering, University of New Orleans), "Buckling, Postbuckling and Imperfection Sensitivity Analysis of Different Type of Cylindrical Shells by Hui's Postbuckling Method" (2013). University of New Orleans Theses and Dissertations. Paper 1781.
ABSTRACT: Buckling and postbuckling has been critical design parameters for many engineering structures. In recent years, this topic has continued to be of major concern due to (1) the discovery of new materials with amazingly superior properties, (2) increasingly more stringent safety requirements, (3) lighter, and more durable requirements. Such applications can be routinely found in aerospace, naval, civil, and electrical, and nuclear engineering structures and especially in the vehicle industries. Koiter is the first one to show that the imperfection-sensitivity of a structure is determined by its initial postbuckling behavior. In Koiter’s 1945 general postbuckling theory, it defines the initial postbuckling behavior and imperfection sensitivity behavior by the postbuckling b coefficient. Hui and Chen (1986) were the first to show that the well-known Koiter’s General Theory of Elastic Stability of 1945 can be significantly improved by evaluating the postbuckling b coefficient at the actual applied load, rather than at the classical buckling load. The reason for such significant improvement in predicting the imperfection sensitivity is due to the fact that for an imperfection-sensitive structure, the slope of the buckling load versus imperfection amplitude curve approaches negative “infinity” as the imperfection amplitude approaches to zero. Thus for “finite” amplitude of the geometric imperfection, the applied load is significantly lower than the classical buckling load, leading to significant overestimate of imperfection using Koiter’s General theory of 1945. Such improvement method was demonstrated to be (1) very simple to apply with no tedious algebra, (2) significant reduction in imperfection sensitivity and (3) although it is still asymptotically valid, there exists a significant extension of validity involving larger imperfection amplitudes. Strictly speaking, Koiter’s theory of 1945 is valid only for vanishingly small imperfection amplitudes. Hence such improved method is termed Hui’s Postbuckling method. This study deals with the Postbuckling and imperfection sensitivity of different kinds of cylinders by using the Hui’s postbuckling method. For unstiffened cylinder and laminate cylinder, the solution of Hui’s postbuckling method is compared with ABAQUS simulation result. A parameter variation of stringer/ring stiffened cylinder is also evaluated. A positive shift of the postbuckling b coefficient has been observed, which indicates a significant overestimate of the imperfection sensitivity by Koiter's general stability theory. More importantly, the valid region is significantly increased by using Hui's postbuckling method compared with the Koiter's general stability theory.
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