Torsion of a cylindrical beam having two linear cracks with different lengths
Cylindrical beam; linear crack, torsion, conform mapping function, stress concentration problems
European Journal of Prosthodontics and Restorative Dentistry (2026) 34(4S), 377-382
AuthorsAbstractIn this study, torsion problem of a circular shaped plate having two cracks with different lengths was studied. The conform function was established and calculated for solving the problem. Elasticity theory has been applied to calculate the stresses emerging at the critical points of the plate. Determination and solution of stress concentration problems occurring around damaged holes and cracks in machine elements (such as shafts, axle shaft etc.) hold great importance. An accurate estimation of the stresses occurring around the damaged regions depends highly on determination of the stress distribution in these areas. Several researches have been performed to understand the stress concentration occurring in the vicinity of such damages. In the literature, these problems have been investigated for elastic areas, mostly having a simple geometry. In previous studies stress concentration was examined on simple polygon areas rather than the complex polygon vertices surrounded by lines, since no conform function was available to conform such complex areas. In this study, conform functions were used for the first time to solve problems involving complex geometric polygons. 1. Introduction Distribution of the stress concentration at the indents of the cracks forming in different types of holes, is crucial for the safety calculations of machine element parts. These machine elements include the shafts, rods, plates, spherical surfaces and cylinders. Service life of these parts is calculated through precise analysis of the stress distribution at the damaged regions. Several studies are available regarding the investigation of such damages [1,2,3]. In these studies, analyses of simple form elastic regions are emphasized. For the regions including the lines with a complex form, particularly the corners are examined less thoroughly. This arises from the inadequacy of conform functions for such regions (linear cracks are also included). Such type of functions was initially described in [4,5]. 2. Problem Definition and Solution In this study, the subject of the torsion-induced-stress analysis is a finite beam with cross-sectional area S, radius R and two different linear cracks surrounded by � contour, as shown in Figure 1. The cracks are positioned on the �� axis. The coordinates of these crack tips are �1 and �2 . Coordinate origin is placed at center of � contour. This problem can be solved by defining the (z) function which satisfies the boundary conditions.
Received-17-05-2026 Revised-23-06-2026 Accepted-27-06-2026
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