The Mechanics of a Cantilever Beam with an Embedded Horizontal Crack Subjected to an End Transverse Force, Part A: Modelling

Abstract

This study addresses the mechanics of a cracked cantilever beam subjected to a transverse force applied at it's free end. In this Part A of a two Part series of papers, emphasis is placed on the development of a four-beam model for a beam with a fully embedded horizontal sharp crack. The beam aspect ratio, crack length and crack centre location appear as general model parameters. Rotary springs are introduced at the crack tip cross sections as needed to account for the changes in the structural compliance due to the presence of the sharp crack and augmented load transfer through the near-tip transition regions. Guided by recent finite element findings reported elsewhere, the four-beam model is advanced by recognizing two key observations, (a) the free surface and neutral axis curvatures of the cracked beam at the crack center location match the curvature of a healthy beam (an identical beam without a crack under the same loading conditions), (b) the neutral axis rotations (slope) of the cracked beam in the region between the applied load and the nearest crack tip matches the corresponding slope of the healthy beam. The above observations led to the development of close form solutions for the resultant forces (axial and shear) and moment acting in the beams above and below the crack. Axial force and bending moment predictions are found to be in excellent agreement with 2D finite element results for all normalized crack depths considered. Shear force estimates dominating the beams above and below the crack as well as transition region length estimates are also obtained. The model developed in this study is then used along with 2D finite elements in conducting parametric studies aimed at both validating the model and establishing the mechanics of the cracked system under consideration. The latter studies are reported in the companion paper Part B-Results and Discussion. Introduction. Over the last two decades, the frequency response [1-5] of a component or a structure has been used to assess structural health as one of several methods used in damage detection [6-13] and structural health monitoring. In most of such studies, the effects of damage on the structural frequency response has been explored using a cantilever beam geometry. With the above in mind, diffused damage detection studies have been developed [7-13] using optimization algorithms that minimize an error estimate . Typically, the employed error estimates are calculated by comparing the experimentally measured frequencies of the structure [8] to the respective frequencies predicted by a physics based model that solves a problem of the same structural geometry but with reduced localized properties at a prescribed location [9]. Often, the optimization algorithms identify the model structure that best matches the experimental results, thus identifying the likely location and degree of structural damage manifested through the EI, structural stiffness reduction. For example, Xu et al. [8] developed a damage detection algorithm that monitors the changes in the first 10 transverse frequencies of a vibrating beam as a means of detecting the location and degree of damage along the axis of the beam as measured through a reduced localized structural stiffness. The promising outcomes of their initial studies have been validated through model experiments and have sparked