Straight Free Edges Research Articles

Tension failure analysis for bolted joints using a closed-form stress solution

Bolted joints are widely used to connect safety-critical parts in the aeronautical industry and require precise structural assessment tools. Focus of this paper is the development of an overall analytical calculation method to determine the stresses of a bolted joint with quasi-isotropic composite material, which is then used in a tension failure analysis. The stress solution is obtained by means of the Airy stress function. To predict crack initiation in the tension failure analysis use is made of the Theory of Critical Distances. Crack initiation occurs in the net section plane and its assessment requires precisely calculated net section stresses. For this purpose, the stress boundary conditions in load direction at the straight free edges must be fulfilled, which is reached by supplementing the field modelling the load introduction with auxiliary functions. This technique is a physically motivated means to deal with symmetric finite geometry problems involving straight free edges. However stress boundary conditions perpendicular to the load direction are not covered and some inaccuracies in the stress solution may arise. A failure analysis is conducted to investigate their impact on the predicted failure load using literature values based on Finite Fracture Mechanics as reference.

Reddy's layerwise laminate plate theory for the computation of elastic fields in the vicinity of straight free laminate edges

Reddy's layerwise laminate plate theory is employed for the closed-form analysis of free-edge effects in layered plates of arbitrary non-orthotropic layups. The approach consists of the subdivision of the physical laminate layers into an arbitrary number of mathematical layers through the plate thickness. In each interface, a set of unknown displacement functions with respect to the inplane width coordinate is introduced while in the thickness direction layerwise linear Lagrangian interpolation polynomials are employed. The unknown inplane functions are determined from the governing Euler–Lagrange differential equations, derived from the principle of minimum potential energy. A solution approach in the form of decaying exponential functions leads to closed-form analytical representations for all free-edge state variables in a straightforward manner while in the inner laminate regions, the stress state corresponding to Classical Laminate Plate Theory is recovered exactly. The present approach is applicable on a standard personal computer, needs insignificant computational effort even in cases of thick laminates consisting of a high number of layers and is in excellent agreement with reference results from the literature.

Efficient computation of order and mode of three-dimensional stress singularities in linear elasticity by the boundary finite element method

The Boundary Finite Element Method (BFEM), a novel semi-analytical boundary element procedure solely relying on standard finite element formulations, is employed for the investigation of the orders and modes of three-dimensional stress singularities which occur at notches and cracks in isotropic halfspaces as well as at free edges and free corners of layered plates. After a comprehensive literature review and a concise introduction to the standard three-dimensional BFEM formulation for the static analysis of general unbounded structures, we demonstrate the application of the BFEM for the computation of the orders and modes of two-dimensional and three-dimensional stress singularities for several classes of problems within the framework of linear elasticity. Special emphasis is placed upon the investigation of stress concentration phenomena as they occur at straight free edges and at free corners of arbitrary opening angles in composite laminates. In all cases, the BFEM computations agree excellently with available reference results. The required computational effort is found to be considerably lower compared to e.g. standard Finite Element Method (FEM) computations. In the case of free laminate corners, numerous new results on the occurring stress singularities are presented. It is found that free-corner problems generally seem to involve a more pronounced criticality than the corresponding free-edge situations.

Asymptotic analysis of stress singularities in composite laminates by the boundary finite element method

The present contribution contains asymptotic studies of stress singularities which occur in the vicinity of interfaces at straight free edges and also at free corners of composite laminates where two straight free edges intersect. The computations are performed by the boundary finite element method (BFEM) which only requires a boundary discretization and which in essence is a boundary element method solely relying on standard finite element formulations. The BFEM does not require the knowledge of a fundamental solution and reduces the spatial dimension of the discretizational effort by one. Anisotropic material behaviour is incorporated without difficulties. As the BFEM requires certain similarity or scaling properties of the given structural situation and involves the closed-form solution of a system of differential equations for the displacement components with respect to the scaling direction, it can also be characterized as a hybrid or semi-analytical method.

Surgical pearl: Readily available cryosurgery shield

C ryosurgery is a very common procedure in dermatology, used to treat benign as well as precancerous and cancerous lesions. Often these lesions are located near sensitive structures. Protection of these structures from stray liquid nitrogen is often necessary. An effective cryosurgery shield can be made by simply folding a piece of paper, such as a prescription sheet, twice. One of the outer free edges so created can be folded over about 1 cm to form a gutter that directs the liquid nitrogen away from the hand of the cryosurgeon. The shield is held in the non-dominant hand by the posterior, straight free edges of the shield. The single creased edge formed at the base is placed against the patient’s skin in front of the delicate structure being protected. Such a shield can also be useful in elevating acrochordons, and a V-fold of the shield around the skin tag restricts freezing to the tag, further protecting surrounding skin (Fig 1). For added comfort a glove can be worn when holding the shield if numerous lesions are to be treated. The paper shield is thrown away after use, and a new one made for each patient.

The “free-corner effect” in thermally loaded laminates

The occurrence of localized high interlaminar stresses is a well-known phenomenon along the free edges of composite laminates, which is called “free-edge effect”. Beyond the generally considered situation of a straight free laminate edge, a more specific but also very important situation is that of a free rectangular laminate corner. Here the localized occurrence of interlaminar stresses can be named as “free-corner effect”. For this free-corner effect an approximate closed-form description is given for a class of relatively simple laminates subjected to thermal loading. The applied approach makes use of an equilibrium stress distribution and the principle of minimum complementary energy to describe the situation in the laminate corner region. The predicted interlaminar stress distribution at the ply interfaces is shown to be in good agreement with comparative 3-dimensional finite element analyses.

Interlaminar Stresses in Composite Notched and Unnotched Laminates

The purpose of this paper is to determine the interlaminar stress distribution at curved and straight free edges for quasi-isotropic and zero-dominated laminates. The interlaminar stresses are calculated using a three-dimensional finite-element code with error control. For straight free edges the results are compared to existing semi-analytical methods. Another objective with this investigation is to evaluate a quadratic failure criterion for prediction of initial load for delamination and the most likely position for the delamination. In this paper it is found, for quasi-isotropic stacking sequences, that it is not possible to tailor a stacking sequence so that o, becomes compressive near hole boundaries or at straight free edges. On the contrary both tensile and compressive stresses appear in a periodic manner through the thickness. It can also be concluded that interlaminar stresses exhibit a thickness effect for repeating bundles. The singularity of the interlaminar components is concentrated to a narrow region, approximately 1-2 fibre diameters in thickness and radial direction. Since the interlaminar stresses are so localised, a continuum approach will not provide useful solutions in this area. Consequently the material cannot be treated as homogeneous in this region, instead the heterogeneous nature within each ply should be modelled. The smallest element size in the present analysis for the refined mesh is less than 1 pum, i.e., far less than a fibre diameter. Utilising a quadratic failure criterion, for the studied quasi-isotropic laminates, in conjunction with obtained FE-results, it is shown that global strain to initiate delamination does not depend on the stacking sequence, but so does the number of possible initiation points. For quasi-isotropic layups with a circular cut-out, initiation of delaminations occurs at approximately the same global strain, 0.25%. For unnotched zero-dominated layups initiation and fracture strain are almost the same, i.e., -1.2%. The investigation also shows that the number of ±450/00 or ±45°/900 alterations should be kept at a minimum to minimise the interlaminar stresses and possible damage initiation points. It is likely that the primary cause for delamination initiation for notched plates are the high interlaminar shear stresses and not the normal stresses. Finally it is demonstrated that available semi-analytical methods do not correctly describe the interlaminar stress distribution.

Interlaminar stresses in general laminates with straight free edges

An efficient analytical method for calculating interlaminar stress distribution near the straight free edges of generally stacked laminates under different types of loading conditions is presented. The variation of stresses for each ply are established according to boundary-layer equilibrium equations. All traction boundary conditions at free edges and traction continuity at ply interfaces are satisfied. Although the emphasis is placed on assessing the behavioral characteristics of interlaminar stresses for unsymmetric laminates, numerical examples for symmetric cases are also presented and compared with available data for demonstrating the efficiency and accuracy of this method. All results show high-stress gradients of interlaminar normal and shear stresses near the free edge.

Verification of a 2-D to 3-D Global/Local Finite Element Method for Symmetric Laminates

A two-dimensional to three-dimensional global/local finite element analy sis technique, as applied to both cross-ply and general symmetric composite laminates, is presented. In particular, the local areas of interest for the laminate are the straight free edge in proximity to a hole and curved free edge of a hole. Verification of this technique was accomplished by comparison to results from complete three-dimensional finite ele ment analyses. Examination of the interlaminar stress fields and the amount of time for their determination indicated that the global/local method developed yielded accurate results and was more efficient.

Method for calculating the interlaminar stresses in symmetric laminates containing a circular hole

An efficient approximate solution for three-dimensional stress distributions around a circular hole in symmetric laminates under a set of far-field in-plane stresses is presented. Stress functions for each ply are assumed according to the boundary-layer equilibrium equations. In addition, all the boundary conditions for each ply and traction continuity at the ply interface are exactly satisfied. Eventually, the unknown parameters in stress functions are determined by the minimization of complementary energy of the whole laminate. Numerical examples are presented in comparison with given literature which shows that the present method is efficient. I. Introduction D UE to stiffness discontinuity between plies, it is easy to cause interlaminar stress concentrations near the freeedge region in composite laminates. Such stresses will commence delamination, matrix crackings, and failure of laminates, especially under fatigue loading. Since 1970, numerous investigations15 have used various methods to determine the interlaminar stresses at the straight free edge of composite laminates. Pipes and Pagano1 adopted the anisotropic elasticity theory in conjunction with the finite difference method to analyze a simple four-ply laminate. Wang and Crossman2 used a finite element approach to investigate this same problem. Wang and Choi3'4 derived an analytical solution based on Lekhnitskii's stress potentials and the theory of anisotropic elasticity to determine the exact order of stress singularity at the free edges of laminate. In an effort to develop an efficient method to deal with thick laminates (say 100 plies), Kassapoglou and Lagace5 used the force balance method in conjunction with the principle of minimum complementary energy to obtain an analytical solution for interlaminar stresses at straight free edges. Owing to the complicated geometry for the curved free edges as compared with the straight free edges, little work has been done for composite laminates with curved free edges. Basically, the analysis of straight free edge may be assumed as a problem with two-dimensional stress and strain variations. However, the curved free edge is a typical three-dimensional problem. This difference has rendered the analysis for the curved free edge more difficult than the straight free edge. A boundary-layer theory based on the perturbation technique for isotropic elastic plates with a circular hole developed by Reiss6 has been extended to composite laminates by Tang.7-8 This approach is based on identifying the boundary-layer problem as two equivalent problems, namely, a modified torsion problem and a modified plane strain problem. However, Tang's solution can only satisfy part of boundary conditions in an average sense, which may result in unreliable stress results near the free edges.9 In addition, Tang's solution will be very tedious for laminates with numerous plies. Zhang and Ueng10 proposed a simplified method to investigate the effect of the ratio of hole radius to laminate thickness on the interlaminar stress distributions around a hole in a (0/90 deg)s laminate under far-field tensile or shear stress; however, it is assumed that the order of stress singularity is prescribed and

Cross-Ply Laminates with Holes in Compression: Straight Free-Edge Stresses Determined by Two- to Three-Dimensional Global/Local Finite Element Analysis

A practical example of applying two- to three-dimensional (2- to 3-D) global/local finite element analysis to laminated composites is presented. Cross-ply graphite/epoxy laminates of 0.1-in. (0.254-cm) thickness with central circular holes ranging from 1 to 6 in. (2.54 to 15.2 cm) in diameter, subjected to in-plane compression were analyzed. Guidelines for full three-dimensional finite element analysis and two- to three-dimensional global/local analysis of interlaminar stresses at straight free edges of laminated composites are included. The larger holes were found to reduce substantially the interlaminar stresses at the straight free-edge in proximity to the hole. Three-dimensional stress results were obtained for thin laminates which require prohibitive computer resources for full three-dimensional analyses of comparative accuracy.

Determination of Interlaminar Stresses in Composite Laminates under Combined Loads

A method to determine the complete state of stress at free edges of com posite laminates under combined loads (uniaxial tension or compression, moment, and out-of-plane shear) is presented. Based on the principle of minimum complementary en ergy, two coupled ordinary differential equations are derived with use of calculus of varia tions. The stresses are determined in closed form and are in good agreement with other solutions published in the literature for uniaxial tension and applied moment. This solu tion is applicable to general laminates with straight free edges and is computationally very efficient.

A three-dimensional analysis of edge effects in composite laminates with circular holes

A three-dimensional (3D) displacement approach is developed in order to account for 3D effects of stress concentration in critical areas of composite laminates (free edges, free and loaded holes, notches, etc.). A finite element procedure is set up both for monodimensional and for bidimensional elements. First the problem of interlaminar stress prediction in a composite laminate with straight free edges under uniaxial loading is critically examined. The stress analysis of a holed composite plate under in-plane action is then studied in the case of a free hole and of a loaded hole. The importance of stacking sequence and of some geometrical parameters, such as the radius to the thickness ratio R/h, on the stress field around the hole are also investigated.

Study of the Onset of Delamination at Holes in Composite Laminates

For laminated composites, delamination can be a controlling failure mode and a major damage mode. The problem of delamination growth has been widely studied, especially for idealized situations such as the edge delamination of specially designed plate specimens. However, comparatively little systematic study has been made of the ini tiation of delamination around notches such as cutouts and holes in laminates of practical interest. Such work is essential to efforts to establish criteria for the design and optimiza tion of laminates to withstand the important practical requirements of notched strength, especially under long term static or cyclic loading. Composite laminates are prone to delamination at free edges, straight edges, or at holes, due to the mismatch of deformations at interfaces where two adjacent plies have different fiber orientations and/or different material properties. The linear analysis of the mismatch at the edge results in a mathe matical singularity. That phenomenon occurs in a boundary layer and has to be treated mathematically and physically as such. It is of great importance to recognize and be able to predict delamination locations at edges prone to such events. The goal of the research presented here was to create a model capable of providing such a prediction and to verify that model. In an effort to generalize the model, the more complicated case of a free edge at holes in a composite laminate was considered rather than the case of a straight free edge. Once the development of the analysis of the free-edge effect at a hole in a composite lami nate was completed, a sequel of two major efforts was employed: 1) performance of an ex tensive experimental program to provide data for the creation of the prediction model, and 2) establishment of the model based on the analysis, and comparison of predictions with the experimental results. The final formulation of the stress field near the hole edge con sists of the determination of the mathematical order of the singularity and a series of near- field parameters. Both terms play an important role in the analysis, and were required for the prediction model. The testing program employed two different material systems and two types of stacking sequences under quasi-static loading. Delamination locations as well as threshold loads for those locations are reported and discussed in terms of common philosophies regarding the subject. Some unexpected contrasts in the behavior of the two material systems are observed.

Closed Form Solutions for the Interlaminar Stress Field in Angle-Ply and Cross-Ply Laminates

The solution to the problem of interlaminar stresses at a straight free edge in a com posite laminate is obtained in closed form for two special and important cases: angle-ply and cross-ply laminates. These solutions are derived using the Force Balance Method and the principle of minimum complementary energy. The results are favorably compared to the predictions of other analyses found in the literature. The solution procedure is found to be simpler and more efficient than other analytical methods. In addition, for some special cases of cross-ply laminates, the solution is shown to be identical to the predictions of a modified plate theory that includes through-the-thickness stretching.

STRIPPING OF THE VOCAL CORDS

The common observation that after laryngofissure tissue replacement produced a structure which objectively looked not unlike a true vocal cord led me in 1932 to studies of cat larynxes.¹I felt that if this regeneration or reformation took place following such an extensive procedure as laryngofissure, it was reasonable to expect complete replacement after a much less formidable operation. With this in mind, a series of experiments were performed on cats: On the basis of these experiments it seemed permissible to state that after removal of a vocal cord structural regeneration would produce a new cord. Grossly, it was impossible to distinguish between the normal and the regenerated cord. Functionally, the results were satisfactory, provided that the basic principle of insuring that a straight free edge would be left after operation was adhered to. The importance of this has been proved by the practical results obtained in

Stripping of the vocal cords

The common observation that after laryngofissure tissue replacement produced a structure which objectively looked not unlike a true vocal cord led me in 1932 to studies of cat larynxes. 1 I felt that if this regeneration or reformation took place following such an extensive procedure as laryngofissure, it was reasonable to expect complete replacement after a much less formidable operation. With this in mind, a series of experiments were performed on cats: On the basis of these experiments it seemed permissible to state that after removal of a vocal cord structural regeneration would produce a new cord. Grossly, it was impossible to distinguish between the normal and the regenerated cord. Functionally, the results were satisfactory, provided that the basic principle of insuring that a straight free edge would be left after operation was adhered to. The importance of this has been proved by the practical results obtained in