R.K.L. Su

Mode I crack problems by fractal two level finite element methods

Abstract A semi-analytical method is suggested to determine the stress intensity factor (SIF) of two-dimensional (2D) crack problems. In it, the singularity is eliminated from the computational domain by the fractal two level finite element method (F2LFEM). In the present method, the fractal geometry concept and two level finite element method (2LFEM) are employed to automatically generate an infinitesimal mesh and transform these large number of degrees of freedom around the crack tip to a small set of generalized coordinates. By taking advantage of the same stiffness of 2D elements with similar shape, one transformation of the stiffness for the first layer of mesh is enough for all. This simple method is very economical in terms of computational time and computer memory. Highly accurate results of SIF and stresses are obtained.

Mixed-mode two-dimensional crack problem by fractal two level finite element method

Abstract Fractal two level finite element method (F2LFEM) has been extended to calculate the mixed mode stress intensity factor in a two-displacement cracked body. The complete eigenfunction expansion of displacement by Williams is employed for the global interpolation function, the factors K I and K II can be easily computed for any arbitrary loading on any boundary. Results are obtained for some slant crack problems in finite sheets and are compared with known results where available.

Numerical solutions of two-dimensional anisotropic crack problems

A complete set of series form solutions of stress and displacement functions, including all higher order terms, around the crack tip for anisotropic crack problems have been newly derived by eigenfunction expansion approach. The analytical solutions of displacement functions were classified into four cases with respect to different types of complex parameters and different corresponding physical meanings. By employing these displacement functions as global interpolation functions, fractal two-level finite element method (F2LFEM) was applied to evaluate the stress intensity factors (SIFs) for various kinds of anisotropic crack problems. In the method of F2LFEM, the infinite number of nodal displacements was transformed to a small set of generalized coordinates by fractal transformation technique. New element matrices need not be generated and the singular numerical integration was avoided completely. Numerical examples of the four cases were studied and high accurate results of SIFs were obtained.

A numerical study of singular stress field of 3D cracks

Abstract We investigate the stress distribution and the variation of the mode I stress intensity factor along a straight three-dimensional (3D) crack by the finite element method. The results are checked against plane strain theory near the mid-crack and against the 3D theory of Zhu at the free surface. Although Zhus formulation is not perfect and has some typographical errors. The surface stress distribution of his results are in line with the present study by the finite element method. The stress intensity factors at the free surface are found to be much lower than that at the mid-crack.

Two-Level Finite Element Study of Axisymmetric Cracks

We extend the two-level finite element method (2LFEM) to the accurate analysis of axisymmetric cracks, where both the crack geometry and applied loads are symmetrical about the axis of rotation. The complete eigenfunction expansion series for axisymmetric cracks developed by us are employed as the global interpolation function such that the stress intensity factors are primary unknowns. The coupled coefficients in the series are solved iteratively. The stress intensity factors are computed directly from the coefficients for any arbitrary axisymmetric loading on the boundary. Engineering applications of 2LFEM to numerical fracture mechanics analysis for stress intensity factors include several examples: Penny-shaped and circumferential cracks in round bars; and internal and edge circumferential cracks in thick wall pipes.

Strength and Ductility of Embedded Steel Composite Coupling Beams

The stringent requirements on dimensions, ductility, energy absorption, strength and stiffness of coupling beams have resulted in much research on various alternative coupling beam designs, which include the use of diagonal reinforcement, rhombic arrangement of main bars and steel composites. Experimental results showed that each of these designs offered better performance than the conventional type but had its own limitations. A new embedded steel composite coupling beam design is therefore proposed. This paper presents the findings from the experimental tests of a coupling beam fabricated with this proposed design and a conventionally reinforced coupling beam, which serves as the reference. The preliminary test results showed that the embedded steel coupling beam with relatively large span-to-depth ratio (l/h = 2.5) had excellent shear capacity (∼10MPa) and very good energy absorption.

Fractal two-level finite element analysis of cracked Reissner's plate

A cracked thick plate subjected to edge moment and transverse loading was customarily analysed either by a fine finite element mesh or by singular elements. In this paper an alternative method is recommended in which conventional finite elements with infinitesimal mesh are used and the number of unknowns is reduced by interpolating the nodal displacements by means of the global interpolating function around the singular region. The global interpolating function is derived by using eigenfunction technique based on Reissners transverse shear plate theory. The crack parameters such as stress intensity factor and moment intensity factor can be evaluated directly from the coefficients of the global interpolating function. New elements need not to be generated and integration is avoided completely. Accurate results with error less than 0.5% are achieved with little computational efforts. Examples on edge cracked plate and central cracked plate subjected to both edge moment and transverse loading are considered.

Body-force linear elastic stress intensity factor calculation using fractal two level finite element method

Fractal two level finite element method (F2LFEM) for the analysis of linear fracture problems subjected to body force loading is presented. The main objective here is to show that by employing the F2LFEM a highly accurate and an efficient linear analysis of fracture bodies subjected to internal loading can be obtained as it is hard to find any analytical and exact values of stress intensity factor (SIF) for any kind of geometry subjected to internal loading. Also in this paper, a fast method to transform the body force to the reduced force vector is presented and has been effectively employed. The problems solved here include both the single mode or mixed mode cracks subjected to internal body-force or external loading. In comparison with other numerical algorithms, it seems that with a small amount of computational time and computer storage, highly accurate results can be obtained.

Fractal two-level finite element method for cracked kirchhoff's plates using dkt elements

A technique for calculating moment intensity factors (MIF) and stress intensity factors (SIF) for through-thickness cracks in thin plates subjected to out-of-plane bending by means of fractal two-level finite element method is proposed. It is based on the nodal displacements transformation near the crack tip in terms of some analytical functions. The similarity characteristic properties of the plate element stiffness are employed. Fractal transformation technique is developed to transform infinitely many nodal displacements around the crack tip to a small set of generalized displacements including the MIF and SIF as direct unknowns. Examples are given on the centre cracked plates in bending. The results are in good agreement with analytical results and with other researchers. Comparison of the results from Kirchhoffs theory and from Reissners plate theory shows large differences up to ca 50%.

NUMERICAL SOLUTION OF CRACKED THIN PLATES SUBJECTED TO BENDING, TWISTING AND SHEAR LOADS

A semi-analytical method namely fractal finite element method is presented for the determination of mode I and mode II moment intensity factors for thin plate with crack using Kirchhoffs theory. Using the concept of fractal geometry, infinite many of finite elements is generated virtually around the crack border. Based on the analytical global displacement function, numerous degrees of freedom (DOF) are transformed to a small set of generalised coordinates in an expeditious way. The stress intensity factors can be obtained directly from the generalized coordinates. No post-processing and special finite elements are required to develop for extracting the stress intensity factors. Examples of cracked plate subjected to bending, twisting and shear loads are given to illustrate the accuracy and efficiency of the present method. The influence of finite boundaries on the calculation of the moment intensity factors is studied in details. Very accuracy results when compare with the theoretical and numerical counterparts are found.