Bar Of Circular Cross-section Research Articles

Dynamic torsional analysis of a functionally graded coated circular shafts weakened by multiple radial cracks

Analytical transient analysis of a functionally graded (FG) coated cylindrical bar weakened by multiple radial cracks under torsional transient loading is presented. The solution corresponding to a Volterra-type screw dislocation is achieved using the Fourier and Laplace transforms in the spatial and temporal domains, respectively. The obtained dislocation stress field is used to develop singular integral equations in terms of the unknown dislocation density function for the evaluation of multiple radial cracks. The resulting Cauchy integral equations are reduced to a set of algebraic equations and are solved numerically to determine the dislocation density functions. The achieved results are employed to evaluate the dynamic stress intensity factors (DSIFs) at the crack tips and the torsional rigidity of the FG coated circular cross-section bar with multiple embedded radial cracks. In addition, several numerical examples are presented to verify the results and to explore the effects of transient loading, FG coating layer, cracks position and interaction between them on the DSIFs and torsional rigidity of the FG coated cracked bars.

Effects of functionally graded coating layer on the torsional behavior of circular cross-section bars with multiple cracks and cavities

This paper presents the reinforcement impact of a functionally graded (FG) coating layer on the induced stress distribution of circular cross-section bars weakened by multiple cracks and cavities under torsional moment. The entire cross-section of the bar is under torsional loading governed by Saint-Venant’s principle. The shear modulus of the FG coating layer is assumed to vary exponentially in the radial direction. The solution procedure begins by finding the stress field corresponding to a Volterra-type screw dislocation in a circular cross-section bar with an FG coating layer using the Fourier transform. After that, the dislocations are distributed along the surfaces of cracks and cavities. The dislocation stress field is employed to derive singular integral equations in terms of the unknown dislocation density function on the surface of cracks and cavities for the evaluation of multiple defects with arbitrary patterns. These integral equations are of the Cauchy singular kinds and are solved numerically to determine the dislocation density functions on the surface of defects. The obtained results are employed to calculate the torsional rigidity of the domain under consideration, and the mode III stress intensity factors (SIFs) at the crack tips. Finally, several numerical examples are presented to study the effects of the FG coating layer and multiple cracks and cavities with arbitrary patterns on the stress distribution, stress intensity factors (SIFs) and torsional rigidity of weakened bar by cracks and cavities.

Determination of stress concentration factors for shafts under tension

Abstract Computer-based design and optimization have become increasingly important in recent years. This paper has investigated the stress concentration factors (SCF) Kt for shoulder filleted shafts with a hole and without a hole. This study contains two types of shoulder filleted shafts, i. e., a stepped bar of circular cross section with shoulder filleted and a tube with filleted shafts under tension stresses. Investigations on SCF that have been carried out in experimental and theoretical studies, were updated and validated for 2 types of shafts. The charts have been converted into numerical value using high precision computer techniques. Dimensional ratios and SCF were determined using previous work charts. This study determines maximum stresses for shoulder filleted shafts by three dimensional finite element analysis (FEA) and artificial intelligence techniques. A set of SCF charts was converted into numerical values and this data was organized and stored in an Excel file. ANSYS models were created and applied the boundary conditions on the models. And also mesh optimizations were performed. Artificial neural networks (ANN) models were designed using previously collected and verified data. Previous works, ANSYS and ANN results were compared to each other. As a result, ANN model and chart results show a good agreement. The usage of ANN model does not require any mathematical formulae or converting the numerical data action for determining the Kt result for shafts. ANN model usage was identified as a very useful and practical method.

Determination of stress concentration factors for shafts under tension

Abstract Computer-based design and optimization have become increasingly important in recent years. This paper has investigated the stress concentration factors (SCF) Kt for shoulder filleted shafts with a hole and without a hole. This study contains two types of shoulder filleted shafts, i. e., a stepped bar of circular cross section with shoulder filleted and a tube with filleted shafts under tension stresses. Investigations on SCF that have been carried out in experimental and theoretical studies, were updated and validated for 2 types of shafts. The charts have been converted into numerical value using high precision computer techniques. Dimensional ratios and SCF were determined using previous work charts. This study determines maximum stresses for shoulder filleted shafts by three dimensional finite element analysis (FEA) and artificial intelligence techniques. A set of SCF charts was converted into numerical values and this data was organized and stored in an Excel file. ANSYS models were created and applied the boundary conditions on the models. And also mesh optimizations were performed. Artificial neural networks (ANN) models were designed using previously collected and verified data. Previous works, ANSYS and ANN results were compared to each other. As a result, ANN model and chart results show a good agreement. The usage of ANN model does not require any mathematical formulae or converting the numerical data action for determining the Kt result for shafts. ANN model usage was identified as a very useful and practical method.

Numerical modeling of waveguides accounting for translational invariance and rotational symmetry

The analysis of high-frequency wave propagation in arbitrarily shaped waveguides requires specific numerical methods. A widely spread technique is the so-called semi-analytical finite element (SAFE) formulation. This formulation enables to account for the translational invariance of waveguide problems and leads to a two-dimensional modal problem reduced on the cross-section. Despite this, solving the problem can still be computationally demanding. In order to further reduce the size of the modal problem, this paper presents a SAFE method for waveguides of rotationally symmetric cross-sections. Such structures are encountered in many applications. Typical examples are bars of circular cross-section, regular polygons, and multiwire cables. Numerical results show that the computational effort required for solving the SAFE modal problem is tremendously reduced by accounting for rotational symmetry.

Modeling and Investigation of Subsystems of Mechatronic Systems by Classic and/or Unclassic Methods as an Introduction to Solution of Their Inverse Task

The problem of analysis within the form of the one differential equation of motion within the mechanical subsystem or of the set of state equations among considered mechatronic model of object has been formulated and solved. Classic and Galerkin's methods to solve the problem presented above have been used. The considered vibrating mechanical subsystems of mechatronic system are continuous bars of circular cross-section, with free ends or clamped on one end. The poles of dynamical characteristic calculated by mathematical exact method and the Galerkin's one have approximately the same values. The results of the calculations were not only presented in mathematical form but also as the transients of examined dynamical characteristic within function of frequency of assumed excitation. This approach is different from the ones considered so far. Using classic and unclassic methods among modeling, analysis and synthesis it can be assumed that the obtained results can have great value for designers of mechatronic system. and the approximate method of analysis - Galerkin's method has been used in order to obtain the frequency -modal characteristics. The exact and Galerkin's method were used in (5) as a comparison of obtaining dynamical characteristics - dynamical flexibilities for mechatronic system. To compare the obtained dynamical characteristics - dynamical flexibilities only for mechanical transverse vibrating beam, being a parts of complex mechatronic systems, an exact method and the approximate methods were used (14-26). Frequency analysis and frequency - modal analysis have been presented in this paper for the mechanical part of mechatronic system. To achieve the aim two methods of analysis have been used - the exact and Galerkin's approximate methods.

Wave propagation in a generalized piezothermoelastic rotating bar of circular cross section

Purpose – The purpose of this paper is to study the wave propagation in a generalized piezothermoelastic rotating bar of circular cross-section using three-dimensional linear theory of elasticity. Design/methodology/approach – A mathematical model is developed to study the wave propagation in a generalized piezothermelastic rotating bar of circular cross-section by using Lord-Shulman (LS) and Green-Lindsay (GL) theory of thermoelasticity. After developing the formal solution of the mathematical model consisting of partial differential equations, the frequency equations have been derived by using the thermally insulated/isothermal and electrically shorted/charge free boundary conditions prevailing at the surface of the circular cross-sectional bar. The roots of the frequency equation are obtained by using the secant method, applicable for complex roots. Findings – In order to include the time requirement for the acceleration of the heat flow and the coupling between the temperature and strain fields, the analytical terms have been derived for the non-classical thermo-elastic theories, LS and GL theory. The computed physical quantities such as thermo-mechanical coupling, electro-mechanical coupling, frequency shift, specific loss and frequency have been presented in the form of dispersion curves. From the graphical patterns of the structure, the effect of thermal relaxation times and the rotational speed as well as the anisotropy of the of the material on the various considered wave characteristics is more significant and dominant in the flexural modes of vibration. The effect of such physical quantities provides the foundation for the construction of temperature sensors, acoustic sensor and rotating gyroscope. Originality/value – In this paper, the influence of thermal relaxation times and rotational speed on the wave number with thermo-mechanical coupling, electro-mechanical coupling, frequency shift, specific loss and frequency has been observed and are presented as dispersion curves. The effect of thermal relaxation time and rotational speed on wave number for the case of generalized piezothermoelastic material of circular cross-section was never reported in the literature. These results are new and original.

Corrosive destruction of prismatic bar of circular cross-section under pure bending

The problem of determining the time before the corrosive destruction of the straight bar of circular cross-section, which is subjected to pure bending in a corrosive medium was solved. The following cases were investigated: bar corrosion process does not depend on the stress caused by its pure bending in a corrosive medium; bar corrosion process essentially depends on its stress. The rate of the bar radius change in the corrosion process is considered to be linearly dependent on its stress during pure bending deformation. Formulas that express the dependence of stress, radius, limit wear time and cracking time of the bar on the specified corrosion process parameters and given bending moment were derived. Using the formulas obtained, the graphs of dependence of full wear time and cracking time on the bar stress at the initial time point were constructed. Also, the graph of the bar radius change (decrease) in time during the corrosion process was plotted.

Fatigue Crack Growth by FEM-DBEM Approach in a Steam Turbine Blade

This study is concerned with the integrity assessment of cracked steam turbine rotor components, made of steel X20Cr13, that operate under cyclic loading. Damage accumulation and growth occurred on the leading edge of turbine blade starting from part-through surface flaws. Tensile tests are performed for assessment of the main mechanical and fracture steel properties: specimens are cut out from critical zones of turbine blade body after a given operating time. The subject for experimental studies is a steel bar of circular cross-section with straight-fronted edge crack. The optical microscope measurement and the COD method are used to monitor crack growth during the tests. An automatic crack propagation simulation on the blade is performed by a coupled approach which, starting from the results of a global finite element method (FEM) analysis, proceeds through the sub modelling of the cracked volume in a DBEM dual boundary element method (DBEM) environment and subsequent propagation analysis.

FEM simulation of a crack propagation in a round bar under combined tension and torsion fatigue loading

An edge crack propagation in a steel bar of circular cross-section undergoing multiaxial fatigue loads is simulated by Finite Element Method (FEM). The variation of crack growth behaviour is studied under axial and combined in phase axial+torsional fatigue loading. Results show that the cyclic Mode III loading superimposed on the cyclic Mode I leads to a fatigue life reduction. Numerical calculations are performed using the FEM software ZENCRACK to determine the crack path and fatigue life. The FEM numerical predictions have been compared against corresponding experimental and numerical data, available from literature, getting satisfactory consistency.

Fatigue surface crack growth in cylindrical specimen under combined loading

The subject for studies is a steel bar of circular cross-section with straight-fronted edge notch undergoing fatigue loads. Both the optical microscope measurements and the crack opening displacement (COD) method are used to monitor and investigate both crack depth and crack length during the tests. The variation of crack growth behavior is studied under cyclic axial and combined tension+torsion fatigue loading. Results show that cyclic Mode III loading superimposed on the cyclic Mode I leads to a fatigue life reduction. In parallel to the experimental activity, numerical calculations are performed based on three-dimensional DBEM analysis to determine the stress intensity factors along curvilinear surface crack front and fatigue life prediction. The experimental fatigue crack growth results obtained from round bar specimens have been compared with the numerical predictions. The computational DBEM results are found to be in satisfactory agreement with the experimental findings.

Simulation of Fatigue Life Behavior of Circular Cross-Section Bar with Straight-Fronted Crack

In this paper, fatigue life circular cross-section elastic bar under pure fatigue axial loading is studied through principles of linear elastic fracture mechanics (LEFM) coupled with the three-dimensional finite element technique for determination of critical crack size and residual lifetime. Three different initial notch depths are discussed. The relations between aspect ratio (b/c) and relative crack depth (b/D) are obtained, and it is shown that there is great difference in the growth of cracks with different front shapes and initial notch depths.

Wave propagation in a piezoelectric solid bar of circular cross-section immersed in fluid

Wave propagation in a piezoelectric solid bar of circular cross-section immersed in fluid is discussed using three-dimensional theory of piezoelectricity. The equations of motion of the cylinder are formulated using the constitutive equations of a piezoelectric material. The equations of motion of the fluid are formulated using the constitutive equations of an inviscid fluid. Three displacement potential functions are introduced to uncouple the equations of motion, electric conduction. The frequency equation of the coupled system consisting of cylinder and fluid is developed under the assumption of perfect-slip boundary conditions at the fluid–solid interfaces. The frequency equations are obtained for longitudinal and flexural modes of vibration and are studied numerically for PZT-4 material bar immersed in fluid. The computed non-dimensional wave numbers are presented in the form of dispersion curves. The secant method is used to obtain the roots of the frequency equation.

An inverse coefficient problem related to elastic–plastic torsion of a circular cross-section bar

An inverse coefficient problem related to identification of the plasticity function g(η) from a given torque τ is studied for a circular section bar. Within the deformation theory of plasticity the mathematical model of torsion leads to the nonlinear Dirichlet problem −∇⋅(g(|∇u|2)∇u)=2φ, x∈Ω⊂R2; u(s)=0, s∈∂Ω. For determination of the unknown coefficient g(η)∈G, an integral of the function u(x) over the domain Ω, i.e. the measured torque τ>0, is assumed to be given as an additional data. This data τ=τ(φ), depending on the angle of twist φ, is obtained during the quasi-static elastic–plastic torsional deformation. It is proved that for a circular section bar, the coefficient-to-torque (i.e. input–output) map T:G↦T is uniquely invertible. Moreover, an explicit formula relating the plasticity function g(η) and the torque τ is derived. The well-known formula between the elastic shear modulus G>0 and the torque is obtained from this explicit formula, for pure elastic torsion. The proposed approach permits one to predict some elastic–plastic torsional effects arising in the hardening bar, depending on the angle of twist.

Stress and Strain Concentrations in an Elasto-Plastic Bar of Circular Cross-Section with Semicircular Groove

The elastic stress and strain fields in an elasto-plastic circular cross-section bar with semicircular groove subjected to uniaxial tension are systematically investigated using the finite element method. It is found that the stress and strain concentrations are different, especially after the partial region near the groove root yielding. The coupled influences of the loading levels and Poisson’s ratios on the stress and strain concentrations are examined. The maximum of strain concentration factor is always at the groove root, but the maximum of stress concentration factor is at the groove root only while the loading levels are lower.

Stress and Strain Concentrations in an Elastic Bar of Circular Cross-Section with Semielliptic Groove Subjected to Tension

The elastic stress and strain fields in a circular cross-section bar with semielliptic groove subjected to uniaxial tension are systematically investigated using the finite element method. It is found that the stress and strain concentrations are different even if the material is in the elastic state. The relation between the stress and the strain concentration factors depends on Poisson’s ratio and the groove’s geometric configuration. The coupled influences of the shape factor of the semielliptic groove and the relative depth of the groove on the stress and strain concentrations are examined. The equations that describe the stress and strain concentration factors for different groove’s geometric configurations and Poisson’s ratios are obtained based on finite element results.

Fatigue crack paths for inclined cracks and surface flaws under biaxial loading

Two approaches are developed for geometrical modeling of crack growth trajectories for the inclined through thickness central cracks and the part-through surface flaw respectively. The principal feature of such modeling is the determination of crack growth direction and the definition of crack length increment in this direction. The damage process zone size concept is employed for calculations of mixed-mode crack growth trajectories and surface flaw shape and positions. Crack front behavior for a straight-fronted edge crack in an elastic bar of circular cross-section is studied through experiments and computations under axial tension loading. The elaborated theoretical model is applied for fatigue crack shape simulation of part-through cracks in a hollow thick- and thin-walled cylinders under different biaxial loading conditions. Suggested approach of crack paths modeling is used for an analysis and prevention of operation failures of existing in-service aircraft gas-turbine engine rotating components.

Stress and strain concentration factors for tension bars of circular cross-section with semicircular groove

In this paper, the elastic stresses and strains in the circular cross-section bars with semicircular groove subjected to uniaxial tension are systematically examined using the finite element method. It is shown that the stress and strain concentration factors are different even if the material is in the elastic state. Poisson’s ratio can have a significant effect on the strain concentration factor. The relation of stress and strain concentration factors is also obtained for stress concentration region in cylindrical coordinates. This relation depends on Poisson’s ratio, the radial stress factor and the triaxial stress constraint factor.

Stress intensity factors for surface fatigue crack in a round bar under cyclic axial loading

In this paper, the surface fatigue crack growth shape for an initial straight-fronted edge crack in an elastic bar of circular cross-section is determined through experiments under pure fatigue axial loading. Three different initial notch depths are discussed. The relations of the aspect ratio (b/c) and relative crack depth (b/D) are obtained, and it is shown that there is a great difference in the growth of cracks with different initial front shapes and crack depths. Further, using the three-dimensional finite element method, the stress intensity factors (SIFs) are determined under remote uniform tension loading. Since the relationship of b/c and b/D changes during the fatigue crack growth, the SIFs are determined for different surface crack configurations.

Fatigue crack growth for straight-fronted edge crack in a round bar

In this paper, firstly fatigue crack growth for a straight-fronted edge crack in an elastic bar of circular cross-section is studied through experiments under pure fatigue axial loading. Three different initial notch depths are discussed. The relations between aspect ratio ( b/ c) and relative crack depth ( b/ D) are obtained, and it is shown that there is great difference in the growth of cracks with different front shapes and initial notch depths. Using the relations, predictions are made of the crack front shape and crack growth rate in the depth direction. Secondly, the variation of crack growth behavior is also studied under cyclic axial loading with steady torsion. Results show that Mode III loading superimposed on the cyclic Mode I leads to a significant reduction in the crack growth rates.