On determination of the natural frequency of transverse and longitudinal vibrations of a cracked beam. Part 1. analytical approach

We consider a perfectly elastic cantilever beam with a crack. The influence of the crack depth and location on the vibration mode of the beam is simulated by a local change in the cross section. This problem is solved by the Rayleigh method. Fairly simple formulas for the determination of the natural frequency of the first mode of bending and longitudinal vibrations of a cantilever beam with an open and closing crack are derived. Institute for Problems of Strength, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 2, pp. 35–44, March–April, 1999.     

Efficiency of the method of spectral vibrodiagnostics for fatigue damage of structural elements. Part 4. Analysis of distortion of harmonicity of vibration cycle of beams with closing transverse cracks

Using an asymptotic method of nonlinear mechanics and concepts of linear fracture mechanics, we derive analytical expressions that relate the constant component and amplitudes of even harmonics, and the ratio of maximum deviations and half-periods of displacements of opposite signs in a vibration cycle of prismatic beams to the length of a closing transverse crack. The analytical solution is compared to the data of experimental determination of these parameters in bending vibrations of rectangular cantilever beams with an edge transverse crack. Analysis of the results of analytic and experimental investigations has shown that harmonic analysis of the strain waveform of beam-type elements in resonance vibrations can be used as an efficient diagnostic method for detecting fatigue damage such as closin cracks. Institute of Problems of Strength, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 1, pp. 5–12, January–February, 2000.     

Efficiency of the method of spectral vibrodiagnostics for fatigue damage of structural elements. Part 3. Analytical and numerical determination of natural frequencies of longitudinal and bending vibrations of beams with transverse cracks. Solution

Based on the concepts of linear fracture mechanics, we derive analytical expressions for the determinational of natural frequencies of longitudinal and bending vibrations of beams which are rectangular in cross section, are fixed in different ways, have variable ratios of the section height to the beam length, and have transverse cracks of various types. The results of the analytical solution are compared with those obtained by the finite-element method as well as with experimental data obtained by the authors and other scientists. The analytical solution under consideration is shown to be quite simple and provide a fairly good accuracy of the results obtained. Using a cantilever beam with one or two symmetrical edge cracks or a central through crack as an example, we consider the possible dependence of the relative change in natural frequencies of vibration on the relative crack length, crack location, and the vibration mode of a beam. We discuss the possible methods of evaluating the crack size and location from the results of experimental determination of the change in natural vibration frequencies of a cracked beam. Institute of Problems of Strength, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 4, pp. 19–31, July–August, 1999.     

Effect of break of the elastic axis of a rotor with transverse crack on its vibrational characteristics

We perform a numerical analysis of the effect of break of the elastic axis on the characteristics of bending vibrations of a multispanned unbalanced rotor with transverse crack mounted on elastic damping supports. The break of the elastic axis in the cross section with crack is caused by local plastic strains. The solution of the problem is sought in the form of a sum of the contributions of the forces of gravity, residual disbalance, and the crack responsible for the break of the axis. The problem is reduced to the numerical analysis of the trajectory of the shaft with its subsequent improvement by the method of iterations. Amplitudes and phases of several first harmonics of vibrational displacements are found as functions of sizes, initial orientation of the crack, and the ratio of vibrational and static displacements. In the presence of the break of the elastic axis, the magnitudes of vibrational characteristics undergo significant changes. The amplitudes of vibrations induced by the crack are comparable with the amplitudes of vibrations caused by disbalance. It is shown that the behavior of the trajectory of motion regarded as a function of principal parameters is nonlinear. Institute for Problems in Mechanical Engineering, National Academy of Sciences of Ukraine, Kharkov, Ukraine. Translated from Problemy Prochnosti, No. 4, pp. 82–89, July–August, 1997.     

On determination of vibration characteristics of a beam with a closing crack in bending vibrations

We consider free vibrations of an elastic cantilever beam with an edge crack, which is simulated as a beam portion with a decreased moment of inertia. The weight of this portion is assumed to be constant, while the dimensions of the portion are determined by the energy criterion of equivalence. We propose an analytical approach to determination of natural frequencies and vibration modes, of a beam with an open or closing crack and to investigation of nonlinear distortions of the displacement wave and acceleration and deformation of various sections of a beam with a closing crack. The solution allows for the possibility that more than one vibration mode of a beam can be generated at the moment of the crack opening and includes the effect of the crack on the strain distribution in the beam volume. It is demonstrated that the approach we propose gives reliable relationships between various vibration characteristics of a beam and the crack parameters and makes it possible to solve an inverse problem of damage diagnostics. Institute of Problems of Strength, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 3, pp. 5–23, May–June, 2000.     

Free vibrations of spherical belts with circular and longitudinal ribs

We propose a procedure for the numerical evaluation of the frequencies of free vibrations of spherical belts with circular and longitudinal ribs, determine those frequencies, and study the influence of the parameters of the ribs on the natural frequencies of free vibrations of shell structures reinforced by ribs. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 3, pp. 124–127, May–June, 1998.     

Longitudinal vibrations of a cross-cracked cantilever beam. Part 1. Small vibrations

Based on the analysis of solutions to differential equations of longitudinal vibrations of a macroelastic body whose nonlinearity is due to “breathing” of the damaged material under tension-compression cycling, we estimate the suitability of the deformation-cycle characteristic of a body under free and forced vibrations for the diagnostics of fatigue damage in structural elements. We consider the following problems: the construction of a simplified model of longitudinal vibrations of mechanical elements with cross cracks, the investigation of free vibrations of a cracked body including resistance forces, the forced vibrations of a cracked body taking into account a perturbation force including and excluding resistance forces, and the study of vibrations of a beam with resistance different from the linear one. We show that the presence of a macrocrack is responsible for the occurrence of representative higher harmonics in the deformation-cycle spectrum. They can serve as a sensitive and efficient vibration-diagnostic parameter to detect cracks in structural elements. Zaporozh'e State University, Zaporozh'e, Ukraine. Translated from Problemy Prochnosti, No. 2, pp. 23–34, March–April, 1999.     

Longitudinal vibrations of a cantilever beam with a transverse crack. Part 2. Piecewise-linear model

A model of vibrations of a cracked body that is based on the difference in the stiffness characteristics in tension and compression is represented as a linear system with piecewise-constant parameters. We determine the parameters of vibrations of this system. We plot the relative crack size as a function of the difference in amplitudes in tension and compression, which is related to the vibration amplitude of a crack-free body with account for the resistance. The approaches described are applicable to evaluation of the “survival” of a structure (i.e., evaluation of the residual durability of a cracked structure). Zaporozh'e State University, Zaporozh'e, Ukraine. Translated from Problemy Prochnosti, No. 5, pp. 78–85, September–October, 1999.     

Solution of the problem of bending of an inhomogeneous curved beam

We propose a new approach to the solution of problems of bending of curved beams consisting of inhomogeneous orthotropic layers. The approach based on the use of equations of the theory of elasticity. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 5, pp. 103–108, September–October, 1999.     

The effect of damping and force application point on the non-linear dynamic behavior of a cracked beam at sub-and superresonance vibrations

The finite element model of an elastic body with a closing crack is presented. Using this model, we show that the non-linear distortions of vibrations at superharmonic and subharmonic resonances are the high sensitive indicators of crack presence. The intensity of these indicators manifestation substantially depends on the level of damping in the system and on the load application point. __________ Translated from Problemy Prochnosti, No. 5, pp. 61–67, September–October, 2006. Report on International Conference “Dynamics, Strength, and Life of Machines and Structures” (1–4 November 2006, Kiev, Ukraine).     

Approximate method for the analysis of stability of beams under the action of a system of longitudinal and transverse forces

We propose an approximate method for the analysis of stability of uniplanar bending of beams under the action of a complex system of longitudinal and transverse forces, obtain formulas for the determination of the critical values of total longitudinal and transverse forces and the factor of stability of flange and rectangular beams that are convenient for practical applications, and present examples of the numerical analysis of actual beams. The critical values of the total forces obtained by the proposed method are compared with known results. The difference between the exact and approximate critical values does not exceed 2–7%. The experimental data enable us to recommend these approximate formulas for practical calculations of the stability of uniplanar bending of beams subjected to the combined action of complex systems of longitudinal and transverse forces. Translated from Problemy Prochnosti, No. 3, pp. 69–77, May–June, 1998.     

Efficiency of the method of spectral vibrodiagnostics for fatigue damage of structural elements. Part 2. Bending vibrations, analytical solution

We discuss the possibility of efficient application of the harmonic analysis of strain cycles formed in an elastic rod in the process of natural or resonance vibrations to the evaluation of the degree of fatigue damage. Fatigue damage is simulated by discontinuities of the material resulting in different values of the modulus of elasticity in the processes of tension and compression. In determining the potential strain energy of a rod in the presence of a crack on the basis of linear fracture mechanics, this model enables us to deduce expressions for relative changes in the stiffness of the rod in the presence of edge and surface cracks. By using the asymptotic method of nonlinear mechanics, we find the second approximation to the solution of the differential equation of vibrations of the rod whose nonlinearity is explained by the effect of “breathing” of the discontinuity formed in the material. It is shown that the discontinuity of the material caused by fatigue damage is responsible both for changes in the natural frequency of vibrations of the rod and for significant contributions of the constant component and even harmonics to the spectrum of strain cycles in the rod. The values of the constant component and the amplitudes of even harmonics (mainly of the second one) may serve as an efficient parameter for the detection of fatigue damage to materials. Institute for Problems of Strength, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 6, pp. 9–22, November–December, 1998.     

Bilateral bending of a plate with nonsymmetric through-thickness arc crack with allowance for the contact of its edges

The problem of bending of plates with nonsymmetric through-thickness arc crack with distributed moments at infinity with allowance for the contact of the crack edges has been examined on the basis of the classical theory of plate bending. The conditions for the existence of the solution of the problem in this formulation have been established. A numerical analysis of intensity factors of forces and moments, contact force between the crack edges has been performed, the results of which have been graphically illustrated. __________ Translated from Problemy Prochnosti, No. 5, pp. 135–141, September–October, 2006. Report on International Conference “Dynamics, Strength, and Life of Machines and Structures” (1–4 November 2006, Kiev, Ukraine).     

Numerical study of vibrations of a nonlinear mechanical system simulating a cracked body

The paper presents the results of a numerical study of forced and decaying vibrations of a system simulating a body with a closing crack under the action of various modes of a nonlinear restoring force and nonlinear viscous friction or the hysteresis-type energy dissipation. We obtained the general patterns of appearance of higher harmonics of the Fourier expansion of time dependence of vibration of a cracked body model. The sensitivity of the higher-harmonic method to the presence of a crack is compared with that of some other vibration damage indicators. Institute of Problems of Strength, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 6, pp. 65–80, November–December, 1999.     

On the procedure of evaluating the damping ability of the materials of specimens subjected to torsional and bending vibrations

We describe a method for the design and numerical analysis of compact mechanical vibrating systems with generators of vibrations on their inertial weights and establish the conditions of damping of unwanted vibrations in these systems aimed at guaranteeing the absence of favorable conditions for the introduction of systematic errors to experimental results. The designed and manufactured vibrating systems are dynamically balanced, and predicted values of the frequencies of their natural vibrations satisfactorily agree with the experimental data. Institute for Problems of Strength, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 4, pp. 145–152, July–August, 1997.     

Description of the kinetics of growth of a fatigue crack on the basis of the criterion of current crack tip opening displacement

We suggest a microprocessor-based data-processing and measuring system and a special device aimed at the determination of current values of the crack tip opening displacement. They are used to establish current values of the crack tip opening displacement at working frequencies of loading of the specimen and describe the kinetics of fatigue crack growth. A decrease in the loading frequency of the specimen from 15 to 0.1 Hz in the course of measuring the current values of the crack tip opening displacement does not affect the reliability of determination of the characteristics of cyclic crack-growth resistance. The reliability of the results is affected only by the inertia of the mass of the gauge case. Institute for Problems of Strength, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 4, pp. 60–67, July–August, 1998.     

Effect of removal of crack lips on the characteristics of the crack resistance of structural alloys

We develop a special procedure and perform experimental investigations of the influence of crack lips on the effect of crack closure under cyclic loading. Results of testing two types of steel and an aluminum alloy for cyclic crack resistance are analyzed for the cases of symmetric and asymmetric loading cycles and the case of symmetric cyclic loading accompanied by removal of material affecting the degree of crack closure from the crack lips. Institute of Problems of Strength, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 5, pp. 70–77, September–October, 1999.     

Closure of a Longitudinal Crack in a Shallow Cylindrical Shell in Bending

We study the two-dimensional problem of contact interaction of the lips of a longitudinal crack in a cylindrical shell subjected to bending under the assumption of contact along a line. The influence of the surface curvature and crack closure on the stressed state and limiting equilibrium of the shell is analyzed. The numerical results are compared with the asymptotic solution of the problem. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 41, No. 2, pp. 45–48, March–April, 2005.     

Prediction of fracture characteristics of metals from high-frequency test data

A method is proposed for the prediction of cyclic crack resistance characteristics of metallic materials under low-frequency loading from high-frequency test data, which is based on a model of development of local plastic deformation regions during the accumulation of fatigue damages and fatigue crack growth with allowance for cyclic loading rate. We performed a comparative analysis of the results of prediction of fatigue fracture diagrams with test data for VT22, VT18U, VNS-25, and AMg6N alloys in a frequency range of 20 Hz–10 kHz. Report on International Conference “Dynamics, Strength, and Life of Machines and Structures” (1–4 November 2005, Kiev, Ukraine). __________ Translated from Problemy Prochnosti, No. 2, pp. 121–128, March–April, 2007.     

Design of SFRC structural elements: post-cracking tensile strength measurement

Utilisation of steel fibre reinforced concrete (SFRC) for designing structural members requires knowledge of the post-cracking tensile response. This paper reviews the experimental characterisation tests and subsequent analysis commonly used for determining the post-cracking tensile properties of SFRC. The experimental program supporting this investigation comprised five different SFRC mixes with fibre volumes ranging from 0.75 to 1.25% used to fabricate a set of characterisation specimens for uniaxial tension tests, notched beam tests and round panel tests carried out in parallel with an extensive experimental program on large scale beams. Characterisation test results allowed a comparison between direct stress–crack opening measurements and the stress–crack openings retrieved from the inverse analysis of bending tests. Discrepancies in post-cracking tensile results obtained with the three types of tests are analyzed and related mainly to test configurations, the presence of a predefined crack, support conditions, fibre orientation, and cracked surface size. Results obtained using material characterisations are then applied to the reproduction of the structural behaviour of large scale beams, documented in a companion paper.