Stress concentration of a cylindrical bar with a V-shaped circumferential groove under torsion, tension or bending

The stress concentration of a cylindrical bar with a V-shaped circumferential groove is analyzed by the body force method. The stress field due to a ring force in an infinite body is used to solve this problem. The solution is obtained by superposing the stress fields of ring forces in order to satisfy the given boundary conditions. The present results for semi-circular notches are in close agreement with Hasegawa's results. As a result of the systematic calculation of a 60° V-shaped notch, it is found that the stress concentration factors obtained by Neuber's trigonometric rule used currently have non-conservative errors of about 10% for a wide range of notch depths. The stress concentration factors are illustrated in charts so they can be used easily in design or research.     

Theoretical stress concentration factors for short flat bars with opposite U-shaped notches subjected to in-plane bending

The values of existing theoretical stress concentration factors for rectangular uniform thickness plates, with opposite U-shaped notches, subjected to in-plane bending do not include the effect of length as a significant parameter. This work demonstrates that below a threshold value, defined as transition length, these stress concentration factors cease to be valid and, notably, also demonstrates that below this threshold the magnitude of the stated factors may be significantly larger than existing values; a fact that may have important consequences for the accurate estimates of fatigue life. The finite element determined, theoretical stress concentration factors for the stated geometry and loading, including length as a parameter, for the existing range of the notch radii values, as well as an extended range of these values are reported and are presented in the standard graphical form. The corresponding values of the transition lengths were computed and are reported as well.     

Stress concentration formulae useful for any shape of notch in a round test specimen under tension and under bending

In this work stress concentration factors, K_t , for a round bar with a circular-arc or V-shaped notch are considered on the basis of exact solutions for special cases and accurate numerical results. Then, a set of K_t formulae useful for any shape of notch is proposed. The conclusions can be summarized as follows. (i) For the limiting cases of deep (d) and shallow (s) notches, the body force method is used to calculate the K_t values. Then, the formulae are obtained as K_td and K_ts . (ii) On the one hand, upon comparison of K_t and K_td it is found that K_t is nearly equal to K_td if the notch is deep or blunt. (iii) On the other hand, if the notch is sharp or shallow, K_t is mainly controlled by K_ts and the notch depth. (iv) The notch shape is classified into several groups according to the notch radius and notch depth. Then, the least-squares method is applied for the calculation of K_t /K_td and K_t /K_ts . (v) Finally, a set of convenient formulae is proposed that are useful for any shape of notch in a round test specimen. The formulae give SCFs with <1% error for any shape of notch.     

Stress concentration formula useful for any dimensions of shoulder fillet in a round bar under tension and bending

In this work stress concentration factors (SCFs), K_t for a round bar with a fillet are considered on the basis of exact solutions, now available for special cases, and accurate numerical results. Then, a convenient K_t formula useful for any dimensions of the fillet is proposed. The conclusions can be summarised as follows: (i) For the limiting cases of deep (d) and shallow (s) fillet, the body force method is used to calculate the K_t values. Then, the formula are obtained as K_td and K_ts. (ii) On the one hand, upon comparison of K_t and K_td, it is found that K_t is nearly equal to K_td if the fillet is deep or blunt. (iii) On the other hand, if the fillet is sharp or shallow, K_t is mainly controlled by K_ts and the fillet depth. (iv) The fillet shape is classified into several groups according to the fillet radius and fillet depth. Then the least squares method is applied for calculation of K_t/K_td and K_t/K_ts. (v) Finally, a convenient formula is proposed that is useful for any dimensions of fillet in a round bar. The formula give SCFs with less than 1% error in most cases for any dimensions of fillet under tension and bending.     

Finite strip with a central crack under tension

In this study, a symmetrical finite strip with a length of 2L and a width of 2h, containing a transverse symmetrical crack of width 2a at the midplane is considered. Two rigid plates are bonded to the ends of the strip through which uniformly distributed axial tensile load of magnitude 2hp₀ is applied. The material of the strip is assumed to be linearly elastic and isotropic. Both edges of the strip are free of stresses. Solution for this finite strip problem is obtained by means of an infinite strip of width 2h which contains a crack of width 2a at y = 0 and two rigid inclusions of width 2c at y = ±L and which is subjected to uniformly distributed axial tensile load of magnitude 2hp₀ at y = ±∞. When the width of the rigid inclusions approach the width of the strip, i.e., when c → h, the portion of the infinite strip between the inclusions becomes identical with the finite strip problem. Fourier transform technique is used to solve the governing equations which are reduced to a system of three singular integral equations. By using the Gauss–Jacobi and the Gauss–Lobatto integration formulas, these integral equations are converted to a system of linear algebraic equations which is solved numerically. Normal and shearing stress distributions and the stress intensity factors at the edges of the crack and at the corners of the finite strip are calculated. Results are presented in graphical and tabular forms.     

Stress intensity factors for an edge cracked circular bar in tension and bending

The variation of the stress intensity factor along the front of an edge crack in an elastic bar of circular cross-section is determined. Both straight and circular-arc crack fronts are modelled using a three-dimensional tinite element analysis of the tension and bending problems. Results are obtained for crack depth to bar diameter ratios ranging from 0.1 to 0.594, in the case of cracks with straight fronts, and from 0.175 to 0.6, in the case of cracks with curved fronts.

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Résumé On détermine la variation du facteur d'intensité de contraintes sur le front d'une fissure latérale dans un barreau élastique de section circulaire. On modélise des points de fissure droits et à arc circulaire à l'aide d'une analyse par éléments finis à trois dimensions des problèmes en traction et en flexion. Les résultats sont obtenus pour des rapports de profondeur de fissure à diamètre de barreaux compris entre 0.1 et 0.594, pour des fissures a front droit, et entre 0,175 et 0,6 dans le cas de fissures à front incurvé.

     

Limiting state of a semiinfinite plate with edge crack in bending with tension

We solve the problem of stressed state and limiting equilibrium of a semiinfinite plate with edge crack under the action of symmetric bending with tension. In the two-dimensional case, the possibility of crack closure is taken into account by using a model of contact along a line. An algorithm for the numerical solution of the mixed problem is proposed. The diagram of limiting equilibrium of the cracked plate shows that the range of safe states enlarges if the effect of crack closure is taken into account.Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 40, No. 2, pp. 73–77, March–April, 2004.     

The infinite sheet with cracked cylindrical hole under inclined tension or in-plane shear

The stress intensity factors are determined at the root of a radial crack emanating from a circular hole in an infinite sheet, under uniform tension in the direction at an arbitrary inclination with angle β and uniform in-plane shear, respectively. The stress analysis is carried out using the Muskhelishvili formulation and the conformal mapping. Numerical results of the stress intensity factors are obtained for varying crack length-to-hole radius ratio, L/R.     

In-plane bending problem of a double edge cracked plate

In this paper the boundary collocation method is used for evaluating the stress intensity factors (SIF) of a double edge cracked plate under in-plane bending. For the case with a large ratio of the plate height to the width, h/b, the results obtained compare very favorably with existing solutions for an infinite strip. Moreover, this method has been used for different finite plates, and a series of conclusions is provided for application.     

Creep stress concentration during loading and unloading processes in a strip with semi circular notches

The stress concentration during the loading and unloading processes of transient creep is investigated for a strip with semi circular notches by means of the photo-viscoplasticity, using specimens of celluloid at 65C. The relations between the stress concentration factor, the geometries of strips, and time during the loading and unloading processes are derived from the time dependent variations of isochromatic fringe orders. Such trends cannot be analysed by the ordinary photoplasticity in which the time effect is not taken into account.     

Plastic loads of elbows with local wall thinning under in-plane bending

Based on systematic three-dimensional (3-D), large strain FE limit analyses using elastic- perfectly plastic materials, this paper quantifies the effect of local wall thinning on plastic behavior and TES (twice-elastic-slope) plastic loads for 90° elbows under in-plane bending. The thinning geometry is assumed to be rectangular rather than circular, but the nonlinear geometry effect is fully considered. Results from systematic analyses lead to simple approximations for TES plastic loads, covering a wide range of elbow and thinning geometries. Although the proposed approximations are developed for the case when wall thinning locates in the center of the elbow, it is also shown that they can be equally applied to the case when thinning exists anywhere within the elbow. Brief discussion is made on application of the proposed approximations to estimate maximum load-carrying capacities of elbows with local wall thinning.     

Fatigue crack propagation behavior in four-points bending specimens with multiple parallel edge notches at regular intervals

Fatigue crack propagation tests were performed using specimens with multiple parallel edge notches at regular intervals. Fatigue pre-cracks of uniform length were successfully introduced by eccentric tension-compression loading. Fatigue crack propagation tests were carried out under four-points bending loading. Obtained crack propagation behavior was simulated by using newly developed stress intensity factor equations for multiple parallel edge cracks with alternately different lengths. Simulated results showed a good agreement with the experiment for specimens with relatively broad crack intervals, while for specimens with narrow crack intervals showed a different tendency from the experiment when crack lengths became long.     

Analytical beam theory for the in-plane deformation of a composite strip with in-plane curvature

The variational-asymptotic-method (VAM) provides a mathematically rigorous way to reduce a three-dimensional elasticity formulation to a one-dimensional beam theory without ad hoc assumptions. In this work, the VAM is employed to develop a beam theory to analyze the in-plane deformation of a laminated strip-beam with initial in-plane curvature. The cross-sectional stiffness constants and recovery relations for stress and strain are presented as analytical expressions. For the case of zero initial curvature, consistency of the expressions with those of plate theory is demonstrated. For strip-beams with initial curvature in the in-plane direction, results obtained show explicit dependence on the curvature. Results are verified by comparison with those obtained from VABS, the accuracy and consistency of which with three-dimensional finite elements has been reported in several published works. In addition to the internal consistency check this work provides and its utility in helping to validate VABS (which is based on the principles of VAM), it is hoped that the results obtained herein, since they are all analytical expressions, will help researchers and engineers validate the effect of initial curvature in their beam theories, whether existing or new.     

Penny shaped crack in a three-dimensional piezoelectric strip under in-plane normal loadings

Summary The singular mechanical and electric fields in a three-dimensional piezoelectric ceramic strip containing a penny shaped crack under in-plane normal mechanical and electrical loadings based on the continuous electric boundary conditions on the crack surface are considered here. The potential theory and Hankel transforms are used to obtain a system of dual integral equations, which is then expressed as a Fredholm integral equation. All sorts of field intensity factors of Mode I are given, and numerical values for PZT-6B piezoelectric ceramic are graphically shown.     

Stress concentration in plates under tension with local thinnings in elastoplastic deformations and creep

The results are given of a calculation-experimental investigation of the stress concentration in the zone of two-sided and one-sided craters in the form of spherical segments. The elastoplastic problem was solved by the method of variable parameters of elasticity with use of the finite element method. The experiments were made on plates of AL2 alloy with the use of short-base strain gauges. The relationship was obtained of the stress concentration factors to the parameters of thinning and loading at characteristic points. The picture is given of the successive transition of the elements into the plastic state. The simulation of creep based on the method of isochronous curves made it possible to obtain curves of the change in stress concentration factors with time for the characteristic points applicable to the steels used in turbine production from the data of experimental investigation of aluminum elastoplastic models.Khar'kov and Perm'. Translated from Problemy Prochnosti, No. 12, pp. 19–23, December, 1989.     

Ductile tearing assessment of high-strength steel X-joints under in-plane bending

This study reports an experimental investigation of a fatigue-cracked, pre-notched circular hollow section X-joints fabricated from high strength steels (with the yield strength higher than 800 MPa) subjected to brace in-plane bending. The circular hollow section X-joint entails a prefabricated V-notch near the weld toe at the crown position. The experimental procedure applies a fatigue pre-cracking cyclic load followed by a monotonic brace in-plane bending, which leads to brittle through-thickness crack propagation after some amount of ductile tearing. The ductile tearing assessment, integrating the fracture resistance curve obtained from the small-scale fracture specimens and the crack extension in the large-scale tubular joint, predicts closely the load level at which unstable crack extension takes place. The generic level 2A curve outlined in the BS7910 provides an un-conservative estimate on the failure load of the X-joint specimen. The parametric numerical investigation reveals that the strength definition for the cracked joints imposes a significant effect on the shape of the failure assessment curve.     

Ductile fracture before localized necking in a strip under tension

Unexpected cracking with a 22° opening angle grew out of a rough-sheared edge before appreciable necking in an 0.79 by 37 mm mild steel strip, which normally fractures after diffuse and then localized oblique necking. From the crack tip, two shear bands formed at 55° to the load direction, consistent with isotropic plane stress characteristics (53° was predicted from anisotropy, but necking in thin strips occurred at 67°). Photomicrographs showed that the 22° crack growth occurred by first tunnelling at mid-thickness, and then spreading along through-thickness shear planes. Springback on unloading caused a 0.038 mm crack closure and local buckling. This form of cracking illustrates a size effect in fracture under macroscopically plane stress. It also gives an example of a local mechanism triggering a fracture mode that can require more total work than an alternative.Analysis of isotropic localized necking shows the equivalent strain at fracture in thin strips to be uniquely related to the Reduction in Squared Thickness (RST). With smooth edges, width and thickness strains before and during necking differed by factors of 1.4 and 1.7; such measures of anisotropy should be routinely found and reported for strips.