Pair Of Concentrated Forces Research Articles

Asymptotic solutions for large fractures growing parallel to a free plane

Straight 2D and disc-like cracks situated close to a parallel free plane are considered and asymptotic solutions for the stress intensity factors (SIFs) and crack opening are developed. Two asymptotic terms with respect to the small ratio of distance from the surface to the crack length/radius are derived. The derivation is based on considering the material between the crack and the free surface as an elastically supported beam or circular plate. The crack is considered to be opened either by uniform tensile stress or a pair of concentrated forces or moments representing the action of central defects (these represent for instance the crack growth from initial inclined crack or pore under compression acting parallel to the free surface). A crack with non-interacting surfaces (the conventional crack) and a crack with linear links between the surfaces are considered. Interaction with the free surface is shown to considerably increase the crack opening and SIFs, while the presence of the elastic links can reduce these crack parameters. In particular, the opening and SIFs of a crack with linear links under a pair of concentrated forces/moments exponentially vanish with the crack dimensions tending to infinity. The obtained asymptotic solutions provide a simple and accurate tool for investigating mechanisms of surface delamination and spallation (including thermal spallation and strain rockbursts).

Antiplane electro-mechanical field of a piezoelectric finite wedge under shear loading and at fixed-grounded boundary conditions

This paper presents the general solutions of antiplane electro-mechanical field solutions for a piezoelectric finite wedge subjected to a pair of concentrated forces and free charges. The boundary conditions on the circular segment are considered as fixed and grounded. Employing the finite Mellin transform method, the stress and electrical displacement at all fields of the piezoelectric finite wedge are derived analytically. In addition, the singularity orders and intensity factors of stress and electrical displacement can also be obtained. These parameters can be applied to examine the fracture behavior of the wedge structure. After being reduced to the problem of an antiplane edge crack or an infinite wedge in a piezoelectric medium, the results compare well with those of previous studies.

Antiplane electro‐mechanical field of a bimaterial piezoelectric wedge under a pair of concentrated forces and surface charges

This paper presents the general solutions of the antiplane electro‐mechanical field for a bimaterial piezoelectric wedge subjected to a pair of concentrated forces and surface charges. Identical wedge angles are assumed here. Using the Mellin transform method, the singularity orders and the generalized stress, strain, electric displacement and electric field intensity factors are derived analytically. The discussion of a semi‐infinite crack at the interface of two bonded piezoelectric half‐planes is also included. Some degenerated cases are investigated to check the accuracy of the obtained solutions.

Impact Factors for Horizontally Curved Composite Box Girder Bridges

This paper presents a method for determining the dynamic impact factors for horizontally curved composite single- or multicell box girder bridges under AASHTO truck loading. The bridges are modeled as three-dimensional structures using commercially available software. The vehicle is idealized as a pair of concentrated forces, with no mass, traveling in two circumferential paths parallel to the curved centerline of bridges. An extensive parametric study is conducted, in which over 215 curved composite box girder bridge prototypes are analyzed. The key parameters considered in this study are: Number of cells, number of lanes, degree of curvature, arc span length, slope of the outer steel webs, number and area of bracing and top chord systems, and truck(s) speed and truck(s) positioning. Based on the data generated from the parametric study, expressions for dynamic impact factors for longitudinal moment, reaction, and deflection are proposed as function of the ratio of the arc span length to the radius of curvature. The results from this study would enable bridge engineers to design horizontally curved composite box girder bridges more reliably and economically. Furthermore, the results can be used to potentially increase the live-load capacity of existing bridges to prevent posting or closing of the bridge.

Antiplane electro‐mechanical field of a piezoelectric finite wedge under a pair of concentrated forces and free charges

This paper presents general antiplane electro‐mechanical field solutions for a piezoelectric finite wedge subjected to a pair of concentrated forces and free charges. The boundary conditions on the circular segment are considered as traction free and insulated. Using finite Mellin transform methods, the stress and electrical displacement in all fields of the piezoelectric finite wedge are derived analytically. Singularity orders and intensity factors of stress and electrical displacement can be obtained too. After being reduced to a problem of an antiplane edge crack or an infinite wedge in a piezoelectric medium, the results compare well with those of previous studies.

The antiplane electro‐mechanical field of a piezoelectric wedge under a pair of concentrated forces and free charges

This paper presents general antiplane electro‐mechanical field solutions for a piezoelectric wedge subjected to a pair of concentrated forces and free charges. Using the Mellin transform method, singularity orders and generalized stress, strain, electrical field and electrical displacement intensity factors are derived analytically. The solutions can be used as a weighting function in other more complicated crack problems. An example of a partially loaded piezoelectric wedge is demonstrated and all intensity factors are obtained accordingly. After being reduced to the problem of a semi‐infinite antiplane crack in a piezoelectric medium or an isotropic elastic body, the results compare well with those of previous studies. The coupling effects, which come from nonzero constant e 15, on the intensity factors are demonstrated graphically. In many cases, the existence of the constant e 15 will reduce the intensity factors.

Evaluation of impact factors for composite concrete–steel cellular straight bridges

This paper presents a method for determining the dynamic impact factors for straight composite concrete deck–steel girder cellular bridges under AASHTO truck loading. The bridges are modeled as three-dimensional structures using commercially available software. The vehicle is idealized as a pair of concentrated forces, with no mass, travelling across the bridge. An extensive parametric study is conducted, in which 120 composite multi-cell bridge prototypes are analyzed. The key parameters considered in this study are: number of cells, number of lanes, span length, number and area of cross-bracing and top-chord systems, and truck(s) speed and truck(s) positioning. Based on the data generated from the parametric study, expressions for dynamic impact factors for moment, reaction, and deflection for such bridges are proposed. The results from this practical-design-oriented study would enable bridge engineers to design new composite cellular bridges more reliably and economically. Furthermore, the results can be used to evaluate the load-carrying capacity of existing composite cellular bridges since even a small increase in strength for live load can make the difference between closing a bridge and leaving it open.

Crack growth under biaxial compression

An experimental and analytical investigation into three-dimensional crack growth under biaxial compression is presented. Tests were carried out on different materials, including transparent resin samples, each with a single embedded disk-like crack. These cracks grew extensively parallel to the load directions causing splitting. This behaviour is markedly different from that observed under uniaxial compression where the crack growth is limited in size, and is not capable on its own to induce failure. The presence of the intermediate principal compressive stress radically changes the mechanism of crack growth. A model is proposed where the growing crack is represented as a disk-like crack oriented parallel to the loading direction and opened by a pair of concentrated forces at its centre. It is shown that the crack growth is stable until it reaches a size comparable to its distance from the free surface.

Asymptotic analysis of crack interaction with free boundary

This paper employs the beam and dipole asymptotic techniques for modelling interaction of a crack with parallel free boundaries. Two configurations are considered: (1) a crack in a half-plane and (2) a crack in the centre of an infinite strip. Both, the stress intensity factors and the areas of the crack opening are calculated. For the crack situated close to the boundary, the part of the material between the crack and the boundary is represented by a beam (plate in plane-strain). This allows calculating the area of the crack opening. The stress intensity factors are calculated by matching the beam approximation with Zlatin and Khrapkov's solution (Zlatin and Khrapkov, 1986) for a semi-infinite crack parallel to the boundary of a half-plane or with Entov and Salganik's solution (Entov and Salganik, 1965) for a central semi-infinite crack in a strip. It has been shown that this asymptotic method allows obtaining two leading terms for the SIFs and the crack opening area. When the distance between the crack and the free surface is large, the problem is treated in the far field approximation. This, dipole asymptotic method allows finding the leading asymptotic terms responsible for the crack–boundary interaction. For intermediate distances between the crack and the boundary, simple interpolating formulas are derived. Particular examples of cracks loaded by pair of concentrated forces and for uniform loading are considered. The obtained results are compared with available numerical solutions.

A contact crack problem in an infinite plate

The problem of an infinite plate with crack of length 2a loaded by the remote tensile stress P and a pair of concentrated forces Q is discussed. The value of the force Q for the initial contact of crack face is investigated and the contact length elevated, while the Q force increases. The problem is solved assuming that the stress intensity factor vanishes at the end point of the contact portion. By the Fredholm integral equation for the multiple cracks, the reduction of stress intensity factor due to Q is found.

Stress intensity factors for two periodical rows of collinear cracks

A problem of equilibrium of a plane with two periodical rows of collinear cracks of equal length is considered. Crack surfaces are loaded either by uniform stresses or by a pair of concentrated forces. The problem is reduced to a system of two real integral equations which kernels contain singularity of the Hilbert type. The system has been solved numerically by using the Gauss-Chebyshev quadrature formula for singular integrals. The stress intensity factors at the crack tips are calculated separately for normal and shear tractions, which allows one to consider a general problem as a composition of obtained solutions. Results of calculations are presented in tables as functions of the distance between rows and the crack overlapping.

Fredholm integral equation for the multiple circular arc crack problem in plane elasticity

An elementary solution for the multiple circular arc problem is obtained in this paper. The elementary solution is defined as a particular case of the single circular arc crack problem, in which remote stresses are equal to zero, and two pairs of concentrated forces are applied at a prescribed point of crack face. By using the principle of superposition, Fredholm integral equation for the multiple circular arc problem in plane elasticity is obtained. The suggested approach is illustrated by several numerical examples. If a smaller arc crack is surrounded by a larger arc crack, the stress intensity factors for the former become rather small. The phenomenon of shielding is illustrated by examples.

CIRCULAR RIGID PUNCH ON A SEMI-INFINITE PLANE WITH AN OBLIQUE EDGE CRACK SUBJECTED TO CONCENTRATED FORCES OR POINT DISLOCATIONS

A circular rigid punch is located on a semi-infinite plane with an oblique edge crack. The punch is acted by an eccentric load to keep the punch vertical, and frictional force is assumed to exist on the contact region. A pair of concentrated forces or point dislocations is assumed to act at arbitrary points in the semi-infinite plane. The analytical solution (Green function) is obtained by a rational mapping function and a complex variable method. To solve the problem, the complex stress functions are divided into two parts; one is the principal part, which corresponds to the fundamental solution of the semi-infinite plane with an oblique edge crack; the other is the holomorphic part of the problem, which can be derived explicitly. The stress intensity factors and resultant moment on the contact region to decide the position of the vertical load on the punch are shown.

Improved weight functions for infinitely long axial and circumferential cracks in a cylinder

The goal of this paper is to derive an accurate analytical representation of the weight functions for infinitely long axial and circumferential cracks in a cylinder, based on available numerical data for the case of a crack face loading by a pair of concentrated forces. The analysis is restricted to inner surface cracks in a cylinder having the ratio of wall thickness to internal radius of 0·1. Although the reference finite-element results are believed to be rather accurate, they are found to deviate from the well-known asymptotic behaviour of the weight function when approaching the crack tip. This inaccuracy of the original numerical solution is eliminated by applying the weight function technique. The resulting weight functions are represented in an analytical form that enables their efficient integration for an arbitrary crack face loading.

Dependence of stress intensity factor on loading condition

The problem of a centrally cracked rectangular plate is taken as a sample problem to evaluate the dependence: of the stress intensity factor on the loading condition. The total loading applied on the upper and lower edges is assumed to be 2 bp (2 b—the width of the plate). With the condition of the same total applied loading 2 bp along the upper and lower edges, the following conditions are imposed: (a) uniform tension with intensity p, (b) a pair of concentrated forces with intensity bp, (c) a type of tension-compression loading. A novel weight function approach is used to evaluate the stress intensity factor at the crack tip. From the numerical solution it is found that, with the condition of h b = 3.0 ( h b being the height width ratio of the plate) the maximum deviation of the stress intensity factor from the uniform tension case is 5.8% in the concentrated force loading case, and with the same condition the dependence of the stress intensity factor on the loading condition is negligible in the tension-compression loading case. However, in the case of h b ≤f 1.5 , the relevant dependence is significant.

An oblique edge crack and an internal crack in a semi-infinite plane acted on by concentrated force at arbitrary position

A semi-infinite plane with an oblique edge crack and an internal crack acted on by a pair of concentrated forces at arbitrary position is studied. The problem can be divided into two particular parts; one is the half plane with the edge crack acted on by the concentrated forces; the other is the half plane with the edge cracked subjected to distributed dislocations on the line of the internal crack. The traction components along the line of the internal crack induced by the distributed dislocations must be of the same magnitude and in the opposite direction to the traction components resulting from the first part, the condition is used to establish the boiunary integral equation of the problem, in which the fundamental solutions of the edge cracked half plane plays an important part, and the dislocation distributions on the line of the internal crack can be determined by solving the boundary integral equation numerically. Since the properties of the half plane and edge crack have been reflected analytically by the fundamental solution of the edge crack have been reflected necessary to perform numerical computation on the surface of the half plane so that the method is presented in this paper is of high precision and less computation.

Fundamental solutions for half plane with an oblique edge crack

An elastic half plane with an oblique edge crack is considered in this paper. A pair of concentrated forces or point dislocations is assumed to act at an arbitrary point in the half plane. The half plane with an edge crack is first mapped into a unit circle by a rational mapping function so that the following analysis can be carried out on the mapped plane analytically. Then the complex stress functions are derived by separating the whole problem into two parts; one is the principal part corresponding to the infinite plane acted on by concentrated forces or dislocations, the other is the holomorphic part, which can be determined by making use of the property of regularity of complex stress functions. The stress intensity factors of the crack can be calculated with different inclined angles of the crack, and the displacement and stress components at an arbitrary position in the half plane can be expressed explicitly.

Methodology for Impact Factor of Horizontally Curved Box Bridges

This paper presents a method for determining the dynamic impact factor of horizontally curved steel box‐girder bridges under vehicle loadings. The two‐dimensional planar grid analogy is used to model box bridges. The vehicle is idealized as a pair of concentrated forces, with no mass, traveling, on circumferential paths with constant velocity. Analysis of multigirder and continuous span bridges indicates a tendency for frequency clustering. Static solutions for these bridges are compared against mode superposition and direct integration. The bridge behavior was studied under several truck loading paths. Static and dynamic bridge behavior was observed under these loadings. A rational methodology for determining the impact factor is developed, and alternate impact‐factor criteria are proposed to replace the current American Association of State Highway and Transportation Officials guide specification.

The sif of the wing crack in the theory of a compression split

THE SIF OF THE WING CRACK IN THE THEORY OF A COMPRESSION SPLIT Hao Tian-hu Department of Basic Sciences, China Textile University 1882 West Yan-an Road, Shanghai 200051 People's Republic of China In the theory of formation of the main crack in a compressed cracked brittle-elastic material, the wing crack (Fig. 1) plays an important role. In order to determine the length of the wing crack we must compute its SIF. For convenience we replace the tortuous crack with a straight crack subjected to the action of two pairs of concentrated forces T and Q (Fig. 2) . The forces T and Q can be determined as follows: T = T o cos 0;Q = T O sin 0;T o = pcos 0(sin 0 I-t cos 0) c > 0 (1)

The Buckling of Flexible Sheets under Tension Part II: Experimental Studies

The paper presents the results of experimental work on the buckling of eighteen suiting fabrics when loaded by a pair of concentrated forces along the centre line. The results support the theoretical analysis given in Part I and show the significance of dimensionless quantities derived by considering the different energies involved.