Singular Loading Research Articles

Boundary element analysis of a plate on an elastic foundation loaded by a twisted annular rigid stamp

The contact problem of an elastic plate on an elastic foundation under torsional surface tractions is solved numerically by the boundary element method (BEM). The BEM solution presented in this paper employs a new Green's function of the bonded dissimilar elastic space under a singular tangential loading source which has been developed by the author recently. Using the proposed technique, no multi-region modelling for the bimaterial plate foundation is necessary. The numerical formulation and results of typical examples are included.

Stability of Deformation and Differential Potentiality of Constitutive Equations of the Theory of Plasticity

We study the problem of stability of the process of deformation and differential potentiality of the constitutive equations specifying the relationship \(\mathop \varepsilon \limits^ \cdot \) in the theories of plasticity with smooth and singular loading surfaces. Special attention is given to the theory of slip taking into account the mutual influence of the mechanisms of inelastic deformation of materials. The author presents his point of view on the correctness of the theory of small strains and the theory of plastic yield with isotropic hardening and the ranges of their applicability.

Similarity Solution of the Flamant Problem by Means of a One-Parameter Group Transformation

The two-dimensional point load problem on a linear elastic half plane is solved using a one-parameter group transformation of the biharmonic equation. This solution technique allows the incorporation of the singular load directly through use of the Dirac delta function. The resulting fourth-order ordinary differential equation, while formidable in appearance, is exact and is readily solved through simple integration.

Analytical Solutions and Numerical Examples for Anisotropic Bending Problems with Elliptic Hole or Rigid Inclusion under Singular Loads.

Analytical Solutions and Numerical Examples for Anisotropic Bending Problems with Elliptic Hole or Rigid Inclusion under Singular Loads.

On applications of generalized functions to beam bending problems

Using a mathematical approach, this paper seeks an efficient solution to the problem of beams bending under singular loading conditions and having various jump discontinuities. For two instances, the boundary-value problem that describes beam bending cannot be written in the space of classical functions. In the first instance, the beam is under singular loading conditions, such as point forces and moments, and in the second instance, the dependent variable(s) and its derivatives have jump discontinuities. In the most general case, we consider both instances. First, we study singular loading conditions and present a theorem by which the equivalent distributed force of a general class of singular loading conditions can be found. As a consequence of obtaining the equivalent distributed force of a distributed moment, we find a mathematical explanation for the corner condition in classical plate theory. While plate theory is not the focus of this paper, this explanation is interesting. Then beams with various jump discontinuities are considered. When beams have jump discontinuities the form of the governing differential equations changes. We find the governing differential equations in the space of generalized functions. It is shown that for Euler–Bernoulli beams with jump discontinuities the operator of the differential equation remains unchanged, only the force term changes so that delta function and its distributional derivatives appear within it. But for Timoshenko beams with jump discontinuities, in addition to changes in the force terms, the operator of one of the governing differential equations changes. We then propose a new method for solving these equations. This method which we term the auxiliary beam method, is to solve the governing differential equations not in the space of generalized functions but rather to solve them by means of solving equivalent boundary-value problems in the space of classical functions. The auxiliary beam method reduces the number of differential equations and at the same time obviates the need to solve these differential equations in the space of generalized functions which can be more difficult.

Classical Versus Improved Thin Shell Theories: A Theoretical Argument or a Design Concern?

Differences in the response of thin nonshallow spherical shells resulting from the choice of the adopted shell theory (classical or improved) are addressed analytically and through a series of representative shell problems. The analytical approach utilized to study the variation between the two theoretical models is based on the response resulting from Singular loads. The differences are quantified in a set of problems that reflect on the assumptions used in formulating the analytical description of the two theories in question. The broad scope of this paper is to examine the impact of shear deformability, introduced by the improved theory on the stress field when amplified under specific loading and geometric conditions, when those are of primary concern to the engineers. Such cases associated with stress concentration around cutouts, interaction of shells with nozzles, stress field in the vicinity of concentrated surface loads, etc. The mathematical formulation is based on the derivation of appropriate Green functions and the computational scheme is formed upon a special type of boundary integral equation. Comparison solutions for stress concentration around circular cutouts of twisted and sheared shells, stress amplification in the junction of shell with nozzles, and local stress field induced by concentrated loads are presented for the two theories.

The analytical calculation of circular rods of variable cross-section by the initial values method

In this work, the stress resultants and displacements (due to in-plane loading) of circular rods with variable cross-sections are calculated by using the initial values method. The principal matrix and its inverse required for the applicability of the method of initial values are obtained. The expression of the solution of a circular-rod loaded by both continuous and singular loads is given in a compact form. Irrespective of the number of singular loads and the degree of redundancy the system of algebraic equations to be solved contains no more than three variables as shown. Two examples exhibiting the advantages and priorities of the method are solved in full detail. The program Mathematica is used throughout.

Effect of t-stresses on the path of a three-dimensional crack propagating quasistatically under type I loading

We consider a three-dimensional semi-infinite almost planar crack, propagating quasistatically under type 1 singular loading. Treating the distortion from planar geometry as a perturbation we evaluate the type II stress intensity factor arising from this distortion including the effect of non-singular T-stresses. Using these results we perform a linear stability analysis by assuming that the crack propagates locally in the direction of pure type I. We obtain a stability diagram as function of Tx and Tz reflecting the interaction between the effects due to the T-stresses and the stabilizing effect which arises from the perturbed geometry of the crack. From this we infer four types of stability of the crack: monotonous stability, oscillatory stability, monotonous instability and oscillatory instability. Additionally we find that for certain values of the ratio Tz/Tx a small interval of wavelengths of the perturbation will be selected as the crack propagates, so that the components corresponding to these wave lengths will grow expontially while all other components are suppressed.

Analytical Solutions for Isotropic Plate Bending Problems with Elliptic Hole or Rigid Inclusion under Singular Loads and Numerical Examples.

We present a unified analysis of plate bending problems based on classical theory containing an elliptic hole or elliptic rigid inclusion. Theoretical closed-form solutions under singular loads as well as bending moments at infinity are shown for cases of arbitrary-shaped elliptic hole or rigid inclusion. The results of stress intensity (singular) factors under the same loads are also derived. Many numerical examples are given by graphic representation.

Basic singular solutions for a cracked circular hole in an infinite medium

This article provides basic elastic solutions for an infinite medium containing a cracked circular hole, subject to singular external loads in form of a dilatation center, a concentrated force and a point couple. The complex function method in conjunction with a rational mapping function is used to solve the problem. Analytic expressions for stress functions, stresses and stress intensity factors are obtained. Numerical values of stress intensity factors are presented in graphic and tabular forms.

Self-Equilibrated Singular Solutions of a Complete Spherical Shell: Classical Theory Approach

The elastic response of a complete spherical shell under the influence of concentrated loads (normal point loads, concentrated tangential loads, and concentrated surface moments) which apply in a self-equilibrating fashion is obtained. The mathematical analysis incorporates the classical uncoupled system of equations for the transverse displacement W and a stress function F. The solution formulae for all three types of singular loading are in closed form and they are expressed in terms of complex Legendre and other elementary functions. The two latter portions of the analysis are associated with a multivalued stress function F which leads to a single-valued stress and displacement formulae. The intricacies of the solutions and their singular character are also discussed. Lastly, some representative shell problems are evaluated.

Singular loadings in elasticity problems and singular solutions of the corresponding integral equations

The method of singular integral equations is an efficient method for the formulation and numerical solution of plane and antiplane, static and dynamic, isotropic and anisotropic elasticity problems. Here we consider three cases of singular loadings of the elastic medium: by a force, by a moment and by a loading distribution with a simple pole. These loadings cause corresponding singularities in the right-hand side function and in the unknown function of the integral equation. A method for the numerical solution of the singular integral equation under the above singular loadings is proposed and the validity of this equation at the singular points is investigated.

Comment on "Analytical Prediction of Vortex Lift"

S comments are warranted concerning the paper by J.W. Purvis entitled Analytical Prediction of Vortex Lift, which appears in the April 1981 issue of the Journal of Aircraft. The first comment is relative to the use of singular loadings in the design of mean camber shapes for wings experiencing leading-edge-vortex separated flow. By comparison, Ref. 1 outlines a low-drag, separated-flow design procedure which uses a more appropriate attached-flow initial loading that is nonsingular. There the final solution for the mean camber surface is obtained by iterating to find the shape-yielding minimum separated-flow drag at the required lift coefficient. The second comment is that the section-induced drag, see Eq. (12), does not have an elliptical distribution over the wing as proposed, but only in the Trefftz plane. This is well known (e.g., Ref. 2), and means that for most wings of interest the section-induced drag has to be determined in the near field. The consequence of this wrong assumption leads to an error in Eq. (26) and, hence, in the magnitude of the leading-edge suction force and vortex flow aerodynamics for wings with varying leading-edge sweep. The third comment deals with the missing multiplication by 2 in Eqs. (27), (30), (33), and (34), the missing division by 2 in Eqs. (28) and (37), and the typographical error in Eq. (32). Equation (32) should read

Fracture of long bones: Rate effects under singular and combined loading states

This study was undertaken to determine the effect of loading rate on bone failure under singular and combined loading states. Using the entire loading range of an Instron Materials Testing Machine, 190 pairs of canine radii were tested. One bone of each pair was subjected to either torsional or a combination of axial and torsional loads, and compared with a control specimen. Compressive loads of 4.6 and 11.5 N were utilized. Torque to failure and energy data versus loading rate were plotted. As loading rate increased, the torque and energy values were found to increase, reach a peak, and then decline at higher rates. Fractures produced under the combined state of loading are representative of those clinically found in human trauma situations.

Admissible Singular Solutions and the Hypercircle Method in Shallow Shell Theory

The object of this study is the construction of geometrically and statically admissible solutions of the basic shallow shell equations in the case of singular loads, especially for the use in the hypersphere theorems. An iterative procedure extends known solutions of plate theory to the classical and an improved shallow shell theory. The results contain all important terms of the exact solution near the singular point.

Wave front analysis of a plane compressional pulse scattered by a cylindrical elastic inclusion

The plane-strain problem of a stress pulse striking an elastic circular cylindrical inclusion embedded in an infinite elastic medium is treated. The method used determines dominant stress singularities that arise at wave fronts from the focusing of waves refracted into the interior. It is found that a necessary and sufficient condition for the existence of a propagating stress singularity is that the incident pulse has a step discontinuity at its front. The asymptotic wave front behavior of the first few P and SV waves to focus are determined explicitly and it is shown that the contribution from other waves are less important. In the exterior, it is found that in most composite materials the reflected waves have a singularity at their wave front which depends on the angle of reflection. Also the wave front behavior of the first few singular transmitted waves is given explicitly.The analysis is based on the use of a Watson-type lemma, developed here, and Friedlander's method [5]. The lemma relates the asymptotic behavior of the solution at the wave front to the asymptotic behavior of its Fourier transform on time for large values of the transform parameter. Friedlander's method is used to represent the solution in terms of angularly propagating wave forms. This method employs integral transforms on both time and θ, the circumferential coordinate. The θ inversion integral is asymptotically evaluated for large values of the time transform parameter by use of appropriate asymptotics for Bessel and Hankel functions and the method of stationary phase. The Watson-type lemma is then used to determine the behavior of the solution at singular wave fronts.The Watson-type lemma is generally applicable to problems which involve singular loadings or focusing in which wave front behavior is important. It yields the behavior of singular wave fronts whether or not the singular wave is the first to arrive. This application extends Friedlander's method to an interior region and physically interprets the resulting representation in terms of ray theory.

Load-induced stress singularities in the bending of cosserat plates

Presented in this paper are the solutions to several plate bending problems as governed by a recent theory developed by Green and Naghdi, into which couple-stress is incorporated. Specifically, each problem considered is subjected to boundary conditions emanating from a singular load distribution acting on the free edge of a semi-infinite plate. The method of integral transforms is applied in the solutions. In general, it is found that the singularities in the shear and moment resultants are of the same order as those given in Reissner's plate theory, however the detailed structures of these singular functions are altered. The present theory also suggests that, in most cases, the maximum magnitudes of the moment resultants are diminished as compared to the corresponding results given in Reissner's theory. Also discussed is the exact relationship between the Green-Naghdi theory and Reissner's theory.

On strain theories of plasticity for singular loading surfaces

On strain theories of plasticity for singular loading surfaces