An analytical method for solving gravity-induced stresses in slope

Stress distribution in an elastic half plane in which body forces are acting

The frictionless contact problem for an elastic layer resting on an elastic half plane is considered. The problem is solved by using the theory of elasticity and integral transformation technique. The compressive loadsPandQ(per unit thickness in direction) are applied to the layer through three rigid flat punches. The elastic layer is also subjected to uniform vertical body force due to effect of gravity. The contact along the interface between elastic layer and half plane is continuous, if the value of the load factor,λ, is less than a critical value, . In this case, initial separation loads, and initial separation points, are determined. Also the required distance between the punches to avoid any separation between the punches and the elastic layer is studied and the limit distance between punches that ends interaction of punches is investigated for various dimensionless quantities. However, if tensile tractions are not allowed on the interface, for the layer separates from the interface along a certain finite region. Numerical results for distance determining the separation area, vertical displacement in the separation zone, contact stress distribution along the interface between elastic layer and half plane are given for this discontinuous contact case.

Fracture problem about anisotropic composite plate of curved crack is researched with the help of appropriate conformal mapping. The crack curve is transformed into a unit circle by the quoted conformal mapping function. Based on the stress boundary conditions, the special stress function is found and the expressions for the stress field and the stress intensity factor near the parabolic crack tip are obtained.

Irregular elastic half-plane gravity problem

—Elastic stresses arising from gravitational loads in a two-dimensional slope of arbitrary shape are calculated easily using a displacement-discontinuity boundary element method (BEM). A long stress-free crack simulates the topographic surface. Gravity-induced stresses (i.e., body forces) in a laterally confined body are simulated by vertical and horizontal “far-field” stresses equal to ρgy and [ν/(1-ν)]ρgy, respectively. Here ρ is material density, ν is Poisson’s ratio, g is gravitational acceleration, and y is elevation, with y = 0 along the surface far from a ridge top or valley floor. BEM stress solutions compare well with analytical solutions for symmetric topography based on conformal mapping. Our analyses indicate that slope failures are likely to initiate near the bases of bedrock ridges and to be widespread along the slopes of gentle valleys cut in bedrock. The BEM method can be applied to the slopes of arbitrary shape and steepness, and it is well suited for evaluating the near-surface propagation of fractures or fracture-like structures, such as dikes and landslide failure surfaces. Our analysis also highlights the critical importance of properly accounting for the boundary conditions in a boundary value problem.

Domain-partition power series in vibration analysis of variable-cross-section rods

Fundamental solutions for half plane with an oblique edge crack

In this paper, the Elzaki transform method is used for solving two-dimensional (2D) elasticity problems in plane polar coordinates. Airy stress function was used to express the stress compatibility equation as a biharmonic equation. Elzaki transform was applied with respect to the radial coordinate to a modified form of the stress compatibility equation, and the biharmonic equation simplified to a fourth order ordinary differential equation (ODE). The general solution for the Airy stress potential function in the Elzaki transform space was obtained by solving the ODE. By inversion, the general solution for the Airy stress potential function was obtained in the physical domain space variables in terms of four unknown integration constants. Normal stresses and shear stress fields were also determined for the general case of 2D elasticity problems. The Flamant problem was solved as a particular illustration of 2D elasticity problems. The stress boundary conditions and the requirement of equilibrium of the internal stress resultants and the external forces were used simultaneously to determine the four constants of integration. The Airy stress potential function and the normal and shear stress fields were thus completely determined. The principle of superposition was used to obtain the elasticity solutions for the stress fields in the elastic half plane due to strip load of infinite extent, and solutions for horizontal stresses on smooth rigid retaining walls due to strip loads and parallel line load of infinite extent acting on the elastic half plane.

A griffith crack at the interface of two bonded dissimilar elastic half planes in which body forces are acting

This paper gives an approach to two-dimensional isotropic elastic theory (plane strain and generalized plane stress) by means of the complex variable resulting in a very marked economy of effort in the investigation of such problems as contrasted with the usual method by means of Airy’s stress function and the allied displacement function. This is effected (i) by considering especially the transformation of two-dimensional stress; it emerges that the combinations xx + yy, xx — yy + 2ixy are all-important in the treatment in terms of complex variables; (ii) by the introduction of two complex potentials Ω(z), ω(z) each a function of a single complex variable in terms of . which the displacements and stresses can be very simply expressed. Transformation of the cartesian combinations u + iv, xx + yy,xx — yy + 2ixy to the orthogonal curvilinear combinations uξ + iun, ξξ + ηη, ξξ - ηη + 2iξη is simple and speedy. The nature of "the complex potentials is discussed, and the conditions that the solution for the displacements shall be physically admissible, i.e. single-valued or at most of the possible dislocational types, is found to relate the cyclic functions of the complex potentials. Formulae are found for the force and couple resultants at the origin z = 0 equivalent to the stresses round a closed circuit in the elastic material, and these also are found to relate the cyclic functions of the complex potentials. The body force has bhen supposed derivable from a particular body force potential which includes as special cases (i) the usual gravitational body force, (ii) the reversed mass accelerations or so-called ‘centrifugal’ body forces of steady rotation. The power of the complex variable method is exhibited by finding the appropriate complex potentials for a very wide variety of problems, and whilst the main object of the present paper has been to extend the wellknown usefulness of the complex variable method in non-viscous hydrodynamical theory to two-dimensional elasticity, solutions have been given to a number of new problems and corrections made to certain other previous solutions.

The dynamics of buildings and structures in an earthquake zone is studied relying on a model boundary value problem describing the dynamics of an elastic punch on an elastic half-plane under diffraction and refraction of waves generated by stress release on a crack. The problem is solved by applying an explicit difference scheme constructed using the method of bicharacteristics combined with dimensional splitting. The problems of wave diffraction and refraction by an elastic inclusion in the release of vertical and shear stresses on a horizontal crack in an elastic half-plane are considered, which model the dynamics of seismic processes in the formation of mode I and II cracks. Computer experiments are performed for media with various elastic parameters and various punch–crack distances. Their results are presented in the form of oscillograms on the ground surface and the punch, as well as diffraction displacement velocity fields and stress tensor invariants in the medium and the punch.

The singular stress field at the vicinity of the apex of an elastic plane indenter of various angles compressing an elastic half plane was studied by using a complex variable technique. Three particular cases where the indenter is perfectly bonded, slips or adheres according to the Coulomb’s law of friction, to the half plane were considered. The characteristic equations for the determination of the order of the stress singularity at the vicinity of the apex of the indenter for the above three cases were defined in terms of the angle of the indenter and the two composite material parameters α and β first introduced by Dundurs, depicting the mechanical properties of the two materials. The order of singularity at the vertex of the indenter was determined for all material combinations of the indenter and the half plane. Valuable and interesting results were derived.

For continuous-time linear time-invariant (LTI) systems with polytopic uncertainties, we develop a robust sampled-data state-feedback control design scheme in terms of linear matrix inequalities (LMIs). Truncated power series expansions are used to approximate a discretized model of the original continuous-time system. The system matrices obtained by using the power series approximations are then expressed as homogeneous polynomial parameter-dependent (HPPD) matrices of finite degrees, and conditions for designing the controller are formulated as a HPPD matrix inequality, which can be solved by means of a recent LMI relaxation technique to test the positivity of HPPD matrices with variables in the simplex. To take care of the errors induced by the remainder terms of the truncated power series, the terms are considered as norm bounded uncertainties and then incorporated into the proposed LMI conditions. Finally, examples are used to illustrate the approach.

The solution for accumulated infiltration using the Green-Ampt equation requires an iterative solution. By using a power series approximtion to a logarithmic term, an equation has been developed to directly calculate the accumulated infiltration for nonuniform soils. A second equation developed primarily for computer models predicts the potential change in the depth of infiltration for discrete time steps. The expression is relatively simple and may be easily programmed for use in hydrologic models. The maximum error resulting from the power series approximation is 8% and will usually be much less. These solutions may be used by engineers and hydrologists to predict infiltration and runoff for vertically nonuniform soils without a time consuming iterative process.

Solution for Eshelby’s elliptic inclusion with polynomials distribution of the eigenstrains in plane elasticity