Critical Wedge Angle Research Articles

Critical Angles in Bending of Rotationally Inhomogeneous Elastic Wedges

An anisotropic rotationally inhomogeneous wedge bent by either a concentrated couple applied at the tip (Carothers problem) or uniform surface loadings (Levy problem) is considered. The existence criteria for homogeneous solutions describing stresses and strains in both problems are established. In the Levy problem there are two types of critical wedge angles, at which homogeneous solutions break down and become infinite. The first type critical wedge angles of Levy’s problem are shown to be critical also for Carothers’problem whatever the rotational inhomogeneity. Particular solutions to both problems are obtained at the critical wedge angle. The form of these solutions is established to depend on two factors: the multiplicity degree of roots of some eigenvalue equation and the number of independent eigenvectors of some real matrix. It is shown also that the eigenvalue equation does not provide an alternative way to calculate the critical angles and in the first-order perturbation theory results in just the same equations for the critical angles.

Geometric optimization of two bonded wedges involving stress singularities

Joints between dissimilar materials can cause stress singularities. Such joints are exemplified by the case of two bonded wedges. The stress singularity can be avoided by reshaping the wedges. An analytical theory for the critical wedge angles of the bonded wedges is developed. The shape is optimized by the geometric strain method, an optimality criterion method. Numerical examples are presented for different materials with/without special conditions on the wedge angles. Optimum wedge shapes are obtained and optimum corner angles are compared with analytical wedge-angle solutions from contact mechanics. Both analytical and numerical approaches are useful tools for reshaping the bonded wedges to avoid a stress singularity.

Solutions for the anisotropic elastic wedge at critical wedge angles

The classical solution for an isotropic elastic wedge loaded by uniform tractions on the sides of the wedge becomes infinite everywhere in the wedge when the wedge angle 2α equals π, 2π or 2α* where tan 2α* = 2α*. When the wedge is loaded by a concentrated couple at the wedge apex the solution also becomes infinite at 2α = 2α*. A similar situation occurs when the wedge is anisotropic except that 2α* is governed by a different equation and depends on material properties. Solutions which do not become infinite everywhere in the wedge are available for isotropic elastic wedges. In this paper we present solutions for the anisotropic elastic wedge at critical wedge angles. The main feature of the solutions obtained here is that they are in a real form even though Stroh's complex formalism is employed.

The critical angle of the anisotropic elastic wedge subject to uniform tractions

The classical solution for an isotropic elastic wedge loaded by a uniform pressure on one side of the wedge becomes infinite when the wedge angle 2θ0 satisfies the equation tan 2θ0 = 2θ0. This is the critical wedge angle which also renders infinite solutions for other types of loadings. In this paper, we study the associated problem for the anisotropic elastic wedge. We first present uniform stress solutions which are possible for symmetric loadings. For antisymmetric loadings, a uniform stress solution is in general not possible and we present a non-uniform stress solution in which the stress depends on θ but not on r. The non-uniform stress solution breaks down at a critical angle. We present an equation for the critical angle which depends on the elastic constants. The Stroh formalism is employed in the analysis. An integral representation of the solution is obtained by using new identities which are derived in the paper.

Bearing Capacity of Strip Footings in Two Layered Cohesive-friction Soils: Reply

A method has been formulated for bearing capacity of footings on two layered soils with varying cohesion, friction, and unit weight based on the second theorem of Drucker and Prager (kinematical consideration). The failure mechanism considered was fundamentally similar to that of Prandtl–Terzaghi but with different wedge angles. The critical wedge angles were found in each case. Bearing capacity charts for footings are presented by varying cohesion in layers, with the same friction angle and unit weight. The charts are readily usable for any combination of c2/c1, d/b, φ, and γ.

Bearing Capacity of Strip Footings in Two Layered Cohesive-friction Soils

A method has been formulated for bearing capacity of footings on two layered soils with varying cohesion, friction, and unit weight based on the second theorem of Drucker and Prager (kinematical consideration). The failure mechanism considered was fundamentally similar to that of Prandtl–Terzaghi but with different wedge angles. The critical wedge angles were found in each case. Bearing capacity charts for footings are presented by varying cohesion in layers, with the same friction angle and unit weight. The charts are readily usable for any combination of c2/c1, d/b, φ, and γ.

Earth Pressures for Bilinear Backfill Surface

An analytical investigation in the evaluation of earth pressures on a retaining wall with a straight back and with a bilinear surface for the backfill is presented. The solution is based on the Coulomb wedge theory. The solution, which includes an expression for the critical wedge angle as well as for the point of action of the earth pressure, is quicker and less tedious than available graphical methods. Several examples are presented to illustrate the solutions.

Hugoniot Curve Determined by Shock Wave Detachment from an Infinite Wedge

A method is presented for the determination of the Hugoniot curve of an arbitrary inviscid compressible fluid flowing at supersonic speed past an infinite wedge. A shock wave is attached to the apex of the wedge provided the wedge angle is less than a certain critical value. The shock wave detaches from the wedge if the wedge angle becomes greater than this critical value. If the Hugoniot curve of the fluid is known, the critical wedge angle can be calculated for a given initial fluid flow velocity. The critical angle also can be determined by an experimental method. This paper treats the theoretical problem of utilizing the discrepancy between theoretical predictions and experimental results to obtain an improved Hugoniot curve of the fluid. A perturbation method is applied to the case of solid copper for which both theoretical and experimental information exists. The actual experimental Hugoniot curve for copper does not differ sufficiently from its theoretical value to yield a fair test of the method; therefore, synthetic experimental data are used to solve a typical numerical example. In the example treated, a 10% discrepancy between theory and experiment was reduced to about 1% by one application of perturbation theory.