Action of a Normal Concentrated Force on an Elastic Right-Angled Infinite Wedge

B. P. Mitrofanov

(Presented by the scientific seminar of the Department of Strength of Materials)

The problem of the action of a concentrated force on an elastic medium is a mathematical abstraction that does not reflect the actual boundary conditions of problems in the theory of elasticity. However, a large number of contact problems [1, 2], which were solved on the basis of the problem of the action of a concentrated force, have shown the practical admissibility and importance of this formal solution.

In the classical theory of elasticity, problems have been solved on the action of a concentrated force on a half-plane, a half-space, a disk, a sphere, and certain other bodies.

It is natural to attempt to extend the limits of applicability of the problem of the action of a concentrated force to bodies of other shapes.

Let us consider the problem of the action of a concentrated normal force on the boundary of an elastic right-angled infinite wedge.

Let a force \(P\) be applied at the point \(C(a, o)\) of the boundary of the wedge under consideration (Fig. 1). The stresses in the wedge arising as a result of the action of the force \(P\) will be determined using the solution for the action of a concentrated force on a half-plane [3].

The force \(P\) in the vertical areas of the points \((o, x)\) of the half-plane will cause the appearance of stresses \(\sigma_y, \tau_{xy}\). If these stresses at the points \((o, x)\) are made equal to zero (of course, without additional loading of the boundary \(x=0,\ y>0\)), then the stresses of the part of the half-plane \(y>0\) will represent the stresses of a right-angled infinite wedge.

The shear stresses vanish when a force \(P\) is applied at the point \(C_1(-a,o)\). We compensate the normal stresses by a distributed pressure of opposite sign, applying it along the vertical boundary of the bounded half-plane. The magnitude of this pressure due to the action of two forces \(P\), according to [3], is equal to

\[ \sigma_y=\frac{4Pa^2}{\pi}\cdot \frac{x}{(a^2+x^2)^2}. \tag{1} \]

The stresses in the bounded half-plane due to the pressure \(\sigma_y\) will be obtained by using the general solution of the plane problem in polar coordi-

... in coordinates [4]. We assign a pole at the point \(O\) (Fig. 1) and represent \(\sigma_y\) in the form of a polynomial.

For this purpose we use the orthogonal Laguerre polynomials and write the function \(\sigma_y\) in the form

Fig. 1.

\[ \sigma_y=\sum_{k=0}^{\infty} c_k L_k^{(\alpha)}(x). \]

If \(e^{-x}\) is used as the weight, then, as is known [5],

\[ c_k=\frac{1}{k!\Gamma(k+1)}\cdot \int_{0}^{\infty} e^{-x}\sigma_y L_k^{(0)}(x)\,dx, \]

where

\[ \Gamma(k+1) \text{ is the gamma function;} \]

\[ L_k^{(0)}(x) \text{ is a Laguerre polynomial.} \]

For convergence of the series in Laguerre polynomials it is sufficient [5] to require that the function \(|\sigma_y|\) be piecewise smooth on \([0,\infty)\) and that the integral

\[ \int_{0}^{\infty} e^{-\frac{x}{2}}\cdot x^{-\frac14}|\sigma_y|\,dx \]

converge.

Representing in the form of a series only the part of the function \(\sigma_y\),

\[ \frac{1}{(a^2+x^2)^2}, \]

one can show that the convergence requirements listed above are fulfilled. Finally,

\[ \sigma_y=\frac{4Pa^2}{\pi}\,x\sum_{k=0}^{\infty} c_k L_k^{(0)}(x). \tag{2} \]

The representation of function (1) in the form of series (2) does not make it possible to determine the horizontal pressure on the boundary of the bounded half-plane (wedge) when

\[ a=0 \quad \text{and} \quad x=0. \]

Using Saint-Venant’s principle, we replace the force \(2P\) by a statically equivalent distributed pressure over a small semicircle. This makes it possible to obtain [3] the horizontal components \(\frac{2P}{\pi}\) (Fig. 2).

Consequently, for small \(a\), the stresses in the wedge may be determined by adding the stresses from the force \(2P\) for the half-plane to the stresses in the right-angled wedge loaded by the force \(\frac{2P}{\pi}\) at the vertex.

Fig. 2.

It is not difficult to verify that the stresses in the wedge due to the load specified by formula (1) can be determined by applying the general solution of the plane problem in polar coordinates, the expansion (2), and the known boundary conditions. The latter serve to compute the constants of the general solution.

Summing the stresses for the half-plane from two forces \(P\) with the stresses for the right-angled wedge from the distributed load (1), we obtain the stresses in the infinite right-angled wedge due to a normal force.

Let us add that, when computing the coefficients, it is advisable to use the tables [6].

REFERENCES

1. Shtaerman I. Ya. The Contact Problem of the Theory of Elasticity. Gostekhizdat, 1949.
2. Galin L. A. Contact Problems of the Theory of Elasticity. Gostekhizdat, 1953.
3. Timoshenko S. P. Theory of Elasticity. ONTI, 1937.
4. Grammel R. B., Biezeno. Technical Dynamics. Moscow, 1950.
5. Berezin I. S. and Zhidkov N. P. Methods of Computation, Vol. 1. Fizmatgiz, 1959.
6. Aizenshtat V. S., Krylov V. I., Metel’skii A. S. Tables for the Numerical Laplace Transform and for Computing Integrals of the Form
   \[ \int_{0}^{\infty} x^{s} e^{-x} f(x)\,dx . \]
   Publishing House of the Academy of Sciences of the BSSR, Minsk, 1962.