Generalize the integral superposition methods used in the examples shown in Section 8.4.9and Exercise 8.40. In particular, show that the stress solution for a half-space carrying general normal and shear distributions p(x) and t(x) over the free surface –a≤ x ≤a is given by:

Data from exercise 8.40

Following a similar solution procedure as used in Section 8.4.9, show that the solution for a half-space carrying a uniformly distributed shear loading t is given by:

Data from section 8.4.9

As a final half-space example, consider the case of a uniform normal loading acting over a finite portion (–a ≥x≥ a) of the free surface, as shown in Fig. 8.24. This problem can be solved by using the superposition of the single normal force solution developed previously. Using the Cartesian stress solution (8.4.36) for the single-force problem

For the distributed loading case, a differential load acting on the free surface length dx may be expressed by dY = pdx. Using the geometry in Fig. 8.25, dx = rdθ/sinθ, and thus the differential loading is given by dY = prdθ/sinθ.

with θ₁ and θ₂ defined in Fig. 8.24. The distribution of the normal and shearing stresses on a horizontal line located a distance a below the free surface is shown in Fig. 8.26. This distribution is similar to that in Fig. 8.20 for the single concentrated force, thus again justifying the SainteVenant principle. The solution of the corresponding problem of a uniformly distributed shear loading is given in Exercise 8.40. The more general surface loading case, with arbitrary normal and shear loading over the free surface (–a≤ s≤ a), is included in Exercise 8.41.

Distributed loadings on an elastic half-space are commonly used to simulate contact mechanics problems, which are concerned with the load transfer and local stress distribution in elastic bodies in contact. Problems of this type were first investigated by Hertz (1882), and numerous studies have been conducted over the last century (see text by Johnson, 1985). Because interest in these problems is

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