1. As a more rigorous alternative to Fig. 2.3, it is proposed to consider the equilibrium of the *finite* disc \(0\!\le \! r\!<\!a,\;0\!<\!z\!<\!h\). The integral in Eq. (2.6) must now have a finite upper limit \(r\!=\!a\) and the shear tractions acting on the curved surface \(r\!=\!a\) will make an additional contribution to the equilibrium equation. Show that these effects are self-cancelling, and also that they both tend to zero as \(a\!\rightarrow \!\infty \).

2. An elastic half-space is loaded by a uniform pressure \(p_0\) inside the square \(-\!a\!<\!x\!<\!a,\; -a\!<\!y\!<\!a\). Use Eq. (2.17) to determine the normal surface displacement at the centre of the square [i.e. at the origin].

3. An elastic half-space is loaded by a uniform pressure \(p_0\) inside the circle \(0\!\le \! r\!<\!a\). Use the field-point integration method of Sect. 2.3.1 to determine the normal surface displacement as a function of *r* for points inside the circle.

4. The flat elliptical rigid punch of Sect. 2.3.2 is loaded by a force *P* whose line of action passes through the point (*c*, 0). This causes the punch to tilt about the *y*-axis. Use Galin’s theorem to determine the corresponding contact pressure distribution, assuming that the entire planform of the punch remains in contact with the half-space. Hence, determine the maximum value of *c* if this assumption is to be correct.

5. Use the potential (2.4) and Appendix A, Eq. ( A.3) to find the radial (tangential) surface displacement \(u_r(r,\theta , 0)\) due to the concentrated normal force *P*. Hence, show that the area of any circle \(r\!=a\!\) inscribed on the surface will be reduced as a result of the deformation, and find the extent of this reduction.

6. Use the potential function solution of Appendix A, Eq. (

A.2) to find a relation between the surface dilatation

and the contact pressure

*p*(

*x*,

*y*), assuming frictionless conditions. Hence show that the radial surface displacement

\(u_r(r, 0)\) and the contact pressure

*p*(

*r*) in an axisymmetric problem are related by the equation

$$ u_r(r)=-\frac{(1-2\nu )}{2Gr}\int _0^r p(s)sds. $$

7. Use a superposition similar to that in Eq. (

2.5) to write expressions for the stress components

\(\sigma _{rr},\, \sigma _{zz}\) on the axis

\(r\!=\!0\) in cylindrical polar coordinates

\(r, \theta , z\), when the circle

\(0\!\le \! r\!<\!a\) is loaded by an axisymmetric contact pressure distribution

*p*(

*r*).

Evaluate the resulting integrals for the case where *p*(*r*) is uniform and equal to \(p_0\), and plot a graph showing how the maximum *shear* stress varies along the axis.

8. Use Eqs. (2.4), (2.50) to find the stress component \(\sigma _{xx}\) at a general point (*x*, *y*, 0) at the surface due to the point force of Fig. 2.2. Hence determine this stress component at a point on the *x*-axis when the half-space is loaded by a total normal force *P* uniformly distributed along the line \(x\!=\!0,\; -a\!<\!y\!<\!a\).