As a criterion of the maximum pressure attainable on the interface between tube and plate when tubes are expanded into boiler drums, a theory is required which will allow for work-hardening and thickening of the plate, and it is also necessary to take into account the elastic as well as the plastic strains. Such a theory is set out in this paper, assuming a uniform hydrostatic pressure, the hole being expanded from zero radius in an infinite plate. The plate varies in thickness proportionately with the radius, and it is shown that the system remains geometrically similar irrespective of the degree of expanding, i.e. of the extent of the plastic region. The axial stress perpendicular to the middle surface of the plate is assumed to be zero for this problem, so that the tube and plate can be considered as a continuous medium. The problem is then one of plane stress in which all variables become functions of a single parameter r/c, where r is any radius and c is the radius of the current plastic-elastic interface. The equilibrium equation, the compressibility relation and the strain relations (through the Reuss equations) are expressed in terms of this parameter and of the velocity v of any element, using c as the time scale. The resulting four equations are solved by taking finite increments of the variables, the appropriate value of the yield stress being inserted for each step using the yield stress curve for a typical boiler steel. The stress and displacement patterns are compared with those for a plate of constant thickness (Taylor 1948; Hill 1950). The residual stresses on release of the expanding pressure are determined assuming no secondary yielding occurs. In addition, the earlier theory due to Nadai is extended to allow an estimate to be made of the effect of secondary yielding owing to the residual stresses reaching the yield stress.