Indentation depth for sphere/limited elastic half-space

Hello,

 My problem is the following:

In clasical Hertz theory I was able to locate indentation depth for the case of sphere/elastic half space, which I will denote as d0. I am interested if there is a relatively simple solution for obtaining indentation depth (d1) if this half-space is limited, that is a sphere in contact with either of these three cases:

1. Semi-infinite cylinder of diameter W (sphere is in contact with the upper cylinder side)

2. Semi-infinite prism of side W (sphere is in contact with the upper prism side)

3. Semi-infinite band of width W

In an article considering indentation of sphere in lamellae of width W I found the following expression, but I do not know where it is derived from:

d1=d0*(1+R/W)^0.5 (R is the sphere radius)

The equation seems logical since upon limiting cylinder/prism/band dimension to infinite d1 becomes d0 and agrees with the Hertz theory for elastic half-space. Derivation, however, is not clear to me.