Tirailleur 

 I can  recommend that you read Ironsides Line. It is a concise history of the development of pillboxes and the thinking behind the designs how they were sited. In short the Stop Line concept, which is what you are referring to, was developed during the invasion threat emergency of 1940 by General Ironsides. The primary stop line is known as the GHQ line which runs from Bristol, East to Surrey and then down around the South of London before swinging north to the Thames. North of the Thames it becomes the Eastern Command line and runs through Esssex, Suffolk and sparsely through Lincolnshire. Along a lot of its length and especially around London and in Essex it is a defence in depth with at least two and sometimes three lines of closely spaced and mutually supporting pillboxes along the expected axes of advance of an invading force. There are a number of other stop lines which consist of single lines of pillboxes both in front of and behind the GHQ line and Eastern Command line. These are generally built along canals, rivers and rail embankments. Outflanking maneuvers would have been hindered by these obstacles making up for the lack of a supporting second line, and although their spacing may seem sparse now, many have been demolished that would have filled in the gaps. The lines were never intended to be garrisoned in their entirety and contrary to their name they were never intended to stop the enemy either, but rather slow them down and box them in to allow what remained of the Army to engage the enemy. 

 The pillbox designs themselves were effective against small arms fire but thin walled versions wouldn't have stood up to a 20mm round or sustained HMG fire. Thick walled versions (3ft 6in of concrete)specifically designed to face tanks such as the Type 28A would have withstood most of what Germans could have fielded at the time apart from the infamous 88mm. 

 However, by 1941 when General Brooke took over from Ironsides and just as the frenzied stop line building phase was coming to completion it was decided that the stop line concept was too inflexible, so the defence strategy was switched to one of Nodal Points, Anti Tank Islands and Defended Localities. Essentially fortifying towns and villages at key road and rail junctions. This would force the enemy to either bypass them and use slower minor roads or reduce each strong point in turn, slowing them down in either case. Many of the pillboxes were re-used in this concept, more were added where required and many, especially around airfields, were upgraded by thickening their walls. 

 If you want more detail I suggest you take a look at the Pillbox Study Group website and forum We are a friendly bunch there with a wealth of information about Pillboxes. The website is a little old fashioned but is being redesigned to be more user friendly and will be upgraded shortly.