# Background

In the 19th Century engineers were unconcerned about live loads on masonry bridges. Any live load was certain to be small in relation to dead load so they contented themselves with computing a line of thrust for the dead loads. Indeed, they set about finding the line of thrust that followed the centre of the ring as closely as possible. In 1848, Barlow showed that to be nonsense, but little changed until Castigliano developed his elastic computation.

Then, in the first world war, heavy, concentrated loads began to appear and a method of calculating their effect was sought. The actual behaviour of masonry bridges is extremely complicated. It is probably beyond our computational ability even today. In the 1930s massive simplifications were needed. The myth of the “effective strip” lingers from those days.

Pippard, in creating the MEXE method had data which amply demonstrated the absence of an effective strip:

He had no tools available for dealing with the complication and so created a fudge. He was concerned about the formation of first crack but that crack was only recorded when observed in real tests so the exact load to cause it was not well known. The model he created was the 1.5m+h strip still specified in BD21. (He said, correctly, that he had made too many assumptions already to allow a further longitudinal distribution. Once computation became easier, however, others introduced such distribution. This note will not venture there, though.)

When British Railways began to worry about live loads, their engineers created a different set of fudges around the basic MEXE model. They wanted transverse distribution to be interchangeable between various bridge types and used their established model of an 800wide by 250 front to back patch under a sleeper. So the railway model became 800+2h from below the sleeper while the highway model is 1500+h from the contact level of road and wheel:

It is demonstrable, from computational results and from tests, including those carried out in the 1930s, that distribution actually takes place largely in the arch as a membrane action. That means that the biggest influence on effective distribution is the span when the fill is shallow, though deep fill will also create substantial distribution before the load reaches the arch.

So, the current distribution models cannot yield worthwhile results but we are constrained to use them.

# Fallacies within the models

The BD21 model says that once the effect of two wheels overlap, they can be treated as one. The distribution lengths and loads concatenated and a rectangular distributed UDL used. Since standard axles are 1.8m between wheel centres, that means that the effect of an axle becomes a single unit (distributing to 1.8+1.5+h) at only 300mm depth between surface and arch. Since the critical load is never at the mid span and the depth off mid span will sensibly always be more than 300mm, Archie has always simply calculated an effective width for an axle.

For railways, the distribution for a wheel truncates where to adjacent distributions meet. Amongst other things, that means centripetal force should be allowed for on railways but need not be on roads.

# Distribution limits

The models presented assume that distribution cannot pass over a crack and must clearly truncate at the point where the bridge ends or where the load from a second lane (or track) begins to encroach. This is a limit that Archie-M cannot understand without the user telling it. Archie-M assumes a default limiting width of 2.5m which is the overall width of a typical standard vehicle. Cracks may reduce that and verges increase it so please DON’T leave the default unless you are just doing a quick run. The limiting width goes into the Archie bridge data on the last page of the wizard (see below).

Since the distributions are so nonsensical, allowing considerable eccentricity between the applied load and the distributed effect on the arch, there is no particular reason to ensure that any particular “bridge width” is symmetrical about the load concerned. If a bridge is 8m wide and has 2 lanes of 3m with verges, the appropriate “bridge width” to put here will be 4m. Note, in particular, that there is no reason to truncate to the width between parapets, though there might be a crack based case to limit distribution to the width between spandrel walls. There are two conditions for that, though. It must be possible to work out where the inside face is and you must be certain that one wheel of a vehicle will not be running on top of the wall itself.

# The distribution calculation within Archie-M

Archie-M offers two distributions for highway bridges. That is about longitudinal not transverse distribution and is of no consequence here. If you select a highway distribution Archie will take the axle width from the load file, add 1.5m plus the depth from the wheel to the arch and check that dimension against the “bridge width” described above. The smaller value will be used for each load.

Archie will display the “Minimum Effective Width” which is the smallest result for the above calculation taking all the different axles into account.

For railway loading the load on the different wheels on an axle may be different because of centripetal forces and nosing. For this reason, we based the load files on wheel loads and the axle length and bridge width must be adjusted accordingly. If you model a flat span with no fill, the distribution shows as 800mm.

# Notes

I think that should all be clear. If you have any doubts, please email support@obvis.com and we will try to clarify anything that is a problem and incorporate that in a new edition of this note.

BH March 2017