Available width, effective width, and transverse distribution
Distribution width
- 2024-08-01, updated to reflect Archie-M version 2.6 terminology and refer to CS454
- First version, 2017-03
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 CS454. (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/CS454 model says that once the effect of two wheels overlap, they can be treated as one. The length between the other limit of the wheel distributions is used, and the sum of the wheel loads, and the result is assumed to act as a rectangular distributed UDL. 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 position 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, if centripetal forces are relevant, the distribution for a wheel truncates where the adjacent distributions meet. Centripetal forces are not considered in CS454.
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 (called "available width") of 2.5m which is the overall width of a typical standard vehicle, and thus the narrowest plausible lane width. Cracks may reduce that and verges increase it so please don't leave the default unless you are just doing a quick run. The available width goes into the Archie bridge data on the last page of the wizard.
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 “available width” is symmetrical about the load concerned. If a bridge is 8m wide and has 2 lanes of 3m with verges, the appropriate “available width” to put here will be 4000mm. Note, in particular, that there is no reason to truncate to the width between parapets, unless there is 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, CS454/BD21 and Archie/Multi. Archie/Multi distribution is obsolete, retained to allow old assessments to be reproduced. That choice is about longitudinal not transverse distribution and is of no consequence here.
If you select a highway distribution, Archie-M 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 “available width” described above. The smaller value will be used for each axle.
Archie-M will display the input available width and the minimum effective width, which shows 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, Archie-M provides both axle and wheel load files. The wheel loads are half of the axle loads, and the axle width is 0. The available width must be adjusted accordingly.
Axle loads remain useful for railway bridges where the wheel loads are equal, and can avoid unnecessary conservatism.
On roads, SV loads have a larger axle width, where wheel loads may not merge on typical depths of fill. The SV load files therefore give wheel loads in general.