CS454 for users of Archie-M

A brief review of CS454 for users of Archie-M

Document history:

• Initial version 2019-10-23
• Updated 2022-04-01 with sections on:
  • Load factors
  • Lift-off
• Correction 2022-05-18:
  • Lump sum factor for masonry bridges was quoted as 3.5, producing a live load factor of 1.95. Actual value from BD21 was 3.4, giving a live load factor of 1.9.
• Correction 2022-08-29:
  • Corrected to refer to ɣfL on live loads, rather than ɣf3.
  • Updated section on skew bridges: revision 1 of CS454 dropped magnification factor for skew bridge assessment.

For load files, see CS454 Loads.

Methods of analysis

I find the order of presentation somewhat surprising and the numbering of clauses positively perverse. I propose to leap first to the clause which lists the available analytical approaches. Surely, all other questions are predicated on the method of analysis to be employed?

7.7.8 Methods of analysis for arch assessment may comprise one or more of the following:

1. mechanism analysis;
2. equilibrium-based analysis;
3. non-linear finite element analysis;
4. the modified MEXE method.

I bang on a lot these days about “Is it a fair test?” it is a question asked of primary school kids in their science classes. The word analysis carries a lot of weight. It implies the use of a model of behaviour that is, at least, capable of representing real behaviour. We are very far from that in masonry bridges. The concept of an effective strip is complete nonsense. Since all but the most complex (and therefore expensive) FE analyses begin by converting the model to 2D using an effective strip, we fall at the first hurdle.

So next we need to ask whether the results are conservative. The answer to that is how can we know? The tests done in the 1930s, on which the effective strip is based, were very good and very effective. The results were interpreted in terms of both the available understanding and the available methods of calculation of the day and the effective strip was the best they could do. The data, though, shows that most of the distribution takes place as membrane shear in the arch and the result of that is dramatically different from an effective strip.

The real tests are carried out every day and are heavy freight vehicles on the railway. They test at the serviceability limit and bridges are failing. The minimum span of 5m for MEXE quoted in clause 7.13(4) is presumably because of the evidence that the effective strip grossly exaggerates capacity at spans shorter than this. What that fails to acknowledge is that the same result applies to Archie and Ring and any other approach that uses an effective strip.

Non-linear FE analysis is only valid if the relative stiffness of all the parts of the structure are in correct proportion. The scale of displacements (and possibly stresses) will be wrong if the stiffnesses are of the wrong value in absolute terms, but if they are not in correct proportion, the very nature of displacement and stress will be wrong. In brick bridges, the stiffness of every part will vary considerably through thickness. The arch is likely to be reasonably well built with full joints and good quality bricks but still the stiffness and strength is likely to vary by an order of magnitude from brick to brick and the intrados layer will be much stiffer than the extrados because the best bricks will have been selected for the shell.

Clause 7.7.3 says that spandrel walls should not be allowed to provide support or strength to the arch barrel. Failure at the joint between these parts is a result of a huge difference in stiffness. If you are going to do an FE analysis it makes no sense to ignore the strength here if the stresses involved are substantially less than sufficient to cause a failure. The mechanism by which failure takes place is complex, though, and very different as between brick and stone bridges. Once a crack has formed, movement in the joint will progressively jack the edge of the arch and the spandrel wall sideways. The contribution of soil pressure to this effect is small. The damage is serious and has to be:

• A)    Anticipated
• B)    Treated

Without understanding why and when it will (or did) happen, any treatment is doomed to do damage rather than improve things.

In any case, what is a complex non-linear FE analysis going to tell you if you don’t analyse the bridge, just selected parts of it.

Mechanism or Equilibrium

There is an awful lot of rubbish talked about this difference. The whole process is based on the plastic theorems in which there are three significant criteria. Mechanism, yield and equilibrium.

If a system of internal forces can be found which everywhere satisfies Equilibrium and in which few enough sections will yield such that a mechanism cannot form, the structure is safe. It happens that the test of a mechanism is the easiest way to check equilibrium and yield. If a mechanism can form (i.e. if the thrust line necessarily passes outside the structure at any point) the load applied is an upper bound to the collapse load. That is, the true collapse load must be less than that applied. The approach within Archie, from the start and in Archie-M since 2000, has been to demonstrate that the thrust can fit within the arch with enough to spare to prevent local compressive yield. In other words, to satisfy the lower bound criteria of the plastic theorems.

There are many different ways of achieving this and that embodied in Ring produces essentially similar results.

The inclusion of the expression “Mechanism Analysis” in BD21 was at the behest of an engineer who seemed unable to understand the issues. There is no such thing. It was part of an argument in favour of an even more dubious (and expensive) “analysis” using a frame analysis program.

Skew arches

There is a good deal of very dubious advice on skew arches and this code does nothing to improve matters. Let’s begin with clause 7.13-8 which says MEXE cannot be used with skew greater than 35 ^o.

When he developed MEXE, Pippard commented widely on the range of assumptions made and the absolute necessity to restrict use to structures within the range for which test results were available. The only useful tests in this sense apply patch loads in realistic wheel patterns and there have been none. The implication that, for skews up to 35 ^o MEXE is valid does not even consider whether the analysis uses the skew or square span.

That takes us to clause 7.7.7 which applies a magnification factor to the strength of an arch which is skew. Such a seriously counter intuitive result can surely not be based on a few tests and definitely not on analysis that doesn’t take real behaviour into account. [This clause 7.7.7 was modified in revision 1 of CS454 to remove the magnification factor. The framing as a formula is now confusing, it would be simpler in prose, but the requirement is to assess bridges with skew angles up to 30deg as a square bridge with the skew span (and presumably, though it is not mentioned, the skew shape).]

Perhaps now is a good place to think about the limits of analysis because it is writ large in skew arches. The argument must begin with skew slabs because skew slabs have big problems in the obtuse corners and that idea has been partly mapped onto skew arches, without proper thought about whether there is a real link.

We need to come back to the relative stiffness issue. The important number in a skew slab is torsional stiffness and it tends to be low. At the same time, the vertical stiffness of the abutments is high enough that treating them as rigid produces conservative results.

In an arch, the big forces are horizontal. The plan stiffness of the arch is huge and the plan stiffness of the abutments is substantially less. For this reason, analysis and tests on skew arches which use sensibly rigid abutments are certain to produce wrong results. I was taught this lesson at Pynes bridge near Exeter. It is skewed at 13 degrees. The piers are four granite columns so the torsional stiffness in plan is essentially zero. That means the resultant of thrust from span to span must coincide and that can only happen if the arches span skew.

But there is another interesting effect. Arches sway under load. They can pretty much only sway in the square direction because the stiff line of the abutment prevents any other result. It is an artefact of stiffness again. The deflected shape tells a different story from the flow of force and limited analytical results from finite elements confirms this.

Do skew arches fail? Emphatically yes, though once more we need to look at the railways for this. I have now seen at least four skew bridges torn into strips. Cracks at about 1m centres. It doesn’t seem to need much skew but does seem to be restricted to shallow arches with relatively shallow fill. For example, East of Birmingham centre there is a loop line that crosses a canal and a road at moderate skew. The arch over the canal has plenty of headroom and is semi-circular. That over the road is much flatter. Unfortunately, my first set of photos of the cracks failed and by the time I went back it was already being lined with reinforced concrete. A similar fate befell a skew span in the COL viaduct in Manchester but at least there we got some photos first, not least because that is where I first saw this phenomenon.

I suspect that the cause is some sort of plane of weakness resulting from interaction of the bond in consecutive rings of the arch and the preferred sway direction of deflection. A starting point in measuring that would be in a skew bridge where a singe track has been widened to two.

If you value your PI you will run whatever “analysis” you use with the skew span. That may be conservative but we have no way of knowing by how much.

Effective width

Clause 7.7.6 simply repeats the advice in the old code. Even the basic engineering lessons seem not to have been learned.

The first joke is the difference between distribution between roads and railways. If the distribution is really in the fill, how can the fill know the difference between a railway and a highway load?

The railways continue with the original Pippard approach of 1:1 each side of the load. They also recognise that centrifugal force and other issues may result in different loads on a pair of wheels. The distribution truncates the distribution where two effects meet.

The rule for roads (Clause 7.7.6) uses 1:2 each side but then adds 1.5m meaning that there is a minimum distribution of 1.5m even if the load is applied direct to the arch. I am yet to see any defence for this.

The code then says that if the effect for two wheels overlaps you simply take the maximum extent of distribution and the total load. But that is nonsense if the two wheels have different loads. The result defies equilibrium.

It is also no use to anyone if the bridge is skew because the two wheels are at different points in the span and therefore have different fill depths but in which direction and how do they combine.

Multiple rings and ring separation

7.7.5 Where an arch barrel comprises multiple rings and there is evidence of ring separation or where there is a risk of ring separation occurring, the effect of the ring separation should be included in the assessment of resistance.

I love code clauses like this. Here is a problem we think exists. We have no idea of the reasons or potential consequences but just deal with it will you. Why bother with a code at all if you are going to cop out on the difficult stuff.

Here is my take:

Rings separate.

They separate with time and essentially dead load only.

If, as is not uncommon, header bricks are included where they fit (this cannot be done in skew bridges) they concentrate load because they are stiffer than mortar and eventually snap, often, apparently, as a result of dead load creep.

If you put a line load across a sound arch and load till it collapses ring separation precedes collapse and can be assumed to have reduced the collapse load but I think, usually not by much.

What we are concerned about is keeping loads at levels that will not cause progressive damage. We have negligible information on any relationship between this load and the so-called ultimate load for a bridge.

Real thrust patterns are completely different from those assumed in analysis. Stresses sufficient to cause shear cracking in even weak mortar are only likely to occur directly under a wheel, or if dead load creep (e.g. of the abutments) has caused the arch to sag, generating substantial stress differentials. Even then, the separation detectable by a hammer test is only going to happen if the effective crown rotation is enough to make the bottom ring too short to meet at the crown.

There are many thousands of bridges out there with ring separation. The question, to me, is, if you can only detect it by hammering does it really matter? I am pretty much certain the answer is no.

If you think it does matter and you feel the need to deal with it stitching alone is of no value. The stitches might have the strength of the mortar they are supposed to replace but there is no chance they would be stiff enough to redistribute stresses.

Using anchors and screwing them up is even worse because you lift the ring back into contact, effectively removing the thrust that clamps the bricks in place. Eventually, the bricks between the anchors will fall out.

The only effective treatment is to stick the layers back together with grout, the stress on the grout will then be negligible so it will work, but only if it is given time to gain strength. So don’t bother grouting up and then letting the traffic run a couple of hours later. You are just wasting the client’s money.

If you want to work out the shear stresses between rings, I can show you how to do it from Archie-M output.

Sliding in joints

Has anyone ever really seen this? It is a necessary condition for sliding that the direct compressive force round the ring is locally less than the shearing force. For that to happen under live load the thrust has to be removed locally by a secondary effect such as mortar loss. Even arch spread should not result in shear failure.

Assessment loading or actions

For the vast majority of arches CS454 seems to have made no changes to loads or factors. For arches over 20m span there is now a requirement to use model ALL model 2 which is closely related to the old HA loading. For arches, we are unlikely to be concerned with spans over 40m except in very special cases and they are beyond reason for the application of simplistic analyses. In the 20m-40m range the complex formulae result in loads that could sensibly be regarded as a falling intensity of UDL at 49 to 31kN/m of lane and a KEL of 82kN

That the loaded length is half the span is defined by the code.

Running that for a 25m span we get:

Compared with the worst case for AWR loads of:

It seems reasonable to suppose that these ALL 2 loads have not actually been tried on arches.

All the other complexities of loads and lanes really don’t apply to arches since there is no sensible way of applying distribution over more than 1 lane. The vehicle lane (or less) will always be the critical width to engage with.

Longitudinal loading

Longitudinal loading is only significant if the deck of a bridge has to be restrained. In the case of masonry bridges, even viaducts, the fill and surfacing are continuous over the length and the abutments and there is no possibility of load being diverted down into the piers as they have negligible stiffness in this context. Heavy braking on a curve will have an interesting effect but there is, as yet, no sensible way of analysing it.

Impact and ɣfL for live load

The second of the three cases in this clause relates this to speed, which seems to me an unlikely issue. If a vehicle hits a bridge likely to cause lift-off at speed the issues would be more with the vehicle than the bridge.

Modification factors

The various modification factors included in the modified MEXE method are complex, strange and undefended. Particularly the condition factor is simply the result of a group of people sitting down together with a pile of photos and saying I will give that 0.5. There is no reason to suppose the results are in any way representative of real bridge behaviour, or that the numbers should be divided by 10 and added to 0.9.

If mortar joints are empty, that must be accounted for. If they are full, the joint width is accounted for in the masonry strength by making due allowance for the construction. The factors given in table 7.5.1b are conjured from the air but might possibly be right. Material strength has little effect on the results from Archie-M unless very low (less than 2) or the arch is very large.

Arch condition factor

Again, these numbers are clearly just made up. The fact that they were made up long ago doesn’t make them right since the engineers had no concept either of the loads to which we now subject these bridges, or of the tools we might have to hand to compute capacities. All rows in the table except the last, have the caveat “unless already included in the analysis model”. I am not aware of any way of including diagonal cracks and the values there are particularly penal. How would one decide within the range 0.3 to 0.7?

It is hard to imagine finding an arch that didn’t warrant the 0.9 factor from the last row of the table.

Before we leave this, though, we need to look at:

Note 2: The value of FcM is selected based on the worst type of defect present, and not by multiplying the factors for several separate defects.

Well isn’t that interesting. For example, lateral cracks and deformation do not make the effect of diagonal cracks worse.

I think what this is saying is “Gee we are getting very conservative here, best back off a bit”.

Bill Harvey

23 October 2019, and revised as detailed at the top.