Somewhat belated but the last two BoMs for 2014 are atNovember, WhangareiandDecember,... [read more]
Seems that the demo uploaded was the wrong one. Sorry. 220.127.116.11 now... [read more]
The Theory behind Archie-M
The theory behind ARCHIE-M is deceptively simple. It is, in general, easy to understand, and therefore reliable. Users will quickly come to understand the response of arches and the reasons for that response. They will also develop a "feel" for the significance of various parts of the makeup of the arch.
Readers are encouraged to read the section on equilibrium first, and then to explore the remainder of this section at their leisure as items rise to prominence in their minds.
ARCHIE carries out a form of equilibrium analysis. That is to say it determines whether an arch will remain stable, without first considering how it will deform under load. Deformations at ultimate load are likely to be measurable whatever the form of the arch. Large span, shallow arches (say spans over 15m with a span depth ratio of 6 or more) are likely to deform substantially, and may collapse as a result of local compression deformation (sometimes called snap through) before a fourth hinge forms. These details are discussed further under Elasticity and Plasticity.
The basis of equilibrium analysis is set out by Heyman (The Masonry Arch, J Heyman, Ellis Horwood, 1980). It is based on the plastic theorems which were probably first developed in Hungary, but which were brought to Britain by Baker. Heyman worked on issues based on the plastic theorems for much of his career. We should, though, not loose sight of the seminal paper by William Barlow (designer of St Pancras) which effectively demonstrated the plastic theorems as applied to arches.
Since the papers and books of Heyman began to appear in the 60s and 70s, it has been regarded as normal to compute a thrust line for an arch on the basis of a collapsing structure or four bar mechanism (one bar is the ground).
The mechanism analysis is often declared to be an upper bound solution in the mistaken impression that the analysis connects directly with the mechanism criterion of the plastic theorems (with which Heyman's name is also closely linked). Heyman himself suggested using the method to determine the level of safety against the formation of a mechanism. In so doing, he satisfied the equilibrium condition. That is why we ALWAYS refer to our analysis as an equilibrium rather than a mechanism analysis. Heyman simplified his solution by using a large factor of safety, suggesting that a geometric factor of 2 would protect against material failure and against deformation. We have made use of the power of a microcomputer to set aside some of Heyman's simplifications and so reduce the need for additional safety factors.
The collapse mechanism is not an entirely satisfactory way of viewing an arch. Despite the suggestions of BD21/97, there is no clear relationship between a load which will cause collapse and one which will cause progressive failure. Since the mid 1980s, our work has moved progressively in the direction of considering the real behaviours of arches.
An arch is built on a (typically timber) support system.
When the support is removed, the arch comes into compression for the first time. Since the compression is not axial, the arch bends as well as shortening. At the same time, the supports receive thrust from the arch and are pushed back into the soil behind them. Thus the arch shortens and the span increases.
The arch cracks in three places to form hinges, thus allowing the changing geometry to be accommodated. The cracks are typically, though not always, at the intrados at the crown and at the extrados near the springings. The thrust in the arch thus rises steeply from intrados to extrados and back again, giving the maximum possible rise within the structure. This yields the minimum possible thrust. Naturally, the thrust cannot actually reach the boundaries of the arch, as some material is need to carry the force.
When a live load is applied to an arch which is stressed in this way, it modifies the shape of the thrust line. Usually a substantial load is required before the hinges begin to move, closing in one place as they open in another.
Work by Chettoe and Henderson showed that as a load approaches a bridge, the abutment is pushed forward, reducing the span. The effect of this will be to raise the crown of the arch very slightly, and force the crown and springing hinges to close. This response in the arch is coupled with an increase in thrust, resisting the movement of the abutment.
At collapse, arches frequently form a mechanism and sway to one side. This sway mechanism is the basis of the original "mechanism" analysis of arches, promulgated by Heyman. He reasoned that though the mechanism was kinematically indeterminate, the stiff/strong nature of arch material was such that it was unlikely there would be any noticeable deformation before failure. The geometry of the arch is therefore known and the statics can be computed in a straightforward way. This is the basis of the old ARCHIE program. Heyman assumed that the arch would form hinges alternately at the intrados and extrados. ARCHIE made some allowance for material strength and formed the hinges slightly in from these arch boundaries(PIC).
ARCHIE also followed Heyman in imagining an arch of reduced overall thickness which might fail under a particular load. The reduction in thickness was expressed as a Geometric Factor of Safety, and was rather similar to the old "Middle Third Rule".
As with most arch assessment regimes, ARCHIE and its users recognised that the middle third rule was not appropriate when coupled with factored ultimate loads.
Equilibrium of 2 spans
The arches in a multi-span bridge behave very much like those in a single span. The main difference is that piers are more flexible than abutments, and that load applied to one span induces movement of the pier, which in turn generates a passive reaction from the next spans. It is important to remember that the combination of a pier and an adjoining span may be considerably stiffer than an abutment
The combination of stiffnesses which makes an arch viaduct is very complex. The arches themselves can respond, and the piers will help to resist the applied overturning forces. Fill above the arch will help to stiffen the response and, especially if there is masonry backing forming part of the fill. Spandrel walls offer two possible contributions to stabilising the arch. They stiffen the arch, but in the upper courses, they often form direct props over the length of the span, offering very great resistance to horizontal movement of the pier head.
The Plastic Theorems
The plastic theorems are of vital importance to all structural engineering. They depend on the concept of plastic redistribution of stresses, and were first applied to that most plastic of materials, steel. Masonry is often regarded as a brittle material, and indeed it is, but it can be built into structures which behave in a plastic way as reinforced concrete does. the plasticity of an arch depends on the ability of the arch to crack deeply, without failing. Most real masonry arches have three natural deep cracks in their dead load state. These are typically at the crown and the springings, though only the crown crack is visible because only it is at the intrados.
In an arch, plastic redistribution of stress is manifested in the movement of the cracks which takes place as loads are increased or moved. Wherever the cracks begin, they will migrate as a live load is increased. Eventually, a fourth hinge will form near the springing which is more remote from the live load (fig).
The mechanism criterion of the plastic theorems, declares that if a mechanism can be found which will allow the structure to collapse under a certain load, then that load is an upper bound to the collapse load of the structure. In terms of arch bridges, that means that if the thrust touches the alternate faces of the arch in four places, the arch is unsafe. The object of our analysis is to demonstrate that a mechanism cannot form.
The plastic theorems were first envisaged in the context of steel structures. In that context, yield of the steel is an important criterion. In bending (and arches usually fail in bending) yield progressively increases until a beam reaches its plastic moment, or until local buckling occurs. (local buckling is not a concern with masonry arches). A simple steel beam will fail when a hinge forms in the span as a result of the moment reaching Mp at some point. If two spans are connected, two hinges are required before a span can fail. Typically one will be at the support and one in the span.
It is vital to the power of the plastic theorems that the formation of the first hinge does not result in failure. This means that the section must exhibit ductile failure, allowing considerable rotation in the plastic hinge before the section actually fails. As load is applied to a two span structure, a hinge is likely to form first at the internal support. If the load is increased further, the moment at the hinge does not increase, but that at the mid span does. This is called moment redistribution.
The middle span of a series of three can only fail when hinges have formed at (or near) both supports and in the span. For a steel ARCH to fail, at least four hinges are needed. When they have formed, the arch is free to sway sideways and so collapse.
In a masonry arch, the equivalent to a plastic hinge is the cracking action which forces the thrust away from the centre of the arch section. The eccentricity of the thrust delivers a moment which is a function of the geometry of the arch,rather than its bending strength. However, the overall arch geometry does not normally change significantly as the thrust increases. The arch is thus able to redistribute moment in the same way as a steel arch or beam. When the moment at one point reaches a limit, it will stop increasing and the moment somewhere else in the structure will increase . This process will continue until there are enough hinges in the arch to allow it to fail.
Buckling is a form of failure which is often thought to be anathema within the plastic theorems and plastic design. In fact, the important issue is that buckling is controlled and does not cause sudden catastrophic loss of strength at any section.
Local buckling in steel structures may prevent a section reaching its full plastic moment. It may, however, leave a residual strength, thus still allowing moment to redistribute before collapse.
Snap through buckling of steel arches occurs when the combination of axial load and moment at a point causes the arch to reverse curvature locally. Eventually, the curvature may mean that the thrust generates enough moment to remove the stiffness of the section all together. at this point, the arch sags through and comes to rest as a tension structure if its supports can sustain it.
Snap thorough can occur in masonry arches when the eccentric thrust becomes sufficient to compress the wedged masonry through the straight line in a local area.
Snap through results in total failure. Moment redistribution cannot then occur. It is vitally important that snap through is not allowed to develop. A much deeper understanding of this phenomenon is desirable. It will only be a problem if the thrust runs close to the extrados of the arch over a significant distance, and this can readily be observed in the ARCHIE graphics. It should be noted that elastic analyses which do not include the effect of changing geometry in their calculations do not protect against snap through buckling.
The yield criterion requires that for a structure to be safe, yield must not occur at a section where plastic hinges have not been assumed. This is readily dealt with in a thrust line analysis of an arch by ensuring that there is always enough material around the line of thrust to sustain the load. The ARCHIE graphics display a zone of thrust. This shows the material required at every point in the span. Provided the zone of thrust is within the arch at every point, the structure (arch) is safe.
The equilibrium criterion is satisfied if a system of internal forces can be found which are everywhere in equilibrium with the external applied forces without violating the yield criterion. In many senses, the opposite of the mechanism criterion. Archie-M works by proving equilibrium negatively. That is, showing that a mechanism cannot form, even having taken yield into account.
The dimensions in which ARCHIE works are important. This is, of course, true of all engineering software.
ARCHIE-M is a 2D analysis system, in which 3D effects are dealt with as part of the input. We expect to extend the powers of ARCHIE-M over the coming year or so, to include some of the 3D effects which contribute dramatically to the performance of a structure. In the mean time, please be conscious of the need to avoid condemning a bridge because a simple analysis shows it to be too weak.
The sections below have a little more to say about arch and viaduct behaviour in various dimensions.
In a 2D analysis, the structure is idealised, usually using a nominal strip. ARCHIE works on the basis of a 1m wide strip through the structure. In this simplified form, it not possible to take account of the edge stiffening provided by spandrel walls, even if that were allowed within the standards being applied (it is not allowed in BD21/03).
We believe that the idealisation presented in BD21 is conservative, and may lead to bridges, particularly viaducts on slender piers, being condemned or "strengthened" unnecessarily. See our description of 3D behaviour for more details.
In ARCHIE-M, we have added to the power of the line of thrust visualisation by incorporating a zone of thrust, which represents the depth of material required to carry the thrust at any point. It is perhaps easiest to imagine a voussoir arch with the mortar cut away to leave just a narrow strip in each joint. If the strip is equally distributed above and below the thrust line and is just big enough to sustain the thrust, the arch will be stable, but on the point of the collapse. a very slight movement of the load, or increase of the load will result in collapse.
The thrust line model of arch behaviour is a very powerful one. It provides both a simple visualisation of behaviour and a simple modelling and calculation scheme. It is, however, inherently limited.
If, for a moment, we consider the distribution scheme proposed in BD21/BA16, we see that a patch load applied at the road surface is distributed to a strip with a minimum width of 1.5m. This carries the implication that the arch has sufficient transverse stiffness to ensure that a kink in the thrust line 750mm from the load can contribute to supporting the load. This is clearly not true.
We believe that a distribution more like a fan, concentrated at the load point and increasing in width towards the springing is more realistic.
We believe that a realistic model of live load thrust is that it is concentrated under the wheel at the point of application, but that it fans out towards the abutments. In a multi span bridge, we believe that the fanning is sufficient to ensure that the whole of the adjacent span can contribute to the stability of the intervening pier.
This model of distribution carries a number of other implications. For example, the stiffening effect of a spandrel wall may not result from its resisting deformation of the arch near the crown, but rather in increasing the resistance to circumferential compression near the springings. If this is true, and we have some reason to believe that it is, The spandrels may be able to reduce the effective span of the arch even when a crack separates them from the arch over the middle half of the span. Indeed, we believe that the reason for spandrel cracks is that the arch tries to deform and concentrates strand, and therefore stress, in the plane below the spandrel. This deformation is inherently much less near the springings, so crack propagation to the end of the arch is much less likely.
Confirmation of this behaviour is now available from tests and FE analysis.
2.5 dimensions is a more realistic view of the fan distribution we are hoping to implement in later versions of Archie-M. It should give much higher predictions of capacity, especially in multi span structures. It will also obviate the need for modifying the effective width for a multi-span mechanism.
It is of paramount importance, that any analytical approach can be demonstrated to be conservative. This statement flies in the face of the suggestion from BA16 annex E that for assessment purposes, analytical results should lie in a band +-20% from the true collapse load established by testing.
We set out in the sections below, our view of the conservatism of ARCHIE-M. Users should satisfy themselves on two counts:
1) That the analysis is sufficiently conservative to leave them confident in the level of safety achieved.
2) The analysis is not so conservative, in a particular case, that a bridge will be failed when it should pass the assessment.
Those actions included in the analysis must be clearly possible, even though the real structure may not deploy them.
By this we mean, for example, that the soil pressures used must be achievable in a real 2D structure such as the model being analysed. In fact, the spandrel walls may stiffen the arch so that the soil pressures are not achieved. The important thing is that they could be achieved if the spandrels were not present.
The included actions are Soil Pressure, Backing Pressure, Live Load Effects, Fill Distribution and Arch Distribution.
The model for soil pressure used within ARCHIE is condemned out of hand by BA/16 97 Annex-E. We believe that the comments made demonstrate a failure to understand both the limitations and the strengths of the model within ARCHIE and those with which it is being compared. The suggestion is that an unrealistic elastic analysis which derives soil pressures within the calculation is inherently better than one which requires the engineer to make a judgement as to what pressure might be achievable in his structure.
The Archie-M model treats granular behaviour only basing the pressures generated on Active, Passive and At Rest values computed using Rankine's theory.
The model of soil pressure used in Archie-M is a simple one. Unless the user chooses otherwise, only at rest soil pressure is applied, based on the values of Phi and soil unit weight given.
If it is found to be impossible to ensure that the thrust remains in the arch without increasing the pressure, it is possible to bring it in by applying passive pressure. In this case Archie-M creates a new hinge, initially at a springing point and increases the pressure sufficiently to ensure that the thrust goes through that point. The proportion of the potential additional passive pressure applied is displayed inside the arch at the appropriate springing.
This may still leave the thrust outside the arch above the new hinge. If so, the user can drag the hinge upwards. As the hinge moves, so the passive pressure below it is removed and that above is increased. It is normally possible to ensure that the thrust eventually stays entirely within the arch, though the passive pressure demand may prove to be higher than the user is prepared to allow.
At rest soil pressure is calculated from K0=(1-sin(Phi)). This is applied as a horizontal pressure on the vertical projection of an arch segment. The pressure varies from top to bottom of the segment. We believe that this representation is conservative. As the arch sways, we expect both the vertical and horizontal component of soil pressure to increase.
Passive pressure is applied to the rising sections of the arch only. The increase from at rest towards passive is assumed to be zero at the static hinge and maximum at the moving hinge. The pressure at any point is also proportional to the depth. This produces a pressure distribution which is a complex curve. The result, though, is very similar to that proposed recently by Hughes et al in (JICE, Structures
Experimental evidence has shown that the soil pressure may be reduced on sections of the arch which are moving away from the fill. Clearly this cannot apply to the sections directly subjected to live load which is causing the movement. The issue is a complex one. We do not believe that an active pressure zone should be implemented by users without detailed discussion with firstname.lastname@example.org.
In many other programs, the soil is modelled as a system of springs which respond to movement of the arch. Since Archie-M does not consider such movement, it is not practicable to use such a model. In any case, we feel that knowledge of particular structures and understanding of behaviour is sufficiently sketchy to make such complexity unjustifiable.
It is impossible to develop a rational value for the presure delivered to the arch by the backing. We have therefore chosen to remove the horizontal component of force from the backing and allow the thrust to pass out of the arch into the backing. It is important when observing this effect to remember that the Zone of Thrust is a very crude (if powerful) representation of the force flow in an arch. The true stress pattern will be much broader. At its simplest, the user may choose to imagine a rectangular stress block which occupies the whole of the space from the zone of thrust to the nearest material edge, and the same distance on the other side. In the backing, this implies very low levels of stress
We believe that the distribution model in BD21 is based on a false premise of distribution through the fill. In most circumstances, this is conservative.
For the distribution model in Archie-M see the section on code rules.
The arch itself is capable of considerable distribution. This does not take the form of bending action, but rather stiff, in plane, shear action.
For the time being, arch distribution is dealt with by the simple expedient of an effective width. This is a very crude representation of real behaviour in which thrust is gathered close under the load and fans out towards the springing. This is best considered on a load by load basis. We have a model based on Boussinesq distribution which works exactly like this and even deals effectively with skew bridges but it will be some time before we are able to incorporate it into Archie.
In the mean time users may wish to adjust loading when considering multi span action (not the loaded span in a group) so that passive action is based on the full available width of the arch.
Some actions which are clearly present in real bridges are specifically excluded from current versions of Archie-M. Particular issues are the stiffening and strengthening actions of spandrel walls and the detailed distribution of live load through the arch barrel. The stiffening action of spandrels is particularly important where there are internal spandrel walls.