The Plastic Theorems

The plastic theorems are fundamental to the Archie-M approach to masonry bridge assessment. This article provides a brief primer.

Origin of the plastic theorems

A series of experiments undertaken by the Structural Steel Research Committee in the 1920s-1930s showed that the stresses in steel frames bore no resemblance to the stresses predicted by analysis. This alarming situation raised the question of how the results of analysis could be trusted, and whether structural designs were safe. It was resolved by the development of the plastic theorems by John (Baron) Baker.

The plastic theorems are of vital importance to all structural engineering; they have operated since before their discovery, and continue to operate now whether the designer is aware of them or not. They depend on the concept of plastic redistribution of stresses, the idea being that if as a structure approaches and reaches the plastic limit at one location, a plastic hinge forms there, and stresses will redistribute. Collapse will not occur until a sufficient number of hinges form to create a mechanism.

At the core of plastic theory is the "master safe theorem", which states that if a system of internal stresses can be found that is everywhere in equilibrium with the imposed (dead and live) loads, and that does not reach yield at enough locations to create a mechanism, then the structure is safe. This is often stated in anthropomorphic terms thus: if you can find a way for the structure to stand up, the structure can find one too. By implication, the way the structure finds is very likely to be different from the one you find. That is why measured stresses do not generally match the results of analysis.

The plastic theorems allowed the results of elastic analysis to be used with confidence. They also provided a new approach to design that allowed greatly improved structural efficiency. 

It is worth noting in passing that standard finite element analysis does not satisfy equilibrium strictly and therefore  does not benefit fully from the master safe theorem. The Sleiper A collapse is probably the highest profile example of this - the master safe theorem is not a theoretical nicety.

The conditions of plastic theory

Equilibrium

The plastic theorems are based on the three conditions of bending moment distributions in a skeletal structure, when shear failure is excluded, at the point of collapse: equilibrium, mechanism and yield.

The equilibrium condition is satisfied when a system of internal forces is found that is everywhere in equilibrium with the external applied forces. This is "table stakes" for structural analysis, though as noted above not satisfied strictly in normal FEM. 

In Archie-M, the internal forces are represented by the line of thrust, and equilibrium is satisfied by way the line of thrust is constructed.

Yield

The yield condition is that the bending moment must nowhere exceed the full plastic moment at that section. If the yield condition is not satisfied, the state of stress could not be realised.

Yield in a thrust line analysis is readily expressed by expanding the line to a zone of thrust. The zone of thrust is the amount of material required to support the axial load without yielding (crushing). If the zone of thrust touches the edge of the ring, the stress cannot increase further and a plastic hinge has formed. 

The zone of thrust must not extend beyond the masonry; this would imply a stress greater than the material strength. If the line of thrust runs outside the masonry the state of stress is incompatible with the assumption that masonry cannot sustain tension. 

Mechanism

The mechanism condition is that, to collapse, the full plastic moment must be reached (forming a "plastic hinge") at enough sections for the structure to become a mechanism. The object of our analysis is to demonstrate that such a mechanism cannot form.

In an arch, at least four hinges are needed for failure. If the zone of thrust touches the alternate faces of the arch in four or more places, the arch is unsafe. 

Plastic moment redistribution

The plastic theorems depend on the fact that plastic structures redistribute moment until they run out of ways to do so. As we will see shortly, masonry structures are plastic in behaviour, but we discussed redistribution first in the context of steel, the behaviour of which may be more familiar.

A simple steel beam will fail when a hinge forms in the span as a result of the moment reaching Mp, the plastic moment, at some point. 

The simply supported beam is statically determinate, and the formation of this single plastic hinge results in collapse. The power of the plastic theorems comes into play if the formation of the first hinge does not result in failure: that is, if the structure is redundant (and thus statically indeterminate). The section must exhibit ductile failure, allowing considerable rotation in the plastic hinge before any local failure occurs. 

If two spans are connected, two hinges are required before a span can fail. As load is applied, a hinge is likely to form first at the internal support (a). If the load is increased further, the moment at the this hinge does not increase, but that at the mid span does. This is moment redistribution. When the moment at mid span reaches Mp, a mechanism forms and failure occurs (b).

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.

The unsafe and safe theorems

The upper bound "unsafe" theorem

One more bit of theory before we get to the application to masonry arches. Formally, all of these deal with "proportional loading". 

If the mechanism and equilibrium conditions are satisfied - that is, if a system if internal forces is found that is in equilibrium with the imposed loads and forms a mechanism - then the imposed loads are an upper bound on collapse load. We refer to this as "unsafe" because it is an upper bound: a mechanism may form at a lower load.

The lower bound "safe" theorem

If the equilibrium and yield conditions are satisfied - that is, a system of internal forces is found that is in equilibrium with the imposed loads, and the material nowhere exceeds yield - then the imposed loads are a lower bound on collapse load. 

The uniqueness theorem

If it possible to find a system of internal forces that satisfies all three conditions, then the imposed loads will cause collapse (but still not necessarily by the mechanism found).

Plastic theorems applied to masonry

Jacques Heyman joined the Department of Engineering at the University of Cambridge as an undergraduate in the 1940s, where he worked with Baker. (Heyman gives a fascinating summary of his career in his technical autobiography, available from the IStructE History Study Group's page under "Resources".)

It had long been recognised by some that an arch, considered in two dimensions, would be safe as long as there existed within it an arch that was in equilibrium with dead weight and external forces. William Barlow's seminal paper to the ICE showed this conclusively using models, leading to some lively and delightful discussion. Heyman's application of the plastic theorems to masonry placed this more intuitive understanding on a sound theoretical footing. 

Interestingly, Barlow does not mention in the text of his paper or show in the figures the possibility of asymmetrical lines of thrust - even in the case of perfectly symmetrical geometry and loading. This model by Bill Harvey shows demonstrates this case. 

Steel is an especially ductile material, but Heyman saw that the plastic theorems were much more broadly applicable, including to masonry. Masonry is often regarded as a brittle material, which indeed it is, but it can be built into structures which behave in a plastic way. The plasticity of an arch depends on the ability of the arch to crack deeply without failing, and for those cracks to move as loads change. 

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.

Consider an arch built over centring.

When the centring is removed, the horizontal thrust increases. The abutments move back slightly and the crown drops. The thrust moves to the intrados at the springings and the extrados at the crown. Cracks - perhaps invisible, in the figure they are greatly exaggerated - open where no compressive stress is present.

If we now add a small load, away from the crown, the line of thrust will change shape slightly, but if the load is small enough the cracks will not move. The thrust acquires a slight kink under the load.

If the load is increased, the cracks will move. First the crown crack will migrate until it is under the load. The the remote springing crack will migrate in the same direction.

With increasing live load, the thrust straightens. Eventually, the crack location and the thrust shape result in the thrust reaching the extrados at the springing away from the load. 

At this point, there are four hinges, alternating between intrados and extrados, forming a mechanism, and the arch will collapse. This is a "four bar mechanism", the fourth bar being provided by the abutments.

This movement of cracks is plastic redistribution of moment in action. 

Archie-M

The essence of assessment with Archie-M is to show that, under a desired loading, the thrust can fit within the masonry with at most three plastic hinges. There are various ways this could be done; Archie-M searches for 3 hinge positions that give the minimum horizontal component of thrust, and then checks, or asks the user to check, for the presence of a fourth hinge.

Considering the conditions of plastic theory:

• The construction of the thrust line ensure that equilibrium is satisfied.
• A line of thrust with one hinge at the extrados, and one each side of this at the intrados, will satisfy the yield condition if and only if the zone of thrust remains within the masonry everywhere. If the zone of thrust moves outside the masonry, then the yield condition is not satisfied.
• If the zone of thrust touches or goes beyond the extrados at one or more locations in the region beyond the intrados hinges, then the mechanism condition is satisfied. 

Consider a 5m span, 1.4m rise elliptical arch with ring thickness 0.3m. With no live load applied, the zone of thrust found by Archie-M is as below.

The circles mark the hinge locations. Archie-M has optimised these locations to minimise the horizontal component of thrust. Since the zone of thrust outside this set of three hinges is entirely within the masonry, it is clear that the structure is stable. (The blue lines are "force vectors", which indicate the direction and magnitude of the resultant of stress - from soil pressure and live load - on the back of the segment.)

If we bring a standard double axle bogie in from the right, there is a point at which the thrust just touches the extrados at the remote end. The situation shown satisfies all three conditions. The uniqueness theorem states that the load applied is exactly the collapse load (at this location). 

Moving the load further onto the span, the thrust now escapes the ring at the left, so yield is not satisfied, but mechanism is. The imposed load is an upper bound on the collapse load.

In practical terms, for masonry bridge assessment, there is no extra value to be had from the upper bound theorem. We simply know in this case that the load imposed is too high (though only because we have assumed no backing, which is unlikely in general). Our interest is to show that we can satisfy equilibrium and yield but not mechanism. 

A note on buckling

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 through 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-M 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.