On Ice Shelf Cracking

Tension Cracks in the Brunt Ice Shelf

Yesterday the BBC news website published images showing a large section of the Brunt ice shelf in Antartica, which has separated from the main body of ice and formed a large iceberg. The associated article notes the giant berg to be 1,290 square km on plan. 

The reasons the ice berg formed are no doubt complex, however that isn’t what caught my eye. The thing that interested me can be seen in the close up images. Again, according to the article, the crack that has formed in the ice is approximately 2 km wide, which is a vast dimension relative to the scale of cracks that are normally considered.....at least in the sort of work I am accustomed to.

The reason this interested me was because, despite the macro scale of cracking on display, it has the appearance of something that is familiar to me at a much smaller scale.

 It follows that my explanation of the BBC’s photos is based on extrapolation from small to big. I don’t really know anything about ice shelves or icebergs so admittedly it’s going to be a bit of speculation on my part; though I hope not too much of a stretch.

The defining characteristics of the crack are; clean edges edges; and what seems like debris in the gap. Almost all of the debris is on the left side of the crack and has accumulated adjacent to the iceberg rather than the shelf. To me that is interesting. 

Clean edges, or more importantly the absence of tearing, indicates that the crack is a tension crack i.e. the iceberg has pulled directly away from the ice shelf and has not slid past it.

If there has been a clean break the question arises as to why there is debris within the crack and why it is clustered on one side. I think the most important clue is to be found in the profile of the debris’ edge. It matches the profile of the ice shelf much closer than the profile of the iceberg. This indicates that the debris must have been in contact with the ice shelf.

From this we can deduce that the original crack must be old and that it must have been dormant for a reasonable period of time. We can also deduce that it must have remobilised more recently. The inferred mechanism would be as follows:

At some point in the past a crack opened in the ice shelf. The images do not convey why, but for some reason the crack stopped becoming larger and was for a time stable. This would have allowed new snow to fall within the crack and over time to pack down and form new layers of ice. The crack was starting to fill. 

For some reason, again the images do not convey why, the crack remobilised and the iceberg started to drift further away from the ice shelf. The forces carrying the berg also carried the debris moving it further away from the shelf. I suppose I shouldn’t really call it debris, as that would imply that it fell from the sides of the ice pack, whereas I think it was new snowfall. If it had fallen from the edge I don’t think the crack edges would be as clean as they are.

I like this explanation, because I think it fits the visual evidence in the photo, however it also suggests that the same principles apply at a human scale as they appear to at a continental scale.

For example, old and dormant cracks in a wall can often be identified by evidence of dust and debris accumulating in the gap. If the debris has become dislodged it could indicate that the crack has started to remobilise. Similarly, cracks with clean edges will generally indicate the presence of tension. This in turn indicates movement perpendicular to the direction of the crack. Together these observations can often be used to infer the underlying cause.

In the case of the Brent ice shelf we can infer that the berg has moved perpendicular to the crack, perhaps due to ocean currents at the water’s edge, or perhaps under the influence of gravity if the underlying shore slopes toward the sea. I don’t know the reason, as it is not evident from the images.

Nevertheless, the visual appearance of the crack in the ice shelf does imply the remobilisation, for some reason, of an older crack that was stable for a period of time. Something has changed.

Or, since I don’t know anything about ice shelves or icebergs, perhaps none of my conjectures are true. I shall allow the reader to decide.

On Lock-down Tyres

Why my car tyres are flat?

I left the house this morning and found that my car tyres had gone soft and one was completely flat. This was a rather curious state of affairs, because I had not driven anywhere to acquire a puncture, due to the covid-19 lockdown. I live in a quiet village and have good neighbours with whom I am friendly; so it seemed unlikely that anyone would have tampered with them. Nevertheless, I did inspect the tyres to make sure that there had been no foul play. There was no visible damage to the tyres and layers of grime on the valve cap indicated that it had not been loosened recently.

I like a mystery and this got me to start thinking about whether there was an engineering reason why my car tyres had become soft. The first potential culprit was the weather. The differing seasons bring with them settled and unsettled weather, which is of course linked to air pressure. It would be rational for the internal pressure in the tyres to be affected by changes in external air pressure. While this may have been a factor it did not seem to be a good explanation, because though the weather has been inclement recently it has not been unremarkable. Scotland regularly experiences inclement weather, often far worse than it is now. This begged the question that if weather was responsible for my tyres’ loss of pressure why had it not happened before?

The only thing that was really different to normal was that the car had not been used, as there was nowhere to go during lock-down. Cars normally deteriorate due to wear and tear; inactivity would therefore seem like and odd, and somewhat ironic, causation. That got me to thinking about what the mechanism for such an outcome could be. I soon found myself reverting to type thinking about materials and load paths.

The weight of my car is split between four wheels, but not evenly. With an empty boot [trunk for American readers] the car engine is responsible for there being more load on the front axle than the rear. This was consistent with the front tyres being flatter than those at the rear, so perhaps I was on to something.

Weight is transferred from the rotor to the wheel hub by four bolts and then from the hub to the tyre by air pressure. When I was growing up tyres had a pressurised inner tube, but today there would seem to be reliance on the joint between the hub and tyre being sealed tight by pressure.

The hub must be exerting a downward force onto the pressurised air at the base of the tyre, which is resisted by an equal and opposite upward reaction from my driveway via the tyre. The air inside the tyre does not like being squeezed and will try to escape out of the way causing the sides and top of the tyre to experience an outward thrust that would cause it to be stretched. Thus, the bottom of the tyre would be experiencing compression while the top and sides experience tension. 

With inactivity seemingly the critical factor I concluded that my tyres did not like this state of affairs for an extended period. I reasoned that this might be because they are intended to be spinning, such that each part of their circumference takes its turn at bearing compressive and tensile load.

Conversely if my car remained stationary then the load experienced would become quasi-permanent rather than temporal. This opened up several possibilities. Perhaps constant tension resulted in the structure of the rubber becoming elongated in such a way that it was more permeable to air, or perhaps constant compression made the rubber at the base less flexible and therefore more permeable. 

Alternatively, perhaps the seal between the tyre wall and the hub starts to slip with the constant application of load. It would seem reasonable to postulate that such an effect would become more pronounced as pressure is lost from the tyre, because the seal is reliant on their being a positive pressure.

I am not sure if any of these potential explanations are correct or if they all play a role, however irrespective of this I shall in future be making sure that my car wheels turn regularly; even during covid lock-down. 

I shall view my theory as proven if my tyre problem does not return....we shall see.

On Breaking Rules

Why codes and standards are not always right

In my page entitled ‘what is structure?’ I began with the following quote attributed to Picasso. 

“Learn the rules like a pro so you can break them like an artist”

I inferred from this that structure does not limit creativity, and went on to argue that structure provides a framework with which to create. From this I deduced that without structure nothing we might seek to create makes sense. In retrospect I realise that I didn’t expand on why engineers, like Picasso, can benefit from breaking the rules that govern structure. I shall try to do that in this post.

This might seem at first to be a tall order, because you might suppose that the rules of engineering are fairly arbitrary and to break them would have unwelcome consequences. To make my case I intend to make a diversion into the subject of racket sports, which I rather enjoy.

After mastering the basic strokes of any racket sport it will soon become clear that there are three possible ways to score points. The first is to hit a shot that your opponent is unable to touch; the clean winner. The second is to hit a shot that stresses their technique and causes them to miss; the forced error. This is normally achieved by making your opponent play his or her shot from an awkward position; perhaps on the run or while stretching. The third way is to hope that your opponent simply makes an unexpected mistake; the unforced error.

While the third case is random, the first two can be induced by clever tactics. Perhaps the simplest, and ultimately most successful, tactic is to maximise the distance your opponent must travel between shots. This will necessitate playing on the run, potentially hitting while stretched and eventually fatigue. 

It follows that if you receive a ball from cross court the rule is to strike back down the line, because your opponent must run the width of the court to play a return shot. Conversely, if you receive a shot from down the line the rule would be to return cross court. Again, your opponent must run the width of the court.

This tactic works really well on a novice player, however it starts to unravel when you find yourself playing another player who knows the same tactical rules that you do. He or she knows that you are going to return cross court from down the line. It follows that as soon as they have hit down the line they will start running to receive your cross court shot. Arriving early on the far side of the court they will quickly turn the tables, by making you run down the other line.

To outmanoeuvre your opponent, and thereby avoid the tables being turned, you must break the rules and hit back down the line from which you received the ball. Your opponent, having set off in the other direction, must turn and run back in the direction from which they came. He or she now finds their technique being stressed again.

Now the rules have changed and you are doing exactly what you learned not to do. This, however, does not negate the rule. If you continue to return in this fashion your opponent will ruthlessly exploit your apparent naivety, as if you did not know the tactical rules to begin with.

The trick is therefore to apply the rules most of the time, but break them just often enough that your opponent cannot take for granted what you will do next and thereby set off early to meet your next shot. 

I accept that this is a rather limited tactical explanation, but it serves as a useful illustration of the cat and mouse game that opposing players must engage in. It also illustrates Picasso’s point that to become an artist you need to know the rules so that you understand how and when to break them.

That is all very well, but how does this principle translate into structural engineering? The structural engineering rule book is provided by the building regulations and by codes of practice. Once a structural arrangement has been selected, a creative process in and of itself, codes and regulations generally provide a fixed methodology for justifying its design, however for several reasons this is not always helpful. You have not mastered structural engineering until you know when and how to depart from their strictures.

In a prior post about pile cap design I demonstrated that a foundation supported by just two piles can be designed in multiple, equally valid, ways. This, however, is not the only reason to think twice about the rule book.

Many current standards have been in use for 20 years or more. It is inconceivable that nothing new has been learned in that time, which could be captured in future standards. It takes time for standards to be revised, however until they are there is no reason why an experienced practitioner cannot take advantage of newer information.

Another reason would be that codes are necessarily general documents intended to cover a broad set of circumstances, however they have evolved to their present form based on what already exists and is common. Consequently, an engineer must know the limits of application for codes of practise and must, when necessary, look to other approaches when something is uncommon.

I think most engineers, though not all, would accept that a design code may not, on some occasions, produce the most efficient design possible, but perhaps fewer would recognise that from time to time codes are habitually applied incorrectly because they are badly written or because they are old and their original meaning has been forgotten. 

Fewer still would recognise that codes can sometimes be plain wrong. While there are some well known examples, identified through catastrophic failure, there are others that are harder to spot, because the circumstances in which they are wrong is rare.

We have learned in this post that a novice tennis player who returns a shot received from down the line back up the same line is not the same as the master who returns up the line to exploit his opponent’s premature movement across the court. 

Similarly, the novice engineer who ignores the requirements of the relevant code or standard through ignorance is not the same as the experienced engineer who sets aside their provisions with good reason.

It is, as Picasso says, necessary to learn the rules so that you can break them.

Postscript

Some may question whether it is legal to ignore codes and standards. It is therefore worth noting what the Building Regulations have to say: 

Approved document A, applicable in England & Wales, says:

‘There is no obligation to adopt any particular solution contained in an approved document if you prefer to meet the relevant requirement some other way’

The technical Handbook, applicable in Scotland, echos similar sentiments when it says:

‘The regulations are mandatory, but……it is quite acceptable to use alternative methods of compliance provided they fully satisfy the regulations’

On Pentre Ifan

Some observations about Dolmen

This evening I watched a documentary presented by Alice Roberts. I don’t think I have watched a programme presented by Dr Roberts that hasn’t been interesting, and this one was no different, except that the most interesting thing wasn’t strictly the topic. The documentary was about the bluestones of stonehenge, but it strayed ever so slightly to Pentre Ifan in Wales were I was introduced to Dolmens. These are Neolithic structures, which predate Stonehenge. There are lots of them around the world and some are thought to be around 7,000 years old. The Dolmen at Pentre Ifan is a spectacular example, whose massive capstone appears to float above the tips of three standing stones. I expect that’s why I noticed it and why Alice Roberts chose that example.

 

Archaeologists believe that Dolmens were ancestral tombs, because scattered human remains are often found between the standing stones. Some also believe that they were originally buried beneath a mound of earth or smaller stones.

Interesting as these conjectures are I can’t get past the graceful form of the structure. I suspect that archaeologists, for the most part, take for granted that the monument stands, while going to great lengths to try and understand how it got there and how it was built. I think I understand why this is, but there is a certain irony that for such an old object so much more effort is expended on the temporal than the permanent [1].

To me the fact that Pentre Ifan is standing at all is more interesting and is the subject of this post. Maybe on another occasion I’ll fall in to line with everyone else and have a go at speculating how such structures are built, but not today.

I should of course pause to note that I have never seen a Dolmen in person, nor do I really know anything about them, other than tonight’s brief introduction. I am keen to remedy that and will aim to visit some examples when the covid-19 lock-down has ended, however that isn’t going to stop me from donning my engineering hat and doing some speculation of my own. I shall be doing so based on a few images I have pulled off the internet; what could possibly go wrong?

I am going to start with the assumption that the stones are igneous rocks, because they have that appearance and because there are outcrops in the the relevant part of Wales. This ought to make the rock relatively strong; though like any rock it will be brittle and weak in tension.

There is evidence of horizontal fractures in the capping stone and equivalent vertical features in the standing stones. Structurally this is not a terribly efficient arrangement. A stronger arrangement would be to align planes of weakness in the standing stones horizontally so that they are squashed together. For the capping stone it would have been better to align them vertically so as to avoid separation due to shear flow generated by bending forces.

I suspect that this was not done, because creating the great slabs of stone required cleaving them from the parent rock by exploiting the noted weaknesses. Without them stone-age workmen would have had difficulty creating such slabs with primitive tools.

The next thing I notice is that the capping stone appears to be fatter at one end than the other and that the soffit appears to have been cleaved as it progresses towards the thin end. I suspect that it was not originally so.

The fat end is supported by two standing stones, while the thin end is carried by one. In the short axis the fat end of the capping stone can bridge laterally between two close supports. It can possibly do this in direct shear and without inducing bending.

Conversely, in the longitudinal direction the capping stone must span almost 5 meters between the single and double support. This almost certainly results in it experiencing bending. As has been seen in prior posts bending causes the top surface of a beam to experience compression and the soffit to experience tension. 

In order for this to happen a beam will also experience a laminar shear flow in the horizontal direction. This can be understood by imagining a beam divided into horizontal slices. For tension to be experienced on the soffit while compression is experienced on top it is necessary for the imaginary slices to slip past each other.

The consequence of these actions appears to be evident in the structure. Since stone does not deal well with tension it is inevitable that small cracks must have developed in the soffit. There would also have been lateral movement along the rock’s natural horizontal weaknesses. It is conceivable that together these effects led to the soffit spalling, however it is more likely that they were abetted by other effects too.

 

Rainwater will have soaked through the top surface of the slab and migrated under gravity to the soffit. While moisture would evaporate quickly from the top the soffit would remain in the shade helping to keep the stone damp and wet. Persistent dampness will have weakened the rock structure and freeze thaw action would have exploited the many small cracks and natural weaknesses. Eventually the fractured rock would spall until it arrived at a horizontal plane of weakness whereupon the process would start again.

Perhaps another aggravating factor would be expansion and contraction due to the cycle of heating and cooling. Since only the top surface is exposed to the sun there is likely to be a thermal gradient in the capping stone as it warms. During the day the top of the stone would expand relative to the soffit and thereby start to close some of the soffit cracks. Conversely, during the night it would start to contract and thereby re-open the soffit cracks. Thus, by repetition the soffit would slowly be fatigued and further cracks induced.

When considering thermal effects it is worth noting that having only three small points of contact between the capping and standing stones is probably beneficial, because the soffit is free to articulate. More severe cracking would be much more likely if the top surface were free to expand and contract while the soffit was held in place by more restrictive contact. 

It would therefore seem that there is a good explanation as to why the capping stone is fatter at one end than the other, and all else being equal, by what mechanism it will eventually fail.

The standing stones appear to carry the capping stone effortlessly. The fact that they do so with such small points of contact would suggest the compressive strength of the stone must be relatively high. That said, it is assumed that the contact surfaces could not have been prepared to a modern standard and therefore the distribution of load will not be entirely even. This will have allowed load concentrations to be formed which may over time pry and fracture the rock locally.

This is important because compression in the standing stones will cause lateral bursting forces to develop due to passion’s ratio [this is another concept we have seen before]. Within the body of the stone compressive stress is generally low and therefore the bursting forces have little effect, however where load is concentrated stresses are higher. If such stresses coincide with a vertical plane of weakness it could encourage the bursting forces to form a split in the rock. As with the capping stones this could be exacerbated, either by moisture penetration, or by bending induced by one side of the rock being warmed faster than the other. There does seem to be some evidence of such processes at work on the surface of the standing stones.

Another facet of the pictured Dolmen is how it maintains lateral stability. It is self evident that there is no rotational resistance at the junctions between the capping stone and its supports, therefore we must conclude that the standing stones must cantilever from ground level. Since the capping stone bears heavily upon them there will be sufficient friction generated to share lateral load between the standing stones according to their stiffness.

Lateral loads would come from the wind and to a lesser extent thermal effects. There also appears to be a slight incline to the capping stone, which would imply there is a resultant lateral load, due to the stone’s self-weight, to be resisted.

Something else that is structurally relevant, though I am unsure whether it was intended, is the orientation of the standing stones. The two stones at one end are orientated perpendicular to the single stone at the other. Strength being proportional to the cube of depth this arrangement presents the full depth of at least one stone in each orthogonal direction, thus maximising cantilever action in both.

It is also interesting that the end supported by two stones is aligned with the noted incline to the caping stone, thus maximising resistance to the permanent lateral load. The possibility that this was intended is intriguing.

A different explanation would be that since one end of the structure has two supports, and the other just one, there could have been differential settlement. Assuming this were the case the narrow supports would again have been beneficial, because they would have allowed the capping stone to rotate and find a new point of equilibrium. Alternatively, more substantial supports would have likely led to fracture. I have no idea what the bearing strata beneath the standing stones is like, but in the absence of further evidence the mechanism for differential settlement seems plausible.

Of course it is also possible that the fractured soffit could have contributed to creating the observed incline too.

While the depth of embedment of the standing stones is not clear from viewing the surface it seems reasonable to assume that it must be substantial to ensure there is sufficient passive resistance to prevent overturning or sliding of the stones. In a uniform soil it would also be reasonable to assume that bearing pressure would increase with depth.

On paper it is perhaps possible that the stones could be mounted near the ground surface with stability being maintained by their shear size and mass. In the real world this does not seem plausible as the surface of the ground is prone to become waterlogged, there is also the possibility of frost action. Either of these effects could be sufficient to topple the stones.

Two further practical matters exist. Firstly, a shallow footing would be vulnerable to digging near the base when bones were to be buried. 

Secondly, and perhaps more importantly, the processes of standing large stones on end without a crane, or other modern equipment, would seem to make it necessary to tip them into a hole. If said hole were then packed tight with backfill it would lock the stones in place allowing them to behave as cantilevers.

Here I run the risk of getting into the question of how the stones were erected and I said I wasn’t going to do that. I best stop here.

[1] I base this thought on working with a few archaeologists and the number of documentaries there are about erecting Stonehenge, rather than any proper search of the archaeological literature.

On Pendentives

A tale of two domes

Perhaps two of the most important domes from antiquity are those belonging to the Pantheon in Rome and the Hagia Sofia of Istanbul. Although the latter is to be found in modern day Turkey, it is of course from the era of Byzantine rule in Constantinople and is therefore of Roman origin.

From their outward appearance the two domes would appear to be similar, indeed how different can the construction of a dome be? Further inquiry would reveal that the materials used to construct the two domes are different. The Pantheon is made of unreinforced concrete, while the Hagia Sofia’s dome is of masonry. Indeed the Pantheon remains the world’s largest unreinforced concrete dome to this day.

This difference is interesting, but in my view is not fundamental to the way in which these two revolutionary structures work. I am perfectly aware of the Pantheon’s oculus, its coffered soffit, the variation in concrete mix over its height, all of which were designed to save weight. These are important details, which are both interesting and worthy of study, maybe I will write about them in some future point, however they are not directly relevant to the subject of this post.

The first thing to note is that classical domes, like later gothic structures, are what I would describe as gravitational or compressive equilibrium structures. That is to say that their structural adequacy is dependent on their shape and not on materials science. This is possible because actual stresses are compressive and sufficiently low that, providing equilibrium is maintained, material strength is unimportant. This makes sense because materials science, at least in the modern sense of stresses and strains, did not exist when they were built. 

Those familiar with the structures in question will no doubt be aware of known cracking in both domes, which might be taken to suggest that there is in fact some material science going on, however as we shall see this is not the case.

To understand the primary difference between the Pantheon and Hagia Sofia, perhaps it is first necessary to explain how a generic dome works. In section domes behave in a similar manner to arches, because their curved profile exerts both vertical and lateral thrust at the seating[1]. Domes are of course unlike arches in the sense that they are 3D structures. This means that the aforementioned vertical thrusts are expressed as compressive meridional stresses extending from the crown of the dome to it’s base. The lateral thrusts push outwards in all directions generating a circumferential or hoop stress that cause domes to spread. It is the way in which the meridional and circumferential stresses are resisted that makes the difference.

Like barrel vaulted structures from the classical period of history the Pantheon is supported on heavy walls that follow the profile of the roof structure, in this case a cylinder, in order to buttress the roof against spreading. Some descriptions I have read speculate a stepped thickening observed at the dome’s base is designed to provide a circumferential tie. Maybe their authors have done more research than me and have data to support this view, however I am disinclined to adopt it based solely on my own intuition that the tensile capacity of concrete, albeit Roman concrete, is too low. Also, if tension were present it would imply materials science is at work to provide the required equilibrium, which is philosophically less satisfying.

It occurs to me that a more elegant solution, which maintains the idea of gravitational equilibrium, would be one where the purpose of the steps was to increase weight at the head of the supporting wall in order to push the dome’s thrust line back into the supporting walls. In essence it would behave, at least in my estimation, like the pinnacle atop a flying buttress.

The dome of the Hagia Sofia is different. It does not find support from heavy buttress walls. Rather it straddles the four corners of a vast open space into which light and air may flood. In character it is a medieval structure whose load paths concentrate the dome’s weight into a carefully defined masonry skeleton. The invention that makes this possible are the inverted triangular masonry panels, known as pendentives, that are located over the four supporting piers. As they spread outwards from their apex a series of four arches are formed on the dome’s perimeter. Together these elements funnel load into the supporting piers where the in plane arch thrusts are buttressed.

It is evident however that there remains unbalanced thrusts perpendicular to the apex of each pendentive arch. Equilibrium is restored by hemispherical domes that lean in the opposite direction to the dome’s thrust in one direction and buttresses in the other.

And so it is that the dome at Hagia Sofia represents a transition between heavy buttress walls and gothic cathedrals of the later medieval period, whose structures had a more clearly defined order of primary and secondary elements and a more sophisticated understanding of load paths.

Now, to the aforementioned cracks in both domes. Many seasoned observers hold the view that these are the result of past seismic events and differential settlements. Indeed the original dome at Hagia Sofia is known to have collapsed during an earthquake leading to the present cupola being constructed with a higher profile in order to reduce the magnitude of lateral thrusts. 

Nevertheless it would appear that in spite of movement to both structures gravitational equilibrium has been restored. They remain stable, or at the very least, are moving very slowly.

[1] For further information I have written several prior posts relating to the behaviour arch structures.

On Ice Walking

Just how safe is it?

Today at work I participated in several online meetings, however while we were waiting for colleagues to tune for one of them we found ourselves doing the stereotypically British thing; we talked about the weather.

To be fair the weather was unusual as much of the country, even in the south, had been covered with a blanket of snow. Indeed it had been reported that a portion of the Thames had frozen, which really was an unusual event. This was the predicate for a discussion about whether it would be safe to walk on said ice [you now know this wasn’t written in the summer].

This intrigued me so I set aside a little time that afternoon to try and work out just how safe it would be. This post is about what I found. Now, I should preface what follows by saying that I have absolutely no practical experience of this subject. Everything that follows is a postulation on my part, based on the engineering principles that I believe to be at work. You therefore shouldn’t base your ice fishing trip or curling match on what I have come up with.  

If you must go walking on a frozen lake, I suggest you ask someone who knows what they are talking about and isn’t making it up as they go. Statistically I imagine that such a person is far more likely to have a Canadian accent than a British one. You have been warned!

Ice floats because it is roughly 10% lighter than water. If we are to stand on a sheet of ice we therefore increase its weight making it more likely the ice will sink rather than float. It therefore struck me that the first part of the puzzle ought to be an assessment of how much ice there needs to be to ensure floatation occurs rather than sinking.

A cube of water with sides 1 m long weighs approximately 1000 kg, which means the same cube made of ice must weigh roughly 900 kg. This would imply that for our cube of ice to remain buoyant in water it must carry no more than 100 kg [1000-900].

Now, the minimum recommended thickness of ice for walking is 4 inches; I know because I googled it. That’s equivalent to 100 mm or 0.1 m. 

This means that a block of ice with a square surface that has sides 1 m long, but a thickness restricted to 0.1 m, can carry 10 kg [0.1 x 100]. It follows that if an average person weight 75 kg, then their weight must be spread over a block of ice with an area of 7.5 meters square [75/10] i.e. a square with sides measuring 2.739 m[1].

Everyone knows that ice is a brittle material that fractures rather than bends so that’s quite a large area for load to spread from our feet without something going wrong. I therefore started to think about how the load gets from our feet to cover such a large area and by what mechanisms it could go wrong.

Perhaps our feet would simply punch through the ice like a stiletto heel on soft ground. Such a mechanism would require the ice to shear on a vertical plane passing through the ice, which extends along a perimeter enclosing our feet. The length of the perimeter and the depth of the ice would therefore define the shear plane.

Punching shear yields a stress in the ice equivalent to 0.001 kg[2] acting on every square millimetre of ice on that plane.

The second potential mechanism could result from a shear plane developing across the full width of our notional 7.5 m square block of ice. This time the shear plane being defined by the width and depth of ice. This mechanism yields a stress of 0.0003.

The final mechanism I considered was bending. To transfer load from the feet to the outer edges of our square block the ice must be capable of bending. This is a bit more tricky to calculate, but it results in a stress equivalent to 0.005 kg acting over every square millimetre of ice.

I didn’t really know if any of these figures were significant or not so I got back on google. It turns out that, according to the people who measure such things, ice has a shear strength of 0.06 and a bending strength of 0.07. 

This means that there is a factor of safety against shear failure of 60 [0.06/0.0001] and a factor of 14 against bending failure [0.07/0.005]. I find this quite reassuring, because I don’t actually believe that ice has the strengths I just quoted.

This is not because the diligence of the researchers is at fault; I’m not saying that their work is wrong. What I am saying is that ice is not a manufactured product like steel. It is not made to possess specified properties.

The strength of ice depends on many things. What is the air temperature, did it rise and then fall again during its formation. Is the water fresh or salty? Is there a current or a flow in the body of water that is busy scouring the underside and weakening its structure. Was there snowfall during its formation. Did someone, or something, step on the ice while it was forming thus inducing cracks in its interior.

These and many other issues that I likely haven’t thought of have the potential to change the strength of a given block of ice; its value is therefore not a fixed thing[3]. If I am to walk on ice I would quite like to know that the existence of one or more adverse factors does not completely undermine the strength of the ice I am going to rely on. A factor of 14 sounds good to me. It’s roughly 10 times what I might use for say concrete. That’s about right because I know with far greater certainty what concrete will do.

Obviously if ice melts we are in trouble, but before that point is reached, I have no idea which combination of ice factors might eventually undermine a safety factor of 14. That’s why you shouldn’t take advice about ice walking from someone that’s making it up as he goes.

That being said, my somewhat crude assessment has yielded an interesting conclusion. I started by trying to work out what amount of ice I would require to mobilise to prevent the average person from sinking. My sums therefore exist on just the right side of not sinking, effectively a factor of safety of 1. This assumption eventually yielded a factor of 14 against the ice breaking i.e. in this scenario I am actually more likely to sink than the ice is to break!

[1] I know it would be more realistic to assume a circular perimeter, but I used a square to keep the sums simple. You’ll get over it.

[2] I know that working out stresses in kg is a bit weird, but unless you have a technical background you won’t know what N/mm2 is, or MPa for my European friends, or psi for my American friends. I didn’t want this post to be an explanation of units.

[3] Ice researchers know the strength of ice doesn’t have a fixed value. They provide ranges of values and couch them in temperature limitations and so forth. I picked from the lower end of the scale.