On Grass & Buckling

Why you shouldn’t walk on a frosty lawn

An interesting thought that hadn’t struck me before is why it is possible to play sport on grass without it being ruined. I am not suggesting that sports pitches don’t suffer from wear and tear; self evidently they do. After two weeks of tennis in the summer the courts at Wimbledon are not in the same condition they were at the start of the tournament. Similarly, after a season of football or rugby pitches around the country need time to recover, albeit pitches today fair better than they did in the past, but that’s not really what I mean. How is it that sport can be played on grass at all? 

It takes no effort at all to pluck a blade of grass and almost none to tear or cut it, how then can we walk on grass without damaging it let alone run and jump? I think the answer is to be found in a structural principle known as Euler[1] or Strut buckling. Euler buckling is a special form of compression failure, which applies to slender structures and is named after Leonhard Euler who sorted out the mathematics. Slender structures are those, which are thin relative to their height.

When a squat structure is subjected to compression it will fracture and split if it is made of a brittle material. Alternatively, if it is made from a ductile material it will bulge and deform, however a slender structure will buckle before any of these states are reached.

Buckling is essentially the point at which a structure subjected to compression gives way due to a rapid increase in lateral deflection. The reason deflection increases rapidly is because the onset of buckling instigates a feedback loop.

When buckling begins the structure is displaced laterally, which causes the compressive load it carries to be applied eccentric to the line of support. As we have learned in earlier posts an eccentric force generates a bending moment, which causes increased lateral displacement. Thus the feedback loop is set in motion.

Euler’s work teaches us that the load at which buckling commences is directly proportional to the stiffness of the structure and inversely proportional to the square of its length. This tells us two important things:

Firstly, doubling the stiffness of a structure will double the buckling load, whereas doubling its length will reduce the buckling load by three quarters. The length of the structure is therefore the most significant factor.

Secondly, providing the material is ductile and will therefore remain elastic, its strength plays no role in Euler buckling. This means that once the compressive load has been removed it will recover and return to its original shape. For those who are interested elasticity is covered in a prior post, On Ductility.

These are the properties which make walking on grass possible. A blade of grass has a low material stiffness and is tall relative to its thickness i.e. it is slender. For this reason when you step on grass the blades simply buckle elastically and then recover when the foot is lifted. It is also worth noting that grass grows from the root rather than the tip so any damage that does occur can be repaired from below the damaged part.

It is often said that you should not walk on a frosty lawn. In light of our buckling logic the reason for this becomes clear. Grass, being a plant, is made or cells which contain water. I am sure biologists would put it better than this, but I think that’s pretty much the case. When a frost sets in water contained in the cells will freeze and make the normally flexible blades of grass brittle. It follows that if you were to step on a lawn in this condition the frozen blades would no longer be elastic and instead of buckling they will fracture. Thus permanent damage is done.

So that’s my answer; sport can be played on grass, because each blade is a small slender structure that behaves according to the rules of Euler Buckling. I suspect that the normal wear and tear that we see on sports pitches is primarily due the lateral movement of feet, which would presumably apply horizontal shears that tear and rip at the buckled blades. That said, this cannot be the dominant behaviour or pitches would become rather more damaged than they actually are.

[1] Pronounced Oiler