Please refer to the attached image.

We know that if an arch is X unit wide then the supporting walls should not be less than X/4 unit width for the structural stability.

  1. I would like to understand how to calculate the maximum possible height of the the supporting walls i.e. Y in terms of X? Is there any relation?
  • If there is no relation then why did gothic architecture relied on pointed arches instead of narrow high opening with a semicircular arch at the top of it, was it just for aesthetics or is there a notion of structural stability?
  1. And to calculate the thickness of the arch (Z) in terms of X (I guess there is a relation between the span and the thickness of a arch).

enter image description here

  • $\begingroup$ Have you found any references for arch construction? $\endgroup$
    – Solar Mike
    Apr 15 at 5:28
  • $\begingroup$ Not a concise one but in bits and pieces in YouTube. $\endgroup$
    – Sarbbottam
    Apr 16 at 6:05
  • $\begingroup$ You reference Gothic - how did they define the ratios? Arches must have a lot of documentation prior to youtube since they go back more than 1000 years... $\endgroup$
    – Solar Mike
    Apr 16 at 6:49
  • $\begingroup$ Do you have any reference? $\endgroup$
    – Sarbbottam
    Apr 16 at 8:25
  • $\begingroup$ A simple Google search for "arch calculations" gives as the first result: structx.com/arches.html with formulae and explanations. Should sort you out ... $\endgroup$
    – Solar Mike
    Apr 16 at 8:29

1 Answer 1


Hold a uniform chain from both ends. Let it sag. That is catenary shape.

The ideal shape of an arc is an inverted catenary shape. That is the ideal. Which means, if you built an arch in that shape, it would just stand by itself by means of internal compression without needing a lateral force to hold it from sides (such as walls on its side). But as soon as you want to increase the freespan ( which is a legitimate need, as people always wanted to pass longer freespans with those historical bridges), then, you deviated from catenary shape and now you will need walls or some form of braces on both sides to prevent the arch from collapsing.

So, in your question, the height of those columns below your "arch" is irrelevant i think. But your arch would fail, if there is no lateral brace because your arch is a semi circle, which significantly deviates from inverted catenary shape. ( although one intuitively thinks at first that semi circle is the best but it is not - remember sagging chain did not naturally sag as semicircle)

As far as thickness, it should be such that, when your arch deviated slightly frkm inverted catenary, if it had enough thickness you might still be able to fit catenary shape into your arch which is slightly off. But a semicircle is way off. Therefore no thickness value would save it to stand on its own. Being a semicircle, your arch needs lateral support. In which case the thickness can vary to any value depending on lateral support force available.

  • $\begingroup$ are you sure about : "without needing a lateral force to hold it from sides"? $\endgroup$
    – user721108
    Apr 18 at 18:13

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