## Hypotheses (as of Oct. 2013)

**Geometric Parameters**Hypotheses of the effects of varying geometric parameters on buckling behaviour have come from an analogy with Euler buckling theory for columns.

- Thickness - Euler analogy represents second moment of area, I, and an increase is expected to increase buckling load
- Roof angle - will change buckling effective lengths, but will also change the direction and length over which the external pressure is applied. Results expected to prove interesting.
- Radius - will likewise increase buckling effective lengths and the length over which the external pressure is applied, greatly reducing buckling strength.

The effect of these geometric parameters on pre-buckling behaviour, for example on stress resultants and on strains is not of primary interest in this research.

*Boundary Conditions (Ring Area)*

- In this research, it is assumed that the vertical restraint is infinite, and of no interest to the behaviour of the roof. This is a reasonable assumption as, under a uniform external pressure, vertical displacement would represent a uniform displacement of the whole roof, resulting in no change in internal structural responses.

It is expected that for a small increment in ring area, a given structural response will change in value from nearer its roller bounded, no ring condition value, to nearer its pin bounded, infinite ring condition. When plotted on a graph an s-curve is expected.

Among the structural responses, it is hypothesised with some confidence that the buckling load will increase as the roof becomes more radially restrained, resulting in a curve similar in appearance to the graph.

It should be noted that a radially unrestrained roof will not have structural responses of trivial values (for example infinite strains, or 0 buckling loads) because as the cone edge expands it goes into circumferential tension, thus forming its own ring.

The phrase structural response is left deliberately vague to illustrate the main hypothesis, the continuum of behaviour. Thus the above graph will be misleading when considering some structural responses, for example meridional strain, which is expected to decrease with radial restraint, resulting in a negative s curve. Effects on pre-buckling peak membrane stress resultants and in particular, buckling modes have only highly speculative hypotheses, and will prove interesting (at least to the author).

Among the structural responses, it is hypothesised with some confidence that the buckling load will increase as the roof becomes more radially restrained, resulting in a curve similar in appearance to the graph.

It should be noted that a radially unrestrained roof will not have structural responses of trivial values (for example infinite strains, or 0 buckling loads) because as the cone edge expands it goes into circumferential tension, thus forming its own ring.

The phrase structural response is left deliberately vague to illustrate the main hypothesis, the continuum of behaviour. Thus the above graph will be misleading when considering some structural responses, for example meridional strain, which is expected to decrease with radial restraint, resulting in a negative s curve. Effects on pre-buckling peak membrane stress resultants and in particular, buckling modes have only highly speculative hypotheses, and will prove interesting (at least to the author).

*Buckling Mode*Since buckling mode has to remain a natural number, while all variables that determine buckling modes are real numbers, some complicated, and hard to articulate behaviour is expected. For example, take a roof geometry for which buckling load and mode is known, change a variable such that the buckling load increases but the buckling mode does not, change the variable even more and the buckling load is "relieved" and a new buckling mode is adopted.

This hypothesis comes from an analogy to plate buckling behaviour, and how it varies with plate dimensions. For a given plate thickness and breadth in this analogy, a buckle will have a "favoured" wavelength equal to the breadth along the length of the plate. Then on the graph below, the x-axis would represent plate length.

In the conical roof, thickness would be its own analogy, while radius and slope would affect both length and breadth, with boundary conditions having an unknown, therefore particularly interesting effect. At higher buckling modes (example n = 50) this effect may be negligible, but in case it isn't, it will be remembered when analysing results.

This hypothesis comes from an analogy to plate buckling behaviour, and how it varies with plate dimensions. For a given plate thickness and breadth in this analogy, a buckle will have a "favoured" wavelength equal to the breadth along the length of the plate. Then on the graph below, the x-axis would represent plate length.

In the conical roof, thickness would be its own analogy, while radius and slope would affect both length and breadth, with boundary conditions having an unknown, therefore particularly interesting effect. At higher buckling modes (example n = 50) this effect may be negligible, but in case it isn't, it will be remembered when analysing results.