Structure of the greenhouse

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The Greenhouse as a Pressure Vessel

Modelling the greenhouse as a thin-walled cylindrical pressure vessel, capped by hemispherical ends, the axial and longitudinal stresses in the greenhouse wall are approximately:

$\sigma_{Long} = {P r \over t}$

and

$\sigma_{Axial} = {P r \over 2 t}$

where:

$\sigma_{Long}$ is the longitudinal stress in the cylindrical section of the greenhouse wall
$\sigma_{Axial}$ is the axial stress in the cylindrical section of the greenhouse wall and also the stress in the hemispherical sections
P is the internal pressure
r is the radius (assuming the greenhouse is symmetric)
t is the wall thickness

Let
r = 2 m
P = 50000 Pa
$\sigma_{allow} = 8.3 MPa$

Where $\sigma_{allow}$ is the allowed stress in an unreinforced polycarbonate plastic sheet with a safety factor of 6. The stresses in the wall must be less than this value, or it will eventually fail under normal wear and tear. If the wall stresses are greater than 6 times this value, there is an unacceptable danger than the pressure vessel will fail outright.

Using these figures, the estimated wall thickness is t = 12 mm for a structure of this size. Using a perfectly spherical design instead of a cylindrical one can reduce this to t = 6 mm, but that is still quite thick – too thick for a single layer inflatable structure using this material. This thickness can be reduced by other means, though.

The two simplest ways to reduce the necessary greenhouse wall thickness are to reduce the internal pressure and/or reduce the radius of the structure. This would reduce the useful volume of a sphere, but a cylindrical greenhouse can simply be extended in length to make up the difference because its wall stresses are largely independent of its length. Reducing the greenhouse radius to r = 1.5 m could reduce its wall thickness to t = 9 mm. Unfortunately, that is still relatively thick, especially for an inflatable, and reduction of radius to less than r = 1 m will make the greenhouse unsuitable for use as part of the crew habitat. The greenhouse can be designed so that only a section of its circumference is window material, allowing the rest of its structure to be built of stronger materials, but the amount of sunlight needed for crops places limits on the utility of this approach. Reduction of pressure can reduce the necessary greenhouse wall thickness in the same proportion, but recent research indicates that plants become stressed at very low pressures (P < 20000 Pa). These pressures also approach the minimum oxygen partial pressure at which the crew can survive without oxygen gear, and represent a practical lower limit to greenhouse operating pressure.

Another approach is to use a multi-walled pressure vessel. This spreads the stresses within the pressure wall, allowing the use of thinner material. In the event of a breach in any one layer, the other layers can still hold. If a double-walled pressure vessel were used, at least one layer would need to be capable of withstanding the full load, although at a reduced safety factor. If a triple-hulled design is employed, the full load can be spread between the two intact layers in the event of a breach in one layer, so that the necessary safety factor can be even smaller for each layer of the pressure wall.

Another approach is to reinforce the pressure vessel with a net or frame. This method distributes wall stress through cables and/or beams rather than through the wall sheeting, allowing thinner walls. In principle, the frame can be made as thick as necessary to absorb the load. However, an extensive framework over the greenhouse windows will block sunlight. Also, a pressure vessel reinforced by a frame no longer behaves as a thin-walled pressure vessel, which means that the approximations used above may no longer apply. Analysis of the effect of using a reinforcing frame requires the use of Finite Element Analysis to locate stress concentrations in the greenhouse pressure vessel.

Multiple layers of plastic which are thin enough to remain flexible can be employed to reinforce each other, allowing the greenhouse structure to be inflatable without introducing other reinforcements. However, this would further increase the necessary safety factor of the greenhouse pressure wall, requiring a thicker wall. This would not necessarily improve transparency or weight.

Greenhouse Windows

Because the windows of a sun-illuminated greenhouse must be made of transparent materials with a specific range of optical and thermal properties, there are only a limited number of high strength materials that they can be made of. Since the windows cannot be of arbitrary thickness but must be thin enough to pass sufficient light, this generally means that they are the limiting component of a pressurized greenhouse, and they will be the weakest structural element unless the greenhouse design makes provisions to prevent this.

To reduce stress in the window material, there should be as little window area between reinforcing elements as possible. The pressure differential across the window should also be kept as small as possible. The window material should be as strong as possible to reduce flexure and stretching. It is also desirable for it to be thin.

These are difficult requirements to fill using any one material. Fortunately, the use of multiple layers allows construction of windows which possess properties of each component material.

For example, polycarbonate plastic is one of the stronger transparent plastics available in film or sheet form, has a very low brittleness temperature, and is not only highly UV-resistant but nearly opaque at ultraviolet wavelengths, making it one of the materials desirable for use in a martian greenhouse. Unfortunately, it is also more susceptible to abrasion and outgasing over time than some materials, and – while it has some opacity at infra-red wavelengths – is not the most capable IR reflecting plastic on the market. Since a greenhouse window opening out on a dusty near-vaccum with an external temperature less than -60 degrees Celsius needs scrath resistance with an IR transmittance as low as possible, it is desirable to introduce another material with these properties.

Transparent PCTFE plastic is also available in film or sheet form, and has a low brittleness temperature. It is abrasion resistant, UV resistant, does not suffer as much outgasing as polycarbonate, and has a lower transmittance at IR wavelengths. Unfortunately, it not as strong as polycarbonate and transparent at UV wavelengths. A double layer of PCTFE + Polycarbonate would have low transmittance at both IR and UV wavelengths, and (properly arranged to have the PCTFE layer between the polycarbonate and the external environment) would resist scratching and outgasing.

Similar mixed material windows of Mylar, Tefzel or other materials may be desirable as well. Wire reinforcement should also be considered, as metal meshes can have relatively high overall light transmission and are not as susceptible to ageing and fatigue as most plastics. Wire meshes also do not require nearly as high a safety factor as unreinforced plastics, and can be used to construct significantly thinner windows.

Greenhouse Frame/Netting

Introducing a frame to support the outside of the greenhouse pressure wall can reduce stresses in the wall by transferring a substantial portion of the load to the frame instead of the pressure wall. The loads in the frame would be primarily tensile, which would allow foldable materials such as rope netting to be used as reinforcement instead of rigid bars, provided that the netting material was strong enough. The pressures involved would be great enough that the pressure wall would expand under the load, pressing itself into position on the frame or netting. The pressure wall could be treated as effectively fixed against the frame by the force of its own expansion, and the unsupported sections of pressure wall could be treated as separate window panes.

The relative rigidity of the reinforcing frame would limit expansion, reducing surface tension and overall wall stress, but would cause the windows to flex more than they would in a simple thin-walled pressure vessel. This will create stress concentrations in the window material.

File:Window Stresses.jpg
Finite Element Analysis Chart of a Sample Greenhouse Window, Showing Expected Stress Concentrations. The pane is reinforced by a steel frame rigidly supported at the corners, and the frame elements are parallel and perpendicular to the greenhouse axis.

The stress concentrations would occur close to the window’s contact point with the frame, where this flexure was greatest. For the relatively large square windows shown in the accompanying analysis results (1m x 1m), the stress is increased by a factor of 2.5 for a square window, but using smaller windows will reduce this to more manageable levels. The window can also be thickened at points to better support the extra stress, or the window material can simply be made uniformly thicker so that there will be no need to worry about exact spacing during assembly.

Alternately, use of square windows can be abandoned for use of round flanged windows with window panes which are spherical shell sections. This would reduce the overall light transmission of the greenhouse in comparison to square framed windows, but would reduce the window stress dramatically, approaching the stresses expected in uniform hemispheres.

File:Greenhouse Foundations.jpg
Examples of various foundations for the greenhouse. The lowest stress is found with the most rigid support.

Support of the frame will also have an important effect on stresses in the greenhouse wall. Rigid supports on the frame which limit free expansion under pressure can reduce loads by promoting transfer of pressure forces from the windows to the frame. However, stresses will tend to concentrate anywhere the frame is allowed to flex. Shoring with soil will reduce flexure relative to using simple rope moorings, with the most load-efficient configuration being a largely soil sheltered design.

Finite Element Analysis

File:Von Mises Stresses without End Constraints.jpg
Sample finite element analysis of cylindrical greenhouse with supporting frame but no end constraints
File:Von Mises Stresses for End Constraints.jpg
Sample finite element analysis of cylindrical greenhouse constrained in a trench

Finite Element Analysis can be used to estimate stresses in a given greenhouse design.

The examples represented here are of roughly cylindrical pressure vessels reinforced by frames and with trench foundations. Both use the same frame design and are exposed to the same internal pressure. One is allowed to expand axially, and the other is restrained axially by soil constrained within the trench. Comparison of the resulting stresses (listed in the tables alongside each image) reveals that soil sheltering may be desirable for the greenhouse, due to the resulting transfer of loads to the soil.

With the unconstrained arrangement, the greatest stresses are found in the window material (the limiting material used in the structure). With the rigidly supported design, the greatest stresses are found in the frame.

Asymmetric arrangement of the soil foundation is undesirable, though. The end-constrained analysis clearly shows a stress concentration at the upper boundary of the trench, where the unrestrained upper section of the greenhouse is flexed about its support. This suggests that both ends of the greenhouse should be completely soil sheltered for the most stress reduction, with only the windows being exposed to the open air.

These models alone do not represent a finalized design, being merely the coarsest meshes that will illustrate the system behavior. These examples show that the assumption of iniformly distributed stress does not apply to this type of structure, and more accurate representations of greenhouse structure using the finite element method of numerical integration will prove useful in finalizing an actual design.