The mathematical study of many dynamic responses of building materials and constructions to climatic variations has been hampered by the lack of a suitable way to model climatic components such as temperature, precipitation, humidity, wind, solar radiation, and cloud cover. It was noticed by the author that a curve called the “companion to the cycloid” seems to be similar in shape to ambient temperature versus time plots. This was the start of the work reported in this paper because, if a successful model of the temperature cycles could be developed, we would have a powerful tool to help us in our understanding of the thermal behavior of materials and systems.
This paper presents methods for the mathematical modeling of the ambient air temperature at any location where statistical atmospheric data are available. The model was checked by comparing the calculated air temperatures with the actual air temperature data at six diverse locations in the continental United States. In addition, the actual hourly air temperatures (recorded at three-hour intervals) were compared with the calculated hourly air temperatures from the model for a six-year period at one location.
The model was expanded, with data from the literature, to model the temperature of white, gray, and black roof surfaces. These calculated surface temperatures are used to illustrate the utility of the model in estimating the sliding potential of several bituminous roofs.
The usefulness of the minimum temperature and the temperature loading parts of the model has been demonstrated by the work reported in this paper. The validity of the surface temperature part of the work is yet to be established.
Finding the proper curve shape for the thermal model opens up many lines of investigation for studying building materials as dynamic components. Other studies currently being performed include the migration of bitumen through the felts within the roofing membrane and the displacement of one-ply systems in response to the normal thermal cycles. Future studies are not limited to roofing; the model can be applied wherever the effects of temperature cycles are important.
The preliminary look at the sliding potential of bituminous built-up roofing membranes reported in this paper may be the start of a study to provide rational limits for the maximum slope that should be permitted for each class of bitumen, instead of using timeworn standards that may not be appropriate.