A Statistical Approach to Biological Factors of Safety: Bending and Shearing inPsilotumAxes

Abstract

A method is presented to calculate the factor of safety (based on any criterion of failure) for populations of equivalent biological structures that normally sustain mechanical loads (e.g. stems of the same order of branching). The method uses the empirically determined mean and standard deviation of the actual loads and the load capabilities provided that both types of loads are normally distributed in the population. The factor of safetySis calculated from the formulaS⩾1+α(sd2ω+sd2η)1/2/Aβfχ L, where α and β are determined from regression analyses,sddenotes standard deviation,ωis load capability (the maximum load a structure can sustain before it mechanically fails), χ is the actual load (the load the structure normally sustains), χL is the mean actual load determined for the population, andAfis the area under the frequency distribution curve for whichω-η<0. The method was used to computeSfor the axes drawn from different orders of branching in the aerial trusses ofPsilotum nudum, a taxonomically problematic pteridophyte. Failure in bending and failure in twisting were used as two criteria of mechanical failure. The actual loads, load capabilities, and the difference between the two loads measured in bending and twisting for representative axes in each branch level were normally distributed. Based on the area of overlap between the frequency distribution curves of the two kinds of loads, the standard deviations of the loads, and the mean actual loads for the axes drawn from each level of branching, the factor of safety in bending and twisting was found to decrease in a basipetal direction: distal axes had the highest factors of safety, while the most proximal axes in trusses had the lowest factors of safety. The data indicated that axes from all levels of branching had factors of safety greater than unity, and thus all axes were mechanically reliable load-bearing members. The limitations and benefits of the method for computingSare discussed in the specific context ofP. nudumand for the vertical stems of vascular plants in general. The principal limitation of the method is the assumption that actual loads and load capabilities have normal frequency distributions; the main benefit of the method is that factors of safety are numerically indexed in an objective way based on the distribution of mechanical properties of a population of otherwise similar organic structures.