Appendix 2.Error analysis forthe approximation of uniformly distributed load
A linear elastic cantilever model is used to determinethe maximum deflections of the cantilever subjected to discrete point and uniformly distributed loads, where the total loads for these two cases are equal.The maximum deflectionsof acantilever subjected to discrete loadscan be determined using,[1]
where is the total load acting on the cantilever, Ncis number of point loads, L is the length of the cantilever, is the distance between a point load and the fixed end, E and I are the Young’s modulusand second moment of area of the cross-section, respectively.
For the case that the spacing between adjacent points loads is equal, i.e. , the discrepancy between the case of discrete loads (a finite value of Nc) and the case of uniformly distributed load () is shown in the second column in Table A1.As can be seen from Table A1, the discrepancy decreases with the number of loading points, and the largest discrepancy is ~ 11.2%.
To consider the variation in the location of the point loads, the standard derivation of spacing between adjacent loading points is a critical parameter. As shown in Figure 6 in the revised manuscript, the spacing between adjacent contacts asperities is not subjected to large variation for all the substrate. A detailed calculation found that the largest standard derivation of value is the spacing between adjacent contacts asperities is 7.8% of the total length of the NW, on substrate No.4.
Assuming the spacing between adjacent loading points is subjected to an uncertainty of 7.8% of L, i.e. , where is a pseudo-random number between -1 to 1, the standard derivation of maximum cantilever deflection is calculated and summarized in Table A1, using the Monte Carlo simulation through 10 repetitions. As seen in the third column in Table A1, the standard derivation of maximum cantilever deflection subjected to the uncertainty of the location of loading points is also insignificant with the largest value being 11.3 %.
Table A1. Number of discrete loading pointand the discrepancy in the deflection obtained from the discrete model and the uniformly distributed model, the second columnlists the results for the cases where the spacing between adjacent loading pointsis equal, and the third column lists the resultsfor the cases where the spacing between adjacent loading points is subjected to a random uncertainty of 7.8% of L
Nc / Discrepancy (%) / Standard derivation in h2 / -11.2 / 11.3%
3 / -8.8 / 9.8%
5 / -5.6 / 5.7%
7 / -4.2 / 5.4%
23 / -1.44 / 3.0%
References
[1] Hibbeler, R.C., 2013. Mechanics of Materials, 9th SI Edition, Pearson Prentice Hall, Singapore.