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At the nano-scale, loads acting on a nanobeam and its material properties are likely to be not known precisely, i.e., uncertain. In the present paper, the deflection of a nanobeam subject to load and material uncertainties is studied by convex modeling of the uncertainties. The level of uncertainty is taken to be bounded and the maximum deflection corresponding to the worst-case of loading or material properties is obtained, that is, the uncertainties are determined so as to maximize the deflection. The sensitivity of the deflection to the uncertainty in the material properties is also investigated. Numerical results are given relating the level of uncertainty to maximum deflection.

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Quite often the values of the elastic constants of composite materials can be estimated with some error due to manufacturing imperfections, defects and misalignments. This introduces some level of uncertainty in the computation of the buckling loads, frequencies, etc. In the present study an ellipsoidal convex model is employed to study the buckling of long cross-ply cylinders subject to external pressure with the material properties displaying uncertain-but-bounded variations around their nominal values. This approach determines the lowest buckling pressure and as such provides a conservative answer. Method of Lagrange multipliers is applied to compute the worst-case variations of the elastic constants and an explicit expression is obtained for the critical buckling pressure for a given level of uncertainty. Expressions for the relative sensitivities of the buckling pressure to uncertain elastic constants are derived.

At the nano scale, the effect of surface stress becomes prominent as well as the so-called small scale effect. Furthermore due to difficulties in making accurate measurements at nano-scale as well as due to due to molecular defects and manufacturing tolerances, there exists a certain degree of uncertainty in the determination of the material properties of nano structures This, in turn, introduces some degree of uncertainty in the computation of the mechanical response of the nano-scale components. In the present study a convex model is employed to take surface tension, small scale parameter and the elastic constants as uncertain-butbounded quantities in the buckling analysis of rectangular nanoplates. The objective is to determine the lowest buckling load for a given level of uncertainty to obtain a conservative estimate by taking the worst-case variations of material properties. Moreover the sensitivity of the buckling load to material uncertainties is also investigated.