A multi-spring model for buckling analysis of cracked Timoshenko nanobeams based on modified couple stress theory

This paper develops a cracked nanobeam model and presents buckling analysis of this de-
veloped model based on a modified couple stress theory. The Timoshenko beam theory and
simply supported boundary conditions are considered. This nonclassical model contains a
material length scale parameter and can interpret the size effect. The cracked nanobeam is
modeled as two segments connected by two equivalent springs (longitudinal and rotational).
This model promotes discontinuity in rotation of the beam and additionally considers di-
scontinuity in longitudinal displacement due to presence of the crack. Therefore, this multi-
-spring model can consider coupled effects between the axial force and bending moment
at the cracked section. The generalized differential quadrature (GDQ) method is employed
to discretize the governing differential equations, boundary and continuity conditions. The
influences of crack location, crack severity, material length scale parameter and flexibility
constants of the presented spring model on the critical buckling load are studied.