Stress-driven nonlocal model on snapping of doubly hinged shallow arches

Abstract

The snap-through analysis of shallow doubly hinged sinusoidal nano-arches subjected to sinusoidal load, considering the nonlocal effects, by using the stress-driven nonlocal model is concerned. The problem is formulated within the framework of Bernoulli–Euler beam theory including geometrical nonlinearity. Numerical simulations are made to investigate the effects of the geometry of the arch, and the nonlocal parameter on the buckling load and on the buckling deflection. Insights and conclusions regarding the effects of various stages of deformation on the stress resultants, and on the buckling including prebuckling and postbuckling are presented. Variations of the stress resultants along the arch are shown.

Keywords:

• Stress-driven
• nano-arch
• Bernoulli–Euler
• buckling
• beam
• imperfect

Notes

1 Since the geometric curvature of a straight beam is zero, the presented forms of Eqs. (1) and (4) are the simplified forms of those given in [45].

2 The negative signs seen at the right-hand sides of Eqs. (1) and (4) are due to the differences in the sign conventions of N and M, defined in the next section of the current study, with those in [43, 45].

3 V is equal to the summation of the vertical components of the normal force N and the shear force Q, known to be equal to ${{\rm M}}^{\prime }$ in the elementary theory [88].

4 The constitutive boundary conditions according to the strain nonlocal model [43] can be checked to be inconsistent with the third and sixth of the support conditions (Eq. ${14}_{3,6}$) of the current problem.

5 Since the concerning arches are shallow and their geometric curvatures are negligibly small, Eq. (18) coincide with Eqs. (1) and (4) which are written for the straight beams.