Buckling of Planar Micro-Structured Beams
Abstract
:1. Introduction
2. Model
- clamp (C) at H: ;
- hinge (H) at H: ;
- horizontal roller (R) at H: ;
- horizontal slider (S) at H: .
- free (F) at H: ;
- hinge (H) at H: ;
- horizontal roller (R) at H: ;
- horizontal slider (S) at H: .
- hinge at A, horizontal roller at B (H-R):
- clamp at A, free at B (C-F):
- clamp at A, horizontal roller at B (C-R):
- clamp at A, horizontal slider at B (C-S):
- finding the general solution of Equation (7)-a, namely the characteristic exponents , , with f a function of P, and the associated eigenvectors, which depends on four arbitrary constants;
- enforcing the boundary conditions (7)-b,c, which provide a (homogeneous) system of linear algebraic equations in the four arbitrary constants;
- zeroing the determinant of the matrix of the coefficients of the linear system, which gives a transcendent equation in P, whose smallest root is .
3. Elastic and Inertial Constant Identifications
- an extensional mode (), in which axially translates;
- a shear mode (), in which transversely translates;
- a flexural modes (), in which rotates around the z-axis and transversely translates.
Analytical Identification of the Elastic Constants
4. Numerical Results
5. Conclusions and Perspectives
- An excellent agreement between the critical loads and modes of the equivalent beam model, with respect to finite-element analyses has been found, providing a sufficient number of cell is considered. This finding is independent from the boundary conditions.
- The geometric effect of the prestress has been introduced both in the equilibrium and constitutive equations of the coarse model. The second aspect, accounting for micro-effects, is shown to be crucial to correctly describe the buckling behavior of the here analyzed micro-structured beams.
- The corrective factors provide a significant correction of the elastic operator, revealing their essential contribution in the proper modeling of planar grid beams.
Author Contributions
Funding
Conflicts of Interest
References
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Ferretti, M.; D’Annibale, F. Buckling of Planar Micro-Structured Beams. Appl. Sci. 2020, 10, 6506. https://doi.org/10.3390/app10186506
Ferretti M, D’Annibale F. Buckling of Planar Micro-Structured Beams. Applied Sciences. 2020; 10(18):6506. https://doi.org/10.3390/app10186506
Chicago/Turabian StyleFerretti, Manuel, and Francesco D’Annibale. 2020. "Buckling of Planar Micro-Structured Beams" Applied Sciences 10, no. 18: 6506. https://doi.org/10.3390/app10186506
APA StyleFerretti, M., & D’Annibale, F. (2020). Buckling of Planar Micro-Structured Beams. Applied Sciences, 10(18), 6506. https://doi.org/10.3390/app10186506