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Project Objectives
Project Objectives
Determining how materials will deform under large deformations.
Using finite element methods in conjunction with the application of Euler-Bernoulli principles.
Expand the work presented in "The Finite Element Solution of a Class of Elastica Problems" by B.W. Golley.
Methodology and results
Methodology and results
The analysis begins with looking at the curvature of a deformed beam.
Assuming pure bending, plane sections remain plane, and a constant cross-section, the moment, shear force, and axial force are determined.
Strain energy equation is determined and minimized, using common classical mechanics techniques.
The finite element methodology is built and the computational algorithm is built in MATLAB.
From the minimized strain energy, the displacement, bending moment, shear force, and axial force can be determined for a variety of beam boundary conditions (e.g. fixed-fixed)
To validate the results a ramped loading condition is applied (1 to 10 N), and the results are compared to the Euler-Bernoulli Method.
The deformation results are optimized using a genetic algorithm (GA) and the "fmincon" function.
References
References
[1] B. W. Golley, “The Finite Element Solution of a Class of Elastica Problems,” Comput. Methiods Appl. Mech. Eng., no. 46, pp. 159–168, 1984.
[2] K. IIT, “Energy Methods Structural Analysis,” Nptl.
[3] Libretexts, “2.3: Curvature and Normal Vectors of a Curve,” Mathematics LibreTexts, 21-Dec-2020. [Online]. Available: https://math.libretexts.org/Bookshelves/Calculus/Supplemental_Modules_(Calculus)/Vector_Calculus/2:_Vector- Valued_Functions_and_Motion_in_Space/2.3:_Curvature_and_Normal_Vectors_of_a_Curve. [Accessed: 19-Feb-2021].
[4] Optics Arizona, “Flexural Stresses In Beams (Derivation of Bending Stress Equation),” Opt. Eng., pp. 48–52.
[5] M. Trabia, “Classical Structural Mechanics II: Slender Beams in Bending.” Department of Mechanical Engineering, UNLV, Las Vegas, pp. 1–13, 2018.
[6] R. C. Juvinall and K. M. Marshek, The Fundamentals of Machine Component Design. 2012.
[7] H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 3rd ed. Pearson, 2001.
[8] J. N. Reddy, Principles and Variational Methods in Applied Mechanics. 2002.
[9] Kelly, “Elastic Strain Energy,” vol. 3, pp. 242–255, 2014.
[10] I. Fried, “Stability and equilibrium of the straight and curved elastica-finite element computation,” Comput. Methods Appl. Mech. Eng., vol. 28, no. 1, pp. 49–61, 1981.
[11] S. Xu, “Gaussian Quadrature Rules.” p. 15, 2016.
[12] Massachusetts Institute of Technology, “More on Finite Element Methods,” pp. 1–7, 2021.
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