AE420/ME471 – Introduction to the Finite Element Method Practice Problems 2b Fall 2022
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AE420/ME471-Introduction to the Finite Element Method
Practice Problems 2b
Fall 2022
Problem 1
The potential energy of a (Euler) beam of length L supported by an elastic foundation (Figure 6. 1) is given by
where w(x) is the beam deflection, E is the Young modulus, I is the moment of inertia,
p(x) is the distributed load acting on the beam, and k is the stiffness of the elastic foundation.
a)What differential equation does the beam deflection satisfy?
b) Derive the local stiffness matrix and local load vector for a 2-node element of length C, stiffness E, moment of inertia I, foundation stiffness k, and transverse load p. Sketch the shape functions to be used with this particular element. Describe how you would derive the expressions of these shape functions (without actually deriving them!). Write the expression of the components of [k] and {r}(without performing the integrations!).
Problem 2
The GDE for an Euler-Bernouilli beam under axial compressive loading P and transverse load q is
where L is the length of the beam, I is the moment of inertia of the beam and E is its stiffness. Assume that the beam is cantilever (fixed displacement and slope) at x=0 and x=L. Using the Galerkin Weighted Residual Method, derive the finite element formulation for a 2-node element of length C. Explain every step of your analysis. Sketch the shape functions. What continuity requirement do you have for this problem and why? What is the expected size of the local stiffness matrix [k] and local load vector {r}?
2023-08-04