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AERSP 301 Finite Element Method Beams

AERSP 301 Finite Element Method Beams. Jose Palacios July 2008. Today. No class Friday Final Rod FEM example Beam Bending Elements. Sample Problem 3: FEM of a more complex system (loaded axially). Beams Under Bending Load. Beams Under Bending Load. Euler-Bernoulli Beam Theory assumes:

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AERSP 301 Finite Element Method Beams

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  1. AERSP 301Finite Element MethodBeams Jose Palacios July 2008

  2. Today • No class Friday • Final Rod FEM example • Beam Bending Elements

  3. Sample Problem 3: FEM of a more complex system (loaded axially)

  4. Beams Under Bending Load

  5. Beams Under Bending Load • Euler-Bernoulli Beam Theory assumes: • Plane sections perpendicular to the mid-plane remain plane and perpendicular to the beam axis after deformation (i.e. no shear) • Long slender beams (Timoshenko theory for short beams) • Consider the displacement of point P (to P’)

  6. Beams Under Bending Load • Thus, we can write the axial and vertical displacements of generic point P as: • Use these displacements to get strains:

  7. Beams Under Bending Load • That leaves us with, • And the stress: • Now consider the Resultant axial force on a cross section:

  8. Beams Under Bending Load • And the Resultant Bending Moment on a cross-section:

  9. Beams Under Bending Load • From the above expressions, it is seen that Extension & Bending are decoupled: • Recall displacement of generic point, P: • But for pure bending problem uo term vanishes, so:

  10. Beams Under Bending Load • Recall, Strain Energy: • This comes from: • For the beam bending problem:

  11. Beams Under Bending Load • External Work, W, for the beam bending problem: • We can use the Finite Element Method to analyze the beam under bending loads. • First, discretize the beam into a number of elements. • Strain Energy and Work can be written for a single element:

  12. Beams Under Bending Load • Before we can obtain the element stiffness matrix, , and element load vector, , we need to decide: • What should the nodal D.O.F.’s be? • What should the assumed displacement with the element be? • To answer the first question, we have to keep in mind that the displacement must be continuous from element to element (i.e. at element boundaries). • For axial (bar) problem, only displacement was u. By having u1 and u2 as elemental NODAL DOF’s and assigning u2k-1 and u1k to the same global DOF (during assembly) we ensured that the displacement was continuous at element boundaries. • For beam bending, both u and w have to be continuous at boundaries.

  13. Beams Under Bending Load • Choose w and  as NODAL DOFs • Then, assign local DOFs of adjacent elements to same global DOFs (during assembly) such that:

  14. Beams Under Bending Load • Second Question: What should assumed displacement be with element? • Recall for bar problem, displacement had to be written in terms of u1 & u2. A linear function was chosen. • For beam bending problem, assumed displacement w (within the element) has to be written in terms of: • You can choose a cubic function: • How do we calculate the a coefficients in terms of nodal displacements (shape functions)? As with the rod elements we must rewrite the assumed displacement function in terms of nodal displacements and shape functions.

  15. Beams Under Bending Load • For a beam bending element, assumed displacement can be written as:

  16. Beams Under Bending Load • Element Strain Energy:

  17. Beams Under Bending Load • Now we need to determine the element load vector

  18. Beams Under Bending Load • The element strain energy was used to derive the element stiffness matrix. • Similarly, the external work (over an element) is used to derive the element load vector: where,

  19. Beams Under Bending Load

  20. Sample Problem 1 • Calculate the tip displacement and rotation of the beam due to tip load P, using a single element. • Calculate the reaction force, R, and the reaction Moment, M, at the clamped boundary.

  21. Sample Problem 2 • Analyze the structure below using 2 elements. Calculate the deflection at the joint of the two beams.

  22. Sample Problem 3 • Look at the previous problem, except now the two beams are connected through a hinge.

  23. Sample Problem 4 • How does the linear spring shown below affect the problem?

  24. Sample Problem 5 • Show the affect of the linear and rotational spring to the finite element method.

  25. Coupled Axial-Bending Problems • We can also combine our rod and beam elements. • Look at elements individually (1 is a rod, 2 is a beam, etc…)

  26. Coupled Axial-Bending Problems • Example 6

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