# Lecture 20: Beam, Plate, and Shell Elements II

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Topics: Beam, plate, and shell elements II

• Formulation of isoparametric (degenerate) beam elements for large displacements and rotations
• A rectangular cross-section beam element of variable thickness; coordinate and displacement interpolations
• Use of the nodal director vectors
• The stress-strain law
• Introduction of warping displacements
• Example analysis: 180 degrees, large displacement twisting of a ring
• Example analysis: Torsion of an elastic-plastic cross-section
• Recommendations for the use of isoparametric beam and shell elements
• The phenomena of shear and membrane locking as observed for certain elements
• Study of solutions of straight and curved cantilevers modeled using various elements
• An effective 4-node shell element (the MITC4 element) for analysis of general shells
• The patch test, theoretical and practical considerations
• Example analysis: Solution of a three-dimensional spherical shell
• Example analysis: Solution of an open box
• Example analysis: Solution of a square plate, including use of distorted elements
• Example analysis: Solution of a 30-degree skew plate
• Example analysis: Large displacement solution of a cantilever
• Example analysis: Collapse analysis of an I-beam in torsion
• Example analysis: Collapse analysis of a cylindrical shell

Instructor: Klaus-Jürgen Bathe

## Related Resources

Study Guide (PDF - 1.2MB)

Section 6.5

Examples

Problems 6.20, 6.21

References

Bathe, K. J., and A. Chaudhary. “On the Displacement Formulation of Torsion of Shafts with Rectangular Cross-Sections.International Journal for Numerical Methods in Engineering 18 (October 1982): 1565-1568.

Dvorkin, E., and K. J. Bathe. “A Continuum Mechanics Based Four-Node Shell Element for General Nonlinear Analysis.Engineering Computations 1 (1984): 77-88.

Bathe, K. J., and E. Dvorkin. “A Four-Node Plate Bending Element Based on Mindlin/Reissner Plate Theory and a Mixed Interpolation.International Journal for Numerical Methods in Engineering 21 (February 1985): 367-383.

Bathe, K. J., and E. Dvorkin. “A Formulation of General Shell Elements: The Use of Mixed Interpolation of Tensorial Components.International Journal for Numerical Methods in Engineering 22 (March 1986): 687-722.

Bathe, K. J., and P. M. Wiener. “On Elastic-Plastic Analysis of I-Beams in Bending and Torsion.Computers & Structures 17 (1983): 711-718.

Bathe, K. J., C. A. Almeida, and L. W. Ho. “A Simple and Effective Pipe Elbow Element: Some Nonlinear Capabilities.Computers & Structures 17 (1983): 659-667.

Lee, P.S., and K. J. Bathe. “Development of MITC Isotropic Triangular Shell Finite Elements.Computers & Structures 82 (May 2004): 945-962.

Bathe, K. J., and P. S. Lee. “Measuring the Convergence Behavior of Shell Analysis Schemes.Computers & Structures 89 (February 2011): 285-301.