tailieunhanh - Ebook Finite element method using MATLAB (2nd edition): Part 2

Presents detailed, step-by-step procedures for understanding and programming of the finite element technique incorporates numerous examples of computer programs for boundary value, initial value, eigenvalue problems, and structural controls, including plate/shell analysis, contains a new chapter of special topics for finite element applications. | CHAPTER EIGHT BEAM AND FRAME STRUCTURES Chapter Overview Beams and frames are very common structures because of their efficient loadcarrying capability. This chapter presents finite element formulations for those structures. In particular several different beam formulations are discussed because each formulation has its own merit. The presentation starts with the Euler-Bernoulli beam equation. The weighted residual technique is applied to derive mass and stiffness matrices and load vectors. Later a shear deformable beam theory is discussed. These formulations have displacements and rotations as the primary nodal variables. Other formulations including mixed elements hybrid elements or elements with only displacements as nodal variables like a continuum element are also explained. Static and dynamic problems are discussed here. With coordinate transformation of the nodal variables beam elements combined with axial and torsional elements are applied to 2-D and 3-D frame structures. Several types of problems are solved using MATLAB programs. They include static eigenvalue transient modal and frequency response analyses of beam and frame structures. Euler-Bernoulli Beam The Euler-Bernoulli equation for beam bending is pdfi d2 where v x t is the transverse displacement of the beam p is mass density per length. El is the beam rigidity. q x t is the externally applied pressure loading and and X indicate time and the spatial axis along the beam axis. Wc apply one of the methods of weighted residual Galerkin 8 method to the beam equation Eq. to develop the finite element formulation and the corresponding matrix equations. 237 Copyrighted material 238 Beam and Frame Structures Chapter 8 y V x 0 2 i Figure Two-Nod cd Beam Element The averaged weighted residual of Eq. is d2v d- i .d2v where L is the length of the beam and w is a test function. The weak formulation of Eq. is obtained from integrations by parts twice for the second .

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