By M Petyt
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Additional resources for Introduction to finite element vibration analysis, second edition
Example text
These loads are normal to the middle surface, which is the plane z = 0. In deriving the energy functions for a thin plate, it is assumed that the direct stress in the transverse direction, σz, is zero. Also, normals to the middle surface of the undeformed plate remain straight and normal to the middle surface during deformation and are inextensible. 6. Plate bending element. where w(x, y) denotes the displacement of the middle surface in the z-direction. 41) for a membrane element. 43). 45). 7 Thick Plate Bending Element As in the case of deep beams, when the wavelengths are less than ten times the plate thickness, shear deformation and rotary inertia effects must be included.
In these situations it is impossible to obtain analytical solutions to the equations of motion which satisfy the boundary conditions. 3). There are a number of techniques available for determining approximate solutions to Hamilton’s principle. One of the most widely used procedures is the Rayleigh–Ritz method, which is described in the next section. A generalisation of the Rayleigh–Ritz method, known as the finite element displacement method, is then introduced. The principlal features of this method are described by considering rods, shafts, beams and frameworks.
The inertia, damping and stiffness matrices are all symmetric matrices. In addition, the inertia matrix is positive definite and the stiffness matrix either positive definite or positive semi-definite. A positive definite matrix is one whose elements are the coefficients of a positive definite quadratic form. The kinetic energy is represented by a positive definite ˙ = 0. 66)) since T > 0 for all possible values of {q} the structural system is supported, then the strain energy is also represented by a positive definite quadratic form since U > 0 for all possible values of {q} = 0.