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Über dieses Buch

This book develops a uniform accurate method which is capable of dealing with vibrations of laminated beams, plates and shells with arbitrary boundary conditions including classical boundaries, elastic supports and their combinations. It also provides numerous solutions for various configurations including various boundary conditions, laminated schemes, geometry and material parameters, which fill certain gaps in this area of reach and may serve as benchmark solutions for the readers.

For each case, corresponding fundamental equations in the framework of classical and shear deformation theory are developed. Following the fundamental equations, numerous free vibration results are presented for various configurations including different boundary conditions, laminated sequences and geometry and material properties. The proposed method and corresponding formulations can be readily extended to static analysis.



Chapter 1. Fundamental Equations of Laminated Beams, Plates and Shells

Beams, plates and shells are named according to their size or/and shape features. Shells have all the features of plates except an additional one-curvature (Leissa in Vibration of Plates (NASA SP-160), US Government Printing Office, Washington, DC, pp. 1–353, 1969, Vibration Of Shells (NASA SP-288), US Government Printing Office, Washington, DC, pp. 1–428, 1973). Therefore, the plates, on the other hand, can be viewed as special cases of shells having no curvature. Beams are one-dimensional counterparts of plates (straight beams) or shells (curved beams) with one dimension relatively greater in comparison to the other two dimensions. This chapter introduces the fundamental equations (including kinematic relations, stress-strain relations and stress resultants, energy functions, governing equations and boundary conditions) of laminated shells in the framework of the classical shell theory (CST) and the shear deformation shell theory (SDST) without proofs due to the fact that they have been well established. The corresponding equations of laminated beams and plates are specialized from the shell ones.
Guoyong Jin, Tiangui Ye, Zhu Su

Chapter 2. Modified Fourier Series and Rayleigh-Ritz Method

Although the governing equations and associated boundary equations for laminated beams, plates and shells presented in Chap. 1 show the possibility of seeking their exact solutions of vibration, however, it is commonly believed that very few exact solutions are possible for plate and shell vibration problems.
Guoyong Jin, Tiangui Ye, Zhu Su

Chapter 3. Straight and Curved Beams

Beams, plates and shells are commonly utilized in engineering applications, and they are named according to their size or/and shape characteristics and different theories have been developed to study their structural behaviors. A beam is typically described as a structural component having one dimension relatively greater than the other dimensions. Specially, a beam can be referred to as a rod or bar when subjected to tension, a column when subjected to compression and a shaft when subjected to torsional loads (Qatu 2004). Beams are one of the most fundamental structural elements.
Guoyong Jin, Tiangui Ye, Zhu Su

Chapter 4. Plates

Plates are one of the most fundamental structural elements which are widely used in a variety of engineering applications. A plate can be defined as a solid body bounded by two parallel flat surfaces having two dimensions relatively greater than the other one (thickness). It can also be viewed as a special case of shells with zero curvature (infinite radii of curvature).
Guoyong Jin, Tiangui Ye, Zhu Su

Chapter 5. Cylindrical Shells

Cylindrical shells are the simplest case of shells of revolution. A cylindrical shell is formed by revolving a straight line (generator) around an axis that is paralleled to the line itself. The surface obtained from the revolution of the generator defined the cylindrical shell’s middle surface. Cylindrical shells may have different geometrical shapes determined by the revolution routes and circumferential included angles. By appropriately selecting the revolution routes, cylindrical shells with desired cross-sections can be produced, such as circular, elliptic, rectangular, polygon, etc. In the literature and engineering applications, cylindrical shells with circular cross-sections are most frequently encountered. In this type of cylindrical shells, each point on the middle surface maintains a similar distance from the axis. The current chapter is devoted to dealing with closed and open circular cylindrical shells.
Guoyong Jin, Tiangui Ye, Zhu Su

Chapter 6. Conical Shells

Conical shells are another special type of shells of revolution. The middle surface of a conical shell is generated by revolving a straight line (generator line) around an axis that is not paralleled to the line itself. Conical shells can have different geometrical shapes. This chapter is organizationally limited to conical shells (both the closed shells and the open ones) having circular cross-sections. In this type of conical shells, the generator line rotates about a fixed axis and results in a constant vertex half-angle angle (φ) with respect to the axis.
Guoyong Jin, Tiangui Ye, Zhu Su

Chapter 7. Spherical Shells

The cylindrical and conical shells considered in Chaps. 5 and 6 are special cases of shells of revolution. Spherical shells are another special case of shells of revolution. A spherical shell is a doubly-curved shell characterized by a middle surface generated by the rotation of a circular cure line segment (generator) about a fixed axis. If the axis of rotation along the diameter of the circle of the line segment, a spherical shell with constant curvature in the meridional and circumferential directions will be resulted and the two radii of curvature are equal. It is noticeable that the spherical shells are very stiff for both in-plane and bending loads due to the curvature of the middle surface, which is also a reason for the analysis difficulties of the shells, especially the exact three-dimensional elasticity (3-D) analysis. The spherical shells can be closed and open. If the generator rotates less than one full revolution about the axis, the spherical shell is open and has four boundaries. If further, the generator rotates one full revolution about the axis and the proper continuity conditions are satisfied along the junction line, a closed spherical shell results, which has only two edges.
Guoyong Jin, Tiangui Ye, Zhu Su

Chapter 8. Shallow Shells

Shallow shells are open shells that have small curvatures (i.e. large radii of curvatures compared with other shell parameters such as length and width).
Guoyong Jin, Tiangui Ye, Zhu Su


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