E-Book, Englisch, 508 Seiten
Reihe: Woodhead Publishing Series in Civil and Structural Engineering
Shanmugam / Wang Analysis and Design of Plated Structures
1. Auflage 2007
ISBN: 978-1-84569-229-2
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: 6 - ePub Watermark
Dynamics
E-Book, Englisch, 508 Seiten
Reihe: Woodhead Publishing Series in Civil and Structural Engineering
ISBN: 978-1-84569-229-2
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: 6 - ePub Watermark
Plated structures are widely used in many engineering constructions ranging from aircraft to ships and from off-shore structures to bridges and buildings. Given their diverse use in severe dynamic loading environments, it is vital that their dynamic behaviour is analysed and understood. Analysis and design of plated structures Volume 2: Dynamics provides a concise review of the most recent research in the area and how it can be applied in the field.The book discusses the modelling of plates for effects such as transverse shear deformation and rotary inertia, assembly of plates in forming thin-walled members, and changing material properties in composite, laminated and functionally graded plates. Various recent techniques for linear and nonlinear vibration analysis are also presented and discussed. The book concludes with a hybrid strategy suitable for parameter identification of plated structures and hydroelastic analysis of floating plated structures.With its distinguished editors and team of international contributors, Analysis and design of plated structures Volume 2: Dynamics is an invaluable reference source for engineers, researchers and academics involved in the analysis and design of plated structures. It also provides a companion volume to Analysis and design of plated structures Volume 1: Stability. - The second of two volumes on plated structures - Provides a concise review of the most recent research in the research of plated structures - Discusses modelling of plates for specific effects
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1 Dynamic behaviour of tapered beams
M A BRADFORD, The University of New South Wales, Australia Publisher Summary
This chapter illustrates that plated members that are tapered lengthwise often provide optimal solutions in structural, mechanical, and aerospace engineering and in engineering mechanics applications when compared with prismatic members; however, quantifying their buckling and dynamic behavior is far more complex than for prismatic members. The use of tapered beams in steel-framed building construction was recognized by Amirikian (1952) as an economically viable alternative to prismatic members because the size of the cross-section could be made to follow the magnitude of the bending moments within the member. The Portland Cement Association (1958) published tables and charts for determining the static deflections and bending moments in frames containing tapered members that could be used with the technique of moment distribution. The tapered members considered in this chapter possess one axis of symmetry that is in the plane in which their motion takes place and modes out of the plane of this axis of symmetry, as well as torsional modes, were excluded. Although there are many solutions for vibration of tapered members in the open literature, until very recently these were restricted to planar vibration modes. 1.1 Introduction
Plated members that are tapered lengthwise often provide optimal solutions in structural, mechanical and aerospace engineering and in engineering mechanics applications when compared with prismatic members, but quantifying their buckling (Bradford 2006) and dynamic behaviour is far more complex than for prismatic members. The use of tapered beams in steel framed building construction was recognised by Amirikian (1952) as an economically viable alternative to prismatic members because the size of the cross-section could be made to follow the magnitude of the bending moments within the member. The Portland Cement Association (1958) published tables and charts for determining the static deflections and bending moments in frames containing tapered members that could be used with the technique of moment distribution (Hall and Kabaila 1986, Hibbeler 1999). Although these treatments were the first major design applications for tapered beams in engineering structures, the more pure research of tapered beams and their response to dynamic excitation has been widespread since the apparent first work of Kirchhoff (1879), in which the solutions of wedge-shaped and cone-shaped beams were treated analytically in terms of Bessel functions (Watson 1956). On a larger scale, tapered cantilevers can be used in an approximate fashion to model tall buildings, and knowledge of their dynamic response is vital for earthquake engineering assessment and applications. In general, the solution of dynamic problems involving tapered members is quite complicated and requires numerical procedures or the use of tabulated Bessel functions. One of the earliest treatments of the problem was that of Ward (1913), and a proliferation of later research followed from the work of Siddall and Isakson (1951), Suppiger and Taleb (1956) and Cranch and Adler (1956). These analytic-based treatments were founded on Bessel function theory (Watson 1956), as were the studies that followed by Conway and his colleagues (1964, 1965), Mabie and Rogers (1964, 1968, 1972, 1974), Wang (1967) and Gorman (1975), with Goel (1976) and Sato (1980) generalizing the theory of Mabie and Rogers to include more complex members with rotational springs at their ends. Other contributions of note are those of MacDuff and Felger (1957), Housner and Keightley (1962), Heidebrecht (1967), Gupta (1985) and Abrate (1995). Finite element-based methods have proven to be a popular numerical tool for many problems in engineering mechanics (Zienkiewicz and Taylor 2000), including the dynamic behaviour of tapered plates. The usual finite element treatment for the free vibration of tapered beams has involved discretising the member into a number of uniform elements whose static stiffness matrices are known, with the mass being lumped usually at the centre of the element (Clough and Penzien 1975). This uniform-element approach has been shown (Bradford 2006) to provide incorrect results for the buckling of tapered members, and it often leads to slow convergence difficulties in the dynamic response of tapered members. Archer (1963) developed a consistent mass matrix for distributed mass systems that improved the accuracy of the solutions for the free vibration of uniform beams, while Lindberg (1963) developed consistent mass and stiffness matrices for linearly tapered beam elements using a cubic interpolation function for the displacements. Later, Gallagher and Lee (1970) derived a more general tapered beam element, again by using a cubic interpolation function, while Thomas and Dokumaci (1973) constructed improved mass and stiffness matrices for tapered beams by using interpolation functions in the form of sixth-order Hermitian polynomials. Rutledge and Beskos (1981) developed stiffness and consistent mass matrices for a linearly tapered beam element of rectangular cross-section with a constant width, which produced superior results to those of Gallagher and Lee (1970), while Karabalis and Beskos (1983) extended the formulation in a more unified fashion to the static, dynamic and stability analysis of tapered beams. These formulations were not founded on specifying interpolation functions a priori, which probably accounted for their favourable attributes in deriving numerical solutions. More recent finite element studies have been directed towards the dynamic response of nonuniform beams comprising advanced composite materials, which find widespread application in the aerospace industry. Kapania and Raciti (1989) described developments in the vibration analysis of laminated composite beams, and Yuan and Miller (1989, 1990) reported the derivation of beam finite elements for the analysis. Oral (1991) developed a shear flexible finite element for nonuniform laminated composite beams, utilising a three-noded finite element with six degrees of freedom per node for a linearly tapered, symmetrically laminated, composite beam utilising first-order shear deformation theory, whilst Ramalingeswara Rao and Ganesan (1997) developed a finite element technique for studying the dynamic behaviour of tapered composite beams subjected to point-harmonic excitation. Laura et al. (1996) used the finite element method, as well as other techniques, to study the vibration of beams with stepped thicknesses. A background to the dynamic response of a plated beam member that is tapered can be illustrated quantitatively by considering the general tapered member shown in Fig. 1.1. This figure shows a typical tapered beam of length l, which is assumed to be supported at each end by translational and rotational springs and subjected to (dynamic) concentrated loads Fi (zi, t) (i = 1, 2, …) and to a (dynamic) distributed load of intensity q(z, t), where t is the time and the z axis is shown in the figure. The vibration is restricted to being in the nominally vertical y–z plane, so that the motion and deformations are in this plane. The stiffness of the translational spring (force per unit length) at the end z = 0 is kT0 and of the rotational spring (moment per radian) at z = 0 is kR0, while the counterpart stiffnesses at z = l are kTl and kRl. The mass density of the beam is ?, so that ? · A(z) is the mass per unit length where A(z) is the area of the cross-section of the tapered beam. Under dynamic excitation, the beam displaces transversely V(z, t). Using the principle of virtual work, when a virtual displacement dV(z) is applied to the dynamic system, the work done by the external forces must equal the change in the internal strain energy of the beam. In a dynamic system, the external forces include both the real loads and inertial forces.
1.1 Tapered beam representation. For the tapered beam in Fig. 1.1, the principle of virtual work may be stated as U=dWF+dWC+dT 1.1 1.1 in which U=?0lEl(z)V?dV?dz 1.2 1.2 is the change internal strain energy due to flexure (the axial and shearing strain energies have been ignored) and in which EI(z) is the flexural rigidity of the beam, where ( )' denotes the differentiation of ( ) with the respect to...