However, the underlying concept remains basically the same in each particular case. Variation of the external load work 23 1. The variational method is particularly well suited to modelling this kind of problem, because it brings out in a logical manner the central importance of nonlinear strains, even if the interest is restricted to the formulation of linear equations. Local and global strains 313 6. Plate loaded by a concentrated in-plane force: spatial attenuation of the local response. Fortunately, such approximations are found to be sufficiently accurate in most engineering applications. Waves in an infinite elastic rod of circular cross-section 53 1.
Plates: out-of-plane motion 311 6. Stresses and convergence rate of modal series. Such a theoretical approach closely follows the common engineering practice of selecting a few appropriate generic shapes to build complex structures. He is the author of more than 50 papers on vibrations, damping and associated subjects and has been involved in numerous international conferences and meetings. The study of fluid-structure interactions in this third volume covers the coupled dynamics of solids and fluids, restricted to the case of oscillatory motions about a state of static equilibrium. Equilibrium in terms of generalized stresses.
This can be checked by rotating a straight segment of length L through a finite angle θ , see Figure 1. As a general result, it will be noted that high order modes are found to be rather insensitive to boundary conditions; this indicates that the ratio of the elastic strain energy of the whole beam over that induced by the elastic supports increases with the wave number of the mode. . They may be considered as the simplest structural element because of the relative simplicity of the equilibrium equations arising from a suitable condensation of the space variables into a single one, namely the abscissa along the beam length. Pure bending mode of deformation 99 2. As a consequence, when waves are triggered in the medium by some excitation process, as the incident waves are interfering with the reflected waves, a steady oscillatory state is established, giving rise to a system of standing waves.
Variation of the kinetic energy. The two principal curvatures have opposite signs and this kind of deformation is termed anticlastic see Figure 6. Before embarking on the presentation of such models, which is the central object of this book, it is appropriate to review first a few fundamental concepts, definitions and laws of continuum mechanics. In particular, an important exception is encountered in the case of thin shells of revolution loaded by concentrated forces. Standing waves and natural modes of vibration.
Finally, the substructuring method using truncated modal bases for describing each substruc- ture is introduced and illustrated by solving a few linear and nonlinear problems involving intermittent contacts. Local stress vector on the beam cross-section Figure 2. Thus the curvature effect inherent in the shell geometry is entirely accounted for by the third equation which links the transverse and the membrane effects. Amongst others, enlightening results concerning the Saint-Venant principle invoked in Chapter 1, are obtained by using the modal analysis method to the response of a rectangular plate to an in-plane point load. The time taken by a wave of a given frequency f to sweep this distance is the same whatever α may be.
The simplest way to prove this important result is to consider the static equilibrium of a rigid body loaded by contact forces only body forces could be included but are not necessary. In terms of phase angle, τ x is replaced by the phase shift between the oscillations located at the source and at a distance x from the source. Variation of the external load work. Elastic-plastic beam A cantilevered beam with a uniform rectangular cross-section is considered depth h, width a. Boundary terms arising from the spatial integrals contribute to define the boundary conditions of the problem.
As indicated in Figure 1. It must be emphasized that the curvature of the sector induces a coupling between the radial and tangential stresses; this is a very important feature which holds for any curved structures beams, plates, shells as will be stressed several times in Chapters 7 and 8. Local formulation The solution of the static problem is immediate. Nevertheless, in the presence of transverse shear stresses, the validity of such models must be revisited, based on Straight beam models: Newtonian approach 117 the distribution of the local shear stresses which are induced by the deformations of the real 3D structure. However, convergence rate of stress or strain series is significantly slower than that of the displacement series because they are obtained through a space derivation of the latter. It can be arbitrarily decided that the cross-sectional surface S separating the two portions belongs to one of them, forming thus a facet of that portion, see Figure 1.
Finally surface and concentrated loads applied on lines or points on the midplane will be included in the analysis. Stresses and convergence rate of modal series. When the wavelength is sufficiently 194 Structural elements small, contribution of shear deformation and rotatory inertia of the beam crosssections must be accounted for. The plane waves correspond to a single index differing from zero. Superposition of several modes of deformation 128 Chapter 3. Generic properties of conservative operators. Solution of a few problems concerning xxii Introduction circular arches or rings and then shells of revolution, brings out that transverse loads cannot be exactly balanced by tensile or compressive stresses in the case of beams but they can in the case of shells.