Free vibrations of composite laminated doubly-curved shells and panels of revolution with general elastic restraints

A unified numerical analysis model is presented to solve the free vibration of composite laminated doubly-curved shells and panels of revolution with general elastic restraints by using the Fourier–Ritz method. The first-order shear deformation theory is adopted to conduct the analysis. The admissible function is acquired by using a modified Fourier series approach in which several auxiliary functions are added to a standard cosine Fourier series to eliminate all potential discontinuities of the displacement function and its derivatives at the edges. Furthermore, the general elastic restraint and kinematic compatibility and physical compatibility conditions are imitated by the boundary and coupling spring technique respectively when the composite laminated doubly-curved panels degenerate to the complete shells of revolution. Then, the desired results are solved by the variational operation. Large quantities of numerical examples are calculated about the free vibration of cross-ply and angle-ply composite laminated doubly-curved panels and shells with different geometric and material parameters. Through the sufficient conclusions obtained from the comparison, it can be seen that highly accurate solutions can be yielded with a little computational effort. To understand the influence of different boundary conditions, lamination schemes, material and geometrical parameters on the vibration characteristics, a series of parametric studies are carried out. Lastly, results for vibration of the composite laminated doubly-curved panels and shells subject to various kinds of boundary conditions and with different geometrical and material parameters are also presented firstly, which can provide the benchmark data for other studies conducted in the future.     

Hamiltonian approach to studying thin shells of revolution made of composite materials

A method for constructing defining relations of the linear theory of shells of revolution in complex Hamiltonian form has been proposed. Based on the Lagrange variational principle, we have constructed a mathematical model of a multilayer orthotropic shell of revolution. We have obtained explicit expressions for the coefficients and right-hand sides of the Hamiltonian complex system of equations describing the statics of shells of revolution in terms of their rigid characteristics and acting loads. The Hamiltonian resolving system of linear differential equations, formulated in the axially symmetric case, has some specific properties facilitating both analytical studies and numerical procedures of their solution.     

Optimal designing of orthotropic cylindrical shells fastened to elastic filler

Conclusions A method has been proposed for optimally designing an orthotropic cylindrical shell rigidly fastened to an elastic and isotropic filler of finite dimensions. The design takes into account simultaneous action of pressure, body forces, and heat on the structure. The optimum design has been calculated for the case of temperature-dependent elastic properties and strength characteristics of the tape. The method allows also for limitation on the strength of the filler. The convergence of the iteration process schematically shown in Fig. 2 is quite fast. Indeed, for the given design variant, the condition of manufacturability (1) is satisfied with a sixfold margin in the third approximation (n=3) already.Translated from Mekhanika Kompozitnykh Materialov, No. 1, pp. 91–94, January–February, 1984.     

Energy inequalities and error estimates for torsion of elastic shells of revolution

Zusammenfassung Das dreidimensionale Problem der Bestimmung der Spannungen und Verschiebungen in einer elastischen, rotations-symmetrischen Schale unter axisymmetrischer Torsion kann auf ein Randwertproblem vom Neumannschen Typ für einen elliptischen Operator zweiter Ordnung zurückgeführt werden. Methoden, die auf Energieungleichungen basiert sind, werden verwendet, um die Fehler abzuschätzen, die entstehen, wenn diese Spannungen und Verschiebungen in einer dünnen Schale mit Hilfe der Ergebnisse der Schalentheorie annähernd berechnet werden.     

Large deformations of bodies of revolution made of elastic homogeneous and fiber-reinforced materials 1. Torsion of toroidal bodies

Equations of a mathematical model for bodies of revolution made of elastic homogeneous and fiber-reinforced materials and subjected to large deformations are presented. The volume content of reinforcing fibers is assumed low, and their interaction through the matrix is neglected. The axial lines of the fibers can lie both on surfaces of revolution whose symmetry axes coincide with the axis of the body of revolution and along trajectories directed outside the surfaces. The equations are obtained for the macroscopically axisymmetric problem statement where the parameters of macroscopic deformation of the body vary in its meridional planes, but are constant in the circumferential directions orthogonal to them. The equations also describe the torsion of bodies of revolution and their deformation behavior under the action of inertia forces in rotation around the symmetry axis. The results of a numerical investigation into the large deformations of toroidal bodies made of elastic homogeneous and unidirectionally reinforced materials under torsion caused by a relative rotation of their butt-end sections around the symmetry axis are presented.     

Nonlinear parametric vibrations of cylindrical shells made from composite materials

The dynamic properties of a hinged shell made from a composite material and subjected to combined loads are investigated by means of an orthotropic model. The problem is solved by means of the geometrically nonlinear dynamic equations of the theory of sloping shells, set up on the basis of the Kirchhoff-Love hypothesis. Various cases of loading are considered, i.e., the combined action of a longitudinal pulsating load and an external static pressure and also of a pulsating external pressure and a static axial compression. The wave processes at the middle surface are not taken into account. The system of resolvents is obtained by consecutive application of the variation and averaging methods. The results of the calculations are presented graphically and are analyzed in detail.Moscow. Translated from Mekhanika Polimerov, Vol. 9, No. 3, pp. 531–539, May–June, 1973.