R. Heuer

Flexural vibrations of elastic composite beams with interlayer slip

SummaryThe objective of the present paper is to analyze the dynamic flexural behavior of elastic two-layer beams with interlayer slip. The Bernoulli-Euler hypothesis is assumed to hold for each layer separately, and a linear constitutive equation between the horizontal slip and the interlaminar shear force is considered. The governing sixth-order initial-boundary value problem is solved by separating the dynamic response in a quasistatic and in a complementary dynamic response. The quasistatic portion that may also contain singularities or discontinuities due to sudden load changes is determined in a closed form. The remaining complementary dynamic part is non-singular and can be approximated by a truncated modal series of fast accelerated convergence. The solution of the resulting generalized decoupled single-degree-of-freedom oscillators is given by means of Duhamel,s convolution integral, whereby the velocity and acceleration of the loads are the driving terms. Light damping is considered via modal damping coefficients. The proposed procedure is illustrated for dynamically loaded layered single-span beams with interlayer slip, and the improvement in comparison to the classical modal analysis is demonstrated.

Static and dynamic analysis of transversely isotropic, moderately thick sandwich beams by analogy

SummaryA flexural theory of elastic sandwich beams is derived which renders quite precise results within a wide range of ratios of dimensions, mass densities, and elastic constants of the core and faces. The assumptions of the Timoshenko theory of shear-deformable beams are applied to each of the homogeneous, linear elastic, transversely isotropic layers individually. Core and faces are perfectly bonded. The principle of virtual work is applied to derive the equations of motion of a symmetrically designed three-layer beam and its boundary conditions. By definition of an effective cross-sectional rotation the complex problem is reduced to a problem of a homogeneous beam with effective stiffnesses and with corresponding boundary conditions. Thus, methods of classical mechanics become directly applicable to the higher-order problem. Excellent agreement of the results of illustrative examples is observed when compared to solutions of other higher-order laminate theories as well as to exact solutions of the theory of elasticity.

A boundary element method for eigenvalue problems of polygonal membranes and plates

SummaryAn advanced Boundary Element formulation for eigenvalue problems of membranes and plates is developed. Polygonal membranes are considered, and are embedded into a proper basic domain in order to satisfy boundary conditions exactly as far as possible. Hence, boundary integrals have to be applied at the not coinciding boundaries only. Eigenvalues of the underlying Dirichlets Helmholtz problem are calculated from frequency response functions evaluated by that “method with Greens functions of finite domains”, and natural frequencies of corresponding membranes and simply supported plates are determined by analogy. A numerical investigation is performed for parallelogram Mindlin plates. Natural frequencies and critical buckling eigenvalues are graphically represented in a nondimensional form, where the influence of skew angle and plate thickness is studied.

Dynamic analysis of polygonal Mindlin plates on two-parameter foundations using classical plate theory and an advanced BEM

Forced vibrations of moderately thick plates on two-parameter, Pasternak-type foundations are considered. Influence of plate shear and rotatory inertia are taken into account according to Mindlin. Excitations are of the force as well as of the support motion type. Formulation is in the frequency domain. An analogy to thin plates without foundations is given. This analogy to classical plate theory is complete in the case of polygonal plan-forms and hinged support conditions. In that case the higher order Mindlin-problem is reduced to two (second order) Helmholtz-Klein- Gordon boundary value problems. An advanced BEM using Greens functions of rectangular domains is applied to the latter, thereby satisfying boundary conditions exactly as far as possible. This problem oriented strategy provides the frequency response functions for the deflection of the undamped Mindlin plate with high numerical accuracy. Structural damping is built in subsequently, and Fast Fourier Transform is applied for calculation of the transient response.

Piezoelectric vibrations of composite beams with interlayer slip

SummaryActuating piezoelectric effects in two-layer beams with interlayer slip are described in detail, and special attention is given to the identification of the piezoelectric actuation as eigenstrains. It is demonstrated that piezoelectrically induced strains conveniently can be interpreted as eigenstrains acting in a background composite beam without piezoelectric actuators. The analogy between the piezoelectric effect and that of thermal strains is utilized in the present paper, where a layer-wise first-order flexural theory is applied to two-layer beams with various boundary conditions. The layers are assumed to be made of piezoelectric materials. Bernoulli-Euler hypothesis is assumed to hold for each layer separately, and a linear constitutive relation between the horizontal slip and the interlaminar shear force is considered. The governing sixth-order initial-boundary value problem is solved by separating the dynamic response in a quasistatic and in a complementary dynamic portion. The quasistatic solution that may also contain singularities or discontinuities due to sudden load changes is determinded in a closed form. The remaining complementary dynamic part is nonsingular and can be approximated by a truncated modal series of accelerated convergence. The proposed procedure is illustrated for piezoelectrically induced flexural deformations, where the forcing function is the piezoelectric curvature.

Thermally induced vibrations of composite beams with interlayer slip

Thermally excited vibrations of composite viscoelastic two-layer beams with interfacial slip are analyzed. Geometrically linearized conditions are considered, and the Bernoulli-Euler hypothesis is applied to each layer. At the interface a linear viscoelastic slip law is assigned. The resulting sixth-order initial boundary value problem of the deflection is solved in the time domain by separating the dynamic response in a quasistatic and a complementary dynamic portions. The quasistatic solution is determined in closed form, and the remaining complementary dynamic part is approximated by a truncated modal series that exhibits accelerated convergence. Numerical results are obtained for single-span composite beams with interlayer slip by means of a time-stepping procedure based on the linear interpolation of the driving terms within the time intervals.Thermally excited vibrations of composite viscoelastic two-layer beams with interfacial slip are analyzed. Geometrically linearized conditions are considered, and the Bernoulli-Euler hypothesis is applied to each layer. At the interface a linear viscoelastic slip law is assigned. The resulting sixth-order initial boundary value problem of the deflection is solved in the time domain by separating the dynamic response in a quasistatic and a complementary dynamic portions. The quasistatic solution is determined in closed form, and the remaining complementary dynamic part is approximated by a truncated modal series that exhibits accelerated convergence. Numerical results are obtained for single-span composite beams with interlayer slip by means of a time-stepping procedure based on the linear interpolation of the driving terms within the time intervals.

Equivalence of the Analyses of Sandwich Beams with or without Interlayer Slip

ABSTRACT Dynamic plane bending of sandwich beams composed of three symmetrically arranged layers is studied. Analyzing two special cases of interlaminar connections, namely, “perfect bond” and “elastic interlayer slip,” a complete analogy between the corresponding initial-boundary-value problems is derived. Hence, the response of the beam with elastic interlayer slip can be determined directly from the solution of the perfectly bonded sandwich beam with moderately thick faces.

Nonlinear random vibrations of thermally buckled skew plates

Abstract Random vibrations in the postbuckling range have chaotic properties superposed. For hard and simply supported polygonal plates a multi-modal projection by the Galerkin-procedure renders as a result of a proper non-dimensional formulation a set of nonlinearly coupled ordinary differential equations. Exact unifying solutions of the stationary F-P-K equation are constructed for that class of problems where the nonlinear restoring forces are derived from a potential function. Assuming an effective white noise excitation, the probability of first occurrence of dynamic snap-through is determined for a single mode approximation. Using a two-mode expansion the probability distribution of the asymmetric snap-buckling is also evaluated.

Large flexural vibrations of thermally stressed layered shallow shells

A dynamic nonlinear theory for layered shallow shells is derived by means of the von Karman-Tsien theory, modified by the generalized Berger-approximation. Moderately thick shells with polygonal planform composed of multiple perfectly bonded layers are considered. The shell edges are assumed to be prevented from in-plane motions and are simply supported. A distributed lateral force loading is applied to the structure, and additionally, the influence of a static thermal prestress, corresponding to a spatial distribution of cross-sectional mean temperature, is taken into account. In the special case of laminated shells made of transversely isotropic layers with physical properties symmetrically distributed about the middle surface, a correspondence to moderately thick homogeneous shells is found. Application of a multi-mode expansion in the Galerkin procedure to the governing differential equation, where the eigenfunctions of the corresponding linear plate problem are used as space variables, renders a coupled set of ordinary time differential equations for the generalized coordinates with cubic as well as quadratic nonlinearities. The nonlinear steady-state response of shallow shells subjected to a time-harmonic lateral excitation is investigated and the phenomenon of primary resonance is studied by means of the “perturbation method of multiple scales’. A unifying non-dimensional representation of the nonlinear frequency response function is presented that is independent of the special shell planform.

Nonlinear flexural vibrations of layered plates

Abstract By means of a higher-order plate-bending theory developed by Whitney and Pagano along with Bergers hypothesis, large amplitude forced vibrations of moderately thick laminated specially orthotropic plates are investigated. The theory includes shear deformation and rotatory inertia in the same manner as Mindlins theory for isotropic homogeneous plates. The in-plane forces due to large deflections are assumed to be constant within the plate domain. Considering time-harmonic forcing a Kantorovich-Galerkin procedure provides the formulation of this problem in the lower band of the frequency domain. In the case of laminates made of isotropic layers an analogy to thin homogeneous plates is given, which is complete in the case of polygonal planforms and hard hinged supports. Furthermore, the plate deflection is determined by the solution of two (second-order) Helmholtz Klein Gordon boundary value problems. Inserting these results into a proper domain integral leads to Bergers normal force. This problem-oriented strategy renders the nonlinear frequency response functions of deflection of the undamped layered plate.