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Many civil and mechanical engineering structures can be simplified as double-beam systems, i.e., a primary beam and a secondary beam connected to the primary beam. Many studies have investi...

This paper presents the formulation of a generalized beam theory (GBT) to analyze the vibration behavior of composite thin-walled prismatic members displaying straight axis, open-section, and arbitrary orthotropy. It accounts for the effects of (1) the cross section in-plane deformation, (2) geometric and material couplings, (3) primary and secondary warping, and (4) rotary inertia. First, the GBT equilibrium equations and boundary conditions are derived, and their terms are physically interpreted, i.e., related to the member mechanical properties. Then, a few remarks on the cross-section mechanical properties appearing in the linear stiffness and inertia terms are presented. Finally, to clarify the concepts involved in the proposed GBT formulation and illustrate its application and capabilities, an in-depth study concerning the local and global vibration behavior of lipped channel members with (1) simply supported end sections and wall cross-ply orthotropy and (2) fixed (clamped) end sections and wall nonaligned orthotropy are presented and discussed in detail.

The nonlinear dynamic analysis of beam-columns undergoing moderate large deflections and partially supported on a nonlinear three-parameter viscoelastic foundation is presented, taking into account the effects of shear deformation and rotary inertia and employing the boundary element method (BEM). The beam’s constant cross section is an arbitrarily shaped, doubly symmetric simply or multiply connected one, while its edges are supported by the most general boundary conditions. In Part I the governing equations have been derived, leading to five boundary-value problems with respect to the transverse displacements, to the axial displacement, and to two stress functions. These problems are numerically solved using the analog equation method, a BEM-based method. In Part II the numerical applications are worked out to illustrate the efficiency, and wherever possible the accuracy and range of applications of the proposed method. Thus, the results obtained from the developed method are presented compared with those obtained from the literature and from finite-element software. More specifically, the linear analysis of a simply supported beam-column on a Pasternak-viscoelastic foundation, the nonlinear analysis of a clamped beam-column on a viscoelastic or nonlinear three-parameter viscoelastic foundation, and the nonlinear analysis of a partially embedded column-pile in a nonlinear three-parameter viscoelastic foundation are presented and discussed through applications of particular interest.

This paper presents dynamic stability analysis of beams with shear deformation and rotary inertia subjected to periodic axial forces by employing a discrete singular convolution (DSC) algorithm with regularized Shannon kernel. The shear effect has been taken into account based on Engesser’s and Haringx’s theories, respectively. A modified algorithm is proposed to solve the governing equations of beam motion. The influence of rotary inertia, shear deformation, and damping on dynamic instability regions has been investigated. The obtained numerical results are compared with those of the existing method. Numerical results indicate that the modified algorithm is a reliable approach for the solutions of this kind of problem in this investigation. The differences between dynamic instability regions of beams based on Engesser’s and Haringx’s theories are also presented.

The paper presents an analysis of the coupled vibration of beams with arbitrary thin-walled open cross section, braced with identical transversal header beams uniformly distributed along their length. The explicit form of analytic solution is derived by directly solving the governing differential equations of motion. The development is based on Vlasov theory which includes the effect of flexural-torsion coupling, the constrained torsion warping, and rotary inertia. The governing differential equations for coupled bending-torsional vibrations are performed using the principle of virtual displacements. In the case of simply supported beam, exact explicit expressions are derived to predict the natural frequencies and the corresponding mode shapes. The frequency equation, given in determinantal form, is expanded in an explicit analytical form, and then solved using the symbolic computing package Mathcad. The expressions are concise and very simple and as such convenient to be used by a practicing engineer who does not need to go into detail of thin-walled beam theory. Also, the use of explicit expressions gives significant savings in computing time compared with the alternative numerical methods [finite-element method (FEM), finite strip method, differential transform method, etc.]. To demonstrate the validity of this method the natural frequencies of braced thin-walled beams, having coupled deformation modes, are evaluated and compared with FEM.

The response of a beam-column frame to impulsive ground shock, such as those induced by an underground explosion, has characteristics of both impact and natural earthquake responses. The critical effects may be governed by the dynamic response of individual elements as continuous mass systems, while to a certain extent the global vibration (as of lumped-mass systems) may also be involved. To incorporate both dynamic features, the present study proposes a combined continua and lumped parameter (CCLP) model, which consists of the basic beam-column element with distributed stiffness and mass, along with concentrated mass-springs at element ends to form the reduced dynamic system. To take into account of the shear deformation and rotary inertia which become important in the impulsive response, the governing equations are formulated based on the Timoshenko beam theory. The nonlinearities are described through three mechanisms, namely the distributed nonlinear flexural and diagonal shear behavior, and the direct sliding shear at the member ends. A generic restoring force model is adopted to describe the hysteretic behavior. Comparison with a scaled model test demonstrates that the CCLP model is capable of representing the primary dynamic features in a frame structure under impulsive ground shock. Extended parametric studies indicate that, with increase of the ground shock frequency, the failure tends to become shear dominant. For ground shocks of frequency at 20 — 30 Hz and above, the failure in a reinforced concrete column will require a peak ground velocity (PGV) on the order of 3 m/s, whereas failure in a beam would occur at PGV of about 1.5 m/s.

In most publications on vibration of annular plates, the natural frequencies are only presented for inner radius as small as one-tenth of its outer radius, but there are hardly any results presented for cores smaller than this inner radius value. This is attributed to severe scaling problems upon using numerical techniques to obtain the fundamental frequency of plates when the inner radius becomes very small. This study aims to solve this class of plate vibration problem where the annular plates are thick and their core radii are much less than one-tenth of the outer radius. The analysis involved using the Mindlin plate theory so as to allow for the effects of transverse shear deformation and rotary inertia and the truncation of the terms in the Bessel functions defining the characteristic equations in order to overcome the scaling problem. Eventually, the fundamental frequency of a thick annular plate could be deduced as finite or zero when the inner radius approaches zero.

A closed-form solution and finite-element formulation are developed for the dynamic analysis of thin-walled members with asymmetric open sections subjected to harmonic forces. The dynamic equations of motion and associated boundary conditions are derived from Hamilton’s principle. The formulation is based on a generalized Vlasov-Timoshenko beam theory and accounts for the effects of shear deformation caused by bending and warping and translational and rotary inertia effects. It also captures the effects of flexural-torsional coupling caused by cross-sectional asymmetry. From this a general closed-form solution is obtained. A family of shape functions is then developed based on the exact solution of the coupled field equations and is used to formulate a beam finite element. The new element has two nodes with six degrees of freedom per node and successfully captures the coupled bending-torsional static and steady-state responses of asymmetric thin-walled members under harmonic forces. Results based ...

For the nonlocal theory of structures, Eringen's small length-scale coefficient e0 may be identified from atomistic modeling or experimental tests. In this study, Eringen's small length-scale coefficients are presented for the vibration and buckling of nonlocal rectangular plates with simply supported edges. The coefficients are calibrated by comparing the vibration frequency and buckling loads obtained from a nonlocal plate and a microstructured beam-grid model with the same characteristic length. The beam-grid model is composed of rigid beams connected by rotational and torsional springs. It is found that the small length-scale coefficient e0 varies with respect to the initial stress, rotary inertia, mode shape, and aspect ratio of the rectangular plate.

A triangular element based on Reissner–Mindlin plate theory is developed and it is applied to free vibration analysis of plates in different situations. The element has three corner nodes, three mid-side nodes and an internal node at the element centroid where each node contains three usual degrees of freedom (transverse displacement and bending rotations). To make the element free from the shear locking problem, the formulation is done in an efficient manner taking transverse displacement and transverse shear rotations as the field variables. The degrees of freedom of the internal node are condensed out to improve the computational elegance. As the condensation cannot be done with a consistent mass matrix, a lumped mass matrix having no mass contribution at the internal node is used. In this context two mass lumping schemes are proposed where the effect of rotary inertia is considered in one of these schemes. All these features have made the element quite elegant, which is tested with numerical examples ...

This paper is concerned with the free vibration problem of Timoshenko beams with an internal hinge. Exact vibration frequencies for axially loaded, clamped-clamped beams and clamped-simply supported beams are determined. The effects of axial force, transverse shear deformation, rotary inertia, and the location of the internal hinge on the fundamental frequency of vibration are investigated. A necessary condition for the optimal location of internal hinge that maximizes the fundamental frequency is also presented. We establish that (a) the optimal location of the hinge is at the point of inflection of an equivalent beam and (b) the maximum frequency value possible for a beam with an optimally positioned internal hinge is equal to the frequency of an equivalent beam without a hinge.

The linear response of a flexible longitudinal pole hitting a column or beam is investigated. Based on Timoshenko beam theory, an efficient analytical solution method using mode superposition for the coupled beam-pole system is presented. Any physical set of boundary conditions can be accommodated, such as a free pole impacting a column with arbitrary boundary conditions or a free-free beam hitting a pole that is either pinned or free at the opposite end. For the case of the free-free beam, it can have arbitrary initial translational and rotational velocities. Impact can occur anywhere within the beam span, although the contact is frictionless. It is shown that, when the beam response governs, the initial impact is likely governed by shear, and therefore, Euler-Bernoulli beam theory is not a good modeling choice in general. Results based on a wood log hitting concrete, steel, and wood columns reveal the behavior governing the impact force time history. A simple design formula for the initial maximum impact force is shown to be quite accurate. The impact duration is governed by either the longitudinal wave speed in the pole or the shear wave speed in the column. In addition, the energy transfer between translational and rotational kinetic energies and strain energies is used to analyze the impact vibrational motion; this analysis reveals both the initial dependence on shear deformation and the transfer of the associated energy to bending energy. The significance of including rotary inertia is also demonstrated.

Rigid body dynamics is used to determine the deformation of a fixed-end, rigid-plastic beam subjected to uniformly distributed impulsive loading. The proposed solution methodology allows calculations of deformations at plastic hinges and can be used to establish rigid-plastic fracture criteria for rigid-plastic beams. Unlike previous solutions to this problem, rotary inertia and the shear deformations at the support are considered. The solution for beam deformations is described in three phases: shear, bending, and membrane. Each phase ends when the corresponding component of the strain rate vector vanishes. The initial shear phase is completed when the transverse shear velocity at the support vanishes. The beam then undergoes only rigid body rotation and axial stretching at plastic hinges in the bending phase. The bending phase ends when the angular velocity vanishes. In the membrane phase, the beam acts like a string until the transverse velocity vanishes. It has been found that beams subjected to low impulse velocity attain permanent deformation in the bending phase, while beams subjected to high impulse velocity reach permanent deformation in the membrane phase. The predictions of the beam deflections using the proposed methodology are within 15% of the experimental results.

A parametric study is performed to investigate influences of the opening angles, the slenderness ratios, the material types, the boundary conditions, and the thickness-to-width ratios of the cross section on the in-plane natural frequencies of symmetric cross-ply laminated circular composite beams. Governing equations are obtained based on the classical beam theory. The transfer matrix method is successfully applied to calculate exact natural frequencies with the help of an effective numerical algorithm, which was previously used for isotropic materials. The effects of the shear deformation, the axial deformation, and the rotary inertia are included in the formulation based on the first-order shear deformation theory. The physical system is considered as a continuous system. To verify the present theory, two examples are worked out for straight beams. A quite good agreement is observed with the reported results.

A powerful and efficient method is presented for finding exact out-of-plane natural frequencies of plane structures composed of curved Timoshenko beams. Initially, exact dynamic stiffnesses are derived from the governing differential equations of motion in a form that can be used directly in the stiffness method of analysis. This enables any appropriate structure to be modeled according to standard techniques, which, in this case, yield a transcendental eigenvalue problem. Then it is shown how any desired natural frequency may be obtained with certainty by employing a modification to a well-established algorithm, which ensures that no natural frequencies can be missed and avoids the usual approximations associated with traditional finite elements. Finally, comparisons are made with published results and an example shows how the natural frequencies of a continuous curved beam are altered when the effects of shear deflection and rotary inertia are considered.

A procedure combining the dynamic stiffness method with the Laplace transform is proposed to obtain accurate transient responses of an arch with variable curvature. The dynamic stiffness matrix and equivalent nodal force vector for an arch with variable curvature subjected to distributed loading are explicitly formulated based on a series solution. The effects of shear deformation, rotary inertia, and damping are considered. As examples, the accurate transient responses of a parabolic and a semielliptic arch subjected to either point loading or base excitation are given. The effects of the shapes of the arches and the phase-shift in the multiple input for base excitation are also discussed.

The influence of rotary inertia, shear, and axis extensibility on the stability boundary of a generalized Leipholz column is analyzed. Namely, we consider the problem of determining the stability boundary for an elastic column, fixed at one and free at the other end, loaded by uniformly distributed tangential forces along its length and a concentrated force at the top having fixed direction. The constitutive equations for the column are taken in the form suggested by Haringx. First, the nonlinear differential equations of motion are derived. These equations are then linearized, around the trivial solution, and the critical (flutter) load is determined numerically. It is found that axis compressibility increases the critical load, while the finiteness of shear stiffness, rotary inertia, and constant compressive force decrease the critical load. The influence of the pulsating component of the compressive force on the stability is also analyzed.

This paper deals with the nonaxisymmetric dynamic behavior of fluid‐filled buried orthotropic cylindrical shells/pipes excited by plane longitudinal waves. A thick shell model including the effects of shear deformation and rotary inertia is taken. A perfect bond between the shell and the surrounding soil is assumed. The linear acoustic equation is used for wave propagation in the fluid inside the pipe. Results are presented for the axisymmetric as well as the flexural mode for different orthotropy parameters of the shell and also for different soil conditions around the pipe/shell. Results of the empty and fluid‐filled shells are compared. Effects of changes in the fluid density are also discussed. The presence of fluid inside the shell significantly alters the shell response. The effect of fluid, in general, is comparable to the effects of variation in the orthotropy parameters of the shell and also to the effects due to changes in the soil condition. In the flexural mode, deformations in the fluid‐fille...

A method of analysis is presented for free vertical vibration of suspension bridges. The method takes into account the effects of shear deformation and rotary inertia, and uses a linearized theory which maintains small amplitudes of vibration. The formulation of the problem is based on the Timoshenko beam theory, and the differential equations of motion and the associated boundary conditions are derived by applying Hamilton's principle. The analysis is conducted by using general solutions for the fourth order differential equation of motion. The objective of the study is to determine a sufficient number of natural frequencies and mode shapes, and to enable an accurate vibration analysis for higher modes. A detailed numerical example, which includes the various boundary conditions of the stiffening girders and the elasticity of the towers, is shown to illustrate the applicability of the analysis and to investigate the dynamic characteristics of vertical vibrating suspension bridges.

This paper is concerned with the geometrically nonlinear axisymmetric static and transient analysis of moderately thick cylindrically orthotropic circular plates subjected to uniformly distributed and discrete central loads. Shear deformation and rotary inertia are included in the analysis. Differential equations in terms of the transverse displacement, w, the rotation of the normal to the middle surface, φ\N, and the stress function, ψ\N, are employed. These three field variables are expanded in finite power series and the discretized equations are obtained by using the orthogonal point collocation method in the space domain and the Newmark-β\N scheme in the time domain. Results are presented for immovable clamped and simply supported plates for static and step function loads. The effct of transverse shear is investigated for isotropic, transversely isotropic and orthotropic plates.

The finite element analysis of an orthogonally stiffened annular sector plate is presented by combining the annular sector plate element and the curved beam element. The plate and the curved beam elements are the isoparametric elements in which the effects of shear deformation and rotary inertia have been taken into account. In addition, both these elements are meant to be used for arbitrarily laminated structures and they are based on the higher order theories previously presented for plates, shells, and beams. Numerical results are presented to demonstrate the accuracy and correctness of the present finite element procedure for stiffened plates. Numerical results obtained on circumferentially stiffened sector plates illustrate the accuracy of an equivalent orthotropic plate (EOP) model and the plate‐stiffener system (PSS) model. The limitations of modeling the plate by thin plate elements and stiffeners by thin beam elements are also discussed.