Generalized stiffness and effective mass coefficients for power-law Euler–Bernoulli beams (original) (raw)
Related papers
In this paper the exact stiffness and mass matrices of a non-uniform Bernoulli-Euler 2d beam resting on an elastic Winkler foundation are computed integrating the exact shape functions. The non-uniformity may result from variable cross-section and/or from inhomogeneous linearly elastic material. It is assumed that there is no abrupt variation in the cross-section of the beam so that the Euler–Bernoulli theory is valid. The shape functions are derived from the solution of the axial deformation problem of a non-uniform bar and the bending problem of a non-uniform beam which are both formulated in terms of the two displacement components. The governing differential equations are uncoupled with variable coefficients and are solved within the framework of the analog equation concept. According to this, the two differential equations with variable coefficients are replaced by two linear ones pertaining to the axial and transverse deformation of a substitute beam with unit axial and bendin...
Power series expansion of the general stiffness matrix for beam elements
International Journal for Numerical Methods in Engineering, 1975
The general stiffness matrix for a beam element is derived from the Bernoulli–Euler differential equation with the inclusion of axial forces. The terms of this matrix are expanded into a power series as a function of the two variables: the axial force, and; the vibrating frequency. It is shown that the first three terms of the resulting series, which are derived in the technical literature from assumed static displacement functions, correspond respectively to the elastic stiffness matrix, the consistent mass matrix, and the geometric matrix. Higher order terms up to the second order terms of the series expansion are obtained explicitly. Also a discussion is presented for establishing the region of convergence of the series expansion for the dynamic stiffness matrix, the stability matrix, and the general stiffness matrix.
Exact Bernoulli–Euler dynamic stiffness matrix for a range of tapered beams
1985
Bernoulli-Euler theory and Bessel functions are used to obtain explicit expressions for the exact dynamic stiffnesses for axial, torsional and flexural vibrations of any beam which is tapered such that A varies as J'" and GJ and I both vary as y(" '), where A, GJ and I have their usual meanings; y = (cx:L) + 1; c is a constant such that c >-1; I, is the length of the beam; and x is the distance from onc end of the beam. Numerical checks give better than seven-figure agreement with the stiffnesses obtained by extrapolation from stepped beams with 400 and 500 uniform elements. A procedure is given for calculating the number of natural frequencies exceeded by any trial frequency when the ends of the member are clamped. This enables an existing algorithm to be used to obtain the natural frequencies of structures which contain tapered members.
LARGE DEFLECTION STATES OF EULER-BERNOULLI SLENDER CANTILEVER BEAM SUBJECTED TO COMBINED LOADING
This work experimentally and numerically studies large deflection of slender cantilever beam of linear elastic material, subjected to a combined loading which consists of internal vertical uniformly distributed continuous load and external vertical concentrated load and a horizontal concentrated load at the free end of the beam. We got equations with the help of large deflection theory, and present the differential equation governing the behaviour of this system and show that that this equation, although straightforward in appearance, is in fact rather difficult to solve due to the presence of a non-linear term. A numerical evaluation is used to evaluate the system and calculate Young`s modulus of the beam material. With simple experiment we show, how a Young`s modulus can be obtained and then the phenomenon of the large elastic sideways deflection of a column under compressive loading is investigated and elastica of buckled column is calculated.
A new approach to modeling the dynamic response of Bernoulli-Euler beam under moving load
Coupled systems mechanics, 2014
This article discusses the dynamic response of Bernoulli-Euler straight beam with angular elastic supports subjected to moving load with variable velocity. A new engineering approach for determination of the dynamic effect from the moving load on the stressed and strained state of the beam has been developed. A dynamic coefficient, a ratio of the dynamic to the static deflection of the beam, has been defined on the base of an infinite geometrical absolutely summable series. Generalization of the R. Willis' equation has been carried out: generalized boundary conditions have been introduced; the generalized elastic curve's equation on the base of infinite trigonometric series method has been obtained; the forces of inertia from normal and Coriolis accelerations and reduced beam mass have been taken into account. The influence of the boundary conditions and kinematic characteristics of the moving load on the dynamic coefficient has been investigated. As a result, the dynamic stressed and strained state has been obtained as a multiplication of the static one with the dynamic coefficient. The developed approach has been compared with a finite element one for a concrete engineering case and thus its authenticity has been proved.
There are many engineering structures that can be modeled as beams, carrying one, two or multi degree-of-freedom spring-mass systems. Examples of such practical applications include components of buildings, machine tools, vehicle suspensions and rotating machinery accessories of machine structures. Because of these wide ranging applications, the vibration behavior of beams carrying discrete structural elements, such as beams carrying a two degree-of-freedom spring-mass system have received considerable attention for many years.
Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 2020
The dynamics of an Euler-Bernoulli beam, with multiple spring/mass attachments, is investigated in this paper. A novel analytical approach for determining the natural frequencies and mode shapes of the system is presented. The formulation is applicable for any parameter value of the attached mass or spring stiffness. The calculation is performed using a minimal binary set of admissible functions for each spring/mass connection, with extremal values (zero or infinity) of the parameter. For example, to study a beam with n intermediate spring/mass connections, only 2 n admissible functions are required. These functions are used in a Rayleigh-Ritz-based energy formulation. The small number of functions used in the formulation leads to an efficient computational procedure. The results from the proposed formulation are compared with those from a finite element simulation, for different boundary conditions. The results obtained by the two methods are in excellent agreement for all boundary conditions as well as for different parameter ranges. Detailed design studies, on the effect of spring stiffness and lumped mass values, as well as their locations, on the natural frequencies and mode shapes, have been carried out. Design charts have also been generated to aid the designer of such structures.
Non-linear dynamic analysis of beams with variable stiffness
Journal of Sound and Vibration, 2004
In this paper the Analog Equation method (AEM), a BEM-based method, is employed to the nonlinear dynamic analysis of an initially straight Bernoulli-Euler beam with variable stiffness undergoing large deflections. In this case the cross-sectional properties of the beam vary along its axis and consequently the coefficients of the differential equations governing the dynamic equilibrium of the beam are variable. The formulation is in terms of the displacement components. Using the concept of the analog equation, the two coupled nonlinear hyperbolic differential equations are replaced by two uncoupled linear ones pertaining to the axial and transverse deformation of a substitute beam with unit axial and bending stiffness, respectively, under fictitious time dependent load distributions. A significant advantage of this method is that the time history of the displacements as well as the stress resultants are computed at any cross-section of the beam using the respective integral representations as mathematical formulae. Beams with constant and varying stiffness are analyzed under various boundary conditions and loadings to illustrate the merits of the method as well as its applicability, efficiency and accuracy.