Study of the bending vibration characteristic of phononic crystals beam-foundation structures by Timoshenko beam theory

2015 ◽  
Vol 29 (20) ◽  
pp. 1550136 ◽  
Author(s):  
Yan Zhang ◽  
Zhi-Qiang Ni ◽  
Lin-Hua Jiang ◽  
Lin Han ◽  
Xue-Wei Kang
Keyword(s):  
Timoshenko Beam ◽  
Beam Theory ◽  
Phononic Crystals ◽  
Timoshenko Beam Theory ◽  
Winkler Foundation ◽  
Vibration Characteristic ◽  
Bending Vibration ◽  
Periodic Composites ◽  
Beam Foundation ◽  
Foundation Structure

Vibration problems wildly exist in beam-foundation structures. In this paper, finite periodic composites inspired by the concept of ideal phononic crystals (PCs), as well as Timoshenko beam theory (TBT), are proposed to the beam anchored on Winkler foundation. The bending vibration band structure of the PCs Timoshenko beam-foundation structure is derived from the modified transfer matrix method (MTMM) and Bloch's theorem. Then, the frequency response of the finite periodic composite Timoshenko beam-foundation structure by the finite element method (FEM) is performed to verify the above theoretical deduction. Study shows that the Timoshenko beam-foundation structure with periodic composites has wider attenuation zones compared with homogeneous ones. It is concluded that TBT is more available than Euler beam theory (EBT) in the study of the bending vibration characteristic of PCs beam-foundation structures with different length-to-height ratios.

scholarly journals THE REACTIONS OF THE «BEAM – FOUNDATION» SYSTEM TO THE SUDDEN CHANGE OF THE BOUNDARY CONDITIONS

2018 ◽  
Vol 188 ◽  
pp. 03008 ◽  
Author(s):  
Vladimir A. Gordon ◽  
Olga V. Pilipenko ◽  
Vladimir A. Trifonov
Keyword(s):  
Mathematical Model ◽  
Boundary Conditions ◽  
Dynamic Process ◽  
Sudden Change ◽  
Static Problem ◽  
Forced Vibrations ◽  
Winkler Foundation ◽  
Initial Condition ◽  
Foundation Parameters ◽  
Beam Foundation

The authors constructed a mathematical model of a dynamic process in a loaded beam on the elastic Winkler foundation in a sudden formation of a defect in the form of a change in the boundary conditions. The solution of the static problem of bending of the beam pinched at the ends served as the initial condition for the process of forced vibrations hinged supported at the ends of a beam, which arose after a sudden break in the connections that prevented the rotation of the end sections. The authors determined the dynamic increments of stresses in a beam for various combinations of a beam and foundation parameters.

Eigenfunction Solution for Beam on Elastic Foundation

1969 ◽  
Vol 36 (4) ◽  
pp. 799-802 ◽  
Author(s):  
M. S. Hess
Keyword(s):  
Exact Solution ◽  
Elastic Foundation ◽  
Nontrivial Solution ◽  
Normal Displacement ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Complex Eigenvalues ◽  
Adequate Accuracy ◽  
Beam Foundation ◽  
Beam Theories

A solution for the end problem of a rectangular beam resting on a simple elastic foundation is obtained as a series expansion in the eigenfunctions of the system. For a beam aligned with the ξ-axis, the eigenfunctions are of the form eγξf(y), where γ is one of the complex eigenvalues. The eigenvalue equation is determined by requiring continuity of the normal displacement and pressure at each point of the beam-foundation interface and seeking a nontrivial solution. In order to evaluate the accuracy and limitations of several approximate beam theories, the eigenvalue predicted by each of these theories is compared with the first eigenvalue of the exact solution. It is shown that the approximate theories give adequate accuracy if the beam modulus, E, exceeds the foundation stiffness, k, and that all tend to the same result as E/k → ∞. From a comparison of the first and second eigenvalues of the exact solution, it is found that the first mode (corresponding to beam behavior) ceases to dominate the higher modes for E/k < 1. Thus the approximate theories are necessarily restricted to E/k > 1 since they predict only the first mode.

scholarly journals MATHEMATIC MODEL OF A BEAM PARTIALLY SUPPORTED ON ELASTIC FOUNDATION

2019 ◽  
Vol 15 (2) ◽  
pp. 144-158
Author(s):  
Vladimir Travush ◽  
Vladimir Gordom ◽  
Vitaly Kolchunov ◽  
Yevgeny Leontiev
Keyword(s):  
Elastic Foundation ◽  
Strain State ◽  
Mathematic Model ◽  
Middle Part ◽  
Stress Strain ◽  
Dimensional Parameter ◽  
Stress Strain State ◽  
Beam Foundation ◽  
Foundation Structure ◽  
Foundation Failure

The paper presents the methodic for analytical determining of stress-strain state of a beam partially supported on elastic foundation at sudden damage of foundation structure (partial failure). Bending equation for a beam is written using dimensional parameter and solved with the initial parameters method. Such approach al­lows to obtain dimensional analytical solution to static and dynamic problems for universal boundary conditions of a beam since it always leads to equations’ system of second order. Using numerical analysis for various values of generalized stiffness parameter of a system “beam - foundation”, we established affecting of the length of failure foundation part to stress-strain state of the beam for two supporting variants: partial supporting and sup­porting by two ends with foundation failure in the middle part of the beam.

Practical Calculus Methods for Flexural Response of Shallow Beam Foundation

2021 ◽  
Vol 118 (4) ◽  
Author(s):  
Giuseppe Campione ◽  
Francesco Cannella ◽  
Maria Zizzo
Keyword(s):  
Flexural Response ◽  
Beam Foundation
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