Formulation of equivalent uniform beam elements from structural stiffness matrices

1992 ◽  
Author(s):  
T. BAUMANN
Keyword(s):  
Structural Stiffness ◽  
Beam Elements ◽  
Uniform Beam ◽  
Stiffness Matrices

Vehicle-Bridge Interaction System with Non-Uniform Beams

2021 ◽  
pp. 2150170
Author(s):  
Judy P. Yang ◽  
Chun-Hsien Wu
Keyword(s):  
Constant Width ◽  
Cross Sectional ◽  
Scanning Method ◽  
Beam Elements ◽  
Uniform Beam ◽  
Numerical Studies ◽  
Interaction System ◽  
Stiffness Matrices ◽  
Frequency Identification ◽  
Sign Convention

Since the bridge is often treated as the uniform beam for simplicity in most numerical studies of vehicle-bridge interaction, this study proposes a non-uniform vehicle-bridge interaction system by incorporating a three-mass vehicle model in a non-uniform bridge for wider applications, in which non-uniform beam elements of constant width and varying depth are considered. For clarity, the inclined ratios of the entire bridge and one beam element are separately defined in order to describe the non-conformity in computation while both mass and stiffness matrices are re-formulated to comply with the finite element sign convention. As the natural frequencies of a non-uniform bridge cannot be accessed directly, the vehicle scanning method is first adopted to obtain the bridge frequencies. Then, the parametric study is conducted by considering vehicle damping, bridge damping, and pavement irregularity. In addition to the vehicle frequency, the numerical results show that the proposed vehicle-bridge interaction system is able to scan the first four bridge frequencies with desired accuracy subject to pavement irregularity. Concerning the pitching effect of the vehicle, it is shown that the locations for installing sensors are actually affected by both the geometry and the cross-sectional geometry of the bridge in the concern of achieving high resolution of frequency identification.

Derivation of New Transcendental Member Stiffness Determinant for Vibrating Frames

2003 ◽  
Vol 03 (02) ◽  
pp. 299-305 ◽  
Author(s):  
F. W. Williams ◽  
D. Kennedy
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Stiffness Matrix ◽  
Infinite Order ◽  
Natural Frequencies ◽  
Dynamic Stiffness ◽  
Structural Stiffness ◽  
Trial And Error ◽  
Stiffness Matrices ◽  
Vibration Problems

Transcendental dynamic member stiffness matrices for vibration problems arise from solving the governing differential equations to avoid the conventional finite element method (FEM) discretization errors. Assembling them into the overall dynamic structural stiffness matrix gives a transcendental eigenproblem, whose eigenvalues (natural frequencies or their squares) are found with certainty using the Wittrick–Williams algorithm. This paper gives equations for the recently discovered transcendental member stiffness determinant, which equals the appropriately normalized FEM dynamic stiffness matrix determinant of a clamped ended member modelled by infinitely many elements. Multiplying the overall transcendental stiffness matrix determinant by the member stiffness determinants removes its poles to improve curve following eigensolution methods. The present paper gives the first ever derivation of the Bernoulli–Euler member stiffness determinant, which was previously found by trial-and-error and then verified. The derivation uses the total equivalence of the transcendental formulation and an infinite order FEM formulation, which incidentally gives insights into conventional FEM results.

Vector Form Intrinsic Finite Element Method for Analysis of Train–Bridge Interaction Problems Considering The Coach-Coupler Effect

2019 ◽  
Vol 19 (02) ◽  
pp. 1950014 ◽  
Author(s):  
Y. F. Duan ◽  
S. M. Wang ◽  
J. D. Yau
Keyword(s):  
Finite Element ◽  
Resonance Phenomenon ◽  
Vector Form ◽  
Beam Elements ◽  
Suspension Systems ◽  
Stiffness Matrices ◽  
Internal Forces ◽  
Dual Resonance ◽  
Multi Body ◽  
Intense Vibration

In this paper, the vector form intrinsic finite element (VFIFE) method is presented for analysis the train–bridge systems considering the coach-coupler effect. The bridge is discretized into a group of mass particles linked by massless beam elements and the multi-body coach with suspension systems is simulated as a set of mass particles connected by parallel spring-dashpot units. Then the equation of motion of each mass particle is solved individually and the internal forces induced by pure deformations in the massless beam elements are calculated by a fictitious reverse motion method, in which the structural stiffness matrices need not be updated or factorized. Though the vector-form equations resulting from the VFIFE method cannot be used to compute the structural frequencies by the eigenvalue approach, this study proposes a numerical free vibration test to identify the bridge frequencies for evaluating the bridge damping. Numerical verifications demonstrate that the present VFIFE method performs as accurately as previous numerical ones. The results show that the couplers play an energy-dissipating role in reducing the car bodies’ response due to the bridge-induced resonance, but not in their response due to the train-induced resonance because of the bridge’s intense vibration. Meanwhile, a dual-resonance phenomenon in the train–bridge system occurs when the coach-coupler effect is considered in the vehicle model.

Vibration analysis of plane frames by customized stiffness and diagonal mass matrices

2011 ◽  
Vol 225 (12) ◽  
pp. 2848-2863 ◽  
Author(s):  
M Rezaiee-Pajand ◽  
R Khajavi
Keyword(s):  
Vibration Analysis ◽  
Simple Structure ◽  
Frame Structures ◽  
Numerical Examples ◽  
Beam Elements ◽  
Stiffness Matrices ◽  
Mass Matrices ◽  
Plane Frames ◽  
Accurate Performance ◽  
Vibration Problems

This article presents a formulation for the vibration analysis of plane frames. The strain gradient notation is utilized to determine the mass and stiffness matrices. The obtained matrices can easily be parameterized due to their simple structure. Both Euler-Bernoulli- and Timoshenko-beam elements are investigated in this study. The parameterized stiffness and mass matrices are optimized for accurate performance in the vibration analysis of frame structures. Some numerical examples are solved to show the advantages of the presented scheme. Results of these sample vibration problems indicate that the proposed technique increases the accuracy of analysis, when these new stiffness and diagonal mass matrices are used.

scholarly journals An Improved Beam Element for Beams with Variable Axial Parameters

2013 ◽  
Vol 20 (4) ◽  
pp. 601-617 ◽  
Author(s):  
Peng He ◽  
Zhansheng Liu ◽  
Chun Li
Keyword(s):  
Temperature Distribution ◽  
Power Series ◽  
Temperature Difference ◽  
Natural Frequencies ◽  
Beam Element ◽  
Elementary Matrix ◽  
Beam Elements ◽  
Stiffness Matrices ◽  
Axial Temperature ◽  
Matrix Series

The traditional beam element was improved to consider the variable axial parameters. The variable axial parameters were formulated in terms of a power series, and the general forms of elementary mass and stiffness matrices which depend on the power order were derived. The mass and stiffness matrices of the improved beam element were obtained in terms of an elementary matrix series. The beam elements for various tapered beams and a beam under linearly axial temperature distribution were derived. The vibrations of the beams with various taper shapes were studied and the variations of natural frequencies and modal shapes were investigated. A uniform beam under linearly axial temperature distribution was modeled and studied. The influences of axial temperature difference on the natural frequencies and modal shapes were investigated. Results show that the improved beam element could consider the variable axial parameters of beam conveniently.

Lie Group Modeling of Nonlinear Helical Beam Elements

2014 ◽  
Author(s):  
Ralph Jödicke ◽  
Uwe Jungnickel ◽  
Andreas Müller
Keyword(s):  
Lie Group ◽  
Beam Model ◽  
Deflection Curve ◽  
Finite Frames ◽  
Beam Elements ◽  
Group Modeling ◽  
Stiffness Matrices ◽  
Torsion Energy ◽  
Shearing Energy ◽  
Helical Beam

A viscoelastic beam model is presented based on SE(3) group theory. We discretize a rod with beams between finite frames on the rod and regard the configurations of these frames as elements of the SE(3) Lie group. Two subsequent frames are connected by a beam. The curvatures and strains are assumed to be constant on the trajectory between them. If the deflection curve of the beam is modeled as a helix, the resulting beam model is geometrical exact for large bending deformations. The stiffness matrices of the discrete beam elements result from the potential extensional and shearing energy as well as from the potential bending and torsion energy. The benefit of this SE(3) modeling for translational elastic coordinates and for translational forces in comparison to an SO(3) × ℝ3 variant is demonstrated.

Stiffness Matrices for Linear and Buckling Analyses of Composite Beams with Partial Shear Connection

2006 ◽  
Vol 22 (4) ◽  
pp. 299-309 ◽  
Author(s):  
L.-J. Leu ◽  
C.-W. Huang
Keyword(s):  
Composite Beams ◽  
Transverse Displacement ◽  
The Finite Element Method ◽  
Shape Functions ◽  
Shear Connection ◽  
Beam Elements ◽  
Geometric Stiffness ◽  
Stiffness Matrices ◽  
Different Types ◽  
Exact Shape

AbstractThis paper is concerned with linear and buckling analyses of composite beams with partial shear connection (partial composite beams) using the finite element method. Two elements derived from different types of shape functions are proposed in this study. The first element, referred to as exact, is based on the exact shape functions obtained by solving the differential equations governing the transverse displacement and the slip of the shear connector layer of a partial composite beam. The second element, referred to as approximate, is based on the conventional linear and cubic shape functions used in conventional axial and beam elements. By making use of these two types of shape functions, the elastic and geometric stiffness matrices can be derived explicitly from the strain energy and the load potential, respectively. Both types of elements can be used to carry out linear static and buckling analyses. As expected, the exact element is more accurate than the approximate element if the same discretization is adopted. However, the approximate element has the advantage of easy implementation since the expressions of its elastic and geometric stiffness matrices are very simple. Also, the solutions obtained from the approximate element converge very fast; with reasonable discretization, say 8 elements per member, very accurate solutions can be obtained.

Reducing the Bandwidth of Structural Stiffness Matrices

1976 ◽  
Vol 4 (2) ◽  
pp. 197-226 ◽  
Author(s):  
Ichiro Konishi ◽  
Naruhito Shiraishi ◽  
Takeo Taniguchi
Keyword(s):  
Structural Stiffness ◽  
Stiffness Matrices
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