scholarly journals An Alternative Approach to Obtaining Stiffness and Mass Matrices for Three-Dimensional Bernoulli-Euler and Timoshenko Beam-Columns

2021 ◽  
Vol 10 (4) ◽  
pp. 253-268
Author(s):  
Ruhi Aydin
Keyword(s):  
Timoshenko Beam ◽  
Mass Matrix ◽  
Tensile Force ◽  
Three Dimensional ◽  
Beam Axis ◽  
Order Effects ◽  
Shape Functions ◽  
Stability Parameters ◽  
Beam Columns ◽  
Mass Matrices

In the static analysis of beam-column systems using matrix methods, polynomials are using as the shape functions. The transverse deflections along the beam axis, including the axial- flexural effects in the beam-column element, are not adequately described by polynomials. As an alternative method, the element stiffness matrix is modeling using stability parameters. The shape functions which are obtaining using the stability parameters are more compatible with the system’s behavior. A mass matrix used in the dynamic analysis is evaluated using the same shape functions as those used for derivations of the stiffness coefficients and is called a consistent mass matrix. In this study, the stiffness and consistent mass matrices for prismatic three-dimensional Bernoulli-Euler and Timoshenko beam-columns are proposed with consideration for the axial-flexural interactions and shear deformations associated with transverse deflections along the beam axis. The second-order effects, critical buckling loads, and eigenvalues are determined. According to the author’s knowledge, this study is the first report of the derivations of consistent mass matrices of Bernoulli-Euler and Timoshenko beam-columns under the effect of axially compressive or tensile force.

Frequency-Dependent Element Mass Matrices

1992 ◽  
Vol 59 (1) ◽  
pp. 136-139 ◽  
Author(s):  
N. J. Fergusson ◽  
W. D. Pilkey
Keyword(s):  
Stiffness Matrix ◽  
Shape Function ◽  
Mass Matrix ◽  
Power Series Expansion ◽  
Shape Functions ◽  
Frequency Dependent ◽  
Mass Matrices ◽  
The Matrix ◽  
Matrix Expansion ◽  
Consistent Mass Matrix

This paper considers some of the theoretical aspects of the formulation of frequency-dependent structural matrices. Two types of mass matrices are examined, the consistent mass matrix found by integrating frequency-dependent shape functions, and the mixed mass matrix found by integrating a frequency-dependent shape function against a static shape function. The coefficients in the power series expansion for the consistent mass matrix are found to be determinable from those in the expansion for the mixed mass matrix by multiplication by the appropriate constant. Both of the mass matrices are related in a similar manner to the coefficients in the frequency-dependent stiffness matrix expansion. A formulation is derived which allows one to calculate, using a shape function truncated at a given order, the mass matrix expansion truncated at twice that order. That is the terms for either of the two mass matrix expansions of order 2n are shown to be expressible using shape functions terms of order n. Finally, the terms in the matrix expansions are given by formulas which depend only on the values of the shape function terms at the boundary.

A Meshfree Higher Order Mass Matrix Formulation for Structural Vibration Analysis

2018 ◽  
Vol 18 (10) ◽  
pp. 1850121 ◽  
Author(s):  
Junchao Wu ◽  
Dongdong Wang ◽  
Zeng Lin
Keyword(s):  
Vibration Analysis ◽  
Mass Matrix ◽  
Meshfree Method ◽  
Higher Order ◽  
Structural Vibration ◽  
Frequency Error ◽  
Shape Functions ◽  
Matrix Formulation ◽  
Order Accuracy ◽  
Mass Matrices

An accurate meshfree formulation with higher order mass matrix is proposed for the structural vibration analysis with particular reference to the 1D rod and 2D membrane problems. Unlike the finite element analysis with an explicit mass matrix, the mass matrix of Galerkin meshfree formulation usually does not have an explicit expression due to the rational nature of meshfree shape functions. In order to develop a meshfree higher order mass matrix, a frequency error measure is derived by using the entries of general symmetric stiffness and mass matrices. The frequency error is then expressed as a series expansion of the nodal distance, in which the coefficients of each term are related to the meshfree stiffness and mass matrices. It is theoretically proved that the constant coefficient in the frequency error vanishes identically provided with the linear completeness condition, which does not rely on any specific form of the shape functions. Furthermore, a meshfree higher order mass matrix is developed through a linear combination of the consistent and lumped mass matrices, in which the optimal mass combination coefficient is attained via eliminating the lower order error terms. In particular, the proposed higher order mass matrix with Galerkin meshfree formulation achieves a fourth-order accuracy when the moving least squares or reproducing kernel (RK) meshfree approximation with linear basis function is employed; nonetheless, the conventional meshfree method only gives a second-order accuracy for the frequency computation. In the multidimensional formulation, the optimal mass combination coefficient is a function of the wave propagation angle so that the proposed accurate meshfree method is applicable to the computation of frequencies associated with any wave propagation direction. The superconvergence of the proposed meshfree higher order mass matrix formulation is validated via numerical examples.

scholarly journals SHAPE FUNCTIONS OF THREE-DIMENSIONAL TIMOSHENKO BEAM ELEMENT

2003 ◽  
Vol 259 (2) ◽  
pp. 473-480 ◽  
Author(s):  
A. BAZOUNE ◽  
Y.A. KHULIEF ◽  
N.G. STEPHEN
Keyword(s):  
Timoshenko Beam ◽  
Three Dimensional ◽  
Beam Element ◽  
Shape Functions ◽  
Timoshenko Beam Element

A consistent mass matrix for three-dimensional beam-columns

1994 ◽  
Vol 51 (5) ◽  
pp. 547-557
Author(s):  
A.H. Namini ◽  
H. Barhoush ◽  
M.W. Fahmy
Keyword(s):  
Mass Matrix ◽  
Three Dimensional ◽  
Beam Columns ◽  
Consistent Mass Matrix

Mean-Axis Conditions for Finite Elements Undergoing Large Rotations and Translations

1997 ◽  
Author(s):  
Venkata R. Sonti ◽  
Om P. Agrawal
Keyword(s):  
Finite Elements ◽  
Mass Matrix ◽  
Three Dimensional ◽  
The Body ◽  
Large Rotation ◽  
Mass Matrices ◽  
Large Rotations ◽  
The Individual ◽  
Rectangular Plate Element ◽  
Individual Mass

Abstract The mean-axis conditions for three types of finite elements that undergo large rotation and translation are derived. The cases include two Lagrangian planar elements (linear triangle and linear rectangle) and a three dimensional rectangular plate element. Each case includes two elements connected at the nodes. Three separate coordinate systems are used in describing the systems. The numerical values of different matrices including the mass matrix are presented for specific cases. It may be seen that the non-linear mass matrix is expressed in terms of a set of time-invariand matrices. The total mass matrix of the body i is obtained by assembling the individual mass matrices of the finite elements of body i.

scholarly journals Numerical Analysis of the Perforated Steel Sheets Under Uni-Axial Tensile Force

Metals ◽  
2019 ◽  
Vol 9 (6) ◽  
pp. 632 ◽  
Author(s):  
Ahmed M. Sayed
Keyword(s):  
Load Capacity ◽  
Tensile Force ◽  
Three Dimensional ◽  
Tensile Load ◽  
Experimental Tests ◽  
Strain Engineering ◽  
Uniaxial Tensile ◽  
Fe Model ◽  
Stress Strain ◽  
Steel Sheets

The perforated steel sheets have many uses, so they should be studied under the influence of the uniaxial tensile load. The presence of these holes in the steel sheets certainly affects the mechanical properties. This paper aims at studying the behavior of the stress-strain engineering relationships of the perforated steel sheets. To achieve this, the three-dimensional finite element (FE) model is mainly designed to investigate the effect of this condition. Experimental tests were carried out on solid specimens to be used in the test of model accuracy of the FE simulation. Simulation testing shows that the FE modeling revealed the ability to calculate the stress-strain engineering relationships of perforated steel sheets. It can be concluded that the effect of a perforated rhombus shape is greater than the others, and perforated square shape has no effect on the stress-strain engineering relationships. The efficiency of the perforated staggered or linearly distribution shapes with the actual net area on the applied loads has the opposite effect, as it reduces the load capacity for all types of perforated shapes. Despite the decrease in load capacity, it improves the properties of the steel sheets.

Static Timoshenko Beam‐Columns on Elastic Media

1988 ◽  
Vol 114 (5) ◽  
pp. 1152-1172 ◽  
Author(s):  
Franklin Y. Cheng ◽  
Chris P. Pantelides
Keyword(s):  
Timoshenko Beam ◽  
Elastic Media ◽  
Beam Columns

Consistent Hydrodynamic Mass for Parallel Prismatic Beams in a Fluid-Filled Container

1984 ◽  
Vol 106 (3) ◽  
pp. 270-275
Author(s):  
J. F. Loeber
Keyword(s):  
Finite Element ◽  
Dynamic Response ◽  
Incompressible Fluid ◽  
Finite Element Methods ◽  
Mass Matrix ◽  
Three Dimensional ◽  
Beam Length ◽  
Step Process ◽  
Beam Bending

In this paper, representation of the effects of incompressible fluid on the dynamic response of parallel beams in fluid-filled containers is developed using the concept of hydrodynamic mass. Using a two-step process, first the hydrodynamic mass matrix per unit (beam) length is derived using finite element methods with a thermal analogy. Second, this mass matrix is distributed in a consistent mass fashion along the beam lengths in a manner that accommodates three-dimensional beam bending plus torsion. The technique is illustrated by application to analysis of an experiment involving vibration of an array of four tubes in a fluid-filled cylinder.

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