Flexibility Matrix and Stiffness Matrix

2020 ◽  
pp. 205-216
Author(s):  
A.I. Rusakov
Keyword(s):  
Stiffness Matrix ◽  
Flexibility Matrix

scholarly journals Crack Detection through the Change in the Normalized Frequency Shape

Vibration ◽  
2018 ◽  
Vol 1 (1) ◽  
pp. 56-68
Author(s):  
Mustapha Dahak ◽  
Noureddine Touat ◽  
Tarak Benkedjouh
Keyword(s):  
Stiffness Matrix ◽  
Electrical Discharge ◽  
Inverse Method ◽  
Crack Detection ◽  
Natural Frequencies ◽  
Electrical Discharge Machining ◽  
Element Model ◽  
Experimental Application ◽  
Flexibility Matrix ◽  
Crack Position

The objective of this work is to use natural frequencies for the localization and quantification of cracks in beams. First, to study the effect of the crack on natural frequencies, a finite element model of Euler–Bernoulli is presented. Concerning the damaged element, the stiffness matrix is calculated by the theory of fracture mechanics, by inverting the flexibility matrix. Then, in order to detect damage, we are going to show that the shape given by the change in the natural frequencies is as function of the damage position only. Thus, the crack is located by the correlation between the shape of the measured frequencies and those obtained by the finite elements, where the position that gives the calculated shape which is the most similar to the measured one, indicates the crack position. After the localization, an inverse method will be applied to quantify the damage. Finally, an experimental application is presented to show the real applicability of the method, in which the crack is introduced by using an Electrical Discharge Machining. The results confirm the applicability of the method for the localization and the quantification of cracks.

scholarly journals Exact Stiffness for Beams on Kerr-Type Foundation: The Virtual Force Approach

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Suchart Limkatanyu ◽  
Woraphot Prachasaree ◽  
Nattapong Damrongwiriyanupap ◽  
Minho Kwon ◽  
Wooyoung Jung
Keyword(s):  
Stiffness Matrix ◽  
Element Stiffness Matrix ◽  
Flexibility Matrix ◽  
Force Equation ◽  
Virtual Force ◽  
Proposed Model ◽  
The Matrix ◽  
Natural Element ◽  
The Individual ◽  
Interpolation Functions

This paper alternatively derives the exact element stiffness equation for a beam on Kerr-type foundation. The shear coupling between the individual Winkler-spring components and the peripheral discontinuity at the boundaries between the loaded and the unloaded soil surfaces are taken into account in this proposed model. The element flexibility matrix is derived based on the virtual force principle and forms the core of the exact element stiffness matrix. The sixth-order governing differential compatibility of the problem is revealed using the virtual force principle and solved analytically to obtain the exact force interpolation functions. The matrix virtual force equation is employed to obtain the exact element flexibility matrix based on the exact force interpolation functions. The so-called “natural” element stiffness matrix is obtained by inverting the exact element flexibility matrix. One numerical example is utilized to confirm the accuracy and the efficiency of the proposed beam element on Kerr-type foundation and to show a more realistic distribution of interactive foundation force.

Analysis of Reinforced Concrete Column Using a Beam-Column Element with a Meshed Section

2011 ◽  
Vol 255-260 ◽  
pp. 1954-1958
Author(s):  
Ling Yuan Zhou ◽  
Qiao Li
Keyword(s):  
Reinforced Concrete ◽  
Stiffness Matrix ◽  
Concrete Beam ◽  
Reinforced Concrete Beam ◽  
Beam Element ◽  
Reinforced Concrete Column ◽  
Flexibility Matrix ◽  
Stiffness Matrices ◽  
Column Element ◽  
Interpolation Functions

A efficient 3D reinforced-concrete beam element based on the flexibility method and distributed nonlinearity theory is proposed, The sections of the beam element are divided into the plane isoparametric elements in this formulation, the section stiffness matrices are calculated through the integration of stress-strain relations of concrete including reinforcing steel effect in the section. The flexibility matrices of the sections are calculated by inverting the stiffness matrices, and the element flexibility matrix is formed through the force interpolation functions. The element stiffness matrix is evaluated through the element flexibility matrix. Finally, the buckling behaviors of a reinforced concrete beam under various eccentric loads are analyzed with the proposed formulation to illustrate its accuracy and computational efficiency.

Vibration Analysis of Rotor With Transverse Surface Cracks

2003 ◽  
Author(s):  
Dan Guo ◽  
Fu-Lei Chu ◽  
Yong-Yong He
Keyword(s):  
Stiffness Matrix ◽  
Vibration Analysis ◽  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Surface Cracks ◽  
Breathing Crack ◽  
Cracked Rotor ◽  
Six Degrees Of Freedom ◽  
Flexibility Matrix ◽  
Vibration Responses

The vibration of cracked rotor is investigated by numerical method. The FEM is used to model the rotor with cracks. Six degrees of freedom are considered in each elemental node. Full 6×6 flexibility matrix is deduced by Papadopoulos and Dimarogonas’ method, and 12×12 stiffness matrix of cracked element is derived. The influence of one or more cracks on the natural frequencies and different modals (including bending modal, torsion modal and longitudinal modal) of cracked rotor is explored. Vibration responses of rotor with open cracks or breathing crack loading by eccentric force and rotor gravity force are obtained and analyzed by numerical integer method and spectral technology. The coupling of lateral, longitudinal and torsion vibrations due to transverse surface crack is studied. It is concluded that the above research is useful in detecting crack in rotor.

scholarly journals Vertical Dynamic Response of Rock-Socketed Piles in a Layered Foundation

2020 ◽  
Vol 173 ◽  
pp. 04002
Author(s):  
Chun Lin Liu ◽  
Shuo Zhang ◽  
Meng Xiong Tang ◽  
He Song Hu ◽  
Zhen Kun Hou ◽  
...  
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Dynamic Response ◽  
Stiffness Matrix ◽  
Dynamic Stiffness ◽  
Matrix Equations ◽  
The Finite Element Method ◽  
Flexibility Matrix ◽  
Simplified Method ◽  
Exact Stiffness Matrix

A simplified method is presented to investigate the dynamic response of rock-socketed piles embedded in a layered foundation. The finite element method is utilized to derive the dynamic stiffness matrix equations of the pile modelled as a 1D bar, and the exact stiffness matrix method is employed to establish the flexibility matrix equations of the foundation modelled as a 3D body. According to the pilesoil interaction condition, these matrices are incorporated together to obtain the solution for the dynamic response of rock-socketed piles. Finally, some numerical results are given to illustrate the influence of rocksocketed depth on the pile vertical impedance.

Spatial combined cable element for cable-supported bridges

2018 ◽  
Vol 36 (1) ◽  
pp. 204-225 ◽  
Author(s):  
Jiandong Wei ◽  
Manyu Guan ◽  
Qi Cao ◽  
Ruibin Wang
Keyword(s):  
Stiffness Matrix ◽  
Design Methodology ◽  
Element Model ◽  
Suspension Bridges ◽  
Force Vector ◽  
Content Type ◽  
Flexibility Matrix ◽  
Arch Bridges ◽  
The Finite Element Model ◽  
Cable Stayed Bridges

Purpose The purpose of this paper is to analyze the cable-supported bridges more efficiently by building the finite element model with the spatial combined cable element. Design/methodology/approach The spatial combined cable element with rigid arms and elastic segments was derived. By using the analytical solution of the elastic catenary to establish the flexibility matrix at the end of the cable segment and adding it to the flexibility matrix at the ends of the two elastic segments, the flexibility matrix at the end of the cable body is obtained. Then the stiffness matrix of the cable body is established and the end force vector of cable body is given. Using the displacement transformation relationship between the two ends of the rigid arm, the stiffness matrix of the combined cable element is derived. By assigning zero to the length of the elastic segment(s) or/and the rigid arm(s), many subdivisions of the combined cable element can be obtained, even the elastic catenary element. Findings The examples in this field and specially designed examples proved the correctness of the proposed spatial combined cable element. Originality/value The combined cable element proposed in this study can be used for the design and analysis of cable-stayed bridges. Case studies show that it is able to simulate cable accurately and could also be used to simulate the suspenders in arch bridges as well in suspension bridges.

Shells With Edge Loads of Rapid Variation—II

1965 ◽  
Vol 32 (1) ◽  
pp. 87-98 ◽  
Author(s):  
C. R. Steele
Keyword(s):  
Stiffness Matrix ◽  
Thin Shell ◽  
Wave Length ◽  
Elastic Shell ◽  
General Character ◽  
Membrane Theory ◽  
Rapid Variation ◽  
Flexibility Matrix ◽  
Shear Beam ◽  
Interior Stress

The previously obtained solution for the thin elastic shell with edge loads whose wave-length is smaller than the shell radii of curvature is shown to provide a transition from “thin-shell” behavior to “flat-plate” behavior. The general character of the interior stress distribution is discussed in this paper. An examination is made of the properties of the “stiffness” matrix, which give the edge stresses due to prescribed edge displacements, and the properties of the inverse (“flexibility”) matrix. A solution is provided for the paradax of the shell of negative curvature with “shear-beam” supported edges, for which the membrane theory indicates a spectrum of edge loads that produce infinite stress.

scholarly journals TWO-NODE CATENARY CABLE ELEMENT WITH RIGID-END EFFECT AND CABLE SHAPE ANALYSIS

2011 ◽  
Vol 11 (03) ◽  
pp. 563-580 ◽  
Author(s):  
Y. B. YANG ◽  
JIUNN-YIN TSAY
Keyword(s):  
Structural Analysis ◽  
Shape Analysis ◽  
Stiffness Matrix ◽  
Suspension Bridge ◽  
Trial And Error ◽  
End Effect ◽  
Flexibility Matrix ◽  
Stay Cables ◽  
The Dead ◽  
Main Cables

The effect of rigid ends is considered in the formulation of a general two-node cable element for the analysis of cable-supported structures. The stiffness matrix of the catenary cable element was derived as the inverse of the flexibility matrix, with allowances for self-weight and pretension effects. In modeling the cables of suspension bridge, distinction is made between single cables (e.g., stay cables and hangers) and multi segment cables (e.g., main cables). The unstressed length of each cable element in terms of the pretension force is determined by a trial-and-error procedure prior to structural analysis. Cable shape analysis was conducted to determine the configuration of main cables for cable-supported bridges under the dead loads. It was demonstrated that the effect of rigid ends cannot be ignored for taut cables, that is, cables with large pretensions. Further, the cable element derived can be reliably used in the analysis of cable-supported bridges, regardless of the sag magnitudes.

scholarly journals Spring Cell Equivalence of Simplex Finite Elements – Exploration of an Iterative Approach

2021 ◽  
Vol 15 ◽  
pp. 222-235
Author(s):  
Ioannis Doltsinis
Keyword(s):  
Finite Elements ◽  
Stiffness Matrix ◽  
Iterative Approach ◽  
Convergence Criteria ◽  
Element Level ◽  
Flexibility Matrix ◽  
Triangular Elements ◽  
Natural Spring ◽  
Set Up

Natural spring cell substitutes of triangular and tetrahedral finite elements at constant strain take advantage of a formalism oriented along the element sides/edges. Two different models in use account just for the diagonal entities of either the flexibility matrix of the element or of its stiffness matrix. Both are incomplete substitutes, and defective to a degree depending on the significance of the off-diagonal parts of the element matrices. The present work discusses an iterative completeness of the substitution accounting for the discarded parts by additives to the spring members of the cell. In this connection, the iteration schemes are set up for either model at the material and at the element level, and convergence criteria are defined in terms of the spectral radii of the iteration operators. The convergence regions are confined for triangular elements, and are demonstrated with reference to a case study.

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