• Title, Summary, Keyword: matrix stiffness method

### A simple method of stiffness matrix formulation based on single element test

• Mau, S.T.
• Structural Engineering and Mechanics
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• v.7 no.2
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• pp.203-216
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• 1999
• A previously proposed finite element formulation method is refined and modified to generate a new type of elements. The method is based on selecting a set of general solution modes for element formulation. The constant strain modes and higher order modes are selected and the formulation method is designed to ensure that the element will pass the basic single element test, which in turn ensures the passage of the basic patch test. If the element is to pass the higher order patch test also, the element stiffness matrix is in general asymmetric. The element stiffness matrix depends only on a nodal displacement matrix and a nodal force matrix. A symmetric stiffness matrix can be obtained by either modifying the nodal displacement matrix or the nodal force matrix. It is shown that both modifications lead to the same new element, which is demonstrated through numerical examples to be more robust than an assumed stress hybrid element in plane stress application. The method of formulation can also be used to arrive at the conforming displacement and hybrid stress formulations. The convergence of the latter two is explained from the point of view of the proposed method.

### An Analysis of Cylindrical Tank of Elastic Foundation by Transfer Matrix and Stiffness Matrix (전달행렬과 강성행렬에 의한 탄성지반상의 원형탱크해석)

• 남문희;하대환;이관희;장홍득
• Computational Structural Engineering
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• v.10 no.1
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• pp.193-200
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• 1997
• Even though there are many analysis methods of circular tanks on elastic foundation, the finite element method is widely used for that purpose. But the finite element method requires a number of memory spaces, computation time to solve large stiffness equations. In this study many the simplified methods(Analogy of Beam on Elastic Foundation, Foundation Stiffness Matrix, Finite Element Method and Transfer Matrix Method) are applied to analyze a circular tank on elastic foundation. By the given analysis methods, BEF analogy and foundation matrix method, the circular tank was transformed into the skeletonized frame structure. The frame structure was divided into several finite elements. The stiffness matrix of a finite element is related with the transfer matrix of the element. Thus, the transfer matrix of each finite element utilized the transfer matrix method to simplify the analysis of the tank. There were no significant difference in the results of two methods, the finite element method and the transfer matrix method. The transfer method applied to a circular tank on elastic foundation resulted in four simultaneous equations to solve completely.

### Structural Dynamics Analysis of a Clamp Jointed Complex Ream by Using the Flexibility Influence Coefficient Method (유연도 영향계수법을 이용한 접촉결합부가 있는 복합구조물의 동적 해석)

• 조재혁;김현욱;최영휴
• Proceedings of the Korean Society of Precision Engineering Conference
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• pp.528-533
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• 1995
• An analyical method is proposed to construct a clamp jointed structure as an equivalent stiffness matrix element in the finite element modal analysis of a complex beam structure. Static structural analysis was first made for the detail finite element model of the clamp joint. Utilizing the results of this analysis, the equivalent stiffness matrix element was buildup by using the flexibility influence coefficient method and Guyan condensation. The proposed method was applied to finite element modal analysis of a clamp jointed cantilever beam. And the finite element analysis results were compared to those experimental modal analysis. Comparison shows doog agreement each other Furthermore the effects of normal contact(or clamping) load on the equivalent stiffness matrix was also examined. The equivalent stiffness matrix showed little change in spite of the remakable increase in the contact load on the clamp joint.

### Improved Numerical Method Evaluating Exact Static Element Stiffness Matrices of Beam on Elastic Foundations (탄성지반위의 보의 엄밀한 강성계산을 위한 개선된 해석방법)

• Kim Nam-Il;Lee Jun-Seok;Kim Moon-Young
• Proceedings of the Computational Structural Engineering Institute Conference
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• pp.589-596
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• 2006
• An improved numerical method to obtain the exact element stiffness matrix is newly proposed to perform the spatially coupled elastic and stability analyses of non-symmetric thin-walled beam-columns with two-types of elastic foundation. This method overcomes drawbacks of the previous method to evaluate the exact stiffness matrix for the spatially coupled stability analysis of thin-walled beam-column. This numerical technique is firstly accomplished via a generalized eigenproblem associated with 14 displacement parameters by transforming equilibrium equations to a set of first order simultaneous ordinary differential equations. Then exact displacement functions are constructed by combining eigensolutions and polynomial solutions corresponding to non-zero and zero eigenvalues, respectively. Consequently an exact stiffness matrix is evaluated by applying the member force-deformation relationships to these displacement functions.

### A simplified geometric stiffness in stability analysis of thin-walled structures by the finite element method

• International Journal of Naval Architecture and Ocean Engineering
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• v.4 no.3
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• pp.313-321
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• 2012
• Vibration analysis of a thin-walled structure can be performed with a consistent mass matrix determined by the shape functions of all degrees of freedom (d.o.f.) used for construction of conventional stiffness matrix, or with a lumped mass matrix. In similar way stability of a structure can be analysed with consistent geometric stiffness matrix or geometric stiffness matrix with lumped buckling load, related only to the rotational d.o.f. Recently, the simplified mass matrix is constructed employing shape functions of in-plane displacements for plate deflection. In this paper the same approach is used for construction of simplified geometric stiffness matrix. Beam element, and triangular and rectangular plate element are considered. Application of the new geometric stiffness is illustrated in the case of simply supported beam and square plate. The same problems are solved with consistent and lumped geometric stiffness matrix, and the obtained results are compared with the analytical solution. Also, a combination of simplified and lumped geometric stiffness matrix is analysed in order to increase accuracy of stability analysis.

### A Study on the Coupled Torsional-Axial Vibration of Marine Propulsion Shafting System using the Energy Method

• Jang, Min-Oh;Kim, Ue-Kan;Park, Yong-Nam;Lee, Young-Jin
• Journal of Advanced Marine Engineering and Technology
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• v.28 no.3
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• pp.482-492
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• 2004
• Recently. the market trend for marine diesel engine has involved the lower running speeds. larger stroke/bore ratio and higher combustion pressure. Consequently, because of the flexible engine shafting system due to the larger mass. inertia and the more elasticity, the complicated coupled torsional-axial vibrations have occurred in the operating speed range. Also, the vibrations act as an excitation on the hull-structural vibration. To predict the vibration behavior with more accuracy and reliability. many studies have proposed the several kinds of method to calculate the stiffness matrix of crankshaft. However, most of these methods have a weak point to spend much time on three dimensional modeling and meshing work for crankshaft. Therefore. in this work. the stiffness matrix for the crankthrow is calculated using the energy method (Influence Coefficient Method, ICM) with the each mass having 6 degree of freedom. Its effectiveness is verified through the comparison with the stiffness matrix obtained by using the finite element method (FEM) and measured results for actual ships propulsion system.

### Dynamic stiffness matrix method for axially moving micro-beam

• Movahedian, Bashir
• Interaction and multiscale mechanics
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• v.5 no.4
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• pp.385-397
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• 2012
• In this paper the dynamic stiffness matrix method was used for the free vibration analysis of axially moving micro beam with constant velocity. The extended Hamilton's principle was employed to derive the governing differential equation of the problem using the modified couple stress theory. The dynamic stiffness matrix of the moving micro beam was evaluated using appropriate expressions of the shear force and bending moment according to the Euler-Bernoulli beam theory. The effects of the beam size and axial velocity on the dynamic characteristic of the moving beam were investigated. The natural frequencies and critical velocity of the axially moving micro beam were also computed for two different end conditions.

### Second-order analysis of planar steel frames considering the effect of spread of plasticity

• Leu, Liang-Jenq;Tsou, Ching-Huei
• Structural Engineering and Mechanics
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• v.11 no.4
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• pp.423-442
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• 2001
• This paper presents a method of elastic-plastic analysis for planar steel frames that provides the accuracy of distributed plasticity methods with the computational efficiency that is greater than that of distributed plasticity methods but less than that of plastic-hinge based methods. This method accounts for the effect of spread of plasticity accurately without discretization through the cross-section of a beam-column element, which is achieved by the following procedures. First, nonlinear equations describing the relationships between generalized stresses and strains of the cross-section are derived analytically. Next, nonlinear force-deformation relationships for the beam-column element are obtained through lengthwise integration of the generalized strains. Elastic-plastic flexibility coefficients are then calculated by differentiating the above element force-deformation relationships. Finally, an elastic-plastic stiffness matrix is obtained by making use of the flexibility-stiffness transformation. Adding the conventional geometric stiffness matrix to the elastic-plastic stiffness matrix results in the tangent stiffness matrix, which can readily be used to evaluate the load carrying capacity of steel frames following standard nonlinear analysis procedures. The accuracy of the proposed method is verified by several examples that are sensitive to the effect of spread of plasticity.

### Vibration Analysis of a Coil Spring by Using Dynamic Stiffness Method (동강성법을 이용한 코일스프링의 진동 해석)

• Lee, Jae-Hyung;Kim, Seong-Keol;Heo, Seung-Jin;Thompson, D.J.
• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• pp.1933-1938
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• 2000
• The partial differential equations for a coil spring derived from Timoshenko beam theory and Frenet formulae. Dynamic stiffness matrix of a coil spring composed of a circular wire is assembled by using dispersion relationship, waves and natural frequencies. Natural frequencies are obtained from maxima in the determinant of inverse of a dynamic stiffness matrix with appropriate boundary conditions. The results of the dynamic stiffness method are compared with those of transfer matrix method, finite element method and test.

### Prediction of the Damage in the Structure with Damping Using the Modified Dynamic Characteristics (동특성 변화를 이용한 감쇠 구조물의 손상예측)

• Lee, Jung Youn
• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.22 no.11
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• pp.1144-1151
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• 2012
• A damage in structure alters its dynamic characteristics. The change is characterized by changes in the modal parameter, i.e., modal frequencies, modal damping value and mode shape associated with each modal frequency. Changes also occur in some of the structural parameters; namely, the mass, damping, stiffness matrices of the structure. In this paper, evaluation of changes in stiffness matrix of a structure is presented as a method not only for identifying the presence of the damage but also locating the damage. It is shown that changed stiffness matrix can be accurately estimated a sensitivity coefficient matrix derived from modifying mode shapes, First, with 4 story shear structure models, the effect of presence of damage in a structure on its stiffness matrix is studied. By using these analytical model, the effectiveness of using change of stiffness matrix in detecting and locating damages is demonstrated. To validate the predicted changing stiffness and its location, the obtained results are compared to the reanalysis result which shows good agreement.