Determinant Of Stiffness Matrix Research Articles

Investigating the effect of edge crack on the modal properties of composite wing using dynamic stiffness matrix

In this study free vibration analysis of a cracked Goland composite wing is investigated. The wing is modelled as a cantilevered beam based on Euler- Bernoulli equations. Also, composite material is modelled based on lamina fiber-reinforced. Edge crack is modelled by additional boundary conditions and local flexibility matrix in crack location, Castigliano's theorem and energy release rate formulation. Governing differential equations are extracted by Hamilton's principle. Using the separation of variables method, general solution in the normalized form for bending and torsion deflection is achieved then expressions for the cross-sectional rotation, the bending moment, the shear force and the torsional moment for the cantilevered beam are obtained. The cracked beam is modelled by separation of beam into two interconnected intact beams. Free vibration analysis of the beam is performed by applying boundary conditions at the fixed end, the free end, continuity conditions in the crack location of the beam and dynamic stiffness matrix determinant. Also, the effects of various parameters such as length and location of crack and fiber angle on natural frequencies and mode shapes are studied. Modal analysis results illustrate that natural frequencies and mode shapes are affected by depth and location of edge crack and coupling parameter.

Assessment of the Effect of Wind Load on the Load Capacity of a Single-Layer Bar Dome

The main purpose of the paper was the assessment of the effect of wind load on the load capacity of a single-layer bar dome. Additionally, which numerical method is appropriate for low-rise single-layer bar domes was checked. In order to explain the effect of the height-to-span ratio on the selection of the appropriate calculation model and method of analysis of the bar dome, an example of the known von Mises truss was proposed. Two cases of von Mises truss differing in the height-to-span ratio were considered. For the shallow structure, a significant change in the value of the stiffness matrix determinant and the current stiffness parameter was observed. A similar tendency in the behavior of the structure can be observed on fragments of larger structures, including shallow single-layer steel domes. These problems are described on the basis of the dome, which is located on top of the building housing the restaurant. This structure is subjected to large displacement gradients and the actual configuration is taken into account in analysis. The analysis showed that there is a change in stiffness for these structures, and, therefore, that such structures should be designed according to geometric nonlinear analysis (GNA).

A Method for System Buckling Analysis of Plane Sway Frame

Presented in this paper is a simple and practical method for buckling analysis of the overall structural system. The method is developed from the idea that the stiffness (for a SDOF system) or determinant of stiffness matrix (for a MDOF system) is getting to zero as the system is loaded to bucking mode, or the loads reaches the buckling load of the system. A negative stiffness (matrix) is introduced to take the effect of axial loads on the stiffness (matrix) of the system into account and the stiffness (matrix) is modified by this negative stiffness (matrix). To get the buckling load of the overall system, first order analysis (P-Δeffect) is performed with a simple method suggested. The second order analysis (P-δ) is performed by the introducing of a force modification factor to modify the buckling load from first order analysis. Application examples are presented and the results are compared with result obtained from system buckling analysis with FEA. The simplicity, effectiveness and the accuracy of the suggested method is demonstrated.

A Software of Pre-Buckling Analysis of Laminated Plates and Shells Based on Stiffness Matrix Determinant

The development of computer technology has provided advanced methods for the analysis of complex mechanics problems. Along with the enhancement of computer speed, the computational time used in a finite element analysis is reduced significantly. The main work in a pre-bucking finite element analysis will concentrate on structuring the analytical model and the software model. A normalized factor of the normalized determinant DET of stiffness matrix is defined in this article.The finite element analysis model used in the pre-bucking analysis of laminated composite plates and shells is presented based on the characteristic of the DET. An algorithm for controlling computational procedure and determining critical load is also presented. Numerical examples are given to validate the proposed method and satisfactory results are obtained.

Co-rotational finite element formulation for thin-walled beams with generic open section

A consistent co-rotational total Lagrangian finite element formulation for the geometric nonlinear buckling and postbuckling analysis of thin-walled beams with generic open section is presented. The element developed here has two nodes with seven degrees of freedom per node. The element nodes are chosen to be located at the shear centers of the end cross-sections of the beam element and the shear center axis is chosen to be the reference axis. The deformations of the beam element are described in the current element coordinate system constructed at the current configuration of the beam element. The element nodal forces are derived using the virtual work principle. The virtual rigid body motion corresponding to the virtual nodal displacements is excluded in the derivation of the element nodal forces. A procedure is proposed to determine the virtual rigid body motion. The way used to determine the element coordinate system and element nodal deformations corresponding to the virtual nodal displacements and that corresponding to the incremental nodal displacement are consistent. In element nodal forces, all coupling among bending, twisting, and stretching deformations of the beam element is considered by consistent second-order linearization of the fully geometrically nonlinear beam theory. In the derivation of the element tangent stiffness matrix, the change of element nodal forces induced by the element rigid body rotations should be considered for the present method. Thus, a stability matrix is included in the element tangent stiffness matrix. An incremental-iterative method based on the Newton–Raphson method combined with constant arc length of incremental displacement vector is employed for the solution of nonlinear equilibrium equations. The zero value of the tangent stiffness matrix determinant of the structure is used as the criterion of the buckling state. Numerical examples are presented to investigate the accuracy and efficiency of the proposed method. The effect of the terms in the element nodal force and tangent stiffness matrix, which will converge to zero with the decrease of element size, on the convergence rate of solution and accuracy for the buckling load and nonlinear behavior of three dimensional beam structures are also investigated through numerical examples.

Exact determinant for infinite order FEM representation of a Timoshenko beam–column via improved transcendental member stiffness matrices

AbstractTranscendental stiffness matrices for vibration (or buckling) have been derived from exact analytical solutions of the governing differential equations for many structural members without recourse to the discretization of conventional finite element methods (FEM). Their assembly into the overall dynamic structural stiffness matrix gives a transcendental eigenproblem, whose eigenvalues (natural frequencies or critical load factors) can be found with certainty using the Wittrick–Williams algorithm. A very recently discovered analytical property is the member stiffness determinant, which equals the FEM stiffness matrix determinant of a clamped ended member modelled by infinitely many elements, normalized by dividing by its value at zero frequency (or load factor). Curve following convergence methods for transcendental eigenproblems are greatly simplified by multiplying the transcendental overall stiffness matrix determinant by all the member stiffness determinants to remove its poles. In this paper, the transcendental stiffness matrix for a vibrating, axially loaded, Timoshenko member is expressed in a new, convenient notation, enabling the first ever derivation of its member stiffness determinant to be obtained. Further expressions are derived, also for the first time, for unloaded and for static, loaded Timoshenko members. These new expressions have the advantage that they readily reduce to corresponding expressions for Bernoulli–Euler members when shear rigidity and rotatory inertia are neglected. Additionally, the total equivalence of the normalized transcendental determinant with that of an infinite order FEM formulation aids understanding and evaluation of conventional FEM results. Examples are presented to illustrate the use of the member stiffness determinant. Copyright © 2004 John Wiley & Sons, Ltd.

Derivation of New Transcendental Member Stiffness Determinant for Vibrating Frames

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.

The member stiffness determinant and its uses for the transcendental eigenproblems of structural engineering and other disciplines

Transcendental stiffness matrices are well established in vibration and buckling analysis, having been derived from exact analytical solutions of the differential equations for many structural members without recourse to finite–element discretization. Their assembly into the overall structural stiffness matrix gives a transcendental eigenproblem, solvable with certainty by the Wittrick–Williams (WW) algorithm, instead of the usual linear (algebraic) eigenproblem. This paper establishes a (normalized) member stiffness determinant , being the value of the stiffness matrix determinant if the member were modelled by infinitely many finite elements and its ends were clamped. It is derived for beams with uncoupled axial and Bernoulli–Euler flexural behaviour, by methods applicable to any member possessing transcendental stiffnesses. Multiplying the product of a structure's member stiffness determinants by the determinant of its transcendental overall stiffness matrix gives a determinant, which, unlike the determinant of the transcendental overall stiffness matrix itself, has no poles when plotted against the eigenparameter, leading to a more secure eigenvalue location. Numerical results demonstrate the resulting advantages. In addition, the member stiffness determinant can give exact determinants for members with various standard end conditions and also provides limits for finite–element solutions as the number of elements approaches infinity. Analogous advantages occur in all other disciplines where the WW algorithm is used, e.g. fluid vibrating in pipes, heat and mass diffusion and Sturm–Liouville problems.

Co-rotational formulation for geometric nonlinear analysis of doubly symmetric thin-walled beams

A doubly symmetric thin-walled beam element with open section is derived using co-rotational (CR) total Lagrangian (TL) formulation. The effects of deformation-dependent third-order terms of element nodal forces on the buckling load and post-buckling behavior are investigated. All coupling among bending, twisting, and stretching deformations for beam element is considered by consistent second-order linearization of the fully geometrically nonlinear beam theory. However, all third-order terms of nodal forces, which are relevant to the twist rate, rate of twist rate and curvature of the beam axis are also considered. An incremental-iterative method based on the Newton–Raphson method combined with constant arc length of incremental displacement vector is employed for the solution of nonlinear equilibrium equations. The zero value of the tangent stiffness matrix determinant of the structure is used as the criterion of the buckling state. A parabolic interpolation method of the arc length is used to find the buckling load. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed element and to investigate the effect of third-order terms of element nodal forces on the buckling load and post-buckling behavior of doubly symmetric thin-walled beams.

A co-rotational formulation for thin-walled beams with monosymmetric open section

A consistent co-rotational total Lagrangian finite element formulation and numerical procedure for the geometric nonlinear buckling and postbuckling analysis of thin-walled beams with monosymmetric open section is presented. The element developed here has two nodes with seven degrees of freedom per node. The element nodes are chosen to be located at the shear centers of the end cross-sections of the beam element and the shear center axis is chosen to be the reference axis. The deformations of the beam element are described in the current element coordinate system, which is constructed at the current configuration of the beam element. In element nodal forces, all coupling among bending, twisting, and stretching deformations of the beam element is considered by consistent second-order linearization of the fully geometrically nonlinear beam theory. However, the third-order term of the twist rate of the beam axis is considered in element nodal forces. An incremental-iterative method based on the Newton–Raphson method combined with constant arc length of incremental displacement vector is employed for the solution of nonlinear equilibrium equations. The zero value of the tangent stiffness matrix determinant of the structure is used as the criterion of the buckling state. A parabolic interpolation method of the arc length is used to find the buckling load. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.

A co-rotational finite element formulation for buckling and postbuckling analyses of spatial beams

A consistent co-rotational finite element formulation and numerical procedure for the buckling and postbuckling analyses of three-dimensional elastic Euler beam is presented. All coupling among bending, twisting, and stretching deformations for a beam element is considered by consistent second-order linearization of the fully geometrically nonlinear beam theory. However, the third-order terms, which are relevant to the twist rate and curvature of the beam axis, are also considered. An incremental–iterative method based on the Newton–Raphson method combined with constant arc length of incremental displacement vector is employed for the solution of nonlinear equilibrium equations. The zero value of the tangent stiffness matrix determinant of the structure is used as the criterion of the buckling state. A bisection method of the arc length is proposed to find the buckling load. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method and to investigate the effect of third-order terms on the buckling load and postbuckling behavior of three-dimensional beams.

Flexural-Torsional Buckling of Variable and Open Cross-Section Members

This work gives the exact solutions for the coupled flexural-torsional buckling loads of variable and open cross-section columns, loaded by variable axial force. Both the cross-section dimensions and the axial load can vary along the column as polynomial expressions. The proposed solution is based on a new finite-element method for getting the exact stiffness matrix for the member, including the effects of the axial loading. The buckling load is found as the load that causes the stiffness matrix determinant to become zero. Several examples are given and compared to published results to demonstrate the accuracy and flexibility of the method. New, exact results are given for several other cases.

Elastic lateral torsional buckling of unbraced and braced planar frames

A general finite element formulation to obtain lateral-torsional buckling solution of braced and unbraced planar structures loaded in-plane, is presented. The effects of in-plane deformations on lateral-torsional buckling are included. In-plane deformations are determined by following the loading path in incremental steps, through a series of equilibrium states and the critical load is found by determining the point at which the determinant of stiffness matrix of out-of-plane deformations becomes zero. The validity of the formulation is verified by testing for convergence and comparing the solutions with available results. The behaviour of unbraced and braced frames are investigated using this formulation.

Post-Buckling Analysis of Cylindrical Shells

A method is presented for the calculation of the critical load and post-buckling displacements of elastic structures, and is demonstrated on circular cylindrical shells. The critical load is found by successive extrapolations of the load with the aid of the tangent stiffness matrix determinant. The buckling mode shape is calculated at the instability point and the prebuckling displacements are perturbed a small amount proportional to this mode shape in order to determine an equilibrium state in the post-buckling range. A modified Newton-Raphson iteration technique is utilized to trace the post-buckling load-displacement curve.