Tangent Stiffness Matrix of Spatial Beam Element Considering Bend and Torsion Coupling Effect

2014 ◽  
Vol 1021 ◽  
pp. 73-78
Author(s):  
Liang Du ◽  
Peng Lan ◽  
Nian Li Lu
Keyword(s):  
Stiffness Matrix ◽  
Coupling Effect ◽  
Beam Element ◽  
High Accuracy ◽  
Fem Model ◽  
Tangent Stiffness ◽  
Tangent Stiffness Matrix ◽  
Spatial Beam ◽  
Element Pattern ◽  
Couple Effect

Based on the spatial beam-column differential equations, the slope deflection equations considering second-order and bend-torsion coupling effect are established. The additional moment caused by torsion and bend deflection are taken into account. Then the finite element pattern of spatial beam-column considering the couple effect of torsion and bend is given. The tangent stiffness matrix and relevant program for nonlinear analysis are further obtained. By the nonlinear calculation and stability analysis of single component which has high accuracy or precise solution, the precision of the FEM model given in this paper is verified by comparing the results with that given in references.

DESIGN OF LATERAL BRACES FOR COLUMNS CONSIDERING CRITICAL IMPERFECTION OF BUCKLING

2004 ◽  
Vol 04 (01) ◽  
pp. 69-88 ◽  
Author(s):  
J. TAKAGI ◽  
M. OHSAKI
Keyword(s):  
Stiffness Matrix ◽  
Load Condition ◽  
Buckling Modes ◽  
Tangent Stiffness ◽  
Tangent Stiffness Matrix ◽  
Total Potential ◽  
Column Type ◽  
Global Buckling ◽  
Imperfect System ◽  
Allowable Stress Design

The present paper discusses the design of column-type structures, which are composed of columns and lateral braces attached perpendicular to the columns. Buckling of the braces of this kind of structures directly leads to global buckling of the columns. The brace-buckling modes are successfully detected by considering higher-order geometrically nonlinear relations and by introducing Green's strain into the total potential energy of the structure. Sensitivity analysis of the eigenvalues of the tangent stiffness matrix under fixed load condition is carried out with respect to imperfections of the nodal locations. Furthermore, the critical imperfection that most drastically reduces the eigenvalues are calculated and buckling loads of the imperfect systems are evaluated. The numerical results show that the second or higher eigenmode of the tangent stiffness matrix of the perfect system should be sometimes used for estimating the buckling load of the imperfect system. Design examples are presented using the proposed method, and they are compared with those in accordance with an allowable-stress design standard. The results show a possibility of reducing the sizes of the brace sections.

scholarly journals Effects of shear deformation on large deflection behavior of elastic frames

2004 ◽  
Vol 26 (3) ◽  
pp. 167-181
Author(s):  
Nguyen Dinh Kien
Keyword(s):  
Shear Deformation ◽  
Stiffness Matrix ◽  
Large Deflection ◽  
Beam Element ◽  
Control Technique ◽  
Arc Length ◽  
The Finite Element Method ◽  
Nonlinear Beam ◽  
Tangent Stiffness Matrix ◽  
Lowest Eigenvalue

In this paper, the effects of shear deformation on the large deflection behavior of elastic frames is investigated by the finite element method. A two-node nonlinear beam element with the shear deformation is formulated and employed to analyze some frame structures. The element based on the energy method is developed in the context of the corotational approach. A bracketing procedure u ed the lowest eigenvalue of structural tangent stiffness matrix as indicating parameter is adopted to compute the critical loads. An incremental iterative procedure with the arc-length control technique is employed to trace the equilibrium paths. The numerical results show that the shear deformation plays an important role in the critical load and the large deflection behavior of the frames constructed from the components having low slenderness. A detail investigation is carried out to highlight the influence of slenderness on the behavior of the frames under large deflection.

scholarly journals On the co-rotational method for geometrically nonlinear topology optimization

2020 ◽  
Vol 62 (5) ◽  
pp. 2357-2374
Author(s):  
Peter D. Dunning
Keyword(s):  
Topology Optimization ◽  
Stiffness Matrix ◽  
Optimization Problems ◽  
Geometrically Nonlinear ◽  
Tangent Stiffness ◽  
Tangent Stiffness Matrix ◽  
Simplified Method ◽  
Nonlinear Topology Optimization ◽  
Consistent Method ◽  
Rotational Method

Abstract This paper investigates the application of the co-rotational method to solve geometrically nonlinear topology optimization problems. The main benefit of this approach is that the tangent stiffness matrix is naturally positive definite, which avoids some numerical issues encountered when using other approaches. Three different methods for constructing the tangent stiffness matrix are investigated: a simplified method, where the linear elastic stiffness matrix is simply rotated; the consistent method, where the tangent stiffness is derived by differentiating residual forces by displacements; and a symmetrized method, where the consistent tangent stiffness is approximated by a symmetric matrix. The co-rotational method is implemented for 2D plane quadrilateral elements and 3-node shell elements. Matlab code is given in the appendix to modify an existing, freely available, density-based topology optimization code so it can solve 2D problems with geometric nonlinear analysis using the co-rotational method. The approach is used to solve four benchmark problems from the literature, including optimizing for stiffness, compliant mechanism design, and a plate problem. The solutions are comparable with those obtained with other methods, demonstrating the potential of the co-rotational method as an alternative approach for geometrically nonlinear topology optimization. However, there are differences between the methods in terms of implementation effort, computational cost, final design, and objective value. In summary, schemes involving the symmetrized tangent stiffness did not outperform the other schemes. For problems where the optimal design has relatively small displacements, then the simplified method is suitable. Otherwise, it is recommended to use the consistent method, as it is the most accurate.

scholarly journals Poroelastoplastic reservoir modeling by tangent stiffness matrix method

Petroleum ◽  
2020 ◽  
Author(s):  
Lijing Zhang ◽  
Hua Zhang ◽  
Yanguang Yuan ◽  
Shunde Yin
Keyword(s):  
Stiffness Matrix ◽  
Matrix Method ◽  
Reservoir Modeling ◽  
Tangent Stiffness ◽  
Tangent Stiffness Matrix
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