Tangent stiffness matrix

About: Tangent stiffness matrix is a research topic. Over the lifetime, 1031 publications have been published within this topic receiving 21140 citations.

Force-Dependent Bilinear Variable Stiffness Design of Continuum Structure.

Abstract: Stiffness is one of fundamental properties in structural design, such as in maximum stiffness design. The nonlinear structural property is an attractive characteristic to be designed although majority of structural design problem deals with the linear elastic property. This article discusses the concept of force-dependent variable stiffness. The mean compliance measures the structural stiffness from the viewpoint of external load and the deformation at the loading sight. A certain gap is introduced as an internal boundary in the continuum structure, and its open/close status may depend on working force condition. Such a gap is expected to work as the switching mechanism, which brings the force-dependent variable stiffness. The attention is forced on the bilinear stiffness design having only one switching point of stiffness in the context of structural topology and geometry determination. First, the topology determination problem is formulated by using SIMP method and solved by sequential linear programming supported by moving limit strategy. Second, the geometry determination problem is formulated based on the topology obtained under the constraints of bilinear stiffness with the specified external load for stiffness switching, and is solved by the traction method. A couple of numerical cases studies demonstrated the feasibility and effectiveness of the proposed design concepts.

Nonlocal Nonlinear Analysis of Composites

Abstract: In this work, we present the behaviour of laminated composite plates, subjected to a static bending load under the influence of varying value of material length scale parameters. Reddy’s (J Appl Mech 51:745, 1984 [1]) third order shear deformation theory (TSDT) is used, which describes the kinematics accurately. The geometric nonlinearity, which prevails under the effect of large deformations, is accounted using von Karman nonlinear strains. Finite element model is developed using four-noded rectangular conforming element. Tangent stiffness matrix is derived to implement Newton Raphson method. The concept of non-locality is adopted from the works of Eringen and Edelen (Int J Eng Sci 10:233, 1972 [2]). Parametric study has been conducted to investigate the effect of non-locality and non-linearity on the behaviour of laminated composites.

A theoretical proof of the invalidity of dynamic relaxation arc-length method for snap-back problems

Abstract: Incorporating the arc-length constraint, the dynamic relaxation strategy has been widely used to trace full equilibrium path in the post-buckling analysis of structures. This combined numerical scheme has been shown to be successful for solving snap-through problems, but its applicability to snap-back problems has been rarely investigated and remains unclear. This paper proposes a direct and more general finite-difference equation to investigate the numerical stability of this combined numerical scheme, which is dominated by the spectral radius of amplification matrix. And a key discovery of this paper is that a first minor of the tangent stiffness matrix is always negative once snap back occurs. Due to this negative minor stiffness, the spectral radius is invariably greater than one, resulting in unconditional instability, which demonstrates the invalidity of dynamic relaxation arc-length method for snap-back problems. These important conclusions are corroborated by the numerical results of three representative examples in one-, two- and three-dimensional spaces.

Tangent Stiffness Equations for Laterally Distributed Loaded Member

Abstract: Synopsis Tangent stiffness equations for a beam-column which is subjected to either uniformly or sinusoidally distributed lateral load are presented. The equations have been derived by differentiating the slope-deflection equations under axial forces for a member. Then, the tangent stiffness equations take into account axial forces, a bowing effect and laterally distributed loads. Elastic buckling behavior of parallel chord latticed beams with laterally distributed loads is investigated, to compare the results of the present method with a conventional method in which the distributed loads are considered as concentrated loads at additional nodes of a member. Furthermore, buckling tests were carried out to confirm the derived equations and to make clear the buckling behavior of space frame structures. As a result, the new equations can lead to a good efficiency of estimating equilibrium paths and a significant savings in the core storage and computing time required for the analysis of space frame structures.

Improved Craig-Bampton method for transient analysis of structures with large-scale plastic deformation

Abstract: This paper reports on the improvement of Craig-Bampton (CB) method for transient analysis of structures with large-scale plastic deformation. As is known, the CB method is effective and accurate in reduced order modeling for linear system. In contrast to this, an improved CB method using tangent modes for nonlinear dynamic analysis has been developed. To do this, the incremental governing equations of nonlinear system are linearized in each time step by using tangent stiffness matrix, and the corresponding tangent modes are proved to be orthogonal with respect to mass matrix as well as with respect to tangent stiffness matrix by incorporating the elastic-plastic material behavior. Thus, the tangent modes can be used to assemble the transformation matrix of CB method in nonlinear dynamic analysis. Using the proposed method, two elastic-plastic beams loaded impulsively are examined. Simulation results show that the improved CB method is valid and accurate for transient analysis of structures with large-scale plastic deformation and has lower computational cost compared with full order model.