Nonlinear Finite Element Formulation for the Large Displacement Analysis of Plates

1990 ◽  
Vol 57 (3) ◽  
pp. 707-718 ◽  
Author(s):  
Bilin Chang ◽  
A. A. Shabana
Keyword(s):  
Finite Element ◽  
Rigid Body ◽  
Coordinate System ◽  
Deformable Body ◽  
Finite Element Formulation ◽  
Deformation Analysis ◽  
Element Formulation ◽  
Rectangular Plates ◽  
Shape Functions ◽  
Intermediate Element

In this investigation a nonlinear total Lagrangian finite element formulation is developed for the dynamic analysis of plates that undergo large rigid body displacements. In this formulation shape functions are required to include rigid body modes that describe only large translational displacements. This does not represent any limitation on the technique presented in this study, since most of commonly used shape functions satisfy this requirement. For each finite plate element an intermediate element coordinate system, whose axes are initially parallel to the axes of the element coordinate system, is introduced. This intermediate element coordinate system, which has an origin which is rigidly attached to the origin of the deformable body, is used for the convenience of describing the configuration of the element with respect to the deformable body coordinate system in the undeformed state. The nonlinear dynamic equations developed in this investigation for the large rigid body displacement and small elastic deformation analysis of the rectangular plates are expressed in terms of a unique set of time invariant element matrices that depend on the assumed displacement field. The invariants of motion of the deformable body discretized using the plate elements are obtained by assembling the invariants of its elements using a standard finite element procedure.

Exact Modeling of the Spatial Rigid Body Inertia Using the Finite Element Method

1998 ◽  
Vol 120 (3) ◽  
pp. 650-657 ◽  
Author(s):  
A. P. Christensen ◽  
A. A. Shabana
Keyword(s):  
Finite Element ◽  
Rigid Body ◽  
Coordinate System ◽  
Deformable Body ◽  
Equations Of Motion ◽  
Body Motion ◽  
Intermediate Element ◽  
Nodal Coordinates ◽  
Body Inertia ◽  
Isoparametric Elements

In the classical finite element literature beams and plates are not considered as isoparametric elements since infinitesimal rotations are used as nodal coordinates. As a consequence, exact modeling of an arbitrary rigid body displacement cannot be obtained, and rigid body motion does not lead to zero strain. In order to circumvent this problem in flexible multibody simulations, an intermediate element coordinate system, which has an origin rigidly attached to the origin of the deformable body coordinate system and has axes which are parallel to the axes of the element coordinate system in the undeformed configuration was introduced. Using this intermediate element coordinate system and the fact that conventional beam and plate shape functions can describe an arbitrary rigid body translation, an exact modeling of the rigid body inertia can be obtained. The large rigid body translation and rotational displacements can be described using a set of reference coordinates that define the location of the origin and the orientation of the deformable body coordinate system. On the other hand, as demonstrated in this investigation, the incremental finite element formulations do not lead to exact modeling of the spatial rigid body mass moments and products of inertia when the structures move as rigid bodies, and such formulations do not lead to the correct rigid body equations of motion. The correct equations of motion, however, can be obtained if the coordinates are defined in terms of global slopes. Using this new definition of the element coordinates, an absolute nodal coordinate formulation that leads to a constant mass matrix for the element can be developed. Using this formulation, in which no infinitesimal or finite rotations are used as nodal coordinates, beam and plate elements can be treated as isoparametric elements.

Full Beam Formulation for Coupled Elastic and Rigid Body Motion

1991 ◽  
Author(s):  
C. Venkatakrishnan ◽  
B. Fallahi ◽  
H. Y. Lai
Keyword(s):  
Finite Element ◽  
Rigid Body ◽  
Coordinate System ◽  
Numerical Solution ◽  
Body Motion ◽  
Finite Element Formulation ◽  
Rotary Inertia ◽  
Element Formulation ◽  
Rotating Beam ◽  
Moving Coordinate

Abstract The need for higher operating speeds has led to the study of flexibility in mechanisms. In most of the previous works, rotary inertia, normal, tangential and coriolis terms are neglected. These assumptions are valid at lower speeds and for slender links. In this paper, a procedure to include all inertia terms in a local moving coordinate system is introduced. It is shown that the inertia terms lead to the introduction of three element matrices in the finite element formulation. The proposed approach is used to model the rotating beam problem. The results of a numerical solution is reported and validated.

B-Spline Finite Element Formulation for Laminated Composite Shells

2008 ◽  
Author(s):  
Antonio Carminelli ◽  
Giuseppe Catania
Keyword(s):  
Finite Element ◽  
Tensor Product ◽  
Shear Deformation ◽  
Laminated Composite ◽  
Finite Element Formulation ◽  
Spline Functions ◽  
Element Formulation ◽  
Shape Functions ◽  
Composite Shells ◽  
B Spline

This paper presents a finite element formulation for the dynamical analysis of general double curvature laminated composite shell components, commonly used in many engineering applications. The Equivalent Single Layer theory (ESL) was successfully used to predict the dynamical response of composite laminate plates and shells. It is well known that the classic shell theory may not be effective to predict the deformational behavior with sufficient accuracy when dealing with composite shells. The effect of transverse shear deformation should be taken into account. In this paper a first order shear deformation ESL laminated shell model, adopting B-spline functions as approximation functions, is proposed and discussed. The geometry of the shell is described by means of the tensor product of B-spline functions. The displacement field is described by means of tensor product of B-spline shape functions with a different order and number of degrees of freedom with respect to the same formulation used in geometry description, resulting in a non-isoparametric formulation. A solution refinement method, making it possible to increase the order of the displacement shape functions without using the well known B-spline “degree elevation” algorithm, is also proposed. The locking effect was reduced by employing a low-order integration technique. To test the performance of the approach, the static solution of a single curvature shell and the eigensolutions of composite plates were obtained by numerical simulation and are then compared with known solutions. Discussion follows.

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