Assumed Displacement Field Research Articles

Application of Composite Plate Theory and the Finite Element Method to the Dynamics and Stress Analysis of Spatial Flexible Mechanical Systems

This investigation concerns itself with the dynamic and stress analysis of thin, laminated composite plates consisting of layers of orthotropic laminae. It is assumed that the bonds between the laminae are infinitesimally thin and shear nondeformable. The finite element formulation presented is sufficiently general to accept an arbitrary number of layers and an arbitrary number of orthotropic material property sets. In the dynamic formulation presented, the laminae is assumed to undergo large arbitrary rigid body displacements and small elastic deformations. The nodal shape functions of the laminae are assumed to have rigid body modes that need to describe only large rigid body translations. Using the expressions for the kinetic and strain energies, the lamina mass and stiffness matrices are identified. The nonlinear mass matrix of the lamina is expressed in terms of a set of invariants that depend on the assumed displacement field. By summing the laminae kinetic and strain energies, the body mass and stiffness matrices are identified. It is shown that the body invariants can be expressed explicitly in terms of the invariants of its laminae. Numerical examples of a spatial RSSR mechanism are presented in order to demonstrate the use of the present formulation.

Analysis of a symmetric laminate with mid-plane free edge delamination under torsion: theory and application to the edge crack torsion (ECT) specimen for mode III toughness characterization

A shear deformation theory is developed to analyze a symmetric laminate with mid-plane edge delamination under torsion loading. The theory is based on an assumed displacement field which includes shear deformation. The governing equations and boundary conditions are obtained from the principle of virtual work. Closed-form solutions for the compliance and strain energy release rate are obtained. The theory is used to analyze a mid-plane Edge Crack Torsion (ECT) specimen used for characterizing the mode III delamination behavior of composites. The prediction of the present theory for a [90, (±45)n, (±45)n, 90]s class of laminates is compared with existing test data, and good agreement is observed. In addition, the Classical Lamination Theory (CLT) is found to be less accurate for the analysis of the edge crack torsion specimen. The layup effect on the behavior of some ECT specimens is analyzed using the present theory.

Least-squares T-elments: Equivalent FE and BE forms of a substructure-oriented boundary solution approach

An efficient solution to boundary-value problems may be based on the application of a suitably truncated T-complete set of Trefftz functions over individual subdomains and on linking the fields by a least-squares procedure. Although it yields a symmetric system of linear equations, this approach as originally presented by Zielinski and Zienkiewicz is not suited for implementation into FE codes. The present paper presents two equivalent formulations, which take respectively the form of the finite (FE) and non-conventional boundary-element (BE) approach. Both allow the resulting simultaneous equations to be assembled following the standard direct stiffness methods and can readily be implemented into existing FE codes. As in the conventional p-method, the accuracy may be controlled within large limits without increasing the number of elements. The present approach allows substantial saving in computer time in comparison with the so-called hybrid-Trefftz (HT) elements, though the assumed displacement fields are identical. The practical efficiency of the new T-element approach is assessed on the problem of stress concentration in a symmetrically compressed perforated panel.

A three‐dimensional multilayer composite finite element for stress analysis of composite laminates

AbstractA three‐dimensional multilayer composite finite element method has been developed based on a composite variational functional which takes three in‐plane strains εx, εx, εxy and three transverse stresses σz, σyz, σxz as the basic variables. The continuity of the transverse stresses σz, σyz, σxz across the laminate thickness is assured a priori by introducing a partial stress field parameter α which is associated with the lower and upper surfaces of a lamina in a laminate. A method has been developed to form the partial stress field based on the assumed displacement field. With this method, a three dimensional (3‐D) multilayer composite finite element is formulated for stress analysis of composite laminates. A numerical example is given, which shows some advantages of this composite element.

Large reference displacement analysis of composite plates part I: Finite element formulation

AbstractThis investigation concerns itself with the dynamic analysis of thin, laminated composite plates consisting of layers of orthotropic laminae that undergo large arbitrary rigid body displacements and small elastic deformations. A non‐linear finite element formulation is developed which utilizes the assumption that the bonds between the laminae are infinitesimally thin and shear non‐deformable. Using the expressions for the kinetic and strain energies, the lamina mass and stiffness matrices are identified. The non‐linear mass matrix of the lamina is expressed in terms of a set of invariants that depend on the assumed displacement field. By summing the kinetic and strain energies of the laminae of an element, the element mass and stiffness matrix can be defined in terms of the set of element invariants. It is shown that the element invariants can be expressed explicitly in terms of the invariants of its laminae. By assembling the finite elements of the deformable body, the body invariants can be identified and expressed explicitly in terms of the invariants of the laminae of its elements. In the dynamic formulation presented in this paper, the shape functions of the laminae are assumed to have rigid body modes that need to describe only large rigid body translations. The computer implementation and the use of the formulation developed in this investigation in multibody dynamics are discussed in the second part of this paper.

An evaluation of equivalent-single-layer and layerwise theories of composite laminates

A review of equivalent-single-layer and layerwise laminate theories is presented and their computational models are discussed. The layerwise theory advanced by the author is reviewed and a variable displacement finite element model and the mesh superposition techniques are described. The variable displacement finite elements contain several different types of assumed displacement fields. By choosing appropriate terms from the multiple displacement field, an entire array of elements with different orders of kinematic refinement can be formed. The variable kinematic finite elements can be conveniently connected together in a single domain for global-local analyses, where the local regions are modeled with refined kinematic elements. In the finite element mesh superposition technique an independent overlay mesh is superimposed on a global mesh to provide localized refinement for regions of interest regardless of the original global mesh topology. Integration of these two ideas yields a very robust and economical computational tool for global-local analysis to determine three-dimensional effects (e.g. stresses) within localized regions of interest in practical laminated composite structures.

Passive and Active Inertia Forces in Flexible Body Dynamics

This investigation is devoted to a discussion on the effect of the coupling between the longitudinal and transverse displacements of kinematically driven rotating beams. To this end, the inertia forces that act on a flexible body as the result of the finite rotation are categorized into two classes. These are the passive and active inertia forces. While the effect of the active inertia forces of kinematically driven systems is recognized in the absence of external disturbances and nonzero initial conditions, the passive inertia forces of such systems are not recognized in the case of zero initial conditions and in the absence of external excitations. Depending on the assumed displacement field, three classes of mechanical systems are defined in this paper. These are the active, partially active, and passive systems. The active system has a mathematical model in which both passive and active inertia forces are fully presented. In a partially active system, a part of the passive inertia forces and the active inertia forces appear in the mathematical model. The vibration of the kinematically driven passive system is governed by homogeneous equations which contain only the passive inertia forces. In the case of zero initial conditions and in the absence of external excitation, the response of the passive kinematically driven system is zero regardless of the value of the angular velocity. The effect of the inertia forces of the passive system appear as a time varying modification of the system parameters. It is shown in this investigation that a rotating beam model in which the axial deformation is neglected is a partially active or passive system. It is also demonstrated that the neglect of the effect of the longitudinal displacement has two significant effects. It decouples the modes of vibration and makes the form of the complementary solution independent of the sense of rotation. The behavior of the active, partially active, and passive systems when they are subjected to driving constraints (specified motion) is examined and it is shown that the response of the passive system converges to the partially active system if the effect of the initial conditions becomes dominant as compared to the effect of the active inertia forces of the partially active system.

Partial mixed 3‐D element for the analysis of thick laminated composite structures

AbstractFor the analysis of thick laminated composite structures this paper proposes a partial mixed 3‐D element. The variational principle of this new element is obtained by modifying the Hellinger–Reissner principle. The functional of the present stationary principle is constructed by treating three displacements (u, v, w) and two transverse shear stresses (τxz, τyz) as independent of each other. Hence the nodal variables of the present mixed element contain three displacements and two transverse shear stresses. The other stresses (σx, σy, τxy, σz) are computed from the assumed displacement field and nodal displacement field and nodal displacements. The present element can satisfy the requirements of (1)transverse shear stress continuity between laminate layers and (2)boundary conditions of free transverse shear stresses on the top and bottom surfaces. These requirements are violated by conventional displacement finite elements. Since the stiffness matrix of the present element is formulated by combining a displacement model and a mixed model, it is definite, rather than indefinite as for the conventional mixed elements. Also, these two transverse shear stresses are part of the solution variables and are solved directly together with displacements. Examples are presented to demonstrate the accuracy and efficiency of this proposed partial mixed 3‐D element in the analysis of thick laminated composite structures.

On first- and second-order moderate rotation theories of laminated plates

The first- and second-order, small strain and moderate rotation, theories of anisotropic laminated plates are developed and numerically evaluated. Beginning with an assumed displacement field and introducing various order-of-magnitude assumptions, the governing equilibrium equations of laminated plates are derived from the principle of virtual displacements. The finite-element formulation for the second-order moderate rotation theory is developed, and numerical results are presented to evaluate the new plate theories in comparison with the von Kàrmàn plate theory with shear deformation. For comparison purposes, the 2-D elasticity theory with full non-linearity is also analysed using the finite element method. It is found that the second-order theory with moderate rotations is closest to the 2-D finite elasticity theory.

Effect of the order of the finite element and selection of the constrained modes in deformable body dynamics

Finite elements with different orders can be used in the analysis of constrained deformable bodies that undergo large rigid body displacements. The constrained mode shapes resulting from the use of finite elements with different orders differ in the way the stiffness of the body bending and extension are defined. The constrained modes also depend on the selection of the boundary conditions. Using the same type of finite element, different sets of boundary conditions lead to different sets of constrained modes. In this investigation, the effect of the order of the element as well as the selection of the constrained mode shapes is examined numerically. To this end, the constant strain three node triangular element and the quadratic six node triangular element are used. The results obtained using the three node triangular element are compared with the higher order six node triangular element. The equations of motion for the three and six node triangular elements are formulated from assumed linear and quadratic displacement fields, respectively. Both assumed displacement fields can describe large rigid body translational and rotational displacements. Consequently, the dynamic formulation presented in this investigation can also be used in the large deformation analysis. Using the finite element displacement field, the mass, stiffness, and inertia invariants of the three and six-node triangular elements are formulated. Standard finite element assembly techniques are used to formulate the differential equations of motion for mechanical systems consisting of interconnected deformable bodies. Using a multibody four bar mechanism, numerical results of the different elements and their respective performance are presented. These results indicate that the three node triangular element does not perform well in bending modes of deformation.

An efficient global-local model for thick composite laminates

An efficient model based on the recently established partial hybrid stress element is proposed in this paper for global-local analysis of thick composite laminated plates. In this study, for displacement field, the higher order plate theory is used throughout the plate while only the transverse shear stress components are assumed in local regions. Thus, in the region of interest, the flexural stresses ( σ x , σ y , τ xy , σ z ) are still obtained from assumed displacement field while the transverse shear stresses ( τ xz , τ yz ) are calculated by the assumed stress version. The variational principle is also presented in this paper. The proposed functional contains two parts. One is the ordinary potential energy used in the global region; the other is the partial hybrid stress model, which contains displacement and transverse shear as independent variables. The interface equilibrium is guaranteed variationally although no stress parameter is assumed in the global region. Three types of global-local analysis are performed in this paper. These are in-plane analysis, through-thickness analysis, and mixed-type analysis. When compared with elasticity solutions, it is found that the agreement is acceptable, although in the so-called ‘transition region’ the results are less accurate.

Effect of the coupling between stretching and bending in the large displacement analysis of plates

AbstractThe effect of the rotary inertia on the non‐linear dynamics of plates that undergo a large reference displacement is examined in this paper. An assumed displacement field that accounts for the coupling between the stretching and bending of the plate as the result of considering the effect of the rotary inertia is used to identify the configuration of the plate. Furthermore, the coupling between the stretching and bending of the plate as the result of finite rotation is also considered in this investigation. Based on the assumed displacement field that accounts for the effect of the rotary inertia, a non‐linear finite element formulation is developed for the large displacement analysis of plates. The element equations of motion are expressed in terms of a set of element invariants that depend on the assumed displacement field as well as the rotary inertia. The use of the formulation presented in this paper is demonstrated using numerical examples.

Dynamics of Flexible Bodies Using Generalized Newton-Euler Equations

A force acting on a rigid body produces a linear acceleration for the whole body together with an angular acceleration about its center of mass. This result is in fact Newton-Euler equations which are used as basis for developing many recursive formulations for open loop multibody systems consisting of interconnected rigid bodies. In this paper, generalized Newton-Euler equations are developed for deformable bodies that undergo large translational and rotational displacements. The configuration of the deformable body is identified using coupled sets of reference and elastic variables. The nonlinear generalized Newton-Euler equations are formulated in terms of a set of time invariant scalars and matrices that depend on the spatial coordinates as well as the assumed displacement field. A set of intermediate reference frames having no mass or inertia are introduced for the convenience of defining various joints between interconnected deformable bodies. The use of the obtained generalized Newton-Euler equations for developing recursive dynamic formulation for open loop deformable multibody systems containing revolute, prismatic and cylindrical joints is also discussed. The development presented in this paper demonstrates the complexities of the formulation and the difficulties encountered when the equations of motion are defined in the joint coordinate systems.

Nonlinear Finite Element Formulation for the Large Displacement Analysis of Plates

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.

Effects of Shear Deformation and Rotary Inertia on the Nonlinear Dynamics of Rotating Curved Beams

In this investigation a method for the dynamic analysis of initially curved Timoshenko beams that undergo finite rotations is presented. The combined effect of rotary inertia, shear deformation, and initial curvature is examined. The kinetic energy is first developed for the curved beam and the beam mass matrix is identified. It is shown that the form of the mass matrix as well as the nonlinear inertia terms that represent the coupling between the rigid body motion and the elastic deformation can be expressed in terms of a set of invariants that depend on the assumed displacement field, rotary inertia, shear deformation, and the initial beam curvature. A nonlinear finite element formulation is then developed for Timoshenko beams that undergo finite rotations. The nonlinear formulation presented in this paper is applied to multibody dynamics where mechanical systems consist of an interconnected set of rigid and deformable bodies, each of which may undergo finite rotations. The equations of motion are developed using Lagrange’s equation and nonlinear algebraic constraint equations that mathematically describe mechanical joints and specified trajectories are adjoined to the system differential equations using the vector of Lagrange multipliers.

On the use of the finite element method and classical approximation techniques in the non-linear dynamics of multibody systems

The dynamic formulations for deformable bodies based on the finite element method and classical approximation techniques are compared. The finite element method and classical approximation techniques such as Rayleigh-Ritz methods are used to develop a set of generalized Newton-Euler equations for deformable bodies that undergo large translational and rotational displacements. In the finite element formulation, a stationary (total) Lagrangian approach is used to formulate the generalized Newton-Euler equations for each finite element in terms of a set of invariants that depend on the assumed displacement field. The deformable body invariants are obtained by assembling the invariants of the finite elements using a standard finite element Boolean matrix approach. This leads to the non-linear generalized Newton-Euler equations for the deformable bodies. These equations are presented in a simple closed form which is useful in developing recursive formulations for multibody systems consisting of interconnected deformable bodies. Both lumped and consistent mass formulations are discussed. Numerical results are presented for comparison of the finite element method and classical approximation methods in the non-linear dynamics of multibody systems.

Partial hybrid stress element for vibrations of thick laminated composite plates

A new element—the partial hybrid stress element—is extended for vibration analysis of thick laminated composite plates. The variational principle of this element can be derived from the Hellinger-Reissner principle to include inertial effects. Equations of motion can then be formulated through variational manipulation. The element stiffness matrix is established by assuming a stress field only for transverse shear, while other stresses are obtained from an assumed displacement field. Traction-free boundary conditions and interface continuity for transverse shear are satisfied exactly. Both consistent and lumped mass matrices are used to investigate the availability of this new element. Examples are illustrated and compared with other published data to demonstrate the accuracy and efficiency of this proposed partial hybrid stress element for vibration analysis.

Total Lagrangian formulation for the large displacement analysis of rectangular plates

AbstractIn this paper, a method for the non‐linear dynamic analysis of rectangular plates that undergo large rigid body motions and small elastic deformations is presented. The large rigid body displacement of the plate is defined by the translation and rotation of a selected plate reference. The small elastic deformation of the midplane is defined in the plate co‐ordinate system using the assumptions of the classical theories of plates. Non‐linear terms that represent the dynamic coupling between the rigid body displacement and the elastic deformation are presented in a closed form in terms of a set of time‐invariant scalars and matrices that depend on the assumed displacement field of the plate. In this paper, the case of simple two‐parameter screw displacement, where the rigid body translation and rotation of the plate reference are, respectively, along and about an axis fixed in space, is first considered. The non‐linear dynamic equations that govern the most general and arbitrary motion of the plate are also presented and both lumped and consistent mass formulations are discussed. The non‐linear dynamic formulation presented in this paper can be used to develop a total Lagrangian finite element formulation for plates in multibody systems consisting of interconnected structural elements.

Distortion (and Torsion) of rectangular thin-walled beams

A study of rectangular cross-section thin-walled beams under torsional, distortional and bimomental loads is presented. The assumed displacement field describes both torsional and distorsional behaviour, the shearing strain in the walls being taken into account. By a variational principle the equilibrium and boundary equations are derived and a physical interpretation of the natural conditions is given. A closed form solution is given with reference to the matrix form of the governing differential system together with a discussion of a sixth-order ‘uncoupled’ differential equation generalizing the BEF analogy. The present formulation shows that, in general, torsion and distortion are pair of coupled problems and a study of distortion without considering torsion would therefore not be legitimate. The comparison of the results predicted by the theory is in good agreement with experimental evidence.

Partial hybrid stress element for the analysis of thick laminated composite plates

AbstractA new element—a partial hybrid stress element—is proposed in this paper for the analysis of thick laminated composite plates. The variational principle of this element can be derived from the Hellinger–Reissner principle through dividing six stress components into a flexural part (σx, σy, σxy, σz) and a transverse shear part (τxy, τyz). The element stiffness matrix can be formulated by assuming a stress field only for transverse shear stresses, while all the others are obtained from an assumed displacement field. Consequently, this new element combines the benefits of the conventional displacement method and the hybrid stress method. A twenty‐node hexahedron element is employed in each layer for the displacement field. For the assumed transverse shear stress field, only the traction‐free boundary conditions and interface traction continuity are satisfied. The equilibrium equation is enforced by the variational principle. Hence, the complicated work of searching an equilibrating stress field for all the six stress components in the hybrid stress method can be avoided. Furthermore, the interlaminar traction discontinuity, especially transverse shear, encountered by the conventional displacement method and higher‐order plate element for laminated plate analysis can also be overcome. Examples are illustrated to demonstrate the accuracy and efficiency of this proposed partial hybrid stress element.