p-Version hierarchical three dimensional curved shell element for heat conduction

This paper presents a finite element formulation for a three dimensional nine node p-version hierarchical curved shell element for heat conduction where the element temperature approximation can be of arbitrary order p , p , and p in the , and directions. This is accomplished by first, constructing one dimensional hierarchical approximation functions and the corresponding nodal variable operators for each of the three directions , and using Lagrange interpolating polynomials and then taking their products (sometimes also called tensor products). The element approximation functions as well as the nodal variables are hierarchical and therefore the element matrices and the equivalent heat vectors are hierarchical also i.e. the element properties corresponding to polynomial orders p , p , and p are a subset of those corresponding to (p +1), (p +1), and (p +1). The element formulation ensures C ⁰ continuity. The curved shell geometry is constructed in the usual way by taking the coordinates of the nodes lying on the middle surface of the element (=0) and the nodal thickness vectors. The element properties i.e. element matrices and the equivalent heat vectors are derived using weak formulation (or quadratic functional) of the three dimensional F ourier heat conduction equation and the hierarchical element temperature approximation. The element formulation is equally effective for very thin as well as extremely thick shells. Numerical examples are presented to demonstrate the accuracy, efficiency, modeling convenience, faster rate of convergence and over all superiority of the present formulation. The h-approximation results are presented for comparison purposes.     

p-Version plate and curved shell element for geometrically non-linear analysis

This paper presents a p-version geometrically non-linear formulation based on the total Lagrangian approach for a nine node three dimensional curved shell element. The element geometry is defined by the coordinates of the nodes located on its middle surface and nodal vectors describing the bottom and top surfaces of the element. The element displacement approximation can be of arbitrary and different polynomial orders in the plane of the element and in the transverse direction. The element approximation functions and the corresponding nodal variables are derived from the Lagrange family of interpolation functions. The resulting approximation functions and the nodal variables are hierarchical and the element displacement approximation ensures C° continuity. The element properties are established using the principle of virtual work and the hierarchical element approximation. In formulating the properties of the element complete three dimensional stresses and strains are considered, hence the element is equally effective for very thin as well as extremely thick shells and plates. Incremental equations of equilibrium are derived and solved using the standard Newton–Raphson method. The total load is divided into increments, and for each increment of load, equilibrium iterations are performed until each component of the residuals is within a preset tolerance. Numerical examples are presented to show the accuracy, efficiency and advantages of the present formulation. The results obtained from the present formulation are compared with those available in the literature.     

Delamination buckling growth in laminated composites using layerwise-interface element

In this paper, a numerical investigation on the buckling of composite laminates containing delamination, under in-plane compressive loads, is presented. For this purpose, delamination propagation is modeled using the softening behavior of interface elements. The full layerwise plate theory is applied for approximating the displacement field of laminates and the interface elements are considered as a numerical layer between any two adjacent layers where the delamination is expected to propagate. A non-linear computer code was developed to handle the numerical procedure of delamination buckling growth in composite laminates using layerwise-interface elements. The load/displacement behavior and the contours of embedded and through-the-width delamination propagation for composite laminates are presented. It is shown that delamination growth can be well predicted using this layerwise-interface elements with decohesive law.     

p-version hierarchical three dimensional curved shell element for elastostatics

This paper presents a hierarchical three dimensional curved shell finite element formulation based on the p-approximation concept. The element displacement approximation can be of arbitrary and different polynomial orders in the plane of the shell (ξ, η) and the transverse direction (ξ). The curved shell element approximation functions and the corresponding nodal variables are derived by first constructing the approximation functions of orders p_ξ, p_η and p_ξ and the corresponding nodal variable operators for each of the three directions ξ, η and ξ and then taking their products (sometimes also known as tensor product). This procedure gives the approximation functions and the corresponding nodal variables corresponding to the polynomial orders p_ξ, p_η and p_ξ. Both the element displacement functions and the nodal variables are hierarchical; therefore, the resulting element matrices and the equivalent nodal load vectors are hierarchical also, i.e. the element properties corresponding to the polynomial orders p_ξ, p_η and p_ξ are a subset of those corresponding to the orders (p_ξ + 1), (p_η +1) and (p_ξ +1). The formulation guarantees C° continuity or smoothness of the displacement field across the interelement boundaries. The geometry of the element is described by the co-ordinates of the nodes on its middle surface (ξ = 0) and the nodal vectors describing its bottom (ξ = ?1) and top (ξ = +1) surfaces. The element properties are derived using the principle of virtual work and the hierarchical element approximation. The formulation is equally effective for very thin as well as very thick plates and curved shells. In fact, in many three dimensional applications the element can be used to replace the hierarchical three dimensional solid element without loss of accuracy but significant gain in modelling convenience. Numerical examples are presented to demonstrate the accuracy, efficiency and overall superiority of the present formulation. The results obtained from the present formulation are compared with those available in the literature as well as analytical solutions.     

An isoparametric quadratic thick curved beam element

A three-noded curved beam element with transverse shear deformation, based on independent isoparametric quadratic interpolations, is designed from field-consistency principles. It is shown that a quadratic element that is field-inconsistent in membrane strain suffers from ‘membrane locking’—i.e. an error of the second kind propagates indefinitely as the element length to thickness ratio and/or the element length to radius of curvature ratio increase, in nearly inextensional bending. However, field-inconsistency in shear strain does not lead to ‘shear locking’ but degrades its performance to exactly that of a field-consistent linear element. It is also seen that field-inconsistency leads to severe axial force and shear force oscillations. Error estimates for locking are derived, wherever possible, and confirmed by numerical experiments. The field-consistent element offered here is the most efficient quadratic curved beam element possible.     

A new higher-order hybrid-mixed curved beam element

The purpose of this work is to show the successful use of nodeless degrees of freedom in developing a highly accurate, locking free hybrid-mixed C⁰ curved beam element. In the performance evaluation process of the present field-consistent higher-order element, the effect of field consistency and the role of higher-order interpolation on both displacement-type and hybrid-mixed-type elements are carefully examined. Several benchmark tests confirm the superior behaviour of the present element. © 1998 John Wiley & Sons, Ltd.     

Locking-free curved beam element based on curvature

The formulation of a curved beam element with 3 nodes for curvature to eliminate the shear/membrane locking phenomenon is presented. The element is based on curvature so that it may represent the bending energy fully, and the shear/membrane strain energy is incorporated into the formulation by the equilibrium equations. To deal with general boundary conditions, a transformation matrix between nodal curvature and nodal displacement vector is introduced. Several examples are presented in order to verify the element formulation and its analytical capability. The solutions obtained reveal that the element describes the curved beam behaviour quite correctly and efficiently, showing no locking phenomena, and that it is also applicable to the analysis of both thin and thick curved beams.     

Analysis of laminated bimodulus composite thin shells of revolution using a doubly curved quadrilateral finite element

This paper presents a finite element analysis of laminated bimodulus composite thin shells of revolution using a 48 d.o.f. doubly curved quadrilateral finite element. All the three displacements of the shell element reference surface are expressed as products of one-dimensional first-order Hermite interpolation polynomials. The constitutive relationship for a bimodulus composite is assumed to depend on the fibre-direction strain experienced by each orthotropic layer. Consequently the true state of strain and the corresponding constitutive relationship in a bimodulus composite structure are to be determined iteratively. The true state of stress/strain is obtained by specifying a maximum error in the locations of the two neutral surfaces (one along each of the orthogonal reference axes) in the shell. The use of the quadrilateral shell finite element is validated by solving the problem of (i) a freely supported single layer (0°) bimodulus composite square plate and (ii) a freely supported single layer (0°) cylindrical panel, which are subjected to sinusoidal transverse loading and for which analytical solutions are available. Next, the problems of a single layer (90°) pinched cylindrical shell and a single layer (0°) open crown hemispherical shell are solved to show the ability of the present program.     

Hierarchical three dimensional curved beam element based on p-version

This paper presents a three node curved three dimensional beam element for linear static analysis where the element displacement approximation in the axial () and transverse directions ( and ) can be of arbitrary polynomial orders p , p and p . This is accomplished by, first constructing one dimensional hierarchical approximation functions and the corresponding nodal variable operators in , and directions using Lagrange interpolating polynomials and then taking the products (also called tensor product) of these hierarchical one dimensional approximation functions and the corresponding nodal variable operators. The resulting approximation functions and the corresponding nodal variables for the three dimensional beam element were hierarchical. The formulation guarantees C ⁰ continuity. The element properties are established using the principle of virtual work. In formulating the properties of the element all six components of the stress and strain tensor are ratained. The geometry of the beam element is defined by the coordinates of the nodes located at the axis of the beam and node point vectors representing the nodal cross-sections. The results obtained from the present formulation are compared with analytical solutions (when available) and the h-models using isoparametric three dimensional solid elements. The formulation is equally effective for very slender as well as deep beams since no assumptions are made regarding such conditions during the formulation.     

Designing laminated composites using random search techniques

A computer program called UWCODA is presented. UWCODA is intended to assist in the design, analysis and optimization of composite plates. UWCODA combines a state-of-the-art global optimization algorithm (Improving Hit and Run) with classical lamination theory. Optimization results are presented for simple loading conditions as well as for complex, biaxial load conditions. The computer code proved to be very effective in the design of composite plates.     

Modeling progressive delamination of laminated composites by discrete element method

Discrete element method (DEM) was used to model progressive delamination of fiber reinforced composite laminates. The anisotropic composite plies were constructed through a hexagonal packing of particle elements. Contacts between the particles were represented by parallel bonds with the verified normal and shear elastic properties. The ply interface was characterized by a contact softening model with a bilinear elastic behavior which is similar to the cohesive zone model in the continuum mechanics. DCB, ELS and FRMM tests were simulated by the DEM model to assess its capability of modeling mode I, mode II and mix mode fracture of delamination, respectively. Good agreements were observed between the DEM and existing numerical and experimental results of loading curves, which confirmed that the DEM model can be used to simulate initiation and propagation of composite delamination, with more insights into microscopic material behavior, such as damage extension and plastic zone.     

A hierarchical multiple plate models theory for laminated composites including delamination and geometrical nonlinear effects

Modeling and predicting the response and failure of laminated composite structures is a very challenging task. On one hand the designer is interested in the overall response of the structure, but on the other he is also interested in modeling phenomena such as damage which have a localized behavior. The ideal solution would be to carry out the analysis at a global level enhancing the model by adding localized information when needed.     

A multi-layer triangular membrane finite element for the forming simulation of laminated composites

Continuous fibre reinforced thermoplastics offer a cost reduction compared to thermosets due to promising fast production methods like diaphragm forming and rubber pressing. Forming experiments of pre-consolidated four-layer 8H satin weave PPS laminates on a dome geometry demonstrated that inter-ply friction is a dominant parameter in forming doubly curved components. Therefore, simulations of this process as sequentially draping the individual layers are invalid. A multi-layer triangular membrane finite element has been developed for efficient simulation of laminated composite forming processes with only one element in the thickness direction. Contact logic between the individual plies is avoided. The simulations were validated by comparison to the experiments mentioned and agree well. The multi-layer membrane element has shown to be capable of predicting the material instabilities during forming. They appeared to be unsuited for realistic wrinkling simulations due to their lack of a bending stiffness.     

Hierarchic models for laminated composites

The principles governing the formulation of hierarchic models for laminated composites are discussed. The essential features of the hierarchic models described herein are: (a) the exact solutions corresponding to the hierarchic sequence of models converge to the exact solution of the corresponding problem of elasticity for a fixed laminate thickness, and (b) the exact solution of each model converges to the same limit as the exact solution of the corresponding problem of elasticity with respect to the laminate thickness approaching zero. Hierarchic models make the computation of any engineering data possible to an arbitrary level of precision within the framework of the theory of elasticity. Examples are presented.     

Micromodel-based simulations for laminated composites

We develop a calculation strategy for the simulation of a complete microscopic model. This strategy enables one to account for damage mechanisms in laminated composites. The model mixes discrete and continuous approaches by introducing potential rupture surfaces and a damageable continuous medium. This approach requires suitable calculation tools unavailable in industrial analysis codes. The strategy presented is multiscale in space and is based on a decomposition of the domain into substructures and interfaces. This strategy enables one to simulate complex problems with multiple cracks. In practice, to use such a model, the strategy must be improved in order to handle very large numbers of substructures and interfaces and to estimate the rupture criteria for the surfaces introduced into the model. We provide simple examples which demonstrate the capabilities of the microscopic model.     

A curved element approximation in the analysis of axi-symmetric thin shells

The stiffness equation is derived for curved elements of orthotropic axi-symmetric thin shells, and equivalent applied loads are found for shells subjected to initial strains, applied surface loads and body forces. The Lure approximation of thin shells and displacement field approximation by polynomials of arbitrary degree are included in the formulae derived.