A four‐node,shear‐deformable shell element developed via explicit Kirchhoff constraints

An efficient, four‐node quadrilateral shell element is formulated using a linear, first‐order shear deformation theory. The bending part of the formulation is constructed from a cross‐diagonal assembly of four three‐node anisoparametric triangular plate elements, referred to as MIN3. Closed‐form constraint equations, which arise from the Kirchhoff constraints in the thin‐plate limit, are derived and used to eliminate the degrees‐of‐freedom associated with the ‘internal’ node of the cross‐diagonal assembly. The membrane displacement field employs an Allman‐type, drilling degrees‐of‐freedom formulation. The result is a displacement‐based, fully integrated, four‐node quadrilateral element, MIN4T, possessing six degrees‐of‐freedom at each node. Results for a set of validation plate problems demonstrate that the four‐node MIN4T has similar robustness and accuracy characteristics as the original cross‐diagonal assembly of MIN3 elements involving five nodes. The element performs well in both moderately thick and thin regimes, and it is free of shear locking. Shell validation results demonstrate superior performance of MIN4T over MIN3, possibly as a result of its higher‐order interpolation of the membrane displacements. It is also noted that the bending formulation of MIN4T is kinematically compatible with the existing anisoparametric elements of the same order of approximation, which include a two‐node Timoshenko beam element and a three‐node plate element, MIN3. Copyright © 2000 John Wiley & Sons, Ltd.     

A penalty finite element approach for couple stress elasticity

There is mounting evidence for size dependent elastic deformation at micron and submicron length scales. Material formulations incorporating higher order gradients in displacements have been successful in modeling such size dependent deformation behavior. A couple stress theory without micro-rotation is considered here as micro-rotations increase complexity and necessitate parameters that are difficult to determine. Higher order gradient theories require continuity in both displacements and their derivatives and direct approaches with both displacements and their derivatives as nodal variables results in a large number of degrees of freedom. Here nodal rotations are applied along with nodal displacements to obtain a simpler finite element formulation with fewer degrees of freedom. The difference in rotation gradients determined with nodal displacements and rotations are minimized by a penalty term. To assess the suggested approach simulations are presented and discussed, where the material parameters have been obtained from experiments of epoxy microbeams in the literature.     

Higher order zig-zag plate theory under thermo-electric-mechanical loads combined

A higher order zig-zag plate theory is developed to refine the predictions of the mechanical, thermal, and electric behaviors partially coupled. The in-plane displacement fields are constructed by superimposing linear zig-zag field to the smooth globally cubic varying field through the thickness. Smooth parabolic distribution through the thickness is assumed in the out-of-plane displacement in order to consider transverse normal deformation and stress. The layer-dependent degrees of freedom of displacement fields are expressed in terms of reference primary degrees of freedom by applying interface continuity conditions as well as bounding surface conditions of transverse shear stresses. Artificial shear correction factors are not needed in the present formulation. Thus the proposed theory has only seven primary unknowns and they do not depend upon the number of layers. Through the numerical examples of partially coupled analysis, the accuracy and efficiency of the present theory are demonstrated. The present theory is suitable in the predictions of deformation and stresses of thick smart composite plate under mechanical, thermal, and electric loads combined.     

A cubic triangular finite element for flat plates with shear

This paper presents the development of a straightforward displacement type triangular finite element for bending of a flat plate with the inclusion of transverse (or lateral) shear effects. The element has twenty two degrees of freedom consisting of ten for the lateral displacement of the midplane and six for rotations of the normal to the undeformed midplane of the plate. The latter are taken as independent of the slopes of the deformed midplane in order to include deformation due to transverse shear. The element is fully conforming and may be orthotropic. At interelement boundaries, the element matches adjacent elements both with respect to lateral displacement of the midplane and the rotations of the normal. The result is an efficient ‘linear moment’ triangular element but with transverse shear deformation included. Numerical computations for a number of examples are presented. The results show the element to be more flexible than most other finite element models and agree closely with those from a numerical solution of the three dimensional elasticity equations. The results also converge to those from thin plate theory when the thickness to length ratio becomes small or when the transverse shear moduli are artificially increased.     

An effective finite rotation formulation for geometrical non-linear shell structures

This paper presents a simple stress resultant 4-node shell element for geometrical non-linear analysis. In order to model smooth surfaces and/or stiffened structures, a simple and efficient technique for finite rotation is adopted. By means of suppressing the component of singular rotation effectivley, convenient use of six degrees of freedom is possible without deteriorating the robustness and the convergence rate of the classical 5-dof formulation. In the formulation of shell element, section eccentricity is also considered to model stiffened structures. Through numerical experiments the effectiveness of the proposed method is demonstrated. Received 16 November 2000     

An assumed-stress hybrid 4-node shell element with drilling degrees of freedom

An assumed-stress hybrid/mixed 4-node quadrilateral shell element is introduced that alleviates most of the deficiencies associated with such elements. The formulation of the element is based on the assumed-stress hybrid/mixed method using the Hellinger-Reissner variational principle. The membrane part of the element has 12 degrees of freedom including rotational or ‘drilling’ degrees of freedom at the nodes. The bending part of the element also has 12 degrees of freedom. The bending part of the element uses the Reissner-Mindlin plate theory which takes into account the transverse shear contributions. The element formulation is derived from an 8-node isoparametric element by expressing the midside displacement degrees of freedom in terms of displacement and rotational degrees of freedom at corner nodes. The element passes the patch test, is nearly insensitive to mesh distortion, does not ‘lock’, possesses the desirable invariance properties, has no hidden spurious modes, and for the majority of test cases used in this paper produces more accurate results than the other elements employed herein for comparison.     

An assumed strain triangular curved solid shell element formulation for analysis of plates and shells undergoing finite rotations

A formulation for 36‐DOF assumed strain triangular solid shell element is developed for efficient analysis of plates and shells undergoing finite rotations. Higher order deformation modes described by the bubble function displacements are added to the assumed displacement field. The assumed strain field is carefully selected to alleviate locking effect. The resulting element shows little effect of membrane locking as well as shear locking, hence, it allows modelling of curved shell structures with curved elements. The kinematics of the present formulation is purely vectorial with only three translational degrees of freedom per node. Accordingly, the present element is free of small angle assumptions, and thus it allows large load increments in the geometrically non‐linear analysis. Various numerical examples demonstrate the validity and effectiveness of the present formulation. Copyright © 2001 John Wiley & Sons, Ltd.     

Finite element analysis of stiffened laminated plates under transverse loading

A finite element model is developed to study the behavior of stiffened laminated plates under transverse loadings. Transverse shear flexibility is incorporated in both beam and plate displacement fields. A laminated plate element with 45 degrees of freedom is used in conjunction with a laminated beam element having 12 degrees of freedom for the bending analysis of eccentrically-stiffened laminated plates. The validity of the formulation is demonstrated by comparing with the available solutions in the literature. The numerical results are presented for eccentrically-stiffened layered plates having various boundary conditions and with stiffeners varying in number.     

A quadrilateral mixed finite element with two enhanced strain modes

An improved plane strain/stress element is derived using a Hu–Washizu variational formulation with bilinear displacement interpolation, seven strain and stress terms, and two enhanced strain modes. The number of unknowns of the four-node element is increased from eight to ten degrees of freedom. For linear and non-linear applications, the two unknowns associated with the enhanced strain terms can be eliminated by static condensation so that eight displacement degrees of freedom remain for the proposed element, which is denoted by QE2. The excellent performance of the proposed element is demonstrated using several linear and non-linear examples.     

Linked interpolation for Reissner-Mindlin plate elements: Part I—A simple quadrilateral

In most plate elements using the Reissner-Mindlin assumptions, the interpolations used for the lateral displacements (w) and the rotation (θ) involve the independent representation of each variable by its nodal values, usually with identical interpolations. To ensure a higher order of expansion for displacement w its representation is linked in the present paper with both sets of nodal variables. Conditions necessary for the use of such expansions are established here and the paper shows the development of a linear quadrilateral element (Q4BL) whose performance and robustness are good (although it possesses one singularity if only three degrees of freedom are prescribed). In Part II we apply the identical formulation to develop a triangular element (T3BL) which performs equally well and is fully robust.     

An Assumed Strain Triangular Solid Element for Efficient Analysis of Plates and Shells with Finite Rotation

A simple triangular solid shell element formulation is developed for efficient analysis of plates and shells undergoing finite rotations. The kinematics of the present solid shell element formulation is purely vectorial with only three translational degrees of freedom per node. Accordingly, the kinematics of deformation is free of the limitation of small angle increments, and thus the formulation allows large load increments in the analysis of finite rotation. An assumed strain field is carefully selected to alleviate the locking effect without triggering undesirable spurious kinematic modes. In addition, the curved surface of shell structures is modeled with flat facet elements to obviate the membrane locking effect. Various numerical examples demonstrate the efficiency and accuracy of the present element formulation for the analysis of plates and shells undergoing finite rotation. The present formulation is attractive in that only three points are needed for numerical integration over an element.     

An extended boundary element method formulation for the direct calculation of the stress intensity factors in fully anisotropic materials

We propose a formulation for linear elastic fracture mechanics in which the stress intensity factors are found directly from the solution vector of an extended boundary element method formulation. The enrichment is embedded in the boundary element method formulation, rather than adding new degrees of freedom for each enriched node. Therefore, a very limited number of new degrees of freedom is added to the problem, which contributes to preserving the conditioning of the linear system of equations. The Stroh formalism is used to provide boundary element method fundamental solutions for any degree of anisotropy, and these are used for both conventional and enriched degrees of freedom. Several numerical examples are shown with benchmark solutions to validate the proposed method. Copyright © 2016 John Wiley & Sons, Ltd.     

A mixed‐enhanced finite‐element for the analysis of laminated composite plates

This paper presents a new 4‐node finite‐element for the analysis of laminated composite plates. The element is based on a first‐order shear deformation theory and is obtained through a mixed‐enhanced approach. In fact, the adopted variational formulation includes as variables the transverse shear as well as enhanced incompatible modes introduced to improve the in‐plane deformation. The problem is then discretized using bubble functions for the rotational degrees of freedom and functions linking the transverse displacement to the rotations. The proposed element is locking free, it does not have zero energy modes and provides accurate in‐plane/out‐of‐plane deformations. Furthermore, a procedure for the computation of the through‐the‐thickness shear stresses is discussed, together with an iterative algorithm for the evaluation of the shear correction factors. Several applications are investigated to assess the features and the performances of the proposed element. Results are compared with analytical solutions and with other finite‐element solutions. Copyright © 1999 John Wiley & Sons, Ltd.     

Improved assumed-stress hybrid shell element with drilling degrees of freedom for linear stress,buckling and free vibration analyses

An improved 4-node quadrilateral assumed-stress hybrid shell element with drilling degrees of freedom is presented. The formulation is based on Hellinger–Reissner variational principle and the shape functions are formulated directly for the 4-node element. The element has 12 membrane degrees of freedom and 12 bending degrees of freedom. It has 9 independent stress parameters to describe the membrane stress resultant field and 13 independent stress parameters to describe the moment and transverse shear stress resultant field. The formulation encompasses linear stress, linear buckling and linear free vibration problems. The element is validated with standard test cases and is shown to be robust. Numerical results are presented for linear stress, buckling, and free vibration analyses.     

Stress analysis of flat plates with shear using explicit stiffness matrix

An explicit expression for the stiffness matrix is worked out for a triangular plate bending element considering the effect of transverse shear deformation. The element has twelve nodes on the sides and four nodes internal to it. The formulation is displacement type and the use of area co-ordinates makes it possible to obtain the shape functions explicitly. Separate polynomials are assumed for transverse displacement and rotations. To obtain the element stiffness matrix no matrix inversion or numerical integration need be carried out and only a few matrix multiplications of low order are necessary. The element, which is initially of thirty five degrees of freedom, can be reduced to a thirty degrees of freedom one by condensation of the internal nodes. An interesting feature of the element developed is that the values of nodal moments computed at a node point, considering different elements surrounding the node, do not vary significantly. Thus the nodal moments can be obtained directly at node points. Also, the element does not give rise to any inconvenience like locking, even for very thin plates. The straightforward approach in formation of the element stiffness will cut down the storage space considerably and will also call for less CPU time, thus making the use of the element well suited to low capacity computers. A number of plate bending problems have been worked out using the present element for different thickness to side ratios and a comparison has been made with the available results. Good accuracy has been observed in all cases, even for a small number of elements.     

Enhanced mixed interpolation XFEM formulations for discontinuous Timoshenko beam and Mindlin‐Reissner plate

Shear locking is a major issue emerging in the computational formulation of beam and plate finite elements of minimal number of degrees of freedom as it leads to artificial overstiffening. In this paper, discontinuous Timoshenko beam and Mindlin‐Reissner plate elements are developed by adopting the Hellinger‐Reissner functional with the displacements and through‐thickness shear strains as degrees of freedom. Heterogeneous beams and plates with weak discontinuity are considered, and the mixed formulation has been combined with the extended finite element method (FEM); thus, mixed enrichment functions are used. Both the displacement and the shear strain fields are enriched as opposed to the traditional extended FEM where only the displacement functions are enriched. The enrichment type is restricted to extrinsic mesh‐based topological local enrichment. The results from the proposed formulation correlate well with analytical solution in the case of the beam and in the case of the Mindlin‐Reissner plate with those of a finite element package (ABAQUS) and classical FEM and show higher rates of convergence. In all cases, the proposed method captures strain discontinuity accurately. Thus, the proposed method provides an accurate and a computationally more efficient way for the formulation of beam and plate finite elements of minimal number of degrees of freedom.     

An inelastic Timoshenko beam element with axial–shear–flexural interaction

An enhanced beam element is proposed for the nonlinear dynamic analysis of skeletal structures. The formulation extends the displacement based elastic Timoshenko beam element. Shear-locking effects are eliminated using exact shape functions. A variant of the Bouc–Wen model is implemented to incorporate plasticity due to combined axial, shear and bending deformation components. Interaction is introduced through the implementation of yield functions, expressed in the stress resultant space. Three additional hysteretic degrees of freedom are introduced to account for the hysteretic part of the deformation components. Numerical results are presented that demonstrate the advantages of the proposed element in simulating cyclic phenomena, in which shear deformations are significant.     

Analyzing static three-dimensional elastic domains with a new infinite boundary element formulation

A new two-dimensionally mapped infinite boundary element (IBE) is presented. The formulation is based on a triangular boundary element (BE) with linear shape functions instead of the quadrilateral IBEs usually found in the literature. The infinite solids analyzed are assumed to be three-dimensional, linear-elastic and isotropic, and Kelvin fundamental solutions are employed. One advantage of the proposed formulation over quadratic or higher order elements is that no additional degrees of freedom are added to the original BE mesh by the presence of the IBEs. Thus, the IBEs allow the mesh to be reduced without compromising the accuracy of the result. Two examples are presented, in which the numerical results show good agreement with authors using quadrilateral IBEs and analytical solutions.     

Active control of forced vibrations in adaptive structures using a higher order model

This paper deals with a finite element formulation for active control of forced vibrations, including resonance, of thin plate/shell laminated structures with integrated piezoelectric layers, acting as sensors and actuators, based on third-order shear deformation theory. The finite element model is a single layer triangular nonconforming plate/shell element with 24 degrees of freedom for the generalized displacements, and one electrical potential degree of freedom for each piezoelectric element layer, which are surface bonded or embedded in the laminate.

The Newmark method is considered to calculate the dynamic response of the laminated structures, forced to vibrate in the first natural frequency. To achieve a mechanism of active control of the structure dynamic response, a feedback control algorithm is used, coupling the sensor and active piezoelectric layers. The model is applied in the solution of illustrative cases, and the results are presented and discussed.