Nodal Rotations Research Articles

Size-dependent finite element analysis of FGMs in thermal environment based on the modified couple stress theory

PurposeThe paper aims to the size-dependent analysis of functionally graded materials in thermal environment based on the modified couple stress theory using finite element method.Design/methodology/approachThe element formulation is developed within the framework of the penalty unsymmetric finite element method (FEM) in that the C1 continuity requirement is satisfied in weak sense and thus, C0 continuous interpolation enhanced by independent nodal rotation is employed as the test function. Meanwhile, the trial function is designed based on the stress functions and the weighted residual method. Besides, the special Gauss quadrature scheme is employed for integrals of matrices in accordance with the graded variation of the material properties.FindingsThe numerical results reveal that in thermal environment, functionally graded materials exhibit better bending performance compared to homogeneous materials, Moreover, the findings also indicate that with an increase in MLSP, the natural frequencies of out-of-plane modes gradually increase, while the natural frequencies of in-plane modes show much less variation, leading to a mode switch phenomenon.Originality/valueThe work provides an efficient numerical tool for analyzing and designing the functionally graded structures in thermal environment in practical engineering applications.

Penalty hexahedral element formulation for flexoelectric solids based on consistent couple stress theory

AbstractThis work proposes a penalty 20‐node hexahedral element for size‐dependent electromechanical analysis based on the consistent couple stress theory. The nodal rotation degrees of freedom (DOFs) are used to approximate the mechanical rotation, meeting the C1 requirement in a weak sense by using the penalty function method, and to effectively enhance the standard isoparametric interpolation for determining the displacement test function. The normalized stress functions that satisfy the relevant equilibrium equation and strain compatibility equation a priori are employed to formulate the stress trial function. Since the element has only three displacement, three rotation and the electric potential DOFs per node, it has relatively simple formulation and can be readily incorporated into the existing finite element program. Several benchmarks are examined and the results demonstrate that the new element has good numerical accuracy and captures the size dependence effectively. In addition, the influence of the micro‐inertia on electromechanical dynamic responses of flexoelectric solids at small scale are analyzed using the proposed element. It is shown that the nature frequency decreases with the increase of micro‐inertia and in general, the influences are more obvious on higher order modes.

Isogeometric degenerated shell formulation for post-buckling analysis of composite variable-stiffness shells

Variable-stiffness (VS) laminated shells with curved fibers have gradually become a promising lightweight structural concept in the aerospace field. However, as the complexity of the shell geometry and the fiber angle increases, the traditional finite element method (FEM) requires mesh refinement to guarantee accurate discretization, which significantly increases the computational cost, especially in the post-buckling analysis with multiple solving requirements. In this study, a non-uniform rational B-Splines (NURBS)-based degenerated solid shell formulation in the total Lagrangian framework is proposed for the post-buckling analysis of VS laminated shells. Since the nonlinear function related to the rotational degrees of freedom (DOFs) is additionally introduced into the displacement field expression, the restriction on the magnitude of the nodal rotations is effectively eliminated. In addition, the paper adopts the NURBS surface reconstruction technology based on the point-line-surface features to build B-spline CAD models from the point cloud with geometric imperfect features. Then, an integrated modeling-analysis framework for predicting the post-buckling behavior of VS structures with geometric imperfections is proposed. A series of numerical examples systematically demonstrate that the framework can effectively conduct the post-buckling analysis of shell structures with geometric imperfections. Meanwhile, the accuracy and robustness of the solutions by IGA are proved to be better than FEM.

Simplified Calculation of Shear Rotations for First-Order Shear Deformation Theory in Deep Bridge Beams

Nodal rotations are produced by bending and shear effects and bending rotations can be easily calculated using Euler–Bernoulli’s stiffness matrix method. Nevertheless, shear rotations are traditionally neglected, as their effects are practically negligible in most structures. This assumption might lead to significant errors in the simulation of the rotations in some structures, as well as the wrong identification of the mechanical properties in inverse analysis. Despite its important role, no other works studying the calculation of shear rotations in deep beams were found in the literature. To fill this gap, after illustrating the errors of commercial software regarding calculating the rotations in deep beams, this study proposed a simple and intuitive method to calculate shear rotations in both isostatic and statically redundant beams. The new method calculates the shear rotation for all segments separately and introduces the result to the total rotation of the structure. This method can be applied to find the shear rotation in a redundant structure as well. A parametric study was carried out to calculate slenderness ratios to determine in what structural systems the shear rotations can be neglected. In addition, the errors in the inverse analysis of deep beams were parametrically studied to determine the role of shear rotation in different structural systems. Finally, to validate the application of the method in actual structures, a construction stage of a composite bridge was analyzed.

Couple stress-based unsymmetric 8-node planar membrane elements with good tolerances to mesh distortion

PurposeThe paper aims to propose two new 8-node quadrilateral membrane elements with good distortion tolerance for the modified couple stress elasticity based on the unsymmetric finite element method (FEM).Design/methodology/approachThe nodal rotation degrees of freedom (DOFs) are introduced into the virtual work principle and constrained by the penalty function for approximating the test functions of the physical rotation and curvature. Therefore, only the C0 continuity instead of C1 continuity is required for the displacement during the element construction. The first unsymmetric element assumes the test functions of the displacement and strain using the standard 8-node isoparametric interpolations, while these test functions in the second model are further enhanced by the nodal rotation DOFs. Besides, the trial functions in these two elements are constructed based on the stress functions that can a priori satisfy related governing equations.FindingsThe benchmark tests show that both the two elements can efficiently simulate the size-dependent plane problems, exhibiting good numerical accuracies and high distortion tolerances. In particular, they can still exactly reproduce the constant couple stress state when the element shape deteriorates severely into the degenerated triangle. Moreover, it can also be observed that the second element model, in which the linked interpolation technique is used, has better performance than the first one, especially in capturing the steep gradients of the physical rotations.Originality/valueAs the proposed new elements use only three DOFs per node, they can be readily incorporated into the existing finite element (FE) programs. Thus, they are of great benefit to analysis of size-dependent membrane behaviors of micro/nano structures.

Soft body impact on composites: Delamination experiments and advanced numerical modelling

Cohesive interface elements have become commonly used for modelling composites delamination. However, a limitation of this technique is the fine mesh size required. Here, a novel cohesive element formulation is proposed and demonstrated for modelling the numerical cohesive zone with equal fidelity but fewer elements in comparison to a linear cohesive element formulation. The newly proposed formulation has additional degrees of freedom in the form of nodal rotations which when combined with the use of multiple integration points per cohesive element, allows for delamination propagation to be modelled with increased stability. This element formulation is introduced with an adaptive modelling method, termed Adaptive Mesh Segmentation (AMS). To demonstrate its effectiveness under impact loading the new model is applied to a soft body beam bending test. This test, containing a delamination pre-crack, uses inertial constraints and results in a dynamic stress state when impacted by a gelatin cylinder.

Trefftz-unsymmetric finite element for bending analysis of orthotropic plates

This work develops a new four-node quadrilateral displacement-based Trefftz-type plate element for bending analysis of orthotropic plates within the framework of the unsymmetric finite element method (FEM). In the present formulation, the modified isoparametric interpolations are employed to formulate the element’s test functions in which the deflection is effectively enriched by the nodal rotation degrees of freedom (DOFs). Meanwhile, the element’s trial functions are determined based on the Trefftz functions that can a prior satisfy the governing equations of orthotropic Mindlin–Reissner plates. Numerical benchmark tests reveal that the new unsymmetric plate element is free of shear locking problem and can produce satisfactory results for both the displacement and stress resultant. In particular, it exhibits quite good tolerances to the gross mesh distortion.

A four-node C$$^{0}$$ tetrahedral element based on the node-based smoothing technique for the modified couple stress theory

In this paper, a four-node $$C^{0}$$ tetrahedral element for the modified couple stress theory is proposed. Since the governing equations are the fourth-order differential equations, the first-order derivative of displacement or rotation should be approximated by a continuous function. In the proposed element, nodal rotations are defined using the node-based smoothing technique. Continuous rotation fields are defined with the shape functions and nodal rotations. Both the displacement field and the rotation field are expressed solely in terms of the displacement degrees of freedom. The element stiffness matrix is calculated using the newly defined rotation field. To prevent the increase of calculation cost due to increase of the bandwidth of the stiffness matrix, the preconditioned conjugate gradient method is introduced. The performance of the proposed element is evaluated through various numerical examples.

8‐node hexahedral unsymmetric element with rotation degrees of freedom for modified couple stress elasticity

SummaryA new C0 8‐node 48‐DOF hexahedral element is developed for analysis of size‐dependent problems in the context of the modified couple stress theory by extending the methodology proposed in our recent work (Shang et al., Int J Numer Methods Eng 119(9): 807‐825, 2019) to the three‐dimensional (3D) cases. There are two major innovations in the present formulation. First, the independent nodal rotation degrees of freedom (DOFs) are employed to enhance the standard 3D isoparametric interpolation for obtaining the displacement and strain test functions, as well as to approximatively design the physical rotation field for deriving the curvature test function. Second, the equilibrium stress functions instead of the analytical functions are used to formulate the stress trial function whilst the couple stress trial function is directly obtained from the curvature test function by using the constitutive relationship. Besides, the penalty function is introduced into the virtual work principle for enforcing the C1 continuity condition in weak sense. Several benchmark examples are examined and the numerical results demonstrate that the element can simulate the size‐dependent mechanical behaviors well, exhibiting satisfactory accuracy and low susceptibility to mesh distortion.

A large rotation finite element analysis of 3D beams by incremental rotation vector and exact strain measure with all the desirable features

Different strategies based on rotation vector and exact strain measure have been proposed over the years for analyzing flexible bodies undergoing arbitrary large rotations. To avoid the singularity of the vector-like parametrization, the interpolation of the incremental rotation vector is the most popular approach in this context, even if this leads to path dependence and numerical instability, i.e. error accumulation. It is also non objective, although both objectivity and path independence are recovered with h and p refinement. Corotational approaches do not have these drawbacks, even though the geometrically exact model is achieved by mesh refinement. In this work, we develop a novel strategy which uses the incremental nodal rotation vectors to define corotational nodal rotations, which are then interpolated for the evaluation of the nonlinear strains. This choice makes the approach singularity-free, allows for additive updates within each increment and preserves all the features of the theoretical problem for any mesh and interpolation: rotational variables, objectivity, exact strain measure, path independence and symmetric stiffness matrix for conservative loads. This last property is a consequence of the direct differentiation of the relation between local and global rotations, whose compact form also makes a simple and general definition of the internal forces and the tangent stiffness for any order of interpolation possible. In addition, we show how the common approach of interpolating incremental vectors can be made stable by a simple updating procedure based on local rotations carried out at the end of each increment in order to avoid cumulative errors. Geometrically exact 3D beams are considered as a demonstrative example. An iterative strategy based on mixed integration points is used to solve the nonlinear discrete equations efficiently.

Dynamic mesh method based on diffusion equation and nodal rotation for high-aspect-ratio composite wings

In this paper a dynamic mesh method based on diffusion equation and nodal rotation is developed for high-aspect-ratio composite wings. Mesh displacement is described by diffusion equation and calculated by finite element method. The mesh was distorted because the displacement of the mesh nodes is uncoupled in different directions, and its quality diminished due to large deformation. The nodal rotational method is introduced to solve this problem. The technique used the same diffusion equation to describe the rotational angle and is solved by the finite element method. The node nearest the wall boundary is the rotational centre and the corresponding rotational vector is the rotational axis. The new position of the mesh node is obtained by rotating transformation. The mesh deformation of two-dimensional NACA0012 airfoil mesh, three-dimensional spherical mesh and high aspect ratio composite wings are verified for various cases. The results show that the present method is applicable to structural and unstructured/hybrid meshes. The mesh quality is improved and its overlap and negative volume are effectively prevented under large deformation.

Generalized micropolar continualization of 1D beam lattices

The enhanced continualization approach proposed in this paper is aimed to overcome some drawbacks observed in the homogenization of beam lattices. To this end an enhanced homogenization technique is proposed and formulated to obtain consistent micropolar continuum models of the beam lattices and able to simulate with good approximation the boundary layer effects and the Floquet–Bloch spectrum of the Lagrangian model. The continualization technique here proposed is based on a transformation of the difference equation of motion of the discrete system via a proper down-scaling law into a pseudo-differential problem; a further McLaurin approximation is applied to obtain a higher order differential problem. The formulation is carried out for simple one-dimensional beam lattices that are, nevertheless, characterized by a rather wide variety of static and dynamic behaviors: the rod lattice, the beam lattice with node rotations and a 1D beam lattice model with generalized displacements. Higher order models may be obtained which are characterized by differential problems involving non-local inertia terms together with spatial high gradient terms. Moreover, the homogenized models obtained by the proposed enhanced continualization technique turn out to be energetically consistent and provide a good simulation of both the static response and of the acoustic spectrum of the original discrete models. The proposed homogenization procedure is first presented for the simple case of monoatomic axial chains. The beam lattice with node rotations and displacement prevented exhibits, in the static regime, decaying oscillations of the nodal rotation in the boundary layer which is well simulated by the homogenized model obtained by the proposed approach. Similar good results are obtained in the simulation of the optical spectrum. It is worth to note that in this case the homogenized model obtained via Padé approximation turns out to be energetically non-consistent. The analysis of the homogenized model derived from the beam lattice with transverse displacement and rotation of the nodes with elastic supports has shown that both the static and the dynamic response are strongly variable on the parameters of the Lagrangian model. Finally, several different cases have been considered and good simulations have been obtained both in describing the static response to prescribed displacements at the end nodes and in representing the Floquet–Bloch spectrum and the polarization vectors.

On the Geometrically Nonlinear Analysis of Composite Axisymmetric Shells

Composite axisymmetric shells have numerous applications; many researchers have taken advantage of the general shell element or the semi-analytical formulation to analyze these structures. The present study is devoted to the nonlinear analysis of composite axisymmetric shells by using a 1D three nodded axisymmetric shell element. Both low and higher-order shear deformations are included in the formulation. The displacement field is considered to be nonlinear function of the nodal rotations. This assumption eliminates the restriction of small rotations between two successive increments. Both Total Lagrangian Formulation and Generalized Displacement Control Method are employed for analyzing the shells. Several numerical tests are performed to corroborate the accuracy and efficiency of the suggested approach.

Rotation‐free triangular shell element using node‐based smoothed finite element method

SummaryIn this paper, a triangular thin flat shell element without rotation degrees of freedom is proposed. In the Kirchhoff hypothesis, the first derivative of the displacement must be continuous because there are second‐order differential terms of the displacement in the weak form of the governing equations. The displacement is expressed as a linear function and the nodal rotation is defined using node‐based smoothed finite element method. The rotation field is approximated using the nodal rotation and linear shape functions. This rotation field is linear in an element and continuous between elements. The curvature is defined by differentiating the rotation field, and the stiffness is calculated from the curvature. A hybrid stress triangular membrane element was used to construct the shell element. The penalty technique was used to apply the rotation boundary conditions. The proposed element was verified through several numerical examples.

An explicit algorithm for geometrically nonlinear transient analysis of spatial beams using a corotational total Lagrangian finite element formulation

An explicit method for nonlinear transient dynamic analysis of spatial beams with finite rotations using a corotational total Lagrangian finite element formulation is presented. The kinematics of the beam element is described in the current element coordinate system constructed in the current configuration of the beam element. The element deformation and inertia nodal forces are derived by the virtual work principle, the d'Alembert principle, and the consistent linearization of the geometrically nonlinear beam theory. A nodal rotation vector is used to represent the finite rotation of a base coordinate system rigidly attached to each node of the discretized structure. A numerical procedure of explicit method is proposed for the solution of the nonlinear equations of motion. The standard central difference method is applied to the incremental displacement vector and the incremental rotation vector, and the time derivatives of displacement vector and rotation vector. The nodal orientations are updated by the incremental nodal rotation vectors. The values of nodal rotation vectors are reset to zero in the current configuration.In order to assess the efficiency and the accuracy of the proposed method, numerical examples are studied and compared with the results obtained using the implicit method based on the Newmark method.

Effects of nodal fillets and external boundaries on compressive response of an octet truss

Additive manufacturing has enabled fabrication of weight-efficient periodic trusses over a wide range of length scales. The principal objective of the present study is to address two coupled aspects of truss design and performance: (i) the extent to which circular nodal fillets enhance node stiffness and alleviate stress concentrations, and (ii) the extent to which external boundaries affect local strut strains. To this end, octet trusses with and without filleted nodes were fabricated out of a hard thermoplastic and tested in compression, with digital image correlation used to monitor axial and bending strut strains and nodal rotations. Complementary finite element simulations were also performed. Macroscopic compressive failure occurs through bending and buckling of certain inclined struts followed by tensile rupture of struts oriented perpendicular to the loading direction. The struts experiencing the highest tensile strain and most prone to rupture are those located along the truss mid-plane and that intersect external edges. Additional boundary effects are manifested in exacerbated nodal rotations and bending of inclined struts intersecting the truss corners. Although buckling begins in the affected struts, large-scale buckling is preceded by tensile rupture. The implications for design and failure prediction of finite truss structures are discussed.

A mixed tetrahedral element with nodal rotations for large-displacement analysis of inelastic structures

Summary A novel mixed four-node tetrahedral finite element, equipped with nodal rotational degrees of freedom, is presented. Its formulation is based on a Hu–Washizu-type functional, suitable to the treatment of material nonlinearities. Rotation and skew-symmetric stress fields are assumed as independent variables, the latter entering the functional to impose rotational compatibility and suppress spurious modes. Exploiting the choice of equal interpolation for strain and symmetric stress fields, a robust element state determination procedure, requiring no element-level iteration, is proposed. The mixed element stability is assessed by means of an original and effective numerical test. The extension of the present formulation to geometric nonlinear problems is achieved through a polar decomposition-based corotational framework. After validation in both material and geometric nonlinear context, the element performances are investigated in demanding simulations involving complex shape memory alloy structures. Supported by the comparison with available linear and quadratic tetrahedrons and hexahedrons, the numerical results prove accuracy, robustness, and effectiveness of the proposed formulation. Copyright © 2016 John Wiley & Sons, Ltd.

A Hybrid Interpolation Method for Geometric Nonlinear Spatial Beam Elements with Explicit Nodal Force

Based on geometrically exact beam theory, a hybrid interpolation is proposed for geometric nonlinear spatial Euler-Bernoulli beam elements. First, the Hermitian interpolation of the beam centerline was used for calculating nodal curvatures for two ends. Then, internal curvatures of the beam were interpolated with a second interpolation. At this point, C1 continuity was satisfied and nodal strain measures could be consistently derived from nodal displacement and rotation parameters. The explicit expression of nodal force without integration, as a function of global parameters, was founded by using the hybrid interpolation. Furthermore, the proposed beam element can be degenerated into linear beam element under the condition of small deformation. Objectivity of strain measures and patch tests are also discussed. Finally, four numerical examples are discussed to prove the validity and effectivity of the proposed beam element.

A New Jacobi-based Iterative Method for the Classical Analysis of Structures

TRADITIONALLY, CLASSICAL METHODS OF STRUCTURAL ANALYSIS SUCH AS SLOPE-DEFLECTION AND MOMENT DISTRIBUTION METHODS (CROSS METHOD) ARE USED FOR PRIMARY ANALYSIS OF STRUCTURES AND ALSO CONTROLLING THE RESULTS OF COMPUTER PROGRAMS. THE MAIN OBJECTIVE OF THIS PAPER IS TO INTRODUCE A NEW METHOD FOR CLASSICAL COMPUTING AND EXTENDING IT TO A MATRIX FORMULATION. THE PROPOSED APPROACH, NAMED THE €œSLOPE DISTRIBUTION METHOD (SDM)€, IS BASED ON A JACOBI ITERATIVE PROCEDURE, IN WHICH WITHOUT FORMING THE SYSTEM OF LINEAR EQUATIONS, STRUCTURAL DISPLACEMENT VALUES ARE OBTAINED. ALSO, TO MAKE THE METHOD APPLICABLE AND TO USE IT IN COMPUTER SOFTWARES, THE MATRIX FORMULATION OF THE APPROACH IS DEVELOPED, WHERE THERE IS NO NEED FOR ITERATIVE PROCEDURES AND THE NODAL ROTATIONS ARE OBTAINED THROUGH SOLVING ONLY ONE MATRIX EQUATION. THE SDM IS ABLE TO ANALYZE FRAMES WITH NON-VERTICAL COLUMNS AND THOSE WITH NODAL VERTICAL DISPLACEMENT. WHEREAS CURRENT ANALYSIS SOFTWARES HAVE SOME ELIMINATION FOR THE ANALYSIS OF NON- PRISMATIC MEMBERS, THE PROPOSED METHOD CAN BE APPLIED TO ANALYZE STRUCTURES WITH ANY NON-PRISMATIC MEMBER. THE SDM PROCESS IS ALSO DEVELOPED FOR THE ANALYSIS OF DUAL LATERAL LOAD RESISTING SYSTEMS (MOMENT RESISTING FRAMES WITH OTHER LATERAL LOAD RESISTING ELEMENTS SUCH AS BRACINGS AND SHEAR WALLS). THE ADVANTAGES OF THE METHOD OVER PREVIOUS ONES AND ALSO, ITS ACCURACY AND RELIABILITY ARE PRESENTED THROUGH THE ARTICLE.

Axisymmetric couple stress elasticity and its finite element formulation with penalty terms

Length scale dependent deformation in polymers has been reported in the literature in different experiments at the micron and submicron length scales. Such length scale dependent deformation behavior can be described using higher order gradient theories. A numerical approach for axisymmetric problems is presented here where a couple stress elasticity theory is employed. For the numerical formulation, a penalty finite element approach is proposed and implemented with C0 axisymmetric elements. In this approach, rotations are introduced as nodal variables independent of nodal displacements, and the penalty term is used to minimize the difference in rotations determined from nodal displacements and nodal rotations. Numerical simulations are performed on different examples to assess the performance of the suggested approach. In particular, a circular cylinder with spherical inclusions and the interaction of the inclusion with the boundary are studied. It is found that with the length scale parameter the distance with which the free boundary affects the stress state of the inclusion increases as well.