Degenerated Shell Element Research Articles

Shape and size optimization framework of stiffened shell using isogeometric analysis

A shape and size optimization framework is developed to improve the structural performance of the stiffened shell using isogeometric analysis. To accurately model the stiffened shell, the Lagrange multiplier method is employed to establish the coupling relationship of isogeometric degenerated shell elements and isogeometric degenerated beam elements. Due to the advantages of the non-uniform rational B-spline (NURBS), the unit normal direction vectors of the shell can be defined analytically, which guarantees stiffeners perpendicular to the skin. Additionally, the stiffeners can be easily generated based on the mapping relationship of the parametric space and the physical space of the shell. Analytical sensitivity is derived in detail, and the gradient-based optimization method can be used to solve the minimum compliance optimization problem owing to the high order continuity of NURBS. Three numerical illustrative examples are presented to verify the effectiveness of the proposed framework for the shape and size optimization of stiffened shells, consisting of a cylindrical shell patch, a hyperbolic shell, and a complex engineering segment. It is worth noting that the proposed framework is especially suitable for optimization problems of the stiffened shell, eliminating the need of complex feature extraction.

Geometrically nonlinear analysis of plates and shells by a cell-based smoothed CS-MITC18+ flat shell element with drilling degrees of freedom

A development of a 3-node triangular flat shell element to analyze geometric nonlinearity of plate and shell structures is presented in this study. The proposed flat shell element has 6 degrees of freedom per node, including the drilling rotational ones, by using the Allman-type approximations for the in-plane displacements and the bending displacement approximations enriched by a cubic shape function at the bubble node located at the element's centroid. The geometrically nonlinear behaviors are described by the von Karman's large deflection assumption. The membrane and bending strains of the presented flat shell elements are averaged over sub-triangular elements based on the cell-based smoothed (CS) technique. To attenuate the shear locking phenomenon, the transverse shear strains are re-interpolated following the mixed interpolation of tensorial components (MITC) technique designed for the 3-node triangular degenerated shell elements enriched by a cubic bubble function (MITC3+). Combined with the Newton-Raphson iterations and load increments predicted by the arc-length method, the suggested 3-node triangular flat shell elements, namely the CS-MITC18+ element, can analyze moderately geometrical nonlinearity, including the snap-thought and snap-back behaviors, of various plates and shells with the different shapes, thickness, and boundaries. The numerical investigations show that the load-displacement curves provided by the CS-MITC18+ flat shell elements well agree with other references.

Establishment and validation of mixed solid degenerate shell elements

Establishment and validation of mixed solid degenerate shell elements

Harmonic in-plane patch loading induced nonlinear dynamic instability of unidirectional and bidirectional stiffened composite plates

A finite element framework is established to study the nonlinear dynamic instability of stiffened plates subjected to in-plane harmonic patch loading. An eight-noded isoparametric degenerated shell element and a three-noded curved beam element are used to model the skin and the stiffener, respectively. The Green–Lagrange strain displacement relationship is adopted to formulate the system matrices for the plate and the stiffener, adopting the total Lagrangian approach. The Bolotin method is adopted to trace the boundaries of linear dynamic instability region. The Incremental Harmonic Balance (IHB) method is used to investigate the nonlinear instability behavior of stiffened plates. The effect of varying loading patches along with the number of stiffeners in unidirectional ( x-directional and y-directional) and bidirectional ( x as well as y-directional) stiffened plates on the nonlinear frequency response is studied. It is observed that the cross-stiffened plates with the higher number of stiffeners are dynamically more stable than the unidirectional stiffened plates. Moreover, the stiffened plates with smaller loading patches are dynamically more stable than those with larger loading patches.

Nonlinear dynamic instability and dynamic response of stiffened laminated composite plates subjected to in-plane pulsating patch loading

In this article, the nonlinear dynamic instability of stiffened laminated composite plates is studied in the finite element (FE) framework subjected to uniform in-plane harmonic patch loading. The harmonic load is applied to the two opposite sides of the stiffened plate. The linear and nonlinear time-history response analysis is also studied. The skin and the stiffener are modeled using an eight-node isoparametric degenerated shell element and a three-node curved beam element, respectively. A system of matrices is developed by considering the Green–Langrange strain–displacement relationship. In the linear case, the Bolotin method is used to analyze the dynamic instability region (DIR). The nonlinear instability behavior of the laminated composite stiffened plate is studied by applying the Incremental Harmonic Balance Method (IHB). The Newmark-β method is used to solve the linear and nonlinear time-history response equations to understand the instability behavior of the stiffened plates. The effect of the parameters such as the length of the in-plane loading patch, varying number of stiffeners in x -direction and the position of the patch on the nonlinear vibrations and nonlinear dynamic response is examined.

Derivation and implementation of one-point quadrature quadrilateral shell element with MITC4+ method (MITC4+R)

In this work, a novel degenerated shell element called MITC4+R is developed, which is based on the MITC4+ (an improved MITC quadrilateral shell element) and reduced integration with one point quadrature. The assumed natural strain method is employed to construct the membrane and shear strains. A physical stabilization term is derivated after the assumed natural strains are decomposed into the linear and a constant term with respect to the natural coordinates. The hourglass modes owing to one point quadrature is controlled in the computation of deformation by introducing this physical stabilization term. The developed degenerated shell element is successfully implemented into the implicit static and explicit dynamic analyses. In order to verify its computational accuracy and efficiency, several standard benchmarks and practical problems are performed. These results show that the MITC4+R element eliminates various locking problems common to shell elements, and it significantly improves the computational efficiency of the original MITC4+ element. Meanwhile, the MITC4+R owns the same accuracy and highly consistent convergence as the original MITC4+ element, making it more promising for practical applications.

A new mixed node-based solid-like finite element method (MNS-FEM) for laminated shell structures

A new solid-shell finite element formulation is investigated in this paper. Departing from the S-FEM concept of node-based stress smoothing scheme, a new pattern that introduces new stress variables linked with an extra smoothing region inside the element is proposed. This finding preserves all the advantageous features of the standard NS-FEM but it makes approximate solutions much more accurate. The assumed natural strain (ANS) and the Discrete shear gap (DSG) techniques are employed to alleviate trapezoidal and shear locking. The simple assumptions made allow the analytical computation of the discrete operators based on the node-based backgrounds making the model particularly efficient. Popular benchmark tests are successfully solved for structured and unstructured triangular meshes in linear and buckling analysis. Numerical results show that the proposed method avoids cross-diagonal meshes, which reduces the generality of three-node degenerated shell elements proposed in the literature. This aspect makes the present formulation much more appealing than the published methods available in the literature. Moreover, in the context of the S-FEMs it is the first time in which geometric nonlinear contributions are presented.

Nonlinear dynamic instability of laminated composite stiffened plates subjected to in-plane pulsating loading

A nonlinear finite element dynamic instability analysis of laminated composite stiffened plates subjected to in-plane harmonic edge loading is presented in this article along with the linear and nonlinear dynamic response study. The eight-noded isoparametric degenerated shell element and a compatible three-noded curved beam element are used to model the stiffened plates. Bolotin method is applied to analyze the dynamic instability regions in linear case. Incremental Harmonic Balance (IHB) method is applied to solve the nonlinear frequency response equations and Newmark-β method is used to solve the linear and nonlinear time history response equations.

Time-Dependent Reliability Assessment of Long-Span PSC Box-Girder Bridge Considering Vehicle-Induced Cyclic Creep

The neglect of the cyclic creep of concrete induced by vehicles may overestimate the deflection reliability of long-span prestressed concrete (PSC) box-girder bridges and increase structural safety risks. In this paper, a novel method considering the combined effects of the shrinkage, static creep, and cyclic creep of concrete and the stress relaxation of prestressed tendons is proposed to assess the time-dependent deflection reliability of PSC box-girder bridges. By employing the Kelvin chain model, the conventional integral-type method for calculating the static creep of concrete is converted to rate-type creep analysis, which has a higher computational efficiency and can consider various nonlinear effects. The cyclic creep of concrete is estimated by the fatigue mechanics-based model and integrated into the quasi-elastic incremental constitutive model. The proposed constitutive model is implemented in a general finite-element program DIANA, in which the mechanical behavior of the thin-walled box girder is described by the composite degenerated shell elements. An importance sampling method is applied when estimating the deflection reliability to balance the computational accuracy and efficiency. The proposed method is applied to a long-span PSC box-girder bridge with a main span of 150 m. The results indicate that the cyclic creep of concrete may accelerate the reduction of the deflection reliability indexes, which may fall below the target level before the expected service life of the bridge. In addition, heavy trucks passing in pairs have a significant impact on the deflection reliability. The research findings can be used for the reliable design and optimal maintenance of long-span PSC box-girder bridges.

Programmed Out-of-Plane Curvature to Enhance Multimodal Stiffness of Bending-Dominated Composite Lattices

Conventional bending-dominated lattices exhibit less specific stiffness compared to stretching-dominated lattices while showing high specific energy absorption capacity. This paper aims to improve the specific stiffness of bending-dominated lattices by introducing elementary-level programmed curvature through a multilevel hierarchical framework. The influence of curvature in the elementary beams is investigated here on the effective in-plane and out-of-plane elastic properties of lattice materials. The beamlike cell walls with out-of-plane curvature are modeled based on three-dimensional degenerated shell finite elements. Subsequently, the beam deflections are integrated with unit cell level mechanics in an efficient semi-analytical framework to obtain the lattice-level effective elastic moduli. The numerical results reveal that the effective in-plane elastic moduli of lattices with curved isotropic cell walls can be significantly improved without altering the lattice-level relative density, while the effective out-of-plane elastic properties reduce due to the introduction of curvature. To address this issue, we further propose laminated composite cell walls with out-of-plane curvature based on the three-dimensional degenerated shell elements, which can lead to holistic improvements in the in-plane and out-of-plane effective elastic properties. The proposed curved composite lattice materials would enhance the specific stiffness of bending-dominated lattices to a significant extent, while maintaining their conventional multifunctional advantages.

A Triangular Shell Element Based on Higher-order Strains for the Analysis of Static and Free Vibration

This research paper proposes a new triangular cylindrical finite element for static and free vibration analysis of cylindrical structures. The formulation of the proposed element is based on deep shell theory and uses assumed strain functions instead of displacement functions. The assumed strain functions satisfy the compatibility equations. This finite element possesses only the five necessary degrees of freedom for each of the three corner nodes. The element's displacement field, which contains higher-order terms, satisfies the requirement of rigid-body displacement. The element's performance is evaluated using various numerical static and free vibration tests for cylindrical shell problems, including an analysis of the effect of shell openings on natural frequencies. The results of the developed element are evaluated in comparison with published analytical and numerical solutions. The new cylindrical element's formulation is straightforward. Compared to the degenerate nine-node shell element and other elements, the results of the present element have shown excellent accuracy and efficiency in predicting static and free vibration of curved structures. This element only requires the use of very coarse meshes to converge. In addition, the triangular shape of this element is more advantageous than the quadrilateral shape when the geometric domain of the structure is deformed or complicated. Doi: 10.28991/CEJ-2022-08-10-06 Full Text: PDF

A NURBS-based degenerated stiffener element for isogeometric static and buckling analysis

Complex shells and curvilinear stiffeners are promising for the lightweight design of aerospace structures. Along with the increasing of structural size and geometric characteristics, traditional finite element analysis (FEA) requires fine grids to content the analysis accuracy, which is extremely time-consuming. In this study, a novel NURBS-based degenerated stiffener element is proposed. Based on the 6 degree of freedom (DOF) degenerated beam element, the proposed stiffener element solves the issue of direct coupling between the beam element and 5 DOF degenerated shell element. Skin and stiffener are characterized by NURBS surface and curve, respectively. Meanwhile, the buckling behavior of stiffened panels is investigated based on isogeometric analysis (IGA), which reduces the geometric discrete error and realizes the integration of modeling and analysis. In addition, since the grids of skin and stiffeners are always unmatched, a strong coupling method based on NURBS interpolation and projection algorithm is constructed to merge non-conforming grids. The degrees-of-freedom (DOFs) of stiffeners is eliminated in the equilibrium equation, which greatly reduces the calculation scale. Furthermore, an improved point-to-surface projection algorithm is proposed, which improves the computational efficiency and solves the problem of unstable convergence of Newton iteration. Finally, the static and buckling analyses are performed for different types of stiffened plates and shells. Displacement solutions, von Mises stress solutions and buckling modes are discussed in detail. The prediction accuracy and meshes are compared with the shell-type model and beam-shell-type model in ABAQUS software, which indicates the high applicability and efficiency of the proposed analysis method for various types of stiffened panels.

Determination of multi-physics effective properties, and actuation response of triply periodic minimal surface based novel photostrictive composites: A finite element analysis

This paper presents the novel triply periodic minimal surface (TPMS) based photostrictive composite as an alternative to naturally occurring photostrictive materials. The non-thermal, light-induced mechanical strain i.e., photostriction involves the simultaneous action of photovoltaic effect and converse piezoelectric effect. All the effective elastic, dielectric, piezoelectric, and pyroelectric properties are evaluated using the finite element analysis and compared with traditional micromechanical models for fibrous composites. Degenerated shell element is used to simulate the actuation response of unimorph cantilever and simply supported beams bonded with TPMS based photostrictive composite. The proposed photostrictive composite consists of poly{4,8-bis[5-(2-ethyl-hexyl) thiophen-2-yl] benzo[1,2-b:4,5-b’] dithiophene-2,6-diyl-alt-3-fluoro-2-[(2-ethyl-hexyl) carbonyl] thieno[3,4-b] thiophene-4,6-diyl} (PTB7-Th) as photovoltaic polymer matrix and Pb(Mg1/3Nb2/3)O3–0.35PbTiO3 (PMN-35PT) as TPMS—based filler material. The proposed TPMS based photostrictive composite induces 10 times larger actuation than 0–3 photostrictive composite. Also, the maximum deflection (3.22×10−5 m) of cantilever smart beam, achieved with Schoen-I-WP based photostrictive composite, is more than that achieved with Schoen-gyroid based photostrictive composite i.e., 2.78×10−5 m. Similar trends are obtained for simply supported smart beam where the center deflection for Schoen-I-WP based photostrictive composite and Schoen-gyroid based photostrictive composite are found to be 4.04×10−6 m and 3.49×10−6 m, respectively. The proposed novel photostrictive composite has potential for future wireless actuation applications.

Nonlinear bending analysis of trapezoidal panels under thermo-mechanical load

The nonlinear bending responses of trapezoidal panels under thermal load are investigated here using the higher order degenerated shell element. First-order shear deformation theory is used here and employed the finite element method with account of von Karman's geometric nonlinearity. The nonlinear governing equations are solved by using Newton Raphson method. First, checked the efficacy of higher order degenerated shell element with available published results of symmetric trapezoidal plates under mechanical load and curved panel under thermal load. In this article authors investigated the effect of boundary conditions, radius to span (R/a) ratio, cord to span (c/a) ratio, aspect (a/b) ratio, and lamination sequence on the nonlinear static response of trapezoidal panels under thermal environment. Moreover, new results for nonlinear bending behavior of symmetric trapezoidal cylindrical panels under uniform temperature are presented here that will be useful for future research and work as bench mark results.

Stiffener layout optimization framework by isogeometric analysis-based stiffness spreading method

An innovative stiffener layout optimization framework is proposed based on the isogeometric analysis-based stiffness spreading method (IGA-based SSM), where the shape and size of stiffeners can be simultaneously optimized. The IGA-based SSM is employed to obtain the stiffness spreading matrices of the stiffeners that are embedded in the flat plate. The flat plate is modeled by high-order continuous isogeometric degenerated shell elements that can provide a smooth sensitivity field. Since the sensitivity required for the optimization can be obtained analytically, a gradient-based optimization algorithm can be employed. Moreover, the ground structure is used to provide the initial layouts of stiffeners and ensure the connectivity of stiffeners. A local geometry control strategy is also adopted to avoid the intersection of stiffeners in the optimization process. To verify the effectiveness and efficiency of the proposed method, typical illustrative examples are discussed in detail, considering the compliance minimization optimization problem. Compared with the continuum topology optimization method, the proposed method can provide a clear stiffener layout with a lower computational cost due to fewer design variables and fewer degrees of freedom of the optimization problem. Besides, the maximum thicknesses of stiffeners can be explicitly controlled, and it can easily solve the optimization problem of various volume fraction constraints. Finally, the proposed method is extended to solve a layout optimization problem of the engine nozzle adjusting plate under pressure loads, which obtains a good stiffener layout under the maximum displacement constraint.

Photostrictive effect in 1-3 composites of photovoltaic and piezoelectric phases: A numerical study

Coalesce of photovoltaic effect with converse piezoelectric effect will turn into a photostrictive phenomenon. The current study conceptualizes a 1-3 photostrictive composite consists of a photovoltaic polymer as matrix and fibers of piezoelectric material. The proposed artificial photostrictive composite is capable of replacing lead-based naturally occurring photostrictive material, not only opening a potential for new applications but also caters to tailor the desired properties. Present study employs poly{4,8-bis[5-(2-ethyl-hexyl)thiophen-2-yl]benzo[1,2-b:4,5-b’]dithiophene-2,6-diyl-alt-3-fluoro-2-[(2-ethylhexyl) carbonyl] thieno[3,4-b] thiophene-4,6-diyl} (PTB7-Th) as organic photovoltaic polymer and Pb(Mg1/3Nb2/3)O3-0.35PbTiO3 (PMN-35PT) as the fibers. A representative volume element technique (RVE) is employed to embrace the local variation of multi-physics properties. The actuation response of cantilever and simply supported beam bonded to photostrictive composite patch is accurately predicted by finite element method, while discretizing the structure with degenerated shell element. Photostrictive composite with 60% volume fraction of fibers, arranged in square pattern have deflected the cantilever tip to 1.95 mm. Therefore, we provide 1-3 photostrictive composite as a solution for future wireless and lightweight vibration control applications.

Free vibration analysis of functionally graded porous plate using 3-D degenerated shell element

The free vibration analysis of porous functionally graded (FG) plates using 8-noded three-dimensional degenerated shell element has been presented in this paper. The Reissner–Mindlin assumptions, which converts 3D field into an equivalent 2D field in terms of mid-surface nodal variables, are employed to develop the degenerated shell elements which are used to model the FG plate. Thus, the computational time is reduced by reducing the number of nodes compared to full three dimensional elements. The three dimensional integration of element provides accurate results and can directly accommodate the variation in material properties across the thickness in case of FG materials.

3-D sloshing of liquid filled laminated composite cylindrical tank under external excitation

A new three dimensional finite element formulation is developed for the analysis of sloshing of liquid inside a partially filled laminated composite cylindrical tank under external excitations. Direct coupling method is adopted for efficient fluid–structure interaction between tank and liquid. The tank structure is modelled using 8 noded 3-D degenerated shell elements and the liquid inside tank is modelled using 20 noded brick elements. Experiments were conducted to validate the developed formulation. The results generated from developed formulation are in close agreement with the results available in open literature and the results obtained from experiment. Parametric studies are conducted by varying the tank thickness, ply orientation, and water level inside the tank to understand their effect on slosh height. Composite tanks are subjected to harmonic and seismic (Koyna earthquake 1967) loading for the parametric study.

Thermo-mechanical nonlinear transient dynamic and Dynamic-Buckling analysis of functionally graded material shell structures using an implicit conservative/decaying time integration scheme

In this paper, the nonlinear time history response and dynamic-buckling of functionally graded material (FGM) shell structures in thermal environments with temperature-dependent material properties is studied. An implicit conservative/decaying time integration scheme and a curved 8-node degenerated shell element are used for time and spatial discretization respectively. Numerical results show that the developed shell element formulation with the implicit time integration scheme is capable of solving nonlinear dynamic-buckling and large displacement dynamic problems of FGM shells in severe thermal environments. Th influence of material constituents, power-law indexes and thermo-mechanical loadings on the transient dynamic response and dynamic-buckling phenomena are considered.

Novel photostrictive 0-3 composites: A finite element analysis

Simultaneous action of photovoltaic effect and converse piezoelectric effect will induce the photostrictive effect. In contrast to the naturally occurring photostrictive material, the present study proposes a 0-3 photostrictive composite. The current photostrictive composite comprises of poly{4,8-bis[5-(2-ethyl-hexyl) thiophen-2-yl] benzo[1,2-b:4,5-b′] dithiophene-2,6-diyl-alt-3-fluoro-2-[(2-ethyl-hexyl) carbonyl] thieno[3,4-b] thiophene-4,6-diyl} (PTB7-Th) as organic photovoltaic polymer matrix and Pb(Mg1/3Nb2/3)O3-0.35PbTiO3 (PMN-35PT) as the particulates. The current study shows that the present 0-3 photostrictive composite is a suitable alternative for the most popular lead-based photostrictive material. While evaluating all the effective elastic, dielectric, piezoelectric, and pyroelectric properties of 0-3 photostrictive composite, the effect of local fluctuations was incorporated in representative volume element (RVE) using the finite element method. A finite element formulation using a degenerated shell element is derived to simulate the actuation response of the proposed 0-3 photostrictive composite. Numerical studies have been performed on cantilever and simply supported beams. The maximum tip and center deflection of cantilever beam and simply supported beam bonded with 0-3 photostrictive patch is mm and mm, respectively. The maximum value of the effective piezoelectric coefficient effective thermal expansion coefficient effective Young’s modulus and effective pyroelectric coefficient of proposed 0-3 photostrictive composite are m/V, 4.59 1.63 N/m2, and 7.57 C/m2 K, respectively, obtained at 25% volume fraction of particulates. This study opens possibilities of photostrictive coupling in composites.