vd.VAISHALI DAGADESTRUCTURAL DESIGN ENGINEER
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Doctoral research / Structural mechanics

Shell finite-element
research.

Understanding a curved structure starts with what happens inside it: how it bends, how its layers share load, and how it vibrates.

Shell geometries

CylindricalSphericalConical

Material systems

CompositeSandwichFGM

FGM / Functionally graded materials

4 nodesQuadrilateral flat-shell element
7 DOFPer node in the DKZigTS1 formulation
3 geometriesCylindrical, spherical, and conical

Selected study: Curved and Layered Structures (2022), covering composite and sandwich shells. The broader research also includes isotropic and functionally graded shell panels.

01 / The structures

Different curves.
Layered behavior.

Composite layers can have different fiber orientations and stiffnesses. A shell model needs to represent those differences as well as the curvature of the structure.

Figure 1 / Shell geometriesCylindrical, spherical, and conical composite panels. Source and reuse credit.

02 / The formulation

A curved surface,
assembled from flat elements.

Figure 2 / Mesh and coordinate systemsLocal translations and rotations must be assembled consistently across the curved surface. Source and reuse credit.

Seven degrees of freedom. A consistent transformation.

The 2022 journal paper presents the development and evaluation of DKZigTS1, a four-node flat-shell element based on zigzag theory. Each node has three translations, two bending rotations, and two shear rotations.

Translations are transformed to global Cartesian coordinates; rotations use a surface coordinate system. This allows the formulation to retain seven degrees of freedom at both the local and global levels.

DKZigTS1 is distinct from the nine-DOF DKZigTS formulation used in some of our other papers.

03 / Free vibration

Not just a frequency.
A pattern of movement.

Validation compares both natural frequencies and their corresponding mode shapes. The figure below places the shell formulation alongside a three-dimensional finite-element benchmark.

Figure 9 / Sandwich conical shell, panel DAll edges simply supported. Top row: 3D FE benchmark. Bottom row: DKZigTS1. Frequencies are nondimensional; the colors depict mode shapes, not stress contours. Source and reuse credit.

Validation across models and boundary conditions. The study compares results with published analytical and elasticity solutions and with ANSYS 3D finite-element models. Mesh refinement, shell thickness, curvature, material lay-up, and support conditions are part of the assessment. Agreement is case-dependent, not a blanket accuracy guarantee.

04 / Through the thickness

What the faces carry.
What the core carries.

For the sandwich panel shown here, the face sheets carry the bending-related normal stresses while the core carries transverse shear. Looking through the thickness reveals behavior that a single surface contour cannot show.

Our formulation was assessed against the 3D FE stress distributions, including the changes across material interfaces. The wider study also examines the zigzag variation of in-plane displacement through layered shells.

Shown case: simply supported sandwich spherical panel G under uniform loading; b/a = 1 and h/a = 0.25. Figure-specific conditions are retained from the paper.

Figure 13 / Layered stress responseNormal stresses above, transverse shear stresses below. Original axes and comparison legends are preserved. Source and reuse credit.

From research to practice

A model is only as useful
as the checks behind it.

Formulate. Connect shell kinematics, material layers, coordinate transformations, and stiffness and mass matrices.

Verify. Compare mesh sizes, boundary conditions, and analytical or independent FE benchmarks.

Interpret. Read stresses and mode shapes in the context of the model's assumptions and the case being studied.

Selected publications

The published work.

  1. 2025

    Static Analysis of Composite and Sandwich Spherical Shells Using a Four-Node Flat Shell Quadrilateral Finite Element

    Recent Advances in Composite Materials and Structures, Volume 2. Springer Proceedings in Materials 74. Book chapter.
  2. 2023

    Four Node Flat Shell Quadrilateral Finite Element for Analysis of Composite Cylindrical Shells

    Recent Trends in Construction Technology and Management. Lecture Notes in Civil Engineering 260. Book chapter.
  3. 2022

    A quadrilateral flat-shell element for the static and dynamic analysis of composite and sandwich cylindrical, spherical and conical shell panels

    Curved and Layered Structures 9, pp. 320-344. Open-access journal article; source of the four figures above.
  4. 2022

    Static and free vibration analysis of sandwich shell panels using quadrilateral flat shell finite element

    Materials Today: Proceedings. Conference proceedings article.
  5. 2022

    Free vibration response of cylindrical and spherical FGM shell panels

    Materials Today: Proceedings. Related functionally graded material research.
  6. 2022

    Shell finite element analysis of FGM spherical panels

    IOP Conference Series: Materials Science and Engineering. Related functionally graded material research.

Earlier coauthored conference work includes static and free-vibration analysis of isotropic cylindrical shell panels, INCAM 2015, IIT Delhi.

Figure credits

Figures 1, 2, 9, and 13 are from A quadrilateral flat-shell element for the static and dynamic analysis of composite and sandwich cylindrical, spherical and conical shell panels, Curved and Layered Structures 9 (2022), 320-344. Copyright © 2022 the authors; published by De Gruyter. Reused under Creative Commons Attribution 4.0.

Full author attribution and publication details are available in the original source paper.

Figures were extracted from their original pages and resized for display. Figure content, labels, and plotted data are unchanged.

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