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Meccanica

, Volume 54, Issue 1–2, pp 205–221 | Cite as

Buckling and post-buckling of arbitrary shells under thermo-mechanical loading

  • M. Rezaiee-PajandEmail author
  • D. Pourhekmat
  • E. Arabi
Article
  • 129 Downloads

Abstract

Thermo-mechanical buckling and post-buckling analysis of arbitrary, smooth and folded shells with different boundary conditions are investigated. A pure displacement-and-theory-based isoparametric curved triangular shell element is introduced. This element is neither hybrid-mixed nor degenerated. Nevertheless, it is free from locking problem. The new element has six nodes while each node has three translational and three rotational (including the drilling) degrees of freedom. Large displacements and rotations are considered by employment of Total-Lagrangian scheme and Euler–Rodrigues formulation. The first-order shear deformation theory is used, and the proposed element is capable of modeling thin to thick shells.

Keywords

Thermo-mechanical buckling and post-buckling Folded structures Nonlinear triangular shell element Shear deformation Large rotations Drilling degrees of freedom 

List of symbols

a

Director vector

\(\bar{b}\)

Body force vector per unit reference volume

Dαβ

Stress–strain tensors

d

Global deformations vector of element

E

Module of elasticity

ei

Orthogonal unit vectors

F

Deformation gradient tensor

Gα

Geometric tensors

h

Thickness

I

Identity tensor

J

Jacobian

kα

Curvature vectors

kG

Element global tangent stiffness matrix

mα

Moment cross-sectional per unit length vectors

\(\bar{m}\)

External moments per unit reference area

N

Interpolation function matrix

n

Shape function vector

nα

Force cross-sectional per unit length vectors

\(\bar{n}\)

External forces per unit reference area

O

Zero tensor

o

Zero vector

P

First Piola–Kirchhoff stress tensor

PG

Element global secant residual force vector

Pext

External power

Pint

Internal power

p

Pressure magnitude

pG

Global deformations vector of nodes

Q

Rotation tensor

\(\bar{q}\)

Generalized external forces vector

r

Effective rotation angle

T

Cauchy stress tensor

\(\bar{t}\)

Surface traction vector per unit reference area

u

Global effective displacements vector

Wext

External virtual work

Wint

Internal virtual work

x

Position vector

z

Mapping vector

α

Thermal expansion coefficient

γα

Strain vectors

ΔT

Temperature change

εα

Strain vectors corresponding to stress-resultant vectors

\(\zeta\)

Thickness coordinate

ηα

Membrane strain vectors

Θ

Global effective angles tensor

θ

Global effective angles vector

ν

Poisson’s ratio

ξα

Surface coordinates

σα

Stress-resultant vectors

τα

Stress vectors

Ψα

Strain–displacement tensors

Ω

Spin tensor

ω

Spin vector

Notes

Acknowledgements

We hereby acknowledge that parts of these computations were performed on the High-Performance Computing (HPC) center of Ferdowsi University of Mashhad.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Civil EngineeringFerdowsi University of MashhadMashhadIran

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