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

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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.

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Abbreviations

a :

Director vector

\(\bar{b}\) :

Body force vector per unit reference volume

D αβ :

Stress–strain tensors

d :

Global deformations vector of element

E :

Module of elasticity

e i :

Orthogonal unit vectors

F :

Deformation gradient tensor

G α :

Geometric tensors

h :

Thickness

I :

Identity tensor

J :

Jacobian

k α :

Curvature vectors

k G :

Element global tangent stiffness matrix

m α :

Moment cross-sectional per unit length vectors

\(\bar{m}\) :

External moments per unit reference area

N :

Interpolation function matrix

n :

Shape function vector

n α :

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

P G :

Element global secant residual force vector

P ext :

External power

P int :

Internal power

p :

Pressure magnitude

p G :

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

W ext :

External virtual work

W int :

Internal virtual work

x :

Position vector

z :

Mapping vector

α :

Thermal expansion coefficient

γ α :

Strain vectors

ΔT :

Temperature change

ε α :

Strain vectors corresponding to stress-resultant vectors

\(\zeta\) :

Thickness coordinate

η α :

Membrane strain vectors

Θ:

Global effective angles tensor

θ :

Global effective angles vector

ν :

Poisson’s ratio

ξ α :

Surface coordinates

σ α :

Stress-resultant vectors

τ α :

Stress vectors

Ψα :

Strain–displacement tensors

Ω:

Spin tensor

ω :

Spin vector

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Acknowledgements

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

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Authors and Affiliations

  1. Department of Civil Engineering, Ferdowsi University of Mashhad, Mashhad, Iran

    M. Rezaiee-Pajand, D. Pourhekmat & E. Arabi

Authors
  1. M. Rezaiee-Pajand
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  2. D. Pourhekmat
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  3. E. Arabi
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Correspondence to M. Rezaiee-Pajand.

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Appendix

Appendix

The algorithm of the Generalaized Displacement Control Method for each iteration j of the increment i can be summarized as below:

  1. 1.

    Compute the tangent stiffness matrix \(K_{j}^{i}.\)

  2. 2.

    Compute internal force \(P_{int}^{i}.\)

  3. 3.

    Obtain residual load using \(R_{j}^{i} = \bar{P}^{i} - P_{int}^{i}.\)

  4. 4.

    Obtain \(\delta u_{pj}^{i}.\) using \(\delta u_{pj}^{i} = (K_{j}^{i} )^{ - 1} P_{ext}.\)

  5. 5.

    Obtain \(\delta u_{rj}^{i}.\) using \(\delta u_{rj}^{i} = (K_{j}^{i} )^{ - 1} R_{j}^{i}.\)

  6. 6.

    Obtain \(\delta \lambda_{j}^{i}.\) according to \(\delta \lambda_{j}^{i} = - \frac{{\delta u_{p1}^{i - 1} .\,\,\delta u_{rj}^{i} }}{{\delta u_{p1}^{i - 1} .\,\delta u_{pj}^{i} \,}} .\)

  7. 7.

    Update load factor as \(\lambda^{i} = \lambda^{i} + \delta \lambda_{j}^{i}.\)

  8. 8.

    Update load vector as \(\bar{P}^{i} = \bar{P}^{i} + \delta \lambda_{j}^{i} P_{ext}.\)

  9. 9.

    Update displacements as \(u^{i} = u^{i} + \delta \lambda_{j}^{i} \delta u_{pj}^{i} + \delta u_{rj}^{i}.\)

  10. 10.

    If convergence is achieved, then

  11. 11.

    Next increment \(i = i + 1.\)

  12. 12.

    Else

  13. 13.

    Next iteration \(j = j + 1.\)

  14. 14.

    End if.

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Rezaiee-Pajand, M., Pourhekmat, D. & Arabi, E. Buckling and post-buckling of arbitrary shells under thermo-mechanical loading. Meccanica 54, 205–221 (2019). https://doi.org/10.1007/s11012-018-00928-7

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  • DOI: https://doi.org/10.1007/s11012-018-00928-7

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