, 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


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.


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

List of symbols


Director vector


Body force vector per unit reference volume


Stress–strain tensors


Global deformations vector of element


Module of elasticity


Orthogonal unit vectors


Deformation gradient tensor


Geometric tensors




Identity tensor




Curvature vectors


Element global tangent stiffness matrix


Moment cross-sectional per unit length vectors


External moments per unit reference area


Interpolation function matrix


Shape function vector


Force cross-sectional per unit length vectors


External forces per unit reference area


Zero tensor


Zero vector


First Piola–Kirchhoff stress tensor


Element global secant residual force vector


External power


Internal power


Pressure magnitude


Global deformations vector of nodes


Rotation tensor


Generalized external forces vector


Effective rotation angle


Cauchy stress tensor


Surface traction vector per unit reference area


Global effective displacements vector


External virtual work


Internal virtual work


Position vector


Mapping vector


Thermal expansion coefficient


Strain vectors


Temperature change


Strain vectors corresponding to stress-resultant vectors


Thickness coordinate


Membrane strain vectors


Global effective angles tensor


Global effective angles vector


Poisson’s ratio


Surface coordinates


Stress-resultant vectors


Stress vectors


Strain–displacement tensors


Spin tensor


Spin vector



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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