Size effects on shear deformable beam structures

Saba Tahaei Yaghoubi

Research output: ThesisDoctoral ThesisCollection of Articles


Bending, buckling and free vibration of isotropic as well as anisotropic centrosymmetric beams are formulated and studied within strain and velocity gradient theory. Different beam theories such as the third-order shear deformable, Timoshenko and Euler-Bernoulli beam theories are considered. A variational approach is applied in order to determine the equilibrium equations together with the classical and nonclassical boundary conditions. For studying the buckling of the beams, the nonlinear von Kármán strain tensor is considered in order to investigate the case of small strains but moderate rotations. The general formulations derived for the anisotropic beams can be easily simplified for more common cases of anisotropy such as orthotropy and transverse isotropy. The analysis of size effect on anisotropic beams is missing in the literature, while the present work enables one to investigate the size effect on ultra-small anisotropic centrosymmetric beam structures. The micro- or nanoscale beams are broadly used in sensors, resonators, actuators, nano- and microelectromechanical systems. In order to investigate the effect of isotropic or anisotropic length scale parameters on the static or dynamic behaviour of the beams, an analytical solution is provided for each case and the results are illustrated in figures. It is observed that the internal length scale parameters in different directions have a great impact on the behaviour of the beams, when the size of the structure is not negligible. This fact emphasizes the importance of considering the size effect in designing the miniature structures. Since the analytical solution is not available for all types of boundary conditions, a numerical method based on isogeometric analysis is developed. The comparison of the results obtained by the two different methods show that the numerical method works properly. Finally, a 3-D formulation for flexoelectric anisotropic materials within strain gradient elasticity is provided briefly. The simplification of the 3-D formulation for specific structures such as beams is the subject of future work.
Translated title of the contributionSize effects on shear deformable beam structures
Original languageEnglish
QualificationDoctor's degree
Awarding Institution
  • Aalto University
  • Paavola, Juha, Supervising Professor
  • Mousavi, S. Mahmoud, Thesis Advisor
Print ISBNs978-952-60-7624-9
Electronic ISBNs978-952-60-7623-2
Publication statusPublished - 2017
MoE publication typeG5 Doctoral dissertation (article)


  • beam
  • strain gradient
  • shear deformable
  • variational approach


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