Locking-free variational formulations and isogeometric analysis for the Timoshenko beam models of strain gradient and classical elasticity

Research output: Contribution to journalArticleScientificpeer-review


Research units

  • Technical University of Munich


The Timoshenko beam bending problem is formulated in the context of strain gradient elasticity for both static and dynamic analysis. Two non-standard variational formulations in the Sobolev space framework are presented in order to avoid the numerical shear locking effect pronounced in the strain gradient context. Both formulations are shown to be reducible to their locking-free counterparts of classical elasticity. Conforming Galerkin discretizations for numerical results are obtained by an isogeometric Cp−1-continuous approach with B-spline basis functions of order p≥2. Convergence analyses cover both statics and free vibrations as well as both strain gradient and classical elasticity. Parameter studies for the thickness and gradient parameters, including micro-inertia terms, demonstrate the capability of the beam model in capturing size effects. Finally, a model comparison between the gradient-elastic Timoshenko and Euler–Bernoulli beam models justifies the relevance of the former, confirmed by experimental results on nano-beams from literature.


Original languageEnglish
Pages (from-to)137-159
Number of pages23
JournalComputer Methods in Applied Mechanics and Engineering
Publication statusPublished - 1 Sep 2018
MoE publication typeA1 Journal article-refereed

    Research areas

  • Isogeometric analysis, Shear locking, Size effect, Strain gradient elasticity, Timoshenko beam, Variational formulation

ID: 21562836