Exact microstructure-dependent Timoshenko beam element

Anssi T. Karttunen*, Jani Romanoff, J. N. Reddy

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

30 Citations (Scopus)
204 Downloads (Pure)


In this study, we develop an exact microstructure-dependent Timoshenko beam finite element. First, a Timoshenko beam model based on the modified couple-stress theory is reviewed briefly. The general closed-form solution to the equilibrium equations of the beam is derived. The two-dimensional in-plane stress distributions along the beam in terms of load resultants are obtained. In addition, the general solution can be used to model any distributed load which can be expressed as a Maclaurin series. Then, the solution is presented in terms of discrete finite element (FE) degrees of freedom. This representation yields the exact shape functions for the beam element. Finally, the exact beam finite element equations are obtained by writing the generalized forces at the beam ends using the FE degrees of freedom. Three calculation examples which have applications in micron systems and sandwich structures are presented.

Original languageEnglish
Pages (from-to)35-42
Number of pages8
JournalInternational Journal of Mechanical Sciences
Publication statusPublished - 1 Jun 2016
MoE publication typeA1 Journal article-refereed


  • Beam finite element
  • Couple-stress
  • General solution
  • Microstructure
  • Timoshenko beam


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