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Eigenvalue analysis of compatible and incompatible rectangular four‐node quadrilateral elements
Author(s) -
Hacker W. L.,
Schreyer H. L.
Publication year - 1989
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1620280315
Subject(s) - quadrilateral , eigenvalues and eigenvectors , hourglass , direct stiffness method , stiffness matrix , element (criminal law) , stiffness , finite element method , mathematics , mathematical analysis , convergence (economics) , node (physics) , structural engineering , monotonic function , matrix (chemical analysis) , bending stiffness , beam (structure) , geometry , engineering , physics , materials science , quantum mechanics , astronomy , political science , law , economics , composite material , economic growth
Abstract Exact analytical expressions for the eigenvalues of the elastic stiffness matrix are obtained for the four‐node, rectangular, quadrilateral element. A procedure is given for identifying alternative hourglass modes and eigenvalues which render the element incompatible but with non‐monotonic convergence assured. A convergence study confirms that for the special case of when the hourglass modes coincide with beam bending the element can serve as a beam element. Analytical expressions are given for the resulting element stiffness matrix.