In this thesis, formulas for the local tangent stiffness matrix of a plane frame member are derived by differentiating the member resistance vector in the displaced position. This approach facilitates an analysis using only one element per member. The formulas are checked by finite difference. The derivation leads to the familiar elastic and geometric stiffness matrices used by other authors plus an additional higher order geometric stiffness matrix. Contributions of each of the three sub-matrices to the tangent stiffness matrix are studied on both the member and structure levels through two numerical examples. These same examples are analyzed three different ways for comparison. First, the examples are analyzed using the method presented in this thesis. Second, they are analyzed with the finite element modeling software ABAQUS/CAE using only one element per member. Third, they are analyzed with ABAQUS using 200 elements per member. Comparisons are made assuming the ABAQUS analysis which uses 200 elements per member is the most accurate. The element presented in this thesis performs much better than the ABAQUS analysis which uses one element per member, with maximum errors of 1.0% and 40.8% respectively, for a cantilever column example. The maximum error for the two story frame example using the ABAQUS analysis with one element per member is 42.8%, while the results from the analysis using the element presented in this thesis are within 1.5%. Using the element presented in this thesis with only one element per member gives good and computationally efficient results for second-order analysis.



College and Department

Ira A. Fulton College of Engineering and Technology; Civil and Environmental Engineering



Date Submitted


Document Type





structural, analysis, one element, second order, tangent stiffness matrix