Minho Kwon

Exact Stiffness for Beams on Kerr-Type Foundation: The Virtual Force Approach

This paper alternatively derives the exact element stiffness equation for a beam on Kerr-type foundation. The shear coupling between the individual Winkler-spring components and the peripheral discontinuity at the boundaries between the loaded and the unloaded soil surfaces are taken into account in this proposed model. The element flexibility matrix is derived based on the virtual force principle and forms the core of the exact element stiffness matrix. The sixth-order governing differential compatibility of the problem is revealed using the virtual force principle and solved analytically to obtain the exact force interpolation functions. The matrix virtual force equation is employed to obtain the exact element flexibility matrix based on the exact force interpolation functions. The so-called “natural” element stiffness matrix is obtained by inverting the exact element flexibility matrix. One numerical example is utilized to confirm the accuracy and the efficiency of the proposed beam element on Kerr-type foundation and to show a more realistic distribution of interactive foundation force.

Improved nonlinear displacement-based beam element on a two-parameter foundation

This paper presents a nonlinear beam element on a two-parameter foundation. A set of governing differential equations of the problem (strong form) is first derived. The displacement-based beam-foundation element with improved displacement shape functions (weak form) is then formulated based on virtual displacement principle. The improved functions are analytically derived based on homogeneous solution to the governing differential equilibrium equation of the problem and are employed to enhance the model accuracy. Tonti’s diagrams are used to conveniently represent the equations that govern both the strong and weak forms of the problem. An averaging technique previously proposed by the authors is employed to determine system parameters needed in evaluating the displacement shape functions. Finally, two numerical simulations are used to verify the accuracy and the efficiency of the proposed beam model. The first simulation is used to perform convergence studies of the proposed model and to show its accuracy in representing both global and local responses. The second simulation is used to address effects of the two-parameter foundation model on system responses when compared to the Winkler foundation model.