APPLICATION OF A GAUSSIAN QUADRATURE ALGORITHM TO CROSS-SECTION CONSTRUCTIONS IN BLUFF DISPLACEMENT MODELING STUDIES

During the past nine years, coastal bluff displacement data have been collected on the southeast shore of Lake Michigan using a pole-and-cable survey system (Chase et al., 2001). In 2004, in-place inclinometers, vibrating wire piezometers, and dewatering wells were installed at three study sites to provide hourly digital information about slope displacements and potentiometric levels during staged dewatering experiments. From a very large displacement and water level data base, current failure geometries and future failure events can be analyzed and predicted. Balancing of cross-sections is an integral part of the kinematic analyses. Polygonal areas between pin lines are defined prior to displacement, and then held constant during deformation recorded from ground surface measurements. Hence near-exact displacement geometries can be established and adapted into slope stability modeling programs This study suggests how a numerical analysis technique called Gaussian quadrature may be applied in the cross-section balancing algorithm that is to be ultimately developed. The technique involves integrating an interpolating polynomial through fixed nodes between a given interval. The integration is of the form:

∫ f(x)w(x)dx≈∑ A_if(x_i), where:

f(x) = Polynomial function through stratigraphic unit boundary between two pinlines, w(x) = applied weight to adjust function, f(x_i) = value of function at fixed point x_i, and A_i = integral of divided difference at x_i

This technique will facilitate the determination of areas between pin lines in a structurally deformed profile. These areas may be compared with those in the original, non-displaced profile. Adjustments can then be made until the difference between the two areas is minimized.