Estimating Pit‐Excavation Volume Using Cubic Spline Volume Formula

Abstract

Current methods used for estimating the volume of pit excavation assumes that the ground profile between the grid points is linear (based on the trapezoidal rule), or nonlinear (based on Simpson's 1/3 and 3/8 formulas). Generally speaking, the nonlinear profile formulas provide better accuracy than the linear profile formulas. However, all the methods mentioned have a common drawback: The joints (grid points) of any two straight lines (the trapezoidal rule), quadratic polynomials (Simpson's 1/3 formula), or cubic polynomials (Simpson's 3/8 formula) form sharp corners. Consequently, these models may not describe the ground profile well. In this paper, formulas based on an extension of the cubic spline polynomial are presented. The grid of this new method is constructed by choosing the variational points of the ground profile and then dividing the pit's area into rectangles of unequal size. Therefore, the new formulas can be used for estimating the excavation volume of a pit divided into a grid with unequal intervals. Further, due to the characteristics of the cubic spline polynomial, a curve modeled by this polynomial can match the ground profile well. The new formulas should provide better accuracy than the other methods mentioned.