Fall Cone Shear Strength Extrapolation via Mathematical Modeling
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Solution Overview
Problem
The existing fall cone apparatus has limitations in measuring shear strength of soil beyond its capability, due to a shortage of empirically tabulated data and lack of graphical representations, restricting its application beyond a certain penetration depth and weight per apex angle ratio range.
Innovation Solution
A method is developed to generate multiple plots on double logarithmic paper to deduce mathematical models based on data from the fall cone apparatus, extending the measurement range by using power functions to predict shear strength values beyond the initial penetration depth and weight per apex angle ratio ranges, achieving high accuracy with a coefficient of determination greater than 0.99.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If the fall cone apparatus is used with existing empirical data and graphs, then shear strength can be determined within the limited penetration depth range of 4-20 mm, but the measurement capability is restricted beyond this range due to shortage of empirically tabulated data
Solution Approach 1:
The patent transforms the limited empirical data into a universal mathematical model by changing the representation parameters from discrete tabulated values to continuous power function relationships. The mathematical model s = k(W/A)^n(d/L)^(m) allows calculation of shear strength for any penetration depth and cone configuration, extending the apparatus versatility while maintaining measurement accuracy through the high coefficient of determination (R² > 0.99).
2Reliability
If the fall cone apparatus uses standard cones with W/A ratios from 0.17 to 13.3, then measurements can be made within this ratio range, but the apparatus cannot measure shear strength for cones with W/A ratios outside this range
Solution Approach 1:
The patent creates a universal mathematical model that makes the fall cone apparatus multi-functional by enabling accurate shear strength determination for any cone weight per apex angle ratio, not just the standard range. The model s = k(W/A)^n(d/L)^(m) with experimentally determined constants allows the same apparatus to reliably measure shear strength for custom cones with W/A ratios from 0.1 to 100, significantly expanding the apparatus adaptability while preserving reliability through validated mathematical relationships.
3Quantity of substance
If empirical tabulated data is used for shear strength determination, then results are available only for specific penetration depths, but interpolation or extrapolation beyond tabulated values is not reliable
Solution Approach 1:
The patent replaces the mechanical lookup system (empirical tables and graphs) with a mathematical computation system based on power functions. Instead of relying on pre-computed tabulated values that require interpolation, the mathematical model s = k(W/A)^n(d/L)^(m) with determined constants enables direct calculation of shear strength for any penetration depth, providing both the quantity of data needed and the precision required for extrapolation beyond traditional ranges through the high coefficient of determination (R² > 0.99).
Data Source
AI summary
A method of determining shear strength of soil using a fall cone apparatus is provided. The method includes generating a first plot having graph lines defined for penetration depth values ranging from 4 to 20 mm on a first logarithmic paper, deducing a first mathematical model based on the first plot, and generating a second plot having the graph lines extended for a predefined penetration depth range based on the first mathematical model. The method further includes generating a third plot having graph lines corresponding to the extended penetration depth values defined for the W/A ratio values ranging from 0.17 to 13.3 on a second logarithmic paper, deducing a second mathematical model based on the third plot, generating a fourth plot having the graph lines extended for a predefined W/A ratio range based on the second mathematical model, and determining shear strength based on the fourth plot.