Geotechnical Engineering Circular No. 9 Design, Analysis, and Testing of Laterally Loaded Deep Foundations that Support Transportation Facilities
Download 6.03 Mb. View original pdf
|
Soldier Rev B
- Navigate this page:
- 12.4.3 Net Resistance (p) Using Piecewise Polynomial Curve Fitting
| 12.4.2 Bending Moment Profiles The flexural stiffness of reinforced concrete in drilled shafts exhibits nonlinear strain degradation it is not a constant. To reduce numerical errors in computing bending moment profiles from the profiles of bending curvature in a drilled shaft, it is therefore important to account for this nonlinear flexural stiffness (Reese and Welch 1975). Put another way, when an elastic beam is loaded, the compressive and tensile strains measured by strain gauges positioned equal distances from the centerline will be equal in magnitude and opposite in sign, at least at small strain. However, fora concrete element, once the concrete cracks in bending, the sitffness is no longer constant, and the elastic beam idealization no longer applies. The neutral axis moves toward the compression side. Field measured moment-curvature relationships for the concrete can be found from instrumentation at the ground line and at some distance above the ground line. Alternatively, nonlinear flexural stiffness relationships for reinforced concrete are available (Wang and Reese 1993). In considering nonlinear flexural stiffness of concrete, it is also important to account for other factors, including Concrete compressive strength will depend on the mix, water cement ratio, age, cure time and conditions of curing Concrete modulus will be different for the cracked and un-cracked sections The as-constructed diameter of the drilled shaft at the strain gauge location may differ from the nominal design diameter The as-constructed position of the reinforcement cage maybe off-center; The actual tensile strength of the steel reinforcement is usually higher than the nominal grade and, 192 If temporary casing is left in place, it will influence the strength, stiffness, and composite action of the shaft concrete. 12.4.3 Net Resistance (p) Using Piecewise Polynomial Curve Fitting Matlock and Ripperger (1956) and Dunnavant (1986) used piecewise cubic polynomial functions to fit discrete moment data. The approach is used to mitigate situations where the data is rather scattered, which is to be expected in natural layered soil profiles that exhibit nonlinear behavior. This polynomial is then subjected to differentiation to obtain the net soil resistance p. Figure 12-6 shows how the polynomial is fitted to five-point intervals along the shaft in a piecewise fashion. Least-square adjustments are used to establish the coefficients. Every five consecutive points along the shaft length where moment was calculated from the curvature found at strain gauge pair locations are fitted to one cubic polynomial curve. Double differentiation of the fitted polynomial curve with respect to the middle point of five provides the resistance p at that point. The resistance of the upper three points and the bottom three points is obtained from the smoothed local cubic polynomial moment curve using the top five points and bottom five points of the fitted equation, respectively. Also, a zero moment at the point of load application or a known moment value at the ground line should be included in the bending moment profiles that are used in this approach. The first polynomial M, is differentiated twice to evaluate p at the 3 moment levels closest to the surface (including the loading level. Other polynomials, such as Mare used to evaluate p at the group center point. The polynomial for the five points closest to the shaft tip is used to evaluate p at the three lowest points. Download 6.03 Mb. Share with your friends:
|