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Soil mechanics - Chapter 17

Tản áp lực nước lỗ rỗng trong quá trình hợp nhất có thể được tính toán rất đơn giản bởi một thủ tục giải pháp số, bằng cách sử dụng các phương pháp khác biệt hữu hạn. Điều này được trình bày trong chương này. Kỹ thuật này được giữ càng đơn giản càng tốt. | Chapter 17 NUMERICAL SOLUTION The dissipation of the pore water pressures during the consolidation process can be calculated very simply by a numerical solution procedure using the finite difference method. This is presented in this chapter. The technique is kept as simple as possible. 17.1 Finite differences The differential equation for one dimensional consolidation is equation 15.17 dp d2pj dt Cv di2 17.1 The time derivative can be approximated by dp Pi t At -Pi t dt At 17.2 where the index i indicates that the values refer to the pressures in the point z Zj. Equation 17.2 can be considered as the definition of the partial derivative dp dt except that the limit t 0 has been omitted. Finite differences will also be used in the z-direction. For this purpose the thickness h of the sample is subdivided into n small elements of thickness Az z i 1 i i - 1 - . p Az h. n The second derivative with respect to z can be approximated by d2p _ pi i t - 2pi t pj-1 t dz2 Az 2 17.3 17.4 Figure 17.1 Second derivative. This relation is illustrated in Figure 17.1. The formula can most simply be found by noting that the second derivative is the derivative of the first derivative. This means that the second derivative is the difference of the slope in the upper part of the figure and the slope in the lower part of the figure 104 Arnold Verruijt Soil Mechanics 17. NUMERICAL SOLUTION 105 divided by the distance Az. It can also be verified from the figure that for a straight line the expression 17.4 indeed gives a value zero because then the value in the center is just the average of the values at the two values above it and below it. Substitution of 17.2 and 17.4 into 17.1 gives Pi t At Pi t o pi 1 t - 2pi t Pi-1 t 17.5 where At a c A 2 . 17.6 The expression 17.5 is an explicit formula for the new value of the pore pressure in the point i if the old values at time t in that point and in the two points just above and just below it are known. The boundary conditions must also be .