{"id":269,"date":"2022-01-13T23:17:35","date_gmt":"2022-01-13T23:17:35","guid":{"rendered":"https:\/\/books.gw-project.org\/land-subsidence-and-its-mitigation\/chapter\/box-1-justification-of-terzaghis-principle\/"},"modified":"2022-01-17T02:06:12","modified_gmt":"2022-01-17T02:06:12","slug":"box-1-justification-of-terzaghis-principle","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/land-subsidence-and-its-mitigation\/chapter\/box-1-justification-of-terzaghis-principle\/","title":{"raw":"Box\u00a01 Justification of Terzaghi\u2019s Principle","rendered":"Box\u00a01 Justification of Terzaghi\u2019s Principle"},"content":{"raw":"
\r\n

To justify Equation\u00a03<\/a>, namely that \u03a3\u00a0<\/em>A<\/em>i<\/em><\/sub>\u00a0<\/em><\/sub><<\u00a01, we can provide the following rough calculation. Assume the solid grains are spherical with radius r<\/em>. According to Hertz\u2019s theory (Hertz, 1881), the contact area A'<\/em> of two spheres pressed by the force P<\/em> reads:<\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle A^{\\prime }=1.23\\pi \\left (0.5\\frac{Pr}{E_{r}} \\right )[\/latex]<\/td>\r\n(Box 1-1)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

with E<\/em>r<\/em><\/sub> the sphere Young modulus (which reflects the stiffness of a solid as the ratio of its tensile stress and axial strain, ML-1<\/sup>T-2<\/sup>). Let\u2019s take a representative porous medium depth (250\u00a0m) and spheres radius equal to, r<\/em>\u00a0=\u00a00.5\u00a0mm, and assume full saturation. Considering the buoyant force exerted by water the weight P<\/em> of a grain column of height h<\/em> is:<\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle P=\\frac{h}{2r}\\gamma ^{\\prime }\\frac{4}{3}\\pi r^{2}[\/latex]<\/td>\r\n(Box 1-2)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

where:<\/p>\r\n\r\n\r\n\r\n\r\n
\u03b3<\/em>\u2032<\/td>\r\n=<\/td>\r\nspecific weight of the spheres minus the upward buoyant force (ML\u22122<\/sup>T\u22122<\/sup>)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

Using Equation\u00a0Box\u00a01\u20112, A<\/em>\u2032 above becomes Equation\u00a0Box\u00a01\u20113:<\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle A^{\\prime }=1.23\\pi r^{2}\\left ( \\frac{\\pi h\\gamma ^{\\prime}}{3E_{r}} \\right )^{2\/3}[\/latex]<\/td>\r\n(Box 1-3)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

Setting h<\/em>\u00a0= 250\u00a0m, \u03b3<\/em>\u2032 = 1.7\u00d7104<\/sup> N\/m3<\/sup> (N is a Newton, the SI unit of force [MLT-2<\/sup>], and 1 N is equal to 1\u00a0kg\u00a0m\u00a0s-2<\/sup>) and E<\/em>r<\/em><\/sub>\u00a0=\u00a01\u00d71011<\/sup>\u00a0N\/m2<\/sup> (corresponding to a volumetric grain compressibility c<\/em>b,r<\/em><\/sub>\u00a0<\/em><\/sub>=\u00a00.16\u00d710\u2011<\/sup>10<\/sup>\u00a0m2<\/sup>\/N and a grain Poisson ratio v<\/em>r<\/em><\/sub>\u00a0<\/em><\/sub>=\u00a00.25, being:<\/p>\r\n

[latex]\\displaystyle c_{b,r}=3\\frac{1-2\\nu _{r}}{E_{r}}[\/latex]<\/p>\r\n

we obtain:<\/p>\r\n

[latex]\\displaystyle A^{\\prime }=1.23\\pi (0.5\\ \\textup{mm})^{2}\\left ( \\frac{\\pi }{3}\\frac{250\\ \\textup{m}\\times 1.7\\times 10^{-4}\\frac{\\textup{N}}{\\textup{m}^{3}}}{1\\times 10^{11}\\frac{\\textup{N}}{\\textup{m}^{2}}} \\right )^{2\/3}[\/latex] [latex]\\displaystyle \\cong 0.00121\\ \\textup{mm}^{2}[\/latex]<\/p>\r\n

that is, equal to 0.121 percent of the horizontal projection area of the spheres (equal to 1\u00a0mm2<\/sup>). Hence the assumption that \u03a3\u00a0A<\/em>i<\/em><\/sub>\u00a0<\/em><\/sub><<\u00a01 is fully warranted.<\/p>\r\n

Return to where text linked to <\/span>Box 1<\/span><\/a><\/p>\r\n\r\n<\/div>","rendered":"

\n

To justify Equation\u00a03<\/a>, namely that \u03a3\u00a0<\/em>A<\/em>i<\/em><\/sub>\u00a0<\/em><\/sub><<\u00a01, we can provide the following rough calculation. Assume the solid grains are spherical with radius r<\/em>. According to Hertz\u2019s theory (Hertz, 1881), the contact area A’<\/em> of two spheres pressed by the force P<\/em> reads:<\/p>\n\n\n\n
<\/td>\n\"\displaystyle A^{\prime }=1.23\pi \left (0.5\frac{Pr}{E_{r}} \right )\"<\/td>\n(Box 1-1)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

with E<\/em>r<\/em><\/sub> the sphere Young modulus (which reflects the stiffness of a solid as the ratio of its tensile stress and axial strain, ML-1<\/sup>T-2<\/sup>). Let\u2019s take a representative porous medium depth (250\u00a0m) and spheres radius equal to, r<\/em>\u00a0=\u00a00.5\u00a0mm, and assume full saturation. Considering the buoyant force exerted by water the weight P<\/em> of a grain column of height h<\/em> is:<\/p>\n\n\n\n
<\/td>\n\"\displaystyle P=\frac{h}{2r}\gamma ^{\prime }\frac{4}{3}\pi r^{2}\"<\/td>\n(Box 1-2)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

where:<\/p>\n\n\n\n
\u03b3<\/em>\u2032<\/td>\n=<\/td>\nspecific weight of the spheres minus the upward buoyant force (ML\u22122<\/sup>T\u22122<\/sup>)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Using Equation\u00a0Box\u00a01\u20112, A<\/em>\u2032 above becomes Equation\u00a0Box\u00a01\u20113:<\/p>\n\n\n\n
<\/td>\n\"\displaystyle A^{\prime }=1.23\pi r^{2}\left ( \frac{\pi h\gamma ^{\prime}}{3E_{r}} \right )^{2/3}\"<\/td>\n(Box 1-3)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Setting h<\/em>\u00a0= 250\u00a0m, \u03b3<\/em>\u2032 = 1.7\u00d7104<\/sup> N\/m3<\/sup> (N is a Newton, the SI unit of force [MLT-2<\/sup>], and 1 N is equal to 1\u00a0kg\u00a0m\u00a0s-2<\/sup>) and E<\/em>r<\/em><\/sub>\u00a0=\u00a01\u00d71011<\/sup>\u00a0N\/m2<\/sup> (corresponding to a volumetric grain compressibility c<\/em>b,r<\/em><\/sub>\u00a0<\/em><\/sub>=\u00a00.16\u00d710\u2011<\/sup>10<\/sup>\u00a0m2<\/sup>\/N and a grain Poisson ratio v<\/em>r<\/em><\/sub>\u00a0<\/em><\/sub>=\u00a00.25, being:<\/p>\n

\"\displaystyle c_{b,r}=3\frac{1-2\nu _{r}}{E_{r}}\"<\/p>\n

we obtain:<\/p>\n

\"\displaystyle A^{\prime }=1.23\pi (0.5\ \textup{mm})^{2}\left ( \frac{\pi }{3}\frac{250\ \textup{m}\times 1.7\times 10^{-4}\frac{\textup{N}}{\textup{m}^{3}}}{1\times 10^{11}\frac{\textup{N}}{\textup{m}^{2}}} \right )^{2/3}\" \"\displaystyle \cong 0.00121\ \textup{mm}^{2}\"<\/p>\n

that is, equal to 0.121 percent of the horizontal projection area of the spheres (equal to 1\u00a0mm2<\/sup>). Hence the assumption that \u03a3\u00a0A<\/em>i<\/em><\/sub>\u00a0<\/em><\/sub><<\u00a01 is fully warranted.<\/p>\n

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