토양 내 응력 분포

토양의 응력 분포는 토양의 종류, 토양의 상대적 강성, 기초와 토양의^[1] 접촉 수준에서의 기초 깊이의 함수이다.건물, 교량 및 압력 침하 예측에는 외부 하중에 의한 토양 질량의 임의의 지점에서 수직 응력을 추정하는 것이 필수적이다.^[2]

경우에 따라서는

최종 적재 영역

지표면의 수직선 하중

지표면의 수직 점 하중

표면에서 수직 점 하중을 받는 탄성 반공간의 응력 계산 문제에 대한 해결책은 하중을 받는 표면의 해당 부분의 치수에 비해 깊이가 큰 토양의 퇴적물에서 유도되는 응력을 추정하는 데 유용할 것이다. $\Delta \sigma _{z}=-{\frac {3P}{2\pi R^{2}}}\cos ^{3}\theta$ z $\Delta \sigma _{z}=-{\frac {3P}{2\pi R^{2}}}\cos ^{3}\theta$ - 3 $\Delta \sigma _{z}=-{\frac {3P}{2\pi R^{2}}}\cos ^{3}\theta$ $\Delta \sigma _{z}=-{\frac {3P}{2\pi R^{2}}}\cos ^{3}\theta$ $r$ R $\Delta \sigma _{z}=-{\frac {3P}{2\pi R^{2}}}\cos ^{3}\theta$ cos 3 $\Delta \sigma _{z}=-{\frac {3P}{2\pi R^{2}}}\cos ^{3}\theta$ ⁡ 、 { $displaystyle$ \ $Delta$ \ $delta$ _ {z } = - { \ $frac$ { $3P }$ \ $cos$ ^ { 2 \ $\Delta \sigma _{z}=-{\frac {3P}{2\pi R^{2}}}\cos ^{3}\theta$ pi $R$ $\Delta \sigma _{r}={\frac {P}{2\pi R^{2}}}(-3\cos \theta \sin ^{2}\theta +{\frac {1-2\mu }{1+\cos \theta }})$ { } } \ $\Delta \sigma _{r}={\frac {P}{2\pi R^{2}}}(-3\cos \theta \sin ^{2}\theta +{\frac {1-2\mu }{1+\cos \theta }})$ $\Delta \sigma _{z}=-{\frac {3P}{2\pi R^{2}}}\cos ^{3}\theta$ $\Delta \sigma _{r}={\frac {P}{2\pi R^{2}}}(-3\cos \theta \sin ^{2}\theta +{\frac {1-2\mu }{1+\cos \theta }})$ $\Delta \sigma _{r}={\frac {P}{2\pi R^{2}}}(-3\cos \theta \sin ^{2}\theta +{\frac {1-2\mu }{1+\cos \theta }})$ = $\Delta \sigma _{r}={\frac {P}{2\pi R^{2}}}(-3\cos \theta \sin ^{2}\theta +{\frac {1-2\mu }{1+\cos \theta }})$ - 3 $\Delta \sigma _{r}={\frac {P}{2\pi R^{2}}}(-3\cos \theta \sin ^{2}\theta +{\frac {1-2\mu }{1+\cos \theta }})$ $\Delta \sigma _{r}={\frac {P}{2\pi R^{2}}}(-3\cos \theta \sin ^{2}\theta +{\frac {1-2\mu }{1+\cos \theta }})$ $\Delta \sigma _{r}={\frac {P}{2\pi R^{2}}}(-3\cos \theta \sin ^{2}\theta +{\frac {1-2\mu }{1+\cos \theta }})$ 2 - $\Delta \sigma _{r}={\frac {P}{2\pi R^{2}}}(-3\cos \theta \sin ^{2}\theta +{\frac {1-2\mu }{1+\cos \theta }})$ （ sin $\Delta \sigma _{r}={\frac {P}{2\pi R^{2}}}(-3\cos \theta \sin ^{2}\theta +{\frac {1-2\mu }{1+\cos \theta }})$ ） $theta$ }}}}} $\Delta \sigma _{t}={\frac {P}{2\pi R^{2}}}(1-2\mu )(\cos \theta -{\frac {1}{1+\cos \theta }})$ $\Delta \sigma _{t}={\frac {P}{2\pi R^{2}}}(1-2\mu )(\cos \theta -{\frac {1}{1+\cos \theta }})$ $\Delta \sigma _{t}={\frac {P}{2\pi R^{2}}}(1-2\mu )(\cos \theta -{\frac {1}{1+\cos \theta }})$ $\Delta \sigma _{t}={\frac {P}{2\pi R^{2}}}(1-2\mu )(\cos \theta -{\frac {1}{1+\cos \theta }})$ $\Delta \sigma _{t}={\frac {P}{2\pi R^{2}}}(1-2\mu )(\cos \theta -{\frac {1}{1+\cos \theta }})$ $\Delta \sigma _{t}={\frac {P}{2\pi R^{2}}}(1-2\mu )(\cos \theta -{\frac {1}{1+\cos \theta }})$ $\Delta \sigma _{t}={\frac {P}{2\pi R^{2}}}(1-2\mu )(\cos \theta -{\frac {1}{1+\cos \theta }})$ - $\Delta \sigma _{t}={\frac {P}{2\pi R^{2}}}(1-2\mu )(\cos \theta -{\frac {1}{1+\cos \theta }})$ ( $\Delta \sigma _{t}={\frac {P}{2\pi R^{2}}}(1-2\mu )(\cos \theta -{\frac {1}{1+\cos \theta }})$ $\Delta \sigma _{t}={\frac {P}{2\pi R^{2}}}(1-2\mu )(\cos \theta -{\frac {1}{1+\cos \theta }})$ - $\Delta \sigma _{t}={\frac {P}{2\pi R^{2}}}(1-2\mu )(\cos \theta -{\frac {1}{1+\cos \theta }})$ + $\Delta \sigma _{t}={\frac {P}{2\pi R^{2}}}(1-2\mu )(\cos \theta -{\frac {1}{1+\cos \theta }})$ $\Delta \sigma _{t}={\frac {P}{2\pi R^{2}}}(1-2\mu )(\cos \theta -{\frac {1}{1+\cos \theta }})$ ) ( $\Delta \sigma _{t}={\frac {P}{2\pi R^{2}}}(1-2\mu )(\cos \theta -{\frac {1}{1+\cos \theta }})$ ）（ $displaystyle$ \ $Delta$ \ $delta _$ { t } = $pi$ R^ { P $}$ { $2$ } ( 1 - $2$ \ $mu$ ) ( $cos$ )} $\cos \theta ={\frac {z}{R}}$ ⁡ $=$ $\cos \theta ={\frac {z}{R}}$ $R {\displaystyle$ \ $cos$ \theta = $scfrac {z}$ {R $\cos \theta ={\frac {z}{R}}$ $R={\sqrt {r^{2}+z^{2}}}$ = $R={\sqrt {r^{2}+z^{2}}}$ + $z$ 2 {\ $displaystyle$ R = $sqrt$ { $r2}$ + z^{ $2$ } $}}}$ $\Delta \sigma _{z}=-{\frac {3Pz^{3}}{2\pi R^{5}}}=-{\frac {3P}{2\pi }}{\frac {z^{3}}{(r^{2}+z^{2})^{5/2}}}=-{\frac {3P}{2\pi z^{2}\left[1+\left({\frac {r}{z}}\right)^{2}\right]^{\frac {5}{2}}}}$ z $\Delta \sigma _{z}=-{\frac {3Pz^{3}}{2\pi R^{5}}}=-{\frac {3P}{2\pi }}{\frac {z^{3}}{(r^{2}+z^{2})^{5/2}}}=-{\frac {3P}{2\pi z^{2}\left[1+\left({\frac {r}{z}}\right)^{2}\right]^{\frac {5}{2}}}}$ - $\Delta \sigma _{z}=-{\frac {3Pz^{3}}{2\pi R^{5}}}=-{\frac {3P}{2\pi }}{\frac {z^{3}}{(r^{2}+z^{2})^{5/2}}}=-{\frac {3P}{2\pi z^{2}\left[1+\left({\frac {r}{z}}\right)^{2}\right]^{\frac {5}{2}}}}$ P $\Delta \sigma _{z}=-{\frac {3Pz^{3}}{2\pi R^{5}}}=-{\frac {3P}{2\pi }}{\frac {z^{3}}{(r^{2}+z^{2})^{5/2}}}=-{\frac {3P}{2\pi z^{2}\left[1+\left({\frac {r}{z}}\right)^{2}\right]^{\frac {5}{2}}}}$ $\Delta \sigma _{z}=-{\frac {3Pz^{3}}{2\pi R^{5}}}=-{\frac {3P}{2\pi }}{\frac {z^{3}}{(r^{2}+z^{2})^{5/2}}}=-{\frac {3P}{2\pi z^{2}\left[1+\left({\frac {r}{z}}\right)^{2}\right]^{\frac {5}{2}}}}$ $\Delta \sigma _{z}=-{\frac {3Pz^{3}}{2\pi R^{5}}}=-{\frac {3P}{2\pi }}{\frac {z^{3}}{(r^{2}+z^{2})^{5/2}}}=-{\frac {3P}{2\pi z^{2}\left[1+\left({\frac {r}{z}}\right)^{2}\right]^{\frac {5}{2}}}}$ $\Delta \sigma _{z}=-{\frac {3Pz^{3}}{2\pi R^{5}}}=-{\frac {3P}{2\pi }}{\frac {z^{3}}{(r^{2}+z^{2})^{5/2}}}=-{\frac {3P}{2\pi z^{2}\left[1+\left({\frac {r}{z}}\right)^{2}\right]^{\frac {5}{2}}}}$ $\Delta \sigma _{z}=-{\frac {3Pz^{3}}{2\pi R^{5}}}=-{\frac {3P}{2\pi }}{\frac {z^{3}}{(r^{2}+z^{2})^{5/2}}}=-{\frac {3P}{2\pi z^{2}\left[1+\left({\frac {r}{z}}\right)^{2}\right]^{\frac {5}{2}}}}$ 5 $\Delta \sigma _{z}=-{\frac {3Pz^{3}}{2\pi R^{5}}}=-{\frac {3P}{2\pi }}{\frac {z^{3}}{(r^{2}+z^{2})^{5/2}}}=-{\frac {3P}{2\pi z^{2}\left[1+\left({\frac {r}{z}}\right)^{2}\right]^{\frac {5}{2}}}}$ - $\Delta \sigma _{z}=-{\frac {3Pz^{3}}{2\pi R^{5}}}=-{\frac {3P}{2\pi }}{\frac {z^{3}}{(r^{2}+z^{2})^{5/2}}}=-{\frac {3P}{2\pi z^{2}\left[1+\left({\frac {r}{z}}\right)^{2}\right]^{\frac {5}{2}}}}$ $\Delta \sigma _{z}=-{\frac {3Pz^{3}}{2\pi R^{5}}}=-{\frac {3P}{2\pi }}{\frac {z^{3}}{(r^{2}+z^{2})^{5/2}}}=-{\frac {3P}{2\pi z^{2}\left[1+\left({\frac {r}{z}}\right)^{2}\right]^{\frac {5}{2}}}}$ $\Delta \sigma _{z}=-{\frac {3Pz^{3}}{2\pi R^{5}}}=-{\frac {3P}{2\pi }}{\frac {z^{3}}{(r^{2}+z^{2})^{5/2}}}=-{\frac {3P}{2\pi z^{2}\left[1+\left({\frac {r}{z}}\right)^{2}\right]^{\frac {5}{2}}}}$ $\Delta \sigma _{z}=-{\frac {3Pz^{3}}{2\pi R^{5}}}=-{\frac {3P}{2\pi }}{\frac {z^{3}}{(r^{2}+z^{2})^{5/2}}}=-{\frac {3P}{2\pi z^{2}\left[1+\left({\frac {r}{z}}\right)^{2}\right]^{\frac {5}{2}}}}$ 3 ( $\Delta \sigma _{z}=-{\frac {3Pz^{3}}{2\pi R^{5}}}=-{\frac {3P}{2\pi }}{\frac {z^{3}}{(r^{2}+z^{2})^{5/2}}}=-{\frac {3P}{2\pi z^{2}\left[1+\left({\frac {r}{z}}\right)^{2}\right]^{\frac {5}{2}}}}$ $\Delta \sigma _{z}=-{\frac {3Pz^{3}}{2\pi R^{5}}}=-{\frac {3P}{2\pi }}{\frac {z^{3}}{(r^{2}+z^{2})^{5/2}}}=-{\frac {3P}{2\pi z^{2}\left[1+\left({\frac {r}{z}}\right)^{2}\right]^{\frac {5}{2}}}}$ + $\Delta \sigma _{z}=-{\frac {3Pz^{3}}{2\pi R^{5}}}=-{\frac {3P}{2\pi }}{\frac {z^{3}}{(r^{2}+z^{2})^{5/2}}}=-{\frac {3P}{2\pi z^{2}\left[1+\left({\frac {r}{z}}\right)^{2}\right]^{\frac {5}{2}}}}$ ) $\Delta \sigma _{z}=-{\frac {3Pz^{3}}{2\pi R^{5}}}=-{\frac {3P}{2\pi }}{\frac {z^{3}}{(r^{2}+z^{2})^{5/2}}}=-{\frac {3P}{2\pi z^{2}\left[1+\left({\frac {r}{z}}\right)^{2}\right]^{\frac {5}{2}}}}$ / $\Delta \sigma _{z}=-{\frac {3Pz^{3}}{2\pi R^{5}}}=-{\frac {3P}{2\pi }}{\frac {z^{3}}{(r^{2}+z^{2})^{5/2}}}=-{\frac {3P}{2\pi z^{2}\left[1+\left({\frac {r}{z}}\right)^{2}\right]^{\frac {5}{2}}}}$ $\Delta \sigma _{z}=-{\frac {3Pz^{3}}{2\pi R^{5}}}=-{\frac {3P}{2\pi }}{\frac {z^{3}}{(r^{2}+z^{2})^{5/2}}}=-{\frac {3P}{2\pi z^{2}\left[1+\left({\frac {r}{z}}\right)^{2}\right]^{\frac {5}{2}}}}$ - $\Delta \sigma _{z}=-{\frac {3Pz^{3}}{2\pi R^{5}}}=-{\frac {3P}{2\pi }}{\frac {z^{3}}{(r^{2}+z^{2})^{5/2}}}=-{\frac {3P}{2\pi z^{2}\left[1+\left({\frac {r}{z}}\right)^{2}\right]^{\frac {5}{2}}}}$ $\Delta \sigma _{z}=-{\frac {3Pz^{3}}{2\pi R^{5}}}=-{\frac {3P}{2\pi }}{\frac {z^{3}}{(r^{2}+z^{2})^{5/2}}}=-{\frac {3P}{2\pi z^{2}\left[1+\left({\frac {r}{z}}\right)^{2}\right]^{\frac {5}{2}}}}$ $\Delta \sigma _{z}=-{\frac {3Pz^{3}}{2\pi R^{5}}}=-{\frac {3P}{2\pi }}{\frac {z^{3}}{(r^{2}+z^{2})^{5/2}}}=-{\frac {3P}{2\pi z^{2}\left[1+\left({\frac {r}{z}}\right)^{2}\right]^{\frac {5}{2}}}}$ $\Delta \sigma _{z}=-{\frac {3Pz^{3}}{2\pi R^{5}}}=-{\frac {3P}{2\pi }}{\frac {z^{3}}{(r^{2}+z^{2})^{5/2}}}=-{\frac {3P}{2\pi z^{2}\left[1+\left({\frac {r}{z}}\right)^{2}\right]^{\frac {5}{2}}}}$ 2 $\Delta \sigma _{z}=-{\frac {3Pz^{3}}{2\pi R^{5}}}=-{\frac {3P}{2\pi }}{\frac {z^{3}}{(r^{2}+z^{2})^{5/2}}}=-{\frac {3P}{2\pi z^{2}\left[1+\left({\frac {r}{z}}\right)^{2}\right]^{\frac {5}{2}}}}$ [ 1 $\Delta \sigma _{z}=-{\frac {3Pz^{3}}{2\pi R^{5}}}=-{\frac {3P}{2\pi }}{\frac {z^{3}}{(r^{2}+z^{2})^{5/2}}}=-{\frac {3P}{2\pi z^{2}\left[1+\left({\frac {r}{z}}\right)^{2}\right]^{\frac {5}{2}}}}$ + ( $\Delta \sigma _{z}=-{\frac {3Pz^{3}}{2\pi R^{5}}}=-{\frac {3P}{2\pi }}{\frac {z^{3}}{(r^{2}+z^{2})^{5/2}}}=-{\frac {3P}{2\pi z^{2}\left[1+\left({\frac {r}{z}}\right)^{2}\right]^{\frac {5}{2}}}}$ z $\Delta \sigma _{z}=-{\frac {3Pz^{3}}{2\pi R^{5}}}=-{\frac {3P}{2\pi }}{\frac {z^{3}}{(r^{2}+z^{2})^{5/2}}}=-{\frac {3P}{2\pi z^{2}\left[1+\left({\frac {r}{z}}\right)^{2}\right]^{\frac {5}{2}}}}$ ) 2 ] $\Delta \sigma _{z}=-{\frac {3Pz^{3}}{2\pi R^{5}}}=-{\frac {3P}{2\pi }}{\frac {z^{3}}{(r^{2}+z^{2})^{5/2}}}=-{\frac {3P}{2\pi z^{2}\left[1+\left({\frac {r}{z}}\right)^{2}\right]^{\frac {5}{2}}}}$ 2 { display \ $Delta$ \ delta _ { { z = - \ $frac$ { 3 $Pi$ } { 3 } { 3 } { 3 $^2$ } { 3 } $\leftfrac {r}{z}}\right)^{2}\right]^{\frac {5}{2}}}}$