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• f~~ ~ a, <br />a~ I ~D"" *d~" I <br />Wf _~ ~'~ t 1 <br />-I '/~ '' <br />s~ ~~ t ~ Ov Tan ~ } <br />Y ~ I I~~ IDr <br />dY <br />PIIRPOSE: FIND DEPTH INTO PILLAR TO PEAR (MFI7CIMIIM) STABLE <br />VERTICAL PILLAR STRESS <br />(1) Force tending to move element is <br /> <br />(2) resistance top plus bottom <br />Frictional <br /> ll <br />20: Tan~,t ~Y <br />(3) Stability requires <br /> m~d~ a 2~v Tan~~d.Y <br /> mc{Ow -zo: Tan~dY <br /> ~ <br />dOM ~ zp ;, Tan o dY <br />(4) ~ <br />Confined strength <br />~~~ <br />• QY = OS + Tan 6 ~ <br />(5) Increment of stress (vertical) increase across <br /> element ' <br />from (3) c{~=Tan6da-„ Constantoa lost during differentiation <br /> dar o Tan 6 an dY 1 <br />m <br /> dQv __ 2 ran / Tan ~QY dY - = <br />/ ~ <br /> ~ <br />dY m dcT 2Tan !Tan I` ~ /I <br />(6) lents <br />Passive Pressure Equiva <br /> S <br />Tan S . t + Sin ~ Tan y~ = Tan ! - 1 <br /> t - Sin f 2Tan! <br />from (5) <br />dY m <br />o ! ~~ <br />d0.. 2Tan! rTan!-1 I` <br />e 2 Tani, <br />RY _ Tan 0 7 _ <br />e~ ian 0 (Tan 0-7) ~ ~ Tan 6 (Tan!-7~ (~ ) <br />T ~ <br />~Y =JTan l~Tan6-1~ ~) OY <br />• <br />