Derivation of Theory

LimitState:GEO

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The work done $W$ on a 180 $o$ arc of a log spiral overcoming cohesion $c$ for a relative body rotation of $ω$ is as follows:

∫ π W = 0 c.ds.(rωcos ϕ),

(21)

where $ds$ is the arc length. Thus

∫ π W = cω cosϕ r.ds. 0

(22)

Now

cosϕ = r.dψ , ds

(23)

where $ψ$ is measured clockwise along the arc from its starting point.

Hence

∫ π 2 ∫ π ψ tanϕ 2 W = cω r .dψ = cα (r0e ) .dψ 0 0

(24)

∫ π [ 2ψtanϕ]π W = r02cω e2ψtan ϕ.dψ = r02cα e------ 0 2tanϕ 0

(25)

-r02cω- ( 2πtan ϕ ) W = 2 tan ϕ e - 1

(26)

cω ( 2 2) cω cωul2 W = 2-tan-ϕ r1 - r0 = 2tanϕ-(r1 - r0) l = tan-ϕ

(27)