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1346Solutiontothefactorofsafety

Ifthemagnitudeoftheanchorloadsisgiven,thesolutiontothefactorofsafetyofthereinforcedslopeisderivedinthissection.Assuming[16]

ωc=uψ c+λp σ1ψ

eqs.[15a]–[15c]arerewrittenas

[17a]ηb 1∫aσ0

s′+ψ1 b1

F dx+η2s ∫aλpσp s′+ψF dx

s =Fx λp∑Px+

1∫b

Fa

ωcdxs[17b]ηb 1∫aσ0

1+s′ψ1 F dx+η∫bλ 2pσp 1

1+s′ψF dx

s as =Fy+λp∑Py+

1b

Fa

s′ωcdxs∫[17c]

Fs=

ηbbb

1aσ0ψrτdx+η2a

λpσpψrτdx a

rτωcdx

η1∫a

σ0rσdx η2∫a

λpσprσdx+Mc+λp∑Mp

Theaboveequationsarerearrangedas

[18a]η 1 A1+1FA′ 1 +η2 A2+

1A′ 2 =A3+1

A′3s Fs Fs[18b]η 1 B1+1FB′ 1 +η 2 B2+1 1

s FB′2 =B3+

B′3s Fs[18c]

FDs=η+Dη+DE1η1+E2η2+E3

inwhich[19a]Ab

b

1= ∫as′σ0dx;

A′1=∫aψσ0dx

[19b]Ab

b

2= ∫a

s′λpσpdx;

A′2=∫a

ψλpσpdx

[19c]

A3=Fx λp∑Px;

A′b

3=∫a

ωcdx

[19d]Bb1=∫a

σ0dx;B′b

1=∫a

s′ψσ0dx

[19e]Bb

b

2=∫aλpσpdx;

B′2=∫a

s′ψλpσpdx

[19f]

B3=Fy+λp∑Py;

B′b

3=∫a

s′ωcdx

[19g]Dbb

1=∫a

σ0ψrτdx;D2=∫a

λpσpψrτdx;

Db3= ∫a

rτωcdx

Can.Geotech.J.Vol.42,2005

[19h]

Eb

b

1= ∫a

σ0rσdx;

E2= ∫a

λpσprσdx;

E3=Mc+λp∑Mp

Equations[18a]–[18c]canbeanalyticallyresolved,resultinginanexplicitsolutiontothefactorofsafety(Fs)asfollows:

[20]

Fts=

3

+

+

wherep,q,andtcanbecomputedwiththeparameters

shownineqs.[19a]–[19h].Thebriefderivationofeq.[20]ispresentedinAppendixA;fordetails,seeZhuetal.(2003).

Solutiontorequiredanchorloads

Inthedesignofmeasuresforstabilizingfailedslopesorslopeshavingunacceptablestabilityconditions,themagni-tudeoftherequiredanchorloadsisoftenneeded.Inthiscase,themagnitudeoftherequiredanchorloadscanbecal-culatedbytrialanderrorusingeq.[20]untiltheslopeat-tainsthespecifiedfactorofsafety.Itcanalsobedirectlycomputedusinganotherexplicitexpression,thederivationofwhichisgivenbelow.Assuming[21a]ωx= s′+ψ

1F;ωy=1+s′ψ

1s

F;s

ωr=rσ+rτψ

1Fs

[21b]ωu=

1

F(uψ c);ωb=

1

F σ1ψs

s

eqs.[15a]–[15c]arewritteninmatrixformas

a12

a13 [22]

a11

η1 a21a22a23

λpη2 c1

= a31

a32

a 33

λ c2 p c3

inwhich

[23a]a11=∫b

b

aσ0ωxdx,

a12=∫a

σpωxdx,

ab

13= ∫a

ωbdx+∑Px

[23b]ab

b

21=∫a

σ0ωydx,

a22=∫a

σpωydx,

ab

23= ∫a

s′ωbdx ∑Py

[23c]ab

31=∫a

σ0ωrdx,

ab

32=∫a

σpωrdx,

a33= ∫b

a

rcωbdx ∑Mp