国土技術政策総合研究所資料 · no.880 december 2015 国土交通省...
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ISSN 1346-7328
国総研資料 第 880 号
平 成 27 年 12 月
国土技術政策総合研究所資料
TECHNICAL NOTE of
National Institute for Land and Infrastructure Managemen t
No.880 December 2015
国土交通省 国土技術政策総合研究所
National Institute for Land and Infrastructure Management
Ministry of Land, Infrastructure, Transport and Tourism, Japan
荷重抵抗係数アプローチによる
レベル１信頼性設計法に関する基礎的研究
～永続状態におけるケーソン式岸壁の滑動および転倒照査を対象に～
竹信 正寛・西岡 悟史・佐藤 健彦・宮田 正史
A Basic Study of the Level 1 Reliability Design Method
Based on Load and Resistance Factor Approach
~Performance verifications of sliding failure and overturning failure
for caisson type quay walls in permanent situation~
Masahiro TAKENOBU, Satoshi NISHIOKA, Takehiko SATO, Masafumi MIYATA
i
No. 880 2015 12
(YSK-N-323)
19
1
2
1
4.5 20.0
1
*
** ***
****
- - -- Fax e-mail: [email protected]
ii
Technical Note of NILIM
No. 880 Dec.2015
(YSK-N-323)
A Basic Study of the Level 1 Reliability Design MethodBased on Load and Resistance Factor Approach
Performance verifications of sliding failure and overturning failure for caisson type quay walls in permanent situation
Masahiro TAKENOBUSatoshi NISHIOKATakehiko SATOMasafumi MIYATA
Synopsis
The level 1 reliability design method (partial factor design method) has been introduced as a
performance verification method for overall stability of breakwaters and mooring facilities, according to
the Japanese design standard for port facilities whose title is "Technical Standards and Commentaries for
Port and Harbour Facilities in Japan (2007)". The purpose of this study is to show a new direction for the
Level 1 reliability design method in preparation for the next revision of the standard, focusing on the
following two points;
1) Adoption of the partial factor design method based on Load and Resistance Factor
Approach instead of the Material Factor Approach, and
2) Readjustment of the target safety level for sliding failure and overturning failure of
caisson type quay walls in permanent situations.
Regarding the first point, the authors came to the conclusion that it’s rational to adopt Load and
Resistance Factor Approach as the partial factor design method, especially for overall stability checks of
port structures which are exposed to strong interaction effects between ground and structures.
Regarding the second point, the authors proposed that the target safety level should be readjusted to
the level of past structures designed by safety factor method. In addition, the authors also proposes two sets
of partial factors based on Load and Resistance Factor Approach, which was calculated by using Monte
Carlo Simulation as a reliability analysis method.
Key Words : Level 1 reliability design method, Load and Resistance Factor Approach, Target safety
level, Caisson type quay wall, Permanent situation, Monte Carlo Simulation
* Senior Researcher, Port Facilities Division, Port and Harbor Department, NILIM ** Exchanging Researcher, Port Facilities Division, Port and Harbor Department, NILIM
(TOA CORPORATION) *** Exchanging Researcher, Port Facilities Division, Port and Harbor Department, NILIM
(PENTA-OCEAN CONSTRUCTION Co., Ltd.) **** Head, Port Facilities Division, Port and Harbor Department, NILIM 3-1-1 Nagase, Yokosuka, 239-0826 Japan Phone +81-468-44-5029 Fax +81-468-44-5081 e-mail: [email protected]
iii
iv
No 880
- 1 -
1)
1
2)
2)
1)
0.1
0.05
3)
ODA
1 FORM First-Order Reliability Method
/
- 2 -
2
3
3
1
4)
5)
No 880
- 3 -
1
1
2
3
3 1) 3Pf 2 ��
1 ��
3
2
1 FORM First-Order Reliability
Method
1
Rd Sd
1
3 2
1
3
3 PfT � Pf 2 �T � � 1 Rd � Sd
1 2
1 2
a)
6)
6)
b)
/
- 4 -
=1.0
γ α μ/Xk
V
1
0.9
0.9
0.9 1.1 1.8
1.1
1.0
w1
w2
w3
w1
w2
w3
No 880
- 5 -
7)
a)
8)
1
b)
8)
1
9)
/
- 6 -
10)
1
1
1
No 880
- 7 -
N
N
N
11)
12 ,13)
/
- 8 -
R S
Fs R /SFs
17)
R S
R S
R = μ W U PV
S = PH PwH
R = W x1 U x2 PV x3
S = PH y1 PwH y2
μ W U PV PH
PwH x1~3 y1~2
2514)
1.2
1.2
3415)
1.0 4216) 1.1
11 17)
No 880
- 9 -
1)
4
4
α μ/Xk V
μd ( Wd PVd Ud ) γa ( PHd PwHd )
μ W U PV
PH
PwH γa
d
μ
μk k
γf
μd γf μk
RC NC
1) 3.1
1.0×10-3 2.7
4.0×10-3
/
- 10 -
27
1.2
0.001
1.2 1.2
kh=0.05
0.05 0.10 0.15
a)
B
H
27kh=0.05
1.0
γγφ
γφ
γ
μ
No 880
- 11 -
/
- 12 -
0.05
0.05
1.3 1.5
kh=0.05
b)
0.0
0.5
1.0
1.5
2.0
0 5 10 15 20 25
H11
H11
H19
H11&H19 (kh=0.05)
4.5m 7.5m 10.0m 16.0m 20.0m
No 880
- 13 -
c)
kh=0.05,0.10,0.15
2.0
kh=0.15
kh=0.10
kh=0.10-0.15
1)
18)
5)
1
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0 5 10 15 20 25
H11
H11 H19 H11&H19 (kh=0.05)H11&H19 (kh=0.10)H11&H19 (kh=0.15)
4.5m 7.5m 10.0m 16.0m 20.0m
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0 5 10 15 20 25
H11
H11 H19 H11&H19 (kh=0.05)H11&H19 (kh=0.10)H11&H19 (kh=0.15)
4.5m 7.5m 10.0m 16.0m 20.0m
/
- 14 -
2
kh=0.10-0.15
3
kh 0.15
11
1.2 1.2
No 880
- 15 -
1 FORM First-Order
Reliability Method
MCS10)
MCS
MCS
MCS
1.3
MCS RC
MCS
R S 300
MCS
6600 0. 22 =6,600 300
R S R S
R S
R S
R S γR 0.76 γS 1.00
MCS
(2952.77, 2962.02)
(3865.34, 2965.42)
R
S
/
- 16 -
MCS
FORM
MCS
MCS
3
MCS
a)
20.0m 1.8m 30kN/m2
5.66m
0.24
7.80m
0.33
0.4019)
b)
Z = R - S
= μ[μ]× W [γc] U [RWL]
+ Pv [γsat1 γt1 γt2 RWL Ka cosδ]
PH [γsat1 γt1 γt2 RWL Ka cosδ] PwH [RWL]
Z = R - S
= W×x1 [γc] U×x2 [RWL]
+ Pv×x3 [γsat1 γt1 γt2 RWL Ka cosδ]
PH ×y1 [γsat1 γt1 γt2 RWL Ka cosδ] PwH ×y2[RWL]
μ
W
γγφ
γφ
γ
μ
No 880
- 17 -
U
PV PH PwH x1~3 y1~2 γc RWL γsat1 γt1 γt2 Ka cosδ
c)
R 20)
Z
Z 0
500,000
Z f(z)
54,921
1.0×10-1 =54,921/500,000
Z
S R
S = R
Z 0
Z 0
MCS
γ sat1
γ t1
γ t2
γ c
μR.W.L.
K a cosδ
0Z
f (Z)
Z 0 Z 0β σZ
Pf
μ Z
/
- 18 -
S R
a)MCS
Z
L = p (γc) × p (μ) × p (γsat1) × p (γt1)
× p (γt2) × p (RWL) ×p (Ka cosδ)
L
p ()
γc
μ
γsat1
γt1
γt2
RWL
Ka cosδ
Ln(L) = Ln (p ( c ))+Ln (p ( ))+Ln (p ( sat1 ))
+ Ln (p ( t1 ))+Ln (p ( t2 )) + Ln ( p(RWL )) +Ln (p (Ka cos ))
Ln
b)a)
FORM
1.0
1.0
nj
XXZ
XXZ
j
j
X
n
j j
Xj
j ,,2,1:2
1
2
1
��
��
��
���
����
�
��
���
����
�
��
�
��
�
�
�
�
�
X α X Z X* X σ j
MCS
X Z
MCS X0 ΔXj 2Z ΔZ
500
700
900
1100
1300
1500
1700
1900
500 700 900 1100 1300 1500 1700 1900
No 880
- 19 -
αXj = ΔZ / ΔXj
αXj Xj ΔXj Xj ΔZ Z ΔXj
j X
MCS
ΔXj
ΔXj
σ ΔXj
ΔXj
0.9350.157 -0.316
0.5210.094
-0.842
a) MCS
MCS
S R
Sk Rk
Sd Rd
Sk Rk
γS = Sd / Sk
γR = Rd / Rk
Sk
Rk
Sd
Rd
γS Sk
γR Rk
Z
XX *j
X
Z
X *j- X/2 X *j+ X/2
Z (X *j X/2)
Z (X *j+ X/2)
Z (X *j)
αXj
-1.0
-0.5
0.0
0.5
1.0
γc μ γsat1 γt2 γt1 R.W.L. Ka2.cosδ Ka1.cosδ
-1.0
-0.5
0.0
0.5
1.0
γc γsat1 γt2 γt1 R.W.L. Ka2.cosδ Ka1.cosδ
/
- 20 -
Sk=946kN, Rk=1138kN
Sd=1000kN Rd=1000kN S
R γS 1.06 γR 0.88
γR γS 1.2
1.2
1.2
b)
a)
3
2
1.2
1.0
1.2
a)MCS
300
500
700
900
1100
1300
1500
1700
1900
2100
300 800 1300 1800
Rk Sk
Rd Sd
SR
WUP V
P H
P wH
γ C
γ sat1
γ t1
γ t2
R.W.L.K a cos
300
500
700
900
1100
1300
1500
1700
1900
2100
300 800 1300 1800
Rk Sk
Rd Sd
α
α
αα
No 880
- 21 -
Z = R -
S 0
Z = R – S + α
Z α
α
b)
α
R S
4
4 Z 0 -100
-200 -300 α
MCS
500
700
900
1100
1300
1500
1700
1900
500 700 900 1100 1300 1500 1700 1900
Rk Sk
Rd Sd
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.0E-03 1.0E-02 1.0E-01 1.0E+00
γ
Pf
/
- 22 -
MCS
FORM First-Order Reliability Method
FORM MCS
β α
MCS
FORM MCS
MCS
MCS
FORM
FORM MCS
FORM MCS
γγφ
γφ
γ
μ
No 880
- 23 -
2 1 3
a)
MCS
3
1.2
1.2
1.2
1.0
4
kh
0.05
0.05
1.0 1.1
17)
b)
1.01 0.03
( ,
1.02 0.04
L.W.L 1.00 0.05
1.00 0.12
1.06 0.15
a)
27
L.W.L.
H
/
- 24 -
LWL/H
1.0×10-1 10%
4.0×10-4 0.04%
1.2 2.0
2
1.0×10-2 1%
1.0×10-3 0.1%
1.2 1.3
1
b)
0
1.2
10%
10%
L.W.L.
H.W.L.
L.W.L 1/3
No 880
- 25 -
-3.2
30kN/m2
30kN/m2
3
w1
w2 w1
1 w1 w2
2 w2 3
21)
( 1
γγφ
γφ
γ
μ
γγφ
γφ
γ
μ
1.0E-06
1.0E-05
1.0E-04
1.0E-03
1.0E-02
1.0E-01
1.0E+00
0 10 20 30
Pf
1.0E-06
1.0E-05
1.0E-04
1.0E-03
1.0E-02
1.0E-01
1.0E+00
0 10 20 30
Pf
/
- 26 -
6
6
6
1.2
0.05
0.05
0.55
1
1
1.2
10%
10%
1.0E-04
1.0E-03
1.0E-02
1.0E-01
1.0E+00
Pf
No 880
- 27 -
kh=0.05
2
1.2
1.2
0.3 0.4 19)
0.35 0.4
1.2 1.2
φ 40
/
- 28 -
9.3 10-2 3.6 10-3
27 27
.
3
3
3
A
A
B
B
A B
LWL/H 0.75
LWL/H 0.75
MCS
R γR
S γS
μ γμ
W γW
U γU
Pv γPV
PH γPH
PwH γPwH
γc γγμ γμ
γsat1 γγsat
γt2 γt2
γt1 γt1
R.W.L. γRWL
Ka.cosδ γKa
No 880
- 29 -
B B
/
- 30 -
No 880
- 31 -
A 27
A
LWL/H =0.75
LWL/H 0.75
0.75 LWL/H 0.75
0.75
A
2
A
1.0
Pf
/
- 32 -
B
H B/H
B/H B/H
26.0m 20m, 4.5m,
+6.0m 15cm
B/H 0.005
10cm
1.0E-02
1.0E-01
1.0E+00
0.0 0.2 0.4 0.6 0.8 1.0
Pf
LWL/H
1.0E-04
1.0E-03
1.0E-02
1.0E-01
0.0 0.2 0.4 0.6 0.8 1.0
Pf
LWL/H
No 880
- 33 -
10cm
3
27
a)3
1.0
27
3 A
B
R γR 0.87 0.99
S γS 1.06 1.23
μ γμ 0.86 -
W γW 1.00 0.96
U γU 1.00 1.00
Pv γPV 1.06 1.27
PH γPH 1.06 1.26
PwH γPwH 1.00 1.02
γc γγ 1.00 0.95
μ γμ 0.86 -
γsat1 γγsat 1.03 1.03
γt2 γt2 1.02 1.04
γt1 γt1 1.02 1.03
R.W.L. γRWL 1.00 1.02
Ka.cosδ γKa 1.04 1.22
/
- 34 -
A B
15cm
26.0m 20m, 4.5m, +6.0m
b)
3
1.0
3
No 880
- 35 -
c)
a) b)
3
10cm
A B
27
3 A B
1.0E-02
1.0E-01
1.0E+00
0.4 0.5 0.6 0.7 0.8 0.9 1.0
Pf
LWL/H
1.0E-02
1.0E-01
1.0E+00
0.4 0.5 0.6 0.7 0.8 0.9 1.0
Pf
LWL/H
1.0E-02
1.0E-01
1.0E+00
0.4 0.5 0.6 0.7 0.8 0.9 1.0
Pf
LWL/H
1.0E-04
1.0E-03
1.0E-02
1.0E-01
0.4 0.5 0.6 0.7 0.8 0.9 1.0
Pf
LWL/H
1.0E-04
1.0E-03
1.0E-02
1.0E-01
0.4 0.5 0.6 0.7 0.8 0.9 1.0
Pf
LWL/H
1.0E-04
1.0E-03
1.0E-02
1.0E-01
0.4 0.5 0.6 0.7 0.8 0.9 1.0
Pf
LWL/H
/
- 36 -
1)
2)
1.5
kh 0.10 0.15
3) 2)
1.21.2
4) MCS
MCS
MCS FORM
5
2
A B
-4.5m -20.0m
A
B
R γR 0.87 0.99
S γS 1.06 1.23
μ γμ 0.86 -
W γW 1.00 0.96
U γU 1.00 1.00
Pv γPV 1.06 1.27
PH γPH 1.06 1.26
PwH γPwH 1.00 1.02
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 5 10 15 20 25
/B/H
H11
H19
A
4.5m 7.5m 10.0m 16.0m 20.0m
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 5 10 15 20 25
/B/H
H11
H19
B
4.5m 7.5m 10.0m 16.0m 20.0m
No 880
- 37 -
1)
2007.
2)
2014.
3) (OCDI) TECHNICAL
STANDARDS AND COMMENTARIES FOR PORT
AND HARBOUR FACILITIES IN JAPAN 2009.
4)
340 1983.
5)
1983.
6) -
2014.
7)Honjo,Y. T.C.Kieu Le T.Hara M.Shirato M.Suzuki and
Y.Kikuchi Code calibration in reliability based design
lebel verification format for geotechnical structures
Geotechnical Safety and Risk(Proc. of Is-Gifu) CRC
press,pp.433-452 2009.
8) -
2014.
9)
1984.
10)
2015
11)
2014.
12)
57 2002.
13)No.448 -19
1992.
14) 1950.
15) 1959.
16)
1979.
17)
1999.
18)
Vol.51A 2005.
19) 1932.
20 R Development Core Team.R A language and
environment for statistical computing. R Foundation
for Statistical Computing, Vienna, Austria. 2007.
21)
No.5
2002.
/
- 38 -
251
10
25 1
34 42
19
4
(1)
251 1.2
34 2
A-1
1.0
F
μ
W
P
(2)
25
A-1
No.880
- 39 -
1 1.2
423 A-2
1.1
F
W
t
P
h
(3)
251
A-4
342
A-3 6
Pai i
Phi i
Pvi i
Kai i
i i
hi i
i i
’ 0
: ’ :
A-3
A-4
A-5
A-2
A-6
/
- 40 -
544
A-7
t
hi i
hj
j
w
h
75
(4)
251 34
2
423
A-8
pdw
k
w
H
y
54 4 6
11 7
A-9
A-10
Pdw
hdw
(A-8)
(5)
25 1
1/2 1/4 342 1/3 2/3 42
3
1/3
1/3
46 8
(A-9)
A-7
A-8
A-9
A-10
No.880
- 41 -
hw R.W.L. L.W.L.
H H.W.L. L.W.L.
(6)
251 0.5
0.6 16
7)
7
1) 1950
2) 1959
3) 1967
4) 1979
5) 1932
6)
1989
7)
1999
8)
No.115 1971
9)
2007
A-9
/
- 42 -
1)
50
4.34m
0.27m
1.3m 2.0m
2)
1992
4.5m
0.4m 1.7m
H.W.L.
L.W.L. 0.3m
1.8m 4.5m
1)
2)
No.880
- 43 -
2)
20m 4 5
3
10
1/3
H.W.L.-L.W.L.
1/3
2)
3) 63
180
-10.0m 26.6% -7.5m
21.2%
-4.5m
-4.5m
-16.0m
-20.0m
4.5m
16.0m 10.0m
7.5m 4 20.0m
5
3)
3) L.W.L.
3.0 4.5m 8 3.5
4.0m 1.5m 6.0m
H.W.L.
L.W.L.
0.3m 1.8m 4.5m
L.W.L.+2.0m L.W.L.+4.0m L.W.L.+6.0m
/
- 44 -
3)
3
3) 1.0m
1.0m
3)
0.5m 6
1 2
0.5m
3)
3)
3)
No.880
- 45 -
3)
sat = 20.0 kN/m3
t = 18.0 kN/m3
4)
Vs
Vc
Vs/Vc
3.03 2.78
3.0
Vs
Vc 3.0
sat = 20.0 kN/m3
c = 24.0 kN/m3
c = 1 24.0+3 20.0 / 4 = 21.0 kN/m3
4)
5)
100kgf/cm2(9.8MN/m2)
1.0kgf/cm2(98kN/m2) 4kgf/cm2(382kN/m2)
40
35 40
/
- 46 -
40
40
=40
N φ
N
N φ
070
1002.325
v
N�
���
��
rD15.028���
070
10021
vr
ND���
�
N1225���
φ
N N
Dr
σ’v0
N 4
φ 30
40
φ=30
15 20
1/2
6)
20cm 30cm 50cm 80cm
20cm
45%
42 51%
30cm tanδ
0.30 =16.7 6)
=15
No.880
- 47 -
6)
10 30kN/m2
30kN/m2
8),9)
50t
2
1 12 15t
3 5
0.6
0.6
μ 0.6
1)
31 3 1992
2)
No.115 1971
3)
No.702 1991
4)
No.716 1991
5)
No.924 1999
6)
No.916 1998
7)
No.268 1977
8)
25 1978
9)
( 2 ) 26 1979
8)9)
/
- 48 -
Z
5 MCS
MCS
MCS
MCS
MCS
20
N=20
MCS 10
50
500
1000
1500
2000
2500
3000
500 1000 1500 2000 2500 3000
1200
1250
1300
1350
1400
1450
1500
1200 1250 1300 1350 1400 1450 1500
No.880
- 49 -
(1)
(2)
0.5 1 3 0.5
1 2 5 0.65
1 3
5
3
0.5 0.5 0.5 0.5 0.5σ=0.03σ
1 2 0.5 0.5 0.5 1.0
1.0σ=0.13σ 3 0.65 0.65 0.65 0.65 0.65σ=0.12σ
0.0
0.1
0.2
0.3
0.4
0.5
-4 -2 0 2 4
0.0
0.1
0.2
0.3
0.4
0.5
-4 -2 0 2 4
12
34
5
6
7
89
1011
12
13
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19
0
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1300 1350 1400 1450 1500
/
- 50 -
-1
-2
-3
-1
-2
-3
1
2
345678910111213141516171819
0
20
1300
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1500
1300 1350 1400 1450 1500
1
2
3456789
0
10
1300
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1300 1350 1400 1450 1500
1
2 34
0
5
1300
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1300 1350 1400 1450 1500
1
23
45
6789
1011
1213
141516171819
0 20
1300
1320
1340
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1380
1400
1420
1440
1460
1480
1500
1300 1350 1400 1450 1500
1
2
3
456
789
0
10
1300
1320
1340
1360
1380
1400
1420
1440
1460
1480
1500
1300 1350 1400 1450 1500
1
2 3
40
5
1300
1320
1340
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1380
1400
1420
1440
1460
1480
1500
1300 1350 1400 1450 1500
No.880
- 51 -
-1
-2
-3
A R
0.004 =0.819 0.814 S
0.005 =1.082 1.077
0.005
1 4 0
0.01
0.01
1
0.005
MCS
-3
1
2
3
4
5
678
9
10111213
1415
16171819
0 20
1300
1320
1340
1360
1380
1400
1420
1440
1460
1480
1500
1300 1350 1400 1450 1500
12
3
4
56789
010
1300
1320
1340
1360
1380
1400
1420
1440
1460
1480
1500
1300 1350 1400 1450 1500
1
23
4
0
5
1300
1320
1340
1360
1380
1400
1420
1440
1460
1480
1500
1300 1350 1400 1450 1500
R S R S γ R γ S
γ R γ S γ S /γ R B/H B/HB/H
/
- 52 -
MCS
ΔXj
ΔZ
ΔXj
ΔXj
ΔXj
αXj = ΔZ ΔX j
αXj Xj
ΔZ Z
ΔXj Xj
ΔXj
ΔXj σ
ΔXj
1000
ΔXj
ΔXj
ΔXj
ΔXj 1.0σ
ΔXj
ΔXj
Z
XX *j
ΔXj
ΔZ
X *j ΔXj / 2 X *j ΔXj / 2
Z (X *j ΔXj / 2)
Z (X *j+ΔXj / 2)
Z (X *j)
αXj
γγφ
γφ
γ
γ sat1
γ t1
γ t2
γ c
μR.W.L.
K a cosδ
-1.00
-0.50
0.00
0.50
1.00γc
μ
γsat1
γt2
γt1
R.W.L.
Ka.cosδ
-1.00
-0.50
0.00
0.50
1.00
γc
γsat1
γt2
γt1
R.W.L.
Ka.cosδ
No.880
- 53 -
MCS FORM
First-Order Reliability Method
FORM
Z X1 X2 … Xn
X*1 X*
2 … X*n
� � � � *
1
***
2
*
1 ,...,, Xj
n
jjjn X
gXXXXXgZ��
�� ��
Z μz σz
� � *
1
*
Xj
n
jjXZ X
gXj �
� ��
�
!!
jX
n
jX
jjZ X
g ��� ��
���
����
�
��
�1
*
αj
β
ZZ �!� �
3
� � 0,...,, **
2
*
1 �nXXXg
njXjj X
Z
ZjX ,...,2,1:* � � ��!
�!
nj
Xg
Xg
n
jXX
j
XXj
j
j
j
,...,2,1:2
1
1
2
2
*
*
�
��
��
���
����
�
��
���
����
�
��
�
��
�
�
�
T
Xj j
njX kj
X
X
XTjj
j
j
j,...,2,1:1
,
���
�
�
��
�
� �
!
!
����
FORM
" # " # " # " #� �" # " #)cos,,,,P(
cos,,,,
211H
211
RWLPKRWLKRWLPRWLUW
SRZ
WHattsat
attsatvC
�$
$� %� �
&���
&����!!
" # " # " #" # " #)cos,,,,P(
cos,,,,
22111H
211321
RWLyPKRWLyKRWLxPRWLUxWx
SRZ
WHattsat
attsatvC
�$
$� � �
&���
&����
FORM MCS
β α
α FORM MCS
FORM
MCS
γγφ
γφ
γ
μ
/
- 54 -
μ γ c γ sat1 γ t1 γ t2 RWL K a cosδ
α j
βX *
Z/ Xα j
βX *
Z/ Xα j
βX *
Z/ Xα j
βX *
Z/ Xα j
βX *
Z/ Xα j
βα jβ
MCS
μ γ c γ sat1 γ t1 γ t2 RWL K a cosδ
α j
β
MCS
FORM
No.880
- 55 -
γ c γ sat1 γ t2 γ t1 RWL K a cosδ
α j
βX *
Z/ Xα j
βX *
Z/ Xα j
βX *
Z/ Xα j
βX *
Z/ Xα j
βX *
Z/ Xα j
βα jβ
MCS
γ c γ sat1 γ t2 γ t1 RWL K a cosδ
MCS
FORM
/
- 56 -
( 1 )
Z = Rd Sd = μ × ( W U + Pv ) ( PH + PWH )
μ
W
U
Pv
PH
PWH
W = H ×B × γC
H = 8.5m
B = 2.7285m
γC
W = 8.5 × 2.7285 × γC = 23.192 × γC
U = B × h1× γw
h1 = 4.5m+RWL
RWL
γw = 10.1 kN/m3
U = 2.7285 × ( 4.5 + RWL ) × 10.1 = 124.010 + 27.558 × RWL
PH = 1 / 2 ×Ka1 cosδ× [w + γt2 × h3 + γt1 × h2 + ( γsat1 γw ) × h1 ] × h1
+ 1 / 2 ×Ka1 cosδ× [w + γt2 × h3 + γt1 × h2 ] × h1
+ 1 / 2 ×Ka1 cosδ× [w + γt2 × h3 + γt1 × h2 ] × h2
+ 1 / 2 ×Ka1 cosδ× [w + γt2 × h3 ] × h2
+ 1 / 2 ×Ka2 cosδ× [w + γt2 × h3 ] × h3
+ 1 / 2 ×Ka2 cosδ× [w ] × h3
Ka1 = 0.2011
Ka2 = 0.3014
δ = 15
w = 30.0 kN/m3
γt2
γt1
γsat1
h2 = 1.8m RWL
h3 = 2.2m
μ
γ
γ
γ
γ
γ
δ
No.880
- 57 -
PH = 1 / 2 × 0.1942 × Ka × [30 + γt2 × 2.2 + γt1 × ( 1.8 RWL ) + ( γsat1 10.1 ) × ( 4.5 + RWL ) ] × ( 4.5 + RWL )
+ 1 / 2 × 0.1942 × Ka × [30 + γt2 × 2.2 + γt1 × ( 1.8 RWL ) ] × ( 4.5 + RWL )
+ 1 / 2 × 0.1942 × Ka × [30 + γt2 × 2.2 + γt1 × ( 1.8 RWL ) ] × ( 1.8 RWL )
+ 1 / 2 × 0.1942 × Ka × [30 + γt2 × 2.2 ] × ( 1.8 RWL )
+ 1 / 2 × 0.2911 × Ka × [30 + γt2 × 2.2 ] × 2.2
+ 1 / 2 × 0.2911 × Ka × [30 ] × 2.2
= 1 / 2 × 0.1942 × Ka × ( γsat1 γt1 10.1 )RWL2
+ 1 / 2 × 0.1942 × Ka × ( 9.0γsat1 9.0γt1 90.9 )RWL
+ 1 / 2 × 0.1942 × Ka × ( 20.25γsat1 + 19.44γt1 + 27.72γt2 + 173.475 )
+ 1 / 2 × 0.2911 × Ka × ( 4.84γt2 + 132 )
Ka
PV = tanδ× PH = 0.268 × PH
= 0.268 × 1 / 2 × 0.1942 × Ka × ( γsat1 γt1 10.1 )RWL2
+ 0.268 × 1 / 2 × 0.1942 × Ka × ( 9.0γsat1 9.0γt1 90.9 )RWL
+ 0.268 × 1 / 2 × 0.1942 × Ka × ( 20.25γsat1 + 19.44γt1 + 27.72γt2 + 173.475 )
+ 0.268 × 1 / 2 × 0.2911 × Ka × ( 4.84γt2 + 132 )
PWH = 1 / 2 ×γw × RWL2 +γw × RWL ×LWL
LWL = 4.5m
PWH = 1 / 2 × 10.1 × RWL2 + 10.1 × RWL × 4.5
= 5.05 × RWL2 + 45.45 × RWL
( 2 )
Z μ γC RWL Ka γsat1 γt1 γt2
∂Z / ∂μ = W U + PV = W U + 0.268 × PH
=23.192 × γC 124.010 + 27.558 × RWL
+0.268 × 1 / 2 × 0.1942 × Ka × ( γsat1 γt1 10.1 )RWL2
+ 0.268 × 1 / 2 × 0.1942 × Ka × ( 9.0γsat1 9.0γt1 90.9 )RWL
+ 0.268 × 1 / 2 × 0.1942 × Ka × ( 20.25γsat1 + 19.44γt1 + 27.72γt2 + 173.475 )
+ 0.268 × 1 / 2 × 0.2911 × Ka × ( 4.84γt2 + 132 )
∂Z / ∂γC = 23.192 × μ
∂Z / ∂γsat1 = (0.268μ 1 ) × 0.1942 / 2 × Ka × ( RWL2 + 9 × RWL + 20.25 )
∂Z / ∂γt1 = (0.268μ 1 ) × 0.1942 / 2 × Ka × ( RWL2 9 × RWL + 19.44 )
γ
γ
γ
γ
δ
δ
/
- 58 -
∂Z / ∂γt2 = 3.396 × Ka × ( 0.268μ 1 )
∂Z / ∂RWL = 27.558μ + (0.268μ 1 )
× {0.1942 × Ka × ( γsat1 γt1 10.1 )RWL + 1 / 2 × 0.1942 × Ka × ( 9γsat1 9γt1 90.9 )} 10.1RWL 45.45
∂Z / ∂Ka = ( 0.268μ 1 ) × {1 / 2 × 0.1942 × ( γsat1 γt1 10.1 )RWL2
+ 1 / 2 × 0.1942 × ( 9γsat1 9γt1 90.9 )RWL
+ 1 / 2 × 0.1942 × ( 20.25γsat1 + 19.44γt1 + 27.72γt2 + 173.475 )
+ 1 / 2 × 0.2911 × ( 4.84γt2 + 132 )}
(1 )
Z = Rd Sd
= ( Wx Ux + PVx ) ( PHy + PWHy )
Wx
Ux
PVx
PHy
PWHy
WX = H × B × γC × B / 2
H = 8.5m
B = 3.2555m
γC
WX = 8.5 ×3.2555 × γC ×3.2555 / 2 = 45.043 × γC
UX = B × h1× γw × B / 2
h1 = 4.5m+RWL
RWL
γw = 10.1 kN/m3
UX = 3.2555 × 4.5 + RWL ×10.1 ×3.2555 / 2
= 53.521 × RWL + 240.846
PHY = 1 / 2 ×Ka1 cosδ× [w + γt2 ×h3 + γt1 ×h2+ ( γsat1 γw ) ×h1] × h1 × 1 / 3 × h1
+ 1 / 2 ×Ka1 cosδ× [w + γt2 ×h3 + γt1 ×h2 ] × h1 × 2 / 3 × h1
+ 1 / 2 ×Ka1 cosδ× [w + γt2 ×h3 + γt1 ×h2 ] × h2 × [h1 + 1 / 3 × h2]
+ 1 / 2 ×Ka1 cosδ× [w + γt2 ×h3 ] × h2 × [h1 + 2 / 3 × h2]
γ
γ
No.880
- 59 -
+ 1 / 2 ×Ka2 cosδ× [w + γt2 ×h3 ] × h3 × [h1 + h2 + 1 / 3 × h3]
+ 1 / 2 ×Ka2 cosδ× [w ] × h3 × [h1 + h2 + 2 / 3 × h3]
Ka1 = 0.2011
Ka2 = 0.3014
δ = 15
w = 30.0 kN/m3
γt2
γt1
γsat1
h2 = 1.8m RWL
h3 = 2.2m
PHY = 1 / 2 ×0.1942 × Ka × [30 + γt2 × 2.2 + γt1 × ( 1.8 RWL ) + ( γsat1 10.1 ) × ( 4.5 + RWL )]
× ( 4.5 + RWL ) × 1 / 3 × (4.5 + RWL )
+ 1 / 2 × 0.1942 × Ka × [30 + γt2 × 2.2 + γt1 × ( 1.8 RWL ) ]
× ( 4.5 + RWL ) × 2 / 3 × (4.5 + RWL )
+ 1 / 2 × 0.1942 × Ka × [30 + γt2 × 2.2 + γt1 × ( 1.8 RWL ) ]
× ( 1.8 RWL ) × [(4.5 + RWL ) + 1 / 3 ×(1.8 RWL )]
+ 1 / 2 × 0.1942 × Ka × [30 + γt2 × 2.2 ]
× ( 1.8 RWL ) × [(4.5 + RWL ) + 2 / 3 ×(1.8 RWL )]
+ 1 / 2 × 0.2679 × Ka × [30 + γt2 × 2.2 ]
× 2.2 × [(4.5 + RWL ) + (1.8 RWL ) + 1 / 3 × 2.2]
+ 1 / 2 × 0.2679 × Ka × [30 ]
× 2.2 × [(4.5 + RWL ) + (1.8 RWL ) + 2 / 3 × 2.2]
= 1 / 6 × 0.1942 × Ka ×[ ( γsat1 γt1 10.1 )RWL3 + 13.5(γsat1 γt1 10.1 )RWL2 + 60.75 ( γsat1 γt1 10.1 )RWL
+ (91.125γsat1 + 261.954γt2 + 158.922γt1 + 2651.737 )] + 1 / 6 × 0.2911 × Ka ×[102.124γt2 + 2930.4]
Ka
Pvx = { 1 / 2 ×Ka1 cosδ× [w + γt2 × h3 + γt1 × h2 + ( γsat1 γw ) × h1 ] × h1
+ 1 / 2 ×Ka1 cosδ× [w + γt2 × h3 + γt1 × h2 ] × h1
+ 1 / 2 ×Ka1 cosδ× [w + γt2 × h3 + γt1 × h2 ] × h2
+ 1 / 2 ×Ka1 cosδ× [w + γt2 × h3 ] × h2
+ 1 / 2 ×Ka2 cosδ× [w + γt2 × h3 ] × h3
+ 1 / 2 ×Ka2 cosδ× [w ] × h3 }
×tanδ × B
= 0.1942 / 2 × 3.2555 × 0.2679 × Ka ×
[(γsat1 γt1 10.1 )RWL2 + 9 ( γsat1 γt1 10.1 )RWL
+ (20.25γsat1 + 27.72γt2 + 19.44γt1 + 173.475 )]
+ 0.2911 / 2 × 3.2555 × 0.2679 × Ka × ( 4.84γt2 + 132 )
γ
γ
γ
δ
y2 y3 y4
γ
γ
γ
δ
/
- 60 -
PWHY = 1 / 2 ×γw × RWL2 × 1 / 3 × RWL + LWL
+ γw × RWL × LWL × 1 / 2 × LWL
LWL = 4.5m
PWHY = 1 / 6 × 10.1 × RWL3 + 1 / 2 × 45.45 × RWL2 + 1 / 2 × 204.525 × RWL
( 2 )
Z γC RWL Ka γsat1 γt1 γt2
∂Z / ∂γC = 45.043
∂Z / ∂γsat1 = ∂Pvx / ∂γsat1 ∂PHY / ∂γsat1
= 1 / 2 × 0.1942 × Ka × 3.2555 × 0.2679 × ( RWL2 + 9RWL + 20.25 )
1 / 6 × 0.1942 × Ka × ( RWL3 + 13.5 RWL2 + 60.75RWL + 91.125 )
∂Z / ∂γt1 = ∂Pvx / ∂γt1 ∂PHY / ∂γt1
= 1 / 2 × 0.1942 × Ka × 3.2555 × 0.2679 × ( RWL2 + 9RWL + 19.44 )
1 / 6 × 0.1942 × Ka × ( RWL3 13.5 RWL2 60.75RWL + 158.922 )
∂Z / ∂γt2 = ∂Pvx / ∂γt2 ∂PHY / ∂γt2
= 1 / 2 × 0.1942 × Ka × 3.2555 × 0.2679 × 27.72 + Ka / 2 × 0.2911 × 3.2555 × 0.2679 × 4.84
1 / 6 × 0.1942 × Ka × 261.954 Ka / 6 × 0.2911 × 102.124
= 10.471Ka
∂Z / ∂RWL = ∂UX / ∂RWL + ∂PVX / ∂RWL ∂PHY / ∂RWL ∂PWHY / ∂RWL
= 53.521 + 1 / 2 × 0.1942 × Ka × 3.2555 × 0.2679 × [2 ( γsat1 γt1 10.1 )RWL + 9(γsat1 γt1 10.1 )]
1 / 6 × 0.1942 × Ka × [3 ( γsat1 γt1 10.1 )RWL2 + 27 ( γsat1 γt1 10.1 )RWL + 60.75 ( γsat1 γt1 10.1 ) ]
5.05RWL2 45.45RWL 102.263
= 5.05RWL2 45.45RWL 155.784 + 1 / 2 × 0.1942 × Ka × 3.2555
× 0.2679 × ( 2RWL + 9 )(γsat1 γt1 10.1 ) 1 / 6 × 0.1942 × Ka × ( 3RWL2 + 27RWL + 60.75 )( γsat1 γt1 10.1 )
∂Z / ∂Ka = ∂PVX / ∂Ka ∂PHY / ∂Ka
+ 1 / 2 × 0.1942 × 3.2555 × 0.2679 × [(γsat1 γt1 10.1 )RWL2 + 9 ( γsat1 γt1 10.1 )RWL
+ (20.25γsat1 + 27.72γt2 + 19.44γt1 + 173.475 )] + 1 / 2 × 0.2911 × 3.2555 × 0.2679 × ( 4.84γt2 + 132 )
1 / 6 × 0.1942 × [(γsat1 γt1 10.1 )RWL3 + 13.5(γsat1 γt1 10.1 )RWL2
+ 60.75 ( γsat1 γt1 10.1 )RWL + (91.125γsat1 + 261.954γt2 + 158.922γt1 + 2651.737 )]
1 / 6 × 0.2911 × ( 102.124γt + 2930.4 )
γ
y'2
No.880
- 61 -
1)
=2.41(t/m3) =0.0448(t/m3) 1)
2 10
=1.019 =0.0376
1)
RC = 24.00 kN/m3
/Xk = 2.41 9.8 / 24.0 = 0.98
= 0.0448 9.8 = 0.44
V = 0.44 / (2.41 9.8) = 0.02
*RC = 24.0 0.98 = 23.52 kN/m3
S = 20.00 kN/m3
/Xk = 1.019 = 1.02
= 0.0376 = 0.04
V =0.04 / 1.02 = 0.04
*S = 20.00 1.02 = 20.4 kN/m3
=0.04 20.4 = 0.816
1)
3 1
C = 1 24.0+3 20.0 / 4 = 21.0
/Xk = (1/4 23.52+3/4 20.40) / 21.0
= 1.01
= {(1/4 0.44)2+(3/4 0.816)2 }1/2
= 0.62
V = 0.62 / (21.0 1.01) = 0.03
/Xk = 1.01
V = 0.03
2)
20cm 30cm 50cm 80cm
20cm
40 45%
/
- 62 -
(Ka cos )
Ka cosδ
Ka cosδ
/Xk = 1.00
V = 0.12
Ka
Ka
cosδ
φ
MCS
2)
3)
19784) 19795)
4) 5)
/Xk = 1.06
V = 0.15
No.880
- 63 -
1/3 2/3
L.W.L.
1/3 L.W.L. 2/3
6)
6)
6)
L.W.L
/Xk = 1.00
V = 0.05
5 25
6)
6)
6)
6)
6)
6)
/
- 64 -
6)
6)
6)
6)
6)
6)
6)
6)
No.880
- 65 -
6)
6)
6)
6)
6)
6)
6)
6)
/
- 66 -
1)
/Xk = 1.02
V = 0.04
1)
No.716
1991
2)
No.811 1995
3)
31 5
1993
4)
25
1978
5)
( 2 ) 26
1979
6)
No.115 1971
No.880
- 67 -
A
B
5 27
MCS
αXj = ΔZ / ΔXj
αXj Xj
ΔXj Xj
ΔZ Z ΔXj
j X
4.3
27
27
γc Ka cosδ
γc
Ka cosδ
B
27
27
W
PH
MW
MH
A
27
27
μ
R
MH
S
/
- 68 -
B
B
27
27
B 27
B 27
-1.0
-0.5
0.0
0.5
1.0
0 5 10 15 20 25 30
μ
γc
γsat1
γt2
γt1
RWL
Ka.cosδ
-1.0
-0.5
0.0
0.5
1.0
0 5 10 15 20 25 30
γc
γsat1
γt2
γt1
RWL
Ka.cosδ
-1.0
-0.5
0.0
0.5
1.0
0 10 20 30
μ
W
U
PH
Pv
PwH
-1.0
-0.5
0.0
0.5
1.0
0 10 20 30
MvMuMHMvMwH
-1.0
-0.5
0.0
0.5
1.0
γc μ γsat1 γt2 γt1 RWL Ka.cosδ
-1.0
-0.5
0.0
0.5
1.0
γc γsat1 γt2 γt1 RWL Ka.cosδ
-1.0
-0.5
0.0
0.5
1.0
μ W U PH Pv PwH
-1.0
-0.5
0.0
0.5
1.0
Mv Mu MH Mv MwH
No.880
- 69 -
A
A
A 27
A 27
-1.0
-0.5
0.0
0.5
1.0
0 10 20 30
R
S
-1.0
-0.5
0.0
0.5
1.0
0 10 20 30
R
S
-1.0
-0.5
0.0
0.5
1.0
R S
-1.0
-0.5
0.0
0.5
1.0
R S
/
- 70 -
1.0
β
A
B
1.0
β
No.880
- 71 -
=1.0
/
- 72 -
=1.0
国土技術政策総合研究所資料
TECHNICAL NOTE of N I L I M
No. 880 December 2015
編集・発行 国土技術政策総合研究所
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