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Slope-Deflection Method 1
4 Slope-Deflection Method
( ) Force Method
Slope-Deflection Displacement Method
Slope-Deflection Method 2
Slope-Deflection (Beam) (Frame) Prismatic
Prismatic
Non-Prismatic
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Slope-Deflection Method 3
4.1 Slope-DeflectionSlope-Deflection Equations
End Distortion + Load
Mab = f (a , b , , Load)Mba = g (a , b , , Load)
(+)- (Mab , Mbc) - ( a , b ) - ()
Mab
Mba
ab
a bLoad
Slope-Deflection Method 4
Mab Mba 4 a
b
Load Fixed-End Moments
LEI2
ML
EI4M aba
aab
=
= .......
LEI4
ML
EI2M bba
bab
=
= .......
2ba2ab LEI6
MLEI6
M
=
= '''''' .......
Maba
a b Mba
b = 0 , = 0
Mab ba b Mbaa = 0 , = 0
Mab a b
Mbaa = 0 , b = 0
MabF a b MbaFLoad
a = 0 , b = 0 = 0
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Slope-Deflection Method 5
Slope-Deflection
Fba2
baFbababababa
Fab2
baFababababab
MLEI6
LEI4
LEI2
MMMMM
MLEI6
LEI2
LEI4
MMMMM
+
+
=+++=
+
+
=+++=
'''
'''
Fbababa
Fabbaab
ML
32LEI2
M
ML
32LEI2
M
+
+=
+
+=
)(
)(
Slope-Deflection Method 6
Moment 1 ( a )
- a Hinge - b Fixed EndConjugate Beam Rotation a (a) Condition 1) Moment Conjugate Beam a = 0
a = 02) Moment Conjugate Beam b = 0
b = 0
baMab
a Mba
a
MabEI Mba
EI
Elastic Load on Conjugate Beam
-
Slope-Deflection Method 7
(1) (2)
... ...
=
=
=
=
20L3L
LEI
M21
3L2
LEI
M21
0M
03L2
LEI
M21
3L
LEI
M21
0M
abaab
b
baab
a
..........)()(
;
)()(
;
= 1M2M baab .............
.......L
EI2M
LEI4
M abaa
ab
=
=
Slope-Deflection Method 8
Fixed-End MomentsP
L
L/28
PL8
PL WL12
WL2
12WL2
W
L
a
)( 2222
a3aL8L6L12
Wa+ )( a3L4
L12Wa
2
3
P
L
a2
2
LPab b
2
2
LbPa
ML
a)( a2b
LMb
2 b
)( ab2LMa
2
WL20
WL2
30WL2
W
L96WL5 2
96WL5 2
L/2
1
3
5
2
4
6
7
-
Slope-Deflection Method 9
EX. Slope-Deflection ABCD
EI EI/2A800 kg.
4 m.
B C D
300 kg./m.
2 m. 2 m. 1 m.
Slope-Deflection Method 10
1) Fixed End Moments
.. mkg40012
430012WL
M22
FAB =
=
=
.. mkg8008
480012
43008
PL12WL
M22
FBC =
=
=
EI EI/2A
800 kg.
4 m.
B C DW=300 kg./m.
2 m. 2 m. 1 m.
MABF MBAF MBCF MCBF
300 kg./m.A4 m.
BMABF MBAF
.. mkg40012WL
M2
FBA ==
300 kg./m.B CMBCF MCBF
800 kg.
2 m. 2 m... mkg800
8PL
12WL
M2
FCB =+=
-
Slope-Deflection Method 11
2) Fixed End Moments Slope-Deflection () = 0
2 B C
4002EI
ML
32LEI2
M BF
ABBAAB =+
+= )(
400EIML
32LEI2
M BF
BABABA +=+
+= )(
80024EI
ML
32LEI2
M CBF
BCCBBC +=+
+= )()(
80024EI
ML
32LEI2
M CBF
CBCBCB ++=+
+= )()(
Slope-Deflection Method 12
3) (2 )
MBA+ MBC= 0 MCB= 150 kg.-m.
4) 2 3
BMBA MBC
FBD. B
A C CMCB 150 kg.-m.
FBD. C
B D
=+++=+ 1080024EI
400EI0MM CBBBCBA ..........])([][...
=++= 215080024EI
150M CBCB .................................])([...
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Slope-Deflection Method 13
(2)
(1)
(3) (4)
=+ 3EI6002
2 CB ...................................................,
=+ 4EI6001
6 CB .......................................................,
., RadEI11
8005B = .
,Rad
EI1120017
C
=
Slope-Deflection Method 14
5) 2
..,, mkg115001
400EI11
80052EI
4002EI
M BAB
===
..,, mkg1120010
400EI11
8005EI400EIM BBA =+=+=
800EI1120017
EI118005
24EI
80024EI
M CBBC =+= ),,()(
.., mkg11
20010
=
800EI1120017
2EI11
80054EI
80024EI
M CBCB +=++= ),,()(
.. mkg150 =
-
Slope-Deflection Method 15
6) FBD.
MA = 1,500/11 kg.-m. RA = 4,425/11 kg.RB = 43,825/22 kg. RC = 24,325/22 kg.
300 kg./m.1,500/11 10,200/11
4,425/11 8,775/11
B8,775/11 26,275/22
300 kg./m.10,200/11 150
26,275/22 17,725/22
800 kg150
300A
1,500/11
RA RB
C17,725/22 300
RC
Slope-Deflection Method 16
EX. Slope-Deflection 1. 2. BMD. ABCD ( EI )
A
100 kg. B C D300 kg./m.
2 m. 2 m.
2 m.
50 kg.-m.
-
Slope-Deflection Method 17
1) Fixed End Moments
,0MM FBAFAB == 0MM
FCB
FBC ==
.. mkg10012
230012WL
M22
FCD =
=
=
.. mkg10012
230012WL
M22
FDC =
==
300 kg./m.C2 m.
DMCDF MDCF
A
100 kg. B C DW
2 m. 2 m.
2 m.
50 kg.-m. MCDF MDCF
W = 300 kg./m.
Slope-Deflection Method 18
2) Fixed End Moments Slope-Deflection (= 0)
2 B C
BF
ABBAAB EIML32
LEI2
M =+
+= )(
BF
BABABA EI2ML32
LEI2
M =+
+= )(
)()( CBF
BCCBBC 2EIML32
LEI2
M +=+
+=
)()( CBF
BCCBCB 2EIML32
LEI2
M +=+
+=
100EI2ML
32LEI2
M CF
CDDCCD =+
+= )(
100EIML
32LEI2
M CF
CDDCDC +=+
+= )(
-
Slope-Deflection Method 19
3) (2 ) B C
MBA+ MBC= 50 MCB + MCD = 0
CMCB
FBD. C
B DMCD
BMBA MBC
FBD. B
A
C100 kg.50 kg.-m.
Slope-Deflection Method 20
4) 2 3
(1) (2)
.RadEI3
20B = .RadEI3
70C =
=++=+ 1502EIEI250MM CBBBCBA ....................)(...
=++=+ 20100EI22EI0MM CCBCDCB ..............)(...
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Slope-Deflection Method 21
5) B C 2
../)/( mkg320EI320EIEIM BAB ===
../)()( mkg3110EI3
70EI3
202EI2EIM CBBC =+=+=
../)/( mkg340EI320EI2EI2M BBA ===
../)()( mkg3160EI3
702
EI320
EI2EIM CBCB =+=+=
../ mkg3160100EI3
70EI2100EI2M CCD ===
../ mkg3370100EI3
70EI2100EIM CDC =+=+=
Slope-Deflection Method 22
6) FBD.
MA = 20/3 kg.-m. RA,x = 10 kg.RA,Y = 45 kg. RC = 310 kg.MD = 370/3 kg.-m. RA,x = 110 kg.RD,Y = 335 kg. 1
A
20/345
10
1045 40/3
B 10100 kg.
50 kg.-m.
40/3
110/3110
45
45
110/3110
45 160/3110
45C
RC=310
11011045 265 300 kg./m.
160/3 370/3265110
335110 D
-
Slope-Deflection Method 23
7) (BMD.)
2
A
C DB
110/3
160/320/3
40/3
370/3
185/3
BMD(kg.-m.)
++
--
+-
Slope-Deflection Method 24
4.2 Slope-Deflection Joint Translation
Joint Translation
......
...Frame ...
=0
=0
-
Slope-Deflection Method 25
Frame Symmetry () Load ( Sway )
=0
Centerline
Centerline
Centerline
Slope-Deflection Method 26
Joint Translation
Frame Symmetry Load Symmetry
...Frame ...
1
2
-
Slope-Deflection Method 27
EX. Slope-Deflection ( EI )
A
P B C
D
10 ft.
20 ft.
Slope-Deflection Method 28
1) Side Sway - 1 B C (1 DOF Joint Translation)
2) Fixed End Moments
0MM0MM0MM FDCFCD
FCB
FBC
FBA
FAB ====== ,
A
P B C
D10 ft.
20 ft.
-
Slope-Deflection Method 29
3) Fixed End Moments Slope-Deflection
3 B , C
)()( =++= 320200EI
ML
32LEI2
M BF
ABAB
BAAB
)()( =++= 3400200EI
ML
32LEI2
M BF
BAAB
BABA
)()( CBF
BCBC
CBBC 4080200EI
ML
32LEI2
M +=+
+=
)()( CBF
CBBC
CBCB 8040200EI
ML
32LEI2
M +=+
+=
)()( =++= 340200EI
ML
32LEI2
M CF
CDCD
DCCD
)()( =++= 320200EI
ML
32LEI2
M CF
DCCD
DCDC
Slope-Deflection Method 30
4) (3 ) B C
MBA+ MBC= 0 MCB + MCD = 0 1
FBD. B
BMBA
MBC
A
CP
0P2
MM2
MM DCCDBAAB =++
++ )()(20MM BAAB /)( + 20MM DCCD /)( +
A
P B C
DMAB MDC
FBD. C
BMCD
MCB
D
C
-
Slope-Deflection Method 31
5) 3 4
=+
=+=+
6EI
P4001126060
5031204040340120
CB
CB
CB
.................................................,......................................................................................................................
=++=+ 104080200EI
340200EI
0MM CBBBCBA .......)()(...
=++=+ 20340200EI
8040200EI
0MM CCBCDCB .......)()(...
201
340200EI
320200EI
0P2
MM2
MMBB
DCCDBAAB +=++
++ )]()([)()(
=+++ 30P201
320200EI
340200EI
CC ...................)]()([
Slope-Deflection Method 32
(4) , (5) (6)
6) 3
Note (Anti-Symmetry) Unknown 1
( Yuan-Yu Hsieh)
.RadEI13P100
CB == .,
ftEI39
P0016=
13P70
EI39P0016
3EI13P100
20200EI
MAB
== ),(
13P60
EI39P0016
3EI13P100
40200EI
MBA
== ),(
13P60
EI13P100
40EI13P100
80200EI
MBC =+= )(
ABDCBACDBCCB MMMMMM === ;;
-
Slope-Deflection Method 33
EX. Slope-Deflection ( EI )
A
300 kg.
B C
D
2.5 m.
4 m.
600 kg./m.
1.5 m.
1.5 m.
1 m.
Slope-Deflection Method 34
1) Sidesway B C
2) Fixed End MomentsA
300 kg.B C
D2.5 m.
600 kg./m.
4 m.1.5 m.
1.5 m.1 m.
A300 kg. B
MABF
MBAF
1.5 m.1.5 m.
C
600 kg./m.
BMBCF MCBF
2.5 m.
.... mkg531212
5260012WL
M22
FBC =
=
=
... mkg531212WL
M2
FCB ==
... mkg51128
33008PL
MFAB =
=
=
... mkg51128
33008
PLMFBA =
==
.. mkg0MM FDCFCD ==
-
Slope-Deflection Method 35
3) Fixed End Moments Slope-Deflection
4 B , C , D
51123EI2
ML
32LEI2
M BF
ABAB
BAAB .)()( =+
+=
511223EI2
ML
32LEI2
M BF
BAAB
BABA .)()( +=+
+=
531225EI4
ML
32LEI2
M CBF
BCBC
CBBC .)()( +=+
+=
531225EI4
ML
32LEI2
M CBF
CBBC
CBCB .)()( ++=+
+=
)()(4
32
2EI
ML
32LEI2
M DCF
CDCD
DCCD
+=+
+=
)()(4
32
2EI
ML
32LEI2
M DCF
DCCD
DCDC
+=+
+=
Slope-Deflection Method 36
4) (4 ) B , C D
MBA+ MBC= 0 MDC = 0 MCB + MCD = 0
1
FBD. BBMBA
MBC
A
C
03004
MM3
450MM DCCDBAAB =++
++ )()(
351300MM BAAB /).( +
4MM DCCD /)( +
MDC
FBD. D
300 kg.
MAB MDC
600 kg./m.
FBD. CB
MCD
MCB
D
C
-
Slope-Deflection Method 37
5) 3 4
=+++ 10531225EI4
511223EI2
CBB ...........)(.)(
=
++++ 204
32
2EI
531225EI4
DCCB ............)(.)(
31
450511223EI2
51123EI2
BB ++ ].)(.)([
=+
++
++ 3030041
43
22EI
43
22EI
DCDC .......)]()([
=
+ 404
32
2EI
DC .........................................................)(
Slope-Deflection Method 38
(5) (8)
=+ 50003EI10EI12EI44 CB ...............................................,
=++ 650012EI20EI15EI104EI32 DCB .............................,
=++ 760021EI54EI91EI54EI96 DCB ...............................,
=+ 80EI8EI3EI4 DC ...........................................
.. RadEI
8244B =
.. mEI
3572=
.. RadEI
6170C
=
.RadEI
300D =
-
Slope-Deflection Method 39
6) B , C , D 3
MAB = -330.9 kg.-m. MBA = 57.34 kg.-m. MBC = -57.34 kg.-m. MCB = 235.3 kg.-m. MCD = -235.3 kg.-m. MDC = 0 kg.-m.
Slope-Deflection Method 40
EX. Slope-Deflection ( EI )
A
BC
D
1.5 T.
/m.5 m.
10 m.E
12 m. 12 m.
-
Slope-Deflection Method 41
1) Sidesway 1 2
2) Fixed End Moments
A
BC
D
1.5 T.
/m.
5 m.
10 m.E
12 m. 12 m.
1 2
A1.5 T.
/m.
B
MABF
MBAF
10 m..... mT512
121051
12WL
M22
FAB =
=
=
... mT51212WL
M2
FBA ==
...sin. mT12531213
13551
M2
FBC =
=
B1.5
T./m.
MABF
MBAF
13 m.C
...sin. mT12531213
13551
M2
FCB =
=
Slope-Deflection Method 42
3) Rotation 1 2
Rotation AB = Rotation BC =
Rotation CD = Rotation DE =
101 /
1013SIN21221 )( =
1013SIN22121 )( =
102 /
A
BC
D
1.5 T.
/m.
5 m.
10 m.E
12 m. 12 m.
1 2
1/10 2/10
-
Slope-Deflection Method 43
4) Fixed End Moments Slope-Deflection
51210
35EI
ML
32LEI2
M 1BF
ABAB
BAAB .)()(
=+
+=
51210
32
5EI
ML
32LEI2
M 1BF
BAAB
BABA .)()( +
=+
+=
125310
32
13EI2
ML
32LEI2
M 12CBF
BCBC
CBBC .])([)( +=++=
125310
32
13EI2
ML
32LEI2
M 12CBF
CBBC
CBCB .])([)( ++=++=
])([)(10
32
13EI2
ML
32LEI2
M 21DCF
CDCD
DCCD
+=+
+=
])([)(10
32
13EI2
ML
32LEI2
M 21DCF
DCCD
DCDC
+=+
+=
Slope-Deflection Method 44
5 B , C , D , 1 2
5) (5 ) - B , C D
][)(10
32
10EI2
ML
32LEI2
M 2EDF
DEDE
EDDE
+=+
+=
][)(10
32
10EI2
ML
32LEI2
M 2EDF
EDDE
EDED
+=+
+=
=+= 10MM0M BCBAB ..........................................................;
=+= 20MM0M CDCBC ..........................................................;
=+= 30MM0M DEDCD ..........................................................;
-
Slope-Deflection Method 45
HA HE
E
D
MED
MDEVD
VE
HE
HD010HMM0M EEDDED =++= ;
)( EDDEE MM101
H +=
1551HH0F EAX =+= .;
A1.5 T.
/m.
B
MAB
MBAVB
VA
HA
HB 05105110HMM0M ABAABB =++= .;
)(. BAABA MM101
57H +=
522MM101
MM101
57 EDDEBAAB .)()(. =++
=+++ 4150MMMM EDDEBAAB ....................................
Slope-Deflection Method 46
VA VE
EDDECDE M241
M81
M121
V =
0VV0F EAY =+= ;
057155112V15HMM0M AACBABC =+++= ..;
CBBAABA M121
M81
M241
68754V ++= .
0M241
M81
M121
M121
M81
M241
68754 EDDECDCBBAAB =+++.
=+++ 505112MM3M2M2M3M EDDECDCBBAAB .............
A1.5
T./m. B
MAB
MCB
VA
HA
HCC
VC
12V15HMM0M EEEDCDC =++= ;
E
D
MED
MCD VC
VE
HE
HC C
-
Slope-Deflection Method 47
6) Slope-Deflection 5
7) (6) (10)
=+ 638609EI3EI90EI10EI46 21CB .....................................
=++ 73120EIEI4EI DCB .........................................................
=+ 80EI90EI3EI46EI10 21DC .............................................
=+ 9250EI20EI20EIEI 21DB ............................................
=+ 1016120EI390EI390EIEI 21DB .....................................
.. RadEI3289
B =
.. mEI5612
1 =
.. RadEI
8655C
= .. Rad
EI8113
D =
.. mEI1010
2 =
Slope-Deflection Method 48
8) 7 Slope-Deflection ( 4) MAB = -69.99 T.-m. MBA = -27.12 T.-m. MBC = 27.12 T.-m. MCB = 11.04 T.-m. MCD = -11.04 T.-m. MDC = 15.06 T.-m. MDE = -15.06 T.-m. MED = -37.82 T.-m.
-
Slope-Deflection Method 49
EX. Slope-Deflection 1. B C 2. B 3. BMD ( EI )
A B
C D
9 m.2I
100 kN./m.
4 m.
6 m.
2I I
Slope-Deflection Method 50
1) Sidesway
- AB CD AB = -CD = -vB
- BC BC = 0
2) 2
- B (B) - C (C)
A B
C D
9 m.2I
100 kN./m.
4 m.
6 m.
2I I
-AB
-CD
A B
C D
9 m.2I
100 kN./m.
4 m.
6 m.
2I I
B
C
-
Slope-Deflection Method 51
3) Fixed End Moments
4) Slope-Deflection ( 6 )
A D = 0
D
100 kg./m.C
MCDF MDCF
9 m.
.. mkg67512
910012WL
M22
FCD =
=
=
.. mkg67512WL
M2
FDC ==
.. mkg0MMMM FCBFBC
FBA
FAB ====
0EIv3330EI6670EI3331M BBAAB +++= ...0EIv3330EI3331EI6770M BBABA +++= ...
0EIv0000EI0001EI002M BCBBC +++= ...0EIv0000EI0002EI001M BCBCB +++= ...
675EIv0740EI2220EI4440M BDCCD += ...675EIv0740EI4440EI2220M BDCCD ++= ...
Slope-Deflection Method 52
5) (3 )
MBA+ MBC= 0 MCB + MCD = 0
FBD. C
CMCB
MCD
B
D
FBD. BA
MCBMBA
C
B
A B
C D
9 m.2I
100 kN./m.
4 m.
6 m.
2I ICDDCCDABBAAB LMMLMM /)(/)( +++
0L1002L100 CDCD =+ )/(
-
Slope-Deflection Method 53
6) 4 5
7) (1) , (2) (3)
1,2
=+++ 100EIv3330EI0001EI3333 BCB .................................
=+ 20675EIv0740EI4442EI0001 BCB ..............................
=+ 30450EIv1280EI0740EI3330 BCB ...........................
.. RadEI
1519B =
.., mEI
49334vB
=
.. RadEI
785C
=
Slope-Deflection Method 54
8) 7 Slope-Deflection 4 MAB = -1,298.4 kN.-m. MBA = -952.4 kN.-m. MBC = 952.4 kN.-m.MCB = 347.7 kN.-m. MCD = -347.7 kN.-m. MDC = 1,021.4 kN.-m.
9)
3-1,298.4
-1,021.4
-347.7-347.7 952.4
952.4
356
BMD.(kN.-m.)
A B
C D
-
Slope-Deflection Method 55
EX. Slope-Deflection 1. C D 2. C D 3. D ( EI )
A BC D
8 m.
60 kN.
4 m.
6 m. 6 m.E
Slope-Deflection Method 56
1) Fixed End Moments
.. mkN0MMMM FCEFEC
FDC
FCD ====
.. mkN908
12608PL
MFAC =
==
MCAFMACFA B
C D
8 m.
60 kN.
4 m.
6 m. 6 m.E
60 kNA
12 m.C
MACF MCAFB .. mkN908PL
MFCA ==
-
Slope-Deflection Method 57
2) Fixed End Moments Slope-Deflection (= 0)
2 C D
90EI1670EI3330ML
32LEI2
M CAF
ACCAAC +=+
+= ..)(
90EI3330EI1670ML
32LEI2
M CAF
CACACA ++=+
+= ..)(
DCF
CDDCCD EI250EI50ML32
LEI2
M +=+
+= ..)(
DCF
DCCBDC EI50EI250ML32
LEI2
M +=+
+= ..)(
ECF
CEECCE EI50EIML32
LEI2
M +=+
+= .)(
ECF
ECECEC EIEI50ML32
LEI2
M +=+
+= .)(
Slope-Deflection Method 58
3) (2 ) B C
MCA+ MCD+ MCE= 0 MDC = 0
4) 2 3
FBD. D
DMDC
FBD. C
MCA MCDC
MCE
=+ 190EI250EI8331 DC ................................................
=+ 20EI50EI250 DC ......................................................
-
Slope-Deflection Method 59
5) (1) (2) 4
1
6) Slope-Deflection MAC = -98.8 kN.-m. MDC = 0 kN.-m. MCA = 72.4 kN.-m. MCE = -52.7 kN.-m. MCD = -19.8 kN.-m. MEC = -26.3 kN.-m. 2
7) D
3
.. RadEI
752C
= .. Rad
EI326
D =
CDYC M8D0M ==
.. kN472D Y =FBD. CD
DMCD CVCD
NCDDY
DX
Slope-Deflection Method 60
EX. Slope-Deflection 1. B 2. B 3. A ( EI )
AB C
4 m.
10 kN./m.
3I
6 m.
I
-
Slope-Deflection Method 61
1) Sidesway
- AB CD AB = BC = -
2) 1
- B (B)
A B C
4 m.
10 kN./m.
3I
6 m.
I
A B C
4 m.
10 kN./m.
3I
6 m.
I
Slope-Deflection Method 62
3) Fixed End Moments
4) Slope-Deflection ( 4 )
B
10 kN./m.A
MABF MBCF
6 m.
.. mkN3012
61012WL
M22
FAB =
=
=
.. mkN3012WL
M2
FBA ==
.. mkN0MM FCBFBC ==
30EI50EIML32LEI2M BFABABABBAABABAB =++= .)/(/
30EI50EI2ML32LEI2M BFBAABABBAABABBA +=++= .)/(/
0EI3750EIML32LEI2M BFBCBCBCCBBCBCBC ++=++= .)/(/
0EI3750EI50ML32LEI2M BFCBBCBCCBBCBCCB ++=++= ..)/(/
-
Slope-Deflection Method 63
5) (2 )
MBA+ MBC= 0 VAB - VCB = (10)(6)
6) 4 5
= 130EI1250EI3 B ........................................................
FBD. BA
MBCMBACB
FBD.A
VCBVABCB
=+ 230EI3540EI1250 B ..................................................
BCCBBCCBABBAABAB LMMVL3610MMV /)(;/)( +=+=
Slope-Deflection Method 64
7) (1) (2)
1,2
8) 7 Slope-Deflection ( 3) MAB = -77.8 kN-m. 3
.. RadEI
66B
= .. m
EI482
=