บทที่ 4 (slope-deflection) - civil method 3 4.1 สมการที่ใช...
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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
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
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 .................................])([...
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 ..............)(...
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