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7/30/2019 SD Method VTU Notes
http://slidepdf.com/reader/full/sd-method-vtu-notes 1/89
Chapter-2: Slope Deflection Method
By Prof. H.P.Sudarshan
Sri Siddhartha Institute Of Tech,Tumkur
7/30/2019 SD Method VTU Notes
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Example: Analyze the propped cantilever shown by using slope defection
method. Then draw Bending moment and shear force diagram.
Solution: End A is fixed hence A =0
End B is Hinged hence B ≠0
Assume both ends are fixed and therefore fixed end moments are
12wLF,
12wLF
2
BA
2
AB
The Slope deflection equations for final moment at each end are
)2(L
EI4
12
wL
2L
EI2FM
)1(L
EI2
12
wL
2L
EI2FM
B
2
ABBABA
B
2
B A AB AB
In the above equations there is only one unknown B .
To solve we have boundary condition at B;
Since B is simply supported, the BM at B is zero
ie. MBA=0.
iseanticlockwisrotationtheindicatessignve-48
wLEI
0L
EI4
12
wLM(2)equationFrom
3
B
B
2
BA
7/30/2019 SD Method VTU Notes
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Substituting the value of BEI in equation (1) and (2) we have end moments
048
wL
L
4
12
wL
M
anticlockwismomentindicatessignve-8
wL
48
wL
L
2
12
wLM
32
BA
232
AB
MBA has to be zero, because it is hinged.
Now consider the free body diagram of the beam and find reactions using
equations of equilibrium.
wL8
3
wL85wLRwLR
wLRR
0V
wL8
5R
wL8
5
2
LwL
8
wL
2
LwLMLR
0M
AB
B A
A
2
AB A
B
7/30/2019 SD Method VTU Notes
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Problem can be treated as
The bending moment diagram for the given problem is as below
7/30/2019 SD Method VTU Notes
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The max BM occurs where SF=0. Consider SF equation at a distance of x
from right support
2
2
Xmax
X
wL128
9
L8
3
2
wL
8
3wL
8
3MM
BsupportfromL8
3atoccursBMmaxtheHence
L83X
0wXwL8
3S
And point of contra flexure occurs where BM=0, Consider BM equation ata distance of x from right support.
L4
3X
02
XwwLX
8
3M
2
X
For shear force diagram, consider SF equation from B
wL8
5SLS
wL8
3S0S
wXwL
8
3S
AX
BX
X
7/30/2019 SD Method VTU Notes
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Example: Analyze two span continuous beam ABC by slope deflection method.
Then draw Bending moment & Shear force diagram. Take EI constant
Solution: Fixed end moments are:
KNM67.4112
520
12
wLF
KNM67.4112
520
12
wLF
KNM89.886
24100
L
bWaF
KNM44.446
24100
L
WabF
22
CB
22
BC
2
2
2
2
BA
2
2
2
2
AB
Since A is fixed 0 A , ,0,0 CB
Slope deflection equations are:
)2(EI3
289.88
62EI289.88
2L
EI2FM
)1(EI3
144.44
6
EI244.44
2L
EI2FM
B
B
ABBABA
B
B
B A AB AB
7/30/2019 SD Method VTU Notes
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)4(EI5
2
5
EI467.41
25
EI267.41
2L
EI2FM
)3(EI5
2EI5
467.41
25
EI267.41
2L
EI2FM
BC
BC
BCCBCB
CB
CB
CBBCBC
In all the above four equations there are only two unknown B and C . And
accordingly the boundary conditions are
i -MBA-MBC=0
MBA+MBC=0
ii MCB=0 since C is end simply support.
)6(0EI5
4EI
5
267.41M
)5(0EI5
2EI
15
2222.47
EI52EI
5467.41EI
3289.88MMNow
CBCB
CB
CBBBCBA
Solving simultaneous equations 5 & 6 we get
EI B = – 20.83 Rotation anticlockwise.
EI C = – 41.67 Rotation anticlockwise.
Substituting in the slope definition equations
M AB = – 44.44 + KNM38.5183.203
1
MBA = + 88.89 + KNM00.7583.203
2
7/30/2019 SD Method VTU Notes
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MBC = – 41.67+ KNM00.7567.415
283.20
5
4
MCB = + 41.67+ 067.415
483.20
5
2
Reactions: Consider the free body diagram of the beam.
Find reactions using equations of equilibrium.
Span AB: ΣM A = 0 RB×6 = 100×4+75-51.38
RB = 70.60 KN
ΣV = 0 R A+RB = 100KN
R A = 100-70.60=29.40 KN
Span BC: ΣMC = 0 RB×5 = 20×5×2
5+75
RB = 65 KN
ΣV=0 RB+RC = 20×5 = 100KN
RC = 100-65 = 35 KN
7/30/2019 SD Method VTU Notes
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Using these data BM and SF diagram can be drawn.
7/30/2019 SD Method VTU Notes
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Max BM:
Span AB: Max BM in span AB occurs under point load and can be found
geometrically
Mmax=113.33-51.38 -
KNM20.4646
38.5175
Span BC:Max BM in span BC occurs where shear force is zero or
changes its sign. Hence consider SF equation w.r.t C
Sx = 35-20x= 020
35x =1.75m
Max BM occurs at 1.75m from C
Mmax = 35 × 1.75 – 20
2
75.1 2
= 30.625 KNM
7/30/2019 SD Method VTU Notes
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Example: Analyze continuous beam ABCD by slope deflection method and thendraw bending moment diagram. Take EI constant.
Solution:
0,0,0 CB A
FEMS MKN44.44-6
24100L
WabF 2
2
2
2
AB
KNM88.886
24100
L
bWaF
2
2
2
2
BA
KNM41.67-12
520
12
wLF
22
BC
KNM41.6712
520
12
wLF
22
CB
MKN30-5.120FCD
Slope deflection equations:
1---------EI3
144.442
L
EI2FM BB A AB AB
2---------EI3
289.882
L
EI2FM B ABBABA
3--------EI5
2EI
5
467.412
L
EI2FM CBCBBCBC
4--------EI5
2EI
5
467.412
L
EI2FM BCBCCBCB
KNM30MCD
7/30/2019 SD Method VTU Notes
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In the above equations we have two unknown rotations CB and , accordingly
the boundary conditions are:
0MM
0MM
CDCB
BCBA
5--------0EI5
2EI
15
2222.47
EI5
2EI
5
467.41EI
3
289.88MM,Now
CB
CBBBCBA
6EI54EI
5267.11
30EI5
2EI
5
467.41MM, And
CB
BCCDCB
Solving (5) and (6) we get
C
B
Substituting value of BEI and CEI in slope deflection equations we have
KNM30M
KNM00.3067.325
275.1
5
467.41M
KNM11.6775.15
267.32
5
467.41M
KNM11.6767.323
289.88M
KNM00.6167.322144.44M
CD
CB
BC
BA
AB
7/30/2019 SD Method VTU Notes
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Reactions: Consider free body diagram of beam AB, BC and CD as shown
7/30/2019 SD Method VTU Notes
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ABSpan
KN31.32R100R
KN69.67R
6111.6741006R
B A
B
B
BCSpan
KN42.57R520R
KN58.42R
11.673052
5205R
BB
C
C
Maximum Bending Moments:
Span AB: Occurs under point load
KNM26.684
6
6111.676133.133Max
Span BC: where SF=0, consider SF equation with C as reference
m13.220
58.42x
0x2058.42SX
MKN26.15302
13.22013.258.42M
2
max
7/30/2019 SD Method VTU Notes
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Example: Analyse the continuous beam ABCD shown in figure by slope
deflection method. The support B sinks by 15mm.
Take 4625 m10120Iandm/KN10200E
Solution:
In this problem A =0, B 0, C 0, =15mm
FEMs:
KNM44.44L
WabF
2
2
AB
KNM89.88L
bWaF
2
2
BA
KNM67.418
wLF
2
BC
KNM67.418
wLF
2
CB
FEM due to yield of support B
For span AB:
KNM61000
151012010
6
2006
LEI6mm
65
2
2baab
7/30/2019 SD Method VTU Notes
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For span BC:
KNM64.81000
151012010
5
2006
L
EI6mm
65
2
2cbbc
Slope deflection equation
5---------KNM30M
4---------EI5
2EI
5
431.50
64.82EI5
241.67
L
EI6)2(
L
EI2FM
3---------EI5
2EI
5
403.33
64.82EI5
2
41.67-
L
EI6)2(
L
EI2FM
2---------EI3
289.82
6EI3
288.89
L
EI6)2(
L
EI2FM
1---------EI
3
144.50
6EI3
144.44-
L
EI62
L
EIF
)L
32(
L
EI2FM
CD
BC
BC
2BCCBCB
CB
CB
2CBBCBC
B
B
2 ABBABA
B
B
2B A AB
B A AB AB
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There are only two unknown rotations B and C . Accordingly the boundary
conditions are
Now,
0EI5
4EI
5
231.20MM
0EI52EI
152286.49MM
0MM
0MM
CBCDCB
CBBCBA
CDCB
BCBA
Solving these equations we get
ockwiseAnticl71.9EI
ockwiseAnticl35.31EI
C
B
Substituting these values in slope deflections we get the final moments:
KNM30M
KNM00.3035.315
271.9
5
431.50M
KNM99.6171.95
235.31
5
403.33M
KNM99.6135.313
289.82M
KNM89.6035.313
144.50M
CD
CB
BC
BA
AB
Consider the free body diagram of continuous beam for finding reactions
7/30/2019 SD Method VTU Notes
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Reactions:
Span AB:RB × 6 = 100 x 4 + 61.99 – 60.89
RB = 66.85
R A = 100 – RB
=33.15 KN
Span BC:
RB × 5 = 20 x 5 x2
5+ 61.99 – 30
RB = 56.40 KN
RC = 20 x 5 - RB
=43.60 KN
7/30/2019 SD Method VTU Notes
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Example: Three span continuous beam ABCD is fixed at A and continuous over
B, C and D. The beam subjected to loads as shown. Analyse the beam by slope
deflection method and draw bending moment and shear force diagram.
Solution:
Since end A is fixed 0,0,0,0 DcB A
FEMs:
KNM30-8
460
8
WlF AB
KNM308
460
8
WlFBA
KNM12.54
MFBC
KNM12.54
MFCB
KNM313.3-12
410
12
wl
F
22
CD
KNM13.3312
410
12
wlF
22
DC
Slope deflection equations:
B A AB AB 2L
EI2FM
04
EI230- B
1--------EI0.530- B
ABBABA 2L
EI2FM
024
EI230 B
2---------EI30 B
7/30/2019 SD Method VTU Notes
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CBBCBC 2L
EI2FM
24
EI212.5 CB
3---------EI5.0EI12.5 CB
BCCBCB 2L
EI2FM
24
EI25.12 BC
4---------EI5.0EI12.5 BC
DCCDCD 2L
EI2FM
24
EI213.33- DC
5----------EI5.0EI33.13 DC
CDDCDC 2L
EI2FM
24
EI213.33 CD
6----------EIEI5.013.33 DC
In the above Equations there are three unknowns, EI DCB EI&EI, ,
accordingly the boundary conditions are:
)hinged(0Miii
0MMii
0MMi
DC
CDCB
BCBA
Now
705.42EI5.0EI2
0EI5.0EI5.12EI30
0MM
CB
CBB
BCBA
8083.0EI5.0EI2EI5.0
0EI5.0EI33.13EI5.0EI5.12
0MM
DCB
DCBC
BCCB
0MDC
90EIEI5.033.13 DC
7/30/2019 SD Method VTU Notes
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By solving (7), (8) & (9), we get
90.18EI
15.11EI
04.24EI
D
C
B
By substituting the values of DcB and, in respective equations we get
KNM090.1815.115.033.13M
KNM63.1190.185.015.1133.13M
KNM63.1104.245.015.115.12M
KNM5.96-11.15.5024.04-12.5M
KNM96.504.2430M
KNM02.4204.245.030M
DC
CD
CB
BC
BA
AB
Reactions: Consider the free body diagram of beam.
Beam AB:
KN015.30R60R
KN985.204
02.4296.5260R
B A
B
Beam BC:
downwardisRKN92.13RR
KN92.134
96.55063.11R
BCB
C
Beam CD:
KN91.22R410R
KN09.174
63.112410R
DC
D
7/30/2019 SD Method VTU Notes
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Example: Analyse the continuous beam shown using slope deflection method.
Then draw bending moment and shear force diagram.
Solution: In this problem fixedis Aend,0 A
FEMs:
MKN53.33-12
810
12
wlF
22
AB
KNM53.3312wlF
2
BA
KNM22.50-8
630
8
WlFBC
KNM22.508
WLFCD
Slope deflection equations:
B A AB AB 2
L
EI2FM
08
I3E253.33- B
1--------EI4
353.33- B
ABBABA 2L
EI2FM
028
I3E253.33 B
2--------EI2353.33 B
7/30/2019 SD Method VTU Notes
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CBBCBC 2L
EI2FM
26
I2E222.5- CB
3--------EI3
2
EI3
4
22.5- CB
BCCBCB 2L
EI2FM
26
I2E222.5 BC
4--------EI3
2EI
3
422.5 BC
In the above equation there are two unknown CB and , accordingly the
boundary conditions are:
0Mii
024MMi
CB
BCBA
50EI3
2EI
6
1783.54
24EI3
2EI
3
45.22EI
2
333.5324MM,Now
CB
CBBBCBA
0EI32EI
345.22Mand BCCB
(6)-----------EI3
125.11EI
3
2BC
Substituting in eqn. (5)
clockwiseantirotation432.1715
658.44EI
0EI6
1544.58
0EI3
125.11EI
6
1783.54
B
B
BB
7/30/2019 SD Method VTU Notes
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from equation (6)
iseanticlockwrotation159.8
432.173
125.11
2
3EI C
Substituting 159.8EIand432.17EI CB in the slope deflection equation
we get Final Moments:
KNM18.27432.172
333.53M
KNM-66.4017.432-4
333.53M
BA
AB
00.0)432.17(32159.8
345.22M
KNM18.51159.83
2432.17
3
45.22M
CB
BC
Reactions: Consider free body diagram of beams as shown
Span AB:
KN87.44R810R
KN13.358
481040.6618.27R
B A
B
Span BC:
KN47.6R30R
KN53.236
33018.51R
BC
B
7/30/2019 SD Method VTU Notes
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Max BM
Span AB: Max BM occurs where SF=0, consider SF equation with A as origin
KNM67.36642
487.410487.487.44M
m487.4x
010x-87.44S
2
max
x
Span BC: Max BM occurs under point load
MKN41.192
18.5145MBC max
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Example: Analyse the beam shown in figure. End support C is subjected to an
anticlockwise moment of 12 KNM.
Solution: In this problem fixedisend,0 A
FEMs:
KNM67.2612
420
12
wlF
22
BC
KNM26.6712
wl
F
2
CB
Slope deflection equations:
B A AB AB 2L
EI2FM
04
I2E20 B
1---------EI B
ABBABA 2L
EI2FM
024
I2E20 B
2---------EI2 B
CBBCBC 2L
EI2FM
24
I5.1E226.67- CB
3---------EI4
3EI
2
326.67- CB
BCCBCB 2L
EI2FM
24
I5.1E226.67 BC
4---------EI4
3EI
2
326.67 BC
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In the above equation there are two unknowns CB and , accordingly the
boundary conditions are
012M
0MM
CB
BCBA
(5)---------067.26EI4
3EI
2
7
EI4
3EI
2
367.26EI2MM,Now
CB
CBBBCBA
(6)---------0EI2
3EI
4
367.38
12EI4
3EI
2
367.2612M,and
CB
BCCB
From (5) and (6)
72.1425
846EI
046EI8
25
033.19EI4
3EI
8
3
067.26EI43EI
27
B
B
CB
CB
From (6)
iseanticlockwrotationindicatessignve-14.33
72.144
367.38
3
2EI C
equationsdeflectionslopeisEIandEIngSubstituti CB
KNM12)72.14(4
3)14.33(
2
367.26M
KNM44.2914.334
3)72.14(
2
367.26M
KNM42.29)72.14(2EI2M
KNM72.14EIM
CB
BC
BBA
B AB
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Reaction: Consider free body diagrams of beam
Span AB:
KN04.11RR
KN04.114
44.2972.14R
B A
B
Span BC:
KN64.29R420R
KN36.504
24201244.29R
BC
B
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Example: Analyse the simple frame shown in figure. End A is fixed and ends B &C are hinged. Draw the bending moment diagram.
Solution:
In this problem ,0,0,0,0 DCB A
FEMS:-
KNM108
WLF
KNM108
420
8
WLF
KNM67.2612
420
12
wlF
KNM67.2612
420
12
wlF
KNM33.53
6
42120
L
bWaF
KNM67.1066
42120
L
WabF
DB
CD
22
CB
22
BC
2
2
2
2
BA
2
2
2
2
AB
Slope deflections are
)1(EI3
267.106
6
I2E267.106
2L
EI2FM
BB
B A AB AB
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)4(EI4
3EI
2
367.262
2
I3
4
E267.26
2L
EI2FM
)3(EI4
3EI
2
367.262
2
I3
4
E267.26
2L
EI2FM
)2(EI3
433.532
6
I2E233.53
2L
EI2FM
BCBC
BCCBCB
CBCB
CBCBBC
BB
BBBABA
)6(EI2
1EI102
4
EI210
2L
EI2FM
)5(EI2
1EI102
4
EI210
2
L
EI2FM
BDBD
BDDBDB
DBDB
DBBDBD
In the above equations we have three unknown rotations B , C , D accordingly
we have three boundary conditions.0MMM BDBCBA
0MCB Since C and D are hinged
0MDB Now
(9)-----0EIEI2
110M
(8)-----0EI23EI
436.672M
(7)-----0EI2
1EI
4
3EI
6
2336.66
EI2
1EI10EI
4
3EI
2
367.26EI
3
433.53MMM
DBDB
CBCB
DCB
DBCBBBDBCBA
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Solving equations 7, 8, & 9 we get
414.14EI
36.13EI
83.8EI
D
C
B
Substituting these values in slope equations
0)83.8(2
1)414.14(10M
KNM38.8)414.14(2
1)83.8(10M
0)83.8(43)36.13(
2367.26M
KNM94.49)36.13(4
3)3.8(
2
367.26M
KNM56.41)83.8(3
433.53M
KNM56.112)83.8(3
267.106M
DB
BD
CB
BC
BA
AB
7/30/2019 SD Method VTU Notes
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Reactions: Consider free body diagram of each members
Span AB:
KN83.91R120R
KN17.286
212056.11256.41R
B A
B
Span BC:
KN515.27R420R
KN485.524
242094.49R
BC
B
Column BD:
20HHKN78.12H
KN92.74
33.8220H
D AB
D
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Example: Analyse the portal frame shown in figure and also drawn bending
moment and shear force diagram
Solution:Symmetrical problem- Sym frame + Sym loading
0,0,0,0 DCB A
FEMS
KNM106.67-6
2480
6
4280
L
cdW
L
abW
F
2
2
2
2
2
2
2
2
2
1
BC
KNM67.106L
dcW
L
bWaF
2
22
2
2
CB
Slope deflection equations:
1--------EI2
10
4
EI202
L
EI2FM BBB A AB AB
2-------EI024
EI202
L
EI2FM BB ABBABA
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3------EI3
2EI
3
467.106)2(
6
I2E267.106
2L
EI2FM
CBCB
CBBCBC
4------EI3
2EI
3
467.106)2(
6
I2E267.106
2L
EI2FM
BCBC
BCCBCB
5-------EI)02(4
EI20
2L
EI2FM
CC
DCCDCD
6-------EI2
1)0(
4
EI20
2
L
EI2FM
CC
CDDCDC
In the above equation there are two unknown rotations. Accordingly the boundary
conditions are
0MM
0MM
CDCB
BCBA
Now (7)-------0EI32EI
3767.106MM CBBCBA
(8)-------0EI3
7EI
3
267.106MM CBCDCB
Multiply by (7) and (8) by 2
Clockwise6445
303.960EI
0EI3
45960.03-
subtracts
0EI3
14EI
3
434.213
0EI3
14EI
3
4969.746
B
B
CB
CB
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Using equation (7)
ckwiseAnticlo6464
3
767.106
2
3-
EI3
767.106
2
3EI BC
Here we find CB . It is obvious because the problem is symmetrical.
aremomentsFinal
KNM-32642
1M
KNM64M
KNM64643
2)64(
3
467.106M
KNM64643264
3467.106M
KNM64M
KNM322
64M
DC
CD
CB
BC
BA
AB
7/30/2019 SD Method VTU Notes
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Consider free body diagram’s of beam and columns as shown
By symmetrical we can write
KNM80RR
KNM60RR
CD
B A
Now consider free body diagram of column AB
Apply
KN24H
32644H
0M
A
A
B
Similarly from free body diagram of column CD
Apply
KN24H
32644H
0M
D
A
C
7/30/2019 SD Method VTU Notes
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Check:
0HH
0H
D A
Hence okay
Note: Since symmetrical, only half frame may be analysed. Using first threeequations
and taking CB
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Example: Analyse the portal frame and then draw the bending moment diagram
Solution:
This is a symmetrical frame and unsymmetrically loaded, thus it is an
unsymmetrical problem and there is a sway
Assume sway to right.
Here 0,0,0,0 DBD A
FEMS:
KNM75.938
3580
L
bWaF
KNM25.568
3580
L
WabF
2
2
2
2
CB
2
2
2
2
BC
Slope deflection equations
2--------EI8
3EI
4
302
4
2EI0
L
32
L
EI2FM
1--------EI8
3EI
2
1
4
30
4
2EI0
L
32
L
EI2FM
BB
ABBABA
BB
B A AB AB
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6---------EI8
3EI
2
1
4
30
4
2EI0
L32
LEI2FM
5---------EI8
3EI
4
302
4
2EI0
L
32
L
EI2FM
4---------EI4
1EI
2
175.932
8
2EI75.93
2L
EI2FM
3---------EI
4
1EI
2
125.562
8
2EI25.56
2L
EI2FM
CC
CDDCDC
CC
DCCDCD
BCBC
BCCBCB
CBCB
CBBCBC
In the above equation there are three unknowns and, CB , accordingly the
boundary conditions are,
0MMMM
04
MM
4
MM,e.i
conditionShear ---0PHH
0MM
conditionsintJo0MM
DCCDBA AB
DCCDBA AB
HD A
CDCB
BCBA
70EI8
3EI
4
1EI
2
325.56
0EI4
1EI
2
125.56EI
8
3EIMM,Now
CB
CBBBCBA
80EI8
3EI
2
3EI
4
175.93
0EI8
3EIEI
4
1EI
2
175.93MM, And
CB
CBCCDCB
7/30/2019 SD Method VTU Notes
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90EI2
3EI2
3EI2
3
EI`8
3EI
2
1
EI8
3EIEI
8
3EIEI
8
3EI
2
1MMMM, And
CB
C
CBBDCCDBA AB
(8)&(7)inSubstitute
EIEIEI(9)From CB
(7)Eqn
10-------0EI8
1EI
8
925.56
0EIEI8
3EI
4
1EI
2
325.56
CB
CBCB
)8(Eqn
11----------0EI8
9EI
8
175.93
0EIEI8
3EI
2
3EI
4
175.93
CB
CBCB
Solving equations (10) & (11) we get 25.41EI B
By Equation (10)
5.3775.7825.41EIEIEI
75.7825.418
925.568
EI8
9
25.568EI
CB
BC
Hence
5.37EI,75.78EI,25.41EI CB Substituting these values in slope deflection equations, we have
7/30/2019 SD Method VTU Notes
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KNM31.255.378
375.78
2
1M
KNM69.645.378
375.78M
KNM69.6475.414
175.78
2
175.93M
KNM31.5575.784125.41
2125.56M
KNM31.555.378
325.41M
KNM69.345.378
325.41
2
1M
DC
CD
CB
BC
BA
AB
Reactions: consider the free body diagram of beam and columns
Column AB:
KN5.224
31.5569.34H A
Span BC:
17.51R80R
KN83.288
38069.6431.55R
BC
B
Column CD:
5.224
31.2569.64HD
7/30/2019 SD Method VTU Notes
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Check:
ΣH = 0H A + HD = 022.5 – 22.5 = 0Hence okay
7/30/2019 SD Method VTU Notes
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Example: Frame ABCD is subjected to a horizontal force of 20 KN at joint C asshown in figure. Analyse and draw bending moment diagram.
Solution:
Frame is Symmetrical and unsymmetrical loaded hence there is a sway.
Assume sway towards right
FEMS
0FFFFFF DCCDCBBCBA AB
Slope deflection equations are
2EI3
2EI
3
4
3
32
3
EI2
L
32
L
EI2FM
1---------EI3
2EI
3
2
3
3
3
EI2
L
32
L
EI2FM
B
B
ABBABA
B
B
B A AB AB
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5EI3
2EI
3
43
32
3
2EI
L
32
L
EI2FM
4--------EI5.0EI
24
EI2
2L
EI2FM
3--------EI5.0EI
24
EI2
2L
EI2FM
C
C
DCCDCD
BC
BC
BCCBCB
CB
CB
CBBCBC
6---------EI3
2EI
3
2
3
3
3
EI2
L
32
L
EI2FM
C
c
CDDCDC
The unknown are &, C,B . areconditionsboundarythey Accordingl
060MMMM
0203
MM
3
MM,e.i
020HH.III
0MM.II
0MM.I
DCCDBA AB
DCCDBA AB
D A
CDCB
BCBA
70EI3
2EI5.0EI
3
7
EI5.0EIEI3
2EI3
4MMNow
CB
CBBBCBA
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80EI3
2EI
3
7EI5.0
EI3
2EI
3
4EI5.0EIMMand
CB
CBCCDCB
9060EI3
8EI2EI2
60EI3
2EI
3
2EI
3
2
EI3
4EI
3
2EI
3
4EI
3
2EI
3
260MMMMand
CB
C
CBBDCCDBA AB
Solving (7).(8) & (9) we get
77.34EI
,18.8EI
,18.8EI
C
B
Substituting the value of and, CB in slope deflection equations
KNM73.1777.343
218.8
3
2M
KNM27.1277.343
218.8
3
4M
KNM27.1218.818.85.0M
KNM27.1218.85.018.80M
KNM27.1277.343218.8
34M
KNM73.1777.343
218.8
3
2M
DC
CD
CB
BC
BA
AB
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Reactions: Consider the free body diagram of the members
Member AB:
KN103
27.1273.17H A
Member BC:
downwardsRof directionindicatessignve-KN135.6RR
KN135.64
27.1227.12R
BCB
C
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Member CD:
righttoleftisHof directiontheindicatessignve-KN103
27.1273.17H DD
Check: ΣH = 0
H A + HD + P = 0
+10 + 10 – 20 = 0
Hence okay
7/30/2019 SD Method VTU Notes
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Example: Analyse the portal frame subjected to loads as shown. Also draw
bending moment diagram.
The frame is symmetrical but loading is unsymmetrical. Hence there is a sway.
Assume sway towards right. In this problem 0,0,0,0 DCB A
FEMs:
KNM13.33-12
410
12
wlF
22
AB
KNM13.3312
410
12
wlF
22
BA
KNM112.5-8
10908wlFBC
KNM112.58
1090
8
wlFCB
Slope deflection equations:
L
32
L
EI2FM B A AB AB
4
30
4
EI213.33- B
1---------EI375.0EI5.013.33-B
L
32
L
EI2FM ABBABA
4
302
4
EI213.33 B
2---------EI375.0EI13.33 B
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CBBCBC 2L
EI2FM
210
I3E2112.5- CB
3---------EI6.01.2EI112.5- CB
BCCBCB 2L
EI2FM
210
I3E2112.5 BC
4---------EI6.01.2EI112.5 BC
L
32
L
EI2FM DCCDCD
4
302
4
EI20 C
5---------EI375.0EI C
L
32
L
EI2FM CDDCDC
4
320
4
EI20 C
6---------EI375.00.5EI C
EIandEIEIunknowns3areThere CB, , accordingly the boundary conditions
are
040HH
0MM0MM
D A
CDCB
BCBA
4
MMH
MM4Hand
4
80MMH
2
4410MM4HHere
DCCDD
BCCDD
BA AB A
BA AB A
080MMMM
0404
MM
4
80MM
DCCDBA AB
DCCDBA AB
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Now MBA + MBC = 0
(8--------0EI375.0EI6.0EI2.25.112
0EI375.0EIEI6.0EI2.15.112
540MMand
(7--------017.99EI375.0EI6.0EI2.2
0EI6.0EI2.15.112EI375.0EI33.13
BC
CBC
DCCB
CB
CBB
---------080EI-1.5EI1.5EI5.1
080EI375.0EI5.0
EI375.0EIEI375.0EI33.13EI375.0EI5.033.13
080MMMMalso
CB
C
CBB
DCCBBA AB
By solving (7), (8) and (9) we get
34.66EI
64.59EI
65.72EI
C
B
Final moments:
KNM70.54)34.66(375.0)64.59(5.0M
KNM52.84)34.66(375.064.59M
KNM52.8465.726.064.592.15.112M
KNM10.6164.596.065.722.15.112M
KNM10.6134.66375.065.72M
KNM-1.8866.34375.065.725.033.13M
DC
CD
CB
BC
BA
AB
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Reactions: Consider the free body diagrams of various members
Member AB:
lefttorightfromisHof directionindicatessignve-KN195.5
4
241088.110.61H
A
A
Member BC:
KN34.38R90R
KN34.4710
59010.6152.84R
CB
C
Member CD
KN81.344
7.5454.84HD
7/30/2019 SD Method VTU Notes
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Check
ΣH = 0
H A + HD +10 × 4 = 0
-5.20 - 34.81+ 40 = 0
Hence okay
7/30/2019 SD Method VTU Notes
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Example: Analyse the portal frame and then draw the bending moment diagram
Solution:
Since the columns have different moment of inertia, it is an unsymmetrical
frame. Assume sway towards right
FEMS:
KNM608
WL
F
KNM608
680
8
WLF
CB
BC
Here 0,0 D A
Slope deflection equations
2--------EI8
3EI
4
302
4
2EI0
L32
LEI2FM
1--------EI8
3EI
2
1
4
30
4
2EI0
L
32
L
EI2FM
BB
ABBABA
BB
B A AB AB
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6---------EI4
3EI
4
30
4
2E2I0
L
32
L
EI2FM
5---------EI4
3EI2
4
302
4
2E2I0
L
32
L
EI2FM
4---------EI3
4EI
3
2602
6
2E2I60
2L
EI2FM
3---------EI3
2EI
3
4602
6
2E2I60
2L
EI2FM
CC
CDDCDC
CC
DCCDCD
CBBC
BCCBCB
CBCB
CBBCBC
In the above equation there are three unknowns and, CB , accordingly the
boundary conditions are,
0MMMM
04
MM
4
MM,e.i
conditionShear ---0HH
conditionsintJo0MM
0MM
DCCDBA AB
DCCDBA AB
D A
CDCB
BCBA
70EI
8
3EI
3
2EI
3
760
0EI3
2EI
3
460EI
8
3EIMM,Now
CB
CBBBCBA
8060EI4
3EI
3
10EI
3
2
0EI4
3EI2EI
3
4EI
3
260MM, And
CB
CCBCDCB
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90EI4
9EI3EI2
3
EI4
3EI
EI4
3EI2EI
8
3EIEI
8
3EI
2
1MMMM, And
CB
C
CBBDCCDBC AB
(7)inEIof valuengSubstituti
EI3EI2
3
9
4EI(9)From CB
10-------060EI6
1EI
12
25
060EI2
1EI
4
1EI
3
2EI
3
7
060EI3EI2
3
9
4
8
3EI
3
2EI
3
7
CB
CBCB
CBCB
Substituting value of EI in (8)
11-------060EI3
7EI
6
1
060EIEI21EI
310EI
32
060EI3EI2
3
9
4
8
3EI
3
10EI
3
2
CB
CBCB
CBCB
Solving (10) & (11) we get EI B =31.03
By Equation (11)
3.27
60EI6
1
7
3EI BC
Now
55.16EI3EI2
3
9
4EI CB
Now
EI B =31.03, 3.27EI C , EI 55.16
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Substituting these values in slope deflection equations,The final moments are:
KNM52.1555.164
393.27M
KNM45.43)55.16(4
3)93.27(2M
KNM43.4393.273
403.31
3
260M
KNM25.3793.273
203.31
3
460M
KNM24.3755.168
3
03.31M
KNM72.2155.168
303.31
2
1M
DC
CD
CB
BC
BA
AB
Reactions: consider the free body diagram of beam and columns
7/30/2019 SD Method VTU Notes
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Column AB:
KN74.144
72.2125.37H A
Beam BC:
03.41R80R
KN97.386
38045.4325.37R
BC
B
Column CD:
KN74.144
52.1545.43HD
Check:
ΣH = 0H A + HD = 014.74-14.74=0
Hence okay
7/30/2019 SD Method VTU Notes
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Ex: Portal frame shown is fixed at ends A and D, the joint B is rigid and joint C ishinged. Analyse the frame and draw BMD.
Solution:
FEM’s:
0,0,0,0,0Here
KNM608
680
8
WLF
KNM608
680
8
WLF
CDCBBD A
CB
BC
Since C is hinged member CB and CD will rotate independently. Also the
frame is unsymmetrical, will also have sway. Let the sway be towards right.
The slope deflections are:
)1(EI8
3EI
2
1
4
30
4
EI20
L
32
L
EI2FM
B
B
B A AB AB
7/30/2019 SD Method VTU Notes
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)6(EI8
3EI
2
1
4
30
4
EI20
L
32
L
EI2FM
)5(EI83EI
4
302
4
EI20
L
32
L
EI2FM
)4(EI3
2EI
3
460
26
I2.E260
2
L
EI2FM
)3(EI3
2EI
3
460
26
I2.E260
L
32
L
EI2FM
)2(EI83EI
4
302
4
EI20
L
32
L
EI2FM
CD
CD
CDDDCDC
CD
CD
DCDCDCD
BCB
BCB
BCBCBCB
CBB
CBB
CBBCBC
B
B
ABBABA
In the above equations areand,, CDCBB unknowns. According theboundary conditions are
I. MBA+MBC = 0,
II. MCB = 0,
III. MCD = 0,
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IV. H A+HD = 0
0MMMM
04
MM
4
MM,e..i
DCCDBA AB
DCCDBA AB
Now using the boundary conditions:
)14(060EI3
2EI
15
26
60EI3
2EI
5
3
3
7
060EI5
8
8
3
EI3
2
EI3
7
MM
)7(EquationinngSubstituti
)13(EI5
8EI
2
3
15
16EIgives)12(Equation
)12(0EI16
15EI
2
3
0EI2
3
EI8
3
2
3
EI2
3
MMMM
)10(inSub
)11(EI8
3EI)9(From
)10(0EI2
3EI
2
3EI
2
3
EI8
3EI
2
1EI
8
3EIEI
8
3EIEI
8
3EI
2
1MMMM
)9(0EI8
3EIM
)8(060EI3
4EI
3
2M
)7(060EI8
3EI
3
2EI
3
7
EI3
2EI
3
460EI
8
3EIMM
CBB
CBB
BCBBBCBA
BB
B
BDCCDBA AB
cD
CDB
CDCDBBDCCDBA AB
CDCD
CBBCB
CBB
CBBBBCBA
7/30/2019 SD Method VTU Notes
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havewe2by(14)equationgmultiplyinand)8(EquationinngSubstituti
29.6442
15180EI
_ __________ __________
0180EI15
42
_ __________ __________
0120EI3
4
EI15
52
060EI3
4EI
3
2
B
B
CBB
CBB
864.10229.645
8EI
5
8EIFrom(13)
B
574.38EI8
3EI(11)From CD
165.77
60864.1028
329.64
3
7
2
3
60EI8
3EI
3
7
2
3EI)7(From BCB
864.102EI,57.38EI,165.77EI,29.64EI CDCBB
Final Moments are
KNM29.19864.1028
3574.38
2
1M
0864.1028
3574.38M
029.643
2165.77
3
460M
KNM72.25165.773
229.64
3
460M
KNM72.25864.1028
329.64M
KNM42.6864.1028
329.64
2
1M
DC
CD
CB
BC
BA
AB
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Reactions: Consider the free body diagram of various members
Column AB:
KN825.44
42.672.25H A
Beam BC:
KN71.3529.4480R
KN29.446
38072.25R
C
B
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Column CD:
KN82.44
28.19HD
Check:
ΣH = 0
H A+HD = 0
Hence okay.
7/30/2019 SD Method VTU Notes
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Example: Analyse the portal frame shown in figure the deflection method andthen draw the bending moment diagram
Fig
Solution:
The frame is unsymmetrical, hence there is a sway. Let the sway be
towards right.
0,0,0,0 DCB A
FEMS:
KNM30215F
KNM67.4112
520F
KNM67.4112
520F
CE
2
CB
2
BC
Slope deflection equations
7/30/2019 SD Method VTU Notes
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2--------EI375.0EI4
302
4
2EI0
L32
LEI2FM
1--------EI375.0EI5.04
30
4
2EI0
L
32
L
EI2FM
BB
ABBABA
BB
B A AB AB
6---------EI375.0EI5.04
30
4
2EI0
L
32
L
EI2FM
5---------EI375.0EI4
302
4
2EI0
L
32
L
EI2FM
4---------EI6.0EI2.167.4125
1.5I2E67.41
2LEI2FM
3---------EI6.0EI2.167.4125
1.5I2E67.41
2L
EI2FM
CC
CDDCDC
CC
DCCDCD
BCBC
BCCBCB
CBCB
CBBCBC
In the above equation there are three unknowns and, CB , accordingly the
boundary conditions are,
0MMMM,e.i
0HH
0MMM
0MM
DCCDBA AB
D A
CECDCB
BCBA
7/30/2019 SD Method VTU Notes
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Now,
7067.41EI375.0EI6.0EI2.2
0EI6.0EI2.167.141EI375.0EI
0MM
CB
CBB
BCBA
8067.11EI375.0EI2.2EI6.0
030EI375.0EIEI6.0EI2.167.41MM, And
CB
CBCCDCB
90EI5.1EI5.1EI5.1
0EI375.0EI5.0EI375.0EI2EI375.0EIEI375.0EI5.0
0MMMM
CB
CCBB
DCCDBC AB
Solving the above equationswe get, EI 98.23B , EI 62.14EI,36.9C
Substituting these values in slope deflection equations, we have
KNM30MKNM16.1062.14375.036.95.0M
KNM84.14)62.14(375.036.9M
KNM83.4498.236.036.92.167.41M
KNM51.1836.96.098.232.167.41M
KNM50.1862.14375.098.23M
KNM50.662.14375.098.235.0M
CE
DC
CD
CB
BC
BA
AB
7/30/2019 SD Method VTU Notes
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Reactions: consider the free body diagram of beam and columns
Column AB:
KN25.64
5.65.18H A
Span BC:
73.44R520R
KN27.555
5.25205.1883.44R
CB
C
Column CD:
25.64
84.1416.10HD
7/30/2019 SD Method VTU Notes
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Check:
ΣH = 0H A + HD = 0Σ = 0
Hence okay
7/30/2019 SD Method VTU Notes
http://slidepdf.com/reader/full/sd-method-vtu-notes 72/89
Example: Analyse the portal frame shown and then draw bending momentdiagram.
Solution:
It is an unsymmetrical problem hence there is a sway be towards right
0,0,0,0 DCB A
FEMs:
KNM41.67-12
520
12
wlF
22
BC
KNM41.6712
520
12
wl
F
22
CB
Slope deflection equations:
L
32
L
EI2FM B A AB AB
3
30
3
EI20 B
1---------EI3
2EI
3
2B
L32
LEI2FM ABBABA
3
302
3
EI20 B
2---------EI3
2EI
3
4B
7/30/2019 SD Method VTU Notes
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L
32
L
EI2FM CBBCBC
25
I5.1E241.67- CB
3---------EI5
3EI5
641.67- CB
L
32
L
EI2FM BCCBCB
025
I5.1E267.41 BC
4---------EI6.01.2EI41.67 BC
L
32
L
EI2FM DCCDCD
4
3
024
EI2
0 C
5----------EI375.0EI C
L
32
L
EI2FM CDDCDC
4
30
4
EI20 C
6----------EI375.00.5EI C
In the above equations there are three unknown and, CB and accordingly the
Boundary conditions are:
0)MM(3)M4(M
04
MM
3
MMi.e
0HH
0MM
0MM
DCCDBA AB
DCCDBA AB
D A
CDCB
BCBA
7/30/2019 SD Method VTU Notes
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Now
)8(0EI375.0EI6.0EI2.267.41
0EI375.0EIEI6.0EI2.167.41
0MM
)7(067.41EI3
2EI5
3EI53.2
67.41EI5
3EI
5
6EI
3
2EI
3
4
0MM
BC
CBC
CDCB
CB
CBB
BCBA
04
MM
3
MM DCCDBA AB
)9(0EI53.7EI5.4EI8
0EI25.2EI5.4EI3
8EI
3
16EI
3
8EI
3
80EI375.0EI5.0EI375.0EI3
EI3
2EI
3
4EI
3
2EI
3
24
CB
CBB
CC
BB
By solving (7), (8) and (9) we get
8.12EI
17.23EI
46.25EI
C
B
Final moments:
KNM65.1680.12375.070.235.0M
KNM50.28)80.12(375.070.23M
KNM50.2846.2060.017.232.167.41M
KNM40.2567.4117.235
346.25
5
6M
MKN40.258.123
246.25
3
4M
KNM8.448.123
246.25
3
2M
DC
CD
CB
BC
BA
AB
7/30/2019 SD Method VTU Notes
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Reactions: Consider the free body diagram
Member AB:
KN28.113
44.840.25H A
7/30/2019 SD Method VTU Notes
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Member BC:
KN36.4864.51520R
KN64.512
2
552030.205.28
R
B
C
Member CD:
HD =4
65.165.28 = 11.28 KN
Check:
ΣH = 0H A + HD = 0Satisfied, hence okay
7/30/2019 SD Method VTU Notes
http://slidepdf.com/reader/full/sd-method-vtu-notes 77/89
Example: A portal frame having different column heights are subjected for forces
as shown in figure. Analyse the frame and draw bending moment diagram.
Solution:-
It is an unsymmetrical problem
0,0,0,0 DCB A , hence there is a sway be towards right.
FEMs:
KNM15-8
430
8
WlF AB
KNM158
430
8
WlFBA
KNM30-8
460
8
WlFBC
KNM308
460
8
WlFCB
CDF = DCF = 0
Slope deflection equations:
L
32
L
EI2FM B A AB AB
4
30
4
I2E215- B
1--------EI75.0EI15- B
7/30/2019 SD Method VTU Notes
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L
32
L
EI2FM ABBABA
4
302
4
I2E215 B
2--------EI75.0EI215 B
CBBCBC 2L
EI2FM
24
I2E230- CB
3---------EIEI230- CB
BCCBCB 2L
EI2FM
24
I2E230 BC
4---------EI2EI30 BC
L
32
L
EI2FM DCCDCD
3
302
3
EI20 C
5---------EI
3
2EI
3
4
C
L
32
L
EI2FM CDDCDC
3
30
3
EI20 C
6---------EI3
2EI
3
2C
There are three unknowns, EI, EI&EI, CB , accordingly the Boundary
conditions are
7/30/2019 SD Method VTU Notes
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0180MM4MM3
030
3
MM
4
60MM,e.i
030HH
0MM
0MM
DCCDBA AB
DCCDBA AB
D A
CDCB
BCBA
Now
7015EI75.0EIEI4
EIEI230EI75.0EI215MM
CB
CBBBCBA
8030EI
3
2EI
3
10EI
EI3
2EI
3
4EIEI230MM
CB
CBCCDCB
90180EI833.9EI8EI9
180EI3
2EI
3
2EI
3
2EI
3
44
EI75.0EI215EI75.0EI153180)MM(4)MM(3
CB
CC
BBDCCDBA AB
By solving (7), (8) & (9) we get
795.20EI
714.7EI
577.9EI
C
B
Substituting these values in the slope deflection equations we get
KNM00.19795.203
2)714.7(
3
2M
KNM15.24795.20
3
2)714.7(
3
4M
KNM15.24577.9714.7230M
KNM18.55-7.714-9.577230-M
KNM55.18795.2075.0577.9215M
KNM01.21795.2075.0577.915M
DC
CD
CB
BC
BA
AB
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Reactions: Consider free body diagrams of the members
Member AB:
KN615.154
23001.2155.18H A
-ve sign indicates the direction of H A is from right to left.
7/30/2019 SD Method VTU Notes
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Member BC:
KN40.3160.28R60R
KN60.284
15.2426055.18R
BC
B
Member CD:
KN38.143
15.2419HD
Check:ΣH = 0
H A + HD + 30 = 0
-15.62 – 14.38 + 30 = 0
Hence okay
7/30/2019 SD Method VTU Notes
http://slidepdf.com/reader/full/sd-method-vtu-notes 82/89
Example: Analyse the frame using slope deflection method and draw the
Bending Moment Diagram.
Solution: Assume sway towards right
7/30/2019 SD Method VTU Notes
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It can be observed from figure in that direction of moments due to sway in
member AB are anticlockwise and that for member CD are clockwise. Wise shall
be taken to incorporate the same in the slope deflection equation.
FEMS
0Here
MKN3212
wIF
MKN3212
424-
12
wIF
D A
2
CB
2
2
BC
Slope deflection equations are:
2EI3
2EI
3
4
3
32
3
EI2
L
32
L
EI2FM
1-------EI3
2EI
3
23
3
3
EI2
L
32
L
EI2FM
B
B
ABBABA
B
B
B A AB AB
7/30/2019 SD Method VTU Notes
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5EI3
2EI
3
43
32
3
2EI
L
32
L
EI2FM
4--------EIEI232
24
I2E232
2L
EI2FM
3---------EIEI232
24
I2E232
2L
EI2FM
C
C
DCCDCD
BC
BC
BCCBCB
CB
CB
CBBCBC
6--------EI
3
2EI
3
2
3
3
3
EI2
L
32
L
EI2FM
C
C
CDDCDC
The unknown are &, C,B areconditionsboundarythey Accordingl
090MMMM
0303
MM
3
MM,e.i
030HH
0MM
0MM
DCCDBA AB
DCCDBA AB
D A
CDCB
BCBA
7032EI3
2EIEI
3
10
EIEI232EI3
2EI
3
4MM,Now
CB
CBBBCBA
7/30/2019 SD Method VTU Notes
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8032EI3
2EI
3
10EI
EI3
2EI
3
4EIEI232MM
CB
CBCCDCB
9045EI3
4EIEI
90EI3
8EI2EI2
90EI3
2EI
3
2EI
3
2
EI3
4EI
3
2EI
3
4EI
3
2EI
3
290MMMM
CB
CB
C
CBBDCCDBA AB
From (7) & (9)
10-------0109EI3EI3
17
045EI3
4EIEI
064EI3
4EI2EI
3
20
CB
CB
CB
By (8) and (9)
11--------0109EI3
17EI3
045EI3
4EIEI
064EI3
4
EI3
20
EI2
CB
CB
CB
By (10) & (11)
071.166EI317
208
071.57EI3EI
17
27
0109EI3EI3
17
B
CB
CB
88.40208
31771.166EI B
7/30/2019 SD Method VTU Notes
http://slidepdf.com/reader/full/sd-method-vtu-notes 86/89
From (10)
88.40EI3
17109
3
1EI BC
From (9)
07.954588.4088.404
3
45EIEI4
3
EI CB
Thus 07.95EI,88.40EI,88.40EI CB
Substituting these values in slope deflection equations
KNM12.3607.953
288.40
3
2M
KNM88.807.953
288.40
3
4M
KNM88.888.4088.40232M
KNM88.888.4088.40232M
KNM88.807.95
3
288.40
3
4M
KNM12.3607.953
288.40
3
2M
DC
CD
CB
BC
BA
AB
7/30/2019 SD Method VTU Notes
http://slidepdf.com/reader/full/sd-method-vtu-notes 87/89
To find the reaction consider the free body diagram of the frame
Reactions:
Column AB
KN153
12.3688.8H A
Beam AB
KN484
2
442488.888.8
RB
KN4848424R
C
Column CD
KN153
12.3688.8HD
7/30/2019 SD Method VTU Notes
http://slidepdf.com/reader/full/sd-method-vtu-notes 88/89
CheckΣH = 0H A + HD +P = 0
-15 – 15 + 30 = 0Hence okay