slope deflection method

Slope Deflection Method

By

Dr. Kota SrinivasuDr. Kota Srinivasu

Shear force at a section: The algebraic sum of the vertical forces acting

on the beam either to the left or right of the section is known as the

shear force at a section.

Sign Convention for shear force

F

F

F

F+ ve shear force

- ve shear force

Bending moment (BM) at section: The algebraic sum of the momentsof all forces acting on the beam either to the left or right of thesection is known as the bending moment at a section

Bending Moment (BM): The moment which causes the

bending effect on the beam is called Bending Moment. It is

generally denoted by M or BM.

F+ ve shear force

Sagging bending moment [Positive bending moment]

Hogging

bending moment

[Negative bending moment ]Dr. Kota Srinivasu

Type of load

SFD/BMD

Between point loads OR for no

load region

Uniformly distributed load

Uniformly varying load

Shear Force Diagram

Horizontal line Inclined line Two-degree curve (Parabola)

Bending Moment Diagram

Inclined line Two-degree curve (Parabola)

Three-degree curve (Cubic-

parabola)

Table: Variation of Shear force and bending moments

Point of Contra flexure [Inflection point]:

It is the point on the bending moment diagram wherebending moment changes the sign from positive to negativeor vice versa.

It is also called Inflection point. At the point ofinflection point or contra flexure the bending momentchanges its sign and magnitude is zero.

Diagram parabola)

Dr. Kota Srinivasu

Dr. Kota Srinivasu

DEFLECTION OF BEAMS

AB

w/unit

2 3

2 3B BWL WLy

EI EI = =

W

3 4wL wLy = =

8

M

6 8B BwL wLyEI EI

= =

2

2B BML MLyEI EI

= =

Dr. Kota Srinivasu

w/unit length

w/unit length

3 4

24 30B BwL wLy

EI EI = =

3 3 3

4 4 46 24 8

11

BwL wL wLEI EI EI

wL wL wL

= =

9

A BCa L-a

4 4 4118 30 120BwL wL wLyEI EI EI

= =

( )C B B B Cy y L a = = +

Dr. Kota Srinivasu

BAa L-a

w

w/unitA B

C

2 3

3 2 2

2 3( )

( ) ( )3 2 2 3

C B C

B C C

Wa WayEI EI

y y L a

Wa Wa Wa aL a LEI EI EI

= = =

= +

= + =

3 4

6 8C B Cwa way

EI EI = = =

10

A

C

MB

AC

B4 3 3

6 8

( ) ( )8 6 6 4B

EI EIwa wa wa ay L a L

EI EI EI= + =

2

2 22

( ) ( )2 2 2

C B C

B

Ma MayEI EI

Ma Wa Ma ay L a LEI EI EI

= = =

= + = Dr. Kota Srinivasu

a L-a

W

3 4

4 3

3

24 30

( ) ( )30 24

( )24 5

B C C

B C C

wa wayEI EI

wa way y L a L aEI EI

wa aLEI

= = =

= + = +

=

2 3WL WLy = = =

11

L/2L/2

a b

W

16 48A B CWL WLy

EI EI = = =

2 2

( ) ( )6 6

( )3 3

A B

C C

Wab L b Wab L aL EI L EI

Wab b a Wa byL EI L EI

+ += =

= =

Dr. Kota Srinivasu

w/unit length

w/unit length

3 3

max

524 384A cwL wLy y

EI EI = = =

3

24BwL

EI =

3 37 8360 360A B

wL wLEI EI

= =

ML MLM3 6A BML MLEI EI

= =

3 6

6 3

A BA

A BB

M L M LEI EI

M L M LEI EI

= +

= +

AM BMAM

BM

3 6

6 3

A BA

A BB

M L M LEI EI

M L M LEI EI

=

= Dr. Kota Srinivasu

FIXED END MOMENTS

STRUCTURE Sagging BM Clockwise momentM MA B MFAB FBAM

L/2

W

L/2 8 8 8 8WL WL WL WL

B C

13

w/unit

a b

W

2 2 2 2

12 12 12 12wL wL wL wL

2 2 2 2

2 2 2 2Wab Wa b Wab Wa b

L L L L

Dr. Kota Srinivasu

STRUCTURESagging BM Clockwise moment

M MA B MFAB FBAM

w/unit length 2 2 2 2

30 20 30 20wL wL wL wL

w/unit length

965 2wL

965 2wL

965 2wL

965 2wL

L/2 L/2

M

4 4 4 4M M M M

b

M2

)2(L

baMb

2

)2(L

abMa

a

2)2(

LbaMb

2

)2(L

abMa

96

96

96

96

Dr. Kota Srinivasu


M MA B MFAB FBAM

312 AE I

L

312 AE I

L

6 6 6 6EI EI EI EI

2 2 2 26 6 6 6A A A AEI EI EI EI

L L L L

15

A=1

4 2 4 2A A A AEI EI EI EIL L L L

2 2 2 26 6 6 6B B B BEI EI EI EI

L L L L

3

12LEI B

3

12LEI B

2

6L

EI A2

6L

EI A

Dr. Kota Srinivasu


M MA B MFAB FBAM

B=1 2 4 2 4B B B BEI EI EI EIL L L L

2

6L

EI B26

LEI B

16

2 2 2 26 ( ) 6 ( ) 6 ( ) 6 ( )A B A B A B A BEI EI EI EI

L L L L

Dr. Kota Srinivasu

Dr. Kota Srinivasu

mkNwLMFAB ====

======== 4012

43012

221

Fixed End Moments:

Slope-Deflection Method

A BC

mkNL

bWaM

mkNL

WabM

mkNM

FCB

FBC

FBA

====

========

====

========

====

====

485

32100

725

32100

4012

4301212

2

2

22

2

2

2

21

2

2

Dr. Kota Srinivasu

(((( )))) (((( ))))

(((( )))) (((( ))))40

24

240225.040

424022

1

1

1

1

B

BBAFBABA

B

BBAFABAB

EI

IELEI

MM

EI

IELEI

MM

++++====

++++====++++++++====

++++====

++++====++++++++====

Slope-Deflection Equations:

(((( )))) (((( ))))

(((( )))) (((( )))))5.0(48

525.124822

)(72

25

25.127222

2

2

2

2

B

BCBFBACB

B

BCBFABBC

B

EI

IELEI

MM

EI

IELEI

MM

++++====

++++====++++++++====

++++====

++++====++++++++====

Dr. Kota Srinivasu

mkNEIM

mkNEI

EIEIM

MomentsFinal

BA

BAAB

=+=

=

++=++=

0.561640

0.32165.0040)5.0(40

:

( )EIEI

EI

EIEIMM

BBB

BBBCBA

16322322

072400

===

=++=+

Equilibrium Equations:

mkNEI

XEIM

mkNEI

EIM

mkNEI

EIM

CB

BC

BA

=

+=

=

+=

=+=

56165.048

0.561672

0.5640

Dr. Kota Srinivasu

Dr. Kota Srinivasu


100 kN20 kN/m

4m 2m 3m 1.5m

I1.25 I

A B C

30 kN/m

D

mkNM

mkNL

bWaMmkNL

WabM

mkNMmkNwLM

CD

FCBFBC

FBAFAB

========

====

============

========

====

========

========

5.2275.05.120

485

32100;72

532100

4012

430;40

12430

12

2

2

22

2

2

2

22

2

2221

Fixed End Moments:

Dr. Kota Srinivasu

(((( )))) (((( ))))

(((( )))) (((( ))))40

24

240225.040

24

24022

1

1

1

1

B

BABAFBABA

B

BABAFABAB

EI

IELEI

MM

EI

IELEI

MM

++++====

++++

++++====++++++++====

++++====

++++

++++====++++++++====


(((( )))) (((( ))))

(((( )))) (((( )))))5.0(48

25

25.124822)5.0(72

25

25.12722240

2

2

2

2

CB

CBCBFBACB

CB

CBCBFABBC

B

EI

IELEI

MM

EI

IELEI

MM

EI

++++++++====

++++

++++====++++++++====

++++++++====

++++

++++====++++++++====

++++====

Dr. Kota Srinivasu

( )( )

EI

EIMMEI

EI

EIEIEIMM

CB

CBCDCB

CBCB

CBBBCBA

644)1(2

)2(5.255.0

05.225.0480

)1(325.02325.02

05.072400

=+

=+

=++=+

=+=+

=+++=+


EIEIEIeqFrom

EI

EIEI

EI

C

B

BCB

CB

285.3857.255.05.25)2(

57.25

5.895.35.255.0)2(

644)1(2

==

=

==+

=+

Dr. Kota Srinivasu

mkNEI

EIM

mkNEI

EIEIM

MomentsFinal

BA

BAAB

=+=

=

++=++=

57.6557.2540

22.27

57.255.0040)5.0(40

:

mkNEIEI

XEIM

mkNEI

XEI

EIM

EI

CB

BC

=

+=

=

+=

5.22285.3857.255.048

57.65285.385.057.2572

Dr. Kota Srinivasu

Dr. Kota Srinivasu

Fixed End Moments:


100 kN20 kN/m

4m 2m 3m 1.5m

I1.25 IA

B C

30 kN/m

E

30 kN

1mD

mkNMmkNMCD

mkNL

bWaMmkNL

WabM

mkNMmkNwLM

CE

FCBFBC

FBAFAB

====

=

===

==

=

==

==

5.2275.05.12030130

485

32100;72

532100

4012

430;40

12430

12

2

2

22

2

2

2

22

2

2221

Dr. Kota Srinivasu

(((( )))) (((( ))))(((( ))))

(((( )))) (((( ))))(((( ))))5.040

24

240225.040

24

24022

1

1

1

1

BA

BABAFBABA

BA

BABAFABAB

EI

IELEI

MM

EI

IELEI

MM

++++++++====

++++

++++====++++++++====

++++++++====

++++

++++====++++++++====


(((( ))))(((( )))) (((( ))))

(((( )))) (((( )))))5.0(48

25

25.124822)5.0(72

25

25.1272225.040

2

2

2

2

CB

CBCBFBACB

CB

CBCBFABBC

BA

EI

IELEI

MM

EI

IELEI

MM

EI

++++++++====

++++

++++====++++++++====

++++++++====

++++

++++====++++++++====

++++++++====

Dr. Kota Srinivasu

( )

( )( )

( )EI

EI

EIEIEIMMEI

EIMM

CBACBA

CBBABCBA

BA

BAADAB

)2(325.025.0325.025.0

05.0725.0400

)1(105.0

0305.0400

=++=++

=++++=+

=+

=+++=+


( )

EIEIEI

andequaionsSolvingEI

EIMMEI

CBA

CB

CBCECB

75.38;

50.26;

25.3)3()2(),1(

)3(5.255.0

05.225.0480

===

=+

=++=+

Dr. Kota Srinivasu

mkNEIEI

XEIM

mkNEIEI

EIEIM

MomentsFinal

BA

BAAB

=

++=

=

++=++=

875.6450.2625.35.040

0.30

50.265.025.340)5.0(40

:

mkNEIEI

XEIM

mkNEI

XEI

EIM

mkNEIEI

XEIM

CB

BC

BA

=

+=

=

+=

=

++=

5.2275.3850.265.048

875.6475.385.050.2672

875.645.040

Dr. Kota Srinivasu

Dr. Kota Srinivasu

30 kN/m

4m

IA B C

6m

1.5 I

40 kN/m

425 000,35;/1022;3;2

cmImmNEmmmmmm CBA

==

===

24

444

282

5

425

107

105.3100

000,35

/1021000

/1000

102

000,35;/102

mkNEI

mmI

mkNmkNE

cmImmNE

=

=

=

=

=

==

Dr. Kota Srinivasu

( )

( );75.13

41000

321076

124306

12

;25.664

1000321076

124306

12

2

42

21

1211

2

42

21

1211

mkNL

EILwM

mkNL

EILwM

BAFBA

BAFAB

=

+

=

+=

=

+

=

+=

Fixed End Moments:


( )

( );50.89

61000

231075.16

206406

20

;50.306

1000231075.16

306406

30

41212

2

42

22

2222

2

42

22

2222

1

mkNL

EILwM

mkNL

EILwM

L

CBFBC

CBFBC

=

+

=

+=

=

+

=

+=

Dr. Kota Srinivasu

(((( )))) (((( ))))

(((( )))) (((( )))) BBBAFBABAB

BBAFABAB

EI

EIIELEI

MM

EI

IELEI

MM

2

75.1324

275.13225.025.66

4225.6622

1

1

1

1

++++====

++++====++++++++====

++++====

++++====++++++++====


(((( )))) (((( ))))

(((( )))) (((( ))))B

BCBFBACB

B

BCBFABBC

EI

IELEI

MM

EI

IELEI

MM

5.05.896

5.125.89225.30

26

5.125.3022

2

2

2

2

++++====

++++====++++++++====

++++====

++++====++++++++====

Dr. Kota Srinivasu

EI

EIEI

EIEIMM

B

BB

BBBCBA

375.8

75.16275.162

05.3075.130

====

========

====++++++++====++++


mkNEI

EIEIM BAB ====

++++====++++==== 06.62375.85.025.665.025.66

:MomentsFinal

mkNEI

EIM

mkNEI

EIM

mkNEI

EIM

mkNEI

EIEIM

CB

BC

BA

BAB

====

++++====

====

++++====

====

++++====

====

++++====++++====

69.93375.85.05.89

125.22375.85.30

125.22375.875.13

06.625.025.665.025.66

Dr. Kota Srinivasu

Dr. Kota Srinivasu

425 000,35;/1022;3;2

cmImmNEmmmmmm CBA

==

===

24

444

282

5

425

107

105.3100

000,35

/1021000

/1000

102

000,35;/102

mkNEI

mmI

mkNmkNE

cmImmNE

=

=

=

=

=

==

Dr. Kota Srinivasu

(((( ))))

(((( ))));75.13

41000

321076

124306

12

;25.664

1000321076

124306

12

2

42

21

1211

2

42

21

1211

mkNL

EILwM

mkNL

EILwM

BAFBA

BAFAB

====

++++

====

++++====

====

++++

====

++++====

Fixed End Moments:

(((( ))))

(((( ))))

;305.120

;50.896

1000231075.16

206406

20

;50.306

1000231075.16

306406

30

2

42

22

2222

2

42

22

2222

1

mkNMM

mkNL

EILwM

mkNL

EILwM

CBFCB

CBFBC

CBFBC

============

====

++++

====

++++====

====

++++

====

++++====

Dr. Kota Srinivasu

(((( )))) (((( ))))

(((( )))) (((( ))))5.122

75.1324

275.13225.025.66

4225.6622

1

1

1

1

BBBAFBABA

B

BBAFABAB

IEEI

EIIELEI

MM

EI

IELEI

MM

++++====

++++====++++++++====

++++====

++++====++++++++====


(((( )))) (((( ))))

(((( )))) (((( )))))5.0(5.89

26

5.125.8922)5.0(5.30

26

5.125.3022

2

2

2

2

CB

CBCBFBACB

CB

CBCBFABBC

EI

IELEI

MM

EI

IELEI

MM

++++++++====

++++

++++====++++++++====

++++++++====

++++

++++====++++++++====

Dr. Kota Srinivasu

(((( ))))(((( ))))

(((( ))))

EI

EIMMEI

EI

EIEIEIMM

CB

CBCECB

CBCB

CBBBCBA

)2(5.595.0

0305.05.890

)1(75.165.0275.165.02

05.05.3075.130

====++++

====++++++++====++++

====++++====++++

====++++++++++++====++++


EIEIEIEIEI

EI

EI

EI

CB

CB

CB

785.7257.265.050.59;57.265.393

5.595.0

5.334)1(2

================

====++++

====++++

Dr. Kota Srinivasu

mkNEIEI

EIM

mkNEI

EIM

mkNEI

EIEIM

MomentsFinal

BC

BA

BAB

=

+=

=

+=

=

+=+=

52.40785.725.057.265.30

52.4057.2675.13

965.5257.265.0405.025.66

:

mkNEIEI

EIMCB =

+= 30785.7257.265.05.89

Dr. Kota Srinivasu

Dr. Kota Srinivasu

wL

mkNWLM

mkNWLM

FBA

FAB

=

==

=

==

430

;108

4208

;108

4208

22

1

1

Fixed End Moments:

4m

4m

3m

30 kN/m

2 I

1.25 I

10 kN

1 mB C

E

20 kN2 m

mkNMMM

mkNwLM

mkNwLM

BE

FDCFCD

FBC

FBC

==

==

=

==

=

==

101100

4012

43012

;4012

43012

222

2224m

2 I

1.25 I

A

D

20 kN

2 m

Dr. Kota Srinivasu

(((( ))))(((( ))))

(((( )))) (((( ))))++++++++====++++++++====

++++====

++++++++====

++++====

++++====

++++++++====

26

224022

75.02104

32275.010

43

42210322

2

2

1

1

11

1

CBCBFBCBC

BBAFBABA

B

BBAFABAB

IELEI

MM

EILEI

MM

EI

IELL

EIMM


(((( )))) (((( ))))

====

++++++++====

====

++++++++====

++++++++====++++++++====

++++++++====

33

325.12322

332

325.12322

24022)2(40

6

33

3

33

3

2

2

2

CDCFDCDC

CDCCDCD

CBCBFCBCB

CB

IELL

EIMM

IELL

EIMM

EILEI

MM

EIL

(((( ))))

(((( ))))

++++++++++++====++++

====++++

====++++++++++++++++++++

====++++++++

332

35.22400

)1(2075.04

010)2(4075.02100

CCBCDCB

CB

CBB

BEBCBA

EIEIMM

EI

EIEIMMM

equationSway

)2(4083333.06667.3 ====++++EICB

( ) ( )

)3(109306.0833333.075.0

0203

33

325.12

332

325.12

422075.021075.010

0

=+

=+

+

++++

=+++

EI

IEIE

EIEI

XL

onABloadstodueBatMomentMMequationSway

CB

CC

BB

AB

BAAB

EIEIEI CB849.7

;68.11

;39.9

============

(((( ))))(((( ))))

(((( ))))(((( ))))

mkNEIMmkNEIMmkNM

mkNEIMMomentsFinal

CB

BC

BA

AB

====++++====

====++++====

====++++====

====++++====

5.203.2668.11239.940

89.32)68.1139.92(4089.2286.775.039.9210

51.686.775.039.910

(((( ))))(((( )))) mkNM

mkNM

DC

CD

========

========

28.16849.768.1135.2

03.26849.768.11235.2

Dr. Kota Srinivasu

Dr. Kota Srinivasu

( ) ( )

( ) ( ))5.0(20

28

222022)5.0(20

28

222022

1

1

1

1

BA

BABAFBABA

BA

BABAFABAB

EI

IELEIMM

EI

IELEIMM

MehodDeflecionSlope

++=

+

+=++=

++=

+

+=++=


( ) ( )

( ) ( ))75.0(16

28

222022)5.1(16

28

322022)5.0(20

1

1

2

2

B

CBCBFBACB

B

CBCBFABBC

BA

EI

IELEIMM

EI

IELEIMM

EI

+=

+

+=++=

+=

+

+=++=

++=

Dr. Kota Srinivasu

EIEIEIMM

EI

EIM

BA

BBABCBA

BA

BA

AB

44)5.25.0(

)5.1(16)5.0(200)1(205.0

0)5.0(200

=+

+++=+

=+

=++

=


EIEI

EIEIEI

EI

EI

BA

BB

BA

BA

111.235.020

2222.625.2141425.2

1025.05.05.01

)2(450.25.0

==

===

=+

=+

Dr. Kota Srinivasu

mkNEI

EIEIM

mkNEIEI

EIM

EIEIEIEIM

MomentsFinal

BBC

BA

BAAB

==+=

=+=

=+=++=

2222.6

33.25)2222.65.1(16)5.1(16

33.25)2222.6111.235.0(20

0)2222.65.0111.23(20)5.0(20:

mkNEI

EIEIM BCB ==+= 33.11)2222.675.0(16)75.0(16

Dr. Kota Srinivasu

Dr. Kota Srinivasu

Symmetric Frame with symmetric Loading

Half Frame

with

Symmetric

loading/

Boundary

Conditions

Half Frame with

Loading

Symmetric beam/Frame

with anti-symmetric Loading

Dr. Kota Srinivasu

Load Decomposition Superposition: Dividing load on symmetric

structure with Symmetric and anti-symmetric loads

Analysis of the half of the structure is sufficient Dr. Kota Srinivasu

slope deflection method

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