slope def vtu

8/20/2019 Slope Def Vtu

1/89

Chapter-2: Slope Deflection Method

By Prof. H.P.Sudarshan

Sri Siddhartha Institute Of Tech,Tumkur


2/89

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


3/89

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


4/89

Problem can be treated as

The bending moment diagram for the given problem is as below


5/89

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


6/89

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


7/89

)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


8/89

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


9/89

Using these data BM and SF diagram can be drawn.


10/89

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


11/89

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


12/89

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

6EI54EI5267.11

30EI5

2EI

5

467.41MM, And

CB

BCCDCB

Solving (5) and (6) we get

[email protected]

[email protected]

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


13/89

Reactions: Consider free body diagram of beam AB, BC and CD as shown


14/89

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


15/89

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


16/89

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


17/89

There are only two unknown rotations B and C . Accordingly the boundaryconditions are

Now,

0EI5

4EI

5

231.20MM

0EI52EI

152286.49MM

0MM

0MM

CBCDCB

CBBCBA

CDCB

BCBA

Solving these equations we get

ockwise Anticl71.9EI

ockwise Anticl35.31EI

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


18/89

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


19/89

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


B A AB AB 2L

EI2FM

04

EI230- B

1--------EI0.530- B

ABBABA 2L

EI2FM

024

EI230 B

2---------EI30 B


20/89

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

8083.0EI5.0EI2EI5.00EI5.0EI33.13EI5.0EI5.12

0MM

DCB

DCBC

BCCB

0MDC

90EIEI5.033.13 DC


21/89

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


22/89


23/89

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


B A AB AB 2L

EI2FM

08

I3E253.33- B

1--------EI4

353.33- B

ABBABA 2L

EI2FM

028

I3E253.33 B

2--------EI2353.33 B


24/89

CBBCBC 2L

EI2FM

26

I2E222.5- CB

3--------EI32

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


25/89

from equation (6)

iseanticlockwrotation159.8

432.173

125.11

2

3EI C

Substituting 159.8EIand432.17EI CB in the slope deflection equationwe 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


26/89

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


27/89

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


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 2LEI2

FM

24

I5.1E226.67 BC

4---------EI4

3EI

2

326.67 BC


28/89

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


29/89

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


30/89


31/89

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


32/89

)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


33/89

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


34/89

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


35/89


36/89

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


1--------EI2

10

4

EI202

L

EI2FM BBB A AB AB

2-------EI024

EI202

L

EI2FM BB ABBABA


37/89

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


38/89

Using equation (7)

ckwise Anticlo6464

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


39/89

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


40/89

Check:

0HH

0H

D A

Hence okay

Note: Since symmetrical, only half frame may be analysed. Using first threeequations

and taking CB


41/89

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


42/89

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 theboundary 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


43/89

90EI23

EI2

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


44/89

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


45/89

Check:

ΣH = 0H A + HD = 022.5 – 22.5 = 0Hence okay


46/89

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


47/89

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


48/89

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


49/89

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


50/89

Member CD:

righttoleftisHof directiontheindicatessignve-KN103

27.1273.17H DD

Check: ΣH = 0

H A + HD + P = 0

+10 + 10 – 20 = 0

Hence okay


51/89

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


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


52/89

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 conditionsare

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


53/89

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


54/89

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


55/89

Check

ΣH = 0

H A + HD +10 × 4 = 0

-5.20 - 34.81+ 40 = 0

Hence okay


56/89

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


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


57/89

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


0MMMM

04

MM

4

MM,e.i

conditionShear ---0HH

conditionsintJo0MM

0MM

DCCDBA AB

DCCDBA AB

D A

CDCB

BCBA

70EI8

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


58/89

90EI49

EI3EI2

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


59/89

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



60/89

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


61/89

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


62/89

)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,


63/89

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


64/89

have we2by(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


65/89

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


66/89

Column CD:

KN82.44

28.19HD

Check:

ΣH = 0

H A+HD = 0

Hence okay.


67/89

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



68/89

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


0MMMM,e.i

0HH

0MMM

0MM

DCCDBA AB

D A

CECDCB

BCBA


69/89

Now,

7067.41EI375.0EI6.0EI2.20EI6.0EI2.167.141EI375.0EI

0MM

CB

CBB

BCBA

8067.11EI375.0EI2.2EI6.0030EI375.0EIEI6.0EI2.167.41MM, 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


70/89


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


71/89

Check:

ΣH = 0H A + HD = 0Σ = 0

Hence okay


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


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


73/89

L

32

L

EI2FM CBBCBC

25

I5.1E241.67- CB

3---------EI53

EI5

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 theBoundary conditions are:

0)MM(3)M4(M

04

MM

3

MMi.e

0HH

0MM

0MM

DCCDBA AB

DCCDBA AB

D A

CDCB

BCBA


74/89

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


75/89

Reactions: Consider the free body diagram

Member AB:

KN28.113

44.840.25H A


76/89

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


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


L

32

L

EI2FM B A AB AB

4

30

4

I2E215- B

1--------EI75.0EI15- B


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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 Boundaryconditions are


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0180MM4MM3

030

3

MM

4

60MM,e.i

030HH

0MM

0MM

DCCDBA AB

DCCDBA AB

D A

CDCB

BCBA

Now

7015EI75.0EIEI4EIEI230EI75.0EI215MM

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


80/89

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.


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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


82/89

Example: Analyse the frame using slope deflection method and draw the

Bending Moment Diagram.

Solution: Assume sway towards right


83/89

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


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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--------EI3

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


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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


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


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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


88/89

CheckΣH = 0H A + HD +P = 0

-15 – 15 + 30 = 0Hence okay


89/89

slope def vtu

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