cen321-02t

Consistent Deformation Method 1

2 (Consistent Deformation)

(Force Method)


) (SI)) (Base Structure)

(Indeterminate) 1 (Determinate)

2 (Determinate)

Consistent Deformation

1Re1

Re1 1


) FBD

+ ( = SI)

) . (Constrain) ()

)


) SI = 2

) Base Structure

) 2

A B C

A B CRe1 Re2

A B C1 2

1 = 0 2 = 0


1 = 0 --------------------------------------------- 1 =F1 + 11Re1+ 12Re2 = 0

2 = 0 --------------------------------------------- 2 =F2 + 21Re1+ 22Re2 = 0

A B CF2F1

AB CRe1

11Re1 21Re1

AB C Re2

12Re2 22Re2

) (Superposition) 1 , 2 = 0

F1 , F2

111Re1 , 21Re1

212Re2 , 22Re2


A B CRe1 Re2R1

R2

R3 - - )

SI = 3 1 =F1 +11Re1+12Re2 +13Re3 = 0 Re1 , Re2 , Re3 2 =F2 +21Re1+22Re2 +23Re3 = 0

3 =F3 +31Re1+32Re2 +33Re3 = 0

(SI = N)1=F1 + 11Re1+ 12Re2 + + 1NReN = 0

2=F2+ 21Re1+ 22Re2 + + 2NReN = 0Re1 , Re2 , ....,ReN

N =FN + N1Re1+ N2Re2 + + NNReN = 0


.. 1 ()

Consistent Deformation A B

50 kNA B

C8 m 4 mI 3I

15 kN /m

SI = 1


1. SI = 12. Base Structure

Redundant = Ry, B Constrain vB= 0

Deflection B = 03. Constrain

Superposition 2

Deflection B = vB

Deflection B = vB

vB = vB + vB = 0

50 kNA B

C8 m 4 mI 3IR y,B

Base Structure

15 kN /m

50 kNA B C

vB

15 kN /m

AB CvB

Ry, B


vB vB ( Virtual Work)

A B C1

1 vB vB

A B C

-600

-480

-

-

Pt.Load

Uniform Load

(M1)

A B C

8 Ry,B+

Ry, B (M2)

A B C

8

+

1 (m)


Constrain Ry,B = 132.5 kN. A Ry,A = 37.5 kN. , R x,A = 0 kN.

MA = -20 kN.-m.

[ ]0MEI31

883

480831

60012

4832

8821

EI1

dxEI

mMv 1

1B +

+=

= )('

EI384067

vB,' =

ByBy2

B REI3512

832

8R821

EI1

dxEI

mMv ,,'' =

=

=

0REI3

512EI384067

vvv ByBB =+

=+= ,''' ,


.. 2 ()

Consistent Deformation A B ( E I )

A B C

8 m4 m

10 kN /m

SI = 1

Z (Redundant)

A B C10 kN /m

Base Structure


1) SI = 12) Base Structure Redundant = ZA= 0 Deflection A = 03)

- A = A- AB M1=0,- CB M1= 40.x2-10.x22/24) Redundant Force

- A = A- AB M2= Z.x1 ,- CB M2= 0.5.Z.x2

B CA

1

x1 x2

m

M1

10 kN./m.B CA

A

M2

B CAZ

x1 x2

A


Virtual Work A- AB m = 1.x1 ,- CB m = 0.5.x2

5) Constrain Z = -13.3 kN.6) AB MB

MB = -53.3 kN.-m.

EI3853dxEIx502x10x40dxEImM 28

0

2221A /.)]/().()/[(]/[

' ===

+==4

0

8

022112A dxEIx50Zx50dxEIx1xZdxEImM ]/).().[(]/)()[(]/[

''

EIZ64 /=

0AAA =+= '''

BAVBZ

MB


EX. Consistent Deformation 1.) A 2.) SBD. BMD. ( EI )

50 kN.A DC

6 m. 3 m.

B E100 kN.

6 m. 3 m.

SI = 2

Base Structure

50 kN.A DCB E

100 kN.

X1 X2


1) SI = 22) Base Structure B ,D = X1 X23) - B B = 0- D D = 04)

B D 5) X1 11 . X1 21 . X1X2 12 . X2 22 . X2

50 kN.A DCB E

100 kN.

B

D

A DCB

E

1 X1

1121

A DCB

E

1 X2

1222


6) Virtual Work ( (M) , 1 B (m1) 1 C (m2))

M

m1 1 B

m2 1 D

BA C D E

-1,800 -900-450 -150

0

60

159 6 0


6.1) B D (B D)

6.2) B D X1(11 21 )

6.3) B D X2(12 22 )

== dxEIL

mMdxEIL

mM 2D1B'' .....................

== dxEIL

mmdxEIL

mm 12211111 .....................

== dxEIL

mmdxEIL

mm 22222112 .....................


7) 6 6.1 , 6.2 6.3 B = -27,000 / EI D= -121,725 / EI11 = 72 / EI 21 = 234 / EI12 = 234 / EI 22 = 1,125 / EI

8) B D = 0

0EIX1251EIX234EI7251210XX

0EIX234EIX72EI000270XX

21

222121DD

21

212111BB

=++=++=

=++=++=

/,//,..................

///,..................

'

'


9) 8 X1 X2

X1 = 72.1 kN. X2 = 93.2 kN.

10)

VA = -15.3 kN. MA = 30.6 kN.-m. 1


11) SFD. BMD.

SFD.(kN.)

BMD.(kN.-m.)

2

A B C D E

15.3 43.2

50.056.8

30.6109.3

61.1150

+ +-

+

-

--+


EX. Consistent Deformation 1.) 2.) FBD. BMD. ( EI )

10 kN./m.30 kN.

A

B C

10 m.

8 m.

SI = 1


1) SI = 1 - (Redundant) 1 - 1

2) Base Structure A C

3) 2

3.1) ( C X1)

3.2) (vC= 0 Deflection C = 0)

10 kN./m.30 kN.

A

B C

X1

Base Structure


1 X1

4) C 4.1) vC4.2) 11 . X1 11 C X1

10 kN./m.30 kN.

A

B C vC

A

BC

1

11


5) vC 11 Virtual Work ( (M) 1 C (m))

A

B C

A

B C

B C

A620

320

320

A

B C

8

8

M m

- +

+-


= dxEI

mMvC'

= dx

EImm

11

5.1) C (vC)

5.2) C X1 (11)

AB BC A B B C

M (kN.-m.) -620 -320 -320 0M (kN.-m.) 8 8 8 0Integrate (M . m) -37,600.0 -5,120.0 -42,720.0 /EIIntegrate (m . m) 640.0 170.7 810.7 /EI

X = 52.7 kN.


6) C = 0

vC 11 (X1) 52.7 kN.

7. HA = -30.0 kN.VA = 27.3 kN.MA = 198.4 kN-m.

0Xvv 111CC =+='

=== 0M0F0F yX ,,10 kN./m.

30 kN.

A

B C

52.7 kN.

HAVA

MA


8) () SFD. BMD.

B B

A A

CC

27.3

30.0 52.7

101.6101.6 138.9

198.4

SFD. (kN.) BMD. (kN.-m.)

++

+-

-+


2.3 Consistent Deformations (Truss)

Indeterminate Truss 3 1. Redundant Forces 2. Redundant Forces (Truss)3. Redundant Forces 2

1


2 Truss 2

- Condition Redundant Forces (1 ) 0

P1 P2

P1 P2

X 1X 1

1= 0


EX. Consistent Deformation (EA / L = 30,000 k / ft.)

24 ft.64 k

a

B C D

bc

d e

24 ft. 24 ft. 24 ft.

32 ft.

SI = 1


1) SI = 1 ( c)

2) Primary Structure Redundant Forces

3) c (1)- 64 k

4) c (11)- 1 X1

B C D

24 ft.64 k

a

B C D

b c d e

24 ft. 24 ft. 24 ft.

32 ft.

X1

Primary Structure

64 ka b c d e

1

1

a

B C D

b c d e

11

== nNAEL

nAENL

1 1'

== AELN

nAENL

12

11


N n N.n n2(k) (k) (k2 ) (k2 )

ab 36 0.375 13.5 0.140625bc 36 0.375 13.5 0.140625cd 12 0.375 4.5 0.140625de 12 0.375 4.5 0.140625BC -24 -0.75 18.0 0.5625CD -24 -0.75 18.0 0.5625aB -60 -0.625 37.5 0.390625Bb 64 0 0 0Bc -20 0.625 -12.5 0.390625Cc 0 0 0 0cD 20 0.625 12.5 0.390625Dd 0 0 0 0De -20 -0.625 12.5 0.390625

122 3.25

5) Condition c

6)

(64 k) (X1) Primary Structure

= N + n . X1 ()

0XL

AE253

LAE

122

X0

1

11111

=+

+==

.

...........'

...

kN5437253122

X1 =

=


EX. Consistent Deformation (EA / L = 30,000 k / ft.)

24 ft.64 k

a

B C D

bc

d e

24 ft. 24 ft. 24 ft.

32 ft.

SI = 2


1) SI = 2 ( e Cd)


3) e (1) Cd (2)

- 64 k Primary Structure

24 ft.64 ka

B C D

bc

de

24 ft. 24 ft. 24 ft.

32 ft. X1

X2

Primary Structure

==

==

222

111

nNAEL

nAENL

1

nNAEL

nAENL

1

'

'

64 ka

B C D

bc

de

1

2

N


4) e (11) Cd (21) 1 X1

5) e (12) Cd (22) 1 X2

=

=

=

=

)(

)(

2121

21

211

111

nnAEL

nAE

Ln1

nAEL

nAE

Ln1

=

=

=

=

)(

)(

222

222

2112

12

nAEL

nAE

Ln1

nnAEL

nAE

Ln1

n1

a

B C D

bc

de

11

1 X1

21

n2

a

B C D

bc

de

12

1 X2

22


N n1 n2 N.n1 N.n2 n12 n1.n2 n22(k) (k) (k) (k2) (k2) (k2) (k2) (k2)

ab 36 1 0 36 0 1 0 0bc 36 1 0 36 0 1 0 0cd 12 1 -0.6 12 -7.2 1 -0.6 0.36de 12 1 0 12 0 1 0 0BC -24 0 0 0 0 0 0 0CD -24 0 -0.6 0 14.4 0 0 0.36aB -60 0 0 0 0 0 0 0Bb 64 0 0 0 0 0 0 0Bb -20 0 0 0 0 0 0 0 Cc 0 0 -0.8 0 0 0 0 0.64cd 0 0 1.0 0 0 0 0 1cD 20 0 1.0 0 20 0 0 1Dd 0 0 -0.8 0 0 0 0 0.64De -20 0 0 0 0 0 0 0

1 = 96 2 = 27.2 11 = 4 12 = 21 -0.6 22 = 4 x L/AE


6) Condition - e ; 1 = 0- Cd ; 2 = 0

Fi = N + n1X1 + n2X2

+

=

=

2

1

2

1

2221

1211

2

1

XX

00

'

'

=

+

00

XX

460604

AEL

22796

AEL

2

1

..

.

=

=

kk

64106025

227096

460604

XX 1

2

1

.

...

..


EX. Consistent Deformation ( EA / L = 30,000 k / ft. , = 6.510-6 in / in /1F)

24 ft.

a

B C D

b c d e

24 ft. 24 ft. 24 ft.

32 ft.

2 50 F

SI = 1


1) SI = 1 ( Cd)


3) Cd (1) BC CD

= )(' Ltn1 11

24 ft.

a

B C D

bc

de

24 ft. 24 ft. 24 ft.

32 ft.X1

Primary Structure

a

B C D

bc

de

1

tL


4) Cd (11) 1 X1 Primary Structure

== )( 211111 nAEL

nAE

Ln1

n1

a

B C D

bc

de

1 X1

11


5) Condition Cd

6)

(64 k) (X1) Primary Structure

= N + n . X1 ()

0X4AEL

004680

X0

1

11111

=

+

+==

.

........'

kips135L

AE4

004680X1 .

.==

L tL n1 (n1)tL n2(ft.) (ft.) (k) (k2 ) (k2)

ab 24 0 0 0 0bc 24 0 0 0 0cd 24 0 -0.6 0 0.36de 24 0 0 0 0bc 24 0.0078 0 0 0CD 24 0.0078 -0.6 -0.00468 0.36aB 40 0 0 0 0Bb 32 0 0 0 0Bc 40 0 0 0 0Cc 32 0 -0.6 0 0.36Cd 40 0 1.0 0 1.0cD 40 0 1.0 0 1.0Dd 32 0 -0.8 0 0.64De 40 0 0 0 0

-0.00468 4.0


EX. Consistent Deformation

4 m.

100 kN.A B C

D

E4 m.

3 m.

1 m.

SI = 2


1) SI = 2 - (Redundant) 2 - 2


3) Primary Structure 2

3.1) ()- C X1- CE X2

4 m.

100 kN.A B C

D

E4 m.

3 m.

1 m.

Primary Structure

X1

X2

100 kN.A B C

D

E

uCE

uC


3.1) - C

(uC = 0 )- CE (CE = 0 )

4) 3.1 3.2 4.1) C (1)

- 100 kN (uC )

- X1 11 . X1- X2 12 . X2

A B C

D

E 1 X1

11

21

1

A B C

D

E 1 X2

12

221


4.2) CE- 100 kN. (BC )- X1 21 . X1- X2 22 . X2

5) uC , BC , 11 , 12 , 21 22 () (N ) 1 C (n1 ) 1 CE (n2 )


L L/EA N n1 n2 Nn1L/EA Nn2L/EA n1n1L/EA n1n2L/EA n2n2L/EA F

AB 4 4/AE -66.67 1 0.24 -266.7/AE -62.9/AE 4/AE 0.9/AE 0.2/AE 0BC 4 4/AE -66.67 1 0.24 -266.7/AE -62.9/AE 4/AE 0.9/AE 0.2/AE 0AD 5 5/AE 83.33 0 -1.18 0 -491/AE 0 0 6.9/AE 40.79CD 5 5/AE 83.33 0 -1.18 0 -491/AE 0 0 6.9/AE 40.79AE 5.66 5.66/AE 0 0 1 0 0 0 0 5.7/AE 36.10CE 5.66 5.66/AE 0 0 1 0 0 0 0 5.7/AE 36.10BD 3 3/AE -100.0 0 0 0 0 0 0 0 -100DE 1 1/AE 0 0 -1.41 0 0 0 0 2.0/AE -51.06

-533.3/AE -1107.8/AE 8/AE 1.9/AE 26.7/AE


5.1) C CE (uC BC )

5.2) C CE 1 (11 , 12 , 21 22 )

AELnn

AE

LnnAE

Lnn 2222

212112

1111

=

==

= ...,

AELnN

AE

LnNu 2BC

1C

=

= '' ...


6) C CE 0

7)

- A = -58.16 kN.- A = 50.00 kN.- C = 50.00 kN.

0XX0XXuu 222121BCBC212111CC =++==++='' ........

................ kN1036XkN1658X 21 ==


8) (F )

(N ), X1 (n1 ) X2 (n2)

2211 XnXnNF ++=

cen321-02t

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