1000 the Stiffness Method

8/13/2019 1000 the Stiffness Method

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Lecture 12: THE STIFFNESS METHOD

Member End-Actions And Reactions

We would now like to develop the matrix equations for determining member end actions

and reactions using stiffness methods. The procedure closely follows the procedure

developed for the flexibility method. First member end-actions due to the external loads,

denoted by {AML}, are determined. Then the contributions of the member end-actions

caused by unit displacements multiplied by the now known actual displacements are added.

Thus

Here:

{AM

}is the vector of member end actions on the actual structure

{AML} is the vector of member end actions due to the external loads on the restrained

structure.

{AMD} is the matrix of member end-actions due to unit values of the displacements onthe restrained structure

DAAA MDMLM


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A similar equation can be written for the reactions, i.e.,

Here:

{AR} is the vector reactions in the actual structure

{ARL} is the vector of reactions due to the external loads on the restrained structure{ARD}is the matrix of reactions due to unit values of the displacements on the

restrained structure

DAAA RDRLR


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Example

Consider again the two span beam previously discussed and determine

The shearing forceAM1 at endB of memberAB.

The bending moment AM2 at endB of memberAB.

The shearing forceAM3 at endB of memberBC.

The bending momentAM4 at endB of memberBC.

The forceAR1 at supportA. The coupleAR2 at supportA.

ForceAR3 at supportB.

ForceAR4 at support C.

PP

PP

PLM

PP

3

2

1 2


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22

2

31

PAP

PA

MLML

84

242

2

2

PLA

PL

L

bPaA

MLML

Once again, member end actions in the restrained structure will be denoted by a vector

{AML}. Keep in mind the beams below are really one restrained beam with the

cantilever support in the middle of the beam. The member end-actions are treated as if

they were support reactions for each beam segment.


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231211

66

L

EIA

L

EIA

MDMD

L

EIA

L

EIA

MDMD

444121

We can use the same approach when we analyze the restrained structure after unit

displacements are applied. The corresponding member end actions, denoted by the

matrix {AMD} are given below when a unit rotation is applied atB.


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23212

60

L

EIAA

MDMD

L

EIAA

MDMD

20 4222

LL

L

L

EIAMD

2

6

0

4

6

4

06

2

The corresponding member end actions associated with a unit rotation is applied at C

are

thus


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L

LP

EI

PL

LL

L

L

EI

L

LPAM

36

64

20

5

565

17

112

24

66

04

06

4

2

8

8

2

2

DAAAMDMLM

5

17

112

2

EI

PLD

The superposition principle leads to the following matrix equation

from a previous solution

which leads to


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22431

PP

PAPAPA

RLRLRL

2432

PA

PLA RLRL

PAAA RLRLRL 2

3333

Turning our attention to beam reactions in the restrained structure, denoted by a vector

[ARL] once again the beams below are really one restrained beam with the cantilever

supports replaced with forces and moments. The two beams are treated as cantilever-

cantilever beams.


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241231211

666

L

EIA

L

EIA

L

EIA

RDRDRD

23121

62

L

EIA

L

EIA

RDRD

031

RDA

We also analyze the restrained structure after unit displacements are applied. The

corresponding reactions, denoted by the matrix [ARD] are given below when a unit

rotation is applied atB.


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0322212 RDRDRD AAA

24223232

66

L

EIA

L

EIAA

RDRDRD

66

60

02

06

2

L

L

EIA

RD

The corresponding reactions associated with a unit rotation is applied at C are

thus


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64

69

31

107

565

17

112

66

60

02

06

2

6

4

4

2

2

LP

EI

PLL

L

EILPAR

DAAA RDRLR

5

17

112

2

EI

PLD

The superposition principle leads to the following matrix equation

from a previous solution

which leads to


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Example 2

For the two span beam previously discussed determine the unknown displacement at joints

B and C. In addition find the member end-actions as well as the reactions.

The unknowns are identified as


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888

211

PLLPLPA

DL

48

LPADL

Using the following restrained structure

The actions in the restrained structure due to applied loads corresponding to thepreviously identified displacements are

thus

22

22

PPA

DL


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221 6LEIS

L

EI

L

EI

L

EISSS

844111111

Applying a unit rotation atB in the restrained structure, i.e.,

leads to the following stiffness coefficients


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212

6

L

EIS

322

12

L

EIS

Applying a unit translation at Cin the restrained structure, i.e.,

leads to the following stiffness coefficients


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63

3422

3L

LL

L

EIS

0DA

DLD AASD 1

LEI

PLLP

LL

L

EI

LD

13

6

2404

843

36

30

2

2

Thus

2

1

43

36

30 LL

L

EI

LS

which leads to

With

and

then


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PPAML 21

1 48

12

PLLP

AML

Once again, member end actions in the restrained structure will be denoted by a vector

[AML]. Keep in mind the restrained beam is treated as two cantilever beams side by

side.

L

PAML

44

thus


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211

6

L

EIA

MD

L

EIAMD

421

We analyze the restrained structure after unit displacements are applied. The

corresponding member end actions, denoted by the matrix [AMD] are given below

when a unit rotation is applied atB.

When a unit translation is applied at C then

012 MDA022 MDA


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L

PDAAA

MDMLM7

23

20

Using superposition and previous results leads to

02

0322 LL

EI

AMD

Thus


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PP

ARL 2

11

48

1

2

PLLPA

RL

88

24

PLLPA

RL

Next we turn our attention to beam reactions in the restrained structure subject to the

applied loads, denoted by a vector [ARL]. Keep in mind the restrained beam is treated

as two cantilever beams side by side.

2

3

22

213

PPPA

RL


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02212 RDRD AA

3332323212120L

EI

L

EIAAARDRDRD

LL

L

L

L

EIA

RD

3

60

0

03

2

2

2

3

2426L

EIARD

When a unit translation is applied at C we obtain

thus


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L

LPDAAA

RDRLR

3

43

4

17

20

Using superposition and previous results leads to

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1000 the Stiffness Method