Download pdf - m4l29 Lesson 29 The Direct Stiffness Method: Beams (Continued)

7/31/2019 m4l29 Lesson 29 The Direct Stiffness Method: Beams (Continued)

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Version 2 CE IIT, Kharagpur

Module4

Analysis of StaticallyIndeterminate

Structures by the DirectStiffnessMethod


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Lesson29

The Direct StiffnessMethod: Beams

(Continued)


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Instructional Objectives

After reading this chapter the student will be able to1. Compute moments developed in the continuous beam due to support

settlements.

2. Compute moments developed in statically indeterminate beams due totemperature changes.3. Analyse continuous beam subjected to temperature changes and support

settlements.

29.1 Introduction

In the last two lessons, the analysis of continuous beam by direct stiffness matrixmethod is discussed. It is assumed in the analysis that the supports areunyielding and the temperature is maintained constant. However, support

settlements can never be prevented altogether and hence it is necessary tomake provisions in design for future unequal vertical settlements of supports andprobable rotations of fixed supports. The effect of temperature changes andsupport settlements can easily be incorporated in the direct stiffness method andis discussed in this lesson. Both temperature changes and support settlementsinduce fixed end actions in the restrained beams. These fixed end forces arehandled in the same way as those due to loads on the members in the analysis.In other words, the global load vector is formulated by considering fixed endactions due to both support settlements and external loads. At the end, a fewproblems are solved to illustrate the procedure.

29.2 Support settlements

Consider continuous beam ABCas shown in Fig. 29.1a. Assume that the flexuralrigidity of the continuous beam is constant throughout. Let the support B settlesby an amount as shown in the figure. The fixed end actions due to loads areshown in Fig. 29.1b. The support settlements also induce fixed end actions andare shown in Fig. 29.1c. In Fig. 29.1d, the equivalent joint loads are shown. Sincethe beam is restrained against displacement in Fig. 29.1b and Fig. 29.1c, thedisplacements produced in the beam by the joint loads in Fig. 29.1d must beequal to the displacement produced in the beam by the actual loads in Fig.29.1a. Thus to incorporate the effect of support settlement in the analysis it isrequired to modify the load vector by considering the negative of the fixed endactions acting on the restrained beam.


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29.3 Effect of temperature change

The effect of temperature on the statically indeterminate beams has already beendiscussed in lesson 9 of module 2 in connection with the flexibility matrix method.Consider the continuous beam ABCas shown in Fig. 29.2a, in which span BC is

subjected to a differential temperature 1T at top and 2T at the bottom of the beam.Let temperature in span AB be constant. Let d be the depth of beam and EI be the flexural rigidity. As the cross section of the member remains plane afterbending, the relative angle of rotation d between two cross sections at adistance dx apart is given by

( )dx

d

TTd 21

= (29.1)

where is the co-efficient of the thermal expansion of the material. When beamis restrained, the temperature change induces fixed end moments in the beam as

shown in Fig. 29.2b. The fixed end moments developed are,

( )d

TTEIMM

TT 2121

== (29.2)

Corresponding to the above fixed end moments; the equivalent joint loads caneasily be constructed. Also due to differential temperatures there will not be anyvertical forces/reactions in the beam.


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Example 29.1

Calculate support reactions in the continuous beam ABC(vide Fig. 29.3a) havingconstant flexural rigidityEI , throughout due to vertical settlement of supportB , by

mm5 as shown in the figure. Assume GPaE 200= and 44104 mI = .


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The continuous beam considered is divided into two beam elements. Thenumbering of the joints and members are shown in Fig. 29.3b. The possibleglobal degrees of freedom are also shown in the figure. A typical beam elementwith two degrees of freedom at each node is also shown in the figure. For this

problem, the unconstrained degrees of freedom are 1u and 2u . The fixed end

actions due to support settlement are,

2

696 kN.m; 96 kN.mF FAB BA

EIM M

L

= = =

96 kN.m ; 96 kN.mF FBC CBM M= = (1)

The fixed-end moments due to support settlements are shown in Fig. 29.3c.

The equivalent joint loads due to support settlement are shown in Fig. 29.3d. In

the next step, let us construct member stiffness matrix for each member.

Member 1: mL 5= , node points 1-2.

[ ]

1

3

5

6

80.024.040.024.0

24.0096.024.0096.0

40.024.080.024.0

24.0096.024.0096.0

'

1356..

= zzEIk

fodGlobal

(2)

Member 2: mL 5= , node points 2-3.

[ ]

2

4

1

3

80.024.040.024.0

24.0096.024.0096.0

40.024.080.024.0

24.0096.024.0096.0

2413..

2

= zzEIk

fodGlobal

(3)

On the member stiffness matrix, the corresponding global degrees of freedomare indicated to facilitate assembling. The assembled global stiffness matrix is of

order 66 . Assembled stiffness matrix [ ]K is given by,


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[ ]

= zzEIK(4)

Thus the global load vector corresponding to unconstrained degrees of freedomis,

{ }

=

=96

0

2

1

p

ppk (5)

Thus the load displacement relation for the entire continuous beam is,

=

6

5

4

3

2

1

6

5

4

3

096.024.00096.0024.0

24.08.0024.004.0

00096.0096.024.024.0

096.024.0096.0192.024.00

0024.024.08.04.0

24.04.024.004.06.1

96

0

u

u

u

u

u

u

EI

p

p

p

pzz

(6)

Since, 06543 ==== uuuu due to support conditions. We get,

=

2

1

8.04.0

4.06.1

96

0

u

uEIzz

Thus solving for unknowns 1u and 2u ,

=

96

0

6.14.0

4.08.0

12.1

1

2

1

zzEIu

u

=

=

3

3

10714.1

10429.0

14.137

285.341

zzEI

radians10714.1;radians10429.0 323

1

== uu (7)

Now, unknown joint loads are calculated by,


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=

14.137

285.341

024.0

04.0

24.024.0

24.00

6

5

4

3

zz

zzEI

EI

p

p

p

p

(8)

=

23.8

71.13

68.24

91.32

Now the actual support reactions543

,, RRR and6

R must include the fixed end

support reactions. Thus,

=

+

=

17.30

29.82

72.13

88.43

23.8

71.13

68.24

91.32

4.38

96

4.38

8.76

6

5

4

3

R

R

R

R

(9)

kN17.30kN.m;29.82kN;72.13kN;88.43 6543 ==== RRRR (10)

Example 29.2

A continuous beam ABCD is carrying a uniformly distributed load of mkN/5 as

shown in Fig. 29.4a. Compute reactions due to following support settlements.

Support B m005.0 vertically downwards.

Support C m010.0 vertically downwards.

Assume GPaE 200= and 44104 mI = .


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SolutionThe node and member numbering are shown in Fig. 29.4(b), wherein thecontinuous beam is divided into three beam elements. It is observed from the

figure that the unconstrained degrees of freedom are 1u and 2u . The fixed end

actions due to support settlements are shown in Fig. 29.4(c). and fixed end

moments due to external loads are shown in Fig. 29.4(d). The equivalent jointloads due to support settlement and external loading are shown in Fig. 29.4(e).The fixed end actions due to support settlement are,

( )L

EIM

F

A

6= where is the chord rotation and is taken ve+ if the

rotation is counterclockwise.

Substituting the appropriate values in the above equation,

9 4

3

6 200 10 4 10 0.00596 kN.m.

5 10 5

F

AM = =

96 96 192 kN.m.FBM = + =

96 192 96 kN.m.FCM = =

192 kN.m.F

D

M = (1)

The vertical reactions are calculated from equations of equilibrium. The fixed endactions due to external loading are,

2

10.42 kN.m.12

F

A

w LM = =

10.42 10.42 0 kN.m.FB

M = =

0FC

M =

10.42 kN.m.FDM = (2)

In the next step, construct member stiffness matrix for each member.

Member 1, mL 5= , node points 1-2.


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[ ]

1

3

5

6

80.024.040.024.0

24.0096.024.0096.0

40.024.080.024.0

24.0096.024.0096.0

'

1356..

=zz

EIk

fodGlobal

(3)


[ ]

2

4

1

3

80.024.040.024.0

24.0096.024.0096.0

40.024.080.024.0

24.0096.024.0096.0

2413..

2

= zzEIk

fodGlobal

(4)


[ ]

7

8

2

4

80.024.040.024.0

24.0096.024.0096.0

40.024.080.024.0

24.0096.024.0096.0

7824..

3

= zzEIk

fodGlobal

(5)

On the member stiffness matrix, the corresponding global degrees of freedomare indicated to facilitate assembling. The assembled global stiffness matrix is of

the order 88 . Assembled stiffness matrix [ ]K is,


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[ ]

=

096.024.000096.0024.00

24.080.00024.0040.00

00096.024.00096.0024.0

0024.080.0024.0040.0

096.024.000192.0096.0024.0

00096.024.0096.0192.024.00

24.040.000024.060.140.0

0024.040.024.00.040.060.1

zzEIK(6)

The global load vector corresponding to unconstrained degree of freedom is,

{ }

=

=96

192

2

1

p

ppk (7)

Writing the load displacement relation for the entire continuous beam,

=

8

7

6

5

4

3

2

1

8

7

6

5

4

3

096.024.000096.0024.00

24.080.00024.0040.00

00096.024.00096.0024.0

0024.080.0024.0040.0

096.024.000192.0096.0024.0

00096.024.0096.0192.024.00

24.040.000024.060.140.0

00375.040.024.00.040.060.1

96

192

u

u

u

u

u

u

u

u

EI

p

p

p

p

p

p

zz

(8)

We know that 0876543======

uuuuuu . Thus solving for unknownsdisplacements 1u and 2u from equation,

=

2

1

60.140.0

40.060.1

96

192

u

uEIzz (9)


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=

96

192

60.140.0

40.060.1

)1080(4.2

13

2

1

u

u

=

3

3

1020.1

1080.1(10)

radians1020.1;radians1080.1 323

1

== uu (11)

The unknown joint loads are calculated as,

( )

=

3

3

3

8

7

6

5

4

3

1020.1

1080.1

24.00

40.00

024.0

040.0

024.0

24.00

1080

p

p

p

p

p

p

=

04.23

40.38

56.34

60.57

56.34

04.23

(12)

Now the actual support reactions 76543 ,,,, RRRRR and 8R must include the fixed

end support reactions. Thus,


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=

+

=

+

=

26.66

02.164

34.16

82.48

64.55

04.48

04.23

40.38

56.34

60.57

56.34

04.23

3.89

42.202

9.50

42.106

2.90

25

8

7

6

5

4

3

8

7

6

5

4

3

8

7

6

5

4

3

p

p

p

p

p

p

p

p

p

p

p

p

R

R

R

R

R

R

F

F

F

F

F

F

(13)

kN.m;82.48kN;64.55kN;04.48 543 === RRR

kN2666kN.m;02.164kN;34.16 876 .RRR === (14)

Summary

The effect of temperature changes and support settlements can easily beincorporated in the direct stiffness method and is discussed in the presentlesson. Both temperature changes and support settlements induce fixed endactions in the restrained beams. These fixed end forces are handled in the sameway as those due to loads on the members in the analysis. In other words, theglobal load vector is formulated by considering fixed end actions due to bothsupport settlements and external loads. At the end, a few problems are solved toillustrate the procedure.