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Sept 2013

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Chapter 16 Stiffness Method

for Frames ، مذكرات شرح وتمارين محلولة

سابقة للعديد من المواد أدناه امتحانات

أدناه الموقعين المذكورين متاحة مجانا على

ال تكن طائرا محبوسا في قفص

.ال يعصصصصر عصصصصن الاصصصصار شصصصصي ا

mailto:[email protected]

http://www.eng-hs.net/

Sept 2013






The concepts in the previous chapters on trusses and beams will be extended

in this chapter and applied to the analysis of frames. It will be shown that the

procedure for the solution is similar to that for beams, but will require the use

of transformation matrices since frame members are oriented in different

directions.

Frame-Member Global Stiffness Matrix:

The member stiffness matrix k' consists of thirty-six influence coefficients

that physically represent the load on the member when the member

undergoes a specified unit displacement.

'

'

'

'

'

'

4

2

60

2

2

60

2

6

3

120

2

6

3

120

0000

2

2

60

4

2

60

2

6

3

120

2

6

3

120

0000

'

'

'

'

'

'

''''''

'

zFd

yFd

xFd

zNd

yNd

xNd

L

EA

L

EA

L

EA

L

EA

L

EA

L

EA

L

EA

L

EA

L

EA

L

EA

L

EA

L

EA

L

EA

L

EA

L

EA

L

EA

L

EA

L

EA

L

EA

L

EA

zFq

yFq

xFq

zNq

yNq

xNq

zFyFxFzNyNxN

dkq

6-61سحب صورة رقم

166صفحة

يا جصام المصاف فصي الصدنيا لوار صه،

.هل أنت بالماف بعد الموت تنتف

(.البغدادي)



Sept 2013






Displacements and Force Transformation Matrices:

By inspection, T transforms the six global x, y, z displacements D into the six

local x', y', z' displacements d. hence T is referred to as the displacement

transformation matrix.

Force Transformation Matrix:

TT transforms the six members loads expressed in local coordinates into the

six loading expressed in global coordinates.

z

y

x

z

y

x

z

y

x

z

y

x

F

F

F

N

N

N

xy

yx

xy

yx

F

F

F

N

N

N

D

D

D

D

D

D

d

d

d

d

d

d

DTd

100000

0000

0000

000100

0000

0000

'

'

'

'

'

'

رقم تينسحب صور

166صفحة 61-2

'

'

'

'

'

'

100000

0000

0000

000100

0000

0000

z

y

x

z

y

x

z

y

x

z

y

x

F

F

F

N

N

N

xy

yx

xy

yx

F

F

F

N

N

N

T

q

q

q

q

q

q

Q

Q

Q

Q

Q

Q

qTQ

لصصو تكلصصم النصصا فيمصصا يعرفونصصه فقصص ،

.لبلصصصصصصص الًصصصصصصصمت حصصصصصصصدا ال يطصصصصصصصا

رقم تينسحب صور

162صفحة 61-3



Sept 2013






Frame-Member Global Stiffness Matrix:

substitute (d = TD) into (q = k'd). we have:

Here the member forces q are related to the global displacements D.

substituting this result into (Q = TT q) yields the final result,

or

where

Here k represents the global stiffness matrix for the member.

Note that this 6 x 6 matrix is symmetric. Furthermore, the location of each

element is associated with the coding at the near end, Nx, Ny, Nz, followed by

that of the far end, Fx, Fy, Fz, which is listed at the top of the columns and

along the rows. Like the k' matrix, each column of the k matrix represents

the coordinate loads on the member at the nodes that are necessary to resist a

unit displacement in the direction defined by the coding of the column. For

example, the first column of k represents the global coordinate loadings at

the near and far ends caused by a unit displacement at the near end in the x

direction, that is DN x.

TkTk

DkQ

DTkTQ

DTkq

T

T

'

'

'

. ! الكلمات هي أعظم ما اخترع اإلنسان من داء ودواء

66-61رقم جدوفسحب

163صفحة



Sept 2013






Notation: By inspection, the frame has two elements

and three nodes, which are identified as shown in Fig.

16-4b. The origin of the global coordinate system is

located at , the code numbers at the nodes are

specified with unconstrained degrees of freedom

numbered first. From the constraints at and , and

the applied loading we have:

Structure Stiffness Matrix: The following terms are common to both

element stiffness matrices:

mkNL

EA

mkNL

EA

kNL

EA

mkNL

IE

mkNL

EA

QD kk

.)10(126

)10()180()10()200(22

.)10(246

)10()180()10()200(44

66

)10()180()10()200(66

/26

)10()180()10()200(1212

/)10(2006

)10()200()10(6

5

4

3

2

1

0

0

0

0

20

9

8

7

6

0

0

0

0

3

66

3

66

2

66

2

3

66

3

3

63

يغريص ال تجعل الثناء

.وال اللصصصصصوم يف عصصصصص



Sept 2013






Member 1:

The rows and columns of this 6 x 6 matrix are identified by the three x, y, z

code numbers, first at the near end and followed by the far end, that is, 4, 6,

5, 1, 2, 3, respectively, Fig. 16-4b.

Member 2:

9

8

7

3

2

1

24061206

0200002000

602602

12062406

0200002000

602602

)10(

987321

16

06,0

6

66

3

2

1

5

6

4

24601260

620620

0020000200

12602460

620620

0020000200

)10(

321564

06

00,1

6

06

3

2

3

1

k

k

yy

kk

16.1تاب إجابة مثاف

رك علي المساحة التصي

.يكون نشاط فيها متقد



Sept 2013






Displacements and Loads:

solving, we obtain

rad

m

rad

m

m

D

D

D

D

D

D

D

D

D

D

D

D

D

D

D

Q

Q

Q

Q

DKQ

)10(243.1

)10(51.17

)10(505.2

)10(47.37

)10(51.17

0

0

0

0

0

2401260

020000200

1204866

6062020

020060202

)10(

0

0

0

0

20

0

0

0

0

24060001206

0200000002000

602000602

000260620

00062401260

0000020000200

120661204866

0200026062020

6020020060202

)10(0

0

0

0

20

987654321

3

3

3

6

3

5

4

3

2

1

5

4

3

2

1

3

5

4

3

2

1

3

9

8

7

6


ليس الكريم بسيد في قومه،

.لكن سيد قومه المتغابي

(.أبو فرا )



Sept 2013






The internal loadings at node can be determined by applying:

mkN

kN

kN

q

q

q

q

q

q

DTkq

mkN

kN

kN

kN

rad

m

rad

m

m

Q

Q

Q

Q

.45

50.7

0

0

50.7

0

3

2

1

5

6

4

3

6

3

3

3

111

3

3

3

6

3

3

9

8

7

6

3

2

1

5

6

4

)10(505.2

)10(47.37

)10(51.17

)10(243.1

0

)10(51.17

100000

010000

001000

000100

000010

000001

24601260

620620

0020000200

12602460

620620

0020000200

321564

.75

50.7

20

50.7

0

0

0

0

)10(243.1

)10(51.17

)10(505.2

)10(47.37

)10(51.1754321

001206

0002000

00602

60620

)10(


ريصصصرة فليتصصص تحلصصصو والحيصصصاة م

وليتصص تر صصى وا نصصام صصا

ا مروليت الذى بينى وبين عصا

وبينصصى وبصصين العصصالمين خصصرا



Sept 2013






Notation: To perform a matrix analysis, the

distributed loading acting on the horizontal member

will be replaced by equivalent end moments and

shears. Then using superposition, the results obtained

for the framed in Fig. 16-5b will be modified for this

member by the loads shown in Fig. 16-5c.

As shown in Fig. 16-5b, the nodes and members are

numbered and the origin of the global coordinate

system is placed at node . As usual, the code

numbers are specified with numbers assigned first to

the unconstrained degrees of freedom. Thus,

Structure Stiffness Matrix:

Member 1:

mkNL

IE

mkNL

EA

QD kk

/1280)5.7(

)10()225()10()200(1212

/)10(2005.7

)10()200()10(7500

3

2

1

150

150

0

9

8

7

6

5

4

0

0

0

0

0

0

3

366

66

3

ير نفس أوال قبل أن

.تحاوف تغيير اآلخرين



Sept 2013






Member 2:

06

5.45.4,1

6

612

.)10(156

)225()200(22

.)10(306

)225()200(44

7500)6(

)225()200(66

/2500)6(

)225()200(1212

/)10(2506

)10()200()10(7500

3

2

1

5

6

4

2484.388.21284.388.2

84.382.7239.9584.384.7239.95

88.239.9546.12888.239.9546.128

1284.388.22484.388.2

84.382.7239.9584.382.7239.95

88.239.9546.12888.239.9546.128

)10(

321564

6.05.7

05.48.0

5.7

06

.)10(125.7

)225()200(22

.)10(245.7

)225()200(44

4800)5.7(

)225()200(66

3

3

22

33

3

66

3

1

3

3

22

yx

yx

mkNL

IE

mkNL

IE

kNL

IE

mkNL

IE

mkNL

EA

k

mkNL

IE

mkNL

IE

kNL

IE


بعصصصل الطصصصصر تصصصص دي

مصا، وبع صها انإلى مك

!إلصصصصى ال مكصصصصان، انتبصصصصه



Sept 2013






Displacements and Loads: Expanding to determine the displacements,

and solving, we have:

rad

mm

mm

D

D

D

D

D

D

DKQ

k

D

D

D

Q

Q

Q

Q

Q

Q

00261.0

76.2

716.0

0

0

0

5466.388.2

66.332.7539.95

88.239.9546.378

10

150

150

0

9

8

7

3

2

1

305.70155.70

5.75.205.75.20

0025000250

155.703000

5.75.205.75.70

0025005.2250

)10(

987321

3

2

1

3

2

1

3

3

2

0

0

0

0

0

0

3

2

1

305.70000155.70

5.75.20000755.20

0025000000250

0002484.388.21284.388.2

00084.382.7239.9584.382.7239.95

00088.239.9546.12888.239.9546.128

155.701284.388.25466.388.2

5.75.2084.382.7239.9566.332.7539.95

0025088.239.9546.12888.239.9546.378

)3

10(

9

8

7

6

5

4

150

150

0

987654321


أكثصصصر لمصصصاالا ال تعصصصي حياتصصص

بسصصاطة ممصصا أنصصت عليصصه اآلن



Sept 2013






These results are shown in Fig. 16-5d.

mkN

kN

kN

mkN

kN

kN

q

q

q

q

q

q

q

q

q

q

q

q

MkN

kN

kN

mkN

kN

kN

Q

Q

Q

Q

Q

Q

.0.50

15.9

6.216

.7.18

15.9

6.216

3

2

1

6

5

4

3

9

8

7

6

5

4

3

2

1

6

5

4

61.2

76.2

716.0

0

0

0

100000

08.06.0000

06.08.0000

000100

00008.06.0

00006.08.0

248.40128.40

8.428.108.428.10

0020000200

128.40248.40

8.428.108.428.10

0020000200

3

2

1

6

5

4

321654

.9.59

5.26

0.179

.7.18

7.122

8.178

0

0

0

0

0

0

)10(00261.0

76.2

716.0

155.70

5.75.20

00250

1284.388.2

84.382.7239.95

88.239.9546.128


.حاوف أنت تنتقد نفس



Sept 2013






Member 1:

3

2

1

9

8

7

484.140244.140

4.1476.504.1476.50

0084000840

244.140484.140

4.1476.504.1476.50

0084000840

)10(

321987

10*245

)10*300(*)10*200(22

10*485

)10*300(*)10*200(44

10*4.145

)10*300(*)10*200(66

10*76.55

)10*300(*)10*200(*1212

10*8405

)10*200(*)10*200(

0,1

3

69

69

69

2

3

6

3

1

6

6

6

2

69

693

k

L

IE

L

IE

L

IE

L

IE

L

EA

yx

.ر مجهوفمًي= إنسان بال أهدا وا حة



Sept 2013






Member 2:

3

2

1

4

6

5

6005.223005.22

010500010500

5.2205.115.22025.11

3005.2260025.22

025.110010500

25.22025.115.22025.11

*)10(

321465

10*304

)10*300(*)10*200(*22

10*604

)10*300(*)10*200(*44

10*5.224

)10*300(*)10*200(*66

10*25.114

)10*300(*)10*200(*1212

10*10504

)10*200(*)10*21(

1,0

62

669

669

6

2

69

2

6

3

69

3

693

k

L

IE

L

IE

L

IE

L

IE

L

EA

yx

16.3تاب إجابة س اف

ن تعمصصصصل عمصصصصال صصصصصحيحا ال

تُشكر عليه خير من أن تعمصل

.عمصصال خاط صصا ال تُعاقصصب عليصصه



Sept 2013






16

9

8

7

6

5

4

3

2

1

484.140000244.140

4.1476.500004.1476.50

0084000000840

000105000010500

000025.115.225.22025.11

00005.22603005.22

244.14005.22301084.145.22

4.1476.501050004.1476.10550

00840025.115.225.22025.851

*6

10

987654321

21

kkK


البصصا تكصصون الواجبصصات التصصي

علي أكثر من الحقو التصي

.تسصصصتحقها، أبصصصدأ بالواجبصصصات



Sept 2013






solving,

kN

DDDDDQ

mDmD

mDmD

DDDD

DDDD

DDDD

DDDD

DKQ

D

D

D

D

Q

Q

Q

Q

Q

mkN

363

)*25.11*5.22*5.22*0*25.11(*10

001603.0,003238.0

0000442.0,0000432.0

)*60*30*0*5.22(*100

)*30*108*4.14*5.22(*1010*300

)*0*4.14*76.1055*0(*100

)*5.22*5.22*0*25.851(*100

54321

6

5

43

21

4321

6

4321

63

4321

6

4321

6

0

0

0

0

0

4

3

2

1

484.140000244.140

4.1476.500004.1476.50

0084000000840

000105000010500

000025.115.225.22025.11

00005.22603005.22

244.14005.22301084.145.22

4.1476.501050004.1476.10550

00840025.115.225.22025.851

*6

10

9

8

7

6

5

0

.300

0

0

استغل وقت فيما

يفيصصصد فقصصصد يًصصصير

.لصصص شصصصأن قريبصصصا



Sept 2013






kN

DDDDDQ

kN

DDDDDQ

kN

DDDDDQ

kN

DDDDDQ

1.77

)*0*0*24*4.14*0(*10

4.46

)*0*0*4.14*76.5*0(*10

3.36

)*0*0*0*0*840(*10

4.46

)*0*0*0*1050*0(*10

54321

6

9

54321

6

8

54321

6

7

54321

6

6

إالا كنت ال تمل القدرة علي

االستفادة مصن أصصاب يصدي ،

.صصصصصاب رجليصصصص فأسصصصصتعمل أ




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