# Direct Stiffness Method: Plane Frame - vsb. · PDF fileMember local stiffness matrix: Direct Stiffness Method: Plane Frame Example 1Example 111 Fixed - Fixed Fixed - Hinged Hinged

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• Direct Stiffness Method:Direct Stiffness Method:

Plane Frame

• Plane Frame Analysis

All the members lie in the same plane.

Members are interconnected by rigid or pin joints.

The internal stress resultants at a cross-section of member

consist of bending moment, shear force and an axial force.

The significant deformations in the plane frame are only

flexural and axial.

Stiffness matrix of the member is derived in its local co-

ordinate axes and then it is transformed to global co-ordinate

system.

Members are oriented in different directions and hence before

forming the global stiffness matrix it is necessary to refer all

the member stiffness matrices to the same set of axes.

This is achieved by transformation of forces and

displacements to global co-ordinate system.

• Plane Frame, Member Stiffness Matrix

The frame members have six degrees of freedom

*au

a b

*au

*

aw *a

*

bu

*bw

*b

=

*

*

*

*

*

b

b

b

a

a

a

ab

w

u

w

*r

b b

• Plane Frame, Member Stiffness Matrix

The forceforceforceforce displacementdisplacementdisplacementdisplacement relationshiprelationshiprelationshiprelationship can be written:

**

aab uX

*abM

*baM

*abX

*abZ

*baX

*baZ

a b

==

=

*

*

*

*

*

*

*

*

*

*

b

b

b

a

a

a

ababab

ab

ab

ab

ab

ab

ab

ab

w

u

w

u

M

Z

X

M

Z

X

**** krkR

member vector of secondarysecondarysecondarysecondary local forcesforcesforcesforces

member vector of local joint displacements

member local stiffness member local stiffness member local stiffness member local stiffness matrixmatrixmatrixmatrix

*Rab

*rab*k ab

• Plane Frame, Member Stiffness Matrix

Member locallocallocallocal stiffness matrix

l

EA

l

EA0000

0000

l

EA

l

EA

FixedFixedFixedFixed FixedFixedFixedFixed connectionconnectionconnectionconnection FixedFixedFixedFixed HingedHingedHingedHinged connectionconnectionconnectionconnection

=

l

EI

l

EI

l

EI

l

EIl

EI

l

EI

l

EI

l

EIl

EA

l

EAl

EI

l

EI

l

EI

l

EIl

EI

l

EI

l

EI

l

EIll

ab

460

260

6120

6120

0000

260

460

6120

6120

22

2323

22

2323

*k

=

000000

03

033

0

0000

03

033

0

03

033

0

0000

323

22

323

l

EI

l

EI

l

EIl

EA

l

EAl

EI

l

EI

l

EIl

EI

l

EI

l

EIll

ab*k

HingedHingedHingedHinged FixedFixedFixedFixed connectionconnectionconnectionconnection HingedHingedHingedHinged HingedHingedHingedHinged connectionconnectionconnectionconnection

=

l

EI

l

EI

l

EIl

EI

l

EI

l

EIl

EA

l

EA

l

EI

l

EI

l

EIl

EA

l

EA

ab

3300

30

3300

30

0000

000000

3300

30

0000

22

233

233

*k

=

000000

000000

0000

000000

000000

0000

l

EA

l

EA

l

EA

l

EA

ab*k

HingedHingedHingedHinged FixedFixedFixedFixed connectionconnectionconnectionconnection HingedHingedHingedHinged HingedHingedHingedHinged connectionconnectionconnectionconnection

• Plane Frame, Member Stiffness Matrix

Member globalglobalglobalglobal stiffness matrix kab

abababab TkTk* = T abababab TkTk =

=

0cossin000

0sincos000

000100

0000cossin

0000sincos

abab

abab

abab

ab

T

x

z

a

b

Tab transformation matrix

100000

0cossin000 abab z angle of angle of angle of angle of

transformation transformation transformation transformation

b

• Plane Frame, Member vector of primary forces

Member vector of primary locallocallocallocal forces is corresponding

to the fixedfixedfixedfixed endendendend rererereactionactionactionaction due to external load.

The fixed end reactions were derived by forceforceforceforce methodmethodmethodmethod The fixed end reactions were derived by forceforceforceforce methodmethodmethodmethod

(as well as secondary forces for member stiffness matrix).

a b*abX

*baX

=*

*

*

ab

ab

ab

M

Z

X

*R

a b*abZ

*abM *

baZ

*baM

=

*

*

*

ab

ab

ab

abab

M

Z

X*R

• Plane Frame, Member vector of primary forces

Member vector of primary locallocallocallocal forces

1n2n

Member connection

+

+

20/)37(

6/)2(

21

21

lqq

lnn

+

+

40/)916(

6/)2(

21

21

lqq

lnn

+

+

40/)411(

6/)2(

21

21

lqq

lnn

( )

+

+

6/2

6/)2(

21

21

lqq

lnn

*

*

ab

ab

Z

X

+

+

+

+

60/)32(

20/)73(

6/)2(

60/)23(

221

21

21

221

21

lqq

lqq

lnn

lqq

+

+

+

0

40/)114(

6/)2(

120/)78(

21

21

221

lqq

lnn

lqq

+

+

+

120/)87(

40/)169(

6/)2(

0

221

21

21

lqq

lqq

lnn

( )

+

+

0

6/2

6/)2(

0

21

21

21

lqq

lnn

=

*

*

*

*

ba

ba

ba

ab

ab

M

Z

X

M

Z

*abR

• Plane Frame, Member vector of primary forces

Member vector of primary locallocallocallocal forces

F

a ba b

Member connection

( )

322 2/)3(

/

lblbF

lbF

z

x

/

/

lbF

lbF

z

x

( )

2/)3(

/

32 lblbF

lbF

z

x

+

32 /)2(

/

lalbF

lbF

z

x

*

*

ab

ab

Z

X( )

( )

( ) ( )

+

2

32

2/

2/)3(

/

0

2/)3(

lalabF

lalaF

laF

lblbF

z

z

x

z

0

/

/

0

laF

laF

z

x

z( )

( ) ( )

( )

+

0

2/)3(

/

2/

2/)3(

322

2

lalaF

laF

lblabF

lblbF

z

x

z

z

+

+

22

32

22

/

/)2(

/

/

/)2(

lbaF

lblaF

laF

labF

lalbF

z

z

x

z

z

=

*

*

*

*

ba

ba

ba

ab

ab

M

Z

X

M

Z

*abR

• Plane Frame, Member vector of primary forces

Member vector of primary locallocallocallocal forces

Member connection

3/6

0

lMab ( ) ( )

2/3

0

322 lblM ( ) ( )

322 2/3

0

lalM

/

0

lM

*

*

ab

ab

Z

X

2

3

2

/)32(

/6

0

/)32(

/6

lalMa

lMab

lblMb

lMab ( ) ( )( )

( ) ( )

0

2/3

0

2/)3(

2/3

322

222

lblM

lblM

lblM ( ) ( )

( ) ( )( )

222

322

2/)3(

2/3

0

0

2/3

lalM

lalM

lalM

0

/

0

0

/

lM

lM

=

*

*

*

*

ba

ba

ba

ab

ab

M

Z

X

M

Z

*abR

• Plane Frame, Member vector of primary forces

Member vector of primary globalglobalglobalglobal forces

*Tababab RTR = ababab RTR =

=

0cossin000

0sincos000

000100

0000cossin

0000sincos

abab

abab

abab

ab

T

x

z

a

b

100000

0cossin000 abab z angle of angle of angle of angle of

transformation transformation transformation transformation

b

• Plane Frame, The load displacement equation

vectorvectorvectorvector, the load displacement relationship can be

written:written:

Global stiffness matrix K is established by the

localization of member global stiffness matrixes kab Global load vector F is taken as difference between

vector of joint loads S and vector of primary global

FKrFrK == 1

v

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