5 ea general beam-column finite element

7/25/2019 5 Ea General Beam-column Finite Element

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Stabilitselmlet2011/2012 MSc

5. elads

The general beam-column finite element

1 General

The methodology getting elements from analytical solution of continuous problems is

restricted. For example, a general beam-column problem with double symmetric thin-walled

cross-section and concentrated end moments and constant compressive force (see Fig.1) leadsto a differential equation system with three equilibrium equations, see for example Vlasov

(1961) or Brezina (1962), such as

000

2

0

00

00

MMMNrGIEI

MMNEI

MMNEI

''''

SV

'''

''

''

=++

=++=++

(1)

where the first two equations express the equilibrium of the bending moments in the (,) andthe (,) initial coordinate planes, respectively. The third equation expresses the equilibriumof the twisting moments. Furthermore,

E is the elastic modulus,

G is the elastic shear modulus,IandI are the moments of inertia,

I is the warping moment of inertia,

ISV is the St. Venant torsion constant,

r0 is the polar radius of gyration,

()and () are the displacement functions,() is the function of torsion,

M0andM0 are the end moments,

N is the constant axial compressive force.

In Eq.(1) denotes the derivative by (for example:( )

=' ).

Figure 1 Beam-column element subjected to end moments and constant compressive force

()

()

-M0 M0

N-N

M0-M0

M0 -M0

()


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The analytical solutions of the problem described above have been solved for some special

cases using different approximation functions for the three deformations (Chen and Atsuta,1977). These solutions relate to simplified loading (for example centric or eccentric axial

compression). The explicit analytical solution of the complex problem is not known.

Therefore, in the engineering practice instead of the analytical element the more generalfinite

element method is used. The idea of the finite element method as well as the practical

application is described in the following sections.

2 The finite element approach

In case of thin-walled element the effect of varying of stresses across the wall of the cross-section may be neglected. In this meaning most of the rolled and welded steel structural

profiles used in buildings may be assumed as thin-walled. The background for engineering

application of thin-walled beam-column element was established by Vlasov (1961) who

assumed that the displacement of an arbitrary cross-sectional point may be considered as a

linear sum of the displacements due to the independent deformations, in other words, the totaldeformation of an element may be given as the linear sum of independent deformations such

as the axial, the flexural about the major and the minor axes and the torsional, respectively.

Barsuom and Gallagher (1970) applied this theory in their early publication, later Rajasekaran

published a general thin-walled beam-column element (Rajasekaran and Murray, 1973; Chen

and Atsuta, 1977). The last work seems to be applicable in modern CAD architecture,

therefore, this type of element will be introduced in this section.

To establish a spatial element, firstly we have to define the local system of the thin-walled

cross-section. Fig.2shows the cross-section in the local system, where Cdenotes the centroid

as the centre of the local coordinate system where is the major and is the minor axis. Ddenotes the shear centre, DandDare its coordinates in the local system. The counter of thewalls defines the cross-section model, and Pdenotes an arbitrary point of the cross-section.

The tangent coordinate system is denoted by sand the starting point is O. Generally, the total

displacement of point Pmay be defined by four independent deformation functions

Figure 2Thin-walled beam-column cross-section in the local system

C

D

D

D

P

P

P

Os

p

a


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Figure 3Total displacement of the cross-sectional point P

such as the axial uC, the lateral uD and uD and the torsional D, where the first is measuredin the centroid and the others in the shear centre system (denoted by CandDin index). Fig.3

shows the cross-sectional point Pafter the total deformation given by the previous functions.

Considering Fig.3 and using the commonly used approximation of small displacement

method such as

1cosandsin (2)

the translation of Pto Pcan be expressed by two components such as

DDD

DDD

)(uu

)(uu

+=

= (3)

Assuming that the cross-section is uniform along the local axis and taking a differentiallength as element, the axial displacement of P can be expressed following the Vlasovs

hypothesis such as

'

Dn

'

D

'

Dc uuuu += (4)

where u is the function of . Eq.(4) expresses a more general assumption than that ofBernoulli-Navierhypothesis:

CD

D

D

P a

P

P

D

P a

P

D

u

u

u D

u D


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- the effect of shear strain in any plate segment may be neglected, and

- the projected shape of any plate segment keeps its initial shape.

However, the above assumptions restrict the cross-section profile to ones that are composed

by straight plate segments. The three displacement components of the cross-sectional point

are the function of and depend on the deformation functions

)(and)(u,)(u,)(u DDDC (5)

where Cdenotes the centroid and Ddenotes the shear centre. The stains in cross-sectional

point Pcan be expressed using the relationships between the strains and displacements given

by the generalised elastic theory

)uuuuuu(uu

)uuuuuu(uu

)uuu(u

,,,,,,,,

,,,,,,,,

,,,,

++++=

++++=

+++= 2222

1

(6)

where the partial derivatives are defined by the general formula: for example = /uu ,

and so on. In Eq.(6) the first non-linear members take the effect of the deformation on

equilibrium into consideration. Using the strain components given by Eq.(6) the equilibrium

of the differential length of element can be expressed using the most general principium of

virtual work:

dd uf =++ dV

dV)( (7)

In Eq.(7)the left hand side expresses the work of the internal stress on the appropriate virtual

strain, while the right hand side expresses the work of the surface forces of the ends on the

appropriate virtual displacements. At left hand side = is the normal stress and is the

corresponding virtual normal strain, and are the components of shearing stress and

and are the corresponding virtual shearing strains at an arbitrary point P on the

counter of the element. At right hand side fd denotes the surface forces, du denotes the

corresponding virtual displacements at the ends of element.

Practically, from the four deformation functions, see Eq.(5), we can derive seven nodal

displacements (seven degrees of freedom) as it is shown in Fig.4:

[ ]i'D

i

D

i'

D

i

D

i

D

i'

D

i

D

i

D

i

C

i ,,u,u,u,u,u ===u (8)


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Figure 4 Degrees of freedom of the arbitrary node of the beam-column element

The seven displacements in Eq.(8) indicate seven stress resultants (see Fig.5) which are

derived from the stresses in the cross-section:

- normal force: ==A

iNdAf (9a)

- shear forces: 'i

A

i

im

d

dmdAf

=== (9b)

'i

A

i

im

d

dmdAf

=== (3c)

- bending moments: =A

idAm (9d)

=A

idAm (9e)

- torsional moment: iSV'ii

Tmm += (9f)

- bi-moment: =A

n

idAm (9g)

Ci

Di

i

Cu

i

i

Du

i

Du

i

D

i

D

i

D

i

D,


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Figure 5 Stress resultants corresponding to degrees of freedom

The stress resultant vector corresponding to the displacement vector, see Eq.(8), can be

written such as

iiiiiiim,m,m,f,m,f,f =

if (10)

Let denote j and k the two nodes of the element. Substituting i by j and k in Eq.(10) the

governing matrix displacement equation of the element can be written such as

=

k

j

kkkj

jkjj

f

f

u

u

KK

KK

k

j

(11)

The Kstiffness matrices depend on how the four deformation functions are assumed. Since

the exact solution is not possible the functions should be approximated by special functions

such as the polynomials. This method is called finite element method. In Example we

introduce the Rajasekarans thin-walled beam-column finite element as a possible and usable

CAD-oriented tool.

Ci

Di

if

im

im

if

if

im

im


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Example: The Rajasekarans thin-walled beam-column finite element

Eq.(11)contains 14 degrees of freedom. Consequently, the four deformation functions

should have also 14 unknown parameters in order to get the stiffness matrices in explicit

form. The solution was published by Rajasekaran (1973) who suggested taking the

following deformation functions:

- axial deformation: Cu linear polynomial 2 unknowns

- bending about major axis: Du cubic polynomial 4 unknowns

- bending about major axis: Du cubic polynomial 4 unknowns

- torsion: D cubic polynomial 4 unknowns

The main idea of the finite element method is just that the deformation functions are

approximating the exact solution dividing the element appropriate number of finite

elements. The method approximates the exact solution from down: it gives the upper

limit of the real stiffness. These conditions of convergence may be satisfied, if the

approximation functions satisfy at least the following:

(1)compatible condition: the deformation function must be continuous within the

element and the displacement must be compatible between adjacent elements;

(2)completeness condition: the deformation function must include rigid body

displacements and constant strain state of the element.

These conservative conditions may be satisfied by the suggested full polynomials above.

To establish the relationship between the deformation functions and the corresponding

nodal displacements Rajasekaran (1973) wrote the second deformation functions in the

following form:

[ ]

=+++=

4

3

2

1

323

4

2

321D 1u

(12)

The third deformation function can be derived from the second such as

[ ]

=++===

4

3

2

1

22

432

'

DD 321032d

duu

(13)

Let be 0= at node j, l= at node k. The vector of nodal displacements may be written

in the following form:


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**

2

32

k

D

k

D

j

D

jD

*

D

l3l210

lll1

0010

0001

l

u

l

u

Au

=

=

= (14)

where*

D

1*

uA = (15)

and (*) denotes the inverse vector. UsingEq.(11)inEq.(12)we can write

[ ]*D3

*D

132D 1u unuA ==

(16)

wheren3is the cubic interpolation function. Let introduce l/= parameter and use it

inEq.(16)getting the following:

( ) ( ) ( ) ( )233232323 232231 ++=n (17)

The other three deformation functions can be written in the same form. Without further

details we can write the deformation functions in the following form:

- approximate deformation functions:

*

D3Du un= *

D3Du un= *

C1Du un= *

D3D n= (18)

- interpolation functions:

( ) ( ) ( ) ( )[ ]233232323 232231 ++=n (19)( )[ ]= 11n

- nodal displacement vectors:

[ ]

[ ]

[ ]

[ ]kD

k

D

j

D

j

DD

k

D

k

D

j

D

j

DD

k

D

k

D

j

D

j

DD

k

C

j

CC

lulu

lulu

lulu

uu

=

=

=

=

u

u

u

(20)

The stiffness matrices in Eq.(11)were expressed by Rajasekaran (1973) who followed

the steps below:


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(i) substitute the strains given byEqs.(9)intoEq.(7)neglecting the products

including derivatives of the deformation functions,(ii) substitute the displacements of cross-sectional point P given byEqs.(3-4)and

use the stress resultants given byEqs.(9),

(iii) substitute the expressions ofEqs.(18-20)into the latest variation form of the

governing equation.

The above procedure results the variation form of the governing equation:

[ ]l0

i

D

i

,

i

D

ii

D

ii

D

ii

D

ii

D

ii

C

i

*

C

1

0

'

1

'*

1C

*

D

1

0

'

3

*

3

1

0

'

3

'*

3

D

D

*

D

1

0

'

3

*

3

1

0

'

3

'*

3

D

D

*

D

1

0

1

0

3

'*

3

SV'

3

'*

3

1

0

''

3

'*'

33D

*

D

1

0

1

0

1

0

3

'*

3

'

3

'*

3

D'

3

'*

3

1

0

''

3

'*'

33D

*

D

1

0

1

0

1

0

3

'*

3

'

3

'*

3

D'

3

'*

3

1

0

''

3

'*'

33D

mmmmufufuf

dl

EA

dfdl

mN

dfdl

mN

dl

GId

l

Kd

l

EI

dfdl

mNd

l

Nd

l

EI

dfdl

mNd

l

Nd

l

EI

++++

=

+

+

+

+

++

+

+++

+

+++

+

++

unnu

nnnn

nnnn

nnnnnn

nnnnnnnnu

nnnnnnnnu

Since the governing equation above contains the N normal force and the m moments, the

following simplification had to be introduced:

(1) N normal force in constant along the element,(2) m bending moments are linear in the function of that means:

fmmandfmmjj +=+= (21)

As a consequence of the second assumption the fand fshearing forces are really not

independent parameters but the results of the m moments. Evaluating the integration by

and using the fact that the virtual displacements may be chosen arbitrary thegoverning equation can be reduced to the following forms:

bending in the plane of major axis:

if 1u =D and 0uu === DCD than (22)


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=

+

++

l

1m

fl

1m

f

)(fl

mNu

l

Nu

l

EI

k

k

j

j

*

D

100

33

110

33

110

33

j

D*

D

110

33

*

D

220

333

KKKKK

bending in the plane of miner axis:

if 1u =D and 0uu === DCD than (23)

=

+

++

l

1m

fl1m

f

)(fl

mNu

l

Nu

l

EI

k

k

j

j

*

D

100

33

110

33

110

33

j

D*

D

110

33

*

D

220

333

KKKKK

torsion:

if 1=C and 0uuu === CDD than (24)

=

++

+

+

+

++

l

1m

ml

1m

m

Kl

KKK

l

K)(f

l

mN

)(f

l

mN

l

GI

l

EI

k

,

k

j

,

j

*

D

111

33

jk110

33

j*

D

100

33

110

33

110

33

j

D

*

D

100

33

110

33

110

33

j

D*

D

110

33

Sv*

D

220

333

uKKK

uKKKKK

axial:

if 1u =C

and 0uu ===DDD

than (25)

=

k

j

*110

11f

fuK

l

EA

In the Eqs.(22-25) the Kfinite element matrices are scalar and they can be written in

the following general form:

=l

0

l

r

*k

p

mklm

pr d nnK (26)


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where p and r are the degree of the interpolation functions, k and l are the degree of

derivations, and m is the power of . Substituting the K matrices into the Eqs.(22-25)and collecting the nodal displacements into one vector, the local equilibrium matrix

equation of the thin-walled beam-column finite element can be written as the following

=

++

++k

j

k

j

g

kk

g

kj

g

jk

g

jj

f

f

u

u

KKKK

KKKKs

kk

s

kj

s

jk

s

jj (27)

where the total nodal displacement vector is

{ }k'Dk

D

k

D

k

D

k

C

k

C

k

C

j'

D

j

D

j

D

j

D

j

D

j

D

j

C uuuuuu =u

and the corresponding total load vector is

{ }kkkkkkkj'jjjjjj mmmmfffmmmmfff =f

Finally, the Ks flexural and the Kg geometric stiffness matrices were written by

Rajasekaran in the excellent textbook of Chen and Atsuta (1977) such as


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flexuralstiffness matrices

=

k

c

g

ji

fe

ba

m

Ks

jj

=

oj

db

hf

ji

fe

ba

m

Ksjk

=

k

c

g

ji

fe

ba

m

Ks

kk

where

l

EAm

lGI30

1

l

EI2olGI

30

4

l

EI4k

GI30

3

l

EI6i

l

GI

30

36

l

EI12i

l

2EId

l

4EIc

l

EI6b

l

EI12e

l

2EI

dl

4EI

cl

EI6

bl

EI12

a

SVSV

SV2

SV

3

23

23

=

=+=

+=+=

====

====

It can be seen that the flexural stiffness matrices are the functions of the cross-section

properties and the length of the finite element.


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geometricstiffness matrices

=

o

hc

'hc

n'i'im

'fb'ea

fbea

Kg

jj

=

pt'tn'ff

jdib

'jd'ib

qs'sm'ee

'gb'ka

gbka

Kg

jk

=

r

wc

'wc

qs'sm

'gb'ka

gbka

Kskk

where

w)andkj,i,h,g,f,e,ofexpressiontheinfofinsteadfandmof

insteadm,ofinsteadusingevaluatedaresexpressionw'andk',j',i',h',g',f',(e'

f30

l)mN(

30

1-tf

30

l3)mN(

30

3s

)KK(60

l3

K30

l4

rK30

4

q)KK(60

l3

K30

l

p

)KK(30

lK

30

l4o)KK(

30

3K

30

3n)KK(

l30

18

l

K

30

36m

f30

l3)mN(

30

4lw

f30

33)mN(

30l

36k)mN(

30

l-jf

30

l6)mN(

30

1-i

f30

l)mN(

30

4lhf

30

l3)mN(

30

3g)mN(

30

3f

f30

3)mN(30l

36eNl30

1-dNl30

4cl

N

30

3bl

N

30

36a

DD

2j

D

j

D

jkjjjkj

jkjjkjjkj

2j

D

j

D

j

D

j

D

2j

D

j

D

j

D

j

D

+=+=

+==

=

+=+=+=

=

==+=

===

=====

It can be seen that the geometric stiffness matrices depend on the stress resultants and

the length of the element. jK and kK symbols in the expressions are the Wagner

coefficients at node j and k, respectively.


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3 Element in global system

Let locate the finite element into the 3D (X,Y,Z) global system. Let assume that in the initial

location the centroid axis of the element coincides with the global axis Xand the local cross-

sectional axis is parallel to the global axis Y(see Fig.6).

Figure 6Initial location of 3D beam-column element in (X,Y,Z) global system.

Figure 7Arbitrary position of the general beam-column element in (X,Y,Z) global system.

Fig. 6 shows the general element in arbitrary position in the global system. The reference axisof the element is denoted by O, the Ccentroid of cross-section is eccentric (yC, zC) to the

X

Z

Y

z

y

z

X

Z

Y

O

C

D

zC

yCyD

zD

D D

X

X

Y

Z

dz

dx

dy

y

j

k

O

C


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reference system. The position ofDshear centre is given by (yD, zD) in the reference system

and by (D, D) in the local system. The angle of the local axis to reference axis yis denotedby . The arbitrary position can be achieved from the initial position due to a tripletransformation (see Fig.7):

transformation: plane (X,Z)turns about axis Zby angle as far as the referenceaxis will be located in plane (X,Z),

transformation: plane (X,Y)turns about axis Yby angleas far as the referenceaxis will be located its required position,

transformation: plane (Y,Z) turns about axisXby angle as far as the cross-section will be located its required position.

The total transformation matrix that gives the global stiffness matrix can be written as

llll

TTTT = (28)

where0

0T

0

0T

0

0l

l

l

l

l

l

l

l

l

=

=

=

T

T

T

T

T

TT (29)

The basic transformation matrices can be written as

=

=

100

cos0sin

010

sin0cos

cos0sin

010

sin0cos

1

100

0cossin

0sincos

100

0cossin

0sincos

ll

TT

The transformation parameters and can be computed easily from the node coordinates ofthe element. The basic transformation matrix relating to the cross-section rotation and

eccentricity can be written as

+

+

=

100

coszsinycossin

cosysinzsincos

1000

cosysinz000

coszsinysincos0

yz0001

DD

DD

DD

DD

CC

l

T


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The above transformation matrices reflect to important modelling features:

(i) The compatibility of the 7th

degrees the equilibrium of bimoment(B) as stress resultant

of the elements connected to the node is satisfied by the hypothetical assumption

such as

=i

i 0B (30)

The assumption theoretically incorrect but from practical point of view it is the way to

take the warping effect into consideration in the beam-column models.

(ii) The element is eccentric to reference system where )z,y( CC takes the eccentricity of the

centroid of cross-section, )z,y( DD takes the position of the shearing centre in the

reference system into consideration (see Fig.7).

4 The solution technique

The basic problem in engineering practice is the solution of the governing matrix equation

where the K stiffness matrix is the linear function of the stress resultants of the elements.

Since the stress resultants are the function of the displacement vector, the governing equation

is non-linear. The required solutions are illustrated in Fig.8. On the actual load combination

level firstly we are looking for the geometrically non-linear (second order) solution, secondlythe generalised critical load factor for the elastic behaviour.

Figure 8Required solutions for engineering design

displacement, u

load factor,

critical laod factor, cr

design load

design displacement


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The simple second order solutions may be achieved by the common direct iteration technique

that should be convergence since the relationship between the displacement and the load isunambiguous. Fig.9illustrates the direct iteration procedure that has the following steps:

Step 1: Solve the geometric linear problem assuming the geometric matrix being zero:

11

1

11 TusFKU ,s1,s k== (31a)

where S1is the stress resultant vector.

Step 2: Solve the governing equation using new stiffness matrix computed by the

previously stress resultant vector:

22

1

22 Tus)F(SKU 1 ,s2,s k== (31b)

Step 3: Repeat Step 2 as far as the norm of the actual displacement vector will be less

than a corresponding limit value:

11

1i

*

i

i

*

i

UU

UU (31c)

According to a large number of numerical examples the number of the iterative steps in

practical problems is normally 2-3. More iterative steps may be needed if the actual load is

close to the critical load level. The main advantage of the direct iteration technique is the

simplicity and the easy programming. The certain disadvantage of the method is that in every

iterative step the factorisation of the stiffness matrix should be repeated. The geometrically

non-linear load-displacement path keeps to a limit value of the load. If the system is perfect

the generalised displacement is zero during the loading this limit is called critical load.

Figure 9Illustration of the direct iteration technique

F

Displacement

Load

U1 Ui U


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In case of imperfect systems this limit may be called generalised critical load. Here we

discuss the problem how to get the generalised critical load. It is a well-established fact thatthe system should be slightly imperfect. The base of the computation method is that the

stiffness matrix becomes singular on the critical load level. This means that the critical load

can be achieved with a simple load step method shown in Fig.10where in every load step a

second order solution should be carried out.

Figure 10Load step method to compute the critical load

critical load

no regular solution (singular stiffness matrix)

last regular solution

load

displacement


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References

Vlasov, V.Z. (1961), Thin-walled elastic Beams, 2nd ed., Washington D.C., 1961

Brezina, V. (1962), Vzpern nosvost kovovych pratn a nozniku, Nakladatelsvi

Ceskoslovenski akademie ved, Praha

Barsoum, R.S. and Gallagher, R.H., (1970), Finite element analysis of torsional and

torsional-flexural stability problems,Int. Journal for Numerical Methods in Engineering,

Vol. 2, Wiley &Sons 1970, pp. 335-352

Rajasekaran, S. and Murray, D.W. (1973), Finite element solution of inelastic beamequations,Journal of the Structural Division, ASCE, Vol. 99., No T6, June, pp. 1024-

1042, 1973.

Chen, W. and Atsuta, T. (1977), Theory of Beam-Columns, Vol.2: Space Behavior and

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5 ea general beam-column finite element

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