6 Finite element methods for the - Aalto€¦ · 288 Contents 1. Strong and weak forms for Euler−Bernoullibeams 2. Finite element methods for Euler−Bernoullibeams Learning outcome

6 Finite element methods for the

Euler−Bernoulli beam problem

287

Contents

1. Modelling principles and boundary value problems in engineering sciences

2. Energy methods and basic 1D finite element methods

- bars/rods, beams, heat diffusion, seepage, electrostatics

3. Basic 2D and 3D finite element methods

- heat diffusion, seepage

4. Numerical implementation techniques of finite element methods

5. Abstract formulation and accuracy of finite element methods

6. Finite element methods for Euler−Bernoulli beams

7. Finite element methods for Timoshenko beams

8. Finite element methods for Kirchhoff−Love plates

9. Finite element methods for Reissner−Mindlin plates

10. Finite element methods for 2D and 3D elasticity

11. Extra lecture: other finite element applications in structural engineering

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Numerical Methods in Structural Engineering

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288

Contents

1. Strong and weak forms for Euler−Bernoulli beams

2. Finite element methods for Euler−Bernoulli beams

Learning outcome

A. Understanding of the basic properties of the Euler−Bernoulli beam problem and

ability to derive the basic formulations related to the problem

B. Basic knowledge and tools for solving Euler−Bernoulli beam problems by finite

element methods – with elements, in particular

References

Lecture notes: chapter 9.1

Text book: chapters 1.16, 3.1, A1.I

6 Finite element methods for the

Euler−Bernoulli beam problem

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1C

6.0 Motivation for the Euler-Bernoulli beam

element analysis

289Rak-54.3200 / 2014 / JN

Beam structures (frames, trusses, beams, arches) are the

most typical structural parts in modern structural

engineering ─ and the Euler−Bernoulli beam element is

the most typical one in commercial FEM software.

6.1 Strong and weak forms for

Euler−Bernoulli beams

290Rak-54.3200 / 2014 / JN

Let us consider a thin straight beam structure subject to such a loading that the

deformation state of the beam can be modeled by the bending problem in a plane.

The basic kinematical dimension reduction assumptions of a thin beam, called

Euler–Bernoulli beam (1751), i.e.,

(K1) normal fibres of the beam axis remain straight during the deformation

(K2) normal fibres of the beam axis do not strech during the deformation

(K3) material points of the beam axis move in the vertical direction only

(K4) normal fibres of the beam axis remain as normals during the deformation

)()(

)(' xdx

xdwxw

)(x

ux,

vy,

)(xw



291Rak-54.3200 / 2014 / JN

Let us consider a thin straight beam structure subject to such a loading that the

deformation state of the beam can be modeled by the bending problem in a plane.

The basic kinematical dimension reduction assumptions of a thin beam, called

Euler–Bernoulli beam (1751), i.e.,

(K1) normal fibres of the beam axis remain straight during the deformation

(K2) normal fibres of the beam axis do not strech during the deformation

(K3) material points of the beam axis move in the vertical direction only

(K4) normal fibres of the beam axis remain as normals during the deformation

come true if the displacements are presented as

)()(

)(' xdx

xdwxw

)(x

ux,

vy,

)(xw

with w denoting the deflection of the beam (central

or neutral axis), appearing as the only variable of

the problem (with the current assumptions) and,

furthermore, depending on the x coordinate only.

),('))('sin(),(

),(:)0,())('cos(1()0,(),(

xywxwyyxu

xwxvxwyxvyxv

292

).(''),(

),( xywx

yxuyxx

Considering linear deformations the displacement field above implies the axial strain

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293

).(''),(

),( xywx

yxuyxx

Linear deformations for the displacement field above implies the axial strain (alone)

Defining the bending moment as (without specifying the stresses at the moment)

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)(

),,(:)(:)(xA

xz dAyzyxxMxM

294

).(''),(

),( xywx

yxuyxx

Linear deformations for the displacement field above implies the axial strain (alone)

Defining the bending moment as (without specifying the stresses at the moment)

the energy balance of the principle of virtual work can be written in the form

Rak-54.3200 / 2016 / JN



t

t

S

y

L

A

xx

SVV

ext

vdStdAdx

dSdVdV

WW

0

int

:

0

utubεσ

)(

),,(:)(:)(xA

xz dAyzyxxMxM

3D elasticity theory!

1D + 2D

295Rak-54.3200 / 2016 / JN



,''

'')(

00

00

wwdxfdxwM

wdxfdxwydA

LL

LL

A

x

1D + 2D

1D, i.e., x alone

296

where the beam is assumed to be subject to a vertical distributed surface loading

acting on the upper and lower surfaces of the beam, defining a

resultant loading

where the integrals are taken along lines and in the z direction for

each x on upper and lower surfaces and , respectively (for collecting the

physical load from surfaces onto the beam axis).

Remark. Other loading types could be considered as well.

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),,( zyxtt yy

,''

'')(

00

00

wwdxfdxwM

wdxfdxwydA

LL

LL

A

x

,),2/,(),2/,(:)(

)()(

xZ

y

xZ

y

tt

dzztxtdzztxtxf

)(xZ t

)(xZ t

tS

tS

297

Integration by parts in the term for the internal virtual work gives the form

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wwdxfMwMwM

wdxfdxwMwM

LLL

LLL

0

00

00

0

)''(')'(

)'(')'(0

298


implying the force balance and boundary conditions, i.e., the strong form, as

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wwdxfMwMwM

wdxfdxwMwM

LLL

LLL

0

00

00

0

)''(')'(

)'(')'(0

L

0

0

L

0

L

0

ww(L)

w(0)

(L)w'

(0)w'

Q(L)M'

Q(0)M'

MM(L)

MM(0)

or

0w(L)

0w(0)

0(L)w'

0(0)w'

0(L)M'

0(0)M'

0M(L)

0M(0)

M)-EB(),0()()(''

w

LxxfxM

L

299


implying the force balance and boundary conditions, i.e., the strong form, as

The shear force is determined by the moment equilibrium:

L

0

0

L

0

L

0

ww(L)

w(0)

(L)w'

(0)w'

Q(L)M'

Q(0)M'

MM(L)

MM(0)

or

0w(L)

0w(0)

0(L)w'

0(0)w'

0(L)M'

0(0)M'

0M(L)

0M(0)

M)-EB(),0()()(''

w

LxxfxM

L

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wwdxfMwMwM

wdxfdxwMwM

LLL

LLL

0

00

00

0

)''(')'(

)'(')'(0

.)(')( xxMxQ



300

Taking into account the linearly elastic constitutive relations in the form

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))(''(),)((),( xEwyyxEyx xx



301


the strong form can be written as a displacement formulation as follows: For a given

loading , find the deflection such that

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Rf : Rw :



.w(L)Q(L)'')'(

w(0)Q(0)'')'(

(L)w'M(L)')'(

(0)w'M(0)')'(

w)-EB(),,0()()(')'''(

LL

00

L

00

wEIw

wEIw

EIw

EIw

LxxfxEIw

L

302


the strong form can be written as a displacement formulation as follows: For a given

loading , find the deflection such that

The moment and shear force are given in terms of the deflection as

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Rf : Rw :




.,:)(:)(),()'''()(

),)(''()(

)(

2

xdAyxIxIxEIwxQ

xEIwxM

xAz

.w(L)Q(L)'')'(

w(0)Q(0)'')'(

(L)w'M(L)')'(

(0)w'M(0)')'(

w)-EB(),,0()()(')'''(

LL

00

L

00

wEIw

wEIw

EIw

EIw

LxxfxEIw

L

303

The corresponding weak form is obtained from the virtual work expressions above

or, as usual, by multiplying the strong form by a test function (variational function),

integrating over the domain and finally integrating by parts:

.ˆ''ˆ'''ˆ)''(ˆ)'''(

'ˆ)'''(ˆ)'''(ˆ')'''(ˆ

000

0000

WwdxwEIwwEIwwEIw

dxwEIwwEIwdxwEIwdxwf

LLL

LLLL

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304




This equation gives the energy balance with respect to the variational space as

and the essential boundary conditions – for a cantilever beam, for instance – as

.ˆ''ˆ'''ˆ)''(ˆ)'''(

'ˆ)'''(ˆ)'''(ˆ')'''(ˆ

000

0000

WwdxwEIwwEIwwEIw


LLL

LLLL

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,ˆˆ''ˆ''00

WwdxwfdxwEIwLL

0.(0)'w0(0)w0;(0)w'0w(0)



305




This equation gives the energy balance with respect to the variational space as

and the essential boundary conditions – for a cantilever beam, for instance – as

Remark. In addition, the trial and test function spaces are determined by the weak

form, as usual – although in this case the space is not an appropriate choice

anymore due to the second order derivatives present in the bilinear form.

.ˆ''ˆ'''ˆ)''(ˆ)'''(

'ˆ)'''(ˆ)'''(ˆ')'''(ˆ

000

0000

WwdxwEIwwEIwwEIw


LLL

LLLL

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,ˆˆ''ˆ''00

WwdxwfdxwEIwLL

0.(0)'w0(0)w0;(0)w'0w(0)

)(1 H



306

The weak form of the Euler−Bernoulli beam problem: Let us consider a

cantilever beam subject to a distributed load . Find s.t.

with the bilinear form, load functional and the variational space

Ww

).(}0)0(',0)0(|)({

,dˆ)ˆ(

,d''ˆ'')ˆ,(

22

HvvHvW

wfwl

wEIwwwa

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,ˆ)ˆ()ˆ,( Wwwlwwa

),0(),(2 LLf

x

0x Lx

IE,

f



307

The weak form of the Euler−Bernoulli beam problem: Let us consider a

cantilever beam subject to a distributed load . Find s.t.

with the bilinear form, load functional and the variational space

Remark. For the first time, the variational space is a subspace of Sobolev space

, which will essentially influence the finite element space. Accordingly,

continuity, coercivity and error estimates will be formulated with respect to the

norm – basic principles for the analysis will remain the same, however.

Ww

).(}0)0(',0)0(|)({

,dˆ)ˆ(

,d''ˆ'')ˆ,(

22

HvvHvW

wfwl

wEIwwwa

Rak-54.3200 / 2016 / JN

,ˆ)ˆ()ˆ,( Wwwlwwa

),0(),(2 LLf

x

0x Lx

IE,

f

)(2 H

2H



308

Break exercise 6

.ˆ,ˆ...d''ˆ'')ˆ,()(

,...d''''),()(

22

22

WuvuvCuEIvuvaii

WvvvEIvvvai

Rak-54.3200 / 2016 / JN

Show that the bilinear form of the Euler−Bernoulli beam problem is (i) elliptic and

(ii) continous with respect to the norm:)(2 H

For which type of values of the cross sectional

quantities E and I the quotient appearing

in the corresponding error estimates will be large/small?

/C



6.2 Finite element formulation for


309Rak-54.3200 / 2014 / JN

Conformity. It is now clear that a piecewise linear continuous finite element

approximation is not an appropriate choice for the current beam problem. Instead,

we have to find out which kind of conditions for the polynomial order and

continuity accross the elements will satisfy the conformity condition

How about a second order (k = 2) piecewise linear continuous approximation?

).(2 HWWh



310Rak-54.3200 / 2016 / JN






Previously, the conformity (subspace) condition was of the form

and it was satisfied by simply defining the discrete space as

).(2 HWWh

)(1 HVVh

}.)(|)({ |1 KPvHvV kKh



311Rak-54.3200 / 2016 / JN






Previously, the conformity (subspace) condition was of the form

and it was satisfied by simply defining the discrete space as

In practice, we have previously used a piecewise linear approximation which is

globally continuous (from element to element). In general, is this a sufficient

property for satisfying the condition (cf. Coffee exercise 5.1)?

).(2 HWWh

)(1 HVVh

}.)(|)({ |1 KPvHvV kKh

)(1 Huh

K nK1K

v

312Rak-54.3200 / 2014 / JN

Continuity. Is a continuous function an function, or even an function?


Euler−Bernoulli beams1H 2H

313Rak-54.3200 / 2016 / JN


It can be shown that continuity accross the element edges is a sufficient condition

for the existence of the weak derivative as long as the funcion has a weak derivative

locally in each element (d = 1, 2, 3):



).()( and )(,R 11

|

d HvCvKHvK

1H 2H

314Rak-54.3200 / 2016 / JN





Since a finite element approximation is often a function which is a polynomial in

each element, , and hence infinitely smooth in each element – due to

the fact that – it means that continuity accross the

element edges is an essential condition to be required from the approximation.



).()( and )(,R 11

|

d HvCvKHvK

)(| KPv kK )()()( 1 KHKCKPk

1H 2H

315Rak-54.3200 / 2016 / JN





Since a finite element approximation is often a function which is a polynomial in

each element, , and hence infinitely smooth in each element – due to

the fact that – it means that continuity accross the

element edges is an essential condition to be required from the approximation.

Accordingly, this means that continuity of the derivative of a function accross the

element edges is a sufficient condition for the existence of the second weak

derivative as long as the function has a second order weak derivative locally in each

element:



).()( and )(,R 11

|

d HvCvKHvK

).()( and )(,R 212

|

d HvCvKHvK

)(| KPv kK )()()( 1 KHKCKPk

1H 2H

316

Conforming finite element method for the Euler−Bernoulli beam problem: Let

us consider a cantilever beam subject to a loading .

Find such thathh Ww

Rak-54.3200 / 2014 / JN

x

0x Lx

IE,

f

),0(),(2 LLf



}.)(,0)0(',0)0(|)({

},0)0(',0)0(|)({

dˆ)ˆ(

d''ˆ'')ˆ,(

,ˆ)ˆ()ˆ,(

3|

1

2

KPvvvCvW

vvHvWW

wfwl

wEIvwva

Wwwlwwa

Kh

h

hh

317

Conforming finite element method for the Euler−Bernoulli beam problem: Let

us consider a cantilever beam subject to a loading .

Find such that

Remark. continuity, i.e., continuity of the function and its derivatives accross the

element edges, will be satisfied by applying the third order Hermite shape functions

which essentially differ from the previously used Lagrange shape functions.

hh Ww

Rak-54.3200 / 2014 / JN

x

0x Lx

IE,

f

),0(),(2 LLf

1C



}.)(,0)0(',0)0(|)({

},0)0(',0)0(|)({

dˆ)ˆ(

d''ˆ'')ˆ,(

,ˆ)ˆ()ˆ,(

3|

1

2

KPvvvCvW

vvHvWW

wfwl

wEIvwva

Wwwlwwa

Kh

h

hh

318

.)(,0

,1)(,)()(

1jdxu

ji

jixdxxu jjhjii

n

i ih

Rak-54.3200 / 2014 / JN

Previously, for the continuous piecewise polynomial finite element approximation of

order k, the degrees of freedom were the nodal values (Lagrange interpolation):



319

.)(,0

,1)(,)()(

1jdxu

ji

jixdxxu jjhjii

n

i ih

Rak-54.3200 / 2014 / JN



Now, a piecewise cubic (third order, i.e., k = 3) finite element approximation will be

used, with nodal values of both the function and its derivatives taken as degrees of

freedom (Hermite interpolation):

,0)('12,0

12,1)(

,)()()(

1212

22121 12

jxji

jix

dxdxxw

jiji

iii

n

i ih

1jxjx

)( jh xw )( 1jh xw



320

.)(,0

,1)(,)()(

1jdxu

ji

jixdxxu jjhjii

n

i ih

Rak-54.3200 / 2014 / JN



Now, a piecewise cubic (third order, i.e., k = 3) finite element approximation will be

used, with nodal values of both the function and its derivatives taken as degrees of

freedom (Hermite interpolation):

.)(',)(

.0)(2,0

2,1)('

,0)('12,0

12,1)(

,)()()(

212

22

1212

22121 12

jdxwdxw

jxji

jix

jxji

jix

dxdxxw

jjhjjh

jiji

jiji

iii

n

i ih

1jxjx

)( jh xw

)(' jh xw )(' 1jh xw

)( 1jh xw



321

)(')()(

)(

)())(2()(

)(

)('))((

)(

),())(2()(

)(

12

12

)1(2

13

12

12

2

21

2

13

21

12

ih

i

iii

ih

i

iiii

ih

i

iii

iiiih

i

iiii

xwh

xxxxx

xwh

xxhxxx

xwh

xxxxx

xxhxwh

xxhxxx

Rak-54.3200 / 2014 / JN

In this approach, four shape functions will be related to each element; two to each

end of each interval :),( 1ii xx

1ixix

:0)(' 1 ix

:1)(' 1 ix

:0)(' ix

:1)(' ix

:0)( ix

:1)( ix

:0)( 1 ix

:1)( 1 ix



322

)(')()(

)(

)())(2()(

)(

)('))((

)(

),())(2()(

)(

12

12

)1(2

13

12

12

2

21

2

13

21

12

ih

i

iii

ih

i

iiii

ih

i

iii

iiiih

i

iiii

xwh

xxxxx

xwh

xxhxxx

xwh

xxxxx

xxhxwh

xxhxxx

Rak-54.3200 / 2014 / JN

In this approach, four shape functions will be related to each element; two to each

end of each interval :),( 1ii xx

1ixix

:0)(' 1 ix

:1)(' 1 ix

:0)(' ix

:1)(' ix

:0)( ix

:1)( ix

:0)( 1 ix

:1)( 1 ix



323Rak-54.3200 / 2014 / JN

Each element will give its contribution to the stiffness matrix and force vector as



.,...,0),,(,)(

2,...,1,,''''),(

1

)()(

)(

1

)(

1

)(

nixxKdxflF

nqpdxEIaK

ii

ex

xpKp

e

p

x

xqpKqp

e

pq

i

i

e

i

i

e

324Rak-54.3200 / 2014 / JN

Each element will give its contribution to the stiffness matrix and force vector as

In practice, only four shape functions are nonzero in each element and hence



.

,

)(

'

)()(

'

)()(

'

')(

''''''

''

''''''

''

)(

)(2

)(2

)(1

)(1

)(

)(2

)(

)(2

)(

)(1

)(

)(1

)(

)(2

)(2

)(

)(2

)(2

)(

)(1

)(2

)(

)(1

)(2

)(

)(2

)(2

)(

)(2

)(2

)(

)(1

)(2

)(

)(1

)(2

)(

)(2

)(1

)(

)(2

)(1

)(

)(1

)(1

)(

)(1

)(1

)(

)(2

)(1

)(

)(2

)(1

)(

)(1

)(1

)(

)(1

)(1

Te

w

e

w

e

w

e

w

e

w

w

w

w

e

wwwwwwww

wwwwwwww

wwwwwwww

wwwwwwww

e

eeee

e

e

e

e

e

e

e

e

e

ee

e

ee

e

ee

e

ee

e

ee

e

ee

e

ee

e

ee

e

ee

e

ee

e

ee

e

ee

e

ee

e

ee

e

ee

e

ee

dddd

F

F

F

F

KKKK

KKKK

KKKK

KKKK

d

FK

.,...,0),,(,)(

2,...,1,,''''),(

1

)()(

)(

1

)(

1

)(

nixxKdxflF

nqpdxEIaK

ii

ex

xpKp

e

p

x

xqpKqp

e

pq

i

i

e

i

i

e

325Rak-54.3200 / 2014 / JN

The global stiffness matrix and force vector can now be assembled in a usual way.

Remark. Hermite type third order continuous deflection approximation

implies, by taking (elementwise) derivatives, that

─ rotation approximation is piecewise quadratic and continuous,

─ moment approximation is piecewise linear and discontinuous,

─ shear force approximation is piecewise constant and discontinuous.



hw

'hw

1C

326Rak-54.3200 / 2014 / JN

The global stiffness matrix and force vector can now be assembled in a usual way.

Remark. Hermite type third order continuous deflection approximation

implies, by taking (elementwise) derivatives, that

─ rotation approximation is piecewise quadratic and continuous,

─ moment approximation is piecewise linear and discontinuous,

─ shear force approximation is piecewise constant and discontinuous.

Remark. In a similar manner, one can deduce that a typical continuous (Lagrange

type) quadratic finite element approximation would lead to piecewise linear

discontinuous rotation. Taking then the (global) derivative of the rotation would lead

on element borders to Dirac delta functions which are not square-integrable. Hence,

the bilinear form of the problem would not be defined for this type of trial functions

and the problem would not be solvable. This argumentation gives a fairly intuitive

justification for the continuity requirement implied by the conformity condition

Due to conformity, the error analysis can be carried out by standard techniques:

1C

1C

).(2 HWWh



hw

'hw

327Rak-54.3200 / 2016 / JN

Error estimates. continuity and coersivity of the bilinear form, together

with Galerkin orthogonality, imply an error estimate following the Cea’s lemma,

,22 hh Wvvw

Cww

2H 2H



328Rak-54.3200 / 2016 / JN



from which we get a more quantitative estimate (assuming a smooth solution)

,22 hh Wvvw

Cww

.3for ,4

2

31

1

2

kwhcwhcwwk

k

kh


Euler−Bernoulli beams2H 2H

329Rak-54.3200 / 2016 / JN




Above – as well as in earlier error estimates – we have used a result from

approximation theory for the interpolation error of polynomials, stating that a

polynomial of order interpolates a function with the following accuracy:

,22 hh Wvvw

Cww

.~1

1

k

mkkm

whcww



.3for ,4

2

31

1

2

kwhcwhcwwk

k

kh

wkw~

2H 2H

330Rak-54.3200 / 2016 / JN




Above – as well as in earlier error estimates – we have used a result from

approximation theory for the interpolation error of polynomials, stating that a

polynomial of order interpolates a function with the following accuracy:

Remark. It can be shown that the Hermite finite element approximation gives

accurate nodal values for the deflection and its derivative (rotation) of the Euler–

Bernoulli beam problem: and for all nodes .

,22 hh Wvvw

Cww

.~1

1

k

mkkm

whcww

wk



)()( ihi xwxw )(')(' ihi xwxw ix

w~

.3for ,4

2

31

1

2

kwhcwhcwwk

k

kh

2H 2H

331Rak-54.3200 / 2014 / JN

Above, continuity was shown to imply an derivative under certain

circumstances. With certain assumptions, the opposite implication holds as well

(Sobolev embedding theorem):

For instance, in 1D case an function is continuous, while in 2D case, instead, a

function has to be regular in order to be continuous.

If a function is regular inside its domain , how regular it is on the boundary of

the domain? If the domain is bounded and its boundary is regular, then the

trace of a function on the boundary is an function (Trace theorem):

where is linear.

If it further holds that , then .

.Rwith ,2/0),()( d dkmCH mk

1H

6.X Sobolev embedding & trace

2H

1H

1H

,)()( 12

HL

vcTv

Tv

1C

v 2L

)()(: 21 LHT

vTv)(Cv

)(mC )(kH

332

6.X Sobolev embedding & trace

Coffee exercise 9

Rak-54.3200 / 2014 / JN

0,0

10),1

sin()(

x

xx

xxv

a

Find the derivative of the function

with . Show that whenever .2/3a)(' 2 Lv1a

QUESTIONS?

ANSWERS”

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6 Finite element methods for the - Aalto€¦ · 288 Contents 1. Strong and weak forms for Euler−Bernoullibeams 2. Finite element methods for Euler−Bernoullibeams Learning outcome