Lecture in Nonlinear FEM on the Building- and Civil...

Lecture in Nonlinear FEM

on

the Building- and Civil Engineering sectors 8.th. semester

for

the Building- and Civil Engineering, B8k, andMechanical Engineering, B8m

AALBORG UNIVERSITY ESBJERG, DENMARK

*****************

Theme:Design of marine constructions.

1

http://www.aaue.dk/bm/dk/b8.html

http://www.aaue.dk/bm/dk/m8.html

http://www.aaue.dk/

Non linear FEM , Dep. of Computational Mechanics, AAU Esbjerg, Denmark

Outline: Updated: 15. februar 2005

1. Introduction Notes2. Geometrical nonlinearity - strain measures Cook 17.1, 17.93. Geometrical nonlinearity - appl. in buckling analysis Cook 17.104. Stress stiffness Cook 18.1-18.45. Buckling Cook 18.5-18.66. Material nonlinearity - introduction Cook 17.3-17.47. Material nonlinearity - solution methods Cook 17.6, 17.28. Contact nonlinearity Cook 17.89. Nonlinear dynamic problems Cook 11.1-11.510. Nonlinear dynamic problems Cook 11.11-11.18

Literature:

Noter → A. Kristensen: http://www.aaue.dk/bm/dk/notes.html

Cook→ Cook, R. D. 2002: Concepts and applications of finite element analysis.John Wiley& Sons

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http://www.aaue.dk/bm/dk/notes.html#m5

http://www.aaue.dk/bm/dk/

http://www.aaue.dk/bm/dk/notes.html


2. Geometrical nonlinearity - strain measures

Programme:

Last time 4

General FEA formulation of geometric nonlinearity 6

Incremental equation of equilibrium 7

The nonlinear strain-displacement matrix 14

Explicit definition of the tangent-stiffness matrix 21

Examples

Assignments

3




Last timeLinear FEA is based on

• linearized geometrical equations (strain-displacement relations):{ε}= [B]{d}

• linearized constitutive equations (stress-strain relations):{σ}= [E]{ε}= [E][B]{d}

• equations of equilibrium: {Ri}= {Re}, linear so that:[K]{D}= {Re}

and suitable boundary conditions, i.e. the assumptions made are often crude.

4




Last timeTypes of structural nonlinearity classifications used in engineering problems:

Geometric nonlinearity

Material nonlinearity

Contact or boundary nonlinearity

The nonlinear behaviour occur as stiffness and loads become functions of displacementor deformation, i.e. in

[K]{D}= {R}both the structural stiffness matrix [K] and possibly the load vector {R} become functionsof the displacements {D}. Therefore it is not possible to solve for {D} immediately as [K]and {R} is not known in advance.

Therefore an iterative process is needed to obtain {D} and the associated [K] and {R}such that [K]{D} with {R}.

5




General FEA formulation of geometric nonlinearityAs basis is selected the Lagrangian formulation (deformations refers to the original config-uration, i.e. undeformated state). The procedure to establish a general FEA formulation ofgeometric nonlinearity is:

1. Derivation of a general expression for the incremental equation of equilibrium, including

a) the relation between strain increments d{ε} and displacement increments d{d}b) introduction of the nonlinear strain matrix [BL(d{d})]c) general (implicit) definition of the tangent-stiffness matrix [KT]

2. Derivation of the nonlinear strain-displacement matrix [BL(d{d})] for iso-parametricsolid elements

3. Explicit definition of the tangent-stiffness matrix [KT], i.e. the incremental equation ofequilibrium can be determined and solved

6




Incremental equation of equilibriumIn general: the load is applied step-wise in load-steps and in each of these load-steps nit is tried iteratively to determine the displacements dn

i , which yield equilibrium betweenthe applied forces fn and the internal forces pn

i , which depend directly of the estimateddisplacements.

The nonlinear equations of equilibrium in the residual formulation is given by:

r(d, f ) = p(d)− f = 0 (1)

The load is applied in load-steps n = 1,2, . . ..

7




Incremental equation of equilibriumIf dn

i is an approximate solution to the exact solution dn for load-step n, then a 1. orderTaylor expansion for a new equation of equilibrium is determined in the next iteration i +1:

r(dni+1, f n)≈ r(dn

i , f n)+∂r(dn

i , f n)∂d

δdni = 0

∂r(dni , f n)

∂d=

∂p(dni )

∂dis the tangent-stiffness KT(dn

i ) evaluated in the point dni . Now the new

equation of equilibrium can be written as:

r(dni , f n)+KT(dn

i )δdni = 0 ⇒ KT(dn

i )δdni =−r(dn

i , f n)

The right-hand-side is the current residual:

r(dni , f n) = rn

i = r(dni , f n) = p(dn

i )− f n

whereby the incremental equation of equilibrium is given by

KT(dni )δdn

i =−rni

When this equation of equilibrium is solved with respect to δdni , the displacement dn

i isupdated by:

dni+1 = dn

i +δdni

8




Incremental equation of equilibriumThe residual {R} is defined as the difference between the internal {Rint} and external forces{Rext}. The internal forces are computed from the stress state in the structure and theexternal forces are given by the vector {F}.

{R}= {Rint}−{Rext}=Z

V[B̄]T{σ}dV−{F}= {0} (2)

The matrix [B̄] is based on the strain definition

d{ε}= [B̄]d{d} (3)

i.e. [B̄] provide the relation between the strain increments d{ε} and the displacement in-crements d{d}.In the linear analysis is given that {ε} = [B0]{d} but for large displacements the strain de-pend nonlinearly of the displacements as stated in the Green-Lagrangian strain definition.Therefore [B̄] is rewritten to a sum of two matrices, i.e. the strain-displacement matrix [B0]from the linear analysis and the matrix [BL(d{d})], which is a function of the displacements{d}. This yield

[B̄] = [B0]+ [BL(d{d})] (4)

9




Incremental equation of equilibriumThe stress-strain relations are defined

{σ}= [E]{ε} (5)

The stresses and strains must be load consistent (work-equivalent).

Differentiation of the residual {R} in the equations of equilibrium 2 yield

d{R} = d

(ZV[B̄]T{σ}dV−{F}

)(6)

(7)

=Z

Vd[B̄]T{σ}dV +

Z

V[B̄]Td{σ}dV (8)

(9)

= [KT]d{d} (10)

as the tangent-stiffness matrix [KT] were introduced asd{R}d{d} .

10




Incremental equation of equilibriumNow the general expression for the tangent-stiffness matrix [KT] can be determined byinserting the found expressions in equation 10.

Combining the equations 5 and 3 yield

d{σ}= [E]d{ε}= [E][B̄]d{d}

Rewriting equation 4 on incremental form yield

d[B̄] = d([B0]+ [BL(d{d})]) = d[BL]

11




Incremental equation of equilibriumThereby equation 10 can be rewritten to

d{R} =Z

Vd[BL]T{σ}dV

︸︷︷︸=[Kσ]d{d}

+Z

V[B̄]T[E][B̄]dVd{d}

= [Kσ]d{d}+Z

V([B0]+ [BL(d{d})])T[E]([B0]+ [BL(d{d})])dVd{d}

= [Kσ]d{d}+Z

V[B0]T[E][B0]dVd{d}

︸︷︷︸=[K0]d{d}

(11)

+Z

V[B0]T[E][BL]+ [BL]T[E][BL]+ [BL]T[E][B0]dVd{d}

︸︷︷︸=[KL]d{d}

= (K0+KL +Kσ)︸︷︷︸=KT

d{d}

= [KT]d{d}

12




Incremental equation of equilibriumNow a general expression for the tangent-stiffness matrix [KT], and the incremental equa-tion of equilibrium

KT(dni )δdn

i =−rni

can be written as

[KT({d}ni )]δ{d}n

i = −{R}ni (12)

Rewriting equation 2 on incremental form yields

{R}ni =

Z

V[B̄({d}n

i )]T{σ}dV−{F}n

=Z

V([B0]+ [BL(d{d}n

i )])T{σ}dV−{F}n = {0} (13)

The incremental equation of equilibrium 12 is solved by a Newton-Raphson procedure. Inorder to define the incremental equation of equilibrium it is necessary to know the nonlinearpart [BL].

13




The nonlinear strain-displacement matrixIn order to derive the expressions in the stiffness matrix [KT], it is necessary to look moreclosely on the strain-displacement relations, i.e. similar to the general derivation of thestiffness matrix for iso-parametric elements.

The components in the Green-Lagrangian strain tensor can generally be written as:

εx =∂u∂x

+12

[(∂u∂x

)2

+(

∂v∂x

)2

+(

∂w∂x

)2]

γxz =∂w∂x

+∂u∂z

+[

∂u∂x

∂u∂z

+∂v∂x

∂v∂z

+∂w∂x

∂w∂z

]

εy =∂v∂y

+12

[(∂u∂y

)2

+(

∂v∂y

)2

+(

∂w∂y

)2]

γxy =∂v∂x

+∂u∂y

+[

∂u∂x

∂u∂y

+∂v∂x

∂v∂y

+∂w∂x

∂w∂y

]

εz =∂w∂z

+12

[(∂u∂z

)2

+(

∂v∂z

)2

+(

∂w∂z

)2]

γyz =∂w∂y

+∂v∂z

+[

∂u∂y

∂u∂z

+∂v∂y

∂v∂z

+∂w∂y

∂w∂z

]

14




The nonlinear strain-displacement matrixThe general Green-Lagrangian strain vector consist of terms from the infinitesimal linearstrain vector and nonlinear terms, which arise from large displacements, i.e.

{ε}= {ε0}+{εL} (14)

where

{ε0}=

εx

εy

εz

γxy

γyz

γzx

=

∂u∂x∂v∂y∂w∂z

∂u∂y

+∂v∂x

∂v∂z

+∂w∂y

∂u∂z

+∂w∂x

(15)

15




The nonlinear strain-displacement matrixThe nonlinear terms can be written as

{εL} =12

[θx]T [0] [0]

[0] [θy]T [0]

[0] [0] [θz]T

[θy]T [θx]

T [0][0] [θw]T [θy]

T

[θz]T [0] [θx]

T

[θx][θy][θz]

(16)

=12[A]{θ} (17)

where

[θx]T =[

∂u∂x

,∂v∂x

,∂w∂x

][θy]T =

[∂u∂y

,∂v∂y

,∂w∂y

][θz]T =

[∂u∂z

,∂v∂z

,∂w∂z

]

16




The nonlinear strain-displacement matrixTaking the variation of equation 17 yield

d{εL}=12

d[A]{θ}+12[A]d{θ}= [A]d{θ} (18)

which can be seen from the definition of [A] and {θ}.Applying iso-parametric elements the formulation for an element with n nodes is:

x =n

∑i=1

Nixi y =n

∑i=1

Niyi

u =n

∑i=1

N̄iui v =n

∑i=1

N̄ivi

(19)

i.e. the same interpolation functions applied to geometry and displacement. Thus

∂u∂x

=∂∑n

i=1N̄iui

∂x=

n

∑i=1

∂N̄i

∂xui =

n

∑i=1

Ni,xui (20)

and similarly for the other components in {θ}.

17




The nonlinear strain-displacement matrixThis can written on matrix form:

{θ}=

[θx][θy][θz]

=

∂u∂x∂v∂x∂w∂x∂u∂y∂v∂y∂w∂y∂u∂z∂v∂z∂w∂z

=

n

∑i=1

Ni,xui

n

∑i=1

Ni,xvi

n

∑i=1

Ni,xwi

n

∑i=1

Ni,yui

n

∑i=1

Ni,yvi

n

∑i=1

Ni,ywi

n

∑i=1

Ni,zui

n

∑i=1

Ni,zvi

n

∑i=1

Ni,zwi

= [[g1], [g2], . . . [gn]]{d}= [G]{d} (21)

where [G] contains Ni,x, Ni,y and Ni,z as the linear strain-displacement matrix [B0].

18




The nonlinear strain-displacement matrixThe components [gi] for i = 1. . .n are obtained from

[gi] =

Ni,x [0] [0][0] Ni,x [0][0] [0] Ni,x

Ni,y [0] [0][0] Ni,y [0][0] [0] Ni,y

Ni,z [0] [0][0] Ni,z [0][0] [0] Ni,z

Ni,x

Ni,x

Ni,x

= [J]−1

Ni,ξNi,ηNi,ζ

and {d}= {{d1}T,{d2}T, . . .{dn}T}T where {di}= {u,v,w}T for i = 1. . .n (n is the number ofelement nodes in the element).

From equation 3, 4, and 14 it is given that

d{ε} = [B̄]d{d} d[B̄] = d([B0]+ [BL(d{d})]) = d[BL] {ε}= {ε0}+{εL}⇓

d{ε} = d{ε0}+d{εL}= ([B0]+ [BL(d{d})])d{d}= [BL(d{d})]d{d}

19




The nonlinear strain-displacement matrixFrom equation 18 and equation 21 it is given that

d{εL} = [A]d{θ} d{θ}= [G]d{d} (22)

⇓d{εL} = [A][G]d{d} (23)

Comparing equation 23 and equation 22 yields

d{εL} = [A(d{d})][G]d{d}and

d{εL} = [BL(d{d})]d{d}⇒ [BL(d{d})] = [A(d{d})][G]

In this manner a general expression for the nonlinear part [BL] of the strain-displacementmatrix [B̄] have been derived. The linear part [B0] of the strain-displacement matrix [B̄]is known from static linear stress analysis. Now the tangent-stiffness matrix [KT] can bedetermined.

20




Explicit definition of the tangent-stiffness matrixFrom equation 12 the tangent-stiffnessmatrix [KT] is given as

[KT] = [K0]+ [KL({d})]+ [Kσ({σ})] (24)

The stiffness matrix [K0] is known from the linear analysis as

[K0] =Z

V[B0]T[E][B0]dV

The matrix [KL] is given as

[KL] =Z

V[B0]T[E][BL]+ [BL]T[E][BL]+ [BL]T[E][B0]dV

where [K0] as well as [KL({d})] can be computed, as [B0] and [BL({d})] are known.

Finally the expression [Kσ] is determined by equation 12

[Kσ]d{d}=Z

Vd[BL]T{σ}dV (25)

21




Explicit definition of the tangent-stiffness matrixApplying 23 yield

d[BL({d})] = d([A({d})][G]) = d[A({d})][G]

Thus

d[BL]T{σ}= [G]Td[A({d})]T{σ}

= [G]T

σx[I3] τxy[I3] τxz[I3]σy[I3] τyz[I3]

σz[I3]

︸︷︷︸=[M]

d{θ}︸︷︷︸=[G]d{d}

= [G]T[M({σ})][G]d{d} (26)

where [I3] is a 3×3 identity matrix.

Finally, by inserting equation 26 in 25 provide

[Kσ] =Z

V[G]T[M][G]dV

Thereby all terms in the tangent-stiffness matrix [KT] are determined, see 24, and theincremental equation of equilibrium 12 can be derived and solved.

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