lecture in nonlinear fem on the building- and civil...
TRANSCRIPT
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
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
2
Non linear FEM , Dep. of Computational Mechanics, AAU Esbjerg, Denmark
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
Non linear FEM , Dep. of Computational Mechanics, AAU Esbjerg, Denmark
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
Non linear FEM , Dep. of Computational Mechanics, AAU Esbjerg, Denmark
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
Non linear FEM , Dep. of Computational Mechanics, AAU Esbjerg, Denmark
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
Non linear FEM , Dep. of Computational Mechanics, AAU Esbjerg, Denmark
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
Non linear FEM , Dep. of Computational Mechanics, AAU Esbjerg, Denmark
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
Non linear FEM , Dep. of Computational Mechanics, AAU Esbjerg, Denmark
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
Non linear FEM , Dep. of Computational Mechanics, AAU Esbjerg, Denmark
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
Non linear FEM , Dep. of Computational Mechanics, AAU Esbjerg, Denmark
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
Non linear FEM , Dep. of Computational Mechanics, AAU Esbjerg, Denmark
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
Non linear FEM , Dep. of Computational Mechanics, AAU Esbjerg, Denmark
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
Non linear FEM , Dep. of Computational Mechanics, AAU Esbjerg, Denmark
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
Non linear FEM , Dep. of Computational Mechanics, AAU Esbjerg, Denmark
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
Non linear FEM , Dep. of Computational Mechanics, AAU Esbjerg, Denmark
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
Non linear FEM , Dep. of Computational Mechanics, AAU Esbjerg, Denmark
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
Non linear FEM , Dep. of Computational Mechanics, AAU Esbjerg, Denmark
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
Non linear FEM , Dep. of Computational Mechanics, AAU Esbjerg, Denmark
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
Non linear FEM , Dep. of Computational Mechanics, AAU Esbjerg, Denmark
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
Non linear FEM , Dep. of Computational Mechanics, AAU Esbjerg, Denmark
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
Non linear FEM , Dep. of Computational Mechanics, AAU Esbjerg, Denmark
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.
22