3.3 Solution-dependent Loading

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In the previous section, the loading terms on the right-hand side of Equation (2.1) were

assumed to be independent of the displacements. This, however, is not necessarily the case.

The effect of the surface traction depends on the size and orientation of the surface it acts

on. Because of the deformation, both the size and the orientation can change. The body

force term too can depend on the displacements. For instance, the centrifugal force depends

on the distance from the rotation axis. This distance can change because of the deformation

of the structure.

3.3.1 Centrifugal forces

In general, the body forces f can be linearly approximated at U = V by

f (X + V +W) = f (X + V ) + ∂f

∂U

____

U=V

・W. (3.22)

The centrifugal body forces f take the form (Equation (2.230))

f = {(q − p1) − [(q − p1) ・ e]e}ω2. (3.23)

Now, we assume that the location of the rotation axis does not change because of the

deformation, that is,

p1 = P1 (3.24)

e = E (3.25)

whereas

q = Q+ U. (3.26)

Q is the original position of q. Accordingly, Equation (3.22) now reads

f (Q+ V +W) = {(Q+ V − P1) − [(Q+ V − P1) ・ E]E} ω2 + [W − (W ・ E)E]ω2.

(3.27)

Notice that f is linear in W. By comparison with Equation (3.22), one observes

f (Q+ V ) = {(Q+ V − P1) − [(Q+ V − P1) ・ E]E} ω2 (3.28)

and

∂f

∂U

____

U=V

・ W = [W − (W ・ E)E_]ω2

= [W − E_(E ・W)]ω2

= [W − (E_ ⊗ E) ・W]ω2

= [I − E_ ⊗ E] ・Wω2. (3.29)

GEOMETRIC NONLINEAR EFFECTS 151

Consequently, the centrifugal term in Equation (3.15) amounts to

_

V0

ρ0f KδWK dV = _

V0

ρ0f K

0 δWK dV + _

V0

ρ0 _WK − (WLEL)EK_ω2δWK dV

(3.30)

where

f K

0

= {(Q+ V − P1) − [(Q+ V − P1) ・ E]E} ω2 ・ GK (3.31)

does not depend on the deformation. The first term on the right-hand side of Equation (3.30)

is the instantaneous force contribution, which has already been taken into account in

Equation (3.15). The second term, however, is new and contributes to the stiffness matrix.

Indeed, writing

WK =

i

ϕjWK

j (3.32)

and similar for δWK leads to

 

e

_

V0e

ρ0

 

j

ϕjW K

j

−

j

ϕjW L

j ELEK

_

 

i

ϕiδWiK

_

ω2 dV (3.33)

or

 

e

 

i

 

j

_

V0e

ρ0ϕiϕj dV _δ K

L

− ELEK_W L

j δWiKω2. (3.34)

The contribution to the stiffness matrix amounts to

            K(iK)(jL)

= −_

V0e

ρ0ϕiϕj dV _δ K

L

− ELEK_ ω2. (3.35)

The minus sign results from bringing the stiffness contribution to the left-hand side. Notice

that, because of the direction of the rotation axis, the contribution to the stiffness matrix is

anisotropic.

3.3.2 Traction forces

The traction term in Equation (3.15) amounts to

I = _

A0t

T

K

(N)δWK dA. (3.36)

Here, T

K

(N) is a function of the deformation. Recall that it is defined by (Equation (1.263)):

T (N) = t (n)

_ da

dA

_

. (3.37)

152 GEOMETRIC NONLINEAR EFFECTS

For a uniform pressure σ = −pg_, one arrives at

T (N) = σ ・ n _ da

dA

_

= −pg_ ・ n _ da

dA

_

. (3.38)

Hence,

T

K

(N)

= T (N) ・ GK

= −pGK ・ g_ ・ n _ da

dA

_

= −pgK

lglknk

_ da

dA

_

= −pgKk _dak

dA

_

. (3.39)

Recall that (Equation (1.66))

dak = JXL

,k dAL. (3.40)

Accordingly,

T

K

(N)

= −pgKk(JXL

,k)NL (3.41)

where

NL = dAL

dA

. (3.42)

Consequently, Equation (3.36) now reads

I = −_

A0t

pgKk(JXL

,k)δWK dAL. (3.43)

            XK

,k is the inverse of        xk

,K . The inverse of a matrix is the transpose of the matrix of its

cofactors divided by its determinant:

XL

,k

= 1

2J

eknm eLNMxn

,Nxm

,M. (3.44)

Assuming that p does not vary over the surface, substitution of Equation (3.44) into

Equation (3.43) yields

I = −p

2

gKkeknm eLNM _

A0t

xn

,Nxm

,MδWK dAL. (3.45)

GEOMETRIC NONLINEAR EFFECTS 153

Since

xm = XMgm

M

+ Um (3.46)

= XMgm

M

+ V m + Wm (3.47)

=: xm + Wm (3.48)

where Equation (3.48) defines xm, and using the shape functions

Wm =

j

ϕjW m

j (3.49)

δWK =

i

ϕiδWiK (3.50)

one obtains

I = −p

2

gKkeknm eLNM _

 

i

_

A0t

xn

,Nxm

,Mϕi dALδWiK

+

i

 

j

_

A0t

ϕj,Nxm

,Mϕi dALW n

j δWiK

+

i

 

j

_

A0t

xn

,Nϕj,Mϕi dALW m

j δWik

+ O(_W_3), _W_ → 0 (3.51)

≈ −

i

_

A0t

pgKkϕi dakδWiK

− p

2

gKkeknm eLNM

i

 

j

_

A0t

_xn

,Nϕj,M − ϕj,Nxn

,M_ ϕi dALW m

j δWiK. (3.52)

The first term in Equation (3.52) is a force term already encountered in Section 3.1, the

second term yields a stiffness contribution:

            K(iK)(jM)

= p

2

gKkgmMeknm eLNP _

A0t

_xn

,Nϕj,P − ϕj,Nxn

,P _ ϕi dAL (3.53)

or, interchanging m, M with l, L,

            K(iK)(jL)

= p

2

gKkglLeknl eMNP _

A0t

_xn

,Nϕj,P − ϕj,Nxn

,P _ ϕi dAM. (3.54)

This stiffness contribution is not symmetric. To reduce the computational costs, a symmetrization

can be performed by replacing             K by 12

_          K +      KT _.

154 GEOMETRIC NONLINEAR EFFECTS

p

p

p

8 h

ν = 0.0

h/10

h/10

Figure 3.6 Slender beam under hydrostatic pressure

−2 −1.5 −1 −0.5 0

0

0.5

0.5

1

1

1.5

1.5

2

2

2.5

105 p

E

103ωh_ρ

E

Without traction stiffness

With traction stiffness

Figure 3.7 Lowest eigenfrequency of the beam

3.3.3 Example: a beam subject to hydrostatic pressure

A slender beam (Figure 3.6) is dropped in the ocean. As it sinks, the pressure steadily

increases and the question arises whether the beam will buckle. Therefore, the eigenfrequencies

are calculated (cf. Section 2.9.3) since buckling will occur at any zero-crossing

of the lowest eigenfrequency. Applying the stress stiffness and large deformation stiffness

leads to the solid curve in Figure 3.7. Buckling occurs for a large enough pressure. However,

taking the traction stiffness also into account yields the dashed curve: no buckling

takes place! Intuitively, as soon as the beam tends to buckle, the deformation-induced traction

forces along the sides of the beam stabilize its state. Other applications can be found

in (Rumpel and Schweizerhof 2003).

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