18. chapter 18- geometric nonlinearities _a4

8/11/2019 18. Chapter 18- Geometric Nonlinearities _a4

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______________________Basics of the Finite Element Method Applied in Civil Engineering

177

CHAPTER 8

GEOMETRIC NONLINEARITIES

The geometric nonlinear behavior is usually associated with thin, flexible

structures, subjected to simultaneous bending and compression. Thus,

structural components as frame columns or diaphragms are dangerously

exposed to the buckling effect. Common materials used in civil engineering,

as masonry, concrete or steel are quite stiff, with rather large values of the

elasticity modulus. Thus, the geometric nonlinearities due to a very

deformable medium are rarely encountered.

Generally, the geometric nonlinearity is an approach based on the

structures geometry changes, occurring when the structure deforms under

the applied loading conditions. When displacements are no longer

negligible, the stiffness matrix K is not a constant matrix any more, but

becomes a function of displacement . The geometric nonlinearities are

associated with shape changes and overall rotations.

Two types of geometric nonlinearities are discussed below:

- large stains the strains are no longer negligible, shape changes and

rotations occur;- large deflections mechanism type of deformation, with large

rotations but small strains (when structural components exhibit a

quasi solid body motion).

1. Large strains

During a large strain deformation, the applied loads make the solid move

from the undeformed position to the deformed one. The loading points are

also following the same displacement. Both initial and final locations are

represented by the corresponding position vector: X for the undeformed

state and xfor the deformed state (see figure 18.1). The displacement vector

is computed as:

Xx = (18.1)


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Chapter 18 Geometric Nonlinearities___________________________________________

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Fig. 18.1Position vectors during the motion of the deforming body

The deformation gradient written in terms of the displacement is defined as:

X

I

X

xF

+=

= (18.2)

The deformation gradient includes the volume change, the rotation and the

shape change. The volume change, with V0and Vthe initial and the current

volumes respectively, is

Fdetdd

0

=VV (18.3)

The deformation gradient can be separated into a rotation and a shape

change:

RUF= (18.4)

with Rthe rotation matrix and Uthe shape change (or stretch) matrix. Once

the stretch matrix is known, a logarithmic strain measure can be defined as:

U ln= (18.5)

Equation (18.5) is determined by spectral decomposition of U:

x

yxX

Undeformed Deformed


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Tiii ee =

3

1ln (18.6)

with ithe eigenvalues and eithe eigenvectors of Urespectively.

The evaluation of is made by incremental approximation, n being the

current load step and n - 1the previous one:

= n d (18.7)

with

nn

U = ln (18.8)

The increment of the stretch matrix Unis computed from the incremental

deformation gradient:

nnn URF = (18.9)

where 11= nnn FFF .

The approximate method starts with

mmn

T

mn RR ,= (18.10)

with Rm the rotation matrix computed from the deformation gradient

evaluated at the midpoint position:

X

IURF

+== mmmm (18.11)

and the midpoint displacement

( )12

1= nnm (18.12)


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180

The midpoint strain increment is also computed from the midpointconfiguration:

nmmn B = , (18.13)

The total strain of the current load step is computed by adding the strain

increment to the previous load step strain:

nnn += 1 (18.14)

On element level, the large strain formulation is represented by a tangent

stiffness matrix with two components: the usual stiffness matrix and thegeometric stiffness contribution:

iii GKK += (18.15)

2. Large deflections

The large deflection hypothesis with large rotations and small mechanical

strain - is seldom encountered in civil engineering, except for assemblage

details of prefabricated structural components. It is more suitable for

machineries or mechanism type of structural parts.

In case of large deflection theory, the logarithmic scale measure (18.5) isreplaced by the small strain measure:

IU = (18.15)

where Uis the stretch matrix and Ithe identity matrix.

To implement large deflections, a corotational approach is used. The

nonlinearities are enclosed in the strain-displacement relationship, which

takes a special form:

nvn TBB =

(18.16)


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181

where Bv is the usual strain-displacement relationship in the originalelement coordinate system and Tn the orthogonal transformation matrix,

relating the original element coordinates to the rotated element coordinates.

Tn can be computed directly or in realationship with the rotation of the

element coordinate system Rn

nvn RTT = (18.17)

with Tvthe original transformation matrix.

The rotated coordinates differ from the original coordinates by the amount

of rigid body rotation. Therefore Tn is computed by separating the rigid

body rotation from the total deformation n using equation (18.4). Thedisplacement field can be divided into a rigid body translation, a rigid body

rotation and a component which causes strains

dr += (18. 18)

where r is the rigid body motion and

d the deformational displacement,

containing both translational and rotational DOF. The translational

component can be computed from the displacement field with:

( ) vvnd xxR += (18.19)

with xvthe original element coordinates in the global coordinate system and

the displacement vector in the same global coordinates. The rotational

components are extracted by subtracting the nodal rotations from the

element rotations given by r.

The elemental tangent stiffness matrix yields:

VnvT

vV

T

ne dTEBBTK = (18.20)

The elastic strain is computed from:

d

nv

el

n B = (18.21)


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182

withd

n the element deformation producing straining.

The modeling of large deflection can be summarized in the following steps:

1. Computation of the updated transformation matrix Tn for eachelement.

2. Extraction of the updated displacements dn from the total element

displacement nfor computing the stresses and the restoring forces.3. Updating the nodal rotations with the appropriate rotational

increments .

18. chapter 18- geometric nonlinearities _a4

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