Spin Softening n Stress Stiffening

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Chapter 3: Structures with Geometric Nonlinearities

This chapter discusses the various geometrically nonlinear options available, including large strain, largedeflection, stress stiffening, pressure load stiffness, and spin softening. Only elements with displacements

degrees of freedom (DOFs) are applicable. Not included in this section are multi-status elements (such

as COMBIN40, discussed in Element Library (p. 411)) and the eigenvalue buckling capability (discussed

in Buckling Analysis (p. 792)).

The following topics are available:

3.1. Understanding Geometric Nonlinearities

3.2. Large Strain

3.3. Large Rotation

3.4. Stress Stiffening

3.5. Spin Softening

3.6. General Element Formulations

3.7. Constraints and Lagrange Multiplier Method

3.1. Understanding Geometric Nonlinearities

Geometric nonlinearities refer to the nonlinearities in the structure or component due to the changing

geometry as it deflects. That is, the stiffness [K] is a function of the displacements {u}. The stiffness

changes because the shape changes and/or the material rotates. The program can account for four

types of geometric nonlinearities:

1. Large strainassumes that the strains are no longer infinitesimal (they are finite). Shape changes (e.g. are

thickness, etc.) are also accounted for. Deflections and rotations may be arbitrarily large.

2. Large rotationassumes that the rotations are large but the mechanical strains (those that cause stresse

are evaluated using linearized expressions. The structure is assumed not to change shape except for rig

body motions. The elements of this class refer to the original configuration.

3. Stress stiffeningassumes that both strains and rotations are small. A 1st order approximation to the rota

tions is used to capture some nonlinear rotation effects.

4. Spin softening also assumes that both strains and rotations are small. This option accounts for the coup

between the transverse vibrational motion and the centrifugal force due to an angular velocity.

All elements support the spin softening capability, while only some of the elements support the other

options. Please refer to the Element Reference for details.

3.2. Large Strain

When the strains in a material exceed more than a few percent, the changing geometry due to this

deformation can no longer be neglected. Analyses which include this effect are called large strain, or

finite strain, analyses. A large strain analysis is performed in a static (ANTYPE,STATIC) or transient (AN-

TYPE,TRANS) analysis while flagging large deformations (NLGEOM,ON) when the appropriate element

type(s) is used.

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The remainder of this section addresses the large strain formulation for elastic-plastic elements. These

elements use a hypoelastic formulation so that they are restricted to small elastic strains (but allow for

arbitrarily large plastic strains). Hyperelasticity (p. 124)addresses the large strain formulation for hypere-

lastic elements, which allow arbitrarily large elastic strains.

3.2.1.Theory

The theory of large strain computations can be addressed by defining a few basic physical quantities

(motion and deformation) and the corresponding mathematical relationship. The applied loads acting

on a body make it move from one position to another. This motion can be defined by studying a position

vector in the deformed and undeformed configuration. Say the position vectors in the deformed

and undeformed state are represented by {x} and {X} respectively, then the motion (displacement)

vector {u} is computed by (see Figure 3.1: Position Vectors and Motion of a Deforming Body (p. 30)):

(3.1)= -

Figure 3.1: Position Vectors and Motion of a Deforming Body

y

x

U n d e f o r m e d D e f o r m e d

{ u }

{ X }

{ x }

The deformation gradient is defined as:

(3.2)=

which can be written in terms of the displacement of the point via Equation 3.1 (p. 30)as:

(3.3)= +

where:

[I] = identity matrix

The information contained in the deformation gradient [F] includes the volume change, the rotation

and the shape change of the deforming body. The volume change at a point is

(3.4)

=

where:

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Structures with Geometric Nonlinearities


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Vo= original volume

V = current volume

det [] = determinant of the matrixThe deformation gradient can be separated into a rotation and a shape change using the right polar

decomposition theorem:

(3.5=

where:

[R] = rotation matrix ([R]T

[R] = [I])

[U] = right stretch (shape change) matrix

Once the stretch matrix is known, a logarithmic or Hencky strain measure is defined as:

(3.6e =

([] is in tensor (matrix) form here, as opposed to the usual vector form {}). Since [U] is a 2nd order

tensor (matrix), Equation 3.6 (p. 31) is determined through the spectral decomposition of [U]:

(3.7e l==

where:

i= eigenvalues of [U] (principal stretches)

{ei} = eigenvectors of [U] (principal directions)

The polar decomposition theorem (Equation 3.5 (p. 31)) extracts a rotation [R] that represents the average

rotation of the material at a point. Material lines initially orthogonal will not, in general, be orthogonal

after deformation (because of shearing), see Figure 3.2: Polar Decomposition of a Shearing Deforma-

tion (p. 31). The polar decomposition of this deformation, however, will indicate that they will remainorthogonal (lines x-y' in Figure 3.2: Polar Decomposition of a Shearing Deformation (p. 31)). For this

reason, non-isotropic behavior (e.g. orthotropic elasticity or kinematic hardening plasticity) should be

used with care with large strains, especially if large shearing deformation occurs.

Figure 3.2: Polar Decomposition of a Shearing Deformation

y

x

x '

x

y

y '

U n d e f o r m e d D e f o r m e d

3.2.2. Implementation

Computationally, the evaluation of Equation 3.6 (p. 31) is performed by one of two methods using the

incremental approximation (since, in an elastic-plastic analysis, we are using an incremental solution

procedure):


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(3.8)e e=

with

(3.9)D De

=

where [Un] is the increment of the stretch matrix computed from the incremental deformation

gradient:(3.10)D D D

=

where [Fn] is:

(3.11)D

= - -

[Fn] is the deformation gradient at the current time step and [Fn-1] is at the previous time step.

(Hughes([156] (p. 929))) uses the approximate 2nd order accurate calculation for evaluating Equa-

tion 3.9 (p. 32):

(3.12) D De e

=

where [R1/2] is the rotation matrix computed from the polar decomposition of the deformation gradient

evaluated at the midpoint configuration:

(3.13)

=

where [F1/2] is (using Equation 3.3 (p. 30)):

(3.14)

= +

and the midpoint displacement is:

(3.15)

= + -

{un} is the current displacement and {un-1} is the displacement at the previous time step. [n] is the

rotation-neutralized strain increment over the time step. The strain incrementDe

[ ] is also computed

from the midpoint configuration:

(3.16)

=

{un

} is the displacement increment over the time step and [B1/2

] is the strain-displacement relationship

evaluated at the midpoint geometry:

(3.17)

= +

This method is an excellent approximation to the logarithmic strain if the strain steps are less than

~10%. This method is used by the standard 2-D and 3-D solid and shell elements.

The computed strain increment [n] (or equivalently {n}) can then be added to the previous strain

{n-1} to obtain the current total Hencky strain:




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(3.18e e e

= +-

D

This strain can then be used in the stress updating procedures, see Rate-Independent Plasticity (p. 64)

and Rate-Dependent Plasticity (Including Creep and Viscoplasticity) (p. 105)for discussions of the rate-

independent and rate-dependent procedures respectively.

3.2.3. Definition of Thermal Strains

According to Callen([243] (p. 934)), the coefficient of thermal expansion is defined as the fractional increase

in the length per unit increase in the temperature. Mathematically,

(3.19a =

where:

= coefficient of thermal expansion

= current length

T = temperature

Rearranging Equation 3.19 (p. 33)gives:

(3.20

= a

On the other hand, the logarithmic strain is defined as:

(3.21e

=

where:

e = logarithmic strain

o= initial length

Differential of Equation 3.21 (p. 33) yields:

(3.22e

=

Comparison of Equation 3.20 (p. 33)and Equation 3.22 (p. 33)gives:

(3.23e a =

Integration of Equation 3.23 (p. 33) yields:

(3.24e e a - = -

where:

e

= initial (reference) strain at temperature ToTo= reference temperature


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In the absence of initial strain ( e = ), then Equation 3.24 (p. 33)reduces to:

(3.25)e a = -

The thermal strain corresponds to the logarithmic strain. As an example problem, consider a line element

of a material with a constant coefficient of thermal expansion . If the length of the line is oat tem-

perature To

, then the length after the temperature increases to T is:

(3.26) = = -

e a

Now if one interpreted the thermal strain as the engineering (or nominal) strain, then the final length

would be different.

(3.27)e a

= -

where:

e= engineering strain

The final length is then:

(3.28) = + = + -

e a

However, the difference should be very small as long as:

(3.29)a

-

because

(3.30)

+

3.2.4. Element FormulationThe element matrices and load vectors are derived using an updated Lagrangian formulation. This

produces equations of the form:

(3.31)

=

where the tangent matrix has the form:

(3.32)

= +

[Ki] is the usual stiffness matrix:

(3.33)

=

[Bi] is the strain-displacement matrix in terms of the current geometry {Xn} and [Di] is the current stress-

strain matrix.

[Si] is the stress stiffness (or geometric stiffness) contribution, written symbolically as:

(3.34)

=




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where [Gi] is a matrix of shape function derivatives and [i] is a matrix of the current Cauchy (true)

stresses {i} in the global Cartesian system. The Newton-Raphson restoring force is:

(3.35

=

Some of the plane stress and shell elements account for the thickness changes due to the out-of-plane

strain zusing an approach similar to that of Hughes and Carnoy ( [157] (p. 929)). Shells, however, do not

update their reference plane (as might be required in a large strain out-of-plane bending deformation);the thickness change is assumed to be constant through the thickness. General element formulations

using finite deformation are developed in General Element Formulations (p. 50)and apply to current-

technology elementsonly.

3.2.5. Applicable Input

NLGEOM,ON activates large strain computations in those elements which support it. NLGEOM,ON also

activates the stress-stiffening contribution to the tangent matrix.

3.2.6. Applicable Output

For elements which have large strain capability, stresses (output as S) are true (Cauchy) stresses in the

rotated element coordinate system (the element coordinate system follows the material as it rotates).

Strains (output as EPEL, EPPL, etc.) are the logarithmic or Hencky strains, also in the rotated element

coordinate system.

An exception is for the hyperelastic elements. For these elements, stress and strain components maintain

their original orientations and some of these elements use other strain measures.

3.3. Large Rotation

If the rotations are large but the mechanical strains (those that cause stresses) are small, then a large-

rotation procedure can be used.

A large-rotation analysis is performed in a static (ANTYPE,STATIC) or transient (ANTYPE,TRANS) analysis

while flagging large deformations (NLGEOM,ON) when the appropriate element type is used.

All large-strain elements also support this capability, as both options account for the large rotations

and for small strains, the logarithmic strain measure and the engineering strain measure coincide.

3.3.1.Theory

Large Strain (p. 29)presented the theory for general motion of a material point. Large-rotation theory

follows a similar development, except that the logarithmic strain measure (Equation 3.6 (p. 31)) is replaced

by the Biot, or small (engineering) strain measure:(3.36 =

where:

[U] = stretch matrix

[I] = 3 x 3 identity matrix


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A corotational (or convected coordinate) approach is used in solving large-rotation/small-strain problems

(Rankin and Brogan([66] (p. 924))). "Corotational" may be thought of as "rotated with". The nonlinearities

are contained in the strain-displacement relationship which for this algorithm takes on the special form:

(3.37)

=

where:

[Bv] = usual small strain-displacement relationship in the original (virgin) element coordinate

system

[Tn] = orthogonal transformation relating the original element coordinates to the convected

(or rotated) element coordinates

The convected element coordinate frame differs from the original element coordinate frame by the

amount of rigid body rotation. Hence [Tn] is computed by separating the rigid body rotation from the

total deformation {un} using the polar decomposition theorem, Equation 3.5 (p. 31). From Equa-

tion 3.37 (p. 36), the element tangent stiffness matrix has the form:

(3.38)

=

and the element restoring force is:

(3.39)

=

where the elastic strain is computed from:

(3.40)

=

is the element deformation which causes straining as described in a subsequent subsection.

The large-rotation process can be summarized as a three step process for each element:

1. Determine the updated transformation matrix [Tn] for the element.

2. Extract the deformational displacement

from the total element displacement {un} for computing

the stresses as well as the restoring force

.

3. After the rotational increments in {u} are computed, update the node rotations appropriately. All threesteps require the concept of a rotational pseudovector in order to be efficiently implemented (Rankin

and Brogan([66] (p. 924)), Argyris([67] (p. 924))).

3.3.3. Element Transformation

The updated transformation matrix [Tn] relates the current element coordinate system to the global

Cartesian coordinate system as shown in Figure 3.3: Element Transformation Definitions (p. 37).




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Figure 3.3: Element Transformation Definitions

C u r r e n t C o n f i g u r a t i o n

O r i g i n a l C o n f i g u r a t i o n

[ T ]

[ R ]

[ T ]

Y

X

X

Y

Y

X

n

n

n

n

v

v

v

[Tn] can be computed directly or the rotation of the element coordinate system [Rn] can be computed

and related to [Tn] by

(3.41

=

where [Tv] is the original transformation matrix. The determination of [Tn] is unique to the type of element

involved, whether it is a solid element, shell element, beam element, or spar element.

Solid Elements. The rotation matrix [Rn] for these elements is extracted from the displacement field

using the deformation gradient coupled with the polar decomposition theorem (see Mal-

vern([87] (p. 925))).

Shell Elements. The updated normal direction (element z direction) is computed directly from theupdated coordinates. The computation of the element normal is given in Element Library (p. 411)

for each particular shell element. The extraction procedure outlined for solid elements is used

coupled with the information on the normal direction to compute the rotation matrix [R n].

Beam Elements. The nodal rotation increments from {u} are averaged to determine the average

rotation of the element. The updated average element rotation and then the rotation matrix [R n] is

computed using Rankin and Brogan([66] (p. 924)). In special cases where the average rotation of the

element computed in the above way differs significantly from the average rotation of the element

computed from nodal translations, the quality of the results will be degraded.

Link Elements. The updated transformation [Tn] is computed directly from the updated coordinates.

Generalized Mass Element (MASS21). The nodal rotation increment from {u} is used to update the

element rotation which then yields the rotation matrix [Rn].

3.3.4. Deformational Displacements

The displacement field can be decomposed into a rigid body translation, a rigid body rotation, and a

component which causes strains:

(3.42 = +

where:


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{ur} = rigid body motion

{ud

} = deformational displacements which cause strains

{ud

} contains both translational as well as rotational DOF.

The translational component of the deformational displacement can be extracted from the displacement

field by

(3.43)

= +

where:

= translational component of the deformational displacement

[Rn] = current element rotation matrix

{xv} = original element coordinates in the global coordinate system

{u} = element displacement vector in global coordinates

{ud

} is in the global coordinate system.

For elements with rotational DOFs, the rotational components of the deformational displacement must

be computed. The rotational components are extracted by essentially subtracting the nodal rotations

{u} from the element rotation given by {ur}. In terms of the pseudovectors this operation is performed

as follows for each node:

1. Compute a transformation matrix from the nodal pseudovector {n} yielding [Tn].

2. Compute the relative rotation [Td

] between [Rn] and [Tn]:

(3.44)

=

This relative rotation contains the rotational deformations of that node as shown in Figure 3.4: Defin-

ition of Deformational Rotations (p. 39).

3. Extract the nodal rotational deformations {ud} from [T

d].

Because of the definition of the pseudovector, the deformational rotations extracted in step 3 are limited

to less than 30, since 2sin(/2) no longer approximates itself above 30. This limitation only applies

to the rotational distortion (i.e., bending) within a single element.




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Figure 3.4: Definition of Deformational Rotations

Y

X

[ R ]

[ T ]

[ T ]

d

n

n

3.3.5. Updating Rotations

Once the transformation [T] and deformational displacements {ud} are determined, the element matrices

Equation 3.38 (p. 36)and restoring force Equation 3.39 (p. 36)can be determined. The solution of the

system of equations yields a displacement increment {u}. The nodal rotations at the element level are

updated with the rotational components of {u}. The global rotations (in the output and on the results

file) are not updated with the pseudovector approach, but are simply added to the previous rotation

in {un-1}.


The large-rotation computations in those elements which support it are activated by the large-deform-

ation key (NLGEOM,ON). Stress stiffening is also be included and contributes to the tangent stiffnessmatrix (which may be required for structures weak in bending resistance).


Stresses (output as S) are engineering stresses in the rotated element coordinate system (the element

coordinate system follows the material as it rotates). Strains (output as EPEL, EPPL, etc.) are engineering

strains, also in the rotated element coordinate system. This applies to element types that do not have

large-strain capability. For element types that have large-strain capability, see Large Strain (p. 29).

3.3.8. Consistent Tangent Stiffness Matrix and Finite Rotation

It has been found in many situations that the use of consistent tangent stiffness in a nonlinear analysis

can speed up the rate of convergence greatly. It normally results in a quadratic rate of convergence. A

consistent tangent stiffness matrix is derived from the discretized finite element equilibrium equations

without the introduction of various approximations. The terminology of finite rotation in the context

of geometrical nonlinearity implies that rotations can be arbitrarily large and can be updated accurately.

A consistent tangent stiffness accounting for finite rotations derived by Nour-Omid and

Rankin([175] (p. 930)) for beam/shell elements is used. The technology of consistent tangent matrix and

finite rotation makes the buckling and postbuckling analysis a relatively easy task. The theory of finite

rotation representation and update has been described in Large Rotation (p. 35)using a pseudovector


Large Rotation


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representation. The following will outline the derivations of a consistent tangent stiffness matrix used

for the corotational approach.

The nonlinear static finite element equations solved can be characterized by at the element level by:

(3.45)

==

where:

N = number of total elements

= element internal force vector in the element coordinate system, generally see Equa-

tion 3.46 (p. 40)

[Tn]T

= transform matrix transferring the local internal force vector into the global coordinate

system

= applied load vector at the element level in the global coordinate system

(3.46)

= Hereafter, we shall focus on the derivation of the consistent tangent matrix at the element level without

introducing an approximation. The consistent tangent matrix is obtained by differentiating Equa-

tion 3.45 (p. 40)with respect to displacement variables {ue}:

(3.47)

= +

=

+

+

ss

It can be seen that Part I is the main tangent matrix Equation 3.38 (p. 36)and Part II is the stress stiff-

ening matrix (Equation 3.34 (p. 34), Equation 3.61 (p. 45)or Equation 3.64 (p. 45)). Part III is another

part of the stress stiffening matrix (see Nour-Omid and Rankin([175] (p. 930))) traditionally neglected in

the past. However, many numerical experiments have shown that Part III of

is essential to the

faster rate of convergence. In some cases, Part III of

is unsymmetric; when this occurs, a procedure

of symmetrizing

is invoked.

As Part III of the consistent tangent matrix utilizes the internal force vector

to form the matrix,

it is required that the internal vector

not be so large as to dominate the main tangent matrix




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(Part I). This can normally be guaranteed if the realistic material and geometry are used, that is, the

element is not used as a rigid link and the actual thicknesses are input.

It is also noted that the consistent tangent matrix Equation 3.47 (p. 40) is very suitable for use with the

arc-length solution method.

3.4. Stress Stiffening

3.4.1. Overview and Usage

Stress stiffening (also called geometric stiffening, incremental stiffening, initial stress stiffening, or differ-

ential stiffening by other authors) is the stiffening (or weakening) of a structure due to its stress state.

This stiffening effect normally needs to be considered for thin structures with bending stiffness very

small compared to axial stiffness, such as cables, thin beams, and shells and couples the in-plane and

transverse displacements. This effect also augments the regular nonlinear stiffness matrix produced by

large-strain or large-deflection effects (NLGEOM,ON). The effect of stress stiffening is accounted for by

generating and then using an additional stiffness matrix, hereinafter called the stress stiffness matrix.

The stress stiffness matrix is added to the regular stiffness matrix in order to give the total stiffness.

Stress stiffening may be used for static (ANTYPE,STATIC) or transient (ANTYPE,TRANS) analyses. Working

with the stress stiffness matrix is the pressure load stiffness, discussed in Pressure Load Stiffness (p. 46).

The stress stiffness matrix is computed based on the stress state of the previous equilibrium iteration.

Thus, to generate a valid stress-stiffened problem, at least two iterations are normally required, with

the first iteration being used to determine the stress state that will be used to generate the stress

stiffness matrix of the second iteration. If this additional stiffness affects the stresses, more iterations

need to be done to obtain a converged solution.

In some linear analyses, the static (or initial) stress state may be large enough that the additional stiffness

effects must be included for accuracy. Modal (ANTYPE,MODAL) and substructure (ANTYPE,SUBSTR)

analyses are linear analyses for which the prestressing effects can be requested to be included

(PSTRES,ON command). Note that in these cases the stress stiffness matrix is constant, so that the

stresses computed in the analysis are assumed small compared to the prestress stress.

If membrane stresses should become compressive rather than tensile, then terms in the stress stiffness

matrix may cancel the positive terms in the regular stiffness matrix and therefore yield a nonpositive-

definite total stiffness matrix, which indicates the onset of buckling. If this happens, it is indicated with

the message: Large negative pivot value ___, at node ___ may be because buckling load has been exceeded.

It must be noted that a stress stiffened model with insufficient boundary conditions to prevent rigid

body motion may yield the same message.

The linear buckling load can be calculated directly by adding an unknown multiplier of the stress stiffness

matrix to the regular stiffness matrix and performing an eigenvalue buckling problem (ANTYPE,BUCKLE)

to calculate the value of the unknown multiplier. This is discussed in more detail in Buckling Analys-

is (p. 792).

3.4.2.Theory

The strain-displacement equations for the general motion of a differential length fiber are derived below.

Two different results have been obtained and these are both discussed below. Consider the motion of

a differential fiber, originally at dS, and then at ds after deformation.


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Figure 3.5: General Motion of a Fiber

Z

X

Y

d S

{ u }

{ u + d u }

d s

One end moves {u}, and the other end moves {u + du}, as shown in Figure 3.5: General Motion of a

Fiber (p. 42). The motion of one end with the rigid body translation removed is {u + du} - {u} = {du}.

{du}may be expanded as

(3.48) =

where u is the displacement parallel to the original orientation of the fiber. This is shown in Figure 3.6: Mo-

tion of a Fiber with Rigid Body Motion Removed (p. 43). Note that X, Y, and Z represent global Cartesian

axes, and x, y, and z represent axes based on the original orientation of the fiber. By the Pythagorean

theorem,

(3.49)= + + +

The stretch, , is given by dividing ds by the original length dS:

(3.50)L = = +

+

+




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Figure 3.6: Motion of a Fiber with Rigid Body Motion Removed

d S

d u

d v

d w

{ d u }

d s

Z

Y

X

x

y

z

As dS is along the local x axis,

(3.51L = +

+

+

Next, is expanded and converted to partial notation:

(3.52L = +

+

+

+

The binominal theorem states that:

(3.53

+ = + - +

when A2< 1. One should be aware that using a limited number of terms of this series may restrict its

applicability to small rotations and small strains. If the first two terms of the series in Equation 3.53 (p. 43)

are used to expand Equation 3.52 (p. 43),

(3.54L = +

+

+

+

The resultant strain (same as extension since strains are assumed to be small) is then

(3.55e

= - =

+

+

+

L

If, more accurately, the first three terms of Equation 3.53 (p. 43)are used and displacement derivatives

of the third order and above are dropped, Equation 3.53 (p. 43)reduces to:


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(3.56)L = +

+

+

The resultant strain is:

(3.57)e

= - =

+

+

L

For most 2-D and 3-D elements, Equation 3.55 (p. 43) is more convenient to use as no account of the

loaded direction has to be considered. The error associated with this is small as the strains were assumed

to be small. For 1-D structures, and some 2-D elements, Equation 3.57 (p. 44) is used for its greater ac-

curacy and causes no difficulty in its implementation.


The stress-stiffness matrices are derived based on Equation 3.34 (p. 34), but using the nonlinear strain-

displacement relationships given in Equation 3.55 (p. 43)or Equation 3.57 (p. 44)(Cook([5] (p. 921))).

For a spar, the stress-stiffness matrix is given as:

(3.58)[ ]=

The stress stiffness matrix for a 2-D beam is given in Equation 3.59 (p. 44), which is the same as reported

by Przemieniecki([28] (p. 922)). All beam and straight pipe elements use the same type of matrix. Forces

used by straight pipe elements are based on not only the effect of axial stress with pipe wall, but also

internal and external pressures on the "end-caps" of each element. This force is sometimes referred toas effective tension.

(3.59) =

- -

-

--

where:

F = force in member

L = length of member




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The stress stiffness matrix for 2-D and 3-D solid elements is generated by the use of numerical integration.

A 3-D solid element (SOLID185) is used here as an example:

(3.60

=

where the matrices shown in Equation 3.60 (p. 45)have been reordered so that first all x-direction DOF

are given, then y, and then z. [So] is an 8 by 8 matrix given by:

(3.61

=

The matrices used by this equation are:

(3.62

=

s s ss s ss s s

where x, xyetc. are stress based on the displacements of the previous iteration, and,

(3.63

=

where Ni represents the ith shape function. This is the stress stiffness matrix for small strain analyses.

For large-strain elements in a large-strain analysis (NLGEOM,ON), the stress stiffening contribution is

computed using the actual strain-displacement relationship (Equation 3.6 (p. 31)).

One further case requires some explanation: axisymmetric structures with nonaxisymmetric deformations.

As any stiffening effects may only be axisymmetric, only axisymmetric cases are used for the prestress

case. Axisymmetric cases are defined as (input as MODE on MODEcommand) = 0. Then, any sub-

sequent load steps with any value of (including 0 itself) uses that same stress state, until another,

more recent, = 0 case is available. Also, torsional stresses are not incorporated into any stress stiffening

effects.

Specializing this to SHELL61(Axisymmetric-Harmonic Structural Shell), only two stresses are used for

prestressing: s, , the meridional and hoop stresses, respectively. The element stress stiffness matrix

is:

(3.64

=

(3.65

=

=


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where [As] is defined below and [N] is defined by the element shape functions. [A s] is an operator

matrix and its terms are:

(3.66)

=

where:

= =

>

The three columns of the [As] matrix refer to u, v, and w motions, respectively. As suggested by the

definition for [Sm], the first two rows of [As] relate to sand the second two rows relate to . The first

row of [As] is for motion normal to the shell varying in the s direction and the second row is for hoop

motions varying in the s direction. Similarly, the third row is for normal motions varying in the hoop

direction. Thus Equation 3.57 (p. 44), rather than Equation 3.55 (p. 43), is the type of nonlinear strain-

displacement expression that has been used to develop Equation 3.66 (p. 46).

3.4.4. Pressure Load Stiffness

Quite often concentrated forces are treated numerically by equivalent pressure over a known area. This

is especially common in the context of a linear static analysis. However, it is possible that differentbuckling loads may be predicted from seemingly equivalent pressure and force loads in an eigenvalue

buckling analysis. The difference can be attributed to the fact that pressure is considered as a follower

load. The force on the surface depends on the prescribed pressure magnitude and also on the surface

orientation. Concentrated loads are not considered as follower loads. The follower effects is a preload

stiffness and plays a significant role in nonlinear and eigenvalue buckling analysis. The follower effects

manifest in the form of a load stiffness matrix in addition to the normal stress stiffening effects. As

with any numerical analysis, it is recommended to use the type of loading which best models the in-

service component.

The effect of change of direction and/or area of an applied pressure is responsible for the pressure load

stiffness matrix ([Spr

]) (see section 6.5.2 of Bonet and Wood([236] (p. 934))). It is used either for a large-

deflection analysis (NLGEOM,ON), for an eigenvalue buckling analysis, or for a modal, linear transient,or harmonic analysis that has prestressing flagged (PSTRES,ON command).

The need of [Spr

] is most dramatically seen when modelling the collapse of a ring due to external

pressure using eigenvalue buckling. The expected answer is:

(3.67)

=

where:




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Pcr= critical buckling load

E = Young's modulus

I = moment of inertia

R = radius of the ring

C = 3.0

This value of C = 3.0 is achieved when using [Spr

], but when it is missing, C = 4.0, a 33% error.

For eigenvalue buckling analyses, all elements with pressure load stiffness capability use that capability.

Otherwise, its use is controlled by KEY3on the SOLCONTROLcommand.

[Spr

] is derived as an unsymmetric matrix. Symmetricizing is done, unless the command NROPT,UNSYM

is used. Processing unsymmetric matrices takes more running time and storage, but may be more

convergent.


In a nonlinear analysis (ANTYPE,STATIC or ANTYPE,TRANS), the stress stiffness contribution is activated

and then added to the stiffness matrix. When not using large deformations (NLGEOM,OFF), the rotations

are presumed to be small and the additional stiffness induced by the stress state is included. Whenusing large deformations (NLGEOM,ON), the stress stiffness augments the tangent matrix, affecting the

rate of convergence but not the final converged solution.

The stress stiffness contribution in the prestressed analysis is activated by the prestress flag (PSTRES,ON)

and directs the preceding analysis to save the stress state.


In a small deflection/small strain analysis (NLGEOM,OFF), the 2-D and 3-D elements compute their

strains using Equation 3.55 (p. 43). The strains (output as EPEL, EPPL, etc.) therefore include the higher-

order terms (e.g.

in the strain computation. Also, nodal and reaction loads (output quantities

F and M) will reflect the stress stiffness contribution, so that moment and force equilibrium include the

higher order (small rotation) effects.

3.5. Spin Softening

The vibration of a spinning body will cause relative circumferential motions, which will change the dir-

ection of the centrifugal load which, in turn, will tend to destabilize the structure. As a small deflection

analysis cannot directly account for changes in geometry, the effect can be accounted for by an adjust-

ment of the stiffness matrix, called spin softening. When large deformations are active (NLGEOM,ON),

or pres-stress effects are active (PSTRES,ON), the spin softening contribution is automatically includedas an additional contribution to the tangent matrix (Equation 3.32 (p. 34)).

Consider a simple spring-mass system, with the spring oriented radially with respect to the axis of rota-

tion, as shown in Figure 3.7: Spinning Spring-Mass System (p. 48). Equilibrium of the spring and centri-

fugal forces on the mass using small deflection logic requires:

(3.68

= w

where:


Spin Softening


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u = radial displacement of the mass from the rest position

r = radial rest position of the mass with respect to the axis of rotation

s= angular velocity of rotation

Figure 3.7: Spinning Spring-Mass System

K

M

r u

s

However, to account for large-deflection effects, Equation 3.68 (p. 47)must be expanded to:

(3.69)

= +w

Rearranging terms,

(3.70)

- =w w

Defining:

(3.71)

= - w

and

(3.72)

= w

Equation 3.70 (p. 48)becomes simply,

(3.73)=

is the stiffness needed in a small deflection solution to account for large-deflection effects. is the

same as that derived from small deflection logic. This decrease in the effective stiffness matrix is called

spin (or centrifugal) softening. See also Carnegie([104] (p. 926)) for additional development.

Extension of Equation 3.71 (p. 48) into three dimensions is illustrated for a single noded element here:

(3.74)= + W

with

(3.75)W

=

- +

- +

- +

w w w w w w

w w w w w w

w w w w w w

where:




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x, y, z= x, y, and z components of the angular velocity (input with OMEGAor CMOMEGA

command)

It can be seen from Equation 3.74 (p. 48)and Equation 3.75 (p. 48)that if there are more than one non-

zero component of angular velocity of rotation, the stiffness matrix may become unsymmetric. For ex-

ample, for a diagonal mass matrix with a different mass in each direction, the matrix becomes non-

symmetric with the expression in Equation 3.74 (p. 48)expanded as:

(3.76

= - +w w

(3.77

= - +w w

(3.78

= - +w w

(3.79

= + w w

(3.80

= + w w

(3.81

= + w w

(3.82

= + w w

(3.83

= + w w

(3.84

= + w w

where:

Kxx, Kyy, Kzz= x, y, and z components of stiffness matrix as computed by the element

Kxy, Kyx, Kxz, Kzx, Kyz, Kzy = off-diagonal components of stiffness matrix as computed by the

element

=

Mxx, Myy, Mzz= x, y, and z components of mass matrix

=

From Equation 3.76 (p. 49) thru Equation 3.84 (p. 49), it may be seen that there are spin softening effects

only in the plane of rotation, not normal to the plane of rotation. Using the example of a modal analysis,

Equation 3.71 (p. 48)can be combined with Equation 15.49 (p. 779)to give:

(3.85- =w

or

(3.86

- - =w w

where:

= the natural circular frequencies of the rotating body.

If stress stiffening is added to Equation 3.86 (p. 49), the resulting equation is:


Spin Softening


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(3.87)

+ - - =w w

Stress stiffening is normally applied whenever spin softening is activated, even though they are inde-

pendent theoretically. The modal analysis of a thin fan blade is shown in Figure 3.8: Effects of Spin

Softening and Stress Stiffening (p. 50).

Figure 3.8: Effects of Spin Softening and Stress Stiffening

X

s

Y

1 0

7 0

6 0

9 0

5 0

4 0

1 0 0

3 0

2 0

8 0

F

u

n

d

a

m

e

n

t

a

l

N

a

t

u

r

a

l

F

r

e

q

u

e

n

c

y

(

H

e

r

t

z

)

0

0 4 0 8 0 1 2 0 1 6 0 2 0 0 2 4 0 2 8 0 4 0 0 3 6 03 2 0

A = N o S t r e s s S t i f f e n i n g , N o S p i n S o f t e n i n g

B = S t r e s s S t i f f e n i n g , N o S p i n S o f t e n i n g

C = N o S t r e s s S t i f f e n i n g , S p i n S o f t e n i n g

D = S t r e s s S t i f f e n i n g , S p i n S o f t e n i n g

A

B

D

C

A n g u l a r V e l o c i t y o f R o t a t i o n ( ) ( R a d i a n s / S e c )

s

On Fan Blade Natural Frequencies

3.6. General Element Formulations

Element formulations developed in this section are applicable for general finite strain deformation.

Naturally, they are applicable to small deformations, small deformation with large rotations, and stress

stiffening as particular cases. The formulations are based on the principle of virtual work. Minimal as-

sumptions are used in arriving at the slope of nonlinear force-displacement relationship, i.e., the element

tangent stiffness. Hence, they are also called consistent formulations. These formulations have been

implemented in PLANE182, PLANE183, SOLID185, and SOLID186. SOLID187, SOLID272, SOLID273, SOL-




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Spin Softening n Stress Stiffening

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