7nonlinear analysis - msc software...

1

77Nonlinear Analysis

7.1 Types of NonlinearityNonlinear Analysis is an optional capability of MSC/NASTRAN for Windows (MSC/N4W) thatenables you to analyze structures that exhibit nonlinear behavior.

One of the considerations in performing any type of analysis is whether the structure willexperience nonlinear behavior. Depending upon the structure, its behavior, and the material it ismade of, many types of nonlinear behavior are possible.

If a structure undergoes displacements of a magnitude on the order of the structure, higher ordereffects may occur which may tend to stiffen the structure. If a structure’s material is loaded aboveit’s yield point, the structure will then tend to be less stiff and permanent deformation will exist.Materials like rubber also exhibit characteristics which are nonlinear in nature. Each type ofnonlinearity is handled differently in MSC/N4W.

Nonlinear behavior can be due to a number of different sources. In some structures, the kinematicrelationship is nonlinear and displacements and rotations can be large. This behavior is known asgeometric nonlinear or large displacement behavior. For this behavior, equilibrium is satisfied inthe deformed configuration.

A related effect to geometric nonlinear analysis is follower force effects. In a small displacementanalysis, the force always maintains its initial orientation. A follower force changes its orientationas the structure deforms.

Nonlinear Analysis 7-2

Large displacement effects are set with a parameter in MSC/N4W by selecting the LargeDisplacement check-box in the Analyze dialog box.

A more obvious nonlinear effect is due to material nonlinearity, where the material constitutiverelation is nonlinear, i.e., there is no longer a linear relationship between forces and displacements.With a nonlinear constitutive relation, the material stiffness can change during the analysis and thematerial may yield, perhaps resulting in permanent deformation. MSC/N4W can representmaterials that exhibit nonlinear elastic and plastic behavior.

The time dependent effects of long term load application can be taken into account using a materialwith creep properties defined. Creep is a time dependent phenomenon where strain changes underconstant stress. It is a material relaxation whose rate is both load and/or temperature dependent.

Types of Nonlinearity

NO

NL

INE

AR A

NA

LY

SIS

7-3

Many nonlinear materials exhibit their nonlinear behavior in the small strain regime. Representingmaterials like rubber, however, require the consideration of element strains that are nonlinearfunctions of element deformation and possibly large strains. MSC/N4W has the capability to modelsuch hyperelastic materials.

A very common form of nonlinear behavior is contact analysis. In contact problems, portions of astructure can have areas of gaps which can open and close or slide in relation to each other.Similarly, boundary conditions can change during a nonlinear analysis. MSC/N4W has slide lineand gap elements for contact problems and multiple boundary condition specification throughAdvanced Case Control.

Any combination of these nonlinear effects can occur simultaneously in an analysis. The analysiscan be static, quasi-static (creep) or transient dynamic. It is important to remember, however, if anyof these effects occur, displacements are no longer directly proportional to loads and the results ofdifferent load cases cannot be superimposed.

The difference in the behavior between linear and nonlinear structures requires a differentnumerical solution approach. In addition, the different nonlinear behaviors may require slightlydifferent numerical solution approaches.

The basic nonlinear solution approach involves a series of incremental solutions. The load isapplied in increments, load increments in a nonlinear static solution, and time increments in anonlinear transient dynamic solution. During each increment, a solution is “predicted” using thecurrent state (stiffness and load increment). Depending on the type of nonlinearity, a forceimbalance or “residual” is created during an iteration where nonlinear behavior occurs. Solutioniterations are required to balance equilibrium (“correct”) for unbalanced forces. The iterationscontinue during an increment until the convergence criteria are satisfied. Once convergence issatisfied, a solution is obtained for the increment and the solution progresses to the next incrementusing this “predictor-corrector” method.

The nonlinear incremental solution is a combination of different advancing schemes, differentiteration schemes, and different convergence criteria. MSC/N4W has the power and flexibility toeasily include the various parameters and solution methods required for nonlinear analysis.

A final type of nonlinear behavior, which is different from those described above, is the nonlinearforce/displacement or force/velocity relationship applied in a linear or nonlinear transient solution.Instead of affecting the stiffness matrix, these nonlinear forces are present in the load vector. These


nonlinear forces permit the user to monitor the displacement or velocity of a particular degree offreedom and apply a load which is a function of that response to the same or different degree offreedom.

Since these nonlinear forces are “right-hand side” quantities, they can easily be used in both linearand nonlinear transient analyses because changes in the force vector are self-equilibrating. Manynonlinear effects can be represented by force nonlinearities. The benefit of nonlinearitiesrepresented by force and not stiffness changes is that they can easily be incorporated into a lineartransient analysis to provide a cost effective solution.

7.1.1 Large Deformation - CantileverExample exercise for large deformation nonlinear analysis.

Large Deformation - Cantilever 7-5N

ON

LIN

EA

R AN

AL

YS

IS

17.2

1Large Deformation - Cantilever


Model Description:

This example uses a simple cantilever beam model to demonstrate the geometric nonlinear effects of large displacements and large rotations. A linear solution method would not be adequate for this case as equilibrium can only be satisfied in the deformed configuration due to the high applied loading.

The model consists of a cantilever beam; 300 mm long, divided into 15 elements, fixed at one end and subjected to an end moment of 1,425,000 Nmm at the free end. The beam has a 8 mm square cross section and the element properties are as follows:

A = 64 mm2

I1 = I2 = 340 mm4

J = 580 mm4

The material is assumed to be an isotropic steel with the following properties:

E = 200,000 N/mm2

υ = 0.3

Large Deformation - Cantilever

NO

NL

INE

AR A

NA

LY

SIS

7-7

Exercise Procedure:

1. Start up MSC/NASTRAN for Windows 4.0 and begin to create a new model.

Start MSC/N4W by double-clicking on the MSC/N4W icon. When the Open Model File dialog box appears; choose New Model.

2. Create the Model.

Use the information given in Model Description (previous page) to create the model.

Fit the model to the display with Ctrl + A or:

Next, create the material.

Finally, the property.

Open Model File: New Model

Geometry/Curve-Line/Coordinates...

X: 0 Y: 0 Z: 0 OK

X: 300 Y: 0 Z: 0 OK

Cancel

View/Autoscale

Model/Material...

Title: Steel

Youngs Modulus, E: 2.0E5

Poisson’s Ratio, nu: 0.3

OK

Cancel

Model/Property...


Click the Shape button at this stage and the Cross Section Definition dialog box can be used to generate most of the element properties from the cross section dimensions. The polar moment of inertia however, is not calculated automatically and has to be entered manually.

Title: Beam Property

Elem/Property Type...

Line Element: ● Beam

OK

Shape...

Shape: Rectangular Bar

Size / H: 8

Size / Width: 8

Orientation Direction (y): ● Up

OK


NO

NL

INE

AR A

NA

LY

SIS

7-9

3. Mesh the Model.

Create mesh seeds along the curve prior to meshing.

Select Curve 1.

Select Curve 1.

Material: 1..Steel

Torsional Constant, J: 580

OK

Cancel

Mesh/Mesh Control/Size Along Curve...

OK

● Number of Elements 15

OK

Cancel

Mesh/Geometry/Curve...

OK

Property: 1..Beam Property

OK

Base: X: 0 Y: 0 Z: 0

Tip: X: 0 Y: 1 Z: 0

OK


4. Apply load and constraint.

Completely fix one end (left end) of the cantilever.

Select Node 1.

Apply the end moment of 1,425,000 Nmm (in the z-direction) to the free end (right end) of the beam.

Select Node 16.

Model/Constraint/Nodal...

Title: Fixed

OK

OK

Fixed

OK

Cancel

Model/Load/Nodal...

Title: End Moment

OK

OK

(highlight) Moment

MZ 1425000

OK

Cancel


NO

NL

INE

AR A

NA

LY

SIS

7-11

Rotate the model to an isometric view.

5. Setting the Nonlinear Solution Parameters.

Select Model/Load/Nonlinear Analysis to obtain the Load Set Options for Nonlinear Analysis dialog box.

View/Rotate...

Isometric

OK

Modal/Load/Nonlinear Analysis...

Solution Type: ● Static

Default...

Basic / Number of Increments: 10


If results are desired at every increment, then the Output Control Intermediate box should be changed to 3..ALL.

It is possible to change the nonlinear solution parameters, to select solution strategy overrides for example, but it is not usually necessary. For certain problems however, the default settings may not lead to a solution convergence.

6. Analyzing the Model.

Run a Nonlinear Static analysis with Large Displacement.

Output Control / Intermediate: 3..ALL

OK

File/Analyze...

Analysis Type: 10..Nonlinear Static

Run Analysis

Large Disp

Output Types: 2..Displacements and Stresses


NO

NL

INE

AR A

NA

LY

SIS

7-13

A dialog box prompting a model save then appears, and the analysis commences.

When prompted, “OK to Read Nonlinear Stresses and Strains?”, select Yes.

7. Processing the Results.

When the results have been read into the database, they can be accessed by selecting View/Select.

OK

Yes

File name: Large_Disp

Save

Yes

Continue

View/Select...

Deformed Style: ● Deform

Deformed and Contour Data...

Data Selection / Category: 1..Displacement

Output Set: 20...Case 20 Time 1

OK


Notice that the automatic iteration control process of MSC/N4W has automatically added some additional load increments on top of the 10 selected in the nonlinear parameter settings. In the anal-ysis phase 10 iterations clearly was not enough to satisfy the convergence criteria so the program automatically added a sufficient number to obtain a solution.

To ensure that the deformed shape is set to the actual scale, do the following steps:

Below shows the actual deformation of the cantilever due to the applied end moment loading.

7.3 Nonlinear Material - Plastic DeformationIn this example a simple rod structure will be used to demonstrate the capabilities of MSC/N4W with elasto-plastic nonlinear material analysis.

OK

View/Option...

Category: ● PostProcessing

Options: Deformed Style

% of Model (Actual)

Scale Act: 1

OK

Nonlinear Material - Plastic Deformation 7-15N

ON

LIN

EA

R AN

AL

YS

IS

17.3

1Nonlinear Material - Plastic Deformation

Nonlinear Material - Plastic Deformation

NO

NL

INE

AR A

NA

LY

SIS

7-17

Model Description:

The rod is fully constrained at one end and has an axial tensile load applied at the other. The applied load is of sufficient magnitude to cause the material to move into the plastic region of the stress-strain curve and undergo permanent deformation. In a second load case the rod will be unloaded leaving the structure in a permanently deformed state.

The model consists of a 100 mm long rod (meshed into 10 rod elements) with a cross sectional area of 1 mm2. The linear part of the material definition is assumed to be an isotropic steel, with the fol-lowing properties:

E = 200000 N/mm2

υ = 0.3

Material Stress-Strain Curve data.

x - stress y - strain

0.0 0.0200.0 0.001250.0 0.003275.0 0.005


Exercise Procedure:



2. Creating the Material.

The desired nonlinear elasto-plastic material is defined using a stress strain function. First create a function using the Model/Function command and create the stress (Type 4..vs.Stress) strain curve.


Model/Function...

Title: Nonlinear Material

Type: 4..vs. Stress

Data Entry ● Single Value

X 0 Y 0

More

X 200.0 Y 0.001

More

X 250.0 Y 0.003

More

X 275.0 Y 0.005

OK

Cancel


NO

NL

INE

AR A

NA

LY

SIS

7-19

Look at a plot of this function.

View/Select...

XY Style: ● XY of Function

Model Data...

Function / Select: 1..Nonlinear Material

OK

OK


Complete the nonlinear material definition enter the Model/Material menu, fill in the required lin-ear material properties (E and υ).

Model/Material...

Title: Steel

Youngs Modulus, E: 2.0E5

Poisson’s Ratio, nu: 0.3

Nonlinear >>

Nonlinearity Type: ● Plastic

Hardening Rule: 0..Isotropic

Function Dependence: 1..Nonlinear Material

Yield Criterion: 0..von Mises

Initial yield Stress: 200


NO

NL

INE

AR A

NA

LY

SIS

7-21

3. Creating the Property.

OK

OK

Cancel

Model/Property...

Title: Rod Property


Line Element: ● Rod

OK

Material: 1..Steel


4. Create the geometry and mesh.

Return to the modeling display.

For simple geometries such as a straight beam or rectangular plate, creating geometry, customizing mesh size, and generating the mesh can all be done in one step.

Fit the model to the display with Autoscale (Ctrl+A).

5. Apply constraint.

Completely fix one end (left end) of the cantilever.

Area, A: 1

OK

Cancel

View/Select...

Model Style: ● Draw Model

OK

Mesh/Between...

Property: 1..Rod Property

Mesh Size / #Nodes: 11

~ / Bias: 1

OK

X: 0 Y: 0 Z: 0 OK

X: 100 Y: 0 Z: 0 OK

View/Autoscale



NO

NL

INE

AR A

NA

LY

SIS

7-23

Select Node 1.

6. Apply the first tensile loading condition.

Apply the load of 300 N (in the x-direction) to the free end (right end) of the beam.

Select Node 11.

Title: Fixed

OK

OK

Fixed

OK

Cancel

Model/Load/Nodal...

Title: 300 N Load

OK

OK

FX 300



OK

Cancel



Default...




NO

NL

INE

AR A

NA

LY

SIS

7-25

7. Apply the second loading condition.

Apply the load of 0.001 N (in the x-direction) to the free end (right end) of the beam.

Select Node 11.

OK

Model/Load/Set...

ID: 2

Title: Zero Load

OK

Model/Load/Nodal...

OK

FX 0.001


OK

Cancel



Default



OK


NO

NL

INE

AR A

NA

LY

SIS

7-27

The final model should look something like this:




The default selection, 0..Standard for Output Types, will only produce displacement output.

File/Analyze...


Run Analysis

Large Disp


Loads...

Select All

OK




9.Processing the Results.

OK

Yes

File name: Plastic_Deform

Save

Yes

Continue


NO

NL

INE

AR A

NA

LY

SIS

7-29

When the results have been read into the database, they can be accessed by selecting View/Select. The load steps of interest are 1 to 20, covering the loading and unloading load cases. Node 11 is at the free end of the rod structure, and the quantity that will be plotted on the XY plot is total transla-tion.

View/Select...

XY Style: ● XY vs Set Value

XY Data...

Output Set: 1..Case 1 Time 0.1

Output Vector: 1..Total Translation

Output Location / Node: 11

Show Output Sets (Blank = All): From: 1

To: 20

OK

OK


Another quantity that can be plotted is plastic strain for an element. Follow the steps in the previous page with 3286..Rod Plastic Strain and Element 10 to get:

Gap Contact - Cantilever

NO

NL

INE

AR A

NA

LY

SIS

7-31

7.4 Gap Contact - CantileverThis example uses a simple cantilever beam model with a gap element to demonstrate the nonlinear gap contact capability within MSC/N4W.

Gap Contact - Cantilever 7-33N

ON

LIN

EA

R AN

AL

YS

IS

17.4

1Gap Contact - Cantilever


Model Description:

The cantilever model, under the applied bending load, deflects until the gap closes, and continues to deflect with the gap closed.

The model consists of a cantilever beam; 10 in long, divided into 10 elements, fully fixed at one end and subjected to a vertical bending load of 30 lb. at the free end. The beam has a square cross section and the element properties are as follows:

A = 0.04 in2

I1 = I2 = 1.333x10-4 in4

J = 2.25x10-4 in4

The material is AISI 4340 Steel, the properties can be obtained from the MSC/N4W material library.


NO

NL

INE

AR A

NA

LY

SIS

7-35

Exercise Procedure:



2. Creating the Material.

Complete the nonlinear material definition enter the Model/Material menu, fill in the required lin-ear material properties (E and υ).

3. Creating the Property.

Click the Shape button at this stage and the Cross Section Definition dialog box can be used to generate most of the element properties from the cross section dimensions. The polar moment of inertia however, is not calculated automatically and has to be entered manually.


Model/Material...

Load...

Library Entry: AISI 4340 Steel

OK

OK

Cancel

Model/Property...

Title: Beam Property


Line Element: ● Beam

OK

Shape...


Shape: Rectangular Bar

Size / H: 0.2

Size / Width: 0.2

Orientation Direction (y): ● Up

OK

Material: 1..AISI 4340 Steel

Torsional Constant, J: 2.25E-4

OK

Cancel


NO

NL

INE

AR A

NA

LY

SIS

7-37

4. Create the geometry and mesh.

For simple geometries such as a straight beam or rectangular plate, creating geometry, customizing mesh size, and generating the mesh can all be done in one step.

Fit the model to the display with Autoscale (Ctrl+A).

5. Apply the 30 lb. load.

Apply the load (in the negative y-direction) to the free end (right end) of the beam.

Select Node 11.

Mesh/Between...

Property: 1..Beam Property

Mesh Size / #Nodes: 11

~ / Bias: 1

OK

X: 0 Y: 0 Z: 0 OK

X: 10 Y: 0 Z: 0 OK

Base: X: 0 Y: 0 Z: 0

Tip: X: 0 Y: 1 Z: 0

OK

View/Autoscale

Model/Load/Nodal...

Title: End load

OK

OK


6. Creating the Gap Element.

Create another node at position (7.0,-0.5,0.0). This will be the position of the "stop" when the gap closes under the applied loading.

The gap properties can be created using Model/Property.

These properties will define a gap which, at the start of the analysis, will be open with the ends of the gap 0.5 in apart. When the gap is closed it has a high stiffness (1000000.), and when it is open it has a very low stiffness (0.0001). Care must be taken when selecting these stiffness values. The dif-ference between the open and closed should be great enough so that the gap functions properly but not too great to cause numerical problems in the solution.

FY -30

OK

Cancel

Model/Nodal...

X: 7.0 Y: -0.5 Z: 0

OK

Cancel

Model/Property...

Title: Gap Property


Line Element: ● Gap

OK

Initial Gap: 0.5

Compression Stiffness: 1.0E6

Tension Stiffness: 0.0001


NO

NL

INE

AR A

NA

LY

SIS

7-39

The gap element can now be created using the menu Model/Element.

This dialog box requests information to create the gap orientation vector, as the gap is connected in the basic y-axis, a suitable vector definition would be the basic x-axis.

OK

Cancel

Model/Element...

Property: 2..Gap Property

Nodes: 8

12

Vector...

Base: X: 7 Y: 0 Z: 0

Tip: X: 8 Y: 0 Z: 0

OK


7. Apply the constraints of the model.

Now that the whole model has been defined use the menus Model/Constraint/Nodal to completely fix one end of the cantilever (node 1) and the bottom of the gap element (node 12).

Select Node 1 and 12.

The model should now look like this:

OK

Cancel


Title: Fixed

OK

OK

Fixed

OK

Cancel


NO

NL

INE

AR A

NA

LY

SIS

7-41

8. Setting the Nonlinear Solution parameters.




Default...







The default selection, 0..Standard for Output Types, will only produce displacement output.

OK

File/Analyze...


Run Analysis

Large Disp



NO

NL

INE

AR A

NA

LY

SIS

7-43



10. Processing the Results.

When the results have been read into the database, they can be accessed by selecting View/Select.

OK

Yes

File name: Gap_Contact

Save

Yes

Continue

View/Select...

Deformed Style: ● Deform


Output Set: 10..Case 10 Time 1.

Output Vectors / Deformation: 1..Total Translation

OK


To ensures that the deformed shape is set to the actual scale, follow these steps.

The deformation shape shows that the gap did close under the applied loading.

Now pick Beam Diagram under Contour Style and plot the Combine Stress at Point 3 of the beam’s loaded end (End A).

OK

View/Option...

Category: ● PostProcessing

Options: Deformed Style

% of Model (Actual)

Scale Act: 1

OK

View/Select...


NO

NL

INE

AR A

NA

LY

SIS

7-45

This beam diagram can be animated right through the loading process.

Contour Style: ● Beam Diagram


Output Vectors / Contour: 3141..Beam EndA Pt3 Comb Stress

OK

OK

View/Select...

Deformed Style: ● Animate-Multiset


Output Set: 1..Case 1 Time 0.1

Final Output Set: 10..Case 10 Time 1.0

OK

OK


The animation can be controlled with the menu View/Advanced Post/Animation, and by stepping through each load step (using the Next > or < Previous buttons) it can be observed that the highest stresses do not occur at the maximum loading position, but at the position just before the gap closes.

7nonlinear analysis - msc software...

Documents