### Applying Preloads with the Differential Stiffness Approach

#### Differential Stiffness Theory

In geometric non-linear problems, displacements are large. Therefore, the equilibrium equations must be written with respect to the deformed geometry. This theory assumes that the geometric stiffness matrix can be added to the structural (or differential) stiffness matrix; therefore, the change in geometry of the structure is only reflected in the stiffness matrix.

#### Differential Stiffness Matrix

A matrix that is necessary to account for the change in potential energy associated with rotation of continuum elements under load. (stiffness effects that depend linearly on displacements)

#### Fundamental Assumption

External loads do not change in magnitude or direction as the structure deflects. However, the loads do travel with the grid points.

Properties of KDGG (Differential Stiffness Matrix)

1. Symmetric
2. Independent of elastic properties
3. Depends on element geometry, displacement field, and state of stress
4. Indefinite and cannot be inverted

#### Setting Up the NASTRAN Data Deck

Using the differential stiffness approach to applying preloads to a NASTRAN finite element model consists of two executions of the model. The first model is set up to obtain the differential stiffness matrix (KDGG) as a results of the preload. The second model is where the differential stiffness matrix is added to the geometric stiffness matrix and desired results are requested.

The DMAPs shown are for use in MSC/NASTRAN Version 67.

#### The First Model

Executive Control

```    Sol 5
Compile sol5, souin=mscsou
Alter 313
Output4 KDGG//-1//11 \$(creates a fort.11 file of KDGG)
Exit \$
```

Case Control

```    Title = 'title'
Subtitle =
SPC = 'boundary points'
'Output requests' - probably want to output elements in which
the preload is being applied to verify the magnitude.
Subcase 1
Deform = 'id'
Subcase 2
```

Bulk Data

Add the necessary number of 'DEFORM' bulk data cards. The 'DEFORM' card requests the deformation of a one dimensional element, i.e. BAR, ROD, TUBE. For a preload on a bar, the deformation is:

delta = PL / EA

where P is the applied preload. You can verify that the correct preload has been applied to the element by reviewing the BAR, ROD, or TUBE, element forces.

#### The Second Model

Executive Control

```    Sol 24
Compile sol24, souin=mscsou
Alter 90
Inputt4 /KDGG,,,,/1/11/-1 \$
Equiv K1X,KGG/Always \$
Endalter

```

Case Control and Bulk Data decks should be set up as any other run would be.

NOTE: The differential stiffness matrix can be very big. Therefore, you should be careful to ensure that the matrix is not sent across the network. This will slow the job execution substantially. If this is a problem, use the ASSIGN card to write the matrix to the scratch directories on the computer that is executing the NASTRAN job.

Cherie Congedo
May 18, 1995

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