1. S3-1 SECTION 3 NONLINEAR NORMAL MODES

2. S3-2 PRE-STIFFENED NORMAL MODES n Section 2 looked at Normal Modes analysis of unloaded structures. In this section, Normal Modes Analysis of structures which have a static pre-load applied to them will be considered. n Typical real-world examples are: u Thin walled pressure vessels, where the pressurization dominates the natural frequencies, such as rocket fuel tanks and satellite oxygen tanks. u Structures subject to significant dead weight loading, such as suspension bridge cables and oil tankers. u Tensioning of strings in musical instruments to achieve required frequencies. u Structures subject to centrifugal loading such as jet engine turbine and fan blades.

3. S3-3 NONLINEAR NORMAL MODES n The objective of applying a static load to a structure prior to carry out a Normal Modes Analysis is to allow the Stiffness Matrix to be updated. n There are three terms that can influence the Stiffness Matrix. u Material Nonlinearity. u Geometric Stiffness due to changes in the structural shape (these can be small displacement theory or large displacement theory). u Follower Force stiffness terms as the loading changes its line of action relative to the displacements (again, small displacement theory or large displacement theory). u Contact  Non-contact

4. S3-4 MULTI-STEP ANALYSIS n Many MSC.Marc analyses require several steps. Usually these will be general (nonlinear) load history steps. n In a series of general analysis steps, the starting condition for each step is the ending condition from the previous step. u In a general step, loads are applied as total values l Example: Cup Forming problem l Step 1: Close Blank-holder l Step 2: Pressurize Blank-holder l Step 3: Move Punch l Step 4: Release Punch l Step 5: Release Holder l Step 6: Release Die

5. S3-5 MULTI-STEP ANALYSIS SELECTION n To select a step, you must click on it once in the step Select panel n To unselect a job step, you must click on it once in the Selected Job Steps panel n The steps will be executed in the order they appear in the Selected Job Steps panel n Don’t forget to click Apply when you are done!

6. S3-6 MULTI-STEP ANALYSIS punch blank die blankholder True Stress Log Strain Many MSC.MARC analysis require several steps. Usually there will be several general analysis steps. Occasionally, these may be punctuated by perturbation steps. In a series of general analysis steps, the starting condition for each step is the ending condition from the previous step. Note that while much of this section discusses perturbation vs non-perturbation steps, THE MOST COMMON USE OF MULTI-STEPPING IS FOR LOADING HISTORY CONTROL In a general step, loads are applied as total values Example: Modified Olson Cup Test  Step 1: close and pressurize blankholder  Step 2: move punch up  Step 3: release punch  Step 4: release blankholder  Step 5: release die The Modified Olson Cup Test is often used to determine the materials properties of a metal for the purpose of stretch forming.

7. S3-7 Linear analysis steps are perturbations about a base state. The base state is the ending condition of the last nonlinear step prior to the linear perturbation. In a linear analysis step, the loads are defined as the magnitudes of the load perturbations only. If a general analysis step follows a linear perturbation step, any perturbation response is ignored. Example: Preloaded Cantilever Beam Step 1: Preload P 1 (Nonlinear Static) Step 2: Natural Frequency Extraction Step 3: Response Spectrum Analysis (Earthquake) Step 4: Preload P 2 > P 1 (Nonlinear Static) Step 5: Natural Frequency Extraction Step 6: Response Spectrum Analysis (Earthquake) Does the earthquake analysis in Step 6 affect the results of Step 1 and Step 4 procedures? MULTI-STEP ANALYSIS (CONT.)

8. S3-8 How does eliminating Steps 2 and 4 change the results of Step 5? MULTI-STEP ANALYSIS (CONT.)  One may combine nonlinear (general) steps with linear (perturbation) steps in the same job.  Example:  Step 1 (pretension): general analysis step (Nonlinear Static)  Step 2 (frequency extraction): linear analysis step performed about the ending condition of step 1 (base state).  Step 3 (pull back): general analysis (Nonlinear Static) continuing from the ending condition of step 1 (last nonlinear step).  Step 4 (another frequency extraction): linear analysis step performed about the ending condition of step 3 (new base state).  Step 5 (dynamic release): general analysis (Nonlinear Transient) continuing from the ending condition of step 3