Load incrementation of initially weak structures

I just worked on an FEA model which was essentially a flat, very thin plate of plastic with a pressure applied. Initially, I always use small displacement and linear contact (bonded) for debugging a model just to make sure that loads and boundary conditions are correct. This saves time because large deformation effects can introduce much longer solve times. The linear model solved fine.

When I turned on the large displacement option (nlgeom), the model failed to converge even after several cutbacks (bisections). I certainly did not expect to solve the model in one substep, but I thought the default bisection algorithm would find a converged solution. I was using automatic timestep control. I reduced the initial substep until I eventually obtained a solution, but by this point the subsequent substeps were so small that it would take many, many substeps to complete the load step.

I noticed that after the first substep converged, convergence for later substeps required only a few iterations. Therefore, I broke the load step into 2 separate steps. In the first step, the load was reduced to the value that converged earlier, about 1/1000th. In the second step, I specified the full load with a reasonably sized initial substep. The model had no trouble converging, even though the load had dramatically increased between steps. The reason is that the initial stiffness of the plate is bending only, because it was flat. Since it was a very thin plastic component, the bending stiffness is very low. The first step established some membrane stiffness as the plate tries to assume a more spherical shape. Once the load generates some membrane stress and there is membrane stiffness, subsequent predictions of displacement are more accurate.

I have also used this 2 step load strategy for preloading of bolts. Sometimes, not always, the contact resisting the preload has trouble converging with the full preload. Before contact is established, there is no stiffness resisting the preload. Then, the contact overcloses so much that the solver cannot resolve the overclosure efficiently.

Engineering Simulation at Olympics

I have been watching Vancouver Winter Olympics 2010 with great deal of attention. My loyalties lie with US but cheer any genuinely great sporting achievement. What fascinates me is the true grit and determination of these athletes to overcome great odds to attain perfection! And knowing our passion "engineering simulation" has helped them along towards these levels of perfection really brings a huge smile on my face.

Images Courtesy: www.fluent.com

I wanted to bring light to some of the engineering simulation such as FEA, CFD etc., that goes on in preparation towards these levels of perfections.

Above images show pressure contours on a simulated skeleton slider, with pathlines colored by velocity magnitude (Postprocessed by Ensight). The story behind the simulation is as interesting as the technology itself. You can read the complete story here. The story in short goes something like this: In preparation for 2006 Winter Olympics at Turin, Italy, Kristan Bromley, the top-ranked skeleton bobsled competitor in the UK, approached the Elite Sports CFD Unit - a part of the Sports Engineering Research Group (SERG) in Sheffield, UK - and asked them to provide CFD flow simulation support to increase his chances of success. Bromley's goal:
Minimize the overall aerodynamic drag by assessing small changes in surface texture of his skin-suit. Bromley ended up with a respectable 5th ranking in Turin Winter Olympics and went on to bring home the first gold medal for Britain since 1965 at the 2008 FIBT World Championships. Bromley maintains philosophy of using advanced technology to enhance on-ice performance. Although, the CFD analysis may have been just a drop in the ocean in terms of the dedication, grit and determination for such olympic athletes, it is still atleast a drop!

More information: http://www.bromley-aet.com/

Below are a few more articles that show the role of engineering simulation that goes into such perfection!
ANSYS Article: Giving Ski Racers an Edge
FLUENT Article: CFD for Bob Sled Team
ANSYS Article: Finite Element Analysis on Mountain Climbing Ice Axe to study crack initiation on serrated blade.

Hats off to all the fantastics athletes and the engineering simulation that is enhancing their performance on ice! :)

Turbine Blade Modal Analysis

I have recently had the opportunity to work on a few steam turbine blade failure investigations and have found a whole new world of engineering simulation that I had not been exposed to. Analysis of turbine blades is its own animal and even though steam turbines have been in use in power generation for a very long time, the physics of power generation is so complex there are many areas that are still not well understood. Being new to turbine analysis I thought my experiences as I lean steam turbine analysis may be useful to others. The first topic I would like to describe has been fundamental to all of the turbine projects that I have been involved with: Modal analysis of a bladed disk row. In each investigation, modal analysis was used to calculate the natural frequencies and mode shapes for a particular stage of blades. This is a valuable tool in determining if the blades are operating near a resonant condition that could be responsible for a failure.

As you can find in any vibrations textbook, modal analysis is an eigenvalue procedure in which the eigenvalues of the equation of motion are the square of the natural frequencies and the eigenvectors are the mode shapes. For blade analysis, SimuTech Group uses an in-house developed code to run a modal Finite Element Analysis on turbine blades. This program is called BLADE.

A portion of the bladed disk is modeled in BLADE which usually consists of a 360°/N sector, where N is the number of blades in the row. The mass and stiffness matrices for the bladed disk sector are then reduced to a superelement containing selected fewer degrees of freedom. These selected degrees of freedom are called master degrees of freedom. They are selected in such a way as to be able to represent and predict the dynamic behavior of the bladed disk.

At operating speed, the rotating system stiffens because of the centrifugal effects. This stress stiffening causes the natural frequencies to be higher than their corresponding values calculated at zero RPM. The effect of the stress stiffening is evaluated and incorporated in the analysis.

If a rotating bladed disk is excited by a forcing which is fixed spatially, there are 3 conditions that need to be satisfied in order to produce a resonant condition:

1. The natural frequency of the blade row is equal to some per-rev forcing frequency

2. The number of nodal diameter of the natural mode equals the forcing harmonic number.

3. The excitation must be able to couple with the blade disk mode shape. For example, the forcing on the blade row must be in a direction that matches the mode shape deflection. If the forcing is along the axial direction of the turbine and the mode shape shows deflection only in the tangential direction, the mode can not be excited. Of course many modes contain components in both axial and tangential directions.

If all of the above conditions are met, resonance will occur. This natural frequency-forcing relationship is usually illustrated as an Interference diagram. In the Interference diagram, the natural frequency is plotted against the number of nodal diameters (harmonic content of a mode). An example of an interference diagram is shown above. The line through the origin is called the Impulse Line which corresponds to the operating speed. Whenever the Impulse Line intersects the natural frequency curves, a resonant condition may exist.

The Interference diagram is used to locate frequencies of interest for a more detailed stress analysis. Resonant stresses are calculated for these conditions and detuned as necessary to estimate the true dynamic stress in each blade. The dynamic stress will also depend on the damping present in the system as well as the stimulus ratio (the ratio of dynamic forcing to the steam bending force in the blade). Dynamic stress analysis is a discussion on its own and will likely be a topic of future entries.

Thanks for reading.

Limit load analysis

I worked on a 2007 ASME B&PV Code, Section VIII, Div 2 analysis of what was essentially an elbow casting with some additional detail. The more traditional approach would be to use stress linearization on an elastic analysis for stress categorization. However, since the elbow was thick-walled relative to the radius, stress linearization can be non-conservative because the stress distribution is non-linear. Think about the difference between a thick walled cylinder and. thin walled cylinder.

I used the limit load analysis method instead. The limit load analysis has established itself as the preferred method, subject to its limitations, to assess primary sizing (Protection Against Plastic Collapse). The limit load analysis eliminates the need for stress categorization because it is a pass-fail criterion. The material definition is elastic-perfectly plastic. So, the limit load analysis is trying to predict when a plastic hinge forms in a plate an uncontrolled deformation with result with any additional applied load.

The basis for the limit load is straightforward. A value of 1.5 is applied to the desired load (i.e. design pressure+static head+dead weight). Recall that a plastic hinge is formed in a rectangular cross-section beam with an elastic-perfectly plastic material when the moment is 1.5 X the moment required for initial yield. You can find this discussion in a Continuum Mechanics textbook in a section on Beams. I have the book by Shames and Cozzarelli, "Elastic and Inelastic Stress Analysis," which has a pretty good description of this derivation with lots of pictures. So, if you enter 1.5*S (in some cases 1.5S is equal to yield strength at temperature) as your FEA yield strength, and the model converges, you will not develop a plastic hinge in the component you are analyzing. There will likely be plastic strain, especially at structural discontinuities.

Bottom Line:
Did the analysis model converge at the desired load (i.e.1.5*Design Pressure)? If yes, Section 5.2, Protection Against Plastic Collapse is satisfied. If not, Section 5.2, Protection Against Plastic Collapse is NOT satisfied.

Advances in the capabilities of computers have enabled the method, since the limit load analysis will take longer to run than an elastic stress analysis. However, post-processing effort is reduced to near zero. Also, there is no question about whether or not the stress categorization line (stress cutline) is in the limiting location.

Jeff

Engineering Simulation blog... a beginning.

First question we asked ourselves is why blog? Does this help our customers? The simulation community? Does it help us?

Being an engineering simulation consulting firm and working with a wide range of customers in various industries, we see several unique simulation requirements, from leading edge to bleeding edge!  We have seen several times how what we learn in simulating for one industry can so easily be transferable to another industry; how simple tricks or having "been there, done that" would have saved us hours and hours of frustration (if not days!).  We also have done some "cool" projects which we love to share with anyone over a cocktail discussion.  We have a love-hate relationship with simulation... years of passion and also sometimes days of frustration!

So, that being said, we thought, blogging about our experiences, analyses and physics in general could be a great way to help the simulation community, our customers and also create our voice!  So, we plan to post interesting analysis stories, tricks, tips, macros, glitches, software patches (or atleast direct to the right resource), FEA vs. Testing, our perspectives and more.... If our experience, macros, resources etc., can save a few hours for a fellow analyst, we will consider our blog a success!

Let us "Engineer, Simulate and Innovate" together.

This blog is maintained by our engineering analysis group out of Rochester, NY office. 
Our blogging team (The SimuSquad) includes:
Jeff Heckman, Jason Zbick, Rolf Orsagh, Mike Sobol, Nick Lynn and Sriram "Rob" Atchutuni.