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Engineering Notes: Curved Area Calcs Using Limited Information

In order to determine how fast the High-Pressure Compressor can safely spin, we need to determine how much stress the blades and hubs are experiencing. If I throw something together and build the CFD (computational fluid dynamics) and FEA (finite entity analysis) models, I would end up with a complete overhaul to the design, and have to repeat the build processes. By employing some basic 2D calculations of selected stress concentrations which the HP compressor will experience can save considerable revision time.

Stress is defined as Force/Area. One such stress that needs to be determined is how much stress is acting on the root of the compressor blade. I tried to approximate the area with a few coefficients, but unfortunately, as the blades grow in size and camber, the approximations lose considerable accuracy.

What we can do is to approximate the blade inner and outer curve parameters, calculate the areas under each, and then subtract these to get the net result thereof. If you are using CAD, such as AutoCAD, you can query radii, areas, and centroid parameters graphically. However if you are generating the shapes using CAD parameters, or linked Excel tables, then you are going to have to do some math.


What we know

From our basic blade flow calculations, we know the centerline (mean camber line) chord length and camber angle of each blade, from root to tip. We also know that the blade thickness factor is 0.1, which indicates that the maximum blade thickness is 10% of the chord length.

Blade Chord Length (C)=0.0194m

Blade Thickness Thick=0.00194m

Mean Blade Camber (or Delta) Angle =51.15°Δ or 0.89274 radians Θ

Note: To avoid confusion of mean camber line with other references to the mean line design of the flow cross section, I’ll refer to the mean camber line as the centerline of blade (c/l).


CAD Curve Area Calculations Centerline

What we need to determine

The equations we will use are basic trigonometry relationships in a circular arc. We would be wise to use some calculus to determine these, but trig will be easier in a spreadsheet or CAD parameter field.

Radius: R =  C/(2*Sin(Θ/2))

Mid Ordinate: M = R(1-Cos(Θ/2))

Camber or Delta angle: Θ = 4*ATan(2 * M/C)

Area under the curve: Area = R^2/2 * (Θ – Sin(Θ))

Note: Area is the area between the curve and the chord.


symbols will include:

R = Radius

Rcl = Radius of centerline

M / Mo / Mi / Mcl = Mid-Ordinate and subscripts for outer, inner, and centerline curves

Θ / Θo / Θi = Delta or Camber angle, in radians, with subscripts for outer and inner curves

AREA = Net area between curves

AREAo / AREAi = Area under respective curves, with subscripts for outer and inner curves


C/l Curve Calculations

Our first step is to determine the radius and mid-ordinate of the c/l curve.

Why the mid-ordinate you may ask. Because, it is easiest to jump to the inner and outer curves as we already know the offset from the centerline at the thickest point. That would approximately be half the thickness of the blade.

CAD Curve Area Calculations Mid Ordinate

R =  C/(2*Sin(Θ/2))

  • Rcl = 0.0194 / (2*Sin(0.89274/2)) = 0.02247m


M = R(1-Cos(Θ/2))

  • Mcl = 0.02247*(1–Cos(0.89274/2)) = 0.00220m

Outer and Inner Curve

As mentioned easrlier, if the blade centerline lies in the middle of the inner and outer curves, then the offset between these is 1/2 the thickness.

CAD Curve Area Calculations Outer

The following calculations are for the outer curve, with subscript _o.

Mid Ordinate:

Mo = Mcl + 0.5*thickness

  • Mo = 0.00220 + 0.5*0.00194 = 0.00317m

Using the Chord length and Mid Ordinate, we can determine the remaining values.

Camber Angle:

Θ = 4*ATan(2 * M/C)

  • Θo = 4 * ATan(2 * 0.00317/0.0194) = 1.26345 rad


R =  C/(2*Sin(Θ/2))

  • Ro = 0.0194/(2 * Sin(1.26345/2)) = 0.01643m

For the inner curve, with subscript _i, we’d subtract the half thickness instead of adding, then repeat the remaining calculations.

Mi = 0.00220 – 0.5*0.00194 = 0.00123m

  • Θi = 4 * ATan(2 * 0.00123/0.0194) = 0.50452r
  • Ri = 0.0194/(2 * Sin(0.50452/2)) = 0.03886m


Now that we have Radius and Camber, we can determine the area under the curve.

CAD Curve Area Calculations Difference

Area = R^2/2 * (Θ – Sin(Θ))

  • AREAo = 0.01643^2 /2 * (1.26345 – Sin(1.26345) = 0.000042 m^2

Now, we can repeat the entire process for the inner curve and get it’s area:

  • AREAi = 0.03886^2 /2 * (0.50452 – Sin(0.50452) = 0.000016m^2
  • Area of blade cross section = AREAo – AREAi
  • Area = 0.000042 – 0.000016 = 0.000026m^2


That seems like a lot of work that some calculus could simplify; very true. However if you are working in Excel or CAD parameters, you need something that’s algebraic (plus I’m not the best at Calculus).

Our old coefficient estimation of this curve was 0.000031m^2, which is about 20% off. That difference applied into three factors of the principle stress calculations should be enough to cause considerable uncertainty. With a safety factor of 3+, 20% starts eating up our usable design room quickly. If each stress estimation is out by 20%, the design stability is overestimated tremendously.

I’ll bring other ways of determining some of this information, as well as centroid calculation, second moment of area, bending moment, stress calculations and more. Keep checking back at Engineering Notes.


While the standard arc trigonometry equations are real, the application for applying these for compressor blade root area cross sections is approximate. If you simply need the trig information for circular arcs, then you are set. If you are applying these factors to DCA airfoil calculations, which can vary shape somewhat, be aware that these are only close estimations intended to get you running quickly.

The entire trigonometry equation for Excel

If you want to simplify things a bit, the following is the whole enchilada, in one equation. You can paste that into Excel if you like, and it only needs you to supply the Chord, and the inner and outer Mid Ordinates.

Area = (((C/(2*SIN(ATAN(2*M)/C)))^2)/2)*((4*ATAN((2*M)/C))-SIN(4*ATAN((2*M)/C)))


Review: solidThinking Inspire 2014 Test Drive

Recently we revisited our New Features Review for solidThiking Inspire 2014. After writing that, we decided that since we love 3D printing, additive manufacturing technology, and topological optimization, we were ultimately responsible to test drive this software. Actually, it just looked so cool… and it is!

Inspire is a topological optimization software that allows users to optimize their parts for mass or strength. Inspire is different in that it can not only optimize existing design concepts to be lighter, it can develop lighter and stronger components as a starting point for a new design process. Don’t miss out on this really amazing technology and a well refined interface in our Full Review of solidThinking Inspire 2014.

Review: solidThinking Inspire 2014

What’s New: solidThinking Inspire 2014

In light of recent and upcoming enhancements from solidThinking (both Inspire and Evolve), I wanted to dust of our recent review of the the new additions of solidThinking Inspire 2014. Inspire is a topological optimizer capable of developing amazing shapes intended to be lighter and stronger than traditionally designed components. The company has increased productivity with more feature enhancements as well as some big ticked additions:

  • Geometry simplification tools
  • Smoothing options
  • Linear static analysis
  • Concentrated mass parts
  • more…


When you are done reading about the changes this year, check out our test drive of solidThinking Inspire 2014 as well.

solidThinking Inspire 2014 Infographic

Autodesk Nastran In-CAD: Test Drive

Recently I reviewed the features that Autodesk Nastran In-CAD offered to inventor users that wanted a lot more simulation power. While I have had some experience with Siemens FEMAP interface for Nastran, I have had very little experience with Autodesk’s new In-CAD UI. I thought that this article would be a good time to get in and try it from the perspective of a new user. (I should mention that I crashed an In-CAD seminar at Autodesk University for about an hour, so I did have an hour jump-start)

It should be noted here that Autodesk is selling In-CAD directly with Autodesk Inventor, as if to say “Here is our Nastran solver with an Inventor front end”. I am however writing this for everyone’s benefit, including existing Inventor users that are considering a Nastran solver.


The setup workflow is the same as one would expect:

  • Establish materials, boundary conditions, and loads.
  • Double-check everything
  • Run the solution which fails [beat head on desk]
  • Adjust the model and rectify and oversights
  • Run the solution – success
  • Review the results
  • … a laundry list of review and comparison to ensure that you are confident in the analysis model and results

Material Properties

Nastran In-CAD sorts components into material categories that are easily defined. It will pull in the material properties from the CAD model with the push of a button. In addition it can import material properties from any of the Inventor or Autodesk material libraries, or allow users to create their own.

Tip: Non-linear material types are supported, but will need to be created as these definitions are not in the existing Inventor Library (as far as I know).


Constraints, Contacts, and Loads all use similar dialogs that permit faces and bodies to be selected and deselected as desired.  Once selected, the particular conditions can be configured and applied as required.

Autodesk Nastran In-CAD Setup Panels

The Constraints dialog also contains buttons that identify limiting concepts (i.e. no rotation, free, symmetry, etc.) that directly relate to the 6 degrees-of-freedom check boxes that most analysts relate to.

Autodesk Nastran In-CAD Constraints Setup

Discerning between various surfaces is handled through the Inventor alternate surface explorer popup.

5 Contact types are available: General, Slide, Welded, Rough, and Offset Weld.

Autodesk Nastran In-CAD Contact Setup

Various limitations to contact proximity, penetration, etc. are available to configure.

Tips: Autodesk suggested using Welded for ‘Bonded’  types, and General for most other applications.

Load applications are equally simple. Load direction is applied by:

  • Individual component coordinate systems (X,Y,Z axes)
  • Normal to face
  • Geometric entity (by edge of selected geometry)

Tip: boundary conditions such as constraints and the like can be applied to different subsets


I have always loved Nastran’s adaptive meshing. It does it well and effortlessly. I typically (not always but often) get mesh concentrations how I needed them without a lot of manipulation.

Meshing is carried out with both global refinement settings and individual component settings. One feature I like is the mesh properties table, where all the component mesh settings are managed in one setting, and are easily editable.

 Autodesk Nastran In-CAD Mesh Table

I liked this a lot. Nastran In-CAD also offers an element check, where In-CAD will inspect the model meshes for inconsistencies, such as Skew, Aspect, and Jacobian limits. The results of these can be highlighted in the model, making detection and adjustments much easier.

Autodesk Nastran In-CAD failed elements

Tip: In-CAD will allow you to return to Inventor to work as normal. Be very aware that once component geometries that are connected with an In-CAD study are altered, anything applied to meshes, including constraints and contacts will be fouled and subsequent runs will require the boundary conditions to be meticulously corrected unless very broad automatic detection settings are imposed (which can be costly in run time)


I chose to setup a simple linear static analysis for this article, so that it would solve quickly and I could get a feel for basic activities. Additionally, we were using a Lenovo P500 Thinkstation CAD platform provided by Lenovo for these “CAD user” type evaluations in order to frame this in a “CAD user” perspective for solve times. While most companies purchasing a Nastran license will be mating that to a serious workstation, I wanted to understand how the solving would fare on their existing workstations, should they want to hold off on an upgrade until later.

My first run was shoddy, and I was getting Jacobian warnings and an unexpectedly long run-time. I found one surface contact that was left to the system to determine, as well as some poorly defined mesh areas. I increased the density slightly, defined the last contact manually (which I prefer to do in Nastran), and added another constraint. The subsequent runs were cleaner with a speed that was on par with an upscale 16 GB CAD platform.

Autodesk Nastran In-CAD Output in Browser

Tip: Watch the convergence in the Output panel. This is a clear indication how Nastran is handling your model setup. If the convergences won’t get close to 100% and the iterations keep rising (within a reasonable amount of time) you may want to cut your losses and stop the analysis (and possibly reduce the model DOF).

Results to Come

Setup was quite easy to pick up on. The UI has been simplified in such a manner that it takes little review in order to setup what you want. Experienced users will get it immediately, and new users should find these methods quite easy to learn.

After the review I realized that I really need to spend a couple hours setting up a simple transient analysis with some alternate elements in order to get a better feel for the setup procedure, accessibility, and capabilities. We’ll do that in the future.

In the mean-time, I will return with the post-processing and results of this review, as well as an overall perspective for adopting this software.

Autodesk Nastran In-CAD: Interface and Features

Last year Autodesk purchased NEI Nastran. This was a good purchase because Nastran is a respected Name in analysis; it’s powerful and used in numerous industries. Furthermore, Nei developed a CAD embedded UI in Solidworks for the Nei Nastran solver, called Nastran In-Cad

Autodesk adapted In-CAD for Inventor, which now gives their Inventor users the ability to perform a large array of studies, including transient and non-linear analyses, right from the comfort of Inventor. Inventor acts as the pre and post processor for the Nastran solver.

We decided to take a look at In-CAD, and see how the UI and processor’s behaved. This article will focus on the basic features, and discussions on using the software, helpful tips, and results will be forthcoming.

User Interface

The UI is divided between a Nastran In-CAD tab, and the Browser.

Autodesk Nastran In-CAD UI

Image courtesy of Autodesk

Nastran In-CAD Ribbon tab

This displays a new set of Ribbon panels dedicated to the analysis model. These panels include settings, meshing tools, boundary condition applications, results plotting, etc.


The Browser is populated not only the Model and Assembly component sets, but three additional Nastran panels.

Nasrtan Model Tree

Nastran Model Tree is divided into the assembly and model groups. The Model group accounts for all boundary conditions and templates defined in the model. The analysis group assign these to individual analyses and subsets as the user dictates to accomplish different studies.

Nastran File

This gives users a look into the Nastran file data rather than trying to interpret everything through the CAD interface.

Nastran Output (Log)

The output window shows the progress of the Nastran run, including real-time convergence data. This browser window is displayed by default during the run process. Once solved, this output log can be reviewed as needed and the contents thereof are saved along-side the other Nastran model and results files.

Autodesk Nastran In-CAD Output

From this window the analysis can be stopped, paused, and resumed.

Autodesk Nastran solver

The Autodesk Nastran (formerly Nei Nastran) solver’s accuracy is routinely tested against the NAFEMS standards at every release. Mitch Muncy, Product Manager for Autodesk’s Nastran product came along with the Nei purchase, and has stated that the company had spent countless hours verifying every detail of the solver, to ensure that it met the advanced requirements of its customers in industries such as aviation and automotive.

Linear, static stress, thermal, and modal analyses

Prestress static – response of complex loading

Static fatigue – repeated loading cycles

Heat transfer – conduction and convection heat in examination of temperature distribution

Linear buckling – compression induced loss of stiffness

Linear statics – stress, strain, and deformation from applied static loads and constraints

Normal modes – component natural frequencies

Advanced analyses

Pre-stress normal modes- capture true stiffness when complex loading is present

Frequency response – structural harmonic response based on frequency-dependent loads

Random vibration fatigue – long-term structural robustness where operation must be characterized by power spectral density (PSD) inputs

Transient response – response through a period of time under the influence of constant or time-dependent loads

Random response – behavior due to imposed of random dynamic loads, such as road vibration, wave cycles, engine vibration, and wind loads

Nonlinear static and transient response – time-varying events including dynamic loading that result in resonant vibration or stress amplification

Superior surface contact, impact analysis, and automated drop tests – includes nonlinearities of large deformations, sliding contact, and nonlinear materials

Advanced material models – complex nonlinear phenomena, including plasticity, hyper-elasticity, and shape-memory effects

Composites – straightforward complex ply data, and progressive ply failure, including Puck and LaRC02 algorithms

Element Types and Modeling

1D, 2D, and 3D element types open a whole new world of modeling capabilities. Beam elements for example offer high performance connectivity in the model at a tiny fraction of the memory and processing power required to study a 3D meshed solid in the same situations. This offers flexibility that linear static analyses never offered.

Automatic bolt connector modeling with preload – simplified setup

Associative intelligent meshing

Autodesk Nastran In-CAD Inventor Meshed Model


We wanted to give a feel for this software to both Inventor users wanting more analysis power, as well as Nastran users that were considering a CAD based environment. The In-CAD interface brings Nastran to Inventor users, and puts a lot more power at their disposal. In our next edition I’ll take Autodesk Nastran In-CAD for a spin with a simple setup and analysis run, in order to try and give all parties a feel for what they might expect.

Autodesk Nastran In-CAD: Test Drive

Autodesk Simulation Nastran Launch : Why NASTRAN? Why NEi?

After returning from the Autodesk Nastran Launch a couple of weeks ago, I discussed the company’s need to upgrade their simulation technologies. Large manufacturing industry segments lean towards software that can handle very complex studies; some specialized industries need tools that can perform studies that are quite complex in nature.

Unless Autodesk decides to invest heavily in developing innovative methods of computing large complex structure arrays as well as numerous solving algorithms, a quick stop at the local stock exchange to pick up some existing intellectual property might be a better way to go.

So Autodesk decided to go the purchase route, and pull in NEi’s Nastran solver variant which is a truly wonderful tool to have in house.

Why Nastran?

Nastran is universally recognized as the defacto Finite-Entity Analysis (FEA) standard, having been developed by NASA in the late 1960’s. The code was quickly released to the public, and as of the spring of 2014, three large companies develop their own enhanced brand of Nastran, namely MacNeal-Schwendler Corporation (MSC), Siemens PLM Software, and NEi Software. These companies have spent decades developing their Nastran variants.

The Nastran solver capabilities include:

  • Linear Static and Steady-State Heat Transfer
  • Normal Modes, Buckling, and Prestress
  • Advanced Dynamics
  • Nonlinear Analysis
  • Nonlinear Transient Heat Transfer

Autodesk’s Simulation Business Line Manager, Vic Vendantham, noted during the briefing that,

“…Nastran is already a standard, a brand, in fact it’s an eco-system that revolves around trusted, accurate, powerful, efficient and robust capabilities. The aerospace industry is already standardized on Nastran as a platform, and we have this unique opportunity … to take this and develop it and offer it to our customers.”

Autodesk Nastran Analysis capabilities

Autodesk’s purchase of a NEi Nastran fills numerous gaps in their simulation offerings, adding more complex and powerful non-linear capabilities, more efficiency, and gives the company a scalable platform from which to move ahead.

Why NEi?

That is probably a better question. The answer is almost as simple.

NEi Nastran is the youngest of the big-three variants, and as such was likely in a better position to be purchased. Better still, the NEi IP includes some nice solutions and integrations that reach beyond the standard analyses you’d expect from Nastran, including*:

  • Tension only cable and shell elements
  • Multi-layer isotropic and orthotropic composite analysis support
  • Progressive Ply Failure Analysis (PPFA™)
  • Automated Surface Contact Generation (ASCG™)
  • Automated Edge Contact Generation (AECG™)
  • Automated Impact Analysis (AIA™)
  • Numerous 3rd party aviation specific software integrations

…and much more.

Autodesk NEi Nastran Acquisition Launch Vic Vendantham

Vic emphasized some advantages NEi offered and noted,

“NEi Software has done an incredible job working to establish themselves as the industry standard for aerospace…Autodesk is in a unique position where we can take that and expand it to a variety of industry segments.”

Additionally, Firehole Composites (or the company formerly known as Firehole Composites), partnered with NEi Software around 2010 in order to deliver some of the Helius:MCT composite technology noted above to NEi Nastran users. That technology is currently owned by Autodesk in their Simulation Composite Analysis software, and many users of Autodesk products including myself, were looking forward to that technology being handed down to other CAD and simulation products. Autodesk now has a great opportunity to deliver these.

Mitch Muncy, Simulation Product Manager for Autodesk (Formerly the Executive Vice-President of NEi Software) stated that one of the important factors in this purchase was NEi having aligned themselves with Autodesk’s efforts in software development. This permitting the extremely fast Nastran rebranding and immediate integration into Autodesk’s existing software. During his discussion, Mitch pointed out:

“We wanted to be industry leaders in composites, so we were focused on putting in advanced technologies…Nastran has strong capabilities in a wide range of uses, but we also wanted to make it easy to use.”

Possibly the most compelling reason for purchasing NEi is their Nastran In-CAD product that was built for the SolidWorks CAD environment. Autodesk has successfully integrated other SolidWorks based purchases into their Flagship Inventor 3D CAD software in the past. This purchase gives Autodesk Inventor users the full power of Nastran including non-linear solutions, right inside their CAD environment. That product is already being marketed as Autodesk Nastran In-CAD 2015.

Vic added,

“One of the big assets that NEi Software brings to the table is the extensive industry expertise that they have put in place. They have already focused on advanced materials… so there is an opportunity for us to very quickly make an impact through advanced materials technology. NEi Software also has expertise in the aerospace industry, and we see this as an opportunity to continue to contribute in that space as well as push into other industries like automotive for example”.

Autodesk Nastran Launch Future Success


Nastran is wonderful. I have enjoyed working with Siemens PLM FEMAP / NX Nastran and am looking forward to seeing how NEi’s (now Autodesk’s) variant handles some problems I have encountered. The benefits from this purchase seem innumerable, and nearly impossible to choose my favorite; Scalability, Non-linear materials in Autodesk Inventor, specialized aeronautical studies integration? Autodesk has already let some of the solver technology loose, which we will look at in our next issue.

References and sources

* Information gathered from the NEi Nastran website

Images courtesy of Autodesk, Inc.

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