Rob Siegwart

World Cup Teams by Club Composition

2026-07-12T00:00:00-05:00

A recent World Cup conversation got me thinking about how national teams at the 2026 World Cup differ in their club composition, that is, the club teams that supply the players on their roster.

My initial hunch was that teams drawing more from one or two clubs (for example, Spain utilizing Barcelona players) might fare better due to pre-existing player familiarity. Of course, that logic depends on the quality of that club — the national team would only be as good as the club team it drew from. To test this, team quality variances would have to be normalized.

Nevertheless, out of curiosity I pulled the rosters for the 2026 World Cup as posted on Wikipedia and created some derivative charts and statistics:

Club Concentration Metrics (Table)
Club Concentration by National Team (Graph)
Top Feeder Leagues (Graph)
Club Count Distribution (Graph)
Top Feeder Clubs by Player Count (Graph)

HHI is a metric used here and refers to the "Herfindahl-Hirschman Index" which is calculated by squaring the share of players for each club and then summing. Squaring makes dominant clubs push the index up faster. For example,

A squad split 10/6/4/3/2/1 across six clubs gives:

(10/26)² + (6/26)² + (4/26)² + (3/26)² + (2/26)² + (1/26)² ≈ 0.246

Takeaways

Talent supply is heavily concentrated at the top. The big 5 leagues (the top leagues of England, Germany, Spain, France, and Italy) account for roughly 39% of all players. At the club level, Manchester City, Bayern Munich, PSG, Arsenal, and Barcelona lead the way above a long list of other feeder clubs.
Most rosters are patchwork, where the median national team draws from 22 distinct club teams for the 26 players.
- Two teams, Sweden and Cape Verde, have no shared club teams among the players: they each have exactly 26 club teams. The United States is not far behind with 24.
High HHI nations like Saudia Arabia and Qatar have a closed-pool effect, where 90% of the players play in their own domestic league which causes the pool to draw from to be smaller.
- The highest HHI European nation is Spain, followed by Germany.
Domestic retention and concentration are correlated across all squads (r=0.83). Teams that keep more players at home do tend to be more club-concentrated because a domestic league is a far smaller pool of clubs to draw from. Spain is an outlier as its league has horizontal breadth -- six different La Liga clubs feed Spain's national team.
The chemistry hypothesis is inconclusive, but is not "no effect." No trend across group stage → semifinals, but with N=4 in the two most advanced buckets and zero control for squad quality, this data can't distinguish "no effect" from "too noisy to tell."

Club Concentration Metrics

In this table:

Top % = percentage of players from the top club
Top-2 % = percentage of players from the top two clubs
HHI = Herfindahl-Hirschman Index
Effective clubs = 1/HHI
Domestic % = percentage of players who play for a club team in their own nation

Country	Distinct Clubs	Top Club	Top %	Top-2 %	HHI	Effective Clubs	Domestic %
South Africa	12	Mamelodi Sundowns	30.8%	61.5%	0.2041	4.9	73.1%
Czech Republic	11	Slavia Prague	38.5%	50.0%	0.1982	5.04	65.4%
Saudi Arabia	9	Al-Hilal	26.9%	50.0%	0.1746	5.73	96.2%
Qatar	8	Al-Duhail	23.1%	42.3%	0.1568	6.38	96.2%
Spain	13	Barcelona	30.8%	42.3%	0.1479	6.76	65.4%
Egypt	14	Al Ahly	30.8%	42.3%	0.142	7.04	65.4%
Turkey	16	Fenerbahçe	23.1%	46.2%	0.1272	7.86	57.7%
Germany	13	Bayern Munich	23.1%	38.5%	0.1243	8.05	73.1%
Jordan	14	Al-Hussein	26.9%	38.5%	0.1213	8.24	46.2%
Iran	14	Tractor	15.4%	30.8%	0.0976	10.24	65.4%
New Zealand	19	Auckland FC	23.1%	34.6%	0.0917	10.9	0.0%
Argentina	19	Atlético Madrid	23.1%	30.8%	0.0888	11.27	7.7%
England	16	Manchester City	15.4%	30.8%	0.0888	11.27	80.8%
France	18	Paris Saint-Germain	19.2%	26.9%	0.0799	12.52	30.8%
Portugal	17	Paris Saint-Germain	15.4%	26.9%	0.0799	12.52	19.2%
Uzbekistan	16	Pakhtakor	11.5%	23.1%	0.0769	13.0	61.5%
Mexico	21	Guadalajara	19.2%	26.9%	0.071	14.08	46.2%
Belgium	18	Club Brugge	11.5%	23.1%	0.068	14.7	11.5%
Netherlands	19	Brighton & Hove Albion	11.5%	23.1%	0.068	14.7	7.7%
Brazil	20	Flamengo	15.4%	23.1%	0.0651	15.36	26.9%
Iraq	20	Al-Talaba	11.5%	23.1%	0.0621	16.1	38.5%
Australia	22	Melbourne City	11.5%	19.2%	0.0533	18.78	19.2%
Canada	22	Los Angeles FC	11.5%	19.2%	0.0533	18.78	7.7%
Norway	22	Bodø/Glimt	11.5%	19.2%	0.0533	18.78	15.4%
Paraguay	22	Palmeiras	11.5%	19.2%	0.0533	18.78	11.5%
South Korea	22	Jeonbuk Hyundai Motors	11.5%	19.2%	0.0533	18.78	26.9%
Uruguay	22	Flamengo	11.5%	19.2%	0.0533	18.78	0.0%
Austria	22	RB Leipzig	7.7%	15.4%	0.0503	19.88	11.5%
Curaçao	22	Miami FC	7.7%	15.4%	0.0503	19.88	0.0%
Ecuador	23	Atlético Mineiro	11.5%	19.2%	0.0503	19.88	7.7%
Ghana	23	Auxerre	11.5%	19.2%	0.0503	19.88	3.8%
Scotland	22	Celtic	7.7%	15.4%	0.0503	19.88	30.8%
Senegal	22	Nice	7.7%	15.4%	0.0503	19.88	0.0%
Japan	23	Sint-Truiden	7.7%	15.4%	0.0473	21.12	11.5%
Panama	23	Saprissa	7.7%	15.4%	0.0473	21.12	7.7%
Algeria	24	Lille	7.7%	15.4%	0.0444	22.53	11.5%
Bosnia and Herzegovina	24	Schalke 04	7.7%	15.4%	0.0444	22.53	3.8%
Colombia	24	Crystal Palace	7.7%	15.4%	0.0444	22.53	3.8%
Croatia	24	Manchester City	7.7%	15.4%	0.0444	22.53	7.7%
DR Congo	24	AEL	7.7%	15.4%	0.0444	22.53	0.0%
Morocco	24	PSV Eindhoven	7.7%	15.4%	0.0444	22.53	7.7%
Switzerland	24	Sevilla	7.7%	15.4%	0.0444	22.53	7.7%
Tunisia	24	Club Africain	7.7%	15.4%	0.0444	22.53	23.1%
United States	24	PSV Eindhoven	7.7%	15.4%	0.0444	22.53	15.4%
Haiti	25	Colorado Springs Switchbacks FC	7.7%	11.5%	0.0414	24.14	3.8%
Ivory Coast	25	Charleroi	7.7%	11.5%	0.0414	24.14	0.0%
Cape Verde	26	Chaves	3.8%	7.7%	0.0385	26.0	0.0%
Sweden	26	Derby County	3.8%	7.7%	0.0385	26.0	11.5%

Tournament Results Metrics

As of 7/12/26 ~ Semi-finals stage

Stage	N	Mean HHI
Group Stage	16	0.090
Round of 32	16	0.062
Round of 16	8	0.069
Quarterfinal	4	0.053
Semifinalist	4	0.101

Squad concentration shows no clean relationship with how far a team advanced — the four semifinalists were actually the most club-concentrated group on average, but with just four teams in that bucket and no control for underlying squad quality, that's not evidence of anything either way.

Club Concentration by National Team

Boxes represent distinct club teams and the darker ones contribute more players than lighter ones.

Club rank within squad (1st = most-represented club)

Top Feeder Leagues

Total players at the 2026 World Cup, by domestic league

Club Count Distribution

Distinct clubs represented in the 26-man squad

Top Feeder Clubs by Player Count

Total players at the 2026 World Cup

The Matrix, the Fall, and Our Addiction to Struggle

2025-07-23T00:00:00-05:00

I recently re-watched the 1999 movie The Matrix, one of my favorite movies. It's rich with themes related to Gnosticism, soul trap theories, the nature of reality, and the tension between fate and free will.

In the scene where Morpheus is captured and interrogated, Agent Smith reveals something interesting about previous versions of the Matrix. He states:

“The first Matrix was designed to be a prefect human world. Where none suffered. Where everyone would be happy.”

But it was a failure, and no one would accept it. In other words, they rejected paradise.

This reminded me immediately of the Garden of Eden. Before the Fall, Eden was a perfect world designed by God. Yet humanity still chose the Tree of Knowledge of Good and Evil. Not for food or power, but to know good and evil, and to experience contrast, to suffer and understand.

Human Nature

This raises deeper questions about the nature of ourselves:

Why do we seem to crave drama and suffering?

Or at the very least, why do we seek it out, over and over?

We …

watch reality TV — human chaos as entertainment
binge on horror films and tragedies
gravitate towards war stories, survival games
get entangled in toxic relationships
watch the news which thrives on fear and shock

It seems we need there to be stakes. Without stakes, things are meaningless. And in the absence of real stakes, we manufacture simulated ones. Because deep down, we equate meaning with adversity.

No resistance = no story.
No story = no self.

In the movie, Agent Smith says as much: that human beings are defined through suffering.

From a biological standpoint, we probably are wired for adversity. Our nervous systems evolved under pressure: danger, famine, betrayal, hardship. And in the absence of a real threat, we create surrogate ones: arguments, obsessions, crises of meaning.

It's like we're wired for Eden not to be enough.

Gnostic Correlation

From a Gnostic or soul-trap perspective, this hunger for struggle becomes a mechanism of control. The Matrix (or world) is structured intentionally around struggle because it keeps you emotionally entangled, energetically drained, and looping.

In that view, this drive for drama isn't noble, it's a trap, a psychic addiction.

Soul trap theory often claims that after death, we're given a life review and enticed to return, either by guilt, emotional manipulation, or the illusion of karmic debt. We ask for “another chance,” thinking we're progressing. But in truth, we're just feeding the machine.

A Christian Parallel

From a Christian point of view, this takes on even deeper weight.

If we are indeed drawn to suffering, if humanity chose to fall, and if we are cyclically trapped in systems of pain, then Christ's suffering becomes uniquely significant.

He didn't just enter the loop, He broke it from within.

He willingly endured suffering, not as drama or karmic payback, but as atonement.
He chose the cross, not to validate our addiction to pain, but to redeem us from it.
His resurrection wasn't a reset — it was a rupture in the system. A path out.

So while The Matrix may suggest the system thrives on suffering, Christianity presents a suffering that redeems, not entraps.

Christ didn't come to perpetuate the cycle.
He came to end it — by entering it fully, and then rising beyond it.

Chess Is Life: Pattern Recognition as the Core Currency

2025-05-13T00:00:00-05:00

The parallels between chess and life are quite astonishing. A recent AI chat session sparked a realization: pattern recognition may be the most critical, and perhaps fundamental, component of intelligence and agency — our ability to effectively navigate the world.

Pattern recognition is central to chess mastery. Non-chess players might assume that calculation and tactics dominate the game, and that grandmasters simply calculate more deeply (which they do, of course). But in truth, what separates them I believe is pattern fluency. High-level players internalize thousands of familiar structures, tactical motifs, and strategic imbalances. With enough practice, a position on the board becomes not a puzzle to be solved, but a known story, an echo of past experience.

Just like in life. We don't gain fluency with the world through rules alone, but through exposure. A young person might begin to navigate life independently around the same age that chess prodigies reach mastery — the early to mid-teens. Competence in both emerges not from memorization, but from immersion.

Pattern recognition also lies at the heart of artificial intelligence. Language models are not programmed with rules. Instead, they are trained on vast datasets and learn by identifying statistical relationships. What appears to be intelligent reasoning is, at its core, a remarkably sophisticated engine of pattern prediction.

This suggests something deeper: pattern recognition isn't just a tool — it's the core currency of intelligence.

The more patterns we can see, understand, and act upon, the more effective we become in any environment.

This also connects to creativity. One could define creativity as the ability to connect seemingly unrelated ideas — another form of pattern recognition. In fact, this article itself emerged from observing a pattern: the presence of pattern recognition itself, across three domains: chess, AI, and human life.

High-level chess players use a method called chunking: processing information in grouped structures instead of individual elements. Familiar formations become perceptual units, allowing players to assess the board rapidly and intuitively.

This is the essence of intuition: subconscious pattern recognition operating beneath the surface of conscious thought. And just like computers have GPUs for fast parallel processing and CPUs for slower sequential tasks, the human mind seems to split decision-making in similar ways. Our "GPU" intuition processes large volumes of familiar input quickly, while our "CPU" logic works more slowly and carefully.

The same principle applies to language learning. A child doesn't begin by learning grammar rules. They absorb patterns (sounds, rhythms, syntax) simply through exposure. By age five, most children speak their native language fluently, without ever studying its mechanics.

So if you want to become more capable, more free — you don't need to memorize more rules. You need to see more patterns.

Chess offers a compressed arena for developing a specific kind of pattern recognition. In it we can see that the process of immersion, repetition, and internalization helps the brain form intuitions and expertise more generally. Mastery in chess doesn't make one universally insightful, but it does reveal how deep familiarity with patterns enables high-level performance.

Pattern recognition is what makes us agents, not just observers.

And in that sense, the more patterns we see, the more engaged we become with the world.

I Know This Steak Doesn't Exist

2025-05-13T00:00:00-05:00

I know this steak doesn't exist. I know that when I put it in my mouth, the Matrix is telling my brain that it is juicy and delicious. After nine years, you know what I realize? Ignorance is bliss.

—Cypher, The Matrix (1999)

This scene from the 1999 movie The Matrix came to mind when I was recently pondering a hypothetical question. If you consider the idea that this world is a trap or a prison—as proposed in systems like Gnosticism and perhaps certain branches of Eastern philosophy or modern esotericism—then the central quest for an individual becomes one of escape: how to get out of this trap, how to ascend, and how to return to our origin or true home. But another angle came to mind:

What if I like it here?

What if someone knows this world's a prison planet, or there's a trap in the afterlife—and still chooses to stay, just like Cypher?

Christian Considerations

The idea of the soul trap isn't explicitly taught or conveyed in traditional Christian theology, though some scriptures hint at the experience of spiritual deception, exile, or entrapment. For example, 2 Corinthians 4:4 speaks of “the god of this world” blinding minds, and Ephesians 6:12 describes a battle not against flesh and blood but against spiritual forces of evil in the heavenly realms. Other passages, like Philippians 3:20 (“our citizenship is in heaven”) and Hebrews 11:13-16 (describing life on earth as exile), suggest that our current world is not the soul's final home.

However, one core idea in Christian theology is the promise of ascending to heaven after death. In this sense, the soul trap concept loosely parallels that framework—it's still about escaping the present world. Modern Christianity often emphasizes the afterlife: heaven or hell. Meanwhile, much of the theological emphasis tends to treat our current existence in this fallen world as something to simply endure—an unfortunate, tainted waiting room for a better, redeemed reality to come.

I've always found that imbalance a little strange. If we were made by God to exist here on Earth, surely there's a purpose to this life beyond simply trying to make it to heaven. Perhaps the Christian counterargument is that this was true before the Fall. Still, the idea that our sole goal now is to escape Earth and reach heaven seems overly reductive—maybe even contrary to the intrinsic incarnational nature of Christian belief.

The Bible does speak to the value of life here. For example, John 10:10 states, “My purpose is to give them a rich and satisfying life.” And in Ecclesiastes 3:12-13:

“So I concluded there is nothing better than to be happy and enjoy ourselves as long as we can. And people should eat and drink and enjoy the fruits of their labor, for these are gifts from God.”

Biological Code and the Drive for Comfort

Regardless of one's religious or philosophical worldview, it seems reasonable to assume that whoever or whatever created this realm also created us—whether through design or evolution—and embedded within us a kind of biological software.

That software is optimized to seek comfort, safety, self-preservation, and to avoid pain and suffering.

So if someone sees the Matrix (or this Earth plane) as their best shot at pleasure, security, or meaning, they may very well choose it.

Not because they're faulty in some way or they hate truth, but because the code is working as designed.

Awakening Is a Bug

In this model, here's the twist:

Awakening isn't a feature of the system. It's a bug.

The desire to break free from comfort, to reject the simulation, to embrace the brutal uncertainty of the “real”, that's aberrant behavior. It's anti-code.

If someone isn't experiencing comfort, safety, or meaning in this world, then awakening or escape might feel like a genuine internal longing. But if one is surrounded by relative pleasure and security, the desire to unplug becomes unnatural—almost irrational.

This suggests that spiritual awakening might not be a common evolutionary outcome, but a rare deviation—an anomaly in the system.

Cypher's Logic

Cypher isn't just a villain. He's a mirror, albiet an uncomfortable one.

He wants what most people want: pleasure, power, peace.
He's self-aware enough to admit it.
He chooses the illusion knowingly

Because deep down, we are all presented with the question,

If you could be happy, would you care that it's not real?

Final Thought

The choice between truth and illusion isn't as simple as red pill vs. blue pill. It cuts down to the deepest layers of who we are and how we were made. Unless we're willing to examine our programming, Cypher's logic might not be betrayal at all.

It might just be the system doing what it was designed to do.

But maybe, something deeper inside of us is whispering:

Comfort isn't enough.

Chess is Life: Conflict

2025-04-27T00:00:00-05:00

In my experience playing chess over the last few years, I've come to see that many of the game's motifs mirror aspects of life. And in this post I will attempt to present some ideas that are related to conflict — whether between individuals, groups, or even within ourselves.

Chess is a confrontational game between you and your opponent. For these analogies we will expand it to one vs many or many vs many as well, however in each case it could be represented with two sides or groups.

Below are some motifs and parallels I've noticed between the mechanics of chess and the dynamics of conflict in the world we live in.

Motifs

Defense through offense

There is a phrase, often used in sports, which is roughly, "the best defense is a good offense." In chess, that's true. You will find as much as possible, you want to keep moving your pieces forward, into your opponents territory, rather than backwards and into your own. This forces your opponent to spend his moves defending — not attacking you — a double win.

Don't let your guard down

When you get a good position, it's tempting to relax — this is a big mistake! Never relax until the game is over. Focus and ruthlessness are required until the conclusion. In chess the hardest position to win is often a winning position.

Never interfere with an enemy making a mistake

There is an old quote, Never interrupt your enemy when he is making a mistake. In chess, this is also true. If your opponent is playing badly, don't get fancy. Just develop, play solidly, and take advantage of his mistakes.

Be an assassin

This one is less about strategy and more about mindset. Chess helps develop — and rewards — what I call an assassin mindset, made of qualities like:

intense focus at the present task and initiative at hand
having a singular goal
ruthless efficiency: filtering your moves to only those that directly advance you toward your goal
- In chess, moves that don't meet this standard are considered passive and can cause you to lose tempo, letting your opponent seize the initiative.
going for the kill shot: playing or taking those actions or moves which have the most devastating impact (checkmate, or the setup for it)

Having fun (Law of Attraction vibes)

In life, when you don't take it too seriously and have fun, you often get better results. Same with chess: obsess over your rating, and you will play worse. Relax and just enjoy the game — you'll see the board better and play stronger.

Others

Motif	Chess	Life
Advance with support	Developing your pieces into supported positions	In soccer, advancing the ball up the field with team support
Harmony, energy balance	Pieces defend each other and are on good squares	Musical harmony; balancing loads in a mechanical design
Tactics	Attack an opponent's piece to distract it from its defensive role	In Muay Thai, a kick attack can be first set up with a low kick to the leg
Kamakaze	Sacrifice a piece to open the position or attack the king. (Also, desperado tactics — grabbing an enemy piece before inevitable capture.)	A kamakaze attack in battle, such as in the movie Lord of the Rings: The Two Towers, the orc carries a bomb to the wall to break it down
Resiliance	Recovering from mistakes and maintaining a fighting spirit	In sports, overcoming mistakes instead of letting them wreck your mindset
Mind Games	Intentionally put youself into a worse position, so that your opponent feels he has an advantage, and therefore might be tempted to lower his defences and play with complacency (not something you would do in an actual game but maybe in a casual game against a lower-rated player)	The Mongols would lure enemies into pursuit with a small force, then ambush them with the main body of troops at a pre-arranged location

In chess as in life, conflict is inevitable — and understanding its mechanics is the beginning of mastery.

But the board holds more lessons still: chaos, order, agency, resilience. The game, and life, are far from over.

FEA Basics and Development of a 2D Link Element

2021-06-17T00:00:00-05:00

In my quest to learn more about the inner workings of the finite element method and its programming implementation, I have started off attempting to program the simplest of FEA models and analyses via the direct stiffness method. (Here is my work so far on the topic of this article - simpleFEA.)

To that end, a 2D link or truss element is probably the simplest finite element there is (maybe besides a point mass).

Concepts

Big Picture

The basics of a linear structural FEA is the stiffness equation:

\begin{equation*} \mathbf{F} = \mathbf{K}\mathbf{U} \end{equation*}

where $\mathbf{F}$ is the global force vector, $\mathbf{K}$ the global stiffness matrix, and $\mathbf{U}$ the global displacement vector. The size of this system (length of vector) is equal to the total number of nodal degrees-of-freedom (DOF). The goal of the linear finite element program therefore is to assemble and solve this system of equations.

The matrix form of this equation represents the interrelation of the connectivity of nodes by elements. If expanded, each row would contain only those nodes that contribute to the force term of that row. And, a node typically is only connected to a few elements and because there are usually a large number of elements, this matrix contains many empty/zero entries. (This leads to the concept of sparse matrices which are specialized ways to store matrix data when much of the matrix is empty.)

Coordinate Systems

All elements have the concept of a local coordinate system where they are aligned in a convenient orientation for deriving equations. When assembled into a finite element model, however, the element may be oriented in any orientation. Thus, a process of transformation is needed to transform the quantities involved in the constituent equations, such as force and displacement, from the local element system to the global coordinate system, and vice-versa. For that, we use a transformation matrix. In a previous post I demonstrated the basic concept of a transformation matrix.

Truss Element

The truss element is represented schematically and can be modeled as a spring: it may only take tension or compression. Each node has two DOF, x and y. The element coordinate system has its x-axis aligned with the element direction vector going from node i to node j. Quantities with respect to the local element coordinate system are indicated by the bar over the variable.

It is considered a discrete element as opposed to a continuum element which has its variables governed by differential equations.

With the element having constant properties A (area), E (elastic modulus), and an inferred length from the nodes, L, the element stiffness in the axial direction is:

\begin{equation*} k = \frac{AE}{L} \end{equation*}

Since the element may only take tension or compression, elemental y axis forces do not contribute to member stiffness. Thus, force equilibrium and force-displacement relations become:

\begin{equation*} \bar{f} = k\Delta x = \frac{AE}{L} (\bar{u}_{x,j} - \bar{u}_{x,i}) \end{equation*}

where $\Delta x$ is the elongation of the bar, equal to $\bar{u}_{x,j} - \bar{u}_{x,i}$

Then, force equilibrium requires:

\begin{equation*} -\bar{f}_{x,i} = \bar{f}_{x,j} \end{equation*}

The local element stiffness equations are then as follows:

\begin{equation*} -\bar{f}_{x,i} = \frac{AE}{L} (\bar{u}_{x,j} - \bar{u}_{x,i}) \qquad \text{or} \qquad \bar{f}_{x,i} = \frac{AE}{L} (- \bar{u}_{x,j} + \bar{u}_{x,i}) \end{equation*}

\begin{equation*} \bar{f}_{x,j} = \frac{AE}{L} (\bar{u}_{x,j} - \bar{u}_{x,i}) \end{equation*}

Which can be combined into matrix form, $\mathbf{\bar{F}} = \mathbf{\bar{K}}\mathbf{\bar{U}}$:

\begin{equation*} \begin{bmatrix} \bar{f}_{x,i} \\ \bar{f}_{y,i}\\ \bar{f}_{x,j}\\ \bar{f}_{y,j} \end{bmatrix} = \frac{AE}{L} \begin{bmatrix} 1 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0\\-1 & - & 1 & 0\\ 0 & 0 & 0 & 0\end{bmatrix} \begin{bmatrix} \bar{u}_{x,i} \\ \bar{u}_{y,i}\\ \bar{u}_{x,j}\\ \bar{u}_{y,j} \end{bmatrix} \end{equation*}

The 4x4 matrix with the coefficient is the element stiffness matrix in the element coordinate system, and for an individual element in the context of a collection of elements is denoted as $\mathbf{\bar{K}}^e$.

Transformation

Transformation is needed to convert element quantities in the local element coordinate system to those in the global element coordinate system so they may be appropriately combined into global stiffness equations. The position or orientation of the element is defined by the nodal coordinates.

\begin{equation*} \mathbf{\bar{u}_{x,i}} \end{equation*}

\begin{equation*} \mathbf{\bar{u}_{y,i}} \end{equation*}

Displacement transformations may be inferred from graphical analysis.

\begin{equation*} \bar{u}_{x,i} = u_{x,i}\cos\theta + u_{y,i}\sin\theta \end{equation*}

\begin{equation*} \bar{u}_{y,i} = -u_{x,i}\sin\theta + u_{y,i}\cos\theta \end{equation*}

The equations for node j are the same as for node i.

\begin{equation*} \bar{u}_{x,j} = u_{x,j}\cos\theta + u_{y,j}\sin\theta \end{equation*}

\begin{equation*} \bar{u}_{y,j} = u_{x,j}\sin\theta + u_{y,j}\cos\theta \end{equation*}

Putting all 4 equations into matrix form, and with the helpers of $c=\cos\theta$ and $s=\sin\theta$, we have:

\begin{equation*} \begin{bmatrix}\bar{u}_{x,i} \\ \bar{u}_{y,i} \\ \bar{u}_{x,j} \\ \bar{u}_{y,j}\end{bmatrix} = \begin{bmatrix} c & s & 0 & 0 \\ -s & c & 0 & 0 \\ 0 & 0 & c & s \\ 0 & 0 & -s & c\end{bmatrix} \begin{bmatrix} u_{x,i} \\ u_{y,i} \\ u_{x,j} \\ u_{y,j} \end{bmatrix} \end{equation*}

Force transformations are the same as for displacement:

\begin{equation*} \begin{bmatrix}\bar{f}_{x,i} \\ \bar{f}_{y,i} \\ \bar{f}_{x,j} \\ \bar{f}_{y,j}\end{bmatrix} = \begin{bmatrix} c & s & 0 & 0 \\ -s & c & 0 & 0 \\ 0 & 0 & c & s \\ 0 & 0 & -s & c\end{bmatrix} \begin{bmatrix} f_{x,i} \\ f_{y,i} \\ f_{x,j} \\ f_{y,j} \end{bmatrix} \end{equation*}

The 4x4 matrix used here is the transformation matrix and is denoted as $\mathbf{T}^e$. Conversion of quantities from the global coordinate system to the element coordinate system is done by:

\begin{equation*} \mathbf{\bar{U}}^e = \mathbf{T}^e \mathbf{U}^e \end{equation*}

\begin{equation*} \mathbf{\bar{F}}^e = \mathbf{T}^e \mathbf{F}^e \end{equation*}

Global Equations

Generation of global equations is developed by first setting equal the equations for the local element on a force basis, which yields:

\begin{equation*} \mathbf{\bar{F}}^e = \mathbf{\bar{K}}^e \mathbf{\bar{U}}^e \end{equation*}

and is equal to:

\begin{equation*} \mathbf{T}^e\mathbf{F}^e = \mathbf{\bar{K}}^e \mathbf{T}^e\mathbf{U}^e \end{equation*}

Next, both sides are multiplied by $(\mathbf{T}^e)^{-1}$, which for this element type is also equal to the transpose, $(\mathbf{T}^e)^T$:

\begin{equation*} (\mathbf{T}^e)^T \mathbf{T}^e \mathbf{F}^e = (\mathbf{T}^e)^T \mathbf{\bar{K}}^e \mathbf{T}^e \mathbf{U}^e \end{equation*}

which becomes,

\begin{equation*} \mathbf{F}^e = (\mathbf{T}^e)^T \mathbf{\bar{K}}^e \mathbf{T}^e \mathbf{U}^e \end{equation*}

therefore, the element's stiffness matrix in global coordinates is:

\begin{equation*} \mathbf{K}^e = (\mathbf{T}^e)^T \mathbf{\bar{K}}^e \mathbf{T}^e \end{equation*}

References

Introduction to Finite Element Methods (ASEN 5007). University of Colorado at Boulder. http://kis.tu.kielce.pl/mo/COLORADO_FEM/colorado/Home.html.
J. N. Reddy, Ph.D. Introduction to the Finite Element Method, Fourth Edition. Analysis of Trusses, Chapter (McGraw-Hill Education: New York, Chicago, San Francisco, Athens, London, Madrid, Mexico City, Milan, New Delhi, Singapore, Sydney, Toronto, 2019, 2006, 1993, 1984). https://www.accessengineeringlibrary.com/content/book/9781259861901/toc-chapter/chapter6/section/section3

Python @staticmethod, @classmethod

2021-06-09T00:00:00-05:00

While regular class methods take in the class instance as the first parameter, @classmethod and @staticmethod are available in Python to create class methods with different behavior and usage.

@staticmethod

does not take in the class or the instance as the implicit first parameter
can be called on either the class or an instance
cannot access class or instance properties

A typical usage might be to store any general utility function to the class, such as when the arguments have nothing to do with the class or its instances.

class Foo:
    @staticmethod
    def add(a,b):
        return a + b

Foo.add(3,12)

@classmethod

takes the class in as the implicit first parameter
can be called on either the class or an instance
has access to class properties

A typical usage might be to create new class instances with some alternative call signature:

class Foo:
    def __init__(self, a, b):
        self.a = a
        self.b = b

    @classmethod
    def bar(cls, a, c):
        return cls(a, c**2)

standard = Foo(2,9)
alternate = Foo.bar(2,3)

Unified Thread Tables

2021-06-09T00:00:00-05:00

Thread tables for Unified coarse and fine threads along with tensile area and equivalent tensile area diameter per equation A-3-7 from Ref [2]. Tap drills are for 75% thread engagement, per Tables 3 and 4 of Ref [1] of section "Threading".

UNC

Size, Number or Inches	Basic Major Diameter (in)	TPI	Tensile Area, in²	Tensile Area Diameter, in	Tap Drill No	Tap Drill Size, in	Clearance Drill (Close)	Clearance Size (Close), in	Clearance Drill (Free)	Clearance Size (Free), in
0	0.0600						52	0.0635	50	0.07
1	0.0730	64	0.0026	0.058	53	0.0595	48	0.076	46	0.081
2	0.0860	56	0.0037	0.069	50	0.07	43	0.089	41	0.096
3	0.0990	48	0.0049	0.079	47	0.0785	37	0.104	35	0.11
4	0.1120	40	0.0060	0.088	43	0.089	32	0.116	30	0.1285
5	0.1250	40	0.0080	0.101	38	0.1015	30	0.1285	29	0.136
6	0.1380	32	0.0091	0.108	36	0.1065	27	0.144	25	0.1495
8	0.1640	32	0.0140	0.134	29	0.136	18	0.1695	16	0.177
10	0.1900	24	0.0175	0.149	25	0.1495	9	0.196	7	0.201
12	0.2160	24	0.0242	0.175	16	0.177	2	0.221	1	0.228
1/4	0.25	20	0.0318	0.201	7	0.201	F	0.257	H	0.266
5/16	0.3125	18	0.0524	0.258	F	0.257	P	0.323	Q	0.332
3/8	0.375	16	0.0775	0.314	5/16	0.3125	W	0.386	X	0.397
7/16	0.4375	14	0.1063	0.368	U	0.368	29/64	0.4531	15/32	0.4687
1/2	0.5	13	0.1419	0.425	27/64	0.4219	33/64	0.5156	17/32	0.5312
9/16	0.5625	12	0.1819	0.481	31/64	0.4844
5/8	0.625	11	0.2260	0.536	17/32	0.5312
3/4	0.75	10	0.3345	0.653	21/32	0.6562
7/8	0.875	9	0.4617	0.767	49/64	0.7656
1	1	8	0.6057	0.878	7/8	0.875
1 1/8	1.125	7	0.7633	0.986	63/64	0.9844
1 1/4	1.25	7	0.9691	1.111	1 7/64	1.1094
1 5/16	1.3125	7	1.0812	1.173
1 3/8	1.375	6	1.1549	1.213	1 7/32	1.2187
1 1/2	1.5	6	1.4052	1.338	1 11/32	1.3437
1 5/8	1.625	6	1.6802	1.463
1 3/4	1.75	5	1.8995	1.555	1 9/16	1.5625
1 7/8	1.875	5	2.2171	1.680	1 11/16	1.6875
2	2	4.5	2.4982	1.783	1 25/32	1.7812
2 1/4	2.25	4.5	3.2477	2.033	2 1/32	2.0312
2 1/2	2.5	4	3.9988	2.256	2 1/4	2.25
2 3/4	2.75	4	4.9340	2.506
3	3	4	5.9674	2.756
3 1/4	3.25	4	7.0989	3.006
3 1/2	3.5	4	8.3286	3.256
3 3/4	3.75	4	9.6565	3.506
4	4	4	11.0825	3.756

UNF

Size, Number or Inches	Basic Major Diameter (in)	TPI	Tensile Area, in²	Tensile Area Diameter, in	Tap Drill No	Tap Drill Size, in	Clearance Drill (Close)	Clearance Size (Close), in	Clearance Drill (Free)	Clearance Size (Free), in
0	0.060	80	0.0018	0.048	3/64	0.0469	52	0.0635	50	0.07
1	0.073	72	0.0028	0.059	53	0.0595	48	0.076	46	0.081
2	0.086	64	0.0039	0.071	50	0.07	43	0.089	41	0.096
3	0.099	56	0.0052	0.082	45	0.082	37	0.104	35	0.11
4	0.112	48	0.0066	0.092	42	0.0935	32	0.116	30	0.1285
5	0.125	44	0.0083	0.103	37	0.104	30	0.1285	29	0.136
6	0.138	40	0.0101	0.114	33	0.113	27	0.144	25	0.1495
8	0.164	36	0.0147	0.137	29	0.136	18	0.1695	16	0.177
10	0.19	32	0.0200	0.160	21	0.159	9	0.196	7	0.201
12	0.216	28	0.0258	0.181	14	0.182	2	0.221	1	0.228
1/4	0.25	28	0.0364	0.215	3	0.213	F	0.257	H	0.266
5/16	0.3125	24	0.0581	0.272	I	0.272	P	0.323	Q	0.332
3/8	0.375	24	0.0878	0.334	Q	0.332	W	0.386	X	0.397
7/16	0.4375	20	0.1187	0.389	25/64	0.3906	29/64	0.4531	15/32	0.4687
1/2	0.5	20	0.1600	0.451	29/64	0.4531	33/64	0.5156	17/32	0.5312
9/16	0.5625	18	0.2030	0.508	33/64	0.5156
5/8	0.625	18	0.2560	0.571	37/64	0.5781
3/4	0.75	16	0.3730	0.689	11/16	0.6875
7/8	0.875	14	0.5095	0.805	13/16	0.8125
1	1	12	0.6630	0.919	59/64	0.9219
1 1/8	1.125	12	0.8557	1.044	1 3/64	1.0469
1 1/4	1.25	12	1.0729	1.169	1 11/64	1.1719
1 5/16	1.3125	12	1.1908	1.231
1 3/8	1.375	12	1.3147	1.294	1 19/64	1.2969
1 1/2	1.5	12	1.5810	1.419	1 27/64	1.4219

References

[1]	Machinery's Handbook, 28th Edition. Industrial Press.

[2]	ANSI/AISC 360-16, Specification for Structural Steel Buildings.

Weld as a Line - Geometric Properties Table

2021-06-02T00:00:00-05:00

Properties

$s$ - weld size
$t$ - weld throat dimension, equal to $0.707s$ for typical fillet welds (joint angles between 80 and 100 degrees)
$d$ - height of weld shape
$b$ - width of weld shape
$A_u$ - unit weld area
$I_{u,x}$, $I_{u,y}$ - unit area moment of inertias, x-x and y-y axes, respectively
$J_u$ - Unit polar moment of inertia

Unit properties are defined in terms of unit weld size. Multiply their values by the throat dimension to obtain the actual value of the property, which is used in calculations when the weld size is known:

\begin{equation*} A = A_ut \end{equation*}

\begin{equation*} I_x = I_{u,x}t \qquad I_y = I_{u,y}t \end{equation*}

\begin{equation*} J = J_ut \end{equation*}

Table

Shape	$$ \bar{x} $$	$$ \bar{y} $$	$$ A_u $$	$$ I_{u,x} $$	$$ I_{u,y} $$	$$ J_u $$
Line	0	$$ \frac{d}{2} $$	$$ d $$	$$ \frac{d^3}{12} $$	$$ \frac{dt^3}{12} \text{*}$$	$$ \frac{d^3}{12} $$
Double Line	$$ b/2 $$	$$ d/2 $$	$$ 2d $$	$$ \frac{d^3}{6} $$	$$ \frac{db^2}{2} $$	$$ \frac{d(3b^2+d^2}{6} $$
L	$$ \frac{b^2}{2(b+d)} $$	$$ \frac{d^2}{2(b+d)} $$	$$ b+d $$	$$ \frac{d^3(d+4b)}{12(b+d)} $$	$$ \frac{b^3(d+4b)}{12(b+d)} $$	$$ \frac{b^4 + d^4 + 4bd(b^2+d^2)}{12(b+d)} $$
C	$$ \frac{b^2}{2b+d} $$	$$ \frac{d}{2} $$	$$ 2b+d $$	$$ \frac{d^2}{12}(d+6b) $$	$$ b^3\left( \frac{b+2d}{3(2b+d)} \right) $$	$$ \frac{d^2}{12}(d+6b) + b^3\left( \frac{b+2d}{3(2b+d)} \right) $$
Box	$$ \frac{b}{2} $$	$$ \frac{d}{2} $$	$$ 2(b+d) $$	$$ \frac{d^2}{6}(d+3b) $$	$$ \frac{b^2}{6}(b+3d) $$	$$ \frac{(b+d)^3}{6} $$
Circle	$$0$$	$$0$$	$$ 2\pi r $$	$$ \pi r^3 $$	$$ \pi r^3 $$	$$ 2\pi r^3 $$

* Typically not defined for this axis, but this allows one to compute a bending stress given a y-y moment.

References

Collins, Busby, Staab. Mechanical Design of Machine Elements and Machines. 2nd Ed.

Easily Zip Ghost Themes with Python

2021-06-01T00:00:00-05:00

Previous iterations of this site used the Ghost blogging platform which utilized a custom theme. In order to upload it to the Ghost application a zipped folder of the theme is required. To save mouse clicks and keyboard strokes, I wrote a Python script to zip the theme files and save out a .zip file with the version included in the filename.

For Ghost, the theme version is set in the package.json and package-lock.json files for it to be recognized in the Ghost admin. The Python script below gets the version value from package.json and includes it in the zipped file name.

The Python script is placed in the directory to be zipped and some variables are set:

BASE_NAME - base name of the output zip file with the output name being [BASE_NAME]-[VERSION].zip
OUTDIR - local output directory name for the resulting zip file
REMOVE_DIRS and REMOVE_FILES - lists of directories and files, respectively, to exclude in the resulting zip file

Then, running this file compiles the files in the current directory and all subdirectories with the exceptions listed in REMOVE_DIRS and REMOVE_FILES and places an named zip file in the directory specified in OUTDIR.

import os
import glob
import zipfile
import json

BASE_NAME = 'my_zip_file'
OUTDIR = 'Zipped'
REMOVE_DIRS = ['Zipped','.vscode','.git']
REMOVE_FILES = ['VERSION','zip.py','routes.yaml']

with open('package.json') as f:
    packagejson = json.load(f)
    version = packagejson['version']

theme_files = set(glob.glob('*'))
theme_files = theme_files - set(REMOVE_DIRS) - set(REMOVE_FILES)

if not os.path.exists(OUTDIR):
    os.mkdir(OUTDIR)

zip = zipfile.ZipFile(
    os.path.join(OUTDIR, f'{BASE_NAME}-{version}.zip'),
    mode='w',
    compression=zipfile.ZIP_DEFLATED
)

def zipdir(directory):
    for root, dirs, files in os.walk(directory):
        for file in files:
            zip.write(os.path.join(root, file))
        for dir in dirs:
            zipdir(dir)

for file in theme_files:
    if os.path.isdir(file):
        zipdir(file)
    else:
        zip.write(file)

zip.close()

GD&T -- Basics

2021-05-31T00:00:00-05:00

Concepts

Feature - A part surface
Feature of size (FOS) - A cylindrical or spherical surface or a set of two opposed elements or parallel surfaces associated with a size dimension.
- Internal FOS - Internal surfaces such as a hole or slot.
- External FOS - External surfaces such as a shaft or tab.

Feature of size dimension - A dimension associated with a FOS.
Non-feature of size dimension - A dimension not associated with a FOS.
Actual Local Size - The value of any individual distance at any cross section of a FOS (for example any two point measurement taken with a pair of calipers).
Envelope, Actual Mating (AME) - A similar perfect counterpart - for an external feature of size is the smallest size that can be contracted about the feature; for an internal feature of size is the largest size that can be expanded within the feature - so that it just contacts the surfaces at the highest points. This surface resides outside the material.
- Related AME - is constrained in orientation to the applicable datums
- Unrelated AME - is not constrained in orientation to the applicable datums and is rotated to align the the axis of the part
Envelope, Actual Mininum Material - a similar perfect counterpart of largest size that can be expanded in a external FOS, and the smallest size that can be contracted about an internal FOS. Can also be related and unrelated. This surface resides inside the material.

Basic dimension - A dimension that represents the theoretically exact size, true profile, orientation, or location of a feature or gage information such as datum targets. When basic dimensions are used, they must be accompanied by geometric tolerances. Typically denoted by enclosing the value in a rectangle.

Material Conditions

Valid for features of size (FOS). Every feature of size has a maximum and minimum material condition.

Maximum Material Condition (MMC)

The condition in which the geometry contains the maximum amount of material within the stated limits of size. Examples:

Smallest hole
Largest shaft
Narrowest slot
Largest tab

Minimum Material Condition (LMC)

The condition in which the geometry contains the minimum amount of material within the stated limits of size. Examples:

Largest hole
Smallest shaft
Largest slot
Smallest tab

Regardless of Feature Size (RFS)

Indicates geometric tolerances apply at any increment of size of the feature within its size tolerance. In other words, the geometric tolerance is valid at whatever size the part is produced at.

No symbol is associated with this condition as it is the default condition.

Modifiers

Term	Abbreviation	Symbol
Maximum material condition	MMC
Least material condition	LMC
Projected tolerance zone	-
Tangent plane	-
Diameter	-
Radius	-
Controlled radius	-
Reference	-	( )

Feature Control Frame

Geometric tolerances specified on a drawing use a feature control frame for their syntax. This consists of a rectangular box divided into three main subdivisions:

geometric characteristic symbol
tolerance value + any modifiers
datums

Rule #1

Where only a tolerance of size is specified, the limits of size of an individual feature prescribe the extent to which variations in its form and its size are allowed.

This is often paraphrased as "perfect form at MMC". At MMC, no further deviation in form is allowed. When the size departs from MMC, some form error is allowed. (Perfect form means all applicable form controls: straightness, flatness, circularity, and cylindricity.)

Can be overridden when a straightness control is applied to a FOS or if a note explicitly stating such is included.

Rule #1 does not affect the location, orientation, or relationship between features of size.

Does not apply to parts that are flexible or to stock parts (e.g. bar stock, tubing, etc.).

Inspection may be performed with a Go gage for FOS at MMC and a No-Go gage for FOS at LMC.

Rule #2

RFS applies, with respect to the individual tolerance, datum reference, or both, where no modifying symbol is specified. MMC or LMC must be specified on the drawing where required.

Where a geometric tolerance is applied on an RFS basis, the tolerance is limited to the specified value regardless of the actual size of the feature.

Geometric Characteristic Symbols

2021-05-25T00:00:00-05:00

Form

Never uses a datum as a reference.

Characteristic	Symbol
Straightness
Flatness
Circularity (Roundness)
Cylindricity

Profile

Sometimes uses a datum as a reference.

Characteristic	Symbol
Profile of a line
Profile of a surface

Orientation

Always uses a datum as a reference.

Characteristic	Symbol
Angularity
Perpendicularity
Parallelism

Location

Always uses a datum as a reference.

Characteristic	Symbol
Position
Concentricity
Symmetry

Runout

Always uses a datum as a reference.

Characteristic	Symbol
Circular runout
Total runout

ASTM A36

2021-05-17T00:00:00-05:00

ASTM A36/A36M - 05 - Standard Specification for Carbon Structural Steel

Mechanical Properties

Strength

Tensile	58 - 80 ksi
Yield	36 ksi

Strain

Plates and Bars

Elongation in 8 in	20%
Elongation in 2 in	23%

Shapes

Elongation in 8 in	20%
Elongation in 2 in	21%

Chemical Properties

All shapes and sizes:

Element	Range, %
Carbon	0.25 - 0.29
Manganese	0.6 - 0.90
Phosphorus, max	0.04
Sulfur, max	0.05
Silicon	0.15 - 0.40
Copper, min	0.20

ASTM A500

2021-05-17T00:00:00-05:00

ASTM A500-03a (2003 revision a) Standard Specification for Cold-Formed Welded and Seamless Carbon Steel Structural Tubing in Rounds and Shapes

Mechanical

Round Structural Tubing

Property	Grade A	Grade B	Grade C	Grade D
Yield (psi)	33,000	42,000	46,000	36,000
Tensile (psi)	45,000	58,000	62,000	58,000
Elongation in 2 in (%)	25	23	21	23

Shaped Structural Tubing


Yield (psi)	39,000	46,000	50,000	36,000
Tensile (psi)	45,000	58,000	62,000	58,000
Elongation in 2 in (%)	25	23	21	23

Tolerance

Round Tubing

Up to 1.900 OD	+/-0.5%
Greater than 2.00+ OD	+/- 0.75%

Square and Rectangular Tubing

Permissible variations in outside flat dimensions for square and rectangular tubing:

Outside Large Flat (in)	Permissible Variation
< 2.5	0.020
2.5 - 3.5	0.025
3.5 - 5.5	0.030
> 5.5	0.01 times the flat dimension

Chemical

All grades:

Element	%
C	0.23 - 0.30
Mn	0.35 - 1.40
P	0.035 - 0.045
S	0.035 - 0.045
Cu	0.18 - 0.20

K-Factor and Bend Allowance

2021-05-17T00:00:00-05:00

Definitions

Bend allowance, $L_b$, is the length of the neutral axis and is used to determine the blank length of a bent part.

The position of the neutral axis depends on the radius and thickness of the bend

\begin{equation*} L_b = \alpha(R + kT) \end{equation*}

where:

$\alpha$ is bend angle in radians

$R$ is the bend radius

$T$ is the thickness of the sheet

$k$ the K-Factor, a constant

$k$ is a factor proportioning the neutral axis in the thickness direction, with zero at the inner surface. Therefore, a smaller $k$ value places the neutral surface closer to the inside of the bend radius.

Thus, when the neutral axis is at the center of the sheet's thickness (midplane), $k = 0.5$. For sharp bends with a small $R/T$ ratio, the neutral axis shifts towards the center of the bend and the factor $k$ decreases. This causes the blank length to be smaller than the case of when it is at the midplane.

The engineering strain in a sheet during bending is:

\begin{equation*} e=\frac{1}{(2R/T) + 1} \end{equation*}

Rules of Thumb

$k$ vales range from 0.33 for $R<2T$ to 0.5 for $R > 2T$.

Minimum Bend Radius for Various Materials at Room Temperature

Values in this table are based on the minimum bend radius, which is that radius which does not produce a crack in the outer fibers when bent. This is dependent on material properties and in particular the reduction of area.

\begin{equation*} R = T\left(\frac{50}{r} - 1\right) \end{equation*}

where $r$ = reduction of area.

Material	Soft	Hard
Aluminum alloys	0	6T
Beryllium copper	0	4T
Brass, low-leaded	0	2T
Magnesium	5T	13T
Austenitic stainless steel	0.5T	6T
Low-carbon, low-alloy, HSLA	0.5T	4T
Titanium	0.7T	3T
Titanium alloys	2.6T	4T

References

Manufacturing Engineering and Technology, Kalpakjian, 3rd Ed.

ASTM A516

2021-05-17T02:26:35+00:00

ASTM A516/A516M-05 Standard Specification for Pressure Vessel Plates, Carbon Steel, for Moderate- and Lower-Temperature Service

Four grades are contained in this specification:

Grade	Tensile Strength, ksi
55	55-75
60	60-80
65	65-85
70	70-90

Mechanical Properties

Property	Grade 55	Grade 60	Grade 65	Grade 70
Tensile strength, ksi	55-75	60-80	65-85	70-90
Yield strength, ksi	30	32	35	38
Elongation in 8 in, %	23	21	19	17
Elongation in 2 in, %	27	25	23	21

Chemical Properties

Values are aggregated across all sizes.

Element	Grade 55	Grade 60	Grade 65	Grade 70
Carbon	0.18-0.26	0.21-0.27	0.24-0.29	0.27-0.31
Manganese	0.60-1.30	0.60-1.30	0.79-1.30	0.79-1.30
Phosphorus	0.035	0.035	0.035	0.035
Sulfur	0.035	0.035	0.035	0.035
Silicon	0.13-0.45	0.13-0.45	0.13-0.45	0.13-0.45

Bolting Materials

2021-05-17T02:19:45+00:00

SAE

Grade	Material	Size Range, in	Min UTS, ksi	Min Yield, ksi	Min Proof, ksi
1	Low or medium-carbon	1/4 - 1 1/2	60	36	33
2	Low or medium-carbon	1/4 - 3/4	74	57	55
		7/8 - 1 1/2	60	36	33
4	Medium-carbon, cold drawn	1/4 - 1 1/2	115	100	65
5	Medium-carbon, Q&T	1/4 - 1	120	92	85
		1 1/8 - 1 1/2	150	81	74
5.2	Low-carbon martensite, Q&T	1/4 - 1	120	92	85
7	Medium-carbon allow, Q&T	1/4 - 1 1/2	133	115	105
8	Medium-carbon allow, Q&T	1/4 - 1 1/2	150	130	120
8.2	Low-carbon martensite, Q&T	1/4 - 1	150	130	120

ASTM

Class	Material	Size Range, in	Min UTS, ksi	Min Yield, ksi	Min Proof, ksi
A307	Low-carbon	1/4 - 1 1/2	60	36	33
A325, type 1*	Medium-carbon, Q&T	1/2 - 1	120	92	85
		1 1/8 - 1 1/2	105	81	74
A325, type 2*	Medium-carbon, Q&T	1/2 - 1	120	92	85
		1 1/8 - 1 1/2	105	81	74
A325, type 3*	Weathering steel, Q&T	1/2 - 1	120	92	85
		1 1/8 - 1 1/2	105	81	74
A354, Gr BC	Alloy steel, Q&T	1/2 - 2 1/2	125	109	105
A354, Gr BD	Alloy steel, Q&T	1/4 - 4	150	130	120
A449	Medium-carbon, Q&T	1/4 - 1	120	92	85
		1 1/8 - 1 1/2	105	81	74
		1 3/4 - 3	90	58	55
A490 Typ. 1*	Alloy steel, Q&T		150	130	120

* Standards A325 and A490 were withdrawn by ASTM in 2016 and replaced by ASTM F3125.

Constant Acceleration Equations

2021-05-17T02:15:52+00:00

acceleration, $a$
velocity, $v$
position, $x$
time, $t$

\begin{equation*} v = v_0 + at \end{equation*}

\begin{equation*} x = x_0 + \frac{1}{2}(v + v_0)t \end{equation*}

\begin{equation*} x ~= x_0 + v_0t + \frac{1}{2}at^2 \end{equation*}

\begin{equation*} 2a(x - x_0) = v^2 - v^2 \end{equation*}

Critical Tip Velocity

2021-05-16T00:00:00-05:00

The following describes an object moving at constant velocity and impacting a rigid obstacle, leading to pivoting and tip-over. The objective is to find the translational velocity that causes the object to rise to the "just-tipping" condition (maximal rise in height).

The equations are based on conservation of momentum and conservation of energy. The linear momentum is converted to an angular momentum by considering its moment arm about the pivot point. After impact, the angular momentum is converted from its initial value to zero at the "just tipping point". Energy is also conserved as the rotational kinetic energy is converted to potential energy.

Variables

$v_0$ - initial translational velocity
$H_O$ - angular momentum about pivot point O
$m$ - mass
$y$ - vertical distance from pivot point O to center of gravity
$I_O$ - moment of inertia about pivot point O
$KE$ - kinetic energy
$PE$ - potential energy
$g$ - gravitational acceleration constant
$h$ - CG height change
$\theta_0$ - resting angle between the CG vector relative to the pivot point and the horizontal plane
$\omega$ - rotational velocity

Equations

Translational momentum moment,

\begin{equation*} H = mv_0y \end{equation*}

Angular momentum,

\begin{equation*} H_O = I_O\omega \end{equation*}

Rotational kinetic energy,

\begin{equation*} KE = \frac{1}{2}I_O\omega^2 \end{equation*}

Potential energy,

\begin{equation*} PE = mgh \end{equation*}

Critical Tipping Velocity

The initial angular velocity, $\omega_0$, can be solved by equating the two momentum quantities.

\begin{equation*} H = H_O \end{equation*}

\begin{equation*} \omega_0 = \frac{mv_0y}{I_O} \end{equation*}

After impact, for the object to rise to the just-tipping point (maximum CG height change), the rotational kinetic energy is completely converted to potential energy:

\begin{equation*} \frac{1}{2}I_O\omega^2 = mgh \end{equation*}

\begin{equation*} \frac{1}{2}I_O \left( \frac{mv_0y}{I_O} \right)^2 = mgh \end{equation*}

Rearranging for the initial velocity, $v_0$:

\begin{equation*} v_0 = \sqrt{\frac{2ghI_O}{my^2}} \end{equation*}

Furthermore, the CG height change $h$ is known and can be expressed in terms of the rest angle, $\theta_0$:

\begin{equation*} h = r - r\sin\theta_0 = r(1-\sin\theta_0) \end{equation*}

$v_0$ then becomes:

\begin{equation*} v_0 = \sqrt{\frac{2gr(1-\sin\theta_0)I_O}{my^2}} \end{equation*}

matplotlib Quick Ref

2021-04-03T00:00:00-05:00

Preliminaries

To import matplotlib, the recommended way is:

import matplotlib.pyplot as plt

Basic Plotting with plt

The imported plt can be used for interactive and basic usage, similar to MATLAB. A typical workflow might be:

Assemble your data

Create a plot using one of the plot types

Add options, legends, axes configurations, annotations using plt.* methods

Call plt.show() to show the plot (may or may not be needed depending on the environment/software you are using)

See here for the full list of methods and API.

Some Common Methods

Plot types

plt.plot
plt.bar
plt.pie

Annotations

plt.axhline, plt.axvline
plt.arrow
plt.annotate, plt.text
plt.title

Axis

plt.xlabel, plt.ylabel
plt.xticks, plt.yticks
plt.xlim, plt.ylim

Object Approach

This approach is initiated by calling the subplots() function with returns separated Figure and Axes objects, allowing direct access to each respective object:

fig, ax = plt.subplots()

This function also allows separate figures and multi-axis figures to be created and assigned to separate variables:

fig1, ax1 = plt.subplots()
fig2, ax2 = plt.subplots()

Additionally, multiple axes (i.e. plots) in one figure can be created with ncols and nrows:

fig, (ax1, ax2) = plt.subplots(nrows=1, ncols=2)

Use the sharey and sharex arguments to share x or y axes between subplots:

fig, (ax1, ax2) = plt.subplots(nrows=1, ncols=2, sharey=True)

Plot calls may then be used on the Axes object:

ax1.plot(x,y)

To save the plot to a file:

fig1.savefig('my_figure.png')

To show the plot, call plt.show() or fig1.show() - which one is preferable depends on the environment and context you are using as explained here2. Also, depending on your environment these may not not be needed at all (such as in iPython/Jupyter).

Configuring

Color, linestyle, linewidth, and other properties of the plot can be specified in the plot call.

See here3 for a full list of options.

Options

Property	Keyword argument
Line color	`color` or `c`
Line width	`linewidth` or `lw`
Line style	`linestyle` or `ls`
Marker style	`marker`

Colors

Colors may be entered in a variety of ways - string name, RGB tuple, HEX string, etc. A range of color names are included in matplotlib by default. Basic colors are available in addition to those named in the X window system and the xkcd color survey.

Basic colors include: blue/ b, green/ g, red/r, cyan/ c, magenta/ m, yellow/ y, black/ k, white/ w.

Legend

Add a legend by calling ax.legend(). For the legend to have names, labels are typically first specified in the plot call with the legend keyword argument:

ax1.plot(t, v, legend='Velocity')

Alternatively, if you store a reference or get the line object from an existing axes, you can call:

line.set_label(str)

Then, when calling ax.legend() the legend generator knows names for each line. These methods are preferred since the names and data are linked together.

Placement of the legend can be set to 9 points forming a grid pattern over the axes, by setting the loc argument to a combination of upper, lower, left, right, center (for example upper left). center only places it in the middle of both directions. best places the legend in the area with the minimum amount of overlap with other elements.

See the API for the many other available options.

Other Things

Grids

Turning on grids can be achieved with:

ax.grid(True, which='both')

The which argument specifies which axis to turn them on for (x, y, or both).

Styles

There are some predefined styles to use in your plots. The style names for these include:

Solarize_Light2
_classic_test_patch
bmh
classic
dark_background
fast
fivethirtyeight
ggplot
grayscale
seaborn
seaborn-bright
seaborn-colorblind
seaborn-dark
seaborn-dark-palette
seaborn-darkgrid
seaborn-deep
seaborn-muted
seaborn-notebook
seaborn-paper
seaborn-pastel
seaborn-poster
seaborn-talk
seaborn-ticks
seaborn-white
seaborn-whitegrid
tableau-colorblind10

A style can be turned in your session via:

plt.style.use(<name>)

To activate a style for just a specific plot, you can use the with mechanism:

with plt.style.context(<name>):
    plt.plot(x,y)

Or,

with plt.style.context(<name>):
    fig,ax = plt.subplots()
    ax.plot(x,y)

Square/1:1 plots

Use Axes.set_aspect('equal') to make the axes scales have a 1:1 ratio.

Remove "Figure" from Cross-References in Word

2020-11-06T00:00:00-06:00

When adding Figure cross-references in Word there are 5 options for the "Insert reference to" field, which determines what the cross-reference displays:

Option	Rendered as
Entire Caption	Figure 1: Lorem ipsum dolor sit amet ...
Only label and number	Figure 1
Only caption text	Lorem ipsum dolor sit amet ...
Page number	(page number the figure is on)
Above/below	above if the figure is above the reference position or below if the figure is below it

Each of these contains some information about the reference. If just the figure number is desired, well, none of these will do the job. This, however can be accomplished by adding additional formatting switches.

The result of issuing Insert > Cross-reference is a REF field code. To edit the field code, press Alt + F9. This will toggle the display of the document to show the contents of all field codes and allow editing. (To return to normal view, press Alt + F9 again.)

A cross-reference for a figure caption might look like:

{ REF _Ref55500687 \h }

To have it just show the number and not the "Figure " portion, add the following to the contents inside the curly braces:

\# "0"

Yielding:

{ REF _Ref55500687 \h \# "0" }

In some instances Word might tell you that picture switch must be the first formatting switch. In this case just move the \# "0" to the left before other switches.

Example:

References

Microsoft community forums answer by Stefan Blom

reStructuredText Elements

2020-10-26T00:00:00-05:00

reStructuredText is a great markup language used in the Python world but can also be used for general text activities. It has more specificity in its design and has more functionality when compared to others in its class such as Markdown.

It is the format for content for this website since I use the Pelican static site generator which natively uses RST format for writing content.

The below is a side-by-side comparison of most of the reStructuredText elements and their equivalent when processed; in this case pandoc was used to convert from RST to html. This list also does not include all forms of each element - for the complete specification see the link above.

inline markup
paragraph
bullet list
enumerated list
definition list
field list
literal block
line block
block quote
table (simple)
table (grid)
footnotes
citation
hyperlink (internal)
hyperlink (external)
hyperlink (embedded)
directive substitution

reStructuredText Result

inline markup

*emphasis*

**strong emphasis**

`interpreted text`

``inline literal text``

emphasis

strong emphasis

interpreted text

inline literal text

paragraph

Lorem ipsum dolor sit amet, consectetur
adipiscing elit. Pellentesque ornare
commodo est eget tempor. Pellentesque
vitae nulla velit.

Praesent non nisl id eros gravida
tempor eget in sapien. Curabitur
vestibulum dapibus placerat.

Lorem ipsum dolor sit amet, consectetur adipiscing elit. Pellentesque ornare commodo est eget tempor. Pellentesque vitae nulla velit.

Praesent non nisl id eros gravida tempor eget in sapien. Curabitur vestibulum dapibus placerat.

bullet list

- Lorem ipsum dolor sit amet
- Pellentesque ornare
- Pellentesque vitae nulla
  gravida tempor eget in sapien
- Nunc euismod tellus turpis,
  efficitur non.

Lorem ipsum dolor sit amet
Pellentesque ornare
Pellentesque vitae nulla gravida tempor eget in sapien
Nunc euismod tellus turpis, efficitur non.

enumerated list

A. Lorem ipsum dolor sit amet,
   consectetur adipiscing elit.
B. Pellentesque ornare commodo
C. Pellentesque vitae nulla
   gravida tempor eget in sapien

a. Lorem ipsum dolor sit amet,
   consectetur adipiscing elit.
b. Pellentesque ornare commodo

  i. Lorem ipsum dolor sit amet,
      consectetur adipiscing elit.
  ii. Pellentesque ornare commodo
      est eget tempor.

Lorem ipsum dolor sit amet, consectetur adipiscing elit.
Pellentesque ornare commodo
Pellentesque vitae nulla gravida tempor eget in sapien

Lorem ipsum dolor sit amet, consectetur adipiscing elit.
Pellentesque ornare commodo
1. Lorem ipsum dolor sit amet, consectetur adipiscing elit.
2. Pellentesque ornare commodo est eget tempor.

definition list

Lorem ipsum
    dolor sit amet

    consectetur adipiscing elit.

Pellentesque
    ornare commodo est eget tempor.

Lorem ipsum

dolor sit amet

consectetur adipiscing elit.

Pellentesque

ornare commodo est eget tempor.

field list

:Lorem ipsum:
    dolor sit amet

    consectetur adipiscing elit.

:Pellentesque:
    ornare commodo est eget tempor.

:Vestibulum:
    - vitae arcu ex
    - Fusce aliquet faucibus rutrum
    - Integer at tortor tellus

Lorem ipsum

dolor sit amet

consectetur adipiscing elit.

Pellentesque

ornare commodo est eget tempor.

Vestibulum

vitae arcu ex
Fusce aliquet faucibus rutrum
Integer at tortor tellus

literal block

Duis imperdiet id orci quis consequat.
Nam in sem consequat, laoreet enim ut,
aliquet nibh. Sed vitae sem ipsum.
Aliquam nec gravida dolor. ::

    Morbi mollis vitae mauris ac interdum.
    Pellentesque id posuere magna, fermentum
    Nam libero lacus, elementum eu faucibus
    aliquet, venenatis quis augue.

    Quisque rutrum arcu elit, sed tempor
    sed. Donec bibendum sit amet libero

Duis imperdiet id orci quis consequat. Nam in sem consequat, laoreet enim ut, aliquet nibh. Sed vitae sem ipsum. Aliquam nec gravida dolor. :

Morbi mollis vitae mauris ac interdum.
Pellentesque id posuere magna, fermentum
Nam libero lacus, elementum eu faucibus
aliquet, venenatis quis augue.

Quisque rutrum arcu elit, sed tempor
sed. Donec bibendum sit amet libero

line block

| Duis imperdiet id orci quis consequat.
| Nam in sem consequat, laoreet enim ut
|
| Sed vitae sem ipsum. Aliquam nec
| mauris ac interdum. Pellentesque
| fermentum dapibus purus.
| eu faucibus aliquet

Duis imperdiet id orci quis consequat.
Nam in sem consequat, laoreet enim ut

Sed vitae sem ipsum. Aliquam nec
mauris ac interdum. Pellentesque
fermentum dapibus purus.
eu faucibus aliquet

block quote

Duis imperdiet id orci quis consequat.
Nam in sem consequat, laoreet enim ut,

    Morbi mollis vitae mauris ac interdum.
    Pellentesque id posuere magna, fermentum dapibus purus.
    Nam libero lacus, elementum eu faucibus
    aliquet, venenatis quis augue.

    -- Morbi mollis

Duis imperdiet id orci quis consequat. Nam in sem consequat, laoreet enim ut,

Morbi mollis vitae mauris ac interdum. Pellentesque id posuere magna, fermentum dapibus purus. Nam libero lacus, elementum eu faucibus aliquet, venenatis quis augue.

-- Morbi mollis

table (simple)

================== =======================
Morbi mollis       vitae mauris
================== =======================
Pellentesque
id posuere magna   fermentum dapibus purus
Nam libero         aliquet, venenatis quis
================== =======================

Morbi mollis	vitae mauris
Pellentesque
id posuere magna	fermentum dapibus purus
Nam libero	aliquet, venenatis quis

table (grid)

+------------------+---------------------------+
| Morbi mollis     | vitae mauris              |
+==================+===========================+
| Pellentesque     |                           |
+------------------+---------------------------+
| id posuere magna | fermentum dapibus purus   |
+------------------+---------------------------+
| Nam libero       | aliquet, venenatis quis   |
+------------------+---------------------------+

Morbi mollis	vitae mauris
Pellentesque
id posuere magna,	fermentum dapibus purus
Nam libero	aliquet, venenatis quis

footnotes

Lorem ipsum dolor sit amet, consectetur
adipiscing elit. Pellentesque ornare
commodo est eget tempor. Pellentesque
vitae nulla velit [1]_.

Praesent non nisl id eros gravida
tempor eget in sapien. Curabitur
vestibulum dapibus placerat [2]_.


.. [1] Lorem ipsum dolor sit amet
.. [2] Duis imperdiet id orci quis consequat.

Lorem ipsum dolor sit amet, consectetur adipiscing elit. Pellentesque ornare commodo est eget tempor. Pellentesque vitae nulla velit¹.

Praesent non nisl id eros gravida tempor eget in sapien. Curabitur vestibulum dapibus placerat².

Lorem ipsum dolor sit amet↩︎
Duis imperdiet id orci quis consequat.↩︎

citations

Lorem ipsum dolor sit amet, consectetur
adipiscing elit. Pellentesque ornare
commodo est eget tempor. Pellentesque
vitae nulla velit [Lorem1]_.

.. [Lorem1] Lorem ipsum dolor sit amet

Lorem ipsum dolor sit amet, consectetur adipiscing elit. Pellentesque ornare commodo est eget tempor. Pellentesque vitae nulla velit [Lorem1].

Lorem1: Lorem ipsum dolor sit amet

hyperlink (internal)

Lorem ipsum dolor sit amet, consectetur
adipiscing elit. Pellentesque ornare
commodo est eget tempor. Pellentesque
vitae nulla velit.

.. _internal1:

The previous line specifies a hyperlink
target which links to this paragraph by
giving it an id.

Later on to link to this paragraph,
use internal1_.

Lorem ipsum dolor sit amet, consectetur adipiscing elit. Pellentesque ornare commodo est eget tempor. Pellentesque vitae nulla velit.

The previous line specifies a hyperlink target which links to this paragraph by giving it an id.

Later on to link to this paragraph, use internal1.

hyperlink (external)

Lorem ipsum dolor sit amet, consectetur
adipiscing elit. Pellentesque ornare
commodo est eget tempor. python_
pellentesque vitae nulla velit.

Praesent non nisl id eros gravida
tempor eget in sapien. Curabitur
vestibulum dapibus placerat
`asme home`_.

.. _python: https://python.org
.. _asme home: https://asme.org

Lorem ipsum dolor sit amet, consectetur adipiscing elit. Pellentesque ornare commodo est eget tempor. python pellentesque vitae nulla velit.

Praesent non nisl id eros gravida tempor eget in sapien. Curabitur vestibulum dapibus placerat asme home.

hyperlink (embedded)

Click `here <https://asme.org>`_ for the ASME
home page.

Click here for the ASME home page.

directive substitution

Lorem ipsum dolor sit amet, consectetur
adipiscing elit. |pellentesque| ornare
commodo est eget tempor. Pellentesque
vitae nulla velit.

Praesent non nisl id eros gravida
tempor eget in sapien. Curabitur
vestibulum dapibus placerat.

.. |pellentesque| math:: a^2 + xb^3

Lorem ipsum dolor sit amet, consectetur adipiscing elit.

a² + xb³

ornare commodo est eget tempor. Pellentesque vitae nulla velit.

Praesent non nisl id eros gravida tempor eget in sapien. Curabitur vestibulum dapibus placerat.

SolidWorks Save-to-PDF Macro

2020-03-08T00:00:00-06:00

Saving drawings as PDF is a common task in SolidWorks. Here we will create a macro and corresponding toolbar button to save an active SolidWorks drawing as a PDF for better productivity. This script saves a PDF file to the current directory and overwrites any pdf of the same name.

Record

Using SolidWorks' macro recorder (Tools > Macro > Record), opening a SolidWorks drawing and choosing File > Save As > PDF, we get the following macro:

' ******************************************************************************
' C:\\Users\\user\\AppData\\Local\\Temp\\swx1900\\Macro1.swb - macro recorded on 03/07/20 by user
' ******************************************************************************

Dim swApp As Object

Dim Part As Object
Dim boolstatus As Boolean
Dim longstatus As Long, longwarnings As Long

Sub main()

Set swApp = _
Application.SldWorks

Set Part = swApp.ActiveDoc
longstatus = Part.SaveAs3(\"<output_pdf_file>\", 0, 0)
End Sub

Edit

In this case the output filename is hardcoded into the script (<output_pdf_file>). We want to change that to be the same name as the active document, just with a .pdf extension.

To do that, we add the following:

Dim strFilename As String

Part.ClearSelection2 True
strFilename = Part.GetPathName
strFilename = Left(strFilename, Len(strFilename) - 6) & \"pdf\"

We obtain the filename from Part.GetPathName and modify the extension with the next line which removes the last 6 characters (slddrw) and replaces it with pdf.

Then, we replace the filename in Part.SaveAs3:

longstatus = Part.SaveAs3(**strFilename**,0,0)

The final script then looks like the following:

Dim swApp As Object

Dim Part As Object
Dim boolstatus As Boolean
Dim longstatus As Long, longwarnings As Long

Dim strFilename As String

Sub main()

Set swApp = _
Application.SldWorks

Set Part = swApp.ActiveDoc
Part.ClearSelection2 True

strFilename = Part.GetPathName
strFilename = Left(strFilename, Len(strFilename) - 6) & \"pdf\"

longstatus = Part.SaveAs3(strFilename, 0, 0)
End Sub

The macro can be run with Tools > Macro > Run, which will open a dialog to choose a macro file. But this is not ideal for something that will be run frequently.

Install

To install the macro in a more usable manner, we can place a shortcut to the macro as a custom button in the SolidWorks interface. To do this we first have to create a new macro by choosing:

Tools > Macro > New

and specifying a filename and save location. Then, the macro editor is shown where we can enter code. Copy and paste the script above into the editor and save.

Then, to add it to the toolbar, choose Tools > Customize, and under the Commands tab, select the Macro category, and then drag the New Macro Button icon to the desired toolbar destination. For this example, we will place it in the top bar for easy access.

After dragging and dropping onto a toolbar, a dialog opens for us to specify what macro to run. For the Macro entry, browse to the macro file just created and click Open. The Method field should auto-populate, and references the subroutine defined as main. Tooltip text can be specified which will display when the cursor hovers over the button. Additionally, a custom icon can be added which is a 16px x 16px bitmap image. I created one here of the Acrobat logo. Click OK and the macro button should be created.

Experimenting with ANSYS ACT in Mechanical

2019-09-07T00:00:00-05:00

ANSYS ACT is the scripting API for various ANSYS products. It is available for Workbench, Mechanical, Fluent, and others. It uses Python 2 and in Mechanical it allows much of the standard click operations to be accessed programmatically.

Usage

In Mechanical, the ACT Console is a pane that allows for running and testing scripts. It is accessed by clicking the white rectangular icon next to the Help menu by default (in pre-tabbed interfaces - in ANSYS 2019R2, the program switched to a tabbed interface).

With the ACT Console open, you have more or less a standard Python interpreter. You issue commands in the command line (darkened area at the bottom), and see results in the area above it. The console features auto-completion and can be used interactively, especially while developing new scripts. New lines, such as with for loops, can be entered with SHIFT + ENTER.

There is a pane on the left which holds Snippets, which are simply files storing code for reuse. You can create your own snippets for actions that you want to remember or perform more than once. You do this by clicking on the Add Snippet button (left most icon in the snippet portion of the toolbar). Then, when you click on the snippet from the Bookmarks pane, the code is simply inserted into the command line and can be executed by pressing ENTER.

A collection of snippets can be grouped together and imported/exported. This might be useful if you or your company has a set of scripts that you use. To use this functionality, create a new group and give it a name. Drag your snippets into it. Then, in the ACT toolbar, export the snippets to an XML file. This exports all of the snippets, including the default ones that came loaded when you opened the console. These can easily be removed by editing the XML file after it is saved. In the XML file, find the group that has the name of your group, and delete everything else, not including the parent xml tags. For example, if I add the following snippet Snippet 1 and group My Snippets:

After exporting I find an xml file with the following:

<!--?xml version=\"1.0\" encoding=\"utf-8\"?-->
<model maxid=\"5\">
<root>
    <folder id=\"1\" label=\"Folder\">
    <bookmark id=\"2\" label=\"ExtAPI\">
    \t<!--[CDATA[ExtAPI.]]-->
    </bookmark>
    </folder>
    <folder id=\"4\" label=\"My Snippets\">
    <bookmark id=\"5\" label=\"Snippet 1\">
    \t<!--[CDATA[print Hello World!]]-->
    </bookmark>
    </folder>
</root>
</model>

In this case I will want to simply delete the <folder> item with the label=\"Folder\" attribute, since that is code not pertaining to my custom folder. ANSYS stores the actual code for the snippet in the bookmark field; in this case the code is print Hello World!.

API

An API is how operations and objects in Mechanical are retrieved and manipulated with code. The default global variable that gives access to all objects in Mechanical is ExtAPI. From this you can access all other objects using Python dot syntax. Listed below are some of the objects in Mechanical and their classes and access points. It is common to assign these to a variable so that they do not need to be fetched each time. For example:

solution = ExtAPI.DataModel.Project.Model.Analyses[0].Solution

Object	Access	Note
The Project	`ExtAPI.DataModel.Project`
List of Analyses	`ExtAPI.DataModel.Project.Model.Analyses`	This is list-like and indexable/loopable
Analysis working directory	`ExtAPI.DataModel.Project.Model.Analyses[0].WorkingDir`	Gets the working directory for the first analysis system (zeroth index)
Connections	`ExtAPI.DataModel.Project.Model.Connections`
Solution	`ExtAPI.DataModel.Project.Model.Analyses[0].Solution`	This is the solution object for the first analysis system (zeroth index)

Object	Class
Force Reaction Probe	`Ansys.ACT.Automation.Mechanical.Results.ProbeResults.ForceReaction`
Moment Reaction Probe	`Ansys.ACT.Automation.Mechanical.Results.ProbeResults.MomentReaction`
Beam Probe	`Ansys.ACT.Automation.Mechanical.Results.ProbeResults.BeamProbe`
Tree Group Folder	`Ansys.ACT.Automation.Mechanical.TreeGroupingFolder`

Change the Python directory to the `user_files` directory for a Project

Scripts that save out to a file might want to take advantage of the user_files directory. The user_files directory exists at the first level of the project files directory and is for storing user information pertaining to a Workbench project. I haven't found the path for this as a parameter in the API directly so the below is a way to obtain it from an analysis working directory. It splits the path at the Workbench project folder, so _files must not come before the Workbench folder in the path or it will return the wrong location.

from os import chdir
MECH_dir = ExtAPI.DataModel.Project.Model.Analyses[0].WorkingDir
user_dir = MECH_dir.split('_files')[0] + r'_files\user_files'
chdir(user_dir)

Usage Example: Results Exporting

One of the major limitations of Workbench / Mechanical is getting result probes out of Mechanical and into an external file (Excel, text file, etc.). If you only have a few result probes this is no problem - copy and paste will work fine. However, if you have a hundred, there is not an easy way to get all of the data and more importantly formatted as you like. Additionally, for a contact, you will likely want a Force Probe and a Moment Probe; these are two separate items in the Solution and you must export both and associate correctly with the contact they are referring to. With ACT, you can programmatically loop through these items and save them to a text file.

Beams

In this case you have some beams in your model defined under the Connections folder and you'd like to export the beam probes. If you've already created a result probe for them, you can retrieve and export them as follows:

solution = ExtAPI.DataModel.Project.Model.Analyses[0].Solution
beam_probes = filter(lambda item: item.GetType() == Ansys.ACT.Automation.Mechanical.Results.ProbeResults.BeamProbe, solution.Children)

f = open('Beam results.txt','w')
f.write('Beam Probes\n\n')
f.write('Name\tAxial Force\tTorque\tShear Force at I\tShear Force at J\tMoment at I\tMoment at J\n\n')

for bp in beam_probes:
    name = bp.BoundaryConditionSelection.Name
    f.write(name + '\t' +
        str(bp.AxialForce).split('[')[0] + '\t' +
        str(bp.Torque).split('[')[0] + '\t' +
        str(bp.ShearForceAtI).split('[')[0] + '\t' +
        str(bp.ShearForceAtJ).split('[')[0] + '\t' +
        str(bp.MomentAtI).split('[')[0] + '\t' +
        str(bp.MomentAtJ).split('[')[0] + '\n' )
f.close()

With the following tree, and after setting the working directory to the user_files directory, running the above code produces Beam results.txt:

Name    Axial Force Torque  Shear Force at I    Shear Force at J    Moment at I Moment at J

Bolt 1  23.464579153988097  -0.69336988094428165    95.214531366676013  95.214531366676013  12.945849648564687  11.227034167745025
Bolt 2  -23.464579153988097     -0.70081092496467057    104.8404773221031   104.8404773221031   14.025457881021863  12.521195678519947

This file is tab-delimited and the bolt names come from the name of the beam items in the Connections folder. Additionally, the unit is stripped from the result.

See full snippets on Github: https://github.com/robsiegwart/ANSYS-ACT-snippets.

Merging and Watermarking PDF Files with Python

2019-08-25T00:00:00-05:00

There are many Python libraries available for working with PDF files. Here we will perform merging and watermarking operations with the Python library pdfrw. This package is bundled in the WinPython distribution.

It is assumed that your path has been set to find the correct Python interpreter such that pdfrw can be imported.

Merging

A merging script can be created in only a few lines of code. Let's create a script that will merge all of the PDF files within a directory, using a command line argument to specify the output file name. Locate the script in the same directory as the files, and it will grab all of the files with a .pdf extension and then combine those (in the same order as if issuing an 'ls' command) into a single output PDF file.

# merge.py

import sys
from glob import glob
from pdfrw import PdfReader, PdfWriter

def main():
    output_file = sys.argv[1]
    files = glob('*.pdf')
    writer = PdfWriter()
    for each in files:
        pdf = PdfReader(each)
        writer.addpages(pdf.pages)
    writer.write(output_file)

if __name__ == '__main__':
    main()

This can then be run with:

python merge.py <output file>

in the terminal or command prompt.

Watermarking

With watermarking, a separate pdf file (the watermark) is overlaid on top of another pdf file (the base file). For the watermark to show up at the correct location the watermark file must be the same size as the base pdf file. With this script the first argument will be the input file to apply the watermark to, and a second argument which is the name of the pdf file to use as the watermark. Again the script is to reside in the same directory as the files.

# watermark.py

import sys
from pdfrw import PdfReader, PdfWriter, PageMerge

def main():
    input_file = sys.argv[1]
    watermark_file = sys.argv[2]

    input_pdf = PdfReader(input_file)
    watermark = PdfReader(watermark_file).pages[0]

    for page in input_pdf.pages:
        merger = PageMerge(page)
        merger.add(watermark).render()

    writer = PdfWriter()
    writer.write(input_file.split('.pdf')[0]+'_w.pdf',input_pdf)

if __name__ == '__main__':
    main()

To run, in a terminal or command prompt issue:

python watermark.py <input file> <watermark file>

Zero Gravity Hinges - An Analysis of Microsoft's Surface Studio Hinge

2019-07-22T00:00:00-05:00

Some time ago (2016), Microsoft released the Surface Studio accompanied by a great promo video. This product is an all-in-one computer featuring an adjustable display that maintains its position at any location set by the user. The hinge used with it - termed a "zero gravity" hinge - allows for this free positioning of the display and here we will attempt to analyze this type of design and determine how it works.

Design Notes

Without actually deconstructing one of these I am using images from the promo video and other publicly available information as reference.

Some observations:

The support arms are not independent - it is a 1 degree-of-freedom (DOF) system
Compression springs are used in the base and torsion springs are used in the display
A four-bar linkage is used in the support arms to connect the base to the display
This can be used to give the desired orientation response of the display by changing the lengths of the connecting members

This YouTube video confirms these observations.

Analysis

Overview

First we construct the overall layout and label relevant parts:

where,

\begin{equation*} m_s = \text{mass of support} \end{equation*}

\begin{equation*} m_d = \text{mass of display} \end{equation*}

At first glance,

At $B$, it appears that the moment needing resisted is only due to the display
At $A$, the moment due to the support arms and the display need resisted

However, since the linkage has only 1 DOF, the display would be supported (i.e. prevented from rotating) by applying a single moment anywhere along this chain such as at location $B$ or $A$. Thus, the moment due to the torsion spring in the display contributes to overall support of deadweight loads.

Mechanism

Based on screenshots from the above linked videos, there is a linkage (brown) internal to the two supporting arms that interfaces with compression springs in the base (purple). This linkage relates the orientation of the display with respect to the base.

Base Compression Springs

At Point A there are a total of four compression springs that act against a rocker arm (cyan). This provides a supporting moment at this joint. This moment is due to the offset height $h$ and the spring force $f_b=k_b\Delta l_b$. Here the moment varies non-linearly as the support arm is rotated since $h$ changes in addition to the spring force changing.

Using the following values and properties (estimates but actually tuned to give the best result - to see if this design could work), and the accompanying Python code, this results in a moment response curve as shown below.

\begin{equation*} r = 0.3 \text{ in} \end{equation*}

\begin{equation*} k_{b} = 4\times40 \text{ lbf/in} \end{equation*}

\begin{equation*} l_{free} = 2.5 \text{ in} \end{equation*}

# Nomenclature:
#   r         = base hinge moment arm length, inches
#   r_fb      = position vector of the base of the compression spring from the
#               pivot point (Point A)
#   k_b       = spring constant of the base compression spring, lbf/in
#   l_free    = free length of the base compression spring
#   start     = start of base hinge rotation, degrees
#   end       = end of base hinge rotation, degrees
#   r_a       = Pivot arm position vector
#   r_afb     = Spring direction vector
#   u_fb      = Spring direction unit vector
#   f_b       = Spring force vector
#   M         = Moment = r x F

r = 0.3
r_fb = np.array([1.53, 0.22, 0])
k_b = 4*40
l_free = 2.5
start = 0
end = 90

spring_M, thetas = [], []

for theta in range(start,end):
    theta_rad = np.radians(theta)
    r_a = np.array([r*np.cos(theta_rad), r*np.sin(theta_rad), 0])
    r_afb = r_a - r_fb
    u_fb = r_afb/np.linalg.norm(r_afb)
    f_b = np.dot(k_b*(l_free-np.linalg.norm(r_afb)),u_fb)
    M = np.cross(r_a,f_b)

    spring_M.append(M[2])
    thetas.append(theta)

fig1, ax1 = plt.subplots()
ax1.plot(thetas, spring_M, c='b', ls='dashed',
    label='Moment due to compression springs')
ax1.grid(True)
ax1.set_xlabel('Rotation, deg')
ax1.set_ylabel('Torque (Moment), in-lbs')

Torsion Spring

The moment due to the torsion spring is simply linear and dependent on the torsion spring stiffness. I am assuming and using some preload on this spring as I did with the compression spring.

The following properties are used and the resulting moments plotted.

k_d = 1.25                  # spring rate, lbf-in/deg
k_d_pre = 75                # Preload amount, lbs-in

tor_sp = []

for i in range(start,end):
    tor_sp.append(-k_d*i + k_d_pre)

fig2,ax2 = plt.subplots()
ax2.plot(thetas, tor_sp, '-.', c='g', label='Moment due to torsion spring')
ax2.plot(thetas, spring_M, c='b', label='Moment due to compression springs',
    ls='dashed')


# Add resisting moments of the compression springs and the torsion spring
combined = np.asarray(tor_sp) + spring_M
ax2.plot(thetas, combined, c='r', label='Combined moment')
ax2.grid(True)
ax2.legend()
ax2.set_xlabel('Rotation, deg')
ax2.set_ylabel('Torque (Moment), in-lbs')

Moment at the Base Hinge From Components Due to Gravity

Now, we calculate the loads due to deadweight. The two components involved are the display and the support arms. Each is assigned a weight:

\begin{equation*} m_d=10\text{ lbs} \end{equation*}

\begin{equation*} m_s=4\text{ lbs, both arms} \end{equation*}

To assess if the display will be supported (or if it will fall or rise), we compare the resisting moments due to the springs to the total deadweight moment. The total deadweight moment is the resultant moment of each of the contributing moments of the display and support arms. Maintaining the display position in any configuration requires that the resisting moments and load moments be exactly equal; though small amounts of difference can be taken up through friction by the system connecting components.

Calculating the deadweight moment requires analysis of the support arm linkage. As an approximation of this we can infer the positions of the support arms and display graphically using the following screenshots:

The center of gravities are taken at the midpoint of both the support arms and display. This allows the combined moment to be estimated:

from scipy.interpolate import interp1d

# Measurement based approach - infer linkage from screenshot measurements
r_d = 3.9              # distance from display pivot point to display CG, inches
r_s = 4.0              # distance from base pivot to support arms CG, inches

m_s = 3.75             # mass of support arms, lbs
m_d = 10               # mass of display, lbs

M_dw = []              # deadweight total moment, in-lbs

# support arm and display angle measurements
disp_rot = interp1d([5.2,14.4,34.4,52.8],[142.7,132.2,115.9,96],
    kind='quadratic')
t2 = []

for theta in np.linspace(min(disp_rot.x),max(disp_rot.x),20):
    theta_rad = np.radians(theta)
    M_dw.append(
        m_s*r_s*np.cos(theta_rad)
        + m_d*(2*r_s*np.cos(theta_rad)
        + r_d*np.cos(np.radians(disp_rot(theta)))) )
    t2.append(theta)

fig3,ax3 = plt.subplots()
ax3.plot(t2, M_dw, c='black', lw=3, label='Deadweight moment')
ax3.set_title('Moment Due to Deadweight')
ax3.set_ylabel('Moment, in-lbs')
ax3.set_xlabel('Base hinge rotation, degrees')
ax3.grid(True)
ax3.legend()
fig3.savefig('fig3.png',dpi=200)

# Make a composite final figure
fig4,ax4 = plt.subplots()
ax4.plot(thetas, combined, c='r', label='Combined supporting moment')
ax4.plot(thetas, tor_sp, '-.', c='g', label='Moment due to torsion spring')
ax4.plot(thetas, spring_M, c='b', ls='dashed',
    label='Moment due to compression springs')
ax4.plot(t2, M_dw, c='black', lw=3, label='Deadweight moment')
ax4.set_ylabel('Moment, in-lbs')
ax4.set_xlabel('Base hinge rotation, degrees')
ax4.grid(True)
ax4.legend()

Moment Summary

Finally, plotting the combined moment with the deadweight moment shows reasonable agreement. More accurate properties - component weights and CG's, in addition to spring properties - would allow for better correlation but the concept confirms that this configuration can result in the desired behavior.

Conclusions

The configuration of springs and linkages show that equal (or near equal) moments are developed to create static equilibrium
- Friction between components could take up small differences in loads or provide for some support for the user
Tuning is accomplished primarily by selection of spring stiffenesses, spring preload amounts (i.e. the load at resting positions), length of the base hinge rocker arm, and potentially also by adjusting the weight and CG of the computer components (though less desirable than adjusting spring parameters)

Layout Sketches and Parameter-Driven Design in Fusion 360

2019-05-06T00:00:00-05:00

Fusion 360 has good support for parameters in CAD documents and the ability to drive designs using them. Here we will create a fully parametric design based on functional design parameters, using the following table as an example. As you can see, nominal dimensions can be specified to create different versions of the design:

Setup

To input parameters open the Parameters dialog which is available under the Modify panel of the Design tab. Here we will enter all of the pertinent values governing our design. Putting as much information here at the outset can make the design more robust and parametric.

Layout Sketches

Layout sketches are then made and dimensions linked to parameters. For the table we will create a rectangle for the overall shape and an offset plane equal to the height. Set the two dimensions for the rectangle equal to the width and height variables. This is accomplished by simply typing their names in the dimension field text. Fusion 360 gives you some suggestions after you start typing to which you can hit Enter to autocomplete.

These two items are sufficient for creating the table top, as seen below, simply by adjusting the Start property of the Extrude feature. Next, we add a sketch for one of the legs. We can add a coincident constraint to the first sketch to keep the leg at the extents of the table, which is applied at each of the lower left corners of the leg and table rectangles. Since we defined a leg dimension for the structural HSS, we link the width and thickness dimensions to the parameters (indicated by the addition of "fx").

Create Components

Legs

Now it is time to start creating components. All of the features created thus far exist at the top level. To add a new component right click on the top of the tree and click "New Component". This adds a "Component1" and activates it. We will create a Leg so we rename it "Leg". By showing only the leg sketch we add a new extrude and extrude the leg profile. Since we will be performing a component pattern later, we add addional features to the component in the feet and casters. This way anything we do here will be copied to the other three legs.

To add a baseplate we create a new sketch on the component origin plane and by only showing the layout leg sketch we created initially we can make sure to constrain to that and not anything else. It is generally a good practice to constrain and create references to upstream sketch geometry rather than to solid features since solid features are more likely to change, and thus result in broken references which must be repaired.

Next we extrude this profile, add some fillets to the HSS extrusion, and add 4 holes in the base plate.

Finally, we add a McMaster-Carr part using the built-in importing feature. We do this while still in the Leg component since it will be included in the rectangular pattern that we are about to do and makes some sense to think of it as part of the leg assembly. However, it could also be added in at the top level since that probably more realisticly reflects its location in the BOM.

After doing that and adding a caster, bolt, and nut, we have a leg assembly ready to be patterned:

We then activate the top level and create a Component Pattern. Select Components as the type and click on the Leg. Click on the global X axis to set the pattern direction and in the distance we can enter in one of our parameters, or an expression using parameters. Our values will be: width - leg_width for the x value and depth - leg_width for the z value, and adding a negative for the z-direction.

Cross Members

Next we add cross members. The procedure is the same as for the legs, making sure to hide created bodies in order to create relations to existing sketch geometry and not solid geometry.

We extrude, add fillets, and pattern components to add additional members.

Side Plates

To create side plates we first add a plane at the front face of the table. This can be set by either keying in an offset equal to depth/2 or by offsetting to a sketch point. Then, we sketch in some geometry for the plate, and make sure to dimension it in a way that allows it to be scaled in the x-direction without error. Additional expressions could be added here, for example to set the length of the horizontal side lines to be some fraction of the overall width.

Top

Lastly, the top plate is added. Here we simply re-use the layout sketches created initially. In the extrude dialog box, set the Start property to From Object and select the construction plane. Then, add a Push/Pull (offset face) feature to extend the sides over the legs, if desired.

Conclusion

To complete the design I add Ground features to each component to prevent components from moving. Apply any physical properties through the Physical Materials context menu and at this point the design should be complete. Scaling and resizing of the table is easy and can be accomplished by changing values in the Parameters dialog box.

The key points to this approach are:

Create parameters specifying all relevant and pertinent properties for your design
Create as much reference geometry in the top level as needed: sketches, planes, axes
Create components for parts
Make sketches at the component level reference top-level sketches and reference geometry
Make sketch and feature dimensions incorporate parameters or parameter expressions
Add Grounds to components at the end of the timeline

Change of Basis and the Transformation Matrix

2019-02-15T00:00:00-06:00

A vector is represented traditionally with respect to a coordinate system. More generally it is represented by a set of basis vectors - two vectors which are linearly independent and form a vector subspace. A vector is therefore a linear combination of these basis vectors. In a global cartesian coordinate system these are the unit vectors $\hat{x}$, $\hat{y}$, and $\hat{z}$. A vector can be represented in another coordinate system by changing its basis vectors.

Definitions

With:

$M$, a basis matrix containing the basis vectors as columns
$\vec{v}$, a column vector in the basis vector space $M$

When written as:

\begin{equation*} \vec{a} = M\vec{v} \end{equation*}

the vector $\hat{a}$ is the vector in global coordinates.

The reverse can be achieved to determine a vector defined by a different basis:

\begin{equation*} \vec{v}=M^{-1}\vec{a} \end{equation*}

This works both for 2-D and 3-D vectors.

Example

Let a set of basis vectors be defined as:

\begin{equation*} \begin{bmatrix}1&3\end{bmatrix} \end{equation*}

and

\begin{equation*} \begin{bmatrix}-1&4\end{bmatrix} \end{equation*}

The basis matrix being:

\begin{equation*} M=\begin{bmatrix}1&-1\\4&4\end{bmatrix} \end{equation*}

These basis vectors result in the following grid when plotted:

Conversion to Global Coordinates

Set a vector in this space equal to $\begin{bmatrix}-2&3\end{bmatrix}$.

This vector using basis coordinates is plotted as the following:

The vector $\vec{a}$ (in global coordinates) is calculated with Eq 1. and is therefore equal to

\begin{equation*} \vec{a}=\begin{bmatrix}1&-1\\3&4\end{bmatrix}\begin{bmatrix}-2\\3\end{bmatrix}=\begin{bmatrix}-5&6\end{bmatrix} \end{equation*}

Converstion to Basis Coordinates

Using a vector in global coordinates of:

\begin{equation*} a=\begin{bmatrix}3&-1\end{bmatrix} \end{equation*}

we have:

And to convert it to the basis space, we multiply the vector by the inverse of the basis matrix.

\begin{equation*} v=\begin{bmatrix}1&-1\\3&4\end{bmatrix}^{-1}\begin{bmatrix}3\\1\end{bmatrix}=\begin{bmatrix}0.571&0.143\\-0.429&0.143\end{bmatrix}\begin{bmatrix}3\\1\end{bmatrix}=\begin{bmatrix}1.571\\-1.429\end{bmatrix} \end{equation*}

Transformation Matrix

Change of basis can be used to derive transformation matices.

Rotation

Counter-clockwise rotation by an angle $\theta$ is developed using unit vectors established by this angle:

\begin{equation*} \hat{x}=\begin{bmatrix}\cos{\theta}&\sin{\theta}\end{bmatrix} \end{equation*}

\begin{equation*} \hat{y}=\begin{bmatrix}-\sin{\theta}&\cos{\theta}\end{bmatrix} \end{equation*}

\begin{equation*} M_{rotate}=\begin{bmatrix}\cos{\theta}&-\sin{\theta}\\\sin{\theta}&\cos{\theta}\end{bmatrix} \end{equation*}

Scale

With scaling, the global unit vectors maintain the same orientation but are scaled in length by a scale factor, $s$. The new basis vectors are then the following:

\begin{equation*} \hat{x}=\begin{bmatrix}s_x&1\end{bmatrix} \end{equation*}

\begin{equation*} \hat{y}=\begin{bmatrix}1&s_y\end{bmatrix} \end{equation*}

\begin{equation*} M_{scale}=\begin{bmatrix}s_x&1\\1&s_y\end{bmatrix} \end{equation*}

Shear

Shearing along a principal axis may be derived as follows, for example when along the x-axis:

\begin{equation*} \hat{x}=\begin{bmatrix}1&0\end{bmatrix} \end{equation*}

\begin{equation*} \hat{y}=\begin{bmatrix}\sin{\theta}&1\end{bmatrix} \end{equation*}

\begin{equation*} M_{shear,x}=\begin{bmatrix}1&\sin{\theta}\\0&1\end{bmatrix} \end{equation*}

Polygon Transformation

As an example of transformation matrices, let's create and transform a generic polygon using Python and matplotlib.

Using the following points as definition:

xy = np.array([[0,0],[2,3],[1,3],[3,6],[5,3],[4,3],[6,0]])

we obtain this polygon:

Rotation

For a rotation of 45 degrees, counter-clockwise about the origin, the transformation matrix becomes:

rotation = np.radians(45)
x = np.array([  np.cos(rotation), np.sin(rotation) ])
y = np.array([ -np.sin(rotation), np.cos(rotation) ])
M = np.hstack( (x.reshape(-1,1), y.reshape(-1,1)) )

>>> print(M)
[[ 0.70710678 -0.70710678]
[ 0.70710678  0.70710678]]

The new polygon points are then calculated with:

xy_new = np.dot(M,xy.T)

Adding the rotated polygon to the plot:

Scale

Using scale factors of 1.9 and 1.4 for the x and y axis, respectively:

sx = 1.9
sy = 1.4

x = np.array([ sx, 0 ])
y = np.array([ 0, sy ])

M = np.hstack( (x.reshape(-1,1), y.reshape(-1,1)) )

>>> print(M)
[[1.9 0. ]
[0.  1.4]]

We calculate the points for the scaled polygon the same way:

xy_new = np.dot(M,xy.T)
poly_scaled = Polygon(xy_new.T, fill=False, edgecolor='blue')

And plotting:

Shear

Here we will shear along the x-axis by 45 degrees. The transformation matrix is:

angle = 45

x = np.array([ 1, 0 ])
y = np.array([ np.sin(np.radians(angle)), 1 ])

M = np.hstack( (x.reshape(-1,1), y.reshape(-1,1)) )

>>> print(M)
[[1.         0.70710678]
[0.         1.        ]]

And the sheared polygon is calculated using matrix multiplication again:

xy_new = np.dot(M,xy.T)
poly_scaled = Polygon(xy_new.T, fill=False, edgecolor='blue')

Interpolating Logarithmic Plots for Fatigue Analysis

2019-01-23T00:00:00-06:00

Here we will develop some classes for interpolating general multi-linear fatigue curves using SciPy's interp1d. interp1d creates a class instance which can then be called with lookup values:

x = [ 1,2,3,...]      # X-axis values
y = [ 20,15,10,...]   # Y-axis values
f = interp1d(x,y)
y1 = f(x1)            # Query at a new value, x1

Semi-Log

Using a linear interpolation function on a semi-log graph requires that the log of the values is used on the log axis. With fatigue curve arrays of $N$ and $S$, interpolating for stress can be done by:

\begin{equation*} f=\text{interp1d}(\log_{10}N, S) \end{equation*}

\begin{equation*} S_{new}=f(\log_{10}N_{new}) \end{equation*}

And interpolating for cycles:

\begin{equation*} f=\text{interp1d}(S, \log_{10}N) \end{equation*}

\begin{equation*} N_{new}=10^{f(S_{new})} \end{equation*}

Log-Log

Log-log calculations are the same as Semi-log but with the exception that both axes use the logarithm of the values.

To obtain stress:

\begin{equation*} f=\text{interp1d}(\log_{10}N, \log_{10}S) \end{equation*}

\begin{equation*} S_{new}=10^{f(\log_{10}N_{new})} \end{equation*}

To obtain cycles:

\begin{equation*} f=\text{interp1d}(\log_{10}S, \log_{10}N) \end{equation*}

\begin{equation*} N_{new}=10^{f(\log_{10}S_{new})} \end{equation*}

Implementation

In Python we can implement the above as classes. First a general class common to both log-log and semi-log:

import numpy as np
import matplotlib.pyplot as plt
from scipy.interpolate import interp1d

class FatigueCurve:
    ''' Base class for a fatigue curve.

        Parameters
        ----------
        points : list
            A list containing (N,S) data point pairs (2+)

    '''
    def __init__(self,points):
        self.N, self.S = list(zip(*points))
        self.N = np.asarray(self.N)
        self.S = np.asarray(self.S)
        self.logN = np.log10(self.N)
        self.logS = np.log10(self.S)

    def _plot(self):
        ''' Plot setup method '''
        fig, ax = plt.subplots()
        ax.set_xlabel('Cycles')
        ax.set_ylabel('Stress')
        ax.grid(True, which='both', axis='both', color='lightgrey')
        return fig, ax

Next we create classes for semi-log and log-log:

class SemiLogCurve(FatigueCurve):
    def __init__(self,points):
        super().__init__(points)
        self.getS_f = interp1d(self.logN,self.S)
        self.getN_f = interp1d(self.S,self.logN)

    def _semilogx(self,ax,N,S):
        ''' Plot an interpolated value on a semilog fatigue curve. '''
        ax.semilogx(self.N,self.S,markersize=6,marker='o')
        ax.semilogx(N,S,markersize=6,marker='o')
        ax.annotate('({:,.2}, {:,.2})'.format(N,S), (N,S))
        return ax

    def getS(self,N,plot=False):
        ''' Interpolate stress from cycles. '''
        S = self.getS_f(np.log10(N))
        if plot:
            fig,ax = self._plot()
            ax = self._semilogx(ax,float(N),float(S))
        return int(S)

    def getN(self,S,plot=False):
        ''' Interpolate cycles from stress. '''
        N = np.power(10,self.getN_f(S))
        if plot:
            fig,ax = self._plot()
            ax = self._semilogx(ax,float(N),float(S))
        return int(N)

    def plot(self):
        fig, ax = self._plot()
        ax.semilogx(self.N,self.S)

    __call__ = getS


class LogLogCurve(FatigueCurve):
    def __init__(self,points):
        super().__init__(points)
        self.getS_f = interp1d(self.logN,self.logS)
        self.getN_f = interp1d(self.logS,self.logN)

    def _loglog(self,ax,N,S):
        ''' Plot an interpolated value on a log-log fatigue curve. '''
        ax.loglog(self.N,self.S,marker='o',markersize=6)
        ax.loglog(N,S,marker='o',markersize=6)
        ax.annotate('({:,.2}, {:,.2})'.format(N,S), (N,S))
        return ax

    def getS(self,N,plot=False):
        ''' Interpolate stress from cycles '''
        S = np.power(10,self.getS_f(np.log10(N)))
        if plot:
            fig,ax = self._plot()
            ax = self._loglog(ax,float(N),float(S))
        return int(S)

    def getN(self,S,plot=False):
        ''' Interpolate cycles from stress '''
        N = np.power(10,self.getN_f(np.log10(S)))
        if plot:
            fig,ax = self._plot()
            ax = self._loglog(ax,float(N),float(S))
        return int(N)

    def plot(self):
        fig, ax = self._plot()
        ax.loglog(self.N,self.S)

    __call__ = getS

Usage

>>> curve = LogLogCurve([(1e3,50e3),(1e6,20e3),(1e7,18e3)])
>>> curve.getS(70e3,plot=True)
28459

>>> curve.getN(34e3,plot=True)
18309

Failure Theories

2015-10-06T00:00:00-05:00

For part stresses that are multi-axial, failure theories relate the failure in a part subjected to arbitrary multi-axial loads to that of a uni-axially loaded test specimen. The general form of these relationships are:

\begin{equation*} M_{part}\geq M_{specimen} \end{equation*}

Where M is a “modulus” chosen by each theory that can be either calculated or measured in the part and test specimen, and then compared to assess failure. Often the modulus is a function of the principal normal or shear stresses.

\begin{equation*} f(\sigma_1,\sigma_2,\sigma_3)\geq \sigma_c \end{equation*}

\begin{equation*} f(\tau_1,\tau_2,\tau_3)\geq \tau_c \end{equation*}

where $\sigma_c$ and $\tau_c$ are the chosen characteristic failure strengths from the uniaxial tension test.

The major theories are:

Maximum Normal Stress Theory

Failure (i.e. yielding, fracture) is expected to occur if the maximum normal stress in the part exceeds the maximum normal stress in test specimen at failure (yielding, fracture).

\begin{equation*} \max(\sigma_1,\sigma_2, \sigma_3)\geq \sigma_c \end{equation*}

\begin{equation*} \min(\sigma_1,\sigma_2,\sigma_3) \leq \sigma_c \end{equation*}

where $\sigma_t$ and $\sigma_c$ are test specimen tensile and compressive strengths, respectively.

This theory is applicable to brittle materials but not to ductile materials.

Maximum Shearing Stress Theory

Here failure is predicted to occur if the maximum shearing stress (the principal shearing stress) is equal to or exceeds the maximum shear stress in a test specimen at failure.

\begin{equation*} \max(|\tau_1|, |\tau_2|, |\tau_3|) \geq \tau_c \end{equation*}

$\tau_c$ is the maximum shearing stress in the test specimen, and for a uniaxial tension test the max shear stress occurs at 45 degrees to the applied load direction and is equal to half of the first principal stress, which is the nominal tensile stress.

This theory applies well to ductile materials and to ductile yielding.

Distortion Energy Theory (von Mises)

This theory postulates that the distortion energy - that which contributes to shape change and not to change in volume - affects the failure of the part. Dilation energy, that which only changes the volume, does not contribute to the failure of the part. The latter is produced by hydrostatic stress, which has been shown to not induce yielding or fracture (none under compression, but fracture under high levels of tensile stress).

The equation for the distortion energy theorem is:

\begin{equation*} \frac{(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2}{2} \geq \sigma_f^2 \end{equation*}

The left side of the equation when (sigma_f) is solved is the von Mises stress:

\begin{equation*} \sigma_{vm} = \sqrt{\frac{(\sigma_1-\sigma_2)^2 + (\sigma_2-\sigma_3)^2 + (\sigma_3-\sigma_1)^2}{2}} \end{equation*}

The yield and failure surface can be plotted, obtaining an infinite cylinder. According to the theory all stress points that lie within the surface do not produce failure and those outside do.

For a 2D stress state, the yield surface produces an elliptical failure curve when intersected with a plane.

This can be verified by setting $\sigma_3$ equal to 0 and plotting. With a constant of 1:

Principal Stresses in 3D

2015-09-14T00:00:00-05:00

When a stress element is characterized by its six stress components and these components are put in stress tensor form, the principal normal stresses are simply the eigenvalues of the stress tensor, and the principal planes are defined by the eigenvectors. This is easily computed using a mathematical program such as MathCAD, Matlab, etc. Here we will use Python and the library NumPy.

Principal Normal Stress

Put the stresses into the stress tensor form:

\begin{equation*} S=\begin{bmatrix}\sigma_x & \tau_{xy} & \tau_{zx} \\ \tau_{xy} & \sigma_y & \tau_{yz} \\ \tau_{zx} & \tau_{yz} & \sigma_z \end{bmatrix} \end{equation*}

import numpy as np
S = np.array([ [S_xx, S_xy, S_zx],
               [S_xy, S_yy, S_yz],
               [S_zx, S_yz, S_zz] ])

Then, compute the eigenvalues and eigenvectors. In Python:

e_val, e_vec = np.linalg.eig(S)

To store the eigenvalues in the variable e_val and the eigenvectors in the variable $e_vec$.

The principal stresses can be assigned by sorting, based on the assignment relation of $\sigma_1 > \sigma_2 > \sigma_3$.

p3, p2, p1 = np.sort(e_val)   # sort smallest to largest

The eigenvectors are stored as column vectors in the e_vec array. To access these, we first need to associate the principal stresses with their positions in the original result in order to assign them the correct vector. To do this we will convert the original eigenvalue array to a regular list and use standard Python indexing. Then, the vectors are extracted from the eigenvector array using these indices.

e_val_l = e_val.tolist()
p1_index, p2_index, p3_index = e_val_l.index(p1), e_val_l.index(p2), e_val_l.index(p3)
p1_vec, p2_vec, p3_vec = e_vec[:,p1_index], e_vec[:,p2_index], e_vec[:,p3_index]

Principal Shear Stress

The principal shearing stresses can be inferred from the usage of a Mohr circle. In constructing a Mohr circle for a 3D stress state, the maximum shearing stress can be shown to be one half of the difference between the maximum and minimum principal normal stresses. From the Mohr circle the principal shearing stresses can be determined as:

\begin{align*} \tau_1=(\sigma_1-\sigma_3)/2 \\ \tau_2=(\sigma_1-\sigma_2)/2 \\ \tau_3=(\sigma_2-\sigma_3)/2 \end{align*}

In Python,

tau1 = (p1-p3)/2
tau2 = (p1-p2)/2
tau3 = (p2-p3)/2

The principal shearing planes bisect the two principal normal planes used in determining the principal shearing stress. The $\tau_1$ plane bisects the $\sigma_1$ and $\sigma_3$ planes, and since all principal normal planes are orthogonal to each other, the principal shearing planes are at 45 degrees to each principal normal plane.

To find the vectors for these planes, we compute the vector that bisects the vectors for the normal principal stresses. Add them vectorally to obtain a rhombus, the diagonal of which splits the rhombus into two equal segments.

In Python we add our normal principal vectors and divide by the magnitude to normalize:

tau1_vec = (p1_vec + p3_vec)/np.linalg.norm(p1_vec + p3_vec)
tau2_vec = (p1_vec + p2_vec)/np.linalg.norm(p1_vec + p2_vec)
tau3_vec = (p2_vec + p3_vec)/np.linalg.norm(p2_vec + p3_vec)