Edwin C. Krupp, Director of Griffith Observatory:

The scientific revolution is rooted in a mentality that doesn’t rely on authority. Galileo had that.

As an astronomer, it’s natural for Dr. Krupp to rave about Galileo. For a man who showed us the planets, and as brave to go against divinity, we are indebted to his science, no doubt, but the part regarding “authority” wasn’t perhaps as bold as Dr. Krupp have us believe. Being a smart scientist, he dedicated his discoveries of Jupiter’s moons to the young and upcoming prince, Cosimo I, of the Medici family to gain prominence. He wrote this flattering letter to the prince:

Your Highness… scarcely have the immortal graces of your soul begun to shine forth on Earth than bright stars offer themselves in the heavens which, like tongues, will speak of and celebrate your most excellent virtues for all time. Behold, therefore, four stars reserved for your illustrious name, and not of the common sort and multitude of the less notable fixed stars, but of the illustrious order of wandering stars, which, indeed, make their journeys and orbits with a marvelous speed around the star of Jupiter, the most noble of them all, with mutually different motions, like children of the same family, while meanwhile all together, in mutual harmony, complete their great revolutions every twelve years about the center of the world…

Indeed, it appears that the Maker of the Stars himself, by clear arguments, admonished me to call these new planets by the illustrious name of Your Highness before all others. For as these stars, like the offspring worthy of Jupiter, never depart from his side except for the smallest distance, so who does not know the clemency, the gentleness of spirit, the agreeableness of manners, the splendor of the royal blood, the majesty in actions, and the breadth of authority and rule over others, all of which qualities find a domicile and exaltation for themselves in Your Highness? Who, I say, does not know that all these emanate from the most benign star of Jupiter, after the God the source of all good? It was Jupiter, I say, who at Your Highnesses birth, having already passed through the murky vapors of the horizon, and occupying the mid-heaven and illuminating the eastern angle from his royal house, looked down upon Your most fortunate birth from that sublime throne and poured out all his splendor and grandeur into the most pure air, so that with its first breath Your tender little body and Your soul, already decorated by God with noble ornaments, could drink in this universal power and authority.

(Excerpt from Sobel, Dava. “Galileo’s Daughter.”)

July 20, 2014.

Kid: There shouldn’t be a right-angled triangle. We already have Isosceles, Scalene and Equilateral types.
Mom: It’s an additional classification.
Kid: Is this something to do with Pythagoras rule?
Dad: Look at it this way: the right-angle triangle is a perfect unit triangle because of the Pythagoras rule. It allows one to divide any odd shaped triangle or even a polygon into many right-angled triangles, which then by Pythagoras rule, allows one to calculate all unit areas and sum them to get the total area of a random shape.
Kid: (Starts doodling for a few minutes, and then looks up.) Yeah, looks right.

July 20, 2014.

A solution to finally purge all .tex compile files using an Automator workflow.

A solution to finally purge all .tex compile files using an Automator workflow.

July 19, 2014.

On allowable von Mises stress

Some months ago, in response to a request for reference on allowable von Mises stress, I replied to a friend and my former colleague with the following.

“Allowable” von Mises stress is tricky, which is why no (college grade or professional) textbook and no standard explicitly advocates one. The reason behind is simple, von Mises stress check is not an universal formulae, because the outcome of this equation is governed by the relative magnitudes of components within.

von Mises stress combines normal and shear stress in a rather simplified manner to obtain a practical (theoretically less robust) approach to a notional limit.

Distortion energy required per unit volume, Ud, for a general three-dimensional case is given in terms of principal stress values as follows:

\begin{aligned} U_{d} = \frac{1 + \nu}{3E}\left({\frac{(\sigma_{1} - \sigma_{2})^2 + (\sigma_{2} - \sigma_{3})^2 + (\sigma_{3} - \sigma_{1})^2}{2}}\right) \end{aligned}

Distortion energy for simple tension case at the time of failure is given as follows:

\begin{aligned} U_{d,simple} = \frac{1 + \nu}{3E} \sigma_{y}^2 \end{aligned}

Above two quantities can be connected using distortion energy, so the condition of failure will be as follows, in which the LHS forms the expression for von Mises stress:

\begin{aligned} \sqrt{\frac{(\sigma_{1} - \sigma_{2})^2 + (\sigma_{2} - \sigma_{3})^2 + (\sigma_{3} - \sigma_{1})^2}{2}} \ge \sigma_{y} \end{aligned}

von Mises attempts to develop a yield criterion for ductile materials that can be applied to any complex three-dimensional loading condition, regardless of the mix of normal and shear stresses. It does this by simplifying a complex stress condition down to a single numerical scalar value, which is determined from a uniaxial tension test — convenient for lab test, and also the easiest. This convenience is why I think most people use it.

However, one should keep in mind that the representative stress for uniaxial tension is not equal to the uniaxial tension stress, but is instead about 81%. This inconvenience is manipulated by factoring it up by 1.22 to make it equal; which is further accepted because the deviatoric strain energy is still proportional.

So, not quite the science, it is an empirical process with inherent errors and deviations — as you can imagine, and there’s no strict rule to conform to saying materials must/will yield in accordance with the von Mises yield criteria. The approximations in the form of results appear to work reasonably well, and for most purposes, people tend to use that as a good indicator. But it’s never a good idea to push safety critical elements to limit using this inconvenient yield criterion — for obvious reasons.

People tend to derive its allowable by substituting allowable limits of normal and shear stresses in the equation, as below:

In the following, fx and fy are maximum allowable principal stresses, where as txy is the maximum allowable shear stress.

fx = 0.6 * Fy
fy = 0.6 * Fy
txy = 0.4 * Fy

von Mises stress, fvm is as follows:

fvm = (1 / sqrt(2)) * sqrt(fx^2 + fy^2 + 6 * txy^2) => 0.92 Fy

The allowable becomes interesting (reduces) when one of the principal stresses is negligible, e.g., if

fy = 0

then,

fvm => 0.81 Fy

If Shear stress is negligible, von Mises stress, fvm becomes:

fy = 0.6 * Fy
txy = 0

fvm => 0.6 Fy

So, one needs to carefully choose how von Mises stress is used, because it depends upon the magnitudes of normal and shear stresses.

July 12, 2014.

Clarence Brown:

Yevgeny Zamyatin’s We is the archetype of the modern dystopia, or anti-utopia; a great prose poem on the fate that might befall all of us if we surrender our individual selves to some collective dream of technology and fail in the vigilance that is the price of freedom.

Somehow We struck me as the book to read instead of its derivatives, viz., The Brave New World or 1984. Films like Gattaca, The Matrix or even the recent The Hunger Games initially appeared too far fetched to be possible, until of course the recent scandal, which turned that thought over its head.

July 12, 2014.

Richard Duffy in Emily Post’s Etiquette:

In the midst of the war, some French soldiers and some non-French of the Allied forces were receiving their rations in a village back of the lines. The non-French fighters belonged to an Army that supplied rations plentifully. They grabbed their allotments and stood about while hastily eating, uninterrupted by conversation or other concern. The French soldiers took their very meager portions of food, improvised a kind of table on the top of a flat rock, and having laid out the rations, including the small quantity of wine that formed part of the repast, sat down in comfort and began their meal amid a chatter of talk. One of the non-French soldiers, all of whom had finished their large supply of food before the French had begun eating, asked sardonically: “Why do you fellows make such a lot of fuss over the little bit of grub they give you to eat?” The Frenchman replied: “Well, we are making war for civilization, are we not? Very well, we are. Therefore, we eat in a civilized way.”

July 8, 2014.

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