More Ligurian Arches

We will complete the stay in Camoglia with another collection of arch structures (great and small, old and new) from the district:

Santuario di Montallegro, near Rapallo

Santuario di Montallegro

Monastry of San Fruttuoso

Monastry of San Fruttuoso

Monastry of San Fruttuoso

Monastry of San Fruttuoso

Monastry of San Fruttuoso

Monastry of San Fruttuoso

Rail viaduct at Recco

Rail viaduct at Recco, at time of construction

The recent photo of the Recco Rail Viaduct is from the Structurae Database.

The photo from the time of construction is from: http://www.miol.it/stagniweb/fs101.htm

The Recco Viaduct (line Genoa-La Spezia) was rebuilt in 1946-48 with 6 parabolic arches on a curve of 400 m radius. Thanks to its design qualities and to the importance of the railway line, it can be assumed as one of the symbols of the reconstruction.

 

 

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They all laughed at Christopher Columbus …

…When he said the World was round (or rather, they didn’t).

Christopher Columbus was born in the town of Genova, in what is now North-West Italy.  The adjacent towns are happy to share the credit:

Christopher Columbus at Santa Margherita

Christopher Columbus as Rapallo

It is (or at least was) a popular myth that the accepted wisdom at the time was that the Earth was flat, until Columbus sailed west with the intention of reaching China, and bumped into the American continent on the way.  In fact the Greeks had believed the Earth to be roughly spherical about 2000 years earlier, and had made an estimate of the Earth’s radius, based on the length of mid-day shadows at different latitudes.  This knowledge had been preserved, and would have been well known to educated people of the era of Columbus (see Wikipedia’s Spherical Earth article for more details).

How widespread this knowledge was amongst the general populace is open to debate, but there is at least some evidence that it was known outside the world of the literate.  for instance from the Wikipedia article:

the Elucidarium of Honorius Augustodunensis (c. 1120), an important manual for the instruction of lesser clergy, which was translated into Middle English, Old French, Middle High German, Old Russian, Middle Dutch, Old Norse, Icelandic, Spanish, and several Italian dialects, explicitly refers to a spherical Earth. Likewise, the fact that Bertold von Regensburg (mid-13th century) used the spherical Earth as a sermonic illustration shows that he could assume this knowledge among his congregation. The sermon was held in the vernacular German, and thus was not intended for a learned audience.

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Ligurian Arches

We have looked previously at Robert Hooke’s statement on the mathematical analysis of arches:

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but of course  people had been constructing large arch structures for many hundreds of years before Hooke made his enigmatic statement, and the masters, if not the inventors, of the arch were the Romans.  In future posts I will be looking at some of the historically important arch bridges in Northern Italy, but for today, here is a collection of arches (natural and constructed) from a walk in the Cinque Terre National Park, and in and around Camogli:

Arch near the harbour at Manarola

The Harbour at Vernazza

The Harbour at Vernazza

The Harbour at Vernazza

The Harbour at Vernazza, Natural Arches

The Harbour as Vernazza

Natural and Railway arches near Vernazza

Camogli Station

Arched strata at Salita Mandrella a Monte
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Galileo and Pisa

Welcome to Galileo Excel Baccusi, which will be replacing Newton Excel Bach for the next four weeks, while I travel in Northern Italy.

Arriving in Rome on Friday morning, we travelled to Camogli by mini-bus, stopping for a quick tour of Pisa, home of the famous Leaning Tower:

The Duomo and Leaning Tower at Pisa

and two magnificent 14th century domes; of The Baptistry:

The Baptistry at Pisa

and the Duomo:

The Dome of the Duomo

This is what Wikipedia says of Galileo’s alleged experiment with dropping balls of different mass from the leaning tower:

According to a biography by Galileo’s pupil Vincenzo Viviani, in 1589 the Italian scientist Galileo had dropped two balls of different masses from the Leaning Tower of Pisa to demonstrate that their time of descent was independent of their mass.[1]. Via this method, he supposedly discovered that the objects fell at the same acceleration, proving his prediction true, while at the same time proving Aristotle’s theory of gravity (which states that objects fall at speed relative to their mass) false. At the time when Viviani asserts that the experiment took place, Galileo had not yet formulated the final version of his law of free fall. He had, however, formulated an earlier version which predicted that bodies of the same material falling through the same medium would fall at the same speed.[2] This was contrary to what Aristotle had taught: that heavy objects fall faster than lighter ones, in direct proportion to weight.[3][4] While this story has been retold in popular accounts, there is no account by Galileo himself of such an experiment, and it is accepted by most historians that it was a thought experiment which did not actually take place.[5][6] An exception is Drake,[7] who argues that the experiment did take place, more or less as Viviani described it.

But whether the real experiment actually happened at Pisa or not, the really important part is the thought experiment that led to it; from Wikipedia again:

Galileo arrived at his hypothesis by a famous thought experiment outlined in his book On Motion[8]. Imagine two objects, one light and one heavier than the other one, are connected to each other by a string. Drop this system of objects from the top of a tower. If we assume heavier objects do indeed fall faster than lighter ones (and conversely, lighter objects fall slower), the string will soon pull taut as the lighter object retards the fall of the heavier object. But the system considered as a whole is heavier than the heavy object alone, and therefore should fall faster. This contradiction leads one to conclude the assumption is false.

This line of thinking led not only to Galileo’s work on falling objects, but leads directly to Newton’s Laws of Motion and Universal Gravitation, which makes it arguably the most important thought experiment in the history of science.

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Composite Beam Spreadsheet 3

The Composite Beam Analysis spreadsheet, previously discussed at: https://newtonexcelbach.wordpress.com/2012/04/27/composite-beam-spreadsheet-2/ has now been updated with a VBA User Defined Function (UDF) in place of the previous macro that used the Excel Goal-Seek command.  This has two main advantages:

  • The UDF is much faster than repeated use of the Goal-Seek command.
  • The UDF is much more flexible, for instance allowing the analysis to be performed at a series of sections along a beam, with the load information in tabular format.

The new spreadsheet may be downloaded from Composite Design Functions .  As usual, the download file includes full open source code.

Not that the previous version is still available from Composite Design Functions-GS.zip

A screen-shot of the input sheet for the new function is shown below:

Input for Composite Design Functions spreadsheet, click for full size view

The basis of the new UDF is the QuadBrent function, previously presented at https://newtonexcelbach.wordpress.com/2010/04/13/the-inverse-quadratic-method-3-brents-method/.  This function uses Brent’s Method (an elaboration of Newton’s Method) to solve equations by iteration.  In this case a two stage iteration was required:

  • The main ElasticComp function calls the QuadBrent function with “FindCurve” as the target function.  FindCurve adjusts the section curvature with a constant Neutral Axis depth so that the resultant bending moment on the section is equal to the applied load.
  • For each iteration of the FindCurve function  the QuadBrent function is called again with “EvalForceMom” as the target function.  This function adjusts the top-face strain, with constant curvature so that the resultant axial force on the section is equal to the applied load.
  • This process is repeated until both bending moment and axial force are equal to the applied loads, within the specified tolerance.

The other main change from the previous version is that the load data may now be entered in a single row, and that the Stage 2 loading may be enetered as 2 or more separate parts, so that if the beam is modelled as two separate elements with a rigid connection in a finite element program, for instance, the loads on the two elements may be entered seperately, and the program will combine them for the analysis.  Examples of alternative layouts of equivalent applied loads are shown below:

Alternative Load Layouts

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