From Galileo's flexural theory to the sandwich theory

GALILEO
The name of the famous physicist has been chosen because Galileo Galilei is considered to be the founder of the study of flexure, and on the basis of the sandwich effect, sandwich elements possess outstanding flexural and torsional rigidity.

Historic development of the flexural theory
To bring us to a point at which we could make efficient use of the outstanding flexural and torsion rigidity of modern sandwich elements in the realm of light construction, two different areas of science had to be established and continually developed over the course of the centuries. The first of these fields was that of architectural statics, specifically the theory of flexure, and the second that of material-specific stress studies.

Today we have mastered both of these disciplines perfectly - but some 250 years ago the theoretical knowledge with respect to architectural statics and the study of flexure was quite scanty. A brief historic trek can show us the tedious road from the flexural theory of Leonardo da Vinci to the founding of modern architectural statics, Claude Louis Navier and the static specifics of the sandwich theory.

Leonardo da Vinci and Galileo Galilei started it all
Very probably it was the great Leonardo da Vinci who first confronted the problem of flexure. But he only arrived at purely qualitative statements in which he compared the supporting capacity of various construction parts with one another.

Fig. 8.3.1 Table VIII from the Theatrum Pontificiale of Jacob Leupold

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Fig. 8.3.2 Leonardo da Vinci

Fig. 8.3.3 Galileo Galilei

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Galileo’s major achievement in the theory of flexure
In the 16th century Galileo Galilei (1564-1642) was the first to study the subject of resistance and in this context also the flexure of solid bodies. But even Galileo could not answer all the questions with regard to flexure. For example he was unable to make quantitative statements with respect to supporting capacity. He only managed to establish that a beam placed on edge could bear more weight than one placed flat down.

Galileo recognised that the sum of left and right-turning moments had to be zero, but the equal weight of inner forces was not yet contained in his theory. Galileo’s major achievement was determining, to begin with, the relation between external load and internal stress. But he fixed the location of the pivotal axis at the bottom edge of a clamped beam rather than the middle of the beam’s cross section. This minor but statically significant error (understandable in historic terms) continued to be upheld into the 19th century.

“Axis of balance” and distribution of tension discovered
After Leonardo da Vinci and Galileo, Hooke, Newton and Mariotte wrote their chapter in the history of flexural studies. In 1668 Robert Hooke (1635-1703) published his significant discovery that tension is proportional to expansion. Hooke also already differentiated between tension, pressure, and flexural requirements of beams.

With regard to flexural requirements, Hooke determined that the fibres were partially lengthened and partially shortened. So this was the basis for the existence of a neutral fibre.

In 1687 Isaac Newton (1642-1727) formulated the axiom concerning the balance of forces and the connection between force and motion.

Edme Mariotte (1620-1684) studied the Galileic problem, was the first to apply the Hookean Hypothesis with respect to the problem of flexure, and arrived at the idea of a triangular distribution of tension. To begin with Mariotte, too, had difficulties placing the neutral axis correctly. But then he correctly established his “axis of equal weight” halfway up the rectangular cross section. Thus the principle of the calculation of flexure was discovered. But Mariotte’s publication met with no significant resonance.

The same applied to Parent (1666-1716) who recognised, as had Mariotte, that the zero line must be at the height of the half rectangular cross-section.

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Fig. 8.3.4 Clamped beams with load grip from Galileo’s Discorsi e dimostrazioni

Fig. 8.3.5 Robert Hooke

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Fig. 8.3.5 Isaac Newton

The great mathematicians lend a hand
Mathematician Gottfried Wilhelm von Leibniz (1646-1716), who fought bitterly with Newton with regard to the discovery rights to calculus, was the first to successfully apply infinity methods to calculate section moduli.

However, since he took over the allocation error of the axis from Galileo, the results of Leibniz were fated to be erroneous, despite his excellent mathematics. While the correct value is 1/6 (b * h2), the calculation of Leibniz of the rectangular cross-section led to twice that value. Galileo’s principle indeed led to three times as great a section modulus.

Load, flexure and elastic lines
In the next step, Jakob Bernoulli (1654-1705) demonstrated the independence of flexure from load.

Leonhard Euler concerned himself intensively with the study of elastic lines that a straight beam assumes after being subjected to flexure.

The integration of his differential comparisons led in 1757 to the well-known Eulerschen buckling comparison.

With such concepts as “absolute elasticity”, Euler already neared the concept of the “elasticity module” championed by Thomas Young in 1807. Nonetheless amongst these great scientists it was not the practical use of their theories for the field of construction that were their inspiration, but above all their love for the newly discovered field of mathematics, calculus.

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Fig. 8.3.6 Gottfried Wilhelm von Leibniz

Fig. 8.3.8 Jakob Bernoulli

Fig. 8.3.9 Leonhard Euler

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Jakob Leupold approaches practical use
In the person of Jakob Leupold (1674-1727), there was a man who was the first to undertake the difficult task of passing on the knowledge of scholars and mathematicians directly to craftsmen and artists. His work encompassed nine books, providing a practical picture of the state of the art of the 17th to 18th centuries. In his Theatrum Pontificale - “Showplace of Bridges and Bridgebuilding”, Leupold also made statements regarding the theory of flexure. The reader will certainly be pleased to enjoy a brief taste of Leupold’s activities.

In addition to various statements with regard to the supporting capacity of beams placed on edge, as well as the dependence of supporting width upon flexural moment in § 127, however, for Leupold the dependency between flexural moment and flexure remained unclear. On the one hand Leupold believed that under normal stress beams demonstrated no flexure, while on the other hand he made the statement that if the straining pieces were too far apart, beams would bend under their own weight.

Leupold’s contradictory explanations with regard to the behaviour of fibres reflected the cognitive problem with regard to the phenomenon of flexure just as it had first been encountered by Galileo and Leibniz.

His determination that beams placed on edge would not bend, while beams lying flat could indeed bend, cannot be sustained from a scientific approach. One of the conclusions of Leupold was that a beam placed on edge could bear four times as much as a beam placed flat.

Fig. 8.3.10 Jacob Leupold

Abb. 8.3.11 Excerpt § 129 from Leupold’s Theatrum Pontificiale

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Abb. 8.3.11 First page of Theatrum Pontificiale by Jakob Leupold published in 1726

Though the first specific quantitative statement with regard to the topic of supporting capacity is made here, we lack the vital specification of what height and width the beam in question possesses. It also remains unclear whether Leupold addressed flexure to determine the cross section of a beam, why he observed flexure in beams lying flat but did not observe flexure with upright beams.

Reading between the lines of Leupold’s arguments that he wished to use them to express a practical threshold, Leupold must have studied a beam cross section with the side ratio height:width = 2: 1. Since Leupold did present flexure as the only parameter for supporting capacity, his theory could not be correct. Because if a beam is bent, its supporting capacity depends upon the tension in the edge fibres. If one compares the tension in the edge fibres, one arrives, with a side ratio of height/width = 2:1, at the conclusion that a beam placed on edge can only bear twice what a flat beam can bear.

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Nor did Leupold yet recognise any boundary loads and no safety value. Apparently Leupold’s attempts were purely flexural attempts with beams that had not been loaded to their breaking limits. It is also amazing that Mariotte’s calculations with respect to the distribution of tension that were already known a half a century before Leupold’s book, were not utilised in his work.

Solution to the flexural problem by Coulomb
The actual final solution to the flexural problem is attributed to a briefly worded publication in 1773 by the French physicist Charles Augustin Coulomb (1736-1806). First of all he dealt with the questions of statics and stress theory according to scientific methods, while secondly he also considered their application to practical construction tasks. Coulomb, whose work included such things as electromagnetic interaction in his Coulomb’s Law, was both an excellent physicist and a practically oriented engineer.

As did Mariotte, Coulomb assumed equally great pull and pressure tensions, and he argued for equality of weight for the inner forces of a beam.

By means of practical attempts, Coulomb then managed to confirm his assumption, and thus the indubitable determination of the zero line location halfway up a rectangular cross-section. Coulomb proceeded in exactly the same way as today’s construction statics studies: he made an imaginary cut through the beam under a load, placed normal and shearing stress on the interfaces, and then arrived with consistent application of the laws of equal weight at a statement concerning the extent of the tension. His theory about the problem of flexure also contains a statement concerning shearing stress.

But what to us sounds so logical in the 21st century was a very difficult, abstract process of understanding in Coulomb’s time, one which many great physicists before and after Coulomb could not clearly recognise and formulate. Unfortunately, in contrast to Jakob Leupold’s “Theatrum Pontificiale”, the activities of Coulomb were left unrecognised by the world of science for the most part. This may have been due to the scantiness of his activities, the meagreness of communication facilities, or an adherence to old theories, but even after Coulomb many scientists returned to the incorrect assumption of Galileo that the location of the zero point played no role at all.

Fig. 8.3.13 Charles Augustin Coulomb

Navier founds modern architectural statics
The Frenchman Claude Louis M.H. Navier (1785-1836), founder of modern architectural statics, to begin with as a branch of mechanical engineering and then as an independent “house of learning”, completed the flexural theory and added to it by working in the realm of elastic engineering. To begin with, Navier too had his problems with the zero line, but corrected his “millennial error” in 1819. He departed from that geometrical nomer and proved that upon pure flexure the stress zero line passes through the axis of the cross section. Navier, who like Coulomb also based his reasoning on the proportionality of stress and flexure, arrived at results that corresponded precisely with the results of Coulomb. Navier’s success was based on the fact that he adapted the theoretical application of flexural theory to practical events. He was the first to make the flexural theory accessible to the realm of construction by means of mathematics and with sufficient accuracy.

Division between material and cross section characteristics completed
To begin with, Navier too “mixed” the material characteristics that were expressed with the “elasticity module” 1) and the cross-section characteristics that were expressed with the “moment of inertia” 2), into a single concept. In 1826, Navier then carried out a consistent division of the material characteristics from the cross-section characteristics. This step, too, was associated at the time with a considerable amount of changes in theory and a high degree of capacity for abstraction.

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Another 150 years until the realisation of the sandwich effect
Another 150 years passed before the discovery of polyurethane and the knowledge needed to combine this universal substance with the ideal characteristics of steel to arrive at a new construction component, the sandwich element. Member companies of GALILEO Creative Construction can make use of this pioneer development in their work.

In sandwich construction components, not only is the insulating layer used to retain warmth, it also integrates the mechanical characteristics of the insulating layer into the supporting capacity of the entire sandwich structural component, with the result that a clear increase in supporting effect is realised.

1) The concept of the “elasticity module” was introduced in 1807 by T. Young (1773-1829) (4.64).
2) The concept of the “moment of inertia” was renamed the “2nd degree surface moment” as per DIN 1080, Part 1.

Fig. 8.3.14 Claude Louis M. H. Navier

Fig. 8.3.14 Modern sandwich elements for roofs

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Sandwich effect - supporting behaviour
If you place an insulating layer and two thin metal cover layers loosely upon two supports, the individual elements, above all the metal cover layers, will bend through simply the force of their own weight. But with sandwich technology, in which these two metal cover layers are permanently bonded to the core insulating polyurethane hard foam, the result obtained is a light construction component with outstanding supporting properties.

A realistic comparison with the familiar substance of wood clarifies the situation: a three metre long piece of wood, that for example might be used in staging for scaffolding group 3 as a scaffolding base, with a point load of 1.5 kN, requires some 50 mm of thickness and gluing in several layers in order to maintain a maximum bend of 20 mm. With a cover width of 500 mm this would lead to a weight of 45 kg. A sandwich element of 50 mm thickness, under the same conditions, would only bend through 16 to 17 mm, plus it only weighs 18 kg. The sandwich element offers a weight saving of 60%. This weight advantage is particularly demonstrated in the realms of transport, assembly, and the dimensioning of supporting structures.

Thanks to the sandwich effect, sandwich elements possess outstanding flexural and torsion rigidity and due to their high inherent rigidity are not merely self-supporting. Despite their light weight, they can bear considerable weights depending on their static systems and dimensioning. For this reason sandwich elements can be used to meet the most demanding façade and roofing requirements in a very wide variety of situations. Basically sandwich elements are suitable for all light and mixed construction methods and can be installed on such varied supporting structures as reinforced concrete, steel, aluminium and wood.

Figure 8.3.16 left: loose layers - right: foamed layers of a sandwich element yield strong supporting properties and inherent rigidity due to the bonding between the insulating core and the cover layers with its great shearing strength.

Figure 8.3.17 Comparison of the flexure of a 50 mm thick sandwich element with a 50 mm thick wooden beam - Source: Koschade, R., Die Sandwichbauweise; Ernst & Sohn, Berlin 2000

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Static particulars of the sandwich theory
In the assessment of supporting capacity, the shearing properties of the core layer is to be considered and thus the “Theory of elastic connection” or the “sandwich theory” to be borne in mind (see in this context Fig. 8.3.17).

As Professor Klaus Berner 1), one of Europe’s leading statics experts regarding the sandwich construction method, explains in detail in Chapter 7 - Statics of “the sandwich construction method”, the “Statement of Bernoulli” of the sustained planar qualities of a cross section can no longer be upheld.

In the sandwich theory, the deformations and the associated tension shifts resulting from shearing movement in the core layer must be borne in mind, something that one can overlook when it comes to “shearing-proof construction components”. This is why for the secure measurement of sandwich construction components the shearing deformation is considered. Mathematical premises that assume a “rigid connection” in sandwich elements would overestimate the supporting capacity of the construction components and result in a measurement on the “uncertain side”.

Thus ends our brief historic foray into flexural science starting with Galileo and ending with the sandwich theory. An intensive involvement with the statics of sandwich elements is offered both by the above-mentioned chapter by Professor K. Berner and the GALILEO Information Documents 8.1 and 8.2.

Figure 8.3.17 Theory of elastic composite construction - Source: Berner, K: Chapter 7 - Statics in Koschade, R.: Die Sandwichbauweise; Ernst & Sohn, Berlin 2000

Authors
Rolf Koschade, Günter Hupe

1) Professor Klaus Berner, Professor at the University of Applied Sciences of Mainz in steel construction, wood construction and statics, promotion (1978) to the Technical University of Darmstadt (theme: sandwich supports), test engineer for construction statics, member of SVA, DIBt, - “Sandwich”, member of WG 7.4 sandwich panels, ECCS (European Convention for Constructional Steelwork) and of W56, sandwich elements, CIB (Conseil International du Batiment pour l’Etude), member of the Construction Standards Committee working group (“mirror committee”) “Metal faced sandwich panels”, member of WG/CEN/TC 128/SC 11 “Metal faced sandwich panels”.

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