An Introduction to the History of

Structural Mechanics

Part II

Edoardo Benvenuto

An Introduction to the History of

Structural Mechanics

Part II: Vaulted Structures and Elastic Systems

With 115 Illustrations

Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona

Edoardo Benvenuto Universita di Genova

Ordinario di Scienza delle Costruzioni Facolta di Architettura di Genova

Genova, Italy

Mathematics Subject Classification: Ol-xx, 73xx, 82xx

Library of Congress Cataloging-in-Publication Data Benvenuto, Edoardo.

An introduction to the history of structural mechanics I Edoardo Benvenuto.

p. cm. Contents: v. 1 Statics and resistance of solids-v. 2. Vaulted

structures and elastic systems. ISBN-13:978-1-4612-7751-4 e-ISBN-13:978-1-4612-2994-0 001: 10.1007/978-1-4612-2994-0 I. Structural analysis (Engineering)-History. I. Title.

TA646.B46 1990 624.1 '71 '09-dc20

Printed on acid-free paper.

89-26230 CIP

This work was originally published in Italian by G.C. Sansoni, 1981. © 1991 by Springer-Verlag New York Inc.

Softcover reprint of the hardcover 1st edition 1991

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag, 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any fonn of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general oescriptive names, trade names, trademarks, etc. In this publication, even if the fonner are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly by used freely by anyone.

Text photocomposed using the LATEX system.

9 8 7 6 5 4 3 2 1

ISBN-13:978-1-4612-7751-4

To my mother, Giovanna

Acknowledgments

This book draws its origin from my textbook on the science of structures and its historical development which I published with the Sansoni Publish­ers in 1981 (La Seienza delle Costruzioni e il suo Sviluppo Storieo, Florence, 1981). I am indebted to Professor Clifford Truesdell, who kindly appreciated my attempt to outline a history of the relation between rational mechan­ics and structural engineering and presented my work to the late Mr. W. Kaufmann-Buhler of Springer-Verlag for an English edition. In fact, this book is not a translation of the original. Mr. Kaufmann-Buhler suggested that I transform the text into a general introduction to the history of struc­tural mechanics concentrating on some specific topics and expanding their historical references. I wrote the new text in Italian. I am very grateful to my colleague and friend Prof. Aurelia V. von Germela for having care­fully interpreted my complex academic style and transformed it into fluent English. I thank very much also Mrs. Molly Wolf for her clever and pre­cious final copy-editing, which made my manuscript more dry and light for American preferences. Very useful to me was Dr. Peter Barrington Jones' kind collaboration, and I am particularly grateful to my assistant, Arch. Massimo Corradi, for his beautiful drawings, hand-made in "old style". Finally, I am glad to thank my colleagues Prof. Gianpietro Del Piero and Prof. Paolo Podio Guidugli for their useful suggestions regarding the topics in the first volume.

Foreword

This book is one of the finest I have ever read. To write a foreword for· it is an honor, difficult to accept.

Everyone knows that architects and master masons, long before there were mathematical theories, erected structures of astonishing originality, strength, and beauty. Many of these still stand. Were it not for our now acid atmosphere, we could expect them to stand for centuries more. We admire early architects' visible success in the distribution and balance of thrusts, and we presume that master masons had rules, perhaps held secret, that enabled them to turn architects' bold designs into reality. Everyone knows that rational theories of strength and elasticity, created centuries later, were influenced by the wondrous buildings that men of the sixteenth, seventeenth, and eighteenth centuries saw daily. Theorists know that when, at last, theories began to appear, architects distrusted them, partly because they often disregarded details of importance in actual construction, partly because nobody but a mathematician could understand the aim and func­tion of a mathematical theory designed to represent an aspect of nature.

This book is the first to show how statics, strength of materials, and elasticity grew alongside existing architecture with its millenial traditions, its host of successes, its ever-renewing styles, and its numerous problems of maintenance and repair.

In connection with studies toward repair of the dome of St. Peter's by Poleni in 1743, on p. 372 of Volume 2 Benvenuto writes

This may be the first case in this history of architecture where statics and structural mechanics are successfully applied to a real problem with maturity and full consciousness bf their im­plications. It marks a turning point between two eras: one in which tradition and prejudice ruled the art of building, and another in which the mathematicians' and physicists' new the­ories, elaborated in academies and laboratories, were allowed to make their contribution. It is somehow pleasant to realize that this anticipation of the great nineteenth-century synthesis of

viii Foreword

science and technology came not from an ordinary bit of build­ing but from one of the most daring and beautiful creations of the Renaissance at the height of its splendor.

On p. xx of the introduction

The division between inspiration and technique is of very re­cent origin and is largely artificial. In building, science and art have always been united in the creative act. Not even the most narrow-minded aesthete or engineer can part the two without losing something. To see Brunelleschi, Michelangelo, Guarini, Wren, Mansart, Souffiot, a hundred others, merely as great artists is to deprive them of credit for their brilliant engineering. Their wonderful technical innovations, their perfect determina tion of the weights that had to be balanced and the mechanisms of collapse that had to be opposed-these give coherence and splendor to their works.

The two paragraphs just quoted provide a kind of summary, indeed par­tial, of what Benvenuto wishes to tell us and to let us learn, step by step, not as philosophy or by journalistic simplisms, but by reading expert ob­servations upon a gradual, not always direct history of the science of con­struction. The last paragraphs of his book read in part as follows:

The long, stormy commotion [about the ideas of Menabrea, Castigliano, Crotti, and Mohr] enlivened scientific literature for more than a century. Persuasive hypotheses, even more persua­sive confutations, fruitful but fallacious intuitions, sterile but unexceptionable verdicts, agreements reached unexpectedly­all have been forgotten. What we remember today are the in­struments of engineers, the formulae in daily use. If we asked an engineer about the origins of the equations he or she uses con­stantly, the reply would be disappointing. They exist; nothing else matters. Why be curious about their derivation?

True, the authors with whom we conclude our historical out­line were able to supply such effective technical solutions that, in their hands, the real meaning of the questions they tackled seems to have been lost. But history has its uses ....

Indeed it does, as the reader will learn. Not only is Benvenuto a man of astonishing erudition and breadth, but

also he loves his science and is humble before it. He thinks clearly, clearly organizes his material, difficult and complicated as it seems, and writes clearly with direct and masterly expression. In leafing over or reading his book, we recognize a great work, one doubtlessly flawed by many small errors among several grand truths. Parts of his matter, bit by bit or lacuna by lacuna, may well be corrected or filled by historians in coming decades,

Foreword ix

but his book can never be replaced as a general, pioneering treatise, a survey of a great field heretofore seen only dimly, from a distance, but never trodden. Never before have I learned so much about the history of mechanics from a single book.

As is often the case with books that start from the foundations of a sub­ject, the beginning of Benvenuto's is the part hardest to understand. The reader accustomed to scientific works could well begin with Chapter 5 of Volume 1, "Galileo and his 'Problem''', or with Chapter 8, "Early Theories of the Strength of Materials". Perhaps, even, he might begin with Volume 2, which opens with "Knowledge and Prejudice before the Eighteenth Cen~ tury". Above all, to get an idea of the spread of the work, every reader should study first of all and carefully the two tables of contents, for the titles of the subsections are fascinating. He who is not already expert in both architecture and mechanics will see there some names he has never before encountered, associated to problems or structures or theories he is unlikely to know. In fact, Benvenuto's clarity and directness are such that a reader might start by fishing out some subsections. Any place you open this book and read in it, you will be fascinated by what is there. Wherever you start, for example at the passage first quoted above, I wager you will end by studying the whole book.

Part I of Volume I, although some may profit best from reading it last, is of great value. Very few readers will know already all of the contents of §1.2, "The Enigma of Force and the Foundations of Mechanics". It be­gins with a resume of what should now be regarded as vague meandering, impotent struggles, foolish attempts at reduction, and justified doubt re­garding the nature of force, the first problem "against which science finds itself powerless." It ends with "one of the most important events in the history of mechanics," namely, Walter Noll's organization of the mechan­ics of continua as a mathematical science. There not only is "system of forces" taken as a primitive term, but also it is clarified by a list of its mathematical properties. The theory of systems of forces makes mathemat­ical sense, just as Hilbert's axiomatization of Euclidean geometry in terms of the undefined objects "point", "line", and "plane" makes mathematical sense. That will not stop philosophers from musing about force and his­torians of science from dilating upon old, obscure, unmathematical ideas about force, but it does make "force" something a modern scientist, be he mathematician or be he architect, can use as he does "point", "line", and "plane" . The intuitive notions, both in geometry and in mechanics, remain; not only that, they help both in applications and in creative thought; but the precise concepts stand behind both.

Of course, Benvenuto makes use of secondary works, but also he studies carefully and analyses meticulously the originals to which they refer. It is not unusual-as I can vouch through reading his treatment of some sources that I described too hastily some thirty years ago-not unusual, I say, that in the end he silently corrects the secondary work he has studied.

x Foreword

Benvenuto rightly refers to many Italian sources which are largely un­mentioned in the general literature. As in many other fields, Italians were the great leaders in architecture, structures, and remedies for the appar­ent beginnings of failure. Architects from other countries studied in Italy, and Italian architects designed castles and palaces from Russia to Spain. The Italians were also second to none in theoretical analyses of architec­tural members and assemblies. Failure to study Italian sources directly is a general malady of the precise history of science.

Occasionally Benvenuto refers to a rule or solution of a problem as "cor­rect" or "incorrect". Even the sociological historians, with their belief that the sciences are no more than ephemeral fads, much as history was called by a famous and once powerful man "the lies that men agree to believe," can not justly cavil here, for in architecture the correctly designed arch is one that does not fall except under conditions it was not intended to withstand.

C. Truesdell

Contents of Part II

III

Foreword .. Introduction.

Arches, Domes and Vaults

vii xvii

307

9 Knowledge and Prejudice before the Eighteenth Century. 309 9.1 "A Strength Caused by Two Weaknesses" . . . . 309 9.2 Viviani's "On the Formation and Size" of Vaults . . 311 9.3 Fr. Derand's Rule . . . . . . . . . . . . . . . . . . . 313 9.4 The First "Scientific" Treatment of the Statics of Arches 315

10 First Theories about the Statics of Arches and Domes . . . . . . 321 10.1 Philippe de la Hire. . . . . . . . . . . . . . . . . . . . . . . 321 10.2 Arches and Catenaries: David Gregory and Jakob Bernoulli 326 10.3 Philippe de la Hire's Memoir of 1712 . 331 10.4 Belidor's Variant . . . . . . . . . . . . . 336 10.5 Couplet's Two Memoirs. . . . . . . . . 338 10.6 Bouguer's First Static Theory of Domes 344

11 Architectonic Debates . . . . . . . . . . . . . . . 349 11.1 The Italians: An Introduction. . . . . . . . 349 11.2 The Case of S. Maria del Fiore in Florence 349 11.3 St. Peter's Dome and the Three Mathematicians 351 11.4 Giovanni Poleni's "Historical Memoirs" . . . 358 11.5 Poleni's Theoretical and Experimental Work 359 11.6 Boscovich and the Cathedral of Milan . . . . 371

12 Later Research . . . . . . . . . . . . . . . . . . . 375 12.1 The "Best Figure of Vaults": Abbe Bossut 375 12.2 Coulomb's Theory of Frictionless Vaults. 386 12.3 Coulomb's Theory: Friction and Cohesion. 394

xii Contents

12.4 Italian Studies on Vaults in the Late Eighteenth Century 399 12.5 Lorgna's Essays ....................... 404 12.6 Fontana's Treatise . . . . . . . . . . . . . . . . . . . . .. 407 12.7 Mascheroni's "New Researches": The Limit Analysis of Arches412 12.8 Mascheroni and Domes of Finite Thickness . . . 420 12.9 Salimbeni's Treatise . . . . . . . . . . . . . . . . 425 12.10 The Nineteenth Century: Further Developments 428

IV The Theory of Elastic Systems . . . . . . . . . . . . . 439

13 The Eighteenth-century Debate on the Supports Problem 441 13.1 Introduction .............. 441 13.2 The Birth of the Question. . . . . . . . . . . . . 442 13.3 Discussion in Eighteenth-century Italy. . . . . . 447 13.4 Volume 8 of the Memorie della Societa Italiana . 455

14 The Path Towards Energetical Principles .... 461 14.1 The Debate Continues. . . . . . . . . . . . 461 14.2 The Nineteenth Century: An Introduction. 466 14.3 The Philosopher Who Understood Everything 470 14.4 From Cournot to Dorna . . . . . . . . . . . . . 476 14.5 Clapeyron and the Case of the Continuous Beam. 479 14.6 Menabrea's Elasticity Principle . . . . . . . . . . . 488

15 The Discovery of General Methods for the Calculation of Elastic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 15.1 Clebsch's Treatise and the "Method of Deformations" 492 15.2 Maxwell's Fundamental Memoir on Frames 499 15.3 Maxwell and the "Method of Forces" 504 15.4 The Goal Attained . . . . . . . . . . . . . . 507

16 From the Theory of Elastic Systems to Structural Engineering 513 16.1 Alberto Castigliano ......... 513 16.2 Some Aspects of Castigliano's Work . . . . . . . . 516 16.3 Francesco Crotti's Clarification . . . . . . . . . . . 523 16.4 Mohr's "Beitrage": Statically Determinate Trusses 530 16.5 Mohr's Solution for Statically Indeterminate Trusses 537 16.6 German Disputes about Castligliano's and Mohr's Methods 542

Author Index . 544 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548

Contents of Part I

Foreword .. Introduction.

-vii xvii

I The Principles of Statics . . . . . . . . . . . . . . . .. 1

1 Methodological Preliminaries . . . . . . . . . . . . . . . . . . . 3 1.1 The Special Objects That Gave Rise to Mechanics . . . . 3 1.2 The Enigma of Force and the Foundations of Mechanics. 7 1.3 Statics as "Science Subordinated to Geometry as Well as to

Natural Philosophy" . . . . . . . . . . . . . . . . . . . 14 1.4 Momentum: Fixed Word, Fluid Concept. . . . . . . . 16 1.5 The Aristotelian Roots of a Vocabulary for Mechanics 20 1.6 A Short Outline of Aristotle's Physical Principles. . . 25 1. 7 Modern Metamorphoses of the Immobile Mover: Towards

the Principle of Conservation . . . . . . . . . . . . . . . .. 30 1.8 The "Mechanical Problems": The Peripatetic Explanation

of the Law of the Lever and the Parallelogram Rule . . .. 34

2 The Law of the Lever . . . . . . . . . . . . . . . . . . . . . . .. 43 2.1 Archimedes'Demonstrations................. 43 2.2 Interpretations (and Improvements) of Archimedes' Proof. 48 2.3 An Alternative Approach: Pseudo-Euclid and Huygens. .. 56 2.4 Marchetti's New Approach and Daviet de Foncenex's

Improvements ................. 61 2.5 De la Hire's Proof, Lagrange's Remarks and

Fourier's Contribution . . . . . . . . . . . . . 64 2.6 Towards the "Dethronement" of the Law of the Lever:

Saccheri and de Maupertuis . . . . . . . . . . . . . . . . 67

xiv Contents

3 The Principle of Virtual Velocities . . . . . . . . . . . . . . . . . 77 77 3.1 Medieval Roots ........................ .

3.2 Guidobaldo del Monte, Galileo, and the Principle

3.3 3.4 3.5 3.6 3.7 3.8

of Virtual Velocities ..................... . 80 Descartes: "Explicatio Machinarum Unico Tantum Principio" 85 Bernoulli and Varignon . . . . . . . . . . 88 Riccati's "Universal Principle of Statics" 91 Lagrange's First Demonstration ..... 95 The Approaches of Fossombroni and Fourier 98 The Principle of Virtual Velocities and Constraints: Poinsot's and Ampere's Contributions and Lagrange's Second Proof .............. . .. 105

4 The Parallelogram of Forces. . . . . . . . . . . . . . . . . 116 4.1 Daniel Bernoulli's Claim. . . . . . . . . . . . . . . . 116 4.2 Daniel Bernoulli's First Geometrical Demonstration 119 4.3 Biilffinger's Paradox . . . . . . . . . . . . . . 122 4.4 Riccati's Solution ............... 123 4.5 Foncenex's Memoir and Lagrange's Criticism 126 4.6 Foncenex's Fundamental Lemma . . . . . . . 127 4.7 Foncenex's and D'Alembert's Functional Equation 130 4.8 D'Alembert's Memoir of 1769 . . . . . . . . . . . . 134 4.9 Further Developments: D'Alembert, Poisson, Cauchy,

Dorna and Darboux 136 4.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 141

II De Resistentia Solidorum . . . . . . . . . . . . . . . . 143

5 Galileo and His "Problem" .. 145 5.1 Introduction .. . . . . . 145 5.2 Galileo: A Short Account 147 5.3 The Subtext: Galileo's Atomism 152 5.4 The Primacy of Geometry over Logic

in the Discorsi . . . . . . . . . . . . . 154 5.5 The First Day of the Discorsi . . . . . 158 5.6 Attempts to Explain the Cause of Resistance 163 5.7 For and Against the Power of the Vacuum .: 166 5.8 First Intimations of an Atomistic Theory of Resistance 169 5.9 Democritus or Plato? 173 5.10 The Second Day . 176 5.11 Opening Remarks . . 179

Contents xv

5.12 Corollaries .................... 183 5.13 The Problem of Solids of Ultimate Dimensions 188 5.14 The Problem of Solids of Equal Resistance 194

6 First Studies on the Causes of Resistance . . . . . . 198 6.1 Experimental Confutations: The Horror Vacui 198 6.2 Mersenne and the Problem of Resistance ... 203 6.3 Descartes' Concept: Stasis as the Best Adhesive 206 6.4 The Atomist Rossetti and His Explanation of Resistance 209 6.5 Atomism and Vacuum: Newton, Leibniz and Clarke . .. 217 6.6 Newton's "vis interna attrahens": Elasticity and Resistance 221 6.7 Boscovich's Reformation of the Old Atomism ........ '223 6.8 Developments of Boscovich's Theory: Early Nineteenth-Century

Research on Elasticity . . . . . . . . . . . . . . . . . . . . . 227

7 The Initial Growth of Galileo's Problem . . . . . . . . . . . . . . 233 7.1 Introduction . . . . . . . . .. . . . . . . . . . . . . . . . . 233 7.2 First Steps in the Controversy about Solids of Equal

Resistance: Blondel's "Evidence" . . . . . . . . . . . . . . . 235 7.3 Marchetti's "Evidence" on Solids of Equal Resistance . . . 241 7.4 Marchetti's Axiomatic Approach to the Resistance of Solids 244 7.5 Viviani's "Evidence" . . . . . . . . . . . . . . . . 246 7.6 Antony Terill and Solids of Ultimate Dimensions 252 7.7 Fabri: Elasticity as an "Intermediate Force" . 254 7.8 Pardies' Statics. . . . . . . . . . . . . . . . . . . 257

8 Early Theories of the Strength of Materials . . . 262 8.1 Elasticity Enters the Theory of Resistance. 262 8.2 Mariotte's Contribution . . . . . . . . . . . 265 8.3 Leibniz's New Demonstrations . . . . . . . 268 8.4 New Problems: Catenaries and Elastic Curves 271 8.5 Jakob Bernoulli's Fundamental Work ..... 274 8.6 Varignon and the Galileo-Mariotte Dichotomy 277 8.7 Musschenbroek and the Imperfections of Matter 280 8.8 The Last of the Eighteenth-Century Treatises on Resistance 284

Author Index . 294 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299