Elastic arch bridges

ELASTIC AKCH BRIDGES

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A rather plain yet pleasing example of rib arch design. A flat parabolic rib sprung from solid rock walls.

Rogue River near Gold Hill, Oregon.)

Frontispiece

(Over the

ELASTIC ARCH BRIDGES

BY

c o n d e b

. M

C

Oregon Stale College

A ND

EDWARD S. THAYER

Designing Engineer, Oregon State Highway Department

I K

N E W YORK

JOHN WILEY & SONS,

Lo n d o n: C H APM A N & HALL, Lim it e d

1931

S. 71

S.

S . S <

S.06

Co p y r i g h t, 1 9 3 1 by c o n d e b. McCu l l o u g h

A ll Rights Reserved T his book or a n y part thereof m ust not be reproduced in a n y form without the written perm ission o f the publisher.

Printed in U. S. A.

Printing

F . H . CILSON CO.

BOSTON

Composition and Plates

TECHNICAL COMPOSITION CO.

CAMBRIDGE

Binding

STANHOPE BINDERY BOSTON

PREFACE

In this volume the writers have endeavored to present, as briefly as is consistent with the rather complex nature of the subject matter, a discussion of the mathematical theory of elasticity as applied to the design of arch bridges, with special reference to certain newer develop­

ments in the field of arch analysis. Chapters I to V inclusive (and perhaps Chapter VII) are arranged with a view to utilization not only as a reference work, but also as a text for undergraduate students in structural engineering, whereas Chapters VI, VIII and IX are suitable for graduate student work, or for a reference or office manual.

The greater portion of the text has been developed by the writers from their own personal experience and observation, and illustrated with arch bridges designed and constructed under their direction. This was done not because of any peculiar inherent excellence in these de­

signs, or any particular suitability for purposes of illustration, but because of the fact that the authors were more familiar with these par­

ticular structures and could, therefore, speak with more authority re­

garding the problems involved in their design.

Throughout the volume it has been the writers’ endeavor to develop a definite and coherent method of structural thought rather than to compile a design manual, for which reason very little space has been accorded to a discussion of details, or such problems as pier and abut­

ment design, the design and construction of foundations, and the like, all of which are already more or less completely covered in current American engineering literature. Matters such as loadings, permis­

sible unit stresses, field control and manipulation of materials, and like specification requirements, have received only the merest mention be­

cause of the extent to which data in reference to these are already avail­

able.

Chapter I comprises a general discussion of the arch as a structural type together with a brief history of its development, while in Chapter II will be found a rather comprehensive treatment of those fundamental structural principles which form the groundwork for the rigorous de­

velopment of the elastic theory. Certain portions of the material in Chapter II are perhaps not absolutely essential from the standpoint of arch formula derivation, but it is believed that a complete résumé of this character will operate to give the student a clearer concept of basic elastic relationships than any other method of approach.

iii

i v PREFACE

Chapter III treats of those preliminary steps necessary to an elastic arch analysis and attempts to develop a more general and rational basis for the determination of the true “ linear arch” or axial curve than any hitherto available to the writers.

Chapters IV and V develop the general theory for both fixed and hinged arches, treating the articulated or hinged structure as a special case of the general (fixed) elastic arch. This is a method of approach somewhat new in American engineering literature and appears to the writers to give the operation of articulation (hinging) a new and clearer significance.

Elastic supports and support displacements are treated in Chapter VI, while Chapter IX considers the question of plastic as well as elastic deformations in connection with a discussion of recent European methods for stress control.

Chapter VII comprises a rather detailed discussion of the newer de­

velopments in mechanical stress analysis (analysis from models), a feature of arch design which has assumed a great degree of importance during the past few years and which hitherto has been accorded scant treatment in text-book form.

Chapter V III treats of the mutual stress effects developed in an arch with rigidly connected superstructure — another problem which is demanding an increasing amount of attention on the part of bridge en­

gineers, particularly since the publication of the Yadkin River Arch test report.

Chapter IX , the final chapter, enters the field of artificial stress control and discusses such recent developments along this line as the Freyssinet system of decentering and arch adjustment and the employment of temporary construction hinges for the purpose of eliminating deformation stress effects.

In addition to the methods of analysis developed in this volume, the designing engineer will doubtless encounter many other special prob­

lems peculiar to arch design and construction, such, for example, as the effect of a variation in the modulus of elasticity of the material on re­

sulting rib stresses, the effect of plastic or time flow in concrete, the analysis of skewed barrel arches, and similar problems. Space has not permitted a discussion of these special cases, but for convenient reference a few of the principal problems likely to be encountered by the designer have been listed in the Appendix, together with a brief digest of recent engineering literature pertinent to the same.

C . B. McCu l l o u g h

E. S. Th a y e r Salem, Oregon, M arch, 1931

CONTENTS CH A PTER I I n t r o d u c t i o n

p a g e

1. The arch defined... 1

2. Brief history of the arch as a structural u n it... 4

3. Brief description of modern arch bridge typ es... 14

4. C onclusion... 21

C H A PTER II RissuMli o p Fu n d a m e n t a l Th e o r y 1. Introductory... 33

2. The theory of structural w ork... 34

3. C asligliano’s theorem ... 36

4. Derivation of expressions for internal w ork... 40

5. Deflection formulas from Castigliano’s theorem ... 43

6. Structural redundancy as applied to arch an alysis... 46

7. Internal redundancy... 52

8. M ulti-span arches on elastic piers... 53

9. Effect of yielding foundations... 56

C H A PTER III Pr e l im in a r y Ar c h De s ig n 1. Introduction... 59

2. Quantities required for a statically indeterminate solution... 59

3. Deadload of superstructure... 60

4. Deadload of arch... 60

5. Variation in section for rib or barrel arches... 63

6. Live loading; highw ays... 65

7. Live loading; steam railroads... 65

8. Live loading; electric railw ays... 66

9. Live loading; sidew alks... 66

10. Im p a ct... 66

11. M oment of in ertia... 69

12. Modulus of e la s t ic it y ... 71

13. Arch alignm ent... 72

14. Curvature of ancient and medieval arches... 74

15. The first principle of arch actio n ... 75

16. Passing a force polygon through given p oin ts... 76

17. Equilibrium polygon: graphic m e th o d ... 78

18. Algebraic method of determining the equilibrium polygon... 80

v

P A G E

19. Placing the arch on the polygon... 82

20. Calculation of the median a x is... 84

21. D evelopm ent of the true axial curve... 85

22. Calculation of the axial curve (multi-parabolic a x is ) ... 86

23. Temperature effects... 89

24. Conclusion and sum m ary... 90

C H A PTER IV An a l y s iso f Fi x e d Ar c h e s 1. Introductory: g e n e r a l... 98

2. Introductory: rib arch es... 104

3. Determination of the elastic center... 105

4. Determination of conjugate redundant a x es... 109

5. Evaluation of deflection term s... 110

6. Special procedure for symmetrical sp an s... 122

7. General observations regarding rib arch d esign ... 123

8. Summary of procedure for rib arch analysis with illustrative arch design. 125 9. Adaptation of fundamental theory... 141

10. Outline of method of an alysis... 145

11. Special procedure for symmetrical spans... 148

12. Effect of web distortions... 148

13. General observations... 153

C H A PTER V Hin g e d Ar c h Br id g e s 1. Introduction... 155

2. Single hinged rib arch (general c a se )... 155

3. Temperature stresses, rib shortening and shrinkage... 161

4. Special method for symmetrical arches... 163

5. Analysis of single hinged framed arches... 163

6. Development of stress influence lin es... 163

7. Effect of axial distortions... 163

8. Ordinary case: Two skewback hinges... 166

9. Temperature, shrinkage and rib shortening... 170

10. Effect of axial distortion... 170

11. Two hinged framed a r c h e s... 171

12. General... 172

13. Determination of load reactions... 173

14. Influence lines for three hinged arches... 176

15. Load divide lin es... 178

16. Arch influence lines from deflection diagram s... 179

17. Influence lines determined from deflection polygons for a three-hinged braced spandrel arch... 182

18. General... 186

19. Arches with two intermediate hinge p oin ts... 187

20. Arches with three intermediate hinges... 191

21. Conclusion... 192

v i CONTENTS

C H A PTER VI

Mtjlti-s p a n Ar c h e s o n El a st ic Pi e r s

P A G E

1. Introductory... 194

2. Derivation of fundamental principles... 195

3. An arch group elastic over three spans... 196

4. An arch group elastic over “ N ” sp an s... 207

5. Temperature, rib shortening and shrinkage stresses... 208

6. Approximate solutions for multi-span elastic groups... 211

7. Complete analysis of a three-span rib arch group on elastic intermediate piers... 211

8. General effect of elastic pier displacem ents... 228

9. R eliability of algebraic method for multi-span analysis... 233

10. The ellipse of elasticity defined... 233

11: General properties of the ellipse... 237

12. Basic theory of the ellipse of ela sticity ... 241

13. Instantaneous centers... 242

14. The instantaneous center as an antipole... 243

15. Analysis of a multi-span arch system on elastic piers (general ca se) 247 16. Graphical method for determination of pier and rib reaction components. 259 17. Application of the ellipse to framed arches... 261

18. Special case of symmetrical end sp an s... 261

19. G eneral... 262

C H A PTER V II Me c h a n ic a l Me t h o d s o f St r e s s An a l y s is 1. Introductory... 269

2. N eed for a mechanical m ethod... 269

3. D evelopm ent of mechanical m ethod... 270

4. Theory of mechanical an alysis... 271

5. Hinged supports... 273

6. Derivation of theory of mechanical analysis from M axwell’s theorem 273 7. Internal stresses determined b y mechanical m ethods... 274

8. Relation between deflection diagrams and influence lin e s... 275

9. Use of models in elastic analysis... 276

10. Types of apparatus in common u s e ... 277

11. The flexible sp lin e... 278

12. Wire m odels... 279

13. The Gottschalk continostat... 280

14. The Beggs deformeter (general)... 282

15. The Beggs deformeter (continued — calibration of the p lu gs)... 286

16. Technique of deformeter operation (general)... 288

17. The same — figuring and making the m odel... 288

18. The same — mounting the m odel... 290

19. T he same — targets and microscopes... 294

20. T he same — readings and tabulations... 295

CONTENTS v ii

PAGE

21. The same — magnitude of errors... 297

22. Temperature stresses... 300

Maxwell’s law of reciprocal displacem ents... 305

C H A PTER V III Ar c h e s w it h Rig id l y Co n n e c t e d Su p e r s t r u c t u r e s 1. Introduction... 307

2. General view of problem... 307

3. Exact solution outlined... 309

4. Articulation of superstructure... 311

5. Mechanical analysis of superstructures... 313

6. Calculation of stresses in a rigidly connected superstructure... 316

7. M ovement of any point on an arch ... 316

8. Column stiffness... 318

9. Illustrative problem ... 322

C H A PTER IX Th e Fr e y s s iNe t a n d Ot h e r Re c e n t Eu r o p e a n Me t h o d s f o r St r e s s Co n t r o l i n El a s t ic Ar c h Ri b s 1. Introductory... 325

2. The method of construction h in ges... 330

3. The method of arch compensation and adjustment (Freyssinet m ethod). . 336

4. Calculation of arch compensations... 338

5. Calculation of arch adjustm ent... 340

6. Deflections and elastic displacements in arch rib s... 343

7. Summary and conclusions regarding Freyssinet method for single sym ­ metrical sp an s... 346

8. The Freyssinet method applied to multi-span arches on elastic piers (two- span group)... 347

9. Adjustment of a three-span group... 352

10. Adjustment of unsymmetrical spans and span arrangements... 354

11. Details of jacking procedure employed for the Rogue River Bridge in Oregon... 357

A P P E N D IX 1. Introduction... 359

2. Skewed arches... 359

3. Approximate arch formulas... 359

4. The elastic modulus of concrete... 362

5. Effect of a variation (with stress) of the elastic modulus of concrete upon arch rib design... 362

6. Effect of assumptions regarding moment of inertia on arch analysis 362

7. Plastic or time flow in concrete... 363

8. Temperature and shrinkage effects... 363

9. Design of arch rib or barrel sections... 364

10. Other arch design and construction problem s... 364

v iii CONTENTS

ELASTIC ARCH BRIDGES

CHAPTER I INTRODUCTION

1. The Arch Defined. — Throughout this volume we are going to discuss arches, and perhaps the logical way to begin is by finding out just what it is that we are talking about. What is an arch? Simple enough — yes, perhaps so; but one of the writers remembers not too long ago listening to the cross-examination of an expert witness who should have known all about arches (and undoubtedly did know a great deal about them) but whose ideas as to just what constituted an arch were apparently somewhat hazy.

What was an arch?

What was the difference between a curved beam and an arch?

Did an arch need to be curved to constitute an arch?

Was there such a thing as a straight arch?

Was a fixed beam in reality a straight arch?

If the thrust in an arch increased in value as the rise decreased, where did it go when the rise became zero?

And so on ad infinitum (nearly).

Some of the questions may appear pointless, meaningless and rather silly. In this particular case, however, the cause of suit was an alleged patent infringement, and the point at issue hinged upon the construction of the term “ arch.” At the outset it appeared that the witness was the only participant party whose conception of the term was at all clear, but at the finish it was conclusively apparent that neither counsel, court nor witness retained any definite ideas on the subject.

The term arch has, in many instances, been rather loosely applied, both in American and European engineering literature. Perhaps this is because the engineer is more interested in facts, figures and results than he is in words. The legal profession, however, deals extensively in words (we do not mean to be sarcastic), for in certain litigation the construction of technical terms becomes a vital issue. Such construction determines whether the instant case is within or without the purview of the statute.

2 INTRODUCTION

Exact definitions in such a case become very necessary, and clear-cut terminology all important.

For this reason, therefore, a little time may well be spent right here in order to evolve a terse, yet comprehensive and clear-cut definition of the term “ arch.”

Perhaps the arch as a structural unit may be best defined or set apart from all other structural members by the following two criteria:

(a) It must be sustained by supports, all of which are capable of developing lateral as well as normal reaction components. >>

(&) It must be of such shape that these lateral reaction com­

ponents are, in fact, developed under load, and these must, in general, constitute thrusts rather than pulls. That is to say, the reaction components must act inwardly, thrusting against the arch, rather than outwardly, pulling away from it.*

Let us look into these cri­

teria just a little further.

Consider the beam (Fig. la) supported at Ri by a fixed shoe and at R 2 by a roller nest.

Under the action of any given load system, the support at R2 moves freely in a plane parallel

Fig s. la to lc. to the roller bed and only nor­

mal reactions can be developed.

Ri will, of course, also be normal unless the loads are inclined. The structure shown at Fig. li>, although curved in axial form, is also a beam and not an arch, since reaction R2 can never assume any direction except that normal to the roller bed. These then are simple beams, straight beams, curved beams or beams of whatever shape, but not arches.

In Fig. lc the roller nest at R2 has been replaced by a fixed shoe.

Under load this structure deflects downwardly — shortens axially and thrusts laterally against its supports. The reactions Ri and R2 are now inclined and may be replaced by their respective horizontal and

* An apparent exception to th is hist sta te m en t occurs in th e case of th e so-called

“ tied arch,” although , as a m atter of fact, th is ty p e of structure can hardly be classi­

fied as a true arch typ e.

THE ARCH DEFINED 3 vertical components Hi, V h H2 and V2 as shown. This structure is clearly an arch — a two-hinged arch type.

In Fig. Id is shown a structure wherein the supports have been fixed against rotation as well as translation. In this case lateral thrusts are again developed, but unlike the

case shown in Fig. lc the point of ,

application of these thrusts is (or may be) at some point other than

the neutral axis of the rib. B y in- x*?

troducing two equal and opposite forces R2 at the neutral axis of the

rib at the right support (Fig. le), the reaction R 2 may be resolved into three components, namely, H2, V2 and M 2 = (R2a). Similarly at the

left support, reaction components H u Vx and M i are developed (Fig.

1/) . This structure is therefore an arch — a fixed or hingeless arch type.

The structure shown at Fig. 1 g is of such shape that as it deflects under load it elongates axially^exerting a pull rather than a thrust against its supports. This structure, therefore, from our definition is not an arch type, but partakes of the nature of a rigid — fixed ended — sus­

pension system. It will be observed that in passing from the shape in­

dicated in Fig. Id to that shown in Fig. lg, the lateral thrusts or hori­

zontal reaction components change direction. There must, therefore, be some point at which these pass through a zero value, as shown in Fig. lh. In this latter case the reaction components constitute only a shear and a bending moment at each support and the structural type developed is a fixed beam — not an arch.

The arch is usually curved in form but from our definition is not necessarily so. Fig. Ik is a two-hinged arch with a polygonal axis.

Fig. 1 m is of a structural unit that looks very much like an arch.

Here we have an inclined reaction at each support, to be sure; but is it really an arch after all? Let us look closely at our definition. Are both supports capable of developing lateral as well as normal reaction components? Obviously no. Ri will, of course, vary as regards its angle of inclination with varying loads, but R2 must always be normal

F ig . le. F ig . If.

4 INTRODUCTION

to the inclined roller bed. There are only three unknown reaction com­

ponents, therefore, and the structure is fully determinate from statics as is seen by the equations (Fig. bn ). It is true, of course, that the

R z = F a /y

internal stresses induced constitute axial thrusts as well as bending moments, but this is not a criterion. The same is true of any inclined beam under vertical loading. N o — Fig. b n is only a simple beam, after all.

The foregoing explanation should completely clarify our definition and criteria for the true arch type, namely:

(а) Supports all capable of sustaining lateral thrusts.

(б) An axial shape such that said thrusts are, in fact, developed under load.

We may now go back and submit to the cross-examination indicated on page 1. Learned counsel will not be able to tangle us up. We now know more about it than he does.

2. Brief History of the Arch as a Structural Unit. — The statement is frequently made that the arch is as old as civilization. According to our definition of the arch, the above statement is not strictly true.

It may be of interest, however, to trace briefly the development of this structural unit, step by step, from earliest historical time down to the present.

BRIEF HISTORY OF THE ARCH AS A STRUCTURAL UNIT 5

Pre-Roman Arches

The ancient Egyptians employed the arch as an architectural rather than a structural unit, the arch construction used in the Pyramids at Gizeh (3000 to 4000 b . c . ) being of the type commonly designated as the

“ corbeled arch,” which in reality is not an arch at all, but a series of superimposed cantilevers with a simple slab span closure. There seems to be one example of an early Egyptian arch (about 1500 b . c . ) that par­

takes of the nature of the true arch (“ voussoir” type), this being dis­

covered in a tomb at Thebes. There is also a record of an arch of the voussoir or block type constructed in Ethiopia which may antedate the Thebes Arch; in fact, the Egyptians may have secured their knowledge of the block arch from the Ethiopians. These early arches, however, were employed only in a minor way, generally for architectural effect in connection with columns and lintels, and with perhaps a few excep­

tions were undoubtedly not true arch types. It is said that the Egyp­

tians feared the arch because of the lateral thrust components developed under dead load. They had a saying “ the arch never sleeps ” from which it is apparent that they possessed at least a crude knowledge of its action and for some reason believed that the steady unceasing applica­

tion of thrust would eventually cause failure of supports.

The Assyrians and ancient Chaldeans also employed the corbeled arch to a considerable extent and perhaps occasionally used the true arch as well. Some of the pointed brick arches built by the Babylonians in connection with their sewers date back to 1300 ^{b . c .} and perhaps several thousand years earlier. The river Euphrates was spanned by a series of brick arches as early as 2000 b . c . , but whether these were of the true arch type or consisted of mud brick corbeled arch construction is not known.

The Hindus employed the arch as did the Chinese, but the dates of their early structures are not known.

The prehistoric Greek tribes, particularly the “ Pelasgi,” a rather virile horde from Syria and Asia Minor (about 1800 b . c . ) , built exten­

sively in huge masonry. Because of the vast proportions of the in­

dividual blocks, the later Greeks, thinking these early structures were (or at least should have been) the work of giants, termed the construction

“ Cyclopean” masonry, which term applied to concrete construction with an interbedding of large stone persists to this day. In this early Greek masonry are several examples of the corbeled arch.

None of the above examples (with the few exceptions noted) consti­

tuted the true arch type, and it remained for the Etruscans, a people of Asiatic origin who entered northern Italy about 1300 B .C ., to use the

6 INTRODUCTION

true arch to an appreciable extent. These early Etruscan arches were perhaps rather crude examples and unimportant in themselves, yet were nevertheless tremendously important from an historical standpoint as a factor of influence in the later architecture of Italy.

The struggle between these Etruscans and the peoples of southern Italy (the forerunners of the Roman empire) makes interesting reading.

One of the earliest Roman bridges (the Pons Sublicius, which inciden­

tally was not an arch bridge) became immortalized as the scene of a battle between the Romans under Horatius Codes and Lars Porsena, the Etruscan leader (598 B .C .).

The Etruscans were finally vanquished, south Italy prevailed and Rome’s architectural- influence spread with her governmental and terri­

torial acquisitions. The Byzantian system — a rather spectacular and truly beautiful arrangement of groined arches and domes of which the Mosque of Hagia Sophia in Constantinople built under Justinian

( a . d . 532) is an outstanding example — is a Roman offshoot.

A dozen centuries later came the Gothic with its beautifully pointed arches and flying buttress construction; later the Renaissance with its reversion to the earlier classical types, but all of them carrying out the ancient Etruscan principle.

The Arch During the Roman Period*

The first real employment of the true arch type was therefore in all probability Roman or pre-Roman (Etruscan), and most assuredly to these great builders of antiquity belongs the distinction of having first utilized the arch to an appreciable extent in bridge construction. One of the very early examples of Roman arch bridge construction and con­

temporary with the Cloaca Maxima (the great arched sewer) may be mentioned the Pons Solarius, built across the Teverone (a tributary of the Tiber) about 600 years before the birth of Christ and in the reign of Tarquin, himself an Etruscan. The great era of Roman bridge building, however, did not begin until two or three centuries later and extended until about a . d . 200. The Pons Palatinus (181 b . c . ) consisting of semicircular arch spans of about 80 feet, the Pons Fabricus (62 b . c . ) —

nearly all of which structure is still standing in its original condition — the Pons Augustus (20 b . c . ) and the Pons Aelius built by order of the emperor Hadrian ( a . d . 138) are examples of great bridges constructed during this period. One of the longest arch bridge spans built during

* Manj' of the dates and data in the paragraphs which follow are extracted from Chas. S. W hitney’s “ Bridges ” (Win. Edwin Rudge, Publisher, N ew York), a most delightful and comprehensive treatment of the history of the Bridge, to which the reader is referred for a more complete story of the progress of the art.

BRIEF HISTORY OF THE ARCH AS A STRUCTURAL UNIT 7

Th e Se g o v ia Aq u e d u c t

An excellent example of the dry stone masonry built b y the Romans.

8 INTRODUCTION

this time was the one at Narni across the Nera during the reign of Augus­

tus. The height is said to be considerably over 100 feet and the longest span about 110 feet in length.

The famous Roman aqueducts also employed the arch effectively and very beautifully. The Pont du Gard at Nîmes, France (19 b . c . ) (a majestic example of this class of construction), is a triple-deck affair of colossal proportions — over 150 feet from deck to waterline; six arches in the lower arcade having span lengths of from 65 to 80 feet, eleven arches of like span in the second arcade, and thirty-five short- span arches in the third or upper one, complete this lasting monument to the skill of these early builders. And there was the double-deck aqueduct at Segovia in Spain so carefully wrought as to give rise to a legend, associated with it at a later date, to the effect that it was super- naturally built — not the product of human hands.

The Roman arch structures were built of cut stone, of brick and of stone shells with concrete filling; but the outstanding examples, those whose resistance to the ravages of time mark them as objects of wonder­

ment and admiration, were of cut stone voussoirs laid without mortar — dry stone masonry — carefully, wonderfully constructed ! It is small wonder that the superstitious Segovians wove tales of m ystery and magic about such construction; small wonder indeed that the repairs and reconstruction work done upon these structures centuries later proved painfully short-lived by comparison; small wonder that even today the excellence of this stonework sets a mark hard indeed to equal ! A happy consummation indeed if those present-day builders whose fetish is low first-cost construction, no matter how tawdry or unbeautiful, could take a leaf from the book of these great construction engineers! B ut we must get back to our story.

The Arch During the Dark Ages

The Roman Empire, top-heavy from expansion and undermined at its foundations by internal intrigue and a general decay of moral fiber, finally collapsed of its own gigantic weight, and with its decline and final fall artistic and intelligent bridge construction came to a stop.

For nearly ten centuries the history of the art is more or less a blank, devoid of interest. In 1176 Peter of Colechurch began the construction of the famous London Bridge — an ugly monstrosity of a thing with short-span arches partially damming up the river. “ London Bridge is falling dovm, etc.” — so goes the old rhyme; but it didn’t — for over six hundred years. Rather it continued to offend the esthetic instinct of the natives, if they had any — a strange contrast indeed to the Pont du Gard of twrelve centuries earlier.

BRIEF HISTORY OF THE ARCH AS A STRUCTURAL UNIT 9 In the twelfth century certain religious orders known as “ Fratres P on tes” or “ Frères du P o n t” (Brothers of the Bridge) came into being.

The first “ Fratres Pontes” was probably a Benedictine order. These were originally bands of devout men who sought to revive the art of bridge building or at least to repair and reconstruct some of the ancient bridges which in many instances were sadly in need of maintenance.

These orders were able, in many instances, to secure charters conferring upon them the right to collect tolls on certain bridges as a method of financing their work, and in certain cases they not only repaired and constructed bridges, but erected upon the adjacent banks hospices or shelters for the care of travelers. These orders, however, were not long to last, being replaced by organizations formed for the purpose of col­

lecting tolls for private profit; fortifications replaced the original hos­

pices, and travelers were oppressed with exacting and exorbitant tolls — were, in fact, in m any cases robbed and outraged until the bridge be­

came a thing of terror, thoroughly consonant with the spirit of the times.

Very little structural development marked this period. The arches of the Romans were of the full centered or semicircular type, but the medieval period witnessed the introduction of the pointed Ogival or semi-Gothic arch, which type is thought by some to be an importation from Persia brought back by the Crusaders. A segmental circular arch curve was also used during this period (the Ponte Vecchio over the Arno, Florence, 1345) and in one case (the Pont d’Avignon), an elliptical curve with its major axis vertical. The fourteenth century also wit­

nessed, for the first time, the construction of a masonry arch bridge having independent ribs supporting a flat masonry cover slab. N ot all the bridges of this period were ugly (as was the London Bridge). In fact, during the latter part of this age a number of rather beautiful structures were erected. The foundation construction was possibly somewhat superior to that of the Romans, although even the best of these foundations were more or less haphazard, judged by our present- day standards, consisting, in general, of piles of stone dumped into the waterway to make a great mound upon which to found the piers.

Cofferdams or scientific underwater construction of any kind was as yet unknown, and it was only because of their great mass that these piers and abutments were at all stable. Typical of the two chief at­

tributes of this age (military activity and religious zeal) we find the medieval bridge characterized by the placement of chapels and profuse works of defense not only at each bank but many times upon the deck of the bridge proper. The engineering skill of the Romans was no­

ticeably absent; the piers were in many cases located hit or miss across the river, apparently the structural plan being developed from day to

10 INTRODUCTION

day as the work progressed. With the possible exception of the founda­

tion work the construction of this period was not at all up to the Roman standard.

We must not pass by, however, without at least mention of the Pont d ’Avignon, begun in 1178 by a young builder named Benoit. He died before the bridge was completed and was later canonized under the title Saint Bénézet. Perhaps even notwithstanding our scientific progress, the bridge engineering profession has not gained in social and spiritual prestige. Eight centuries ago he was sainted and sent to Heaven;

today he is very frequently requested to take a diametrically opposed path, if our theological geography is correct — but again we must get on.

The Arch of the Renaissa?ice

The Renaissance, marking, as it did, a general revival in art and architecture, saw the construction of many really beautiful stone masonry arch bridges. This era was characterized by architectural rather than engineering development, the application of scientific design principles coming at a later date. In general the period witnessed the adoption of much lighter piers than formerly (about one-fifth the span length). Even these piers, however, were proportioned to carry the entire arch thrust from one single adjacent arch acting alone, the bal­

ancing of lateral thrust components and the simultaneous striking of centers for two arches meeting at a pier being a development of a later time. The use of the fiat elliptical arch, the employment of lower roadway gradients avoiding the high humped-back effect of a former age, the employment of foundation piles, the use of cofferdams and, in one case (the Pont Royal, 1685-1687) possibly the use of the caisson, the employment of dredging and the use of Puzzolana cement are among the more notable innovations of this period.

Among the important structures of the Renaissance mention may be made of the Ponte S. Trinita over the Arno, Florence (1567—1570) ; the Ponte de Pierre at Toulouse (1543-1632) and the Pont N euf at Paris (1607).

The Modern Period

The modern period in arch bridge building may be said to have had its beginning in the formation (in 1716) of the French “ Département des Ponts et Chaussées.” In 1763 Perronet, one of the greatest bridge engineers of the time, discarded the use of abutment piers and, for the first time in the history of the art, made use of the principle that the lateral thrust components for two arches meeting at a pier m ay be made to balance (or nearly so), leaving the pier under the action of vertical

BRIEF HISTORY OF THE ARCH AS A STRUCTURAL UNIT 11 loads only; thus the piers of the Renaissance were cut from a width of one-fifth to about one-tenth of the adjacent span length. During the early years of this period the names of Chezy and Gauthey occur as noted bridge engineers.

Some of the stone arches built at this time were unusually flat, one arch having a rise of only one-seventeenth the span. Foundations were carried to rock by means of cofferdam construction, and Gauthey em­

ployed a method of placing concrete under water around the tops of cut-off piles somewhat similar to present-day practice. In 1755 an iron arch bridge was cast, but not erected, at Lyons, France, and in 1776 the first cast-iron arch bridge was built over the river Severn at Coalbrookdale, England. In 1819 a cast-iron arch of 240-foot span was built across the Thames. In 1783 iron was first rolled into struc­

tural shapes and in 1828 steel was first utilized for bridge work.

The Advent of the Elastic Arch

All the arch types above described were of the “ voussoir” or arch block type as distinguished from the modern monolithic or elastic struc­

ture. There has been a certain amount of confusion as regards the term “ elastic arch,” for which reason a few words of explanatory matter may be in order. The early voussoir arch consisted of a series of blocks either with or without mortar joints. The individual arch blocks were termed “ voussoirs,” and, as will be seen from Fig. 2, were held in equi­

librium under the action of a lateral crown thrust T, which was un­

known in direction, in amount and in point of application. Once this crown thrust were determined it could be combined with the various active external forces (dead or live) to form a line of axial pressure (as indicated in Fig. 2). The value, direction and point of application of the crown thrust T were determined by means of one of several empiri­

cal or semi-empirical formulas. This, in substance, constituted the

“ voussoir” or fine of pressure theory. The early stone arches were undoubtedly built purely by rule of thumb in accordance with no theory at all but as bridge building emerged from the condition of a skilled trade to that of a science, it was this sort of a theory which was first introduced and used.

In contradistinction to this method of analysis, the “ elastic theory”

assumes the entire structure as a monolithic elastic unit, and the true crown thrust is determined by means of equations which take into con­

sideration the deflection or elastic distortion of the material under load.

The line of pressure theory was independent of the particular structural material used; in other words, a cast-iron arch analyzed by this theory

12 INTRODUCTION

•would show the same pressure line under equivalent load as one of concrete or stone, whereas the elastic theory, depending as it does upon the distortion of the arch under load, can not be divorced from a con­

sideration of the elastic properties of the component structural material.

Stone arches of the voussoir type are equally susceptible to analysis by means of the elastic theory, and this theory is the correct one provided the tensile stresses at the joints do not exceed the ultimate. If this latter condition occurs, however, there results a spreading at the joints, the moment of inertia of the rib is decreased at these points, the elastic distortion under load becomes indeterminate and the method fails.

The elastic theory is today the universally accepted method for arch analysis. Probably the first arch span of monolithic type was a wrought- iron foot-bridge built over the river Crou at St. Denis in 1808; however, it was not until 1879 that Weyrauch demonstrated the fundamental equations upon which the elastic theory is based. About eleven years later the Austrian Society of Engineers and Architects conducted an exhaustive series of arch tests, publishing the results thereof in the form of a very complete report, the sum and substance of which was that this new elastic theory furnished the only rational and correct basis of design for arch structures.

The advent of reinforced concrete made possible a great advance in the art of arch bridge construction. Here was a material capable of

BRIEF HISTORY OF THE ARCH AS A STRUCTURAL UNIT 13 withstanding tensile as well as compressive stress, and the uncertainty regarding temperature and shrinkage effects no longer operated to limit the span length and restrict the scope of utility of masonry arch spans.

The invention of this type of construction is generally conceded to Joseph Monier of Paris, who built his first arch bridge in 1867. In the year 1894 F. Von Empergcr introduced the “ M elan” system (which em­

ployed rolled I beams for reinforcing) into the United States and built the first reinforced-concrete arch bridges of considerable span. Edwin Thatcher was also a pioneer in this work, building concrete arch struc­

tures utilizing bar reinforcement. One of the early examples of rein­

forced concrete arch construction in the United States was an arch bridge constructed in 1893 near Rock Rapids, Iowa. This span was of the Melan type with concrete body and a facing of rough-cut Sioux Falls quartzite.

From its somewhat modest beginning, the reinforced-concrete arch has rapidly extended its field of utility until it constitutes today one of the most important masonry bridge types in use, particularly for high­

way bridge structures. The early concrete barely attained a com­

pressive strength, at 28 days, of 2000 pounds per square inch. Today it is a comparatively easy matter (owing to the great strides made in the art of proportioning, mixing and general field manipulation) to obtain concrete which Mil run uniformly above^4000 pounds per square inch at the same age. Tests and experimental work have established values for temperature and shrinkage effects which were hitherto matters of conjecture, and the art of cold-weather concreting has been brought to a stage where no difficulty need be apprehended the year around. The development of mechanical methods for stress analysis, such as the Beggs “ Deformeter ” and the “ Continostat ” of Gottschalk, has re­

moved, to a certain extent, the difficulties surrounding the determination of the effect of slender elastic piers and has also made it entirely possible and feasible to evaluate the effect of the superimposed superstructure (spandrel columns, beams and deck) upon the elastic distortion of the arch rib proper.

Lastly, the development of construction hinges and the art of stress control by means of hydraulic jacks bids fair to increase still further the economic lim it of span lengths. Freyssinet, the noted French bridge engineer, is very optimistic as regards the possibilities along this line.

Speaking of his own method of decentering by means of a battery of hydraulic jacks, he states as follows:

“ As a whole, the methods which I have explained permit a con­

siderable reduction in the maximum stresses in arches . . .

14 INTRODUCTION

I consider that the construction of a concrete arch of more than 500 meters [1625 feet] opening presents no difficulties which cannot now be overcome.”

The modern period marked the use, for the first time, of iron (and later steel) in arch bridge construction, and during the latter part of this period great advancement has been made in the art.

The increasing development in the manufacture of high-strength alloy steels and recently the possibility of an aluminum alloy of com­

bined high strength and light weight, the possibility of weight saving through the use of arc-welded joints at least on secondary connections

— all these bid fair to extend greatly the economic and feasible limit of span length in metal arch design.

In 1910 Charles Worthington proposed a single-hinged voussoir type arch for the St. Lawrence River crossing at Quebec. The voussoirs were of steel and the span length 1800 feet. This structure was not built, but the design appears entirely feasible. The Kill van Kull design has a span length of 1650 feet, and it seems quite probable that even these limits can be greatly exceeded.

The Freyssinet system of decentering and the newer and more scien­

tific methods of stress analysis are not confined to masonry structures, but will undoubtedly play an important part in extending the field of utility of metal arch structures as well.

And thus from the days of Cheops and Chephren to the present, a span of fifty centuries or perhaps more, has the arch paralleled the development of civilization.

3. Brief Description of M odem Arch Bridge Types. — In the fore­

going article we have briefly outlined the development of the arch as a structural unit. The present article will be devoted to an enumeration and brief discussion of the various arch bridge types in common use at the present time. We shall group this discussion as follows:

(а) Materials of Construction

(б) Structural Arrangement of Arch Proper (c) Stress Distribution in Arch Proper (d) Methods for Supporting Superstructures (e) Shape or Curve of Arch.

Ma t e r i a l s o f Co n s t r u c t i o n

As stated in the preceding article, arch bridges have been built in timber, brick and stone masonry, cast iron, wrought iron, steel and concrete.

BRIEF DESCRIPTION OF MODERN ARCH BRIDGE TYPES 15 Timber arch bridges are rather rare at the present time although now and then a rib arch is built utilizing either heavy dimension timbers or laminated ribs composed of deep, narrow plank material. Braced spandrel or open frame arches may be constructed in timber, utilizing the same general arrangement of framing details as those for the ordinary Howe truss. All in all, however, the timber arch can hardly be classified as a common modern bridge type. Typical of this class of construction, although of interest to us only from an historical standpoint, is the record of a timber arch bridge built about ^{a . d .} 104 over the Danube in Hungary by order of the Emperor Trajan. The span length was somewhat over 150 feet, and the structure was sprung from cut stone piers.

The early brick and stone arches have rapidly given way to the modern concrete structure. Where stone or brick facing is considered desirable for architectural effect the same is now generally added in the form of a veneer placed upon and supported by a concrete skeleton frame designed to carry the entire load.

Arches of iron have been superseded by the modern steel structure, which type both in the form of ribs (fixed and hinged) and the spandrel braced or open frame design is utilized to a very great extent in the bridge construction of today. One of the early American examples and a struc­

ture that still ranks as one of America’s great bridges is the Eads Bridge at St. Louis, built out of chrome steel. This structure, which was opened to traffic July 4, 1874, consists of three fixed arch spans (one at 520 feet and two at 502 feet) carrying a double-track railway on the lower deck and highway traffic over the upper one.

The concrete arch constitutes the latest development in the art and a type of extreme importance in the bridge construction of the present period. It has supplanted the stone and brick voussoir types because of the inability of these latter types properly to resist tensile stresses and also because of the great reduction in first cost. The advancement made in the art of surface treatment for concrete renders it possible to secure remarkably pleasing and harmonious architectural effects, and the great accumulation of scientific data concerning the distribution of rib stresses together with the development in the art of controlling and caring for the same render it quite probable that this type will be able to compete with other structural types for increasingly longer span lengths.

In summary, therefore, it appears that the concrete arch and the steel arch constitute the only arch types remaining to any considerable ex­

tent in the bridge construction of today. For short-span structures, the concrete arch has nearly an exclusive field; for intermediate span lengths the types are competitive; whereas for very long spans the steel

16 INTRODUCTION

arch, particularly when utilizing a high-strength alloy, will doubtless have the field pretty much to itself for many years to come.

S k e w b a c k

indicates these two arrangements and also some others that are some­

times employed in concrete.

In the solid rib type (Fig. 3b) the load stresses are carried to the sup­

ports by two or more independent solid ribs generally connected to-

St r u c t u r a l Ar r a n g e m e n t o f Ar c h Pr o p e r

(Arrangements in Concrete Arches)

The arch proper, in concrete construction, usually consists either of a series of independent solid ribs or else a single solid barrel. Fig. 3

R ib A r c h S e c tio n B a r r e l A r c h S e c ti o n F ig . 3 - c

H o llo w R ib S e c tio n F ig . 3 - d

Fig. 3a.

F i g . 3 - e O p e n W e b C o n c r e t e R i b

Figs. 3b to 3e.

gether by means of suitable cross struts or lateral bracing. At Fig. 3c the independent ribs have been replaced by one single solid section.

BRIEF DESCRIPTION OF MODERN ARCH BRIDGE TYPES 17' This is the “ barrel” type and is generally less economical of material, although it is apt to prove more rigid under load. The above types are the most usual arrangements in concrete. In a few instances a hollow, box section, rib or barrel has been employed (Fig. 3d), and sometimes the solid rib section is replaced by a webbed or latticed concrete girder with solid block flanges (Fig. 3e). These latter types are, however, rather rarely employed.

In all the above types the arch rib or barrel acts as the principal structural unit supporting the roadway through a system of columns and girders (Fig. 3a), or, in

the case of barrel arches, the column and girder construc­

tion may be replaced by side walls termed “ spandrel walls.” (Fig. 3/.) These spandrel walls retain a filling of earth, gravel or other ma­

terial which in turn supports the roadway.

In each of the above cases the arch rib or barrel carries the entire load to the supports; Fig. Zg, however, indicates a different arrangement.

In this type of design the columns terminate in an upper chord member

F ig . 3 <7^.

which, in turn, supports the deck. This upper chord is braced and connected to the lower rib by means of a suitable web or diagonal sys­

tem and the entire frame constitutes the arch proper. This arrangement is known as the “ spandrel braced” or “ braced spandrel” or “ framed spandrel ” arch, and though frequently employed in metal arches is not

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