Philosophy of Structures [3rd printing, Reprint 2020 ed.] 9780520328457

Citation preview

philosophy off structures

by Eduardo Torr o ja English Version by J. J. Polivka and Milos Polivka

philosophy

1967 University of California Press Berkeley and Los Angeles

University of California Press Berkeley and Los Angeles, California Cambridge University Press London, England © 1958 by The Regents of the University of California Third Printing, 1967 Library of Congress Catalog Card Number: 58-6523 Designed by John B. Goetz Manufactured in the United States of America

preface

Eduardo Torroja is one of the few great engineers and architects of our time. He has been mentioned in the same breath with Maillart, Freyssinet, Le Corbusier, and Wright for his revolutionary achievements in structural design. He has written a great deal in his native Spanish. The present book has already appeared in Spain under the title Razón y ser de los tipos estructurales (Reason and Use of Structural Types). This English version refers to more American examples of modern structures. When Torroja came to the United States in 1950 for a lecture tour he told us of his projected book to be entitled Philosophy of Structures. The first Spanish manuscript was sent to us in 1951, and we were fortunate to be able to start the translation into English with the assistance of Elizabeth Kendall Thompson, A.I.A., senior West Coast editor of Architectural Record, and several graduate students from Latin America who were at that time attending J. J. Polivka's class in "Contemporary Structures" at Stanford University. The following year, however, Torroja sent us a completely revised manuscript, and the work on the English version had to be done over from the beginning. We wish to extend our thanks to Mrs. Elizabeth Houdek for her generous help on the final English version. [J. J. Polivka died February 9, 1960, and Eduardo Torroja died June 15, 1961. This book, one of their last major works, pays tribute to two men who contributed to the development of a new period in the history of structural design. (July, 1962—M. P.)] J. J.

POLIVKA

M I L O S POLIVKA

contents

x

General Approach to the Problem

1

2

Phenomena of Stressing

3

Classical Materials

24

4

Timber and Steel

39

5

Reinforced and Prestressed Concrete

56

8

6

Supports, Walls, and Foundations

68

7

The Arch

80

8

The Vault and the Dome

94

The Beam and the Slab

114

10

9

Trusses

134

11

Retaining Structures

150

12

The Roof and Other External Covering

164

13

The Floor and the Building

191

14

Bridges and Aqueducts

207

15

Static-Resisting Functionalism

230

16

Construction Methods

246

17

The Beauty of Structures

268

18

Line and Surface

290

19

Genesis of the Structural Scheme

313

20

The Calculation

331

21

The Designer and the Organization

347

List of Illustrations

357

Index of Names

363

chapter 1 general approach to the problem

In order successfully to conceive and to plan a structure or building of any kind it is necessary to investigate and to know well its reasons for existence, its major and minor capacities to resist and to bear. The technical literature on structural engineering abounds with theoretical works of a mathematical nature, but few publications are concerned with the various kinds of structures or the fundamental reasons for their existence. Structural design is concerned with much more than science and techniques: it is also very much concerned with art, common sense, sentiment, aptitude, and enjoyment of the task of creating opportune outlines to which scientific calculations will add finishing touches, substantiating that the structure is sound and strong in accordance with the requirements. Mathematics is merely a convenient tool by which the designer determines the physical proportions and details of a planned structure in order to transform his ideas from the lines of a blueprint to the actuality of a finished structure. The nineteenth and twentieth centuries have produced such astounding mechanical advances in the structural field that ontological studies of stress morphology have been overshadowed and bypassed. The present-day student has to learn so many facts that his thought processes have little opportunity for development. At the same time, it should be pointed out that any designer who disregards the principles of stress morphology may be in danger of serious failures. Before a man can successfully plan a structure of any kind, he must study, from every possible angle, the ultimate purpose of his building. Attention must be directed to the basic structural concept before the mathematical process of calculation is undertaken. All too frequently an architect will begin determining the dimensions of Beam Number One

1

Fig. 1:1. Roman aqueduct in Segovia, Spain. Photograph, M. Garcia Moya. even before he has finally decided that beams are to be used in the building. This book makes no attempt to offer anything new on the subject of structural design. Rather, its purpose is to offer an informal discussion that will stress ideas and concepts at the expense of anything mathematical or theoretical. We must disregard extraneous details—and especially eliminate

mathematical procedures and numerical values—to concentrate on the problems rather from a more general and qualitative point of view. It is absurd to enter into quantitative concretion without the assurance that there will be a definite connection with ideas already being established. It is difficult to find in modern literature—many things a few decades old would be useless now—authors who have made a study of the question involved in this book (or the way it is presented). An attempt will be made to discuss the problem of structural design in its full generality, nakedness, and purity. To that end, we should first formulate the primary purpose of structure. In the past this has largely been an intuitive process in the mind of the architect or engineer. The primary functions of all structure can be summarized as follows: To enclose a certain space and to protect it from the natural elements of wind, rain, and snow, from changes in temperature, and from noise. This function is achieved by the use of walls and roofs. To provide passageways for the movement of persons and vehicles. Floors, staircases, and ramps of buildings, and bridges and viaducts are used for these functions. To resist the lateral thrust of earth, water, or other fluids. Included in this category are dams, dikes, reservoirs, storage tanks, silos, and retaining walls. In addition to the primary function of any structure, other equally essential considerations must be taken into account. For example, the roadway of a vehicular bridge must have a smooth surface and a proper slope to permit the easy passage of vehicles; a dwelling will require windows or similar provisions to admit light and air. Then too, every structure has a resistant function to fulfill. In the present context the word "resistant" is used in its broadest sense and not in its more limited technical connotations. Here the word refers to the entire complex of conditions necessary to ensure total or partial immobility—in other words, the static equilibrium of the structure for a long period of time. This resistant function must do more than merely ensure against structural failure; it must also achieve stability and immobility. Even more broadly, the term should perhaps be "static" function. Every constructional problem is conditioned essentially by a final purpose, secondly by certain essential conditions to be met, thirdly by secondary requirements, and finally by the material means available for its accomplishment. The ultimate purposes vary enormously in each individual case. In order not to get lost in this labyrinth, these possible conclusive features and conditions will be treated in separate, more or less homogeneous sections.

3

There are many properties of materials necessary to fulfill the requirements of static resistance. Among the many problems encountered in structural design, those properties are fundamental. Yet, in reducing the problem to its essential parts, many diversified types of mechanical properties must be specified. First of all, the materials should resist mechanical forces and other effects to which any part of the structure is subjected. It is necessary to know these effects and actions. Investigation of them, starting with all types of loading and external forces (which usually are assumed to occur and act) and with the mechanical properties of materials (e.g., elasticity, plastic flow, etc.), constitutes the part given most emphasis in technical books and schools. Here, only the fundamentals and general aspects of this investigation will be discussed. W e all know that a structure should comply with conditions and limitations of economy. Certainly, there are reasons for sumptuary buildings and structures. It is difficult to evaluate the human and social reasons for luxury. Extremes can always be criticized; yet they rest in human nature. Always there is the problem of determination of decent limits to expenditure, which in every case will be different. There are exceptional cases, but in general it can be stated that, in any given circumstances, the condition of the least cost or the greatest economy should always be observed and respected. However, the solution of this problem is seldom clear and easily determined. Economic factors depend on the degree of safety needed in the structure, its life, the possible future uses planned or intended, aesthetic appearance, etc. If the variable costs can be numerically determined, the advantages and the inconveniences involved usually cannot be determined quantitatively in all pertinent details. For this reason, it is necessary to consider all the factors whose importance is more or less subjective (since they cannot be precisely evaluated) and to take account of conflicting implications, before the final estimate of cost is made in each individual case. W e shall see ultimately how in many of these problems logic and mathematics can be instrumental to the common sense and the balanced consideration by which our judgment should always be guided. The cost of a certain type of structure or structural element can be influenced by various factors such as climate, expansion, and density of population, conditions of transportation, industrialization of the country, availability and efficiency of labor present, conditions of employment, simultaneous construction activity, and similar conditions of employment and prospects in the near future. The factor of economy commands special consideration in the present era of materialistic viewpoints and habits. But it is not always the deter-

4

mining factor at all. In any case, it is never the only determining factor. It very often happens that a small cost increase results in considerable gain of strength in construction and can be accepted as justified and reasonable because of the advantage obtained. This should be kept in mind also in other fields where, for relatively little additional expense, greater value and improvement are possible. More apt to our subject, and certainly of fundamental importance, is the aesthetic aspect of construction. There are monumental buildings in which aesthetic considerations govern the basic design; and there are others, like industrial buildings or ultilitarian structures that are out of sight, for which the aesthetic factor can be neglected or omitted altogether. To what extent this aesthetic factor can be sacrificed to factors of economy will be a matter for consideration in each individual structure. But even so the effect of aesthetics must be initially considered in every case, even if it is later decided to disregard it. Aesthetic considerations should be discussed separately because they have their special characteristics and specific relations with the finality of the construction. In most constructions of today, the aesthetic considerations are not so concrete as other functions. Aesthetic considerations take place in a more abstract way—at least today—in the visible parts of the structure. It is difficult to determine how far aesthetic exigencies are purely of a visual or of a sensible character and how much they are of an intellectual order in our present-day requirements that the external appearance should forcibly convince us of internal phenomena, both functional and structural. These aspects will require special consideration. Out of this heterogeneous complex of considerations and factors, the study of the problem to be resolved by the designer should emerge. It must be realized that in order to arrive at a solution of the problem the designer must deal with certain specific materials and with construction methods and procedures. Another important factor is the construction site, which might be suitable or in conflict with economic conditions. Every structure to be erected on a given site and at a certain time will require a definite procedure in order to achieve the greatest economy and the lowest cost. To this end it will be necessary to consider also other possibilities or modifications of construction methods or of design if it can be proved that the alternatives increase economic and structural advantages. Proper financing and budgetary arrangements are essential to avoid delays in the progress of the construction with the resultant extra costs. The interest on the capital successively invested during construction and the loss resulting from postponement of the income a building is supposed to

5

Fig. 1:2. Eiffel Tower. Photograph, Boucher Adep. render after completion, might justify the increase of construction costs to achieve the hastening of the time of completion. The characteristic properties of materials used will influence the structural type to be selected. Stone can effectively resist compression but is relatively weak in tension. Because of its mass and weight it may be used

6

advantageously in structural types that can be made stable by the proper weight (dead load, gravity) and are but slightly exposed to lateral forces. Construction methods are also variable for each specific material; and the appearance of the structure and its resistance to external factors (e.g., weather conditions) will largely vary with the type of material used. Some material may be economical in one region, yet prove expensive in another. The number of variables and conditions which may exert influence is unlimited. Finally, the construction methods to be used should not be overlooked. Evidently, they will depend upon material used. In the selection of materials the conditions already mentioned should be taken into account as well as availability and economy of the common and skilled labor trained for the work, availability of necessary mechanical equipment, a suitable site that will permit rapid construction, the number of identical structural parts which will make possible more economical installation. In conclusion, every building will have its own course of creation influenced by its bearing capacity and resistance, its economy, its construction site, and last but not least, a more or less pronounced aesthetic interpretation and presentation. This review of various subjects or viewpoints encountered in the solution to the problem of structural design should serve only as a guide for the reader, showing him how to answer pertinent questions one by one. Yet, even when it is necessary to differentiate them in order to analyze the problem, we shall find that all of them are continually interconnected, so that it will be essential, when one question is discussed, to refer simultaneously to others. Only when all are integrated will it be possible to achieve the best conclusion or solution to the problem at hand.

7

chapter 2 phenomena off stressing

An attempt to create a structure without taking into account principles of stresses upon which all phenomena of structural resistance are fundamentally based would be as vain as the attempt of a physician to prescribe and arrange treatment for his patients without knowing the physiology of the human organism. It is not enough to study all theories of resistance and all calculation methods. One must absorb all details and experiments until he becomes completely familiar in a natural and intuitive way with all phenomena of stress and deformation. Then, he can visualize immediately how a structure is strained and in which way it finally will fail if overstressed first as he can clearly comprehend a stone falling in space or the inevitable impulse given to an arrow by the arc of a crossbow. Complex and abstruse mathematical calculations are not alone sufficient to lead to conception of a structure or to guide the hand in tracing its outline: intimate and intuitive comprehension of its working forms is also needed. One should become so familiar with the structure as to have the feeling of being, in full vitality and sentiment, part of it and of all of its elements. As a German would express it, it is necessary to achieve a sincere Einfühlung of the process of resistance, a process we are made aware of through the deformation that is always essentially united with the process of stressing. We could express it in a more concise and academic language: the comprehension of a structure requires intuitive knowledge of the ethology of its resistance and of its constituent materials. Long before the development of our techniques of today, men could conceive and build structures adapted to requirements of resistance eternally satisfactory in aesthetic forms, because he had observed with intimate intuition the branches of a tree bending under the weight of fruits and the

8

tensioned cords of swings in which children have rocked from time immemorial, long before the Mycenaean era (fig. 2:1). It is felt that a few pages should be devoted to this subject. Although nothing basically new will be revealed, a simple commentary will permit appreciation of a different view of the question; and thanks to thinking and experimenting, a reassurance will be insinuated into the mind, to guide it subconsciously in the research of the new structural forms. There are three different but interconnected conceptions to be considered in every structure, and in every structural element involved: equilibrium, resistance, and stability. Equilibrium is easiest to understand. Equilibrium is of static character, or, in other words, it must secure the immobility not only of the structure as a whole but also of all its parts and individual members, with consideraFig. 2:1. Small prehistoric figure from Crete, now in Candia Museum, Crete (from Summa Artis, by M. B. Cossio and J. Pijoan; Madrid, Espasa Calpe).

tion of connections tying them together. Equilibrium requires that the whole of the structure, the form of its elements, and the means of interconnection be so combined that at the supports there will automatically be produced passive forces or reactions that are able to balance the forces acting upon the structures, including the force of its own weight. It is easy to understand at first glance, except in some special cases, whether or not the system of supports and connections satisfies the conditions of equilibrium. Figure 2:2 demonstrates in a simple way how a structure consisting of certain members can be made stable or can be statically sound or not under various conditions of support and connections, when subjected to external forces. Whether the structure or its elements are sufficiently resistant to acting forces is but a question of calculation. However, before these calculations are made it is important to be certain that the selected structural scheme will not permit free movement of connected members. Certain connections permit reactions in only one sense and not in the opposite direction, and the structure can react under certain loading as if no supports existed. In a similar way, a bulk of masonry representing an assembly with practically no tensional resistance, or in a plane of masonry above the grade, will require such arrangement that the resultant of all proper weights will fall within the supporting cross-sectional area of the masonry in such a way as to produce compression in it. If the resulting force has greater inclination than the angle of friction between the foundation masonry and the soil, the possibility of sliding can be eliminated. Not only must this condition be maintained for the structure as a whole, but any layer of the assembled structure must comply with this requirement. Dunes on a desert are difficult to translate as a whole, but a slight breeze will gradually move and displace the grains of sand from one declivity to the other and will produce the same effect of slow displacement as if all the massif had slid upon the grade. The equilibrium, moreover, to be static—perhaps it would be more convenient to say: in order to become static—should be steady, permanent, lasting: it would be useless if indifferent or unstable. The structure shown in figure 2:2 is in equilibrium as long as it is not pushed horizontally—the slightest horizontal force would cause its collapse. All these phenomena are clear and can easily be understood; a technician can hardly be misled by them because all of them are verified by daily experience. This type of equilibrium exists between acting forces and reactions, and is, therefore, independent of any scale. A reduced model will show the same effects as the proper structure. Experiments on models are simple and of

10

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@ Fig. 2:2. Various structural schemes. great educational importance, and instrumental for understanding such structural problems. Independent of these primary conditions of equilibrium, which can be easily evaluated after experimenting, are the problems of strength or resistance of a structure. The material in all elementary parts of a structure must have properties of resistance to all internal forces produced by general loading conditions and by action of any exterior force. The designer who deals with current structures consisting of linear members is accustomed to consider in most cases only typical internal forces and stresses (i.e., tension, compression, and shear) which can and should be evaluated separately in various effects of failure in lattice struts, supporting members, posts, beams, arches, etc. However, these forces can be, in the majority of structural problems, presented by the three principal stresses acting orthogonally, one with respect to another. The enveloping curves of these three directions form a network of so-called isostatic lines (or simply isostatics) which give a good representation of the state of stress. They can

11

be better understood if we imagine that along any of these lines a solid volume unit contracts or expands in proportion to compression or tension exerted. Certainly the stresses in one direction produce not only a deformation along this direction but also transversally, the relation of both being expressed by the so-called Poisson ratio. However, the consideration of these transversal deformations is of no great importance for the first, approximate judgment. It is impossible to express all these phenomena in a few pages; and the only aim now is to remember these things already known by the technician trying to visualize the strength phenomenon. This kind of knowledge and representations should always be emphasized instead of learning formulas without thinking as to what they represent. For better understanding of this effect let us restrict the stress problem to a plane. Imagine a grid, following the orthogonal network of isostatics, made of struts interconnected with hinges in which each vertex or node represents a point of the stressed body, and, under deformation, the individual struts will be extended or shortened without changing the angles of intersection. It is evident that, in general, such a network would be in an unstable and incomplete equilibrium and would need diagonal bracing to make it stable. In a solid body, however, these diagonal struts can be considered as substituted by the continuity of the material. One should remember also that the slippage produced by shearing stresses is maximum on lines crossing this network at 45°; and the importance of this shear depends on the difference between the two principal stresses. If, however, these isostatics are represented by segments of different length and different thickness (fig. 2 : 3 c ) , corresponding to the intensities of principal stresses, the discussed phenomenon becomes more clear and instructive. Fig. 2:3. Stress distribution in a gravity dam (lines of principal stresses).

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Each elementary part (volume) of a solid body with its faces oriented in conformity with the aforementioned gridwork can be imagined as maintained in equilibrium by four forces (or better by stresses expressing the acting forces per unit area), represented by the four semibranches of the isostatic network acting normally on the faces of the imaginary cube of the solid element. Such "plexo-tensional representation" (an imaginary network representing a tri-dimensional state of stress) with lines of certain direction and thickness is instrumental in visualizing the transmission and distribution of stresses inside the solid body; variations in direction and intensity of stresses; how they increase in certain zones, thus indicating weaknesses in those particular parts; and how finally they vanish and lose intensity in their continuation within an indefinite bearing mass of homogeneous material. Many effects are revealed by observing this spatial network; for example, it is shown that, if the fibers following a concave periphery are in tension, necessarily other tension, in a normal direction, will be produced to prevent these fibers from separating from the rest of the solid. Contrarily, if they are in compression, compression in the perpendicular direction will be necessary. The sharper the curvature of isostatics in a certain location the greater will be the variation of transverse stresses along the lines of the other stresses normal to the latter. Thorough study, investigation, and interpretation of the plexo-tensional device are useful to arrive at a rapid qualitative judgment of the state of stress produced within a certain part of a solid body under the action of direct forces and their reactions. The continuity of the mass and of its deformations and the necessary orthogonality of the spatial network are helpful in the drawing of an approximate scheme of the stress distribution and hence in correcting the forms in order to arrive at the best solution of the resistant structure. This is not the appropriate place to give particular details and commentary with more examples pertaining to the plexo-tensional representation. Therefore, we should emphasize those things which are practical and of educational interest and importance, things not usually emphasized in textbooks and courses of instruction. The ability to imagine how a structure is deformed under given loads is undoubtedly a great help in realizing not only the state of stress in a body but also the location and conditions of a possible failure. Daily experience gives us opportunity to observe how a bar deflects and breaks because of tension or bending. It is possible to become familiar with, and informed about, other more complex cases of stress, achieving a better understanding of the failure of the material in simple cases. It will always be worthwhile to analyze the deflection curves. Any time devoted to discussion of these effects will always tap a fertile fountain of inexhaustible knowledge.

13

A good master used to recommend to his disciples at the beginning of their study of similar stress problems that they always carry in their pockets a rubber eraser with a network and some circumferences traced thereon, so that they may observe its deformations. It can be seen how certain circular circumferences are converted into ellipses (fig. 2:4a) and how directions originally perpendicular change their angles, except in instances where the directions of the network coincide with those of principal stresses. The experience immediately becomes clearer when corroborated by observation of models made of plastic material, like wax or the clay used for pottery. In such models it can clearly be seen that the material breaks in a plane normal to the maximum tension (fig. 2 : 4 b ) or fails by successive slippage ( b ' ) in a plane under 45° with the principal directions, due to the maximum tangential stress and to maximum shear. Similar slippage effects can be observed in specimens tested for compression (subjected to crushing strength), in which case, however, the angle of the plane of slippage can vary ( c ) due to the internal friction. Also, rupture in certain materials can occur parallel to the direction of compression ( c ' ) because of expansion corresponding to Poisson's effect. Under pure shearing stress material will break, due to its intrinsic properties, by slippage along the plane of maximum shear ( d ' ) or by separation under 45° with respect to the plane of maximum shear ( d ) . Because this type of strain is caused only by the effects of two principal stresses, equal and of opposite sign (one being tension and the other compression), there will be produced a shearing stress of equal value that will act in planes bisecting the planes of those principal stresses. In relation to resistance, the designer (especially if he is dealing with details of a structure, types of connections, etc.) should know that failure depends not only on the maximum principal stress to which the structural member is primarily subjected but also on the other two principal stresses acting along normal directions, and which effect is not always depreciable. In effect, the different curves of intrinsic resistance of various materials demonstrate the importance not only of the intensity and the direction of the maximum principal stress, but also of the difference between maximum and minimum internal stresses, factors which will affect the resistance of the structure and the type of failure. As it is certain that some structural materials are brittle and others ductile, it is well known that the majority, if not all of them, break suddenly in pieces if subjected to tension in all directions. However, materials become ductile and fail by important slipping, but without falling apart, when subjected to high compression in all three principal directions. In this field much elucidation would result from study of liquids and

14

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Fig. 2:4. Deformations and types of failure. granular materials, which in certain aspects demonstrate quite opposite behavior. On first consideration, it would seem that liquids do not sustain tension. However, this is not quite true. Practically, a liquid is not able to resist flow or slippage (which is of the same character as shearing stress) or difference of principal stresses. A liquid behaves as an extremely ductile solid if there are some differences among the three principal stresses. However, it has been shown that water freed from dissolved air and subjected to a centrifugal traction of perfect isotropy—which is not easy to achieve—can resist tension up to 13,000 pounds per square inch, because there is a tendency to slippage in any direction. It can be concluded after thorough investigation that liquids cannot resist simple compression (in only one direction). If we try to compress a liquid with a piston which does not fit exactly in the pipe, the liquid will escape; but if the adjustment of the piston and pipe is made, there will be a counteraction of the pipe shell which will exert lateral compressions of

15

equal intensity in all directions. If the pipe is filled with sand instead of liquid the leaking piston will nevertheless resist because the internal friction of the sand will balance with the resistance to sliding; and if it is a solid with certain cohesion, the difference in principal stresses can increase as a function of the intensity of the cohesion. It should be realized that resistance is not a simple magnitude which can be expressed by a number. The syndrome of a failure with all symptoms of weakness in resistance is more complex and has features more interesting than the proper stresses. Whether possible rupture can be foreseen as brittle or ductile is important in terms of the preceding effects, and especially important in respect to the safety factor and to precautions necessary in order to prevent failure. If a ductile rupture is preceded by considerable deformations, it is generally possible to adjust and decrease the loading in the endangered part of the structure while the deformations are still small; and a disaster can be prevented. In case of a brittle rupture caused suddenly by loss of cohesion, no sign appears to warn of catastrophe, and this is the case under an isotropic tension. Proper consideration of these particulars of a material's resistance or its capacity to resist failure, in connection with the process of stress, will elucidate many phenomena of importance for a sound design of a structure. Therefore these characteristics should be known and adequately utilized by the designer. Such grooves in a plate or in a bar under compression as shown in figure 2:5a act as kneepan under high compression and small rotation. The curvature of the isostatic network is such that compression in that particular section is tripled and pasticizes the material, thus increasing its resistance to rupture and permitting deformations and rotations which, under other

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circumstances, would be impossible without rupture. The opposite effect would occur if the grooved bar should be subjected to a tension. Similarly, a small circular opening in a plate subjected to tension (fig. 2:5b) will triple the average tension close to the interior boundary, and will produce also tensile stresses in the perpendicular direction. If, instead of a circular hole, an elliptical opening (c) is provided with the major axis perpendicular to the traction or a simple horizontal cut is made, the tensile stress along the perimeter of the opening will increase in proportion to the flatness of the elliptical or other cut, theoretically becoming infinite when the ends of the cut are fine fissures. So reentrant angles of tensioned boundary are most often responsible for initiation of cracks and fissures. If, in such cases, the material does not break under minor force, it will be because the high tension is concentrated along the opening and on a relatively small area; and in such small zones showing great difference of principal stresses, the material undergoes plastic deformations that can be much greater than elastic deformations without increase of tensile stress. These plastic deformations occur always as ductile slippage, which can be observed in certain plastic materials, such as the wax and the pottery clay already mentioned. All these stress phenomena are, however, not independent of the scale, as was the case in problems pertaining to simple external static equilibrium. Because the specific forces indicating their intensity per unit of the area, or simply stresses, determine the failure of the material, and because with increasing dimensions of a structural element the volume and therefore the proper weight increase more rapidly than its cross-sectional areas, the proper weight alone of the real structural member could cause failure, whereas a model reduced to scale would be proportionally much lighter and its proper weight would not be decisive for rupture. Many ductile materials (e.g., semiductile steel used in typical steel structures ) remain elastic to a certain limit of stress, and become plastic beyond that limit. Consideration of these stress equilibria in the plastic zone—now being introduced and used in analysis of our structures—offers without doubt certain inconveniences. Plastic equilibria are not so simply reversible as elastic equilibria. Under repeated or alternated stresses, fatigue or failure of the material will occur after a certain number of repetitions and under much lower stresses. The number of stress variations will be much lower when plastic deformations pass certain limits. In fact, the same phenomenon can be observed in a region apparently elastic. The material can fail under stresses lower than those produced at the first loading. However, this can occur only after an enormous number of repetitions (in the range of hundreds of thousands or millions); but if the

17

stresses exceed the average value, the material can endure under a very limited number of loadings. This question has its importance most particularly in cases where the structure must sustain a large number of stress alternations as in foundations of vibrating machines or in certain railroad and highway bridges. The problem is different for materials that do not follow Hook's law (e.g., concrete), materials in which proportionality between stresses and deformations does not exist. This raises a question of secondary significance which will be discussed in connection with each specific material. The discussed complex of phenomena shows how interesting and important are the investigations and study of these questions. This type of stress analysis replacing the simple method of calculation has a great importance for some contemporary structures and new materials. If a technician, who is accustomed to design frameworks of his buildings by using formulas derived from the most simple theory of strength of materials, believes that many questions pertaining to failure and to experimental stress analysis (plexo-tensional investigation) involve inaccessible problems, and, besides, have little, if any, influence on his creative imagination, he should be aware that there are many problems that cannot be understood or explained by terms of old methods; he should realize that there are many details and important factors in forms, interconnection of structural members, arrangement of elements, reinforcement, etc., which although only details, are of fundamental importance in the strength of the assembly. Finally he will learn that it is not so difficult to draw a logical sketch of an isostatic network if a little thought is given to the theory and interest taken in the approach to such problems. Structures can be formed of three-dimensional comparable blocks, as dams; of superficial elements or shells that are characteristic for reinforced concrete; or of individual members in which one dimension is predominant as compared with the other dimensions, as bars, beams, girders, arches, and similar elements. Little is known and not very much can be said about the first category (three-dimensional structures). Unfortunately, the theory of elasticity did not establish in practical form principles to be used in analysis of this type of structure. A three-dimensional boundary does not afford a simple analytical interpretation; even in forms of revolutions the exact solution is limited to very few cases. The theories based upon elasticity, viscosity, and plasticity of materials are very intricate, and still less application of them has been made to three-dimensional bodies. More advanced is the theory of elasticity in the field of flat slabs or of straight members with constant section, in which all sections are under the same state of stresses, for example, tubes (fig. 2:6). Restricting the study

18

to stress problem in a plane, the isostatic network and the values of principal stresses can be obtained—although in a more difficult way—by theoretical analysis or experimental stress analysis as, for example, photoelasticity, which is taking and analyzing photographs of isochromatics (lines indicating equal differences of principal stresses and of isostatics ( lines showing the directions of principal stresses). Adequate representation of the actual state of stress under given loading enables us to determine the most suitable structural form. As will be shown later, this experimental investigation of stresses in a plane structure is of great interest and merit as applied to shells with single curvature ( barrel shells, pipes ) and to shells with double curvature ( domes, etc.). In these structures similar states of stress are produced and can be determined at any point as acting in a plane tangential to the center layer of the shell. However, most of our structures in the past consisted of individual members having relatively small transverse dimensions as compared with their length, and in such members stress analysis is much more simple. Still today, many modern structures consist of frameworks of this type, and frequently the word "structure" refers specifically to assemblies consisting of linear elements.

Fig. 2:6. Net of isostatics in a tunnel, one side with revetment and the other without. Photoelastic investigation made at the Madrid Central Laboratory by C. Benito and A. Moreno.

a

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Fig. 2:7. Stress distribution and deformations of a freely supported beam. The behavior of these members (bars, struts, beams, posts, columns, etc.) is so well known and so often encountered in daily practice that it needs no special commentary. The deformation of a structural member due to bending is more important than that due to axial tension or compression: everybody knows how a branch of a tree deflects under wind pressure or can be bent by his hands. Even children know that a drafting rule is easier to bend flat than across the edges; and although they never have thought about the reason for it, they will not be much surprised if told that for the same width of the rule its resistance is proportional to the square of the thickness and the deflection is inversely proportional to the cube of the thickness. Nevertheless some modern designers seem to be unaware of these basic relationships, since they require that specialists design beams of such slenderness that they resemble springboards, and confuse the sensation of static lightness with that of painfully strained effort. The stress network of a beam subjected to uniformly distributed load (fig. 2:7a) can be roughly represented by a system of bars in compression and in tension which shows clearly why beams need longitudinal fibers at the bottom to resist tensile stresses and similar fibers at the top to resist compression; and how these fibers, when they are strained, are subjected to action of two groups of diagonal forces, one in tension and the other in compression. The beam will deflect, as shown in figure 2:7c, and it can be seen that the diagonal forces are necessary to prevent slipping of longitudinal fibers ( d ) or shearing in transverse plane (e), both effects being caused by contraction and expansion of the fibers due to deformation.

20

As mentioned already, the shear resulting from tension and compression, both acting under 45°, is resisted by the diagonal members, whereas the bending moment produces longitudinal stresses in tension and in compression which are resisted by longitudinal members of the imaginary framework. As is seen in figure 2:7/, the lever arm of the internal bending moment is much smaller than that of the external moment; to maintain the balance between external and internal forces the extreme fiber stresses will increase with the decreasing depth of the beam. For this reason it is logical to increase the depth and to shorten the width of the beam, in order to obtain a greater bearing capacity by maintaining the same cross-sectional area or volume of the beam. However, the shear stress does not depend, in general, on the lever arm of direct forces and will require rather greater width of the cross section. These principles are practically utilized in the I-shaped beams. The torsion effect is less often utilized and somewhat more complex in calculation. It can easily be visualized by following the deformations. It can be shown that the resistance and rigidity in torsion of a beam as described will increase with the value of the polar moment of inertia of the cross section, which, for the same area, is greater in a hollow section, as for example, of a pipe or box beam. We should not, however, overlook the fact that the danger of failure is not the only one to which a structure is exposed. There is another typical danger that affects more the morphology of individual structural elements and the structure as a whole: the buckling or, more general, the instability. In the same way as it is not enough to attain an equilibrium of the acting forces and reactions without stability, the equilibrium in the state of internal stresses in all elemental parts of a solid would not be sufficient without conforming to laws of elastic or plastic stability. If you lean on a straight stick (as a cane), that stick can resist your weight, despite its slenderness, as long as it remains straight; once it starts to bend, it will easily break. The danger of breaking increases with the deflection or with the lever arm of the bending moments; and as in all instances where bending prevails, the danger of failure will diminish with the increased moment of inertia of the cross section, or if the same area of cross section should be maintained, by distributing the material in a hollow section. Similar effects of instability can be observed in other instances; for example, a thin bracket will bulge due to torsion produced by lateral flexure and warp of its original plane. In general, these are phenomena very easy to perceive and to understand, but difficult to calculate. The danger of failure increases rapidly with decreasing depth or thickness of the structural member or with increasing

21

slenderness beyond a certain limit depending on the specific properties of each material. And the importance of these phenomena is precisely greatest with the strongest materials that would permit the greatest slenderness, were it not for this danger of buckling. In designing these members it is necessary to consider that danger of bulging and to counteract it by providing adequate anchorage, thus shortening the free length of the member and reducing its relative slenderness. Advantages and inconveniences of isostatism and hyperstatism in structures should be considered and discussed as well as methods of determining at the first glance whether a structure is in equilibrium to sustain the loads and whether the sustentation is of isostatic or hyperstatic character. The latter is characterized by having more reactions or more simultaneous actions in connections than would be necessary for a simple equilibrium. But the understanding of the importance of hyperstatic reactions goes much further; a problem of indeterminate relationship arises as to the possible solution of equilibrium which disappears only when consequent deformations are considered. Keeping this in mind, one can realize how complex the state of stresses and reactions can be. For example, in an isostatic beam having two equal spans (fig. 2:8a) subjected to uniform load, the center support will carry twice the load carried by each end support. However, if this beam becomes hyperstatic owing to continuity (fig. 2:8b), the previous solution resulting from the equilibrium of the acting forces is no longer compatible with the deformation. In order to maintain the continuity of the beam, negative moments over the center support will be produced which necessarily will increase the reaction at the center support and consequently, owing to a statical equilibrium, will decrease the reactions at the end supports. The actual solution will be found by calculation under the condition that the work performed by the acting forces during deformation becomes a minimum (method of least work). The loads will have the tendency to produce a type of deformation permitting the maximum of work; and the reactions balancing these forces will be distributed in such a way that a minimum of work will be performed. The existing state of stress will be changed only by failure. However, all this can be deduced from purely qualitative consideration of the phenomena we are dealing with. The determination of all these hyperstatic reactions, both internal and external with respect to the structure, is one of the primary problems to be solved by structural analysis; and the complexity of these analytical calculations will depend on the order of hyperstatism or the number of statically indeterminate reactions (redundant forces). To estimate the intensities of these reactions is much more difficult than

22

Fig. 2:8. Principle of freely supported and continuous beams.

in the case of isostatic reactions; they will depend on proper deformations of the material, and are always more diversified than the reactions of isostatic character. For this reason they will always occur within a certain margin of permissible error. Usually, statically indeterminate structures with continuous connections of individual members have the advantages of greater slenderness and rigidity, although, in certain instances, the hyperstatic conditions present disadvantages to be discussed in other chapters.

23

chapter 3

classical materials

"Each material has its own message to the creative artist," says Frank Lloyd Wright. To understand this statement it is necessary to ponder the properties of each individual material until we are imbued with its proper nature. Consequently each material presents a different and specific problem. At present, we are not interested in materials in the sense of merely matter, or mass, as the word is used by builders. We shall consider, for example, concrete or reinforced concrete, as structural material, not cement, and masonry, etc., not stone. It is possible to subdivide working structural materials into various groups: masonry, metals, timber, and reinforced or prestressed concrete. These four large groups present diversified characteristics and properties that have a decisive influence upon the fundamentals already discussed. In the first group, the following contemporary types of masonry are included: dry masonry, stone masonry with various arrangements of joints, mud masonry, adobe, brick masonry, masonry of plain and hollow bricks, tiles, etc. One of the oldest types of masonry still preserved is the Cyclopean dry masonry (fig. 3 : 1 ) which, although without any technical significance for contemporary structures and of only archaeological interest, can be considered as the forerunner of all stone masonry. Its monolithic mass, silent and immobile, sustains loads and forces as does a great hero, as Lin Yutang has told us. As a virginal creation resulting from human struggle it is the last extant message of the first task aiming toward permanency of art and construction. Yet today, vicarious children of this technique appear again in simple and modest dry walls and enclosures of our gardens and groves, in supporting embankments and terraces, and in abrupt cliffs in the mountains.

24

Fig. 3:1. Cyclopean wall at Tarragona, Spain, sixth century B.C. (from Summa Artis, by M. B. Cossio and J. Pijoän; Madrid, Espasa Calpe).

When man was able to make tools that permitted him to cut stone, stone masonry was born. Extant old Roman walls (fig. 3:1) show that stones were assembled in layers one above the other without mortar in the joints. Comparing this type of masonry with that built with mortar, and realizing that dry masonry offered but small resistance and stability necessitating considerable thickness, man immediately became aware of advantages to be obtained by binding the stones with mortar. Much greater resistance is obtained with mortar joints, and although the joints are always the weakest points owing to the relatively small adherence between stone and mortar, this type of masonry shows improved mechanical properties. The greater the thickness of the wall with respect to its size and the smaller the possibility of a separation in the joints or of sliding in oblique planes with respect to the direction of compression, the greater will be the resistance to failure. The prevailing direction or inclination of the layers of masonry work should be as normal as possible to the direction of the resulting pressure. Classic masonry in adjusted hewn stones in general is the most expensive type and is restricted mostly to monumental or luxury buildings. It is practically everlasting if good quality of stone is used, and its magnificent aspect and the artistry of its workmanship can surpass all possibilities of spectacular classic structures (fig. 3:3). Its form was given by the artist, but its quality depends on the properties of the material which in construction work have always been decisive and independent of human labor. Under blows, chips flew from the wounded stone, but soon time began to restore its surface and to cover it with an attractive patina that is appreciated as is the aroma of an old wine.

25

Stone masonry necessitates a relatively great mass; and any attempt to reduce the thickness over a certain limit will result in increased costs. Its resistance to compression is high, but its strength against tension is slight, not only because stone in itself has relatively low resistance to tension, but much more because the mortar in the joints is for all practical purposes unable noticeably to resist any tension. In order to increase this resistance (a practical and convenient objective seldom considered in theoretical calculations), vertical joints should be alternated in contiguous rows to reduce the danger of cracking to a minimum, whereas horizontal joints may be continuous, on the assumption that they will always be compressed and little shear or slipping along ;heir planes is possible. These conditions occur

Fig. 3:2. Frieze over the Puerta del Sol, Toledo, Spain, sixteenth century. Photograph, M. Garcia Moya.

Fig. 3:3. Quince Ojos Viaduct, Madrid. Photograph, S. v. Kaskel. in very heavy structures in which the dead load represents the maximum loading. Hewn stone can have various forms, but angular or curved surfaces should be avoided because they involve prohibitive labor costs. Stone masonry is entirely appropriate for embellishment of exteriors of monumental structures in which the stone represents the reality and the existence of the mass. It is a material suitable for structural elements (e.g., retaining walls) in which mass and weight are the basic characteristics. In general, stone masonry can be used in structures subjected exclusively to compression (e.g., piers, columns, arches, vaults); however, this material is practically without value in structures subjected to tension.

At this point stone-veneered masonry should be mentioned since it has the same appearance as stone masonry. However, here relatively thin stone slabs face the masonry, which has a different structural character. Anchorage and other mechanical devices permit us to use economically very thin facing slabs or tiles in stone, and the whole has the appearance of solid stone masonry. This is, from the classic point of view, an unnatural, false, and absurd solution; here, beyond doubt, modern techniques have created, by changing functional utilization, a substitute material with different properties and bearing capacity. In this way, solid masonry in its noble, massive, and immovable assembly of hewn stones has been converted into a simple cover, apparent and expletive, giving to the structure only its face, adhering to other materials to be supported, like a vine upon a tree. The technician should remember that in general these slabs of facing have greater rigidity and less deformability than the bulk of the wall masonry to which they adhere and they will tend to loosen, as in concrete masonry veneered with stone. For this reason, it will be necessary to provide secure anchorage and, in extreme cases, to allow for certain movement between these two materials. Bricks are considered to be the first material created by human intelligence from the four elements: earth, air, water, and fire. This material, so close to the human spirit and need, being laboriously assembled, cast with skill, dried with patience, and transformed into stone in the heat of fire grievously kindled, presents characteristics and morphology in its manufacture which are genuinely specific and totally different from those of natural stones. One basic difference is that bricks can be manufactured in quantity and by assembly-line methods; all units of any certain type are identical, and the number of types is necessarily limited. The great variety of designs and effects that artists of the past, especially the Arabs (fig. 3:2), were able to create in their brickwork, assembled with an element so monotonous and so restricted in its dimensions, can be compared only with the beauty and attractiveness a romantic poet attained by adjusting his verses to the rigidity of a formal meter. Difficulties of drying and burning bricks—which modern technology has been able to overcome to some extent—make it necessary to manufacture them in small size, and in shapes suitable to the producing machinery. Hollow bricks with thin walls can be manufactured; but there is no economical method of providing for space completely enclosed within the bricks, which without doubt would improve the properties of the structural type, as well as its thermal insulation. The small volume and weight of a brick made its placement and assembly

28

simple and inexpensive, since it is manipulated by one hand only. However, its size is limited not only for that reason but also with regard for easy manufacturing. If suitable methods were used, large hollow pieces would permit development of other, more economical, methods, techniques, and procedures. This is already true of large prefabricated units of brick masonry which offer definite advantages where suitable equipment is available. The small size of bricks results in a relatively large area of joint surface per unit volume of masonry. For this reason the tendency is to reduce the thickness of joints to a minimum. There are certain types of bricks with inner surface recesses which permit assembly without mortar on the very thin visible joints, but this type of assembled structure is not rational. Its resistance is much lower and its impermeability to air and water is practically nil; furthermore, its aesthetic effect will always be disputable. Certain minimum thickness of the mortar and inevitable small variations in form and size of the bricks, plus contraction and warping due to burning of clay bricks will make it necessary to maintain the thickness of the joints within reasonable limits. On the other hand, thickness of the joint cannot exceed a certain limit because excess mortar would flow out under the weight of the superimposed bricks. The mortar is subjected to shrinkage and to thermal and hygroscopic deformation which are greater than those of the bricks. Like stone materials, bricks have a relatively low expansion coefficient, and if they are burned in a good kiln, the hygroscopic effect becomes practically negligible. In thick layers the influence of the movement of the mortar is most noticeable in the direction normal to the layers; and, if the brick masonry is connected with another more rigid structure, mischievous cracks and fissures will occur. As thick joints are usually horizontal, the masonry will shrink mainly in the vertical direction. The effects of thermal differences and hygroscopic changes are of minor importance, yet they are still appreciable if the water tends to penetrate the masonry throughout. The shrinkage effect normal to the plane of the layers is especially noticeable in the course of time in lime mortars. Along the horizontal direction the mortar is much less exposed to the aforementioned effects, because of lower proportion of joints and because the artful assembly of the bricks hinders these effects, and at the most causes microscopic fissuration of the mortar. Longitudinal thermal and hygroscopic movements in vertical joints can for all practical purpose be assimilated with those of the brick material and, therefore, they always will be unimportant. Thickness and other dimensions of this masonry are not arbitrary; they

29

are governed by the typical size of bricks, with consideration of the thickness of the joints, which, with their proper variation, make possible some adjustment of the dimensions of the structure to the typical size of the bricks. The thickness can be very small (e.g., some curtain partitions built up of hollow bricks no more than one inch thick). This type of masonry naturally has limitations because of its low bearing capacity and resistance to lateral forces. Work can be considerably speeded up by using, instead of cement mortar, plaster of Paris which sets practically instantaneously. However, effects of humidity cancel many advantages of this material, and, for this reason, its use is restricted to interior structures or to masonry sufficiently protected from humidity. The color of brick should not be overlooked; although bricks do not offer such richness in color as stone, they nevertheless present a great variety of shades, from neutral and light ocher through a whole scale of rose colors to lively red, so that it is easy to incorporate in a structure, by proper selection of the shade, a specific personal character of enjoyable reality and a delicate vibration that would be difficult to attain with other materials. Adobe (sun-dried clay bricks) is a material usable in regions where sandy clays suitable for fabrication are found. Adobe masonry is, however, useful only for walls and structural elements subjected to lower values of tension and compression, and is, therefore, of little interest as structural material. Concluding: structures in stony materials, either natural or of ceramic origin, resist compression well, but are weak in tension. Such materials, which could be called "frail," are basically different from those which resist tension (e.g., steel or timber), and require that a greater mass be used in a structure. On the other hand, thermal and acoustic insulation, appearance, and resistance to deterioration of such stony materials are good, and all these known properties can influence the selection of a structural type, and therefore should be taken into account. Permeability will depend basically on the type of mortar used, but it is very difficult to assure an absolute impermeability in these jointed materials. Mud or clay masonry must be treated separately, not only because it is, like adobe, a material excessively weak for structural purposes, but also because it requires a special method of building in situ in forms that permit molding in various shapes, although in some cases the structure can be built without forms, in much the same way that a sculptor uses clay. However, this type of clay is the antithesis of stone in respect to permanency beyond the lifetime of those who build it and requires special care and treatment to prevent erosion. It is perhaps because of this that those country people, much attached to their traditional environment and with an

30

intimate, human feeling for its clay soils, have a strong incentive periodically to protect and repair their old-type, homely dwellings. Clay masonry, like concrete, is a "plastic" material that permits the pouring in situ of large monoliths of any shape or form, with certain limitations and by using special methods. Its economical possibilities when used in walls of popular housing, its excellent heat and sound insulation, as well as its aesthetic possibilities, should not be underestimated. Its resistance and durability can be substantially improved by using modern techniques and methods, for example, admixture of small amounts of Portland cement in accordance with granulometric investigation, petrographic composition, and rheological behavior of all constituents. Every new material provokes by its appearance strong opposition in minds accustomed to classify its value exclusively in terms of forms used by our ancestors. This was true of concrete. There are people still living who learned in school how morbidly mad it would be to try to substitute for noble stone a repulsive chemical mixture, a sickly and mechanical product of our artificial era, that was already infected by such substitutes. But now nobody would dispute the great utility and specific adaptation which concrete affords in certain structural types. Although concrete is specifically a modern material in its development and application, it can be considered old and classical, because it was used, although poorly, in many villages of antiquity, and by the Old Romans in its pozzuolanic variety. (Ed. note.—Pozzuolana is a volcanic ash or dust, first discovered near Pozzuoli, Italy, which has the effect, when mixed with lime, of enabling the latter to harden in air or under water. Pozzuolanic materials have been used in recent times for better quality of concrete.) The paste of cement and water, which sets first and hardens later, binds gravel and sand to make a compact artificial conglomerate resembling certain types of natural stones. The variety of exterior forms in concrete is limited only by the cost of forming; and its thickness is limited only by the size of aggregates, gravel, and sand. We shall leave the subject of reinforced concrete for further discussion and shall consider here plain concrete with its characteristic properties. From the construction point of view, concrete necessitates forms and also scaffolds strong enough to support the forms with the weight of the poured concrete, not only vertically but also in lateral directions, since fresh concrete behaves like a liquid. The lateral forces disappear as the concrete sets; but most structural elements should continue to be supported by scaffolding or centering for several weeks, until the concrete has reached its specified strength. This property has basically influenced the morphology of concrete structures. In terms of pure resistance, concrete belongs, like the

31

stony materials discussed previously, to the general group of brittle or frail materials, because it is strong in resisting compression and weak in tension. When concrete is subjected to compression, failure will be caused either by separation along planes parallel to the direction of force or by shearing along planes that will have the tendency to incline about 30° with respect to the direction of the acting force, the inclination corresponding to the effect of internal friction. If concrete is exposed to tension, failure will occur along planes normal to the force. For a builder with a classical bent, concrete is a "plastic" material converted into stone—not in independent layers to be assembled but as a monolithic mass having certain properties of stone which improve with time, slowly with the agglomerate materials of antiquity and more rapidly with the cement and admixtures of today. Concrete has been thoroughly investigated, and essential differences between concrete and natural stones, especially those created in the original fire, are known. To say that concrete is not a solid and is not claimed to be such would appear too strong and somewhat insulting to those who would take the trouble to consider the idea; but those who have devoted themselves a little to the delicate experimental study of its laws of deformation know very well that these laws correspond closely to the deformation laws of the so-called pseudosolid material. A discussion of both chemical and physical phenomena of cement paste would require hundreds of pages and, even then, all the involved problems would not be clearly resolved. Yet, for our purpose, it would be sufficient to evolve a scheme that would permit presentation of the ideas in a form suitable for general comprehension in common terms, so that these phenomena could react in the mind of the designer in his attempt to substantiate a certain structural type. Concrete, as a pseudosolid, can be defined as composed of inert particles (gravel and sand) and a paste in which one part (which we can call solid) is surrounded and then infiltrated by water containing salts and air which fills the remaining pores. Thermodynamic equilibrium with ambient air requires that the water should evaporate or condense when the hygroscopic degree of the ambient changes, following complex laws that depend on its surface tension, form of pores, etc. Once the cement has set, important variations of different character will be produced in the interior—especially variations in volume similar to those of a gel. The phenomenon will depend not only upon the ambient humidity but also on continuous chemical action extending over months and even years, during which the phenomena of dissolution of the primitive cement and formation of new amorphous and crystalline substances continue.

32

/

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Fig. 3:4. Stress-strain diagram for concrete. Consequently concrete presents, besides a coefficient of thermal expansion, a coefficient of hygroscopic dilatation which is more appreciable. While concrete is kept wet, either by immersion or by simple sprinkling (if the concrete members are not too thick), it expands; and as it dries, it contracts, in inverse proportion to the hygroscopic degree of the ambient air. Fortunately, the effect will diminish with increasing thickness of the member, despite the porosity of the concrete. During its hardening period, in the first months exposed to the normal ambient air, concrete is subject to considerable shrinkage (in the order 1 to 10,000) which generally will be determined by the amount of cement in the mix and the dryness of the surrounding air. This phenomenon is due, to a great extent, to the absorption of water in chemical processes and partly to the evaporation of water into the atmosphere. The shrinkage, consequently, will vary with the porosity of the concrete and with the thickness used in a structural member. In members of greater thickness and mass, the effect will be less or at least much slower and less apparent than in thinner members of less mass. Shrinkage is much greater in structures consisting of thin elements subjected to the intemperateness of dry climate, and disappears or becomes negligible in underground structures (e.g., foundations). Perhaps specialists will be surprised by the short superficial treatment of the subject here; but the main interest is only to explain how these properties affect the design of the structure. Under action of various states of stress to which it is exposed, concrete undergoes different types of deformation caused by the aforementioned phenomena. Under the initial load, within low limits there will be certain proportionality between stress and deformation (fig. 3 : 4 a ) . With increasing stresses the deformations increase rapidly, especially when the stresses approach the ultimate strength of the material and leave certain low permanent deformations. Hook's law of proportionality between stress and strain can therefore be assumed for any safe state of stress.

33

However, under loading maintained for a long period of time, as under proper load, the deformation will continue, following certain laws, and will fade away after a certain time (fig. 3 : 4 h ) . Yet, when the most important part of stressing occurs in the first months of hardening, during successive years these gradual deformations will increase continually and can exceed the initial deformations by two or three times. The longer the interval before the full load is applied, the smaller will be the slow deformation, which undoubtedly will be influenced by conditions of the environment. Finally, these deformations are partly reversible and will be recuperated as the loading is removed. All these conditions, together with low resistance to tension, give to concrete characteristics and types of behavior to be carefully considered in design. Failure in tension is automatically produced as soon as the external rigidity excessively reduces the possibility of retraction; and the thermal and hygroscopic variations frequently produce the same effect, causing spontaneous cracking. Concrete is much more heterogeneous than any other structural material. Shrinkage of the paste around its embedded aggregates produces tension in the paste which is resisted only by its plasticity and the series of nonelastic deformations already mentioned. The fresh paste, especially during the first days after setting, is more propitious and accessible to those adaptations. Thanks to this and to the continuous recrystallization and refilling of its fissures, the concrete maintains its cohesion; but the resistance to tension not only diminished in respect to that which it might attain were it not for the shrinkage, but becomes in a way more uncertain and less dependable. Failure in concrete is of the brittle type, as it is in other stone and ceramic materials already discussed. This is very important because it means that failure can occur without the warning of preceding strong deformation, which gives time to correct the danger and to avoid the catastrophe. After such an explanation, one might think that concrete is a poor, brittle, and miserable material; but, on the contrary, this complex of behaviors permits possibilities of adaption and regeneration, and the rehealing of its own wounds. Concrete structures have been preserved up to our day after thousands of years, side by side with stonework, holding their own as to durability and permanent resistance. In some Roman ruins, concrete still can be seen in good condition after the stone veneer has crumbled away, revealing material that was hidden as if unworthy. Concrete is as adequate as brickwork and even natural stone for the types of construction men of antiquity accomplished with these materials, and has advantages over them in more advanced constructions of today. Among normal ecologic agents, only frost can affect it unfavorably, if the concrete is

34

not compact enough or if the aggregates are not sufficiently resistant. Frost damage also occurs in certain natural stones, which for this reason are less suitable for utilization as resistant elements in a structure. Among the properties of various order usually required of concrete is the resistance to deterioration, which is of interest here only incidentally. In some instances the structural material is exposed to pedestrian and vehicular traffic, and for protection the surface is frequently reinforced with additional layers of special coating. This property and pertaining technological tests of resistance to deterioration and abrasion are of little interest in this discussion, but we should remember that overlooking them in certain cases may result in considerable damages and possible failure. Specific weight can decisively influence the selection of the constituent materials of concrete. Sometimes, as in gravity dams, high specific weight is of greater value to assure increased stability of the masonry against the overturning forces of water. In other cases, proper weight of concrete should be minimal for specified strength and other properties, because gravity produces excessive dead-load stresses (e.g., in long-span girders for bridges and roofs). Lightweight concrete (weight two-thirds that of ordinary concrete and compressive strength 4,000 pounds per square inch) is available today, and much progress has been made in recent years in this field. Lightweight concrete obtained with special admixtures producing bubbles throughout the mass shows lower resistance, which makes it less suitable for structural purposes; its practical application is limited to smaller structural elements adequately reinforced in which thermal insulation is required. Permeability and capillarity, which depend on the type and proportion of the aggregates, form, and size of the pores of the aggregate, should always be low in a good concrete, not only for the proper scope of the structure, but also for their influence on the durability of material. The prejudicial effect of frost, in general, becomes all the more noticeable as greater amounts of water are retained in the concrete and have no outlet when volume increases due to congelation. Today, we use special resins to reduce this peril. Porosity in concrete dams is of interest because it originates in the skeleton of the solid mass internal stresses, which are to be considered in combination with direct stresses acting upon the mass, and also in regard to the influence upon the stability of the whole structure or upon the tensional system in each individual part. A designer, weighing various properties of a material, should consider its durability, not only under normal ecologic conditions but also under the action of chemically agressive elements (e.g., sea water, sulphuric ingredients of soil, and smoke and gases resulting from certain manufacturing

35

processes). Natural stones and ceramic products resist these actions fairly well, although not always and not completely; under such circumstances concrete must have greater impermeability, and in many cases special cements are used which effectively resist such aggressive actions. Modern techniques and research have created new types of concrete, for instance, so-called Prepakt-concrete, in which forms are filled with loose aggregates and a cement mortar as a paste in a collodial state is injected. Its application in certain foundations, tunnel linings, repairs of dams, etc., considerably widens the possibilities for the use of concrete. This method improves the properties of concrete, especially reduces the detrimental shrinkage and is highly economical. Other techniques—like artificial extraction of water from concrete (vacuum concrete), treatment by heat, vibration, etc.— make immediate stripping of forms possible. There are many other modern methods and processes, but this broad field of research and improvement is beyond the limited scope of our discussion. Concrete presents more complex behavior than does stone or brick and has both advantages and disadvantages as compared with them. Its economy, especially in great masses, its easy adaptation to various forms, and its character of plasticity give to concrete a specific value without substitute in many cases. This value steadily increases, due to the continued progress resulting from experiments with different, more advanced techniques. In any event the cost of forms requires the adoption of simple surfaces, and concrete structures should be designed with simple polyhedral forms, or with surfaces of little curvature, if possible of the type generated by a straight line moving in space. This naturally does not mean that other, more complicated and more attractive shapes could not be used (e.g., shells with double curvature, domes, or complicated ornamental moldings, recesses, and projections). But such complication would increase the costs of forms and of labor and would reduce the economic advantages of concrete. However, in general, a structural element in concrete will cost less than a similar one in hewn stone. Sometimes it is desirable to give concrete surfaces certain textures, appearances, colors, etc. In such cases two alternatives are possible: either to apply suitable surface coatings, or else to select an appropriate surface treatment similar to those given to natural stones. In certain cases concrete seems inexpressive in terms of appearance. Artists are accustomed to treat it as a material of second order as compared with stone. Some deny to it the right to enter into the sanctuary of Art, thus affirming the lack of gracefulness of concrete. Perhaps this explains why Roman divinities preferred timber to concrete in bridges built by their worshippers. But otherwise, its suitability and even expressiveness have

36

been recognized, and new methods for surface treatment have been developed to improve its appearance or aspect. When the concrete dries, and the formwork is stripped, a gray, uninspiring surface is revealed. Only sometimes does the framework leave its imprint marked upon it, and it is perhaps this unbeautiful appearance that is mainly the cause of the horror that many feel toward it. Certainly it must be admitted that its appearance is prosaic, or at least not too elegant. Stone and timber show their inner structure through integrity and express its beauty. Concrete, even as steel, suffers because it cannot express its inner structure. But aggregates may be permitted to appear on the surface; the surface may be pricked or roughened; or special types of paint may be applied on the forms to avoid the setting of the cement near the surface; and so, leaving the aggregates visible. These are relatively inexpensive methods that improve the appearance of concrete. In some cases, prefabricated slabs of concrete mixed with white cement are used as forms and serve simultaneously as veneer. In other cases, interior forms or plates are used to separate layers of special surface concrete from the bulk concrete of the structure, thus permitting use of only a relatively thin layer of the more expensive facing concrete, which can have various patterns and colors. The interior forms are removed before the concrete begins to set. Besides, in such facing it is not only the color but also the texture and surface quality that are significant. It might be thought that concrete being softer than many natural stones, should have a more gentle surface (whether in actual fact, or imagined). But rather is it the opposite. It has sharp edges, and the marks along the joints of the formwork give it a rough, unkind appearance, unwelcome to the touch. The time will come when it will be possible to manufacture cements of warmer coloration and with properties that will permit a softer texture to be obtained from molding; then concrete will have more artistic possibilities. Another disadvantage, the monotony of large smooth walls, can be much more easily corrected by appropriate forming (e.g., by changing the direction of forming boards or arranging them in artistic patterns, or by increasing the joints with triangular battens) which can give to concrete surface a pleasant, attractive, and lively appearance. The ground, the very soil on which a structure rests, should also be considered as material. It has the same importance for a structure as water for a ship or air for an airplane. The bearing soil can be continuously solid (rock), partly or pseudosolid (clay), or a loose conglomerate (sand and gravel). In the latter the soil has no cohesion and offers only friction to act as the bearing and resistant element.

37

Fig. 3:5. Failure of the soil beneath a foundation. When the soil is composed of various strata, the stratification is responsible for anisotropics, deformability, and resistances in different directions. Sliding along certain planes is one of the more serious dangers, especially in layers of clay where water acts as lubrication. Independently of this phenomenon, the problem becomes more complicated in clayey soils, because interstitial and intracrystalline water is accompanied by many physicochemical phenomena. As the foundations affect the state of stress in the soil (e.g., increasing the compressive load), water will slowly escape under the foundation, following very complex laws. This evacuation of water will depend on various factors (e.g., geological consolidation of clayey materials, thickness of bearing layers and of surrounding soil), and will influence the size of the foundations and the time necessary for stabilization and consolidation. The contractions and expansions (swelling of the foundations soil) will also depend on variations of humidity and original biological pressures. In all sorts of soil, even if cohesion is small or nonexistent, failure will seldom be produced by separation, because foundations basically exert compression. Failure is usually caused by sliding, and the soil can resist only by cohesion of its particles, which is generally small, and by internal friction. If these limits are overpassed, failure along surfaces of minimum resistance to sliding occurs, producing the displacement of a certain volume of soil and with that movement a yielding of the foundation (fig. 3:5). When there are sand and gravel without cohesion, friction alone makes the ground stable and resistant to failure. For this reason it is advisable to provide foundations in greater depth, in order to increase the area of possible surfaces of sliding, and the resistance to it by greater depth and weight of the superimposed soil. Determination of characteristics that influence all these phenomena (e.g., by oedometric and other diagrams), especially those of cohesion and internal friction, is made on undisturbed samples obtained by improved methods of boring. Such samples, however, are not always easy to obtain. Also, we cannot forget that the other materials are man made, hence it is relatively easy to control their manufacture and quality, to attain uniformity. Conversely, except in rare cases, there is no certainty about the uniform quality of soil, and geological investigations do not help too much in this, because the characteristics of foundation soils are local and heterogeneous.

chapter 4 timber and steel

Wood is chronologically the first structural material having the capacity to resist equally tension and compression in the direction of its fibers. It is the only living material used on a large scale in construction, and like all materials delivered by the forces of Life, it is rather more adaptable and less rigid and schematic than other materials. There are no two pieces of wood having the same arrangement of grain and knots, even as there are no two identical fingerprints. Its attractiveness is largely due to its vital properties. Nevertheless, compared with other materials previously discussed, lumber has minor durability. Although there are types of wood which have resisted, under certain climatic conditions, centuries of intemperateness, in general, alternations of humidity and dryness rapidly affect wood, especially through the influence of bacteria, parasitic fungi, or insects. Here, under the eternal laws of Nature, a substance, once living, resigns itself to passing into the inorganic world, submitting to other living modes, which we think inferior. Definitely, modern techniques have introduced ways of greatly prolonging the life of lumber; but, all in all, wood must look upon stone, brick, and even concrete with the same envy with which we look upon Methuselah. After its life processes have stopped, wood becomes much more susceptible to ecological agents. Wood never stops dying; when it is exposed to drying and movements after being felled the ecologic agents continue their work, Every change of température or humidity will change the volume of the wood, especially across its fibers. Nature offers us wood in prismatic and straight workable pieces, the length of which is much greater than the width. The cross section of wood

39

is limited; some trees have stems of large diameter but the quality changes across the section; smaller sizes have better uniformity and are easier to sell. Wood is a material strongly anisotropic, consisting of longitudinal fibers glued together by resins or by tissues less resisting than grains (parenchyma: tissues composed of blunt-ended cells). The fibrous quality of wood gives it an attractive appearance, resistant expressiveness, and a vital structural characteristic, which are reasons enough for its use. The compressive strength of wood and its resistance to tension normal to the grain are much lower than those along the fibers. And, similarly, the resistance to slipping between fibers is smaller than that across the fibers. Veining often breaks down this form of strength, even in well-treated timbers. It is difficult to shear off a living tree trunk; it is easier to lop off branches by exerting tension and slipping between individual fibers. In wood types without prevailing knots the strength in compression and tension are practically the same. Another characteristic of wood is its great deformability under stress, which not only is elastic and proportional to loading, but also is plastic and increases as failure approaches. Wood is, however, subjected to deformations that are permanent and not totally reversible when strained by sustained load. In thin and long prismatic pieces under compression these deformations become dangerous due to bulging, because they are equivalent to lowering the modulus of elasticity, thus facilitating bulging.

Fig. 4:1. Failure of timber, according to Professor F. Stüssi.

Failure due to compression under permanent or sustained load is usually caused by excessive deformation and transverse bulging of fibers and along the walls of the cells (fig. 4:1). This type of failure takes time and is not abrupt, even as in other plastic materials. It can occur after the member is subjected to the same compression for months. This phenomenon should be considered in evaluating the factor of safety. In wood, failure in tension can be abrupt, although not so abrupt as in stony materials. The most important factors in timber structures are the connections that transmit forces from one member to another. The weakness of a connection in classical buildings is especially noticeable in members subjected to tension. Connection between two pieces in compression will require a more perfect cut of the ends in contact in order to transmit forces in as uniform distribution as possible over the whole contact area. This condition can be artificially obtained by grouting the inaccurate joint with cement mortar. In classical buildings connections exposed to pure tension were made possible only by converting the traction into shear or compression, as shown in figure 4:2a, b. This type of joint was called thunderbolt of Jupiter. The traction was resisted by compression along the surface a-a and by shear along the surface a-b. This connection is not able to transmit the total force that the two interconnected members can resist with their full cross section. It will be necessary to use more cuts of the previous type, as show in figure 4:2b, in order to increase the resistance of the joint, and it is evident that for such arrangement exact performance of mill work is necessary, otherwise all inclined cuts would not work simultaneously. Nailed connections of individual members made either by straight nailing or by using nailed splices increase their resistance somewhat by friction of wood against wood due to pressure on the driven nails. However, this type of resistance vanishes with time owing to shrinkage of the wood. Nails or spikes transmit shearing only with difficulty since—timber being comparatively very soft—they deflect easily. Thus they create a complex state of stress which permits certain distribution of the force among individual nails. There will always be appreciable slipping as the force increases. Nail connections become still weaker if the force changes their intensity, especially when compression is reversed to tension and vice versa. To be effective, nailing requires reduction in thickness of connected members in order to provide greater nailed contact areas as compared with their cross sections. Despite all deficiencies, nailed connections have been used for many masterworks of engineering (e.g., the centering of the Plougastel Bridge). Other types of connectors are bolts with metal rings

41

SHEARING f W N t t

MtTAl 6vssIT VJiT4f &OLTS

SECTiON

DIAGONAL

BOLT

TIE MEMBER ROUGH METAL WASHERS

Fig. 4:2. Various types of timber joints (connections,

a-d).

(fig. 4:2a, d ) in various forms. Connections always represent weak points in the structure with respect to breakage and sliding along fibers. These weaknesses, owing to stress concentrations in the timber around the bolts, begin to develop when the wood shrinks and its friction with the plates lessens. New techniques bring about new structural trends; an example is the encouraging development in glue-laminated material—large boards assembled of thin layers, glued together with synthetic materials into structural members of any cross section. This is basically a material different from common wood cut out of the tree trunk. Its structure is monoxylic; it has some properties of other materials and behaves as a whole quite differently from plain wood both in techniques and structural processes. Synthetic glues and the drying of them by infrared rays or by diathermic processes are revolutionary in modern techniques of timber structures (fig. 4:4). Plywood actually was the prototype of glue-laminated members, being a material with isotropic characteristics in the plane and having the same mechanical properties in two directions. Its application in the structural field is growing from year to year.

42

Fig. 4:3. Arched centering of the Longeray Viaduct, France. Limousin Construction Co. (from Cent Ans de Béton Armé; Paris, Science et Industrie ).

Fig. 4:4. Glue-laminated timber arch. Most of the important properties of wood as a structural material are due to new treatments improving its resistance to ecologic agents and its elastic behavior and to improvements in mechanical connections in proportion to the increased resistance, stability, and durability of the material. Much work has been done in development of new techniques. But much still remains to be done until these techniques are ready for effective and wide industrial application. Furthermore, structural methods will have to evolve which will exploit the full possibilities of this new material. Relatively recent times have witnessed the development and use of structural materials other than stone and wood and with them new structural principles. These materials are cast iron, weldable steel, and reinforced concrete. Cast iron, which preceded steel, is a material of great specific weight and great resistance to compression. Its tensile strength is much lower than its compressive strength, but still exceeds that of either of the classical materials, stone and wood. Its elastic behavior is not perfect, and deformations in-

44

crease more rapidly than corresponding compressions accompanied by creep, i.e., showing residual deformations after a load is removed. Cast iron has a high thermal expansion and no hygroscopic coefficient; its spontaneous deformations are appreciable during the first months of its life. Pig iron is molded in forms according to certain rules and limitations imposed in principle by the process of melting at high temperatures followed by a cooling process during which retraction is not uniform throughout the casting, so that residual stresses originate. These residual stresses considerably affect the tensile strength of casting, and may even weaken them to the point where they cannot take any practical external load in tension. For these reasons, the thickness and other dimensions of casting have their technical and economical limitations. Cast iron is somewhat similar to concrete, although of greater homogeneity and strength. The principal difference between these two materials is that cast iron is molded in forms in the workshop and not in situ as is most of the concrete used. Although cast iron is used in certain structures with elements of dimensions much smaller than those of stony materials, it can never compete with steel as to slenderness, and cast iron members will always appear robust and heavy. Bolted connections in cast-iron structures will have the same deficiencies as the material itself, and will have greater resistance in compression than in tension. Modern welding techniques have not been able to make this brittle material more usable for structural purposes. It was used in arch bridges and columns in the last century, but more recently, it has been replaced by the more economical concrete and rolled steel. Molded and forced steel are not considered as structural materials and are used with advantage only in such auxiliary or local elements as pivot pins and rollers in bearings. Its good resistance both to tension and compression, which is equal or superior to the strength of rolled steel, do not compensate for higher costs and difficulties involved in its préfabrication process. Typical structural steel, with its many rolled shapes, is now replacing practically all other types of materials produced from iron ore; and although exclusive use is not yet attained, it continues to be irreplaceable, in realization of high structures and great spans in construction. Steel has much more technical character than classical materials, and its application in structures has different features and style, since its fundamental property is high strength. Before going into details, we should be aware that steel is not likely in our day to be given adequate expression of its qualities such as that given stone, wood, and bricks. Except in large engineering works where steel is permanently protected by painting, this perfect structural material is shame-

45

Fig. 4:5. Quebec Bridge. Photograph, E. Torroja. fully hidden in current structures, behind foreign elements alien to its character. And it is not only lack of natural color which causes this strange situation; it is rather the fixed, hard, and inflexible character of its rolled shapes. Steel cannot express variable, natural textures as does wood with its fibers showing the character of growth. The tendons of steel flanges emerge nakedly in a structure giving it an essentially tendinous character, which asks frequently to be wrapped in a skin of other materials. In steel, tenacity and resistance predominate, its edges and the contours of the assembly are impressive; its potent lightness is overwhelming. The beauty of this material can be seen in the completed skeleton, though not in its individual structural members or in the final facing. The morphology of steel elements is always unpleasant, no matter whether their shapes are closed (e.g., in box girders and columns) or open, with flanges and legs visible; these elements—unfortunately—are not always considered for orderly aesthetic effect and often leave openings that are not harmonically disposed. 46

Steel looks a rude, rough, primitive material like the inhospitable cactus; but it is light and strong, and is, besides, much more ductile and trustworthy than it appears. Gradually we learned to understand it and to adjust it to our aesthetic exigencies, but it never presents such adaptability and such variable possibilities of forms as other materials do. Per unit of volume, steel is the most expensive structural material and has the highest stress capacity. Simultaneously, it represents the most wanted morphological determinism as a consequence of the process of lamination. We have now developed various shapes of thin plates and sheets, corrugated or extruded, which represent new and promising techniques not only of creating structural elements of extreme lightness but also of opening new ways to the evolution of resistive forms. It may be that these forms will make it impossible to escape the limitations imposed by the shapes of rolled steel, especially in the small elements. Like wood, rolled steel for structural purposes is delivered in straight pieces (sheets and plates may be compared with plywood). In wood and in rolled steel resistance to tension and compression is practically equal, hence the connections of both materials were based upon the same principles. The main difference between these two materials pertains to cohesion, which in wood is much greater along the fibers than across the grain; in rolled steel there is no such big difference. In wood, the difference between resistance to axial stressing (to tension or compression) and to shear is great, the latter being only a small fraction of the axial strength. In steel, however, this difference is very small, and this explains why I-shapes and channels with thin webs—where the shear usually attains its maximum—have been adopted and have become typical. Tubular and box shapes, based upon the same principle, are not so common because of cost of fabrication. Since the strength of steel is so high, the rolled member can be quite slender. Flanges are added to the web, as in 1-, L-, and Z-shaped profiles, to obtain lateral rigidity and to simplify connections. There is a certain limited number of types and sizes of steel shapes that the designer must accept. He has no freedom to create others. In prismatic members having relatively small cross section in comparison with the length, the flexural behavior of the steel is of special interest, since the bending produces simultaneously both tension and compression. Mild steel as used in typical rolled shapes maintains its elasticity proportional to strain to a certain limit. Beyond this limit, deformations increase under practically the same stress or with small variation, as demonstrated in a stress-strain diagram (fig. 4:6). Finally, increasing load produces parasitic flow, which ends in failure.

47

600

500 '•si

10

cj^ 400 % in 300 Uj

200

if

"i II if

if

100 Fig. 4:6. Stress-strain diagram for mild steel.

0.05 0.10 0.15

o.zo 0.25 O.JO 0.35

5TRAIN

After the elastic limit is reached—naturally leaving permanent deformation—and the load is removed for a certain period of time under normal temperature, the material begins again to behave elastically under renewed loading to approximately the same stress intensity it has been obliged to accept before. This phenomenon is utilized for increasing the period or range of elastic behavior and has certain importance for reinforcing bars embedded in, and protected by, concrete, for which usually the permissible stresses are higher than those specified for steel shapes. There is no place here to discuss the causes of this peculiar behavior. The polycrystalline nature of the material and the coexistence of crystalline amorphous phases, with special arrangement in intercrystalline surfaces and possibly under migration of certain atoms due to the effect of stresses, would require study and explanation far beyond the purely technical problems. From a broader point of view, the fundamental facts are that after the elastic period up to certain stress values the material can accept irreversible deformations, much greater than the elastic deformations under sensibly constant stress; and finally, the material progressively yields before it breaks. Moreover, steel shows neither hygroscopic nor delayed deformations when submitted to permissible loading under normal temperature. On the contrary, its thermal expansion coefficient is much higher than that of any stony material. Forces inherent in the rolling process of steel and the consequent variations of temperature in the cross section of the shape, especially when the cooling process is not uniform, cause permanent residual stresses, which, in very large shapes, are sometimes of such intensity that spontaneous failure occurs (fig. 4:7). These are rather exceptional cases, but this phenomenon can produce serious consequences in welded structures. Causes of brittle failure—due to disintegration and loss of cohesion, and characterized by absence of the warning period of plastic flow—have recently been thoroughly studied in connection with serious and spectacular

48

Fig. 4:7. Spontaneous failure of a steel beam caused by parasitic stresses (Professor F. Campus). accidents (e.g., failure of certain welded, and in some instances even riveted, ships and bridges). Temperature is an important factor. Transition temperature, under which the material loses its ductility, varies from one type of steel to another, but in some climates this temperature range falls within the limits for which the usual types of steel and customary techniques of construction are used. The brittle rupture of steel is influenced by such factors as metallographic composition, temperature, rapid increase of stresses, superposition of residual stresses, etc.; however, of decisive importance is the type of stress distribution. States of biaxial and triaxial tensions always produce in a structural member marked tendency to fail by separation rather than by slippage. W e shall not discuss such questions as how far biaxial stress in the plane of thick plates, in collaboration with the anisotropy resulting from the rolling process, can produce the same effect, or how experimental stress analysis can detect this phenomenon. It is, however, of extreme importance to the designer to know that any triaxial state of stress (as produced in certain interior cuts and openings, occlusions, reentrant angles, etc.) is dangerous and can cause failures of this brittle type; that is, failures without previous indication or sign of weakening (e.g., large deformations that appear before failure in the ductile materials used today). Upon such ductility is based our confidence in the safety of most arrangements and types of connections. Besides such special and abnormal conditions that we always try to eliminate and avoid, strength exhaustion in typical structural steel follows certain laws of ductility before the materials begin to flow and to yield. Only

under increased loading, accompanied by large deformations, will the material begin to yield and finally fail. In steel structures the problem of connections is as important and delicate as in structures of timber, since the materials have so many common properties. Not long ago, the problem of connections in steel construction was solved only by riveting, the rivets serving the same function as nails and screws in wood. A rivet, however, provides a better surface friction than a nail, as well as greater resistance to slipping, because the connected and connecting materials have the same resistant characteristics. However, the state of stress in and around a rivet is much more complex, especially if rivets are used in groups, as shown in figure 4:8. Along the perimeter of a rivet, as around any orifice, the stress intensity varies considerably. The network of isostatics changes from point to point, due to holes and concentrated forces transmitted by individual rivets, and the stress distribution over various rivets is influenced by deformation of the plates under the acting forces. In general, a certain cross-sectional part of the member is under no stress, since the stresses are concentrated in the immediate proximity of the rivets. If the connection proves effective, it is because the material undergoes, before failure, considerable deformations of plastic character in zones where maximum stresses occur, thus relieving stresses and producing a more uniform distribution of them in other regions. This is the principal and frequently the only advantage of mild steel: its ductility and tenacity which make shaping possible, and permit it to undergo deformations that equalize and remove stress concentrations. The most suitable steel for our structures is not the type of noble tempered steel such as that used in the swords of Toledo, whose hard temper makes them willing to bend with elegant grace under reasonable forces, but will break suddenly rather than submit, under permanent deformation, to a

Fig. 4:8. Stress distribution in a riveted joint (from Résistance des Matériaux et Elements de la Technique de l'Elasticité et de la Plasticité des Corps solides, Vol. I, by L. Baes; Brussels, Maurice Lamertin, Editeur, 1930-1934).

50

cruel imposition. In contrast with such highly tempered material, steel for structural purposes must maintain its strength, be filed, punched, perforated, and shaped without exaggerated effort to forms permitting most favorable stress distribution for carrying maximum load and must firmly retain its tenacity up to the last moment before failure. If the material were perfectly elastic up to the point of failure, the aforementioned effects would not be possible, and riveted connections would lose their efficiency. For this reason, hard or tempered steel is not very suitable for this type of connection, and in semisoft steel we should avoid all fissures normal to the tension (such as occur around punched holes, which are not scarified). Along such fissures the material suffers strong stresses and loses a great part of its capacity to deform plastically. In a brittle material the even stress concentration caused by perforation alone can produce sudden failure, as many examples have shown. All this discussion does not indicate that rivets are bad for connecting structural members; on the contrary, they have been used for a long time and experience is still proving their efficiency. However, these successful applications were possible only because of special characteristics of the material's mechanical resistance; and the type of stress distribution in a riveted joint, as shown in figure 4:8 by isostatical fringes, cannot be accepted as the best, because of the great stress concentrations. Consequently, despite all regulations and rules of thumb given in textbooks in regard to arrangements of rivets, their size and spacing, etc., it is always recommended that riveted connections be designed with special care that the curvature of these isostatic lines is not too sharp, high concentrations and discontinuities of stresses are eliminated, and bi- and triaxial states of stress are avoided as much as possible. During recent years, methods of welded connections have been highly developed by using special electrodes and techniques. Welding, however, presents not only advantages but also disadvantages worthy of study. Welding permits butted connections that cannot be accomplished by riveting. Welding also makes possible, as riveting does not, lapped connections with transverse and longitudinal fillets with respect to acting forces and many other varieties of connections, even in oblique planes (fig. 4:9). Butt welding undoubtedly has many theoretical and practical advantages. Stresses can be transmitted through them adequately, without affecting the isostatics, so that the connection acts as a monolithic part of the member without any mechanical and, therefore, theoretical discontinuity. Under such conditions material can be much better utilized and the welded structures are more economical.

51

Fig. 4:9. Welded joint on the Tordera Bridge, Spain. Photograph, Ferriz. Yet, this is not true in every instance, especially when certain structural difficulties arise, or when basic abnormal states of stress are produced by thermal deformation caused by welding processes. As the welding material is deposited, at the temperature of fusion, in fillets upon the basic material of normal temperature, thermal expansion and contraction occur, varying from point to point not only during the welding process but also later during the cooling period. Expansion and contraction produce stresses of great complexity which later greatly disappear due to the ductility of material under high temperature. In some part, however, these stresses will be "frozen" into the connection under normal temperature. There have been many investigations and much discussion of the importance of these residual stresses and especially of their effect on the resistance of welds and the resistance of members in the proximity of welds. Actually, two types of stresses must be distinguished: (1) stress produced at the joint due to contraction of the weld in instances where structural members are more or less unable to contract because of connections with the rest

52

of the structure; ( 2 ) local residual stresses produced when the other end of the structural member can move freely. Residual stresses of the first group produce effects external to the weld, which is not true in the second group. The first group pertains to secondary stresses due to the welding and the second to parasitic or residual stresses ( already mentioned in the discussion of rolled shapes), which are secretly hidden inside the member without being transmitted to the rest of the structure. Both types of stresses are of hyperstatic character; however, only the secondary stresses can be introduced in structural analysis if the contraction coefficient is known. The socalled parasitic stresses are difficult to determine, and calculation of them requires complex and delicate study by specialists who, to date, cannot claim to have reached the final object of their work. It is said that this type of residual stress has no great importance because, under the superposition of forces due to exterior loads, as long as the elastic limit is surpassed, plastic deformation will rearrange the stresses into equilibrium so that in the end no extreme stresses will exist. However, that is not so general as it would seem to be and has been demonstrated by tests of F. Campus (University of Liège, Belgium). If a circular opening in a steel plate is closed by an adequate piece of plate and welded in order to reestablish the continuity, the weld will warp, twist, and break. It is therefore necessary to avoid any excessive rigidity facing the inevitable contraction of a weld which would cause triaxial stresses, or biaxial stresses in the plane of the plate, in some instances excessive and isotropic. This does not mean that there are cases where parasitic stresses (such as the secondary stresses produced by welding) can have certain importance and can be utilized to compensate part of the forces to which the structure may be later subjected, or to reduce them to a certain extent by the method of welding and the type of bead used. Recently there have been introduced several methods and processes in the techniques of welding which regulate and reduce parasitic stresses in some rigid structures (e.g., steel pipes). These processes include cold forging of welds (to a certain degree resulting in plastic deformations) and heating and low cooling after welding. No failure or serious damage due to the parasitic eflFect of welding has occurred in linear or framed structures where there has been good workmanship and an adequate method of fabrication, in which semimild steel of good quality and suitable electrodes have been used, as long as the structural members and their connections have not been subjected to excessive bi- and triaxial stresses and as long as there was no great structural rigidity. However, we should not forget that, as shown in figure 4:10, if a crossbar is welded to a bar in tension, the isostatics will enter, after welding,

53

Fig. 4:10. Stress distribution in a plate in tension near a stiffener.

TTTTTTT

1=1 tension

into both parts of the jointed crossbar thus producing in them tensile stresses normal to the principal force in the main bar. Although this arrangement is not prohibitive, nevertheless, critical stress combination could take place, and such cases should be cautiously investigated. It is more convenient to weld along horizontal or slightly inclined edges. In general, the designer and the fabricator should be thoroughly acquainted with new welding techniques in order to avoid the difficulties and failures which often occurred in the past. Furthermore, the problem will be also affected by the weldability of the basic material, properties of the electrodes, and by continually improving welding methods. Steel is a very valuable structural material because of its great elasticity and its strength both in tension and in compression, and, in certain categories of structures, it can hardly be replaced. But it has a very serious disadvantage: it oxidizes easily, even in relatively dry surroundings. Because stainless steel is very expensive and its fabrication is difficult, it cannot normally be considered as a structural material for larger buildings. Generally, it is necessary to rustproof steel by paint or by other protective coating. Because of the combined action of wind, rain, sun, etc., the protective coating must be renewed periodically, which means increased costs caused by constant necessary maintenance and curtailment of its practical use and life-

54

time, as compared with other materials (e.g., stone, clay products, concrete, etc.). Present methods of analysis attempt to utilize tenacity, maleability, or ductility of the material derived from its rheologic behavior, and various modifications are introduced in calculating the stresses, as, for example, the elimination of the extreme parts of diagrams representing bending moments and direct stresses over the supports of continuous beams. We meet another problem in the fatigue of structural materials; that is to say, the reduction of their strength due to cycles of loading and unloading or to repeated changes of stresses, which are especially important if the stresses alternate in sign, changing from tension to compression or vice versa (reversible stresses). Beyond the elastic limit, failure will occur under fewer repetitions of alternating stresses. For this reason the postelastic utilization of semimild steel—as it is being discussed at present—is possible only under the action of dead load and not under frequently repeated live load. Nevertheless, before the elastic limit is reached, the strength of the material is often reduced to half when it is subjected to a large number (more than a million) of repeated stresses, and then the failure occurs suddenly. Since the reduction of strength and the danger of failure depend on the ratio between maximum and minimum stresses during the aforementioned cycles, this problem of fatigue becomes more important in mechanical than in structural members. However, this problem is of serious concern in certain structures exposed by their nature to variable loading (e.g., foundations of machines and mechanical equipment under steady vibrations or oscillations, railroad bridges under repeated vibrations caused by passage of trains plus shocks produced at the joints of rails) in which the number of repeated stresses is very high and is responsible for considerable reduction in the required strength and in the period of use. These facts should be kept in mind by the designer in his choice of structural material and of structure type. Designers of steel structures, especially those with welded connections, are considering more and more the resilience of the material, which is currently measured by the total energy elastically or plastically absorbed before failure by a certain type of specimen cut out of the material and subjected to bending under impact loading. This is of special value for utilization of certain types of high-strength steel, which permit greater lightness and economy of structures. Selection of the material and of the structural type is fundamental for design. In general, the higher the yield point, and therefore the ultimate strength of steel, the less ductile is the material and the more susceptible to abrupt failure, especially when the structure is very rigid.

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chapter 5 reinforced and prestressed concrete

Reinforced concrete is a new material having characteristics basically different from those of its constituents, concrete and steel, although these materials maintain their properties. It may properly be said that in reinforced concrete steel gives tenacity to stone and concrete gives mass to steel. Reinforced concrete is an organically constituted stone in the mass of which the tensional function of the reinforcement is effectively distributed and arranged in such a way that the concrete at every point will resist tension in accordance with the existent network of stresses. It is therefore technically the most nearly perfect material and is unique in deserving the designation "adequately resistant structural material." This antithesis to the brittle behavior of ordinary mass concrete, which has the same appearance, took many years for assimilation by artists and even by the technicians who were in charge of improving forms and shapes of concrete structures, the evolution of which is not yet complete and will culminate in still more original conceptions. Reinforced concrete is a unique material whose structural behavior cannot be judged by its apparent organic values, because the essential resistive element—reinforcement—embedded in its interior is not visible and must be imagined, thus giving to classical stone a vigor and tenacity never before known among the products of inorganic creation. Reinforcing bars in concrete are not tied together as cables, welded members, or other similar interconnected structural elements. Both materials—concrete and steel—work together by adherence, including fric-

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Fig. 5:1. Projecting canopy of the Ramiro Maeztu Institute, Madrid. Architects, M. Domínguez and C. Arniches. Engineer, E. Torroja. Photograph, M. García Moya. tion. Usually reinforcement consists of round bars with diameters seldom exceeding 1% inches, which have normally deformed surface to reduce the danger of slippage. Under such conditions and with sufficient coverage and spacing of individual bars, reliable transmission of stresses from concrete to steel and vice versa can be attained. Simultaneously, the relative impermeability of concrete and the alkalinity of the ambient air in the concrete make the steel practically inoxidizable, and give to the whole compound material better durability, if, as has been already mentioned, the embedment and the quality of concrete comply with the ecologic conditions to which it is subjected and as long as the reinforced concrete is not exposed to aggressive chemical actions. Originally, reinforced concrete was not a product of science and technology. It would hardly have been possible to imagine and to believe in its effectiveness, and the results of experiments surprised the inventors, who had no conception of the actual values and the application possibilities of concrete. Even today, it is difficult exactly to explain the interaction of the concrete and the steel. When Joseph Monier started his flower pots with cement mortar in which he incorporated wires, he did not realize he was creating a structural material which several decades later would be used for construction of bridges and buildings surpassing in dimensions, beauty, and economy everything that had been achieved in the past. Since the expansion coefficients of concrete and steel are sensibly different, and since concrete possesses hygroscopic deformations and initial

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shrinkage (properties that do not appear in steel at all), and since steel behaves quite differently under stress (as has been demonstrated), it will be necessary to resort to other phenomena of adaptability and other explanations, more subtle and obscure, to demonstrate that the steel will not deform, break, or slip independently of the concrete. The fact is that concrete and steel remain united as a solid material due to the effect of nonelastic properties (e.g., conditions of adaptability and the possibility of local relaxation in stresses, caused by microscopic fissurations and recrystallizations of the paste without rupture, etc.). It is evident that shrinkage of concrete has a tendency to produce tensile stresses in concrete and to put the reinforcement partly in compression, thus increasing compressive stresses of steel under loading. Yet, gradual and slow deformation will amortize the prejudicial effect, and no dangerous cracking will occur as long as the amount of material, the method of anchorage, the embedment of reinforcement, and other requirements are fulfilled within the limits and regulations based upon practical experience and tests. The designing engineer must stick to these regulations and specifications strictly and must have a good knowledge of these properties—their origins, contributing causes, and counteracting effects. All this knowledge is necessary if he would design large buildings efficiently. It is essential before he would dare to attempt modification of these regulations or extrapolation of new ideas and dispositions. Sometimes the concrete will break owing to shrinkage when the reinforcing bars are too crowded or their diameters too large, because continuity of the structural members, which makes it more resistant, is interrupted. A slab might crack because the reinforcing bars, although sufficient for the intended bearing capacity, are too far apart. Many other such examples could be presented. However, this does not mean that reinforced concrete is so definite and regular that a designer would not have any freedom for his own ideas and modifications. On the contrary, a distinguishing characteristic of reinforced concrete as a structural material is the freedom it allows to the designer to select various structural schemes. Unlike rolled steel, concrete is not available in definite shapes listed in a catalogue; its forms and dimensions must be designed. Reinforcement can be increased or decreased, distributed or concentrated in a certain way in one part of the structure and continued in another way in the adjacent part (e.g., in continuous beams). Every bar can change its position in accordance with the directions of stress. A steel shape changes its bearing capacity with variable thickness and with additional plates connected with flanges and webs; a reinforced concrete mem-

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Fig. 5:2. Open-web girder (Vierendeel girder) at University City, Madrid. Photograph, Férriz. her changes its bearing capacity with any arbitrary shape of the cross section, with the variable amount of the concrete and the steel. This freedom of dimensioning makes the design of a reinforced concrete structure very difficult because there are so many details to be considered and correctly evaluated. It is also necessary to consider adequate construction method, scaffolds and forms, the placement of steel, and the pouring of concrete. The primary purpose of reinforced concrete is to obtain a material in which tension is resisted by the reinforcement, and compression, by the concrete, in the most economical way. It is true that the reinforcing bars embedded in concrete and connected with transverse reinforcement in order to avoid its buckling are able to resist compressions. However, in general, it is more expensive to use additional steel in compression than to use adequate amounts of concrete. The ratio of these two comparable costs per unit of load is called the "economic-resistant coefficient." An example should clarify the concept of this coefficient. Let us assume the working stresses of concrete and steel to be 800 and 20,000 pounds per square inch, respectively. It can easily be proved that if the price of one pound of steel is not lower than % 0 o of the price of a cubic yard of concrete, both materials in place, it will not be economical to use steel in compression. Naturally, this has no general meaning, and it is not necessary to emphasize that the use of steel in compression might be economical in spe-

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cial cases where it would prevent considerable and abrupt changes in crosssectional areas or where other factors necessitate reduction of these areas. However, in tension zones, even if the low value of tensile strength of concrete, which is about one-tenth of its compressive strength, would be permissible, the coefficient of economy and resistance is reversed, and preference must be given to the steel. Tensile reinforcement can be subjected to maximum permissible value, with the restriction that the corresponding elongation (expansion) of steel would not provoke dangerous cracking of the surrounding concrete. As has already been noted, theoretically it is difficult to justify or assure the absence of fissures and cracks in reinforced concrete. It is equally difficult to foresee the importance of these fissures. So far as the reinforcement is concerned, usually its good arrangement in placing—preferably in smaller diameters—is just as important as the amount of steel. Suitable distribution of rods connected with ties and stirrups can eliminate visible cracks. Fissures not exceeding to %oo of a n inch are acceptable in most structures in which the tension in concrete has not been considered in calculation and in instances where the surrounding atmosphere would not cause oxidation of the reinforcement. If ordinary steel with relatively low working stresses is used and if its quantity is within the limits specified by regulations, fissures seldom exceed the aforementioned limit. Otherwise, it will be necessary to grout the cracks to prevent oxidation of the reinforcement. In any case, such cracks should be avoided as much as possible, be it only for a better appearance. For this reason, high-tensile steel, which has a much better economicresistant coefficient, could not be used in ordinary concrete. As its modulus of elasticity is practically the same as that of mild steel, its higher working stress would cause large cracks due to its greater expansion. High-tensile steel could be used in ordinary concrete up to a certain limit of stress only under special conditions (e.g., with deformed bars or twisted bars of small diameter). The problem of connections, which is inevitable in structures of timber and metals, practically has no existence in structures in reinforced concrete. Welding or simple overlapping solves the continuity of reinforcement, and pouring in situ assures the monolithic character, if desired, of any structure. In reinforced concrete, there are certain structural details where metallic elements, like hinges and expansion joints, are used; however, since the angles of possible rotations and movements are relatively small, various types of hinges without metallic elements have been proposed, for example by Mesnager and Freyssinet (fig. 5:3). They work satisfactorily and are inexpensive.

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'Hinge faesnatper)

Fig. 5:3. Hinges in reinforced concrete structures (type Mesnager and Freyssinet). In recent years, prefabricated (precast) concrete elements have been more extensively used. They are more economical because costs of forms and scaffolds are considerably reduced. This method has been especially adopted for structural members in prestressed concrete—a new material of rapidly growing interest and importance. Among new developments in the field of reinforced concrete, of special interest is the combination of concrete with inserted glass units—glass concrete—which gives to resistant concrete slabs, shells, and domes the character of skylights. Contrary to what might appear at first sight, the glass, more than the concrete itself, is the load-carrying medium. Shrinkage and creep finally transfer to the glass most of the compressive loads, and the concrete remains in the end as mere bond between glass and metal. For this reason the structure is calculated as if it were of reinforced glass, on the basis of the proper ratio of moduli of elasticity for steel and glass. But it is necessary to take account of the special properties of the glass: this can break after a long period under smaller loads than those resisted in a brief test. These disadvantages and the different thermal coefficients of both materials make it advisable that the amount of concrete be reduced to the minimum necessary to embed the glass units and reinforcing bars. Prestressed concrete is being used in two basically different schemes: (1) with pre-tensioning, in which reinforcement is stressed before the concrete is poured, and (2) with post-tensioning, in which the concrete, sufficiently hardened, is stressed by tensioned wires or cables with adequate anchorage in the concrete. The stressed reinforcement is being placed either inside or outside the concrete member, and, in some way, independent of the concrete during the whole lifetime of the structure, being attached to it by addi-

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tional concrete or grouted with cement mortar inside the structural member. Both methods are widely used in various applications of great interest. Among structural materials, prestressed concrete is the most fascinating, yet also the most complicated invention of today (fig. 5:4). With it, new types of structures and construction methods were introduced, basicallv different from other techniques. When I explained my design of a prestressed concrete bridge to a typical contractor in charge of its construction, he said: "I suppose that you proposed and calculated all this in order to make the music sound when you cross the bridge. Otherwise I don't see the reason for complicating our work so much with so many pre-tensioned wires." As a matter of fact, the harp of a suspension bridge is quite a simple problem when compared to the great complexity of a prestressed bridge, in which every single cable must be stretched to the correct tension within its separate cavity. Those who are not very well acquainted with the disadvantages of a new apprenticeship always affirm that the extra cost of labor surpasses all the savings of materials as the designer is figuring on his drafting board. The basic economic disadvantages attributed to structures in prestressed concrete will go on disappearing day by day because of technical progress and industrialization, and applications of the material will increase extensively, thus substantiating advantages logically foreseen. One of the principal advantages of prestressing is the high economicresistant coefficient of high-tensile wires, both isolated and in strands or cables, due to the strength which enormously increases with decreasing diameter of the wires. Prestressed concrete, with pre-tensioned wires, actually is used almost exclusively for members prefabricated in workshops and requires reinforcement of much higher elastic limit and strength in order to obtain greater initial tension than required by design, thus counteracting the losses of tension produced by shrinkage and creep of concrete under compression

Fig. 5:4. Prestressed concrete bridge at Esbly, France, by Freyssinet method. Photograph, H. Baranger.

exerted by prestressing. However, the loss of prestressing is only a small part (approximately 10 per cent) of the initial stressing to which structural members are subjected. The intensity of this relief is to be watched carefully, for reduction of prestressing, partial or total, due to retraction and visco-plastic flow of concrete, is the principal enemy of this new material. For this reason, a new factor, time, must be introduced into the designs and calculations of prestressed concrete. If this new variable is not considered along with all these effects of viscosity and plasticity, the stress conditions can become totally different from those intended. Furthermore, besides the aforementioned deformations of concrete, it is necessary to account for special properties of high-tensile steel wire. Much greater deformations of microcrystals in steel wires are produced during its fabrication than by simple lamination. The tendency of their orientation in the surface layers along the axis of the wire explains why sometimes stresses below the conventional elastic limit can produce important creep and considerable stress relaxation in the wires. It is of great importance to have the wires used for pre-tensioning without anchors cleaned from grease or any rust-preventing coating. Therefore it is a major task of specialists to investigate the properties of steel and its treatment and arrangement in order to reduce these unfavorable effects to a minimum. In prestressed concrete, the tension of steel and the compression of concrete, as well as parasitic phenomena (independent of external solicitation to which the materials will be exposed), are maintained at least in a certain proportion by the bond between the steel and the concrete. It is therefore necessary to use thin reinforcing elements with relatively greater surface and even use corrugated wires to increase the bond and to prevent slippage. In the practice of today, it is recommended that prestressed reinforcement be provided with special anchors. However, these variegated reinforcements, very thin and properly distributed, make concrete more plastic in its tensional deformations. This property is especially important in pre-tensioned concrete. Another advantage of prestressed concrete is that, as long as the reinforcement is tensioned within the elastic limit, cracks, which can appear under excessive loading, will disappear as soon as the excessive loading is reduced, so perfectly that not the slightest fissure is any longer visible. The designer must take into account specific limitations of certain structures, either pre- or post-tensioned. One is that the wires or cables cannot be placed in curved tubular cavities in the concrete with the same ease as in ordinary poured concrete, because the friction of tensioned wires in a curved tube reduces the effect of tautening. Another limitation is set by the fact that the initial tension is not final, because there will be changes

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under variable live load and due to certain slippage in the anchorage. In many beams the initial state of stress produces bending moments of the opposite sign than moments under full live load, due to the eccentricity of prestressed reinforcement. Sometimes it will be necessary to change the shape of the beam, for example, reverse the position of the T-beam to provide greater area for the reinforcement and for the excessive eccentric compression that appears in that zone before the full load is applied. Prestressed concrete, cast in the workshop from the best materials, makes possible the manufacture of slender structural members of great flexibility and high resistance which appear to be made of a quite different material and have perfect elasticity and impermeability and are not subject to cracking. More important is that prestressing allows the complete utilization—without fissures in concrete—of high-tensile steel of a better economicresistant coefficient under normal and commercial conditions of the country. Under these effects, tension in the reinforcement can be raised closer to the ultimate strength because during manufacture higher tension has already been applied in order to compensate for the effects of creep and shrinkage; the material has already undergone sufficient testing. Prestressed concrete is normally used with reinforcement running in only one direction, although two- or three-way, mutually perpendicular reinforcement can be used. In the latter cases concrete of very high strength and ductility is obtained, since failure by separation as in a brittle material becomes less likely to occur. Prefrabrication of prestressed members in the workshop permits better mixing, curing, and special treatment (e.g., curing under higher temperature). This treatment also has industrial and commercial importance because of the speedy hardening which results in better properties as greater strength, uniformity, economy, and in lower weight in relation to the bearing capacity, which is especially important for transportation. In their recent development and progress, precast and prestressed concrete were mutually influential, because they present the same problems with regard to suitable connections, problems that are not presented by ordinary concrete (concrete in situ). However, these problems can be solved easily, as will be explained later. Connections in precast members of reinforced concrete present weak points, even as in members of other materials, because the bond between hardened and fresh concrete is always much lower than the adhesion and the shear in a precast unity. This deficiency increases with time due to unequal retraction and different deformations in both types of concrete. It is necessary to allow the reinforcing bars to protrude in the direction of internal forces and to bend them at the ends for better anchorage in the new

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concrete, thus assuring the required continuity in tension. Or, in the concrete members to be connected, suitable cavities, holes, or recesses must be left, into which bars are inserted with proper anchorage. Such difficulties and deficiencies disappear completely in poststressed concrete. Plane joints between contiguous blocks with cement mortar and maintained in constant compression by stressed reinforcement installed across them are absolutely secure against bending and cracking and are even highly resistant in shear. Under sufficient compression the friction in the plane of the joint is very high and prevents slippage. These ideas and arrangements created new construction methods which yielded unprecedented achievements in longspan bridges consisting of assembled and post-tensioned concrete girders (fig. 5:4). Although stressing of reinforcement can be performed directly in situ, this requires special devices, and other difficulties arise, making it a very expensive proposition. Because of this, nowadays prestressed parts are largely made in workshops and not at the working site. They are almost always relatively small parts and they require very rigid and expensive molds or other devices, whose cost can only be amortized by using them many times. Hence, prestressed parts become a workshop production, whereas post-tensioning has found its natural place in the actual site of large engineering projects. It seems that we should not speak of "post-tensioned concrete," but rather should use a term that actually expresses a structure consisting of metallic parts and of concrete or reinforced concrete. However, in this material the metallic part has different characteristics than the metal in typical metallic structures which will be discussed later. It consists of round bars similar to those used in ordinary reinforced concrete, wires with high strength especially used for pre-tensioning, strands, and cables, and other special types of reinforcement, always compact and at the same time flexible in order to be easily inserted and incorporated into the structure. The most usual current method of post-tensioning provides continuous tubular holes or cavities into which wires are inserted and anchored at their ends by special devices (fig. 5:5). The anchorage and tensioning devices, appropriate to the method of stressing, and installation of hydraulic jacks and strain gauges present the most important and difficult problems of post-tensioning. Post-tensioning is followed by injection of cement grout into the tubes to prevent oxidation of the reinforcement, to produce sufficient bond between both materials, and to assure the compound action as in ordinary reinforced concrete. Twisted cables have been used with good success in Spain, and in other

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countries, but have certain limitations. Their installation is very simple, as has been demonstrated in special structures to be discussed later. In Germany, use has been made principally of plain bars of relatively large diameter, with special anchorage in concrete but without grouting, so that additional stressing can be made at any time, if necessary. In this case the reinforcing is an independent part of the reinforced concrete, both as to stresses and to deformations. The initial compression of concrete can be achieved in certain cases without tensioned reinforcement, as, for example, by jacks and other ingenious devices. The procedure embracing displacement, opening, and spreading of concrete arches with hydraulic jacks is based upon the same principle as post-tensioning. This general description of various phenomena and behavior that characterize the constituent materials serves only to remind the designer of their existence and their importance in the conception of modern structural design. Extensive study of them, although never exhaustive, could fill many volumes and still be incomplete. However, the more familiar the designer is with all the factors affecting his design, the better he can understand, substantiate, and appreciate various advantages and possible disadvantages and difficulties. Many of these factors could easily be overlooked in following the older standard design procedures; and in each individual case, different relative importance affects the most appropriate design solution. Continued study of all these factors in detail will also aid in foreseeing the possibilities and the optimum forms of application of new structural materials still to be brought to the market by manufacturers, and in appraising them with a greater judgment. Thus aluminum and its alloys became triumphant in the realization of great buildings, thanks to its lightness. Many other materials will compete with standard materials and probably will win supremacy in the field of modern structural materials—when industry succeeds in reducing the costs of production.

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chapter 6 supports, walls, and foundations

In general, any structure consists of a combination of various typical structural elements or members. It will therefore b e profitable to discuss the characteristics of these elements and their types. A support is, and in all times has been, one of the most fundamental structural elements. Among supports, the column is the most genuine and frequent element. How much work and thought of humanity are resting upon it! One can feel the delightful touch of the greatest artists upon its decor. If the column were not a monument in itself, humanity would have to erect a special one in its honor. The mission and function of a column are to support something on top of its capital. The column, whether disengaged and solitary or in a row of companions, is beautiful as it stands over the soil in its anxiety of preeminence. Its erect verticality is symbolic of the human figure yearning upward toward the skies. Rising beyond its surroundings, it expresses proudly its inflexible permanency: the motive of its ideal and its being. It has the pristine and mysterious enchantment of the first achievement: the paternity of all building existing in all times. When Jacob had the vision of that infinite flight of stairs ascending to Eternity, he posted on that spot a column and marked it with his indelible sign in oil to record his vision for all future generations. And when one column is united with another by the perpetual yoke of a lintel, its gracefulness does not vanish; rather it is enriched in the acceptance of the function for which it was created. Column and lintel represented the emblem of all classic constructions until the maximum consecration achieved by Hellenic art. The mission of the column synthesizes all our structural purpose: to

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Fig. 6:1. Roman columns at Merida, Spain, second century B.C. Photograph, M. Garcia Moya. support. This is a term that in our language has in it something of conformity and humble renunciation of vain rights when it is voluntarily accepted or guided by the ideal of service, and which attains sublime limits of the best virtues. To support means simultaneously to bear and therefore the column is also an emblem of strength. The first column undoubtedly was built of wood—a column, still a tree, supported the shelter of our early forefathers. There is an old saying: "He who gets under a good tree has a good shelter." Then came a stone column, since stone is a most suitable material for this member because of its high strength in compression. The most logical shape is trunce-conical with circular cross section widening toward the base, since the load transmitted to the capital is increasing by the proper weight of the column. The base of the column has the function of transmitting and distributing these loads over the supporting soil when it is soft and less resistant. The function of a capital is not so clear. So far as wood as structural material is concerned, its strength perpendicular to the grain is much lower than that parallel to the grain. Consequently, if the fust (body of a column) were to press directly on the lintel beam, it would tend to dig into the latter,

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crushing its transverse fibers. So the only means of making full use of the strength of the material in both fust and lintel is to sandwich between them a piece of another, harder wood, or if the same wood is used, a large piece of wood must be used sandwiched between these two structural elements. This arrangement may have become a permanent practice when applied to stone. But this is not a matter for discussion here, after all that has been opined and written about it. When the arch made its first appearance the columns were attached to a broader structure behind, thus being more stable to withstand unequal lateral thrusts from two contiguous arches. In Roman art, the function of a capital is clearer because it widens the body of the column and forms a wider support for the arches resting upon it. This also occurred in Arabic art, where the thickness of the column was abruptly reduced toward the top often much more than could be expected from mechanical requirements. Finally, the slenderness of the column culminated in the Mediterranean Gothic art. When the slenderness is exaggerated, the supported part is usually made to appear lighter (as, for example, in Arabic structures in which the wall is filled with carvings and filigree openings). Even so, the slenderness of the arched colonnade contrasts strongly with the massiveness of the wall, and the stone is put to a use not proper to its nature. In the last century stone columns were replaced by columns of cast iron in which, at the beginning, the same basic forms were used. These columns are usually hollow in order to gain lightness without appreciably reducing the resistance to bending. As compared with stone, lighter sections are possible due to higher allowable stress. Moreover, the mechanical characteristics of both materials are analogous. To timber supports having in many cases insufficient cross section with only one log, several ones are joined in one support in order to attain the needed bearing capacity. Because of the slenderness of these independent members, failure normally will occur due to partial bending or buckling, and therefore, it is necessary to tie them together in order to increase their total rigidity, which is in direct proportion to the moment of inertia of the composite section. This can be seen in certain buildings of antiquity, where several logs, or the peripheral parts of the same trunk, were tied together. The ornamental motif, deriving from it, was later imitated by people who had not freed themselves from the influence of their earlier timber constructions. With steel, theoretically, the tubular shape is the most economical. But if the wall of the tube is very thin, typical bending of cylindrical shapes combined with buckling will occur so that the column will have a lower

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Fig. 6:2. Arab-Spanish columns at the Alhambra, Granada, Spain, fourteenth century (from Ars Hispaniae, Vol. IV, by L. Torres Balbas; Madrid, Editorial Plus-Ultra).

Fig. 6:3. Palacio de la Audiencia, Barcelona, Spain, fifteenth century (from Summa Artis, by M. B. Cossio and J. Pijoan; Madrid, Espasa Calpe).

bearing capacity than desired. Rolled sections are less expensive than tubular ones and can be assembled into sections having great resistance to bending. Their appearance of composite and riveted sections is marred by the complexity of their form. To permit access for riveting equipment, open sections or sections with exposed wings must be used. Welding, however, makes the assembly of enclosed sections much more attractive. But the high strength characteristic of steel makes the danger of buckling more immediate than that of direct failure, and this necessitates the design of composite sections with a larger radius of gyration or of assembled members having individually shorter lengths. It is also necessary in order to avoid the buckling. This is a classical type of steel column consisting of a number of longitudinal members connected transversely by lattice work or cross bracing. Rectangular plates can also replace cross bracing, if the longitudinal members are not too far apart. The distance between each bracing is determined by the requirement that the members shall form a stable strut between two joints. Also, the flanges of the members in compression and the free edges of the plates must be prevented from buckling. Since steel has a high tensile strength, it is highly resistant to eccentric compression, bending, and shear. Considering the simple and fundamental function of a supporting member, namely resistance to direct compression along the central axis, it would seem that resistance to tension and to shear is superfluous. However, these properties are advantageously used in continuous and rigid structures (e.g., columns restrained with continuous beams). This type of structure will be discussed later. In trussed supports various types of lattice work are frequently used, as shown in figure 6:4. In choosing the proper type it will be necessary to consider, besides aesthetic aspects, the distinct type of deformation that will develop if direct compressive stress is predominant. In triangular latticework (fig. 6:4a, d), such deformation is symmetrical. The choice between type a and type d will depend on the relation between the width of the support and the vertical free length between the joints. In type b there is a slight tendency to swaying because the vertical eleFig. 6:4. Trellis types of supporting structural members.

^

t

T 0'

ments are submitted to contractions which do not act on the diagonals. Type c can be more economical because the diagonals do not need to resist compression; however, its theoretical advantage lies in the fact that the diagonals do not need to resist compression because one diagonal in each panel always resists the tension, thus giving the panel sufficient rigidity. But reduction in length along vertical axes tends to relent all diagonals and the assembly can take initial and unforeseen slight lateral movements, and also disagreeable vibration stresses caused by lateral forces as wind or earthquake. Reinforced concrete has the same basic properties as stone, although, in general, reduced depth of structural elements results due to the higher resistance and rigidity of concrete acquired by the steel reinforcement. The most frequent cross section is rectangular because of its suitability and economy of forms. Longitudinal and transversal reinforcement increases the resistance of concrete and diminishes the possibility of cracking, as well as other defects. Longitudinal reinforcement can be arranged in accordance with the distribution of loads, since the additional bars not only increase the moment of inertia and consequently the bearing capacity but can also resist bending moments. It is also important to eliminate excessive deflection and sagging caused by the plastic flow of concrete. Reinforced concrete is not very suitable for hollow sections, especially for constructional and economic reasons. Reinforced concrete of good quality resists considerable tension and shear due to its reinforcement; its monolithic character makes it applicable to continuous and framed structures. Stirrups and transverse reinforcement are necessary to increase concrete's resistance to failure and to prevent sagging of the longitudinal reinforcement which can cause cracking of the covering concrete layer, a phenomenon preceding failure. When the available section of supporting members (e.g., columns) is not sufficient to carry the imposed load, the most economical solution is to provide additional spiral reinforcement instead of increasing the amount of longitudinal reinforcement. However, one should not overlook the greater danger in such columns and pilasters of plastic flow which is caused by higher compressive stress. Normally, the supporting members transmit the load to the base of the structure, and this is the simplest way to assure its stability. The connection with the base can be hinged, in which case tapered columns are used (fig. 6:5) with stiffening lintels at the top. Hinged connections can be used both at the top and at the base of the column when the structure is prevented from swaying laterally by other rigid supports. Such hinged supports allow

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Fig. 6:5. Concrete supports at the Madrid Hippodrome. Photograph, S. v. Kaskel. the structure the required freedom cf thermal shrinkage and expansion. Maximum slenderness in supporting members under compression is achieved with steel and maximum economy with concrete, because, as has been explained before, the economic-resistant coefficient of the latter is always greater than that of steel, when compression forces prevail. We shall not discuss specifically the brick masonry because its field of application is more restricted than that of stone masonry due to its lesser strength. In general, bricks are more suitable for thick pilasters than for slender columns. Old brick masonry columns that are still preserved, including twisted columns, are rather proof of ingenuity and artifice, than significant prototypes. Even so, brickwork is a very apt medium to construct walls reinforced by pilasters, and in many places it is a cheap type of construction. In any case it should be remembered that this type is not capable of resisting any bending without special reinforcement and is in general less economical than columns in concrete, especially when smaller sections are required. When speaking of supports, one always has in mind vertical members. Yet, inclined supports, such as braces of cantilevers or brackets, can have important functions. To use them in stone would not be very practical, be-

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cause even the proper weight produces bending in any stone member of greater length and the loads producing compression are not great enough to compensate for dangerous tensions caused by bending. In any such instance where the direct stress does not undergo great variations, as when the dead load is great compared with the live load, the supporting member should have a curved form that follows the line of compression. In response to this necessity, the flying buttress was created. Cantilevers are used much more in structures of timber than in those of any other structural material (fig. 6 : 6 ) , and mostly as auxiliary elements to reduce the bending of supported girders. It can be noted that in many contemporary structures, both in reinforced concrete and in steel, inclined and curved supports have been used with more liberty because of their great aesthetic and structural values. Usually it is not necessary to use arch shapes in these materials, since their dead load is relatively light compared with the total load supported. Naturally, if bending is predominantly in the vertical plane of the member, it may become necessary to change its cross section from square to rectangular, or from circular to elliptical. Nor is the wall a recent constructional element. For we must remember how the ancient peoples stopped in front of the insurmountable barriers of those walls made up of Cyclopean blocks. More than once they made use of the wall surface as a parchment upon which to essay their early calligraphy and leave lasting records of their glorious deeds. All these important functions of a wall—enclosure, support, and content—have been known and utilized since the most remote antiquity. Normally walls are built of stone, brick, or concrete. From the oldest type of Cyclopean masonry without mortar in the joints to the newest very light tile walls, many kinds of walls have been built (e.g., those of rubble and ashlar masonry, those with various jointing, those of massive and hollow blocks, molded walls). All these types, each with its distinct resistance and durability are based on the same principle. Special attention should be given to very thin walls, since the vertical dead load is relatively small to compensate for small resistance to tension and to bending provoked by lateral forces. The classic wall is a structural element of exterior continuity and of certain thickness and mass which resists simple or combined compression without tension. These properties are characteristic for stony materials, mass concrete, etc. The potent appearance of the wall attains its most noble appearance through the ageless medium of stone. Structurally, the resistance of these types of wall depends on the quality of mortar in the joints. There is practically no resistance of tension, and for this reason the jointed surface should be rough and normal to the in-

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Fig. 6:6. Centering of the Salgina-Tobel Bridge, Switzerland (from EMPA Bericht, 99). Engineer, R. Coray. ternal forces. Inclined surfaces in the joints can cause tilting or sliding of individual elements and can produce forces that instead of assuring stability can contribute to failure, as has very often been observed in the ashlar masonry of antiquity and in veneered walls backfilled with loose stones or with earth fill inside. Since a masonry wall must maintain its equilibrium only by its proper

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weight, the danger of bulging is normally eliminated. In many cases, a wall must resist horizontal forces that originate either by backfill or by arches resting upon the wall, or, if the wall is very high, by lateral forces such as wind or earthquake. Under such circumstances and without any restraint, fixation at the base or counteraction along its height, the thickness of a wall must be increased considerably if materials without tensile strength are used. Otherwise, the combined compression resulting from the external horizontal forces and the proper weight will induce bending, causing the wall to fail under tensile stresses, or, if there is no real rupture, to crack along some joint and overturn. This can be prevented by producing a condition in which the overturning moment about any joint due to external forces is smaller than that caused by the proper weight. Such walls require greater thickness and also materials of higher specific weight. Furthermore, since the overturning moment becomes greater with the depth, the wall should have increasing thickness toward the base. For this reason, Egyptian architects originated trapezoidal walls, magnificent in appearance with their upward tapering sections. Also a profile inclined toward the back of the wall can be the solution in some cases, although this shape can lose stability if the external forces disappear, as can happen, for example, in an empty reservoir. To overcome danger of overturning in walls built of materials without tensile strength—where the proper weight is not able to counteract external forces—piers or buttresses (counterforts) can be arranged in such a way that the proper weight of the wall becomes more stabilizing, with its increased moment counteracting the overturning moment. The combination of tapered, buttressed walls, and a lintel or vault across the top of the buttresses—so that the edge of the lintel lies vertically over the feet of the buttresses—enable to reduce the mass of the wall without reduction on the top platform, and to achieve an interplay of mass and design, a design so attractive to the later medieval builders (fig. 3 : 2 ) and to those who inspired them in the Middle East, that their patterns have survived, even as strictly decorative forms, during the whole Romanic period; and even these were neoclassical and Renaissance revivals of this treatment. A massive wall in reinforced concrete can develop the resistant function of a bracket or cantilever anchored at the base of a foundation slab. It can also consist of panels or slabs subject to bending between cantilevers and horizontal beams of greater depth, thus forming complex structures (to be discussed later) which in their initial concept are basically different from classical walls. In this sense, neither timber nor steel is actually appropriate and suitable for building walls. Old palisades, which protected villages from marauders of the forests, were built of wood. In recent times, steel has had a limited

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utilization in construction of walls consisting of plates reinforced with rolled shapes and is almost exclusively restricted to temporary containers of earth fill in cofferdams or to underground water conduits. Today, for the sake of lightness, simple walls consist of thin skin or panels attached to vertical supporting members with fill materials between them adequate to assure various properties and specific functions (e.g., impermeability, and heat and sound insulation). But these fill materials are something specifically separate from the structure, and from the structural function with which we are dealing here. The foundation is an important part of a wall, since it transmits all forces and loads to the ground. In its classic form it is a massif or a slab which distributes the pressure of a structure over the bearing soil. These slabs in reinforced concrete can be eccentrically arranged in order to increase the moment counteracting the overturning, with less material than that necessary in gravity foundation without any reinforcement. A foundation can be large or small, deep or shallow, and its construction can be complicated, requiring auxiliary equipment (e.g., caissons with compressed air, or the more primitive Indian caissons). Basically, its function is to distribute the loads over a surface of the bearing soil in whatever depth is necessary in each particular case to comply with given conditions. The problem is different when the bearing surface required by the nature and the bearing capacity of the soil is so large—compared with the distances between each two supports—that it becomes preferable to join the isolated footings into one single slab. The problem is then identical with that of a loaded floor slab resting on columns, supposed to be turned upside down. Just as in a flooring the applied loads act downward, and the reactions of the columns act upward; in a foundation slab the upward reaction of the ground is balanced by the concentrated loads on the columns. Hence we shall postpone further comment until we deal with flooring structures. The only essentially different type is the pile foundation. These naillike members, driven into the soil under the platform that links them together, or extended upward to form a palisade (wooden fencing), were already in use by ancient builders, and have been continuously used to our own times by inhabitants of muddy or soft clayey regions. The length of the piles has greatly increased with the use of steel or reinforced concrete. Pile-driving methods and equipment have been perfected, but the structural type and the principle remain the same. Summarizing, it is not the foundation itself that is basically significant for structural design, but the influence that the type and properties of the soil may have on the foundations and their cost, and through them on the whole structure. But this matter goes beyond the scope of this chapter; we shall return to it later.

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chapter 7 the arch

An arch is very similar to a column from the standpoint of stressing, because the prevailing stress is compression. I t is a primary discovery in the field of applied mechanics perfectly developed in classical art. The arch has always been impressive to man, but for a long time he was not thoroughly aware of its resistant capacity. The simplest proof of its uncomprehended impressiveness is found in the many legends attributing the original construction of an arch to a devil. If the column is presented as the creation of pure architecture, then it can be said that the arch was surely a creation of engineering. Or better said (to avoid all professional prejudice), if the column is art, the arch is technique. This does not mean that a column would not require technical skill and an arch cannot have the highest aesthetic expression. As the beloved favorite of art, the column was replaced by the arch—"the arch that never is dormant," to use the phraseology of an Arabian proverb. Although a column points to the sky, an arch, with its curvature spanning space, is unquestionably more dramatic and impressive. Like a crossbow, the arch in stone will always be connected with the idea of powerful stress, of a leap to dominate distance. Therefore, an arch, always of monumental character, was predestined to proclaim the honors of victory. The construction of an arch is not so simple as the erection of a column over its base. When erecting a column the problem was to apply forces and other adequate means in order to place one block of stone upon another. In constructing an arch all elements must be supported until they close at the crown. Only then is an equilibrium attained, and only then can the action of an arch begin. The earliest arches were actually false arches constructed by extending horizontal layers of supporting walls farther out into the gap as cantilevers

Fig. 7:1. False vault at Uxmal, Mexico, twelfth century (from Summa Artis, by M. B. Cossio and J. Pijoan; Madrid, Espasa Calpe). which were closed by a stone at the crown. But this is not easy to do, nor is there always a wall that would permit this procedure. Often the builder tends to build a higher arch using this method, since the overhanging is limited to avoid overturning. And he may finally arrive at a trapezoid or ogival arch (fig. 7:1). The difference between a real arch and a false arch must be clearly understood. For the technician the arch is, or pretends to be, a structural element, the shape of which follows the pressure line of acting internal forces. A real arch is characterized as a curved member, resisting only (or principally)

Fig. 7:2. Bridge of San Martin at Toledo, Spain, thirteenth century. Photograph, J. Ortiz Echague. compression, which transmits proper weight and superimposed loads to the two end supports. If the span is built of two cantilever members projecting from the supports with a central part in the form of a beam, the arch action is eliminated. This is a false arch. Arches, like columns, must be built up of materials strong in compression. But we shall distinguish between two types of arches: the built-up arch and the spandrel arch (figs. 7:2 and 7:3). In the first type, the facing wall on both sides of the arch prevents bending, and the arch may be no more than an archivolt. In this case the distribution of compressive stresses has the pattern of discharging arches (fig. 7 : 4 ) . Classical and medieval arches were of this type. From the standpoint of stressing, the spandrel arch is a typical true arch, used in one of the most beautiful early examples: the Mezquita arch in Cordova. If the loads and their distribution are exactly known, the axis of the arch will coincide with the resulting compression line so that each section will be under direct stress. Naturally, this can be accomplished only with material of sufficient strength, as is currently found in stone arches with relatively small spans. If, as is common today, an arch is built of materials of high strength and higher permissible stresses, important bending will occur as the span increases. When the live load is applied in only one-

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Fig. 7:3. Sando Bridge, Sweden. Engineer, S. Haggbon. Photograph, K. W. Gullers.

HUM*

COMPRE 55/Ofi

Fig. 7:4. Stress distribution in a full mandrel arch.

1=1 TEflSIOn

half of the arch, this part will deflect, but the opposite part of the arch will rise; and the deflections will follow the curve indicated in figure 7:5a, changing the sign from positive to negative. Although the initial axis of the arch exactly follows the compression line, its contraction due to compression will change the form, cause some eccentricity in various sections, and consequently will produce bending. It is not worthwhile to try to correct such relatively small deflection and to avoid this bending. The only practical way to do so is to produce an additional expansion of the arch which will compensate for the elastic retraction. Freyssinet has applied this principle successfully in long-span concrete arches by spreading the crown section with hydraulic jacks and adding an adequate keystone or concrete fill. This has become a standard method for long-span concrete arches, which has been used in many countries, including the United States. In any case, the live load takes various positions so that the compression line varies from case to case, and the structure should be designed for additional bending moments. A true arch (without any stiffening) may dilate or contract, bend or bulge—may breathe—better than other structural members in compression. This possibility is greater in slender arches, especially if the arches are hinged. The secondary bending due to these deformations are appreciable and must be maintained under certain limits. If both ends of the arch are restrained in the abutments, the deflections will be smaller and therefore the resistance of the arch greater than if the ends are hinged; but the influence of thermal dilation is also greater. If the arch is narrow, there is danger of lateral bulging with torsion (fig 7 : 5 b ) , which can be intensified by lateral forces such as wind or earthFig. 7:5. Lateral and vertical buckling of an arch.

a

b

quake. All these accidental deformations and forces should be taken into consideration in the design and in the selection of the structural type. It is not only the greater specific strength of steel and concrete which, in many cases, makes the free arch preferable to the built-up one. However, deformations caused by thermal and hygroscopic changes, small and negligible in structures of stone and brick masonry, become very significant in concrete structures and can cause undesirable cracking. These circumstances lead to the choosing of structural types that have great deformability, and among them the true arch with light spandrel columns is one of the most suitable and economical. The same type can be used in steel structures, naturally with more slender dimensions. If the true arch is built in stone, bricks, or plain concrete, greater thickness will be necessary to eliminate excessive tensile stresses that such materials cannot resist. In shorter spans the proper weight of stone arches can considerably reduce the unfavorable effect of moving loads. However, in longspan arches of such material the necessary thickness would become so great that the cost of material, labor, and other items involved would be practically prohibitive. Furthermore, long-span arches of stone require much heavier foundations than arches built of reinforced concrete or steel. High costs of hewing and cutting stones can be considerably reduced if the interior part of the arch is constructed of irregular stone or brick masonry, in which case joints in the masonry must be practically normal to the compressive forces. This is an altogether logical solution. The different contraction and shrinkage in both types of masonry should be considered, especially if the veneer masonry is of stone and the other material of the arch is concrete or bricks. In such cases,, the veneer can separate from the other material, endangering the stability of the whole structure. For arches, especially in large spans, concrete is definitely more economical than stone or brick masonry, even in countries where these materials abound and labor is cheap. Nevertheless, the concrete, which is being constantly studied and improved, involves an increasing complex of rheological problems, even in relatively flexible, open spandrel arches. Contraction of concrete results primarily from shrinkage and creep under permanent compression (due to the dead load), and these contractions produce additional secondary stresses that are much greater than those in stone masonry. A suitable opening of the crown by means of jacks can serve to compensate for, and counteract, the effects of contraction in an arch. Plastic flow can be counteracted by various consecutive operations during the first months after completion of the concrete work. There are concrete bridges such as, for example, the prestressed concrete bridge over the Marne River at Esbly, France, in which, at any time during the life of the structure, com-

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pensation can be made with hydraulic jacks for variations in span length caused by plastic flow, temperature, and the yielding of abutments. In steel there is neither shrinkage nor plastic flow within the margin of permissible stresses, but, since the relatively small mass of steel is exposed directly to solar radiation, the effect of temperature changes becomes very important. However, steel appears to be less adaptable to this type of structure where compression prevails, being more suitable for structures subjected to bending. A very important property of steel is its relatively low proper weight (dead load), especially advantageous in long spans which, if made of reinforced concrete, would require large sections with heavy dead load. This is a fundamental point and should always be considered. Long spans produce great forces and require, especially for economy, materials of great strength and high resistance in relation to proper weight. Omission of this consideration in designing can double the cost of a bridge. The Sando Bridge (fig. 7 : 3 ) has the longest span in reinforced concrete (866 feet between abutments). Steel bridges, however (except suspension bridges), can exceed 1,700 feet in span (Firth of Forth Bridge, fig. 14:3). There are other reasons for limitation of span length in bridges of reinforced concrete. Concrete bridges often require expensive centering, whereas steel structures of considerable spans can be erected by use of relatively simple lifting devices or with inexpensive partial centering or supports. Furthermore, concrete with its heavy weight and greater thrust in arches requires much heavier foundations than does steel. Modern techniques developed in recent years make reinforced and prestressed concrete more and more competitive with steel in bridge construction. Also, concrete arches with stiff reinforcement in the form of light-steel skeleton—serving simultaneously as initial permanent centering used by Joseph Melan—are very economical because the expensive temporary centering is eliminated (e.g., Cappellen Bridge over the Mississippi River, Esla Bridge, etc.). There are very few permanent timber arches because of certain defects of the material already discussed and because heavier members are difficult to bend in the form of an arch unless it has a large radius of curvature or smaller sections are glued together (glue-laminated arches). There are some very interesting arches built of very heavy timber, but they are rare. In general they have the form of the untrue arch, or of a beam with multiple brackets, in which shapes the members can be more easily assembled. Glue-laminated arches have been recently considerably improved and developed and can be assembled with any cross section, desired shape, and curvature without difficulty and with relative economy. Freyssinet has developed the idea of assembling thin boards with large numbers of small nails and filling the compression joints between the boards with concrete

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Fig. 7:6. Floating centering of the famous Plougastel Bridge in France. Engineer, E. Freyssinet, Limousin Construction Co. Photograph, J. Moalic. mortar. This method was used for the centering of the famous Plougastel Bridge, with its span of 612 feet (fig. 7:6). The failure of this new technique in the construction of the centering for other bridges occurred for other reasons which do not affect the soundness of this method. Glue-laminated timber arches and frames with spans as great as 200 feet have been executed. If it is possible to make them simultaneously fireproof and to protect them against decay, there are great possibilities for use of timber arches in many other structural fields. One typical function of an arch is to produce horizontal thrust at the supports. If the axis of the arch exactly follows the compression line of loads (directrix), the reaction at the abutments will be tangential to the directrix. This indicates that the reaction will have greater inclination at the supports, and the lower the arch or the smaller its rise, the greater the intensity of the reaction will be. The horizontal thrust is approximately determined by multiplying the total weight, including all loads of the arch, by one-eighth of the ratio between the span and the rise. This holds exactly for a three-hinged arch. If the arch is restrained, the intensity of the horizontal thrust will be lessened because the restraint increases the rigidity of the arch. However, this difference is not very large, nor is there serious interest in decreasing the thrust by increasing the rigidity, which results in greater bending moments, since to allow for bending is always more expensive than to allow for compression, when planning the design.

It can be concluded that an arch always requires very good foundations, or anchorage, to resist the horizontal thrusts. The same effect can, of course, be obtained by tying together the buttresses of the arch, but in this way the main advantage (i.e., economy) is lost, since in the arch with abutments, instead of with the tie, resistive soil is used, which is an inexpensive structural element. The tension in a tie is of about the same order as the maximum compression in the arch, and the total economy obtained avoiding the tie can be evaluated at approximately 50 per cent; and if the tensioned material of the tie in a steel arch is more expensive per ton of the acting force than that in the arch in compression, the economy can be even greater, theoretically. It can be said that an arch is a gamble, a jugglery with structural functions that are passed to elements external to the structure proper— elements that existed before construction started, such as the resisting soil. If the supports of an arch yield and do not react, no horizontal thrust can be produced and the arch is converted into a curved beam that resists by bending as a straight beam under the loads instead of being compressed. The arch can be restrained or articulated with one, two, or three hinges. Three-hinged arches are easy to analyze because they are statically determinate (isostatic), and, therefore, have the advantage that it is not necessary to consider, in design, secondary forces, such as those produced by temperature changes, hygroscopic effects, and plastic flow. However, in general, hinges increase the weakness and cost with expensive connections. On the other hand, bending stresses diminish and finally vanish in the proximity of articulations, yet increase in the intermediate zones. The advantage gained by eliminating bending at the abutments is evident, and may determine the preference of this type when the soil is not very resistant and makes the foundations expensive. Since the rotation in hinges is small, plastic articulation can be used (fig. 5 : 3 ) . This is more economical, especially in cases where compression is not very high. One- and two-hinge arches are intermediate types having many of the advantages and drawbacks of three-hinge arches. They are more deformable than arches with rigid end attachments and will have comparatively smaller secondary stresses, such as those due to temperature variations, within normal ranges of stresses. But double-hinged arches, unless they are of the bowstring arch type, have no clearly defined advantages, and are not really in use. For bowstring arches, or where foundations cannot be subjected to large eccentricities of reaction, double-hinged arches are standard design. Three-hinge arches can be stiffened and made more rigid by spandrels carrying the superstructure, provided they do not restrict the rotations around the hinges (fig. 7 : 7 ) . For this reason, the three-hinge arch in steel with triangulated tympans is more rational and economical. The classical cross section of the arch is of rectangular shape, and there

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Fig. 7:7. Bridge with hinged vaults. is no reason for changing it unless specific conditions make the change necessary. Sometimes the thickness of the arch increases toward the abutments to counteract the danger of bulging, and to secure greater stability against lateral forces. In fixed arches the axial compression for vertical loading increases inversely as the cosine of the angle between the tangent to the arch and the horizontal. If the arch itself is made to vary in thickness in a similar manner a satisfactory effect is achieved. From the standpoint of the flexural resistance of an arch, its thickness will depend principally on bending moments that usually reach their maximum at the abutments. The final solution will depend on the proper shape of the arch axis determined on the basis of all aforementioned conditions and requirements. In hinged arches it is logical that the thickness should decrease toward the hinges, where bending moments vanish. Then it is most logical, from the standpoint of rational stress distribution, to provide variable thickness diminishing toward the hinges and simultaneously increasing the width of the arch at these hinges; in this way, the reaction is distributed over a longer edge, which compensates for smaller thickness, thus maintaining approximately constant compressive stress in all sections of the arch, since the compression forces vary but slightly whereas the bending moments reach their maximum in the proximity of the quarter points and vanish in hinges. The possibilities and advantages of two- and three-hinge arches resting upon heavy cantilever abutments—to reduce the span between hinges— have not been sufficiently considered, evaluated, and utilized. The tendency in actual practice is to avoid hinges because of difficulties of installation and higher cost. However, these disadvantages do not exist if plastic hinges in

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reinforced concrete are used in relatively short spans, and, if loads are not excessive, this type of hinge has considerable advantages. The rectangular section is not the only one used. In steel arches, hollow sections will almost always be necessary for attainment of greater rigidity without increasing the weight of the structural material beyond the economical minimum. In steel cross sections consisting of four adequate steel shapes connected together in all four planes with stiffening and bracing lattice work prove to be most efficacious. Box sections consisting basically of two flange plates, riveted by means of corner angles or directly welded to two other steel plates forming webs, not only are, in most cases, an economical solution but also have a very neat and satisfactory appearance. In reinforced concrete, it will be necessary to use box sections in rectangular or similar shape when the spans are exceptionally great. Disadvantages of such sections are higher cost and the difficulty of removing the internal forms, so that in many cases these forms are simply left in the structure. Much has been written about the best form of the arch axis, and its determination becomes especially important in long-span arches with great dead load. Apart from the rise-span ratio (which is usually determined by conditions other than strength requirements), the form of the directrix is not significantly influenced by thermal or shrinkage factors, but rather by the dead load and type of live load to be supported by the arch. The funicular for dead load only acting upon an arch of constant thickness is a catenary. For loading uniformly distributed over the whole span, it takes the shape of a parabola of second degree. In bridge engineering, where one must consider many different and irregular types of loading (e.g., the weight of the superstructure with the roadway, spandrel columns, more or less heavy, variable thickness of the arch, etc.), the parabola will be of the fourth to the sixth degree. However, when the live load is heavy as compared with the dead load, the form of the directrix loses its importance, and one must examine various forms under the most unfavorable loading conditions and attempt to select the most suitable. When the flatness ratio between the rise and the span of an arch exceeds certain limits, there will be an increase of the compressive forces, and the horizontal thrust acting upon the abutments will become greater. Greater rise for a constant span increases the transverse bending due to lateral forces such as wind and ear-thquake. If there are no functional, constructional, or other factors to be considered, the selection of the proper flatness will always require thorough study. Trial and error methods are recommended. Typical flatness is between 1:5 and 1:7. To reduce it to 1:10, it will be necessary to investigate the precise effect of increased shrinkage and temperature changes upon the stresses, especially in arches of relatively

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great rigidity. Of course, by augmenting the flatness, the eccentricities of the force polygon become less, compression forces being larger while the bending moments are not much affected. Finally, a definite limit to flatness must be set, to avoid not only excessive increase of horizontal thrust, but also considerable contraction of the arch axis which accompanies excessive bending and bulging and can be still further aggravated by plastic flow. For precisely these reasons Freyssinet introduced artificial correction of the arch axis, especially in bridges of low flatness, by temporary hydraulic jacks inserted at the crown. On the other hand, if the flatness increases starting with approximately 1:4, the horizontal thrust of the arch decreases only slightly, and there is nothing to be gained by increasing the rise further, because lateral bending and bulging become more pronounced. Greater rise of an arch normally will be determined by factors other than pure structural (e.g., the shape of the valley or canyon spanned, aesthetic considerations, etc.). All these conditions discussed here are of fundamental importance for bridge arches of extraordinary dimensions, where the economic factors, such as reduction in materials and weight, are of primary importance. It is not true of arches of smaller span combined with facing walls or massive tympans. The latter give the arch greater rigidity and therefore greater possibility of shaping. Both the intrados and extrados become independent of the compression line of the arch which can even disappear, so that it can be said that an arch of this type completely loses its personality. Stress distribution is no longer that of an arch, in stress parlance. Rather must it be interpreted as a problem of plane elasticity in which the isostatic pattern is the only real representation of stress distribution. This stress distribution can vary considerably in stone masonry due to arrangement of the joints in the arch, especially at points of intersection with contiguous tympanums. Only in reinforced concrete, which can be considered as a homogeneous material, do the isostatic lines give true pictures of stress distribution, and only in such case can it be used for determination of the arrangement of reinforcement following maximum tension and for checking compression zones as to the stresses allowable in concrete (fig. 7:4). But we have seen that the resulting type of reinforcement has no relation with the classical arrangement in an open spandrel arch. In this way, the arch is reduced to an ornamental element, not superfluous or arbitrary, yet not arising in answer to mechanical requirements because its existence is essentially a symbiosis with the wall that strengthens and raises it. Classic art developed to full plenitude and half-circular arch, for if no other reasons intervened the most simple and uniform shape is the best. The raised elliptical arch definitely reduces maximum stresses, and it is strange that it has not been more extensively used in architecture. The shape of our doors induces rectangular

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Fig. 7:8. Stone dentil of Torre de los Lujanes, Madrid. Photograph, M. Garcia Moya.

openings, which have made it necessary to stretch the arch to a very flat arch resting upon vertical cornices and buttressed by the walls; or, in extreme cases, to a straight lintel (fig. 7:8), which is nothing but an arch masked with straight lines of its extrados and intrados; the obliquity of its dowels clearly demonstrating the erroneous expression of stresses. Finally, construction methods will have their own requirements. This becomes important in long-span arches, which, for all practical purposes, are reserved for large bridges, and will be discussed later in connection with the design of such bridges. The portico or portal frame is an arch with a polygonal or irregular directrix (fig. 7:9). It is an arch designed without consideration of internal forces, in which, in general, the directrix deviates very much from the funicular of internal forces. Consequently, compression is relatively small as compared with bending, but still is greater than in straight beams on free supports, and this yields a certain advantage.

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©

© Fig. 7:9. Various types of portal frames: a is portico without tie rod; c, the restrained areh portico with tie rod, is an intermediate solution between a and b.

The most elementary portico consists of a beam and two supporting members, all monolithically united. The beam is rigidly connected with the supports, so that they are forced to deflect when the beam is loaded, and owing to the restraint the beam will simultaneously be relieved in deflecting. Since compression is small in raised porticos, it may be supposed that the area of the bending moment diagrams, or their integral, along the three members, should be made a minimum to achieve most saving in material. But, in general, other reasons are more important in deciding which of the two solutions, portal frame or simply supported beam, is better. It depends on which part of the structure can most easily be thickened, or strengthened in bending. Thus, for instance, if it is sought to decrease the depth of the beam in the center to increase the clearance, a portal frame will be preferable. Conversely, if there is little lateral space for the supports, it may be preferable to provide simple supports at the tops of the supporting columns, to free them from bending, and so reduce their width. To eliminate as much bending as possible, the shape of a portico, where no other factors interfere, should always follow the line of internal forces (funicular polygon of loads). Thermal effects can slightly influence the design of porticos, especially if they are flexible. Like an arch or vault, a portal frame can be fitted with one, two, or three hinges, and may be fitted with one or more tie rods to reduce bending, and bring to closer coincidence the shape of the structure and the force polygon. In such cases, the designation "simple portico" no longer applies and we are dealing with a more complex structure. The portico without the tie rod, shown in figure 7:9a, is evidently subject to considerable bending; if it is fitted with a tie rod and two hinges it acquires a restful, "unstrained" aspect. A restrained arch portico with a tie rod (c) is an intermediate solution between a and b. The portico, like any other structural member subjected to bending, requires materials that resist tension as well as compression. It should be built of steel or reinforced concrete, since timber joints, although suitable, in most instances do not fit in the whole assembly. Since proticos usually do not have excessive spans, concrete can very easily compete in economy with steel. The T-section, so typical for reinforced concrete, can be changed and adapted in accordance with the changing sign of the bending moments. The portico should be laterally braced, otherwise there is the same danger of bulging under combined transverse and torsional bending, as in an arch. Within its plane, it is less likely that a portico will buckle than an arch, since its sections have usually relatively larger moments of inertia, owing to its greater eccentricities and consequently greater bending moments. Perhaps this is, mechanically, the main difference between a portico and an arch.

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chapter 8

the vault and the dome

The vault is one of the oldest structural elements in the history of building techniques, the first type being, of course, the cylindrical vault. It seems that the Greeks granted to Democritos the honor of its invention, but it must have been only a local discovery, because the Egyptians had been using it many hundreds of years earlier. A continuous vault spanning parallel walls may be considered as a succession of independent arches assembled together. However, it took a long time to change this simple concept and to consider the continuity along the longitudinal lines or the generatrix of the vault which makes the structure resistant to bending also in that direction. In this way each vault element can be partly supported by the adjacent one, transmitting to it a part of the load acting on it. The concept of the vault was slow to develop because of a certain confusion at the very beginning. Yet, once understood in basic masonry forms, it was developed with increasing clarity and efficiency up to modern shell structures in which the original function of the arch disappears. In the primitive vaults of antiquity, the curved shape was in conformity with the simple idea of a continuous assembly of arches with complete joints between them. Very interesting and instructive is the type shown in figure 8:1a which was developed by early builders of the Orient who used segmental flat bricks. Slight inclination of individual arches having relatively small thickness and inclined joints (normal to the plane of inclination) facilitated placement of the bricks without centering or other temporary supports. It is a typical case demonstrating the influence of construction method and of material on the resulting structural type. Enchainment of bricks and stones was used in the Old Roman architecture and is still practiced today.

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CHALDEAN YAUIN

Fig. 8:1. Masonry arrangement in Chaldean (a) and Roman (b ) vaults.

It prevents separation and possible uneven settlement of individual sections (fig. 8:1 b). In each row of stones along the generatrix normal joints are offset from those of the row above or below, thus making one continuous vault of greater strength. The advantage of this type of vault masonry is especially evident when perpends are used for strengthening. Perpends are not merely ornamental features, which break up the monotonous continuity of the vault, improving its appearance and dividing up the space. They are also structurally important, increasing the strength of the whole vault, along its total length (owing to the longitudinal rigidity). In masonry vaults, the stiffening effect of the perpends does not extend longitudinally so far as in vaults made of stronger and more efficient materials under bending (as will be discussed later). Consequently masonry perpend arches should be disposed at more frequent intervals to make them more beneficial. Vaults in general, of any shape, produce inclined forces upon supporting parts of the building, and if these supports are vertical walls, their thickness must be considerable to counteract the horizontal thrust by greater proper weight of the wall and to maintain the resulting reaction close to the center of the supporting base. Simultaneously, the horizontal thrust of the vault produces shearing forces that tend to displace the supporting layers along their horizontal joints, and this danger must be avoided. In a vault with perpend arches, forces tend to concentrate upon these more rigid ribs. This makes it easier to relieve these thrusts by means of buttresses, placed precisely in the planes of the perpends. Cylindrical vaults are most suitable to cover rectangular spaces, and can be single across the width or in several parallel rows, in which case the horizontal thrusts over interior supports counteract and balance each other. For this reason, these interior supports are much lighter, and in many cases, slender columns will be sufficient. Nevertheless it should be realized that the equilibrium becomes unstable if the columns and the proper vault do not have sufficient flexural resistance. Small differences of loading upon two adjacent vaults can produce considerable distortion or failure. In such instances, stability depends on rigidity against horizontal movement (swaying) of the vault, as will be demonstrated in detail in our discussion of cylindrical shells in reinforced concrete. Enclosing walls along extreme directrixes of the vault and any intermediate wall (as, e.g., a partition) will increase the lateral rigidity and thus the stability of the whole assembly. Yet, efficient interconnection of two vaults can be attained not only along a common generatrix but also by many other arrangements. Combination of various intersecting cylindrical vaults has yielded results interesting from many points of view, for example, groined vaulting and the cloister vault. If

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the latter seems to be most naturally suited to cover polygonal spaces enclosed by massive walls, the groined vault (fig. 8:2) has its functional origin in the attempt to distribute reactions among various isolated points. In the case of a single-vaulted element, the inclined reactions will require buttresses (counterforts). If there are two or more vaults crossing each other, the reactions can be balanced, thus resulting in vertical forces acting upon intermediate columns, with all the attractive features of medieval architecture. The value of a groined vault with ribs culminates finally in its complete structural development as achieved in the Plantagenet style with double curvature vaults between groin ribs. It is difficult to make a general statement about the exact stress distribution in a groined vault. If the vault is of stone masonry, the stress distribution will largely depend on the type of bond and the system of joints. Even when a vault is made of a monolithic material, it has not been possible to determine exactly the pattern of stress distribution (isostatics). But it can be asserted, none the less, that a ribbed vault is not the same as merely a series of vault segments supported by the arched ribs. These ribs are really no more than the intersections between adjacent vault segments, and its essential sense as structural parts can disappear. This is not the appropriate place to analyze the fascinating wealth of methods of treatment arising from the mutual encounter of various types of vaults, door vaults, lunettes, and many others. Stone masonry applied to these architectural forms has been replaced currently by new material and techniques. Even so, the study and contemplation of classical examples from all ages still is a source of great information and enjoyment. Cylindrical shells built of reinforced concrete are structures totally different from the old masonry vaults built essentially of materials (e.g., stone or brick), strong in compression but without tensile strength. Once reinforced concrete had been invented, it was possible to use its tensile resistance for much lighter structures, which of course involve more complex stress distribution. A completely new type of structure was thus made possible: the cylindrical shell supported by rigid arches, or transversal walls, arranged at certain distances in planes normal to the cylinder axis. That is, the cylindrical shell is unsupported along its longitudinal edges. If one tries to support a sheet of paper in a horizontal position along two parallel edges, as shown in figure 8:3a, he will observe a considerable deflection, and, if the paper is not thick enough, it will slip and fall between the supports. However, if the sheet is supported only on the two middle points of two opposite edges, as shown in figure 8:3b, the unsupported edges will move downward, but the curved sheet will resist not only its own weight but even some additional load. In the same way the cylindrical

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Fig. 8:2. Groined vaults in underground station in Madrid. Engineer, E. Torroja. Photograph, Cartagena.

Fig. 8:3. Stress patterns in cylindrical vaults and shells.

surface works like a rigid beam with a cross section following the directrix of the cylinder. This simple experiment clearly demonstrates the fundamental principle and the advantage of this structural type. The shell, though shaped as a vault, has an entirely different system of stress distribution: for more than to a vault, it could be compared to a beam. If the sheet of cylindrical shape is supported only along the curved edges either by walls or stiffeners, as indicated in figure 8:3c, or, in other words, if the shell is restrained or fixed along the end directrixes, resistance and rigidity increase considerably. If the lowest part of the lamina is not vertical, this part near the longitudinal edges, and the edge itself, bend slightly, overcoming the smaller flexural rigidity of the paper. What is the actual stress distribution of this shell structure? What forces act, and how are they in equilibrium? For better understanding of the bearing of the structural element let us consider the interior forces and stresses as arranged in two different groups (see fig. 8:3c): longitudinal forces, T, acting more or less as those in a beam, and shear forces, S, acting in planes tangent to the shell. We must also consider direct stresses (tensions and compressions), Q, acting along the directrixes. Furthermore, there are bending moments around the generatrices, particularly in the central zones of the vault. And, as already stated, these deformations are responsible for sagging along the edges. Bending moments, M, normal to the shell, are only nil along the two supported edges. These moments, and the corresponding shear forces, N, have sinoidal variation, that is, change their intensity along both directrix and generatrix. The magnitude of these moments is not trivial and may even cause the failure and collapse of the shell, since this is so thin. Their variation along the directrix depends largely on the shape of this, and on the manner in which the longitudinal edges are attached to other contiguous structures. The primary internal forces—T, S, and Q—acting in tangential planes, produce principal stresses, like those in structural elements of bidimensional elasticity. These stresses act along a network of isostatic lines which, on each side of the symmetrical lobe, is similar to the isostatic network of a plane beam. Such a network, spread over the cylindrical surface of a twodimensional shell, is shown in figure 8:3d. Longitudinal forces, T, increase with the decreasing rise of the cylindrical shell in relation to its span, L. Shear stresses also increase in proportion to the increasing flatness of the shell. Both conditions, therefore, determine the final proportions of the shell: the minimum slenderness (1/k ratio) and curvature required for the safety of the structure. The minimum thickness

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Fig. 8:4. "Seagull" type of beam at the Fronton Recoletos (Sports Hall), Madrid. Photograph, S. v. Kaskel.

necessary to resist transversal deflection is usually determined by aforementioned bending moments. However, it is not advisable to increase the ratio between the span and the thickness too much beyond 500:1 to avoid buckling; for example, for a span of approximately 100 feet, the thickness of the shell should not be less than 2% inches. The amazing slenderness of a thin shell in reinforced concrete is demonstrated in figure 8:4, the roof structure of the sporting hall Frontón Recoletos, which had a span of 180 feet and was only 3.2 inches thick, resulting in ratio cf 680:1. In any case, transversal stiffness can be considerably increased by small ring beams arranged in adequate and economical distances—inside or outside the shell -—following the directrixes and interconnected monolithically with the shell. It is very interesting to observe that in cases of full cylindrical shells (fig. 8:5a), that is, shells with closed directrix—circular, elliptical, etc.— theoretically, equilibrium of forces can be attained without necessity of flexural resistance of the shell, and, therefore, the only limiting factor for

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Fig. 8:5. Types of cylindrical shells. the thickness of the shell would be the danger of buckling caused by compressive stresses acting in the direction of generatrices. These cases are referred to as equilibrium of so-called pure membranes. This does not imply that there is no bending at all, since deformations in transversal planes induce secondary bending moments along the directrixes, and these increase with the thickness of the shell. None the less, these forces are not significant in the general equilibrium of the system. If such a closed cylindrical shell is cut in a horizontal diametral plane (fig. 8:5b) and supports only its proper weight, the antisymmetry of this loading will cause the stresses Q along the directrixes to vanish along both edge generatrices. The original equilibrium can be reestablished by introducing in these edges post-tensioned reinforcement, neutralizing the shear stresses S which result from the removal of the lower half of the shell. But nowadays, in modern design, it is preferred to do away with the steeper edges of the shell vault. Such a shape presents some difficulty in concreting and is more expensive due to double forming. The most suitable

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shapes are flat cylindrical shells intersecting along parallel generatrices, with additional vertical beams under the longitudinal edges (fig. 8:5d) or without them (fig. 8:5e), the latter being aesthetically more pleasant and more economical in forming. In this instance the compressive forces, Q, produced at intersection lines of the shells, in a complete cylindrical shell, are no longer equal to zero as it would be if these intersection lines were in the horizontal diametral plane. It is evident that on removing these forces, which kept it in equilibrium, the "skirts" (i.e., the lower, lateral sides of the shell), lacking support, will tend to fall. To counter this, the central zone of the shell will have to support the outer part as cantilevers. The shell structurally, or the isostatic lines, then behave as a series of bows (curved in three dimensions), whose central portion will be near the crown of the shell, and whose extremities will converge upon the corners of the lamina. The reinforcement along the longitudinal edges will then act as the tensional tie member of these bowed strips. Figure 8:5f shows the deflection lines in both directions. These deflections diminish and vanish toward the supports of the shells where the arched strips following the generatrices are rigid. The classic cylindrical vault, uniformly loaded, is no more than a particular case of the cylindrical shells, in which all transversal sections are identically loaded and are equally unrestrained to deflect. Each deforms freely according to its own loading, behaving as an independent arch. But Fig. 8:6. Vaults of the underground vestibule at the Madrid Hippodrome.

Fig. 8:7. Saint Engelbert's Church, Cologne, Germany. Architect, D. Böhm. Photograph, H. Schmölz. if the vault, being elastic, rests on rigid walls along its two longitudinal edges, the end sections will be rigidly restrained, and thus not all transversal sections will deform equally (the shell will no longer remain cylindrical). Intermediate sections will deform, whereas the extreme ones will remain unchanged, so that longitudinal strips will bend. If the lamina is thin, this bending will provoke negligible stresses. The important fact is that if the central longitudinal strips bend, shearing must occur in the plane of the lamina. But such shells are very resistant and rigid to shearing in their own tangent plane in each point. And actually, it is these shearing forces, S (owing to the relative movement of parallel strips), that differentiate the structural behavior of a cylindrical shell and a cylindrical vault, unrestrained transversally throughout its length. Three-dimensional shells result from combinations of two or more barrel shells in mutual intersection (fig. 8:7), and classic cloister vaults with groins belong to this category of polylobular shells. There are interesting applications in the new field of space structures, one of the most original being the Butterfly Bridge proposed by Frank Lloyd Wright and J. J. Polivka, as the second crossing of the San Francisco Bay (fig. 8:8).

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In such structural types the mechanically resistant assembly is not conceived as an arrangement in which the main arches support secondary arches along common directrixes. It is rather an assembly of primary arched elements in the space having directrixes of double curvature rooted in the supports but adjusted to, and joining, one family of isostatic lines, whereas isostatic lines of another family establish counteracting forces between each primary arch and its contiguous elements in order to align the network of pressures over the warped directrix. Only in this fundamental and, at the same time, experimental and developmental way of thinking will the designer, technician, and artist be able to deduce the many possibilities that have previously been hidden. The general comments made for structures of the membrane type need not be limited to those built of solid materials (e.g., reinforced concrete) but also apply to those consisting of a network or grid substituting for the ideal continuous membrane. In the roof structure designed by the author, and shown in figure 8:4, a part of the cylindrical shell was replaced by a triangular gridwork of reinforced concrete providing skylights, and in figure 8:9 similar arrangement is shown for a cylindrical roof of steel or timber, which then can be covered with adequate and appropriate sheathing. The slightness of the welded latticework, which can be prefabricated and lifted in parts or as an entire assembly can result in a great economy for this structural concept.

The dome is another of the simple and most favored structural types of classic architecture. It is a most natural answer to a need for a structure that overs a given space without intermediate supports. It can be light, yet sinultaneously very strong and resistant, and can be constructed with £ minmum amount of material, which therefore can make it most economOrijinallv, domes were built in circular shape with a pointed crown, and were supported along the whole periphery. Designers of domes in the Renaiisance era were confronted by very difficult problems. The materials then a/ailable did not resist tension, and so their domes had to be built with pointel crowns, because in a spherical dome the horizontal layers are subjected to tension and require very heavy buttressing to secure equilibrium. This i: especially true if the domes are built upon cylindrical walls of considerate height. Hovever. the pointed directrix of the dome structure was first developed in Balylonian times in more or less primitive constructions. The prehistoric cupoli of a dolmen at Romeral (fig. 8:10) is interesting precisely because the anh has been built up by means of horizontal layers (instead of arranging thj joints radially about the arch center) and finally crowned with a great Jab. Only by this arrangement could shearing along the horizontal joints ->e reduced.

Fig. 8:10. Prehistoric dolmen at Romeral, Antequera, Spain (from Summa Artis, by M. B. Cossio and J. Pijoan; Madrid, Espasa Calpe).

Most interesting are domes built in clay by African Negroes in Tchad (fig. 8:11). Never has man been able to build with such primitive means a structure more rational or any better adapted to the properties of the material used and to the economic exigencies of structural lightness. Tensile stresses in these structures are so small that even clay can resist them. These huts can be built in successive circular layers, and the structure remains stable throughout the whole process of erection, without need of scaffolding or formwork. Actually the builders can work on the higher part of the structure by using as footrests the promontories left for this purpose on the outer surface. Whole settlements of these paraboloidlike dwellings possess an undoubted aesthetic quality, and if the dome is in its majesty the symbol of monarchy, these assemblies of ovoloid homes would rather suggest the healthy sense of democracy; each small dome rising no higher than its ubiquitous neighbor. But it is with stone that man has developed, up to recent times, the most

Fig. 8:11. Mud huts in the Tchad region (from Summa Artis, by M. B. Cossio and J. Pijoan; Madrid, Espasa Calpe).

technical and aesthetic values of the dome. Along the parallels and meridians, its arched form corresponds perfectly with the stress distribution. Compression along the meridians acts normally to the joints of horizontal layers, and in these horizontal rows the compression in the parallels is smaller or converts into tension. For masonry walls an important technical problem involves good distribution of this compression in vertical planes and the elimination of the tension in the parallel layers. In a spherical dome this tension begins to occur in the parallels below the angle of 51° with the vertical axis. For this reason it is recommended, to avoid tensions, that spherical domes in stone masonry be designed with a smaller angular opening or with ogival profiles in meridian planes. Domes of the same span but of greater flatness have, however, the disadvantage of greater horizontal thrust at the supports. On the other hand, pointed shape of domes (e.g., that of Saint Peter's Cathedral, fig. 17:1) is statically necessary because of the interrupted compression line due to the additional load of the lantern. All these difficulties are eliminated by use of reinforced concrete, which permits realization of domes with enormous spans and of minimal thickness, the latter being nearly always limited only by danger of bulging. Reinforced concrete resists both tension and compression, eliminates statical exigencies of a directrix as it is in stone or brick, and widens the conception of a dome to a more general thin shell of revolution. Structures already designed, such as the Stockholm Hall of 500-foot span and only 6-inch thickness, show the enormous possibilities of this structural type, which can easily supersede the proportions of an eggshell (fig. 8:12). The basic stress distribution and deformation of a dome structure can be interpreted and understood if one imagines that the dome consists of meridian elements whose deflection, under given loading, is counteracted or lessened by horizontal ring elements. In sections where the meridian strips have a tendency to bend downward, there will be compression in the hori-

Fig. 8:13. Circumferential and radial stresses in a symmetrical cupola. zontal rings, and where the strips tend to rise, there will be tension. W e are now confronted by the problem of how to take care of tensions and resulting deformations. In the first place the deformations of the shell are not always so small that they can be neglected in analysis. The dome surface is continuously attached to the outer ring. This reacts the radially outward thrusts owing to the weight of the dome. Consequently the ring stretches, increasing in diameter (fig. 8:13). Hence the shell's lower part must also stretch circumferentially, to adjust itself to the greater size of the ring. This causes the shell's lower sector to bend outward. Furthermore, the circumferential elongation tends to crack it along its meridians. A similar phenomenon would result if the dome, which is elastically deformable, were to rest on a rigid ring—unless the edge were to correspond to a latitude at which stress and strain are nil (at an angle of 51° as noticed before). But the usual practice for flattened domes is that the lowest circular section tends to be compressed, while the encompassing hoop is in tension. This aggravates the difficulty, and the shell bends considerably along the lower part of its meridians. In this type of problem, bending may not be a primary stress consideration (since the shell, if very thin, would have negligible bending stresses, the bending moments tending in that case to be very small). But it is still significant, as the shell may either crack in bending, or fail in compression, when the normal end load along the meridians is increased by the compression due to bending. Shrinkage of concrete is responsible for similar effects. This is easy to appreciate, and must not be overlooked. Post-tensioning of the supporting ring is the ideal solution of new techniques used in such situations, since any circumferential compression can be introduced which will counteract the resulting tension and will simultaneously eliminate or considerably reduce the bending moments along the meridians.

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One important question is the effect of unequal heating of certain parts of the cupola in relation to other parts, because these effects, like an unequally distributed live load, are different from the symmetry of revolution. In thin-shell domes, the dead load is comparatively low, so that the live load (e.g., wind or snow), if not evenly distributed, can produce considerable secondary stresses. This uneven distribution not only makes stress analysis very difficult, but also may considerably reduce the advantages of this type of structure. Finally, the danger of bulging in thin-shell domes may require a stiffening network of ribs following the meridians, or in any grid shape that satisfies aesthetic requirements, as exemplified in the past by the work of HispanoArabian artists. The design of a dome must be intimately connected with its construction method, especially if the dome is to be erected without scaffolds. In such case, the supporting ring makes it possible to build up the dome with overlapping horizontal layers, each projecting over the previous one. In certain cases, a horizontal stiffening ring, temporary or final, may be justified, and will definitely be necessary if the dome is to be topped with a lantern. Very often, a dome is supported at spaced points instead of resting on a continuous wall along the whole periphery. One of the classic solutions is a dome erected over four arches in planes forming a square in the plan and with corner supports, as shown in figure 12:7. The most suitable material for such domes is obviously reinforced concrete, although there are certain limitations of span as compared with dome shells supported along the whole periphery. The Market Hall in Algeciras (fig. 8:15), of 156-foot span, and only 3%-inch thickness, designed by the author, has eight supports interconnected by a polygonal post-tensioned ring. The dome shell is cut between the supports by lowered cylindrical shells that project as marquees under which are installed lunettes to illuminate the interior. These cylindrical shells simultaneously strengthen the dome shell, as shown by isostatics (fig. 8:14). The octagonal periphery ring is contained by adjustable ties in Fig. 8:14. Market Hall at Algeciras, Spain.

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Fig. 8:15. Market Hall at Algeciras, Spain. Architect, M. S. Areas. Engineer, E. Torroja. Photograph, Granada.

Fig. 8:16a

Fig. 8:16. Church of Saint Felix and Church of Saint Regula in Zurich, Switzerland. Architect, Metzger. Engineer, Schubiger. Consulting engineer, E. Torroja. Fig. 8:16b order to maintain the radial forces in equilibrium. This effectiveness of tensioning these ties was clearly noticeable when the forms were ready to be removed after completion and curing of the dome. In this operation, the center part of the dome shell was actually raised slightly and the scaffolding then was easily removed. A surface of revolution permits many forms of structures in which the characteristics of a dome completely disappear, such as the hyperboloid, generally used for thermodynamic installations (fig. 15:1). This is a perfect

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Fig. 8:17. Dome at Festival of Britain, London, 1951. Architect, R. Tubbs. Consulting engineers, Freeman, Fox and Partners.

structural type not only from the standpoint of rational design but also with respect to the efficiency of the aerial flow. Elliptical dome shells have similar characteristics (fig. 8:16) and their stress distribution can be derived directly or by analogy from that of a circular shell. There are many applications of similar structural principles: cupolas as enclosures of cylindrical shells, metal sheets with spheroidal corrugations which increase the bearing capacity and reduce deflections, cylindrical retaining walls, and multiple arch dams. Domes, even as vaults, can be built of metal or other materials as well as of concrete, with a grid work substituting for the continuous surface. The grid work following the meridians and the horizontal rings is usually stiffened with diagonals. Most logical appears to be an arrangement with triangles, such as was used for the huge aluminum dome of the Festival of Britain (fig. 8:17). It has always been classical in construction, until our time, to place the concave part of a curved surface inward. This was not only because the enveloping surface is thus less for a given volume. The more fundamental reason was that up to recently the materials used could not take tensile forces. It is only in recent years that some tendency has developed to adopt 112

Fig. 8:18. Municipal Public Hall at Karlsruhe, Germany. Architect, E. Schelling. Engineers and builders, Dyckerhof & Widman Co., Weiss & Freytag Co., M. Jordan. Photograph, G. Weiss. surfaces whose convex side faces toward the interior. Such surfaces work in tension and are anchored along their periphery (this periphery is not normally in one plane), which is also the upper edge of the vertical enclosing surface, or walls. Examples of this type of design are the Municipal Public Hall at Karlsruhe, Germany (fig. 8:18), and the Raleigh Stadium (fig. 10:12). In both these structures the top lamina works basically in tension, and the double curvature of its surface ensures the rigidity of the whole assembly. In the second example the structure consists of a network of cables or steel rods. In the other, the prestressed shell is in itself waterproof. This indicates the wide variety of solutions that are possible with these tension-stressed surfaces.

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chapter 9 the beam and the slab

Following our practice of looking to the origin and evolution of each structural member, it seems to be logical to assume that the first practical beam was made from the trunk of a tree, the second was a monolith of stone without trimming, and the third, a lintel consisting of a large hewn stone. Much later, small stones, hewn to wedge shape and acting actually as an arch, were given the exterior pattern and shape of a rectangular beam (fig. 7:8). It required centuries of experience and study before builders achieved these types of beams; and many more centuries yet were necessary to pass from that stage of progress to the development of long-span beams and girders such as the 666-foot girders of the steel bridge across the Rhine River at Cologne, Germany (fig. 9:1). A monolithic lintel over the pilasters or jambs of a Cyclopean wall was the first triumph of the human builder in providing for, and securing, an open space bridged over by a permanent structure. Those first builders had no idea that such a beam carries its load only because of its resistance to bending, as was later discovered by Galileo, Napier, Euler, and Saint Venant. They learned by failure that if the depth of the lintel was small in relation to its span, the beam split easier and collapsed. They also learned that if a beam was prevented from rotating over its supports by proper fixation of its ends in the masonry, the danger of breakage became less serious. They discovered, too, that a greater resistance was obtained if the depth of the beam was increased in the center where the maximum bending moments occur. Finally, they recognized the limits of the bearing capacity of such beams, and in instances where these limits were to be approached or exceeded, they provided supplementary strength in the form of relieving arches, thus preventing collapse of a monolithic lintel. All this was necessary

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Fig. 9:1. Steel bridge over the Rhine River at Cologne, Germany. Engineer, F. Leonhardt. Architect, G. Lohmer. Builder, Gute-Hoffnungshiitte Corp, Photograph, Stein. since stone has practically no resistance to bending due to its low tensile strength. However, a wooden beam distinctly shows its resisting capability: it has a considerable flexibility and its fibers are subjected to both compression and tension. It helps also to resist the shear that in many timber elements is decisive. But as timber dries, its cohesion between parallel fibers becomes weaker and therefore less resistant to shearing forces. Shearing failures always occur in slippage of the fibers. To prevent this kind of failure between two beams arranged one upon the other, inclined blocks were inserted along the connecting face where maximum shear occurs. These blocks or connectors can transform shear into compression, which acts at an angle upon parallel grains (fig. 9:2a). In quite recent times,-built-up girders have been used. These consist of flanges and webs in an arrangement similar to that of steel-plate girders, in which individual members of various cross sections and boards are nailed together, as shown in figure 9:2b. This structural principle has mechanical

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Fig. 9:2. Composite timber beams. and economical advantages and permits the building up of timber girders of bearing capacity and dimensions greater than any which could be attained with even the largest piece of solid wood. Steel is the ideal material for structural elements subjected to bending. The rolled sections, especially the typical I-shapes, permit a good utilization of the material, although with certain economical limitations. The section is constant throughout the length of the beam, whereas the internal forces vary in accordance with the loading scheme. The maximum bending moment usually occurs near the center of the span where shear stresses vanish and the maximum shear is produced near the supports of the beam where the bending moment is zero when no continuity is considered. The ideal section would vary from the I-shape in the middle of the beam, with decreasing width of flanges toward the supports where the simple web would be sufficient, or, maintaining the I-shape throughout, with increasing thickness of flanges toward the center of the span, and with decreasing thickness of the web in the same direction. It is evident that this type would be impractical, due to the difficulty of rolling shapes with a variable thickness. For smaller sizes the excessive weight of steel is usually compensated for by the lower unit price of constant sections. For heavier loads and longer spans,

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plate girders are used, in which a more economical distribution of material is attained by riveting or welding additional steel sections to the web plate and to the flanges in accordance with the variable internal forces that require variable moments of inertia of the cross sections. Steel today offers great possibilities in girder structures of long spans. The arrangement of flanges, consisting of angles and of single or multiple plates and the connections, is of great importance (fig. 9 : 3 ) . Sections of great slenderness always require careful consideration to resist any possible bulging that can be caused by torsion. Bulging can be eliminated either by widening the flange in compression or by transverse bracing and vertical stiffeners in the web which restrict torsional effects. Stiffeners may also be used at the edge of the plates, and special shapes are rolled for this purpose, as is typical for light-weight beams. Stiffeners welded or riveted to the web plate of heavy girders should be arranged at intervals required by internal forces and by the thickness of the web plate: the thinner the plate the closer the spacing. Many studies and experiments have been made in this field. If the elements are riveted together, the rivet holes not only reduce the section in tension but also cause stress concentrations, and it has been shown that in cases of failure these stress concentrations are usually responsible. In welded structures the danger should be sought rather in the stiffeners, since there is a sudden change of the section. The most serious danger may be seen in the tension zones if the welding of stiffeners causes tensional stressing in the other two directions (i.e., in the vertical sense and normal to the plane of the web), and for this reason special attention should be given to the welding details of these elements. Fig. 9:3. Types of composite metal sections.

I

• I

In heavy girders, the box shape, similar to that previously described and used for arches and heavy columns, either with single or multiple enclosures, can also advantageously be used for heavy and long-span girders. They have a greater bearing capacity with smaller weight, a greater compactness, and a more classical and pleasing appearance than simple I-beams. The use of box girders is fully justified when the thickness of the web in an I-beam is not sufficient, or when a wider section is necessary to resist torsion and to prevent bulging. Very often there is a great disproportion between thickness of the web and that of flanges which may seriously affect the strength of the welding owing to different heating and cooling conditions that depend so much on the thickness of the plate. However, these are details of minor importance for the main subject discussed. Despite all these difficulties long full-web girders have been built, for example, the Düsseldorf Bridge, spanning 650 feet. For smaller spans and lower loading, however, modern techniques tend to become independent of all these limitations of built-up beams by developing numerous shapes consisting of thin sheets and plates so formed and arranged that maximum bearing capacity with least material is attained (fig. 9:3). Among these shapes were some that, not many years ago, no designer would have dared to propose. These possibilities, not yet exhausted, demonstrate clearly what can be achieved with typical materials, not to mention possibilities offered by quite new materials (e.g., lightweight alloys, plastics, etc.). Reinforced concrete has different characteristics and possibilities in bending. Whereas the I-shaped section is rational for homogeneous materials having equal strength in both compression and tension, the section of a typical beam in reinforced concrete has the shape of a simple T, since the top section, being in compression in a simply supported beam, requires a greater area due to the lower permissible stress of concrete as compared with that of the steel. Concrete in the tension zone is needed only for embedding of the reinforcing bars with a certain minimum spacing required to secure the coaction of these two materials of such different properties. Besides, the concrete in the web must resist shearing stresses that transmit the increments of tension in the steel reinforcement, and these shear stresses also determine the thickness of the web. For better understanding of the state of stress and its effect on the strength of the structural element in reinforced concrete, reference is made to the network of isostatic lines (stress trajectories) shown in figure 2:7. If the concrete is able economically to resist the compressions, and the steel, to resist the tensions, it is logical to arrange the reinforcing bars along the lines of maximum tension. However, since the loading is not always the same, the designer can only hope

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to guide himself approximately by the network of isostatics in deciding on the best possible distribution of the reinforcement. Although the isostatic lines are curved, it will be more economical to bend the reinforcement bars in polygonal shapes rather than in smooth outlines of varying curvature. Thus the main reinforcement is placed on the tension side, increasing in number from the supports to the mid-span in a freely supported beam in accordance with the magnitude of the bending moment. The distribution must also aim at providing the maximum lever arm and strength in bending. Reinforcing bars near the supports are bent up at about a 45° angle, following approximately the inclination of the lines of maximum tension. Consequently they help to resist the shear, which increases toward the neutral fiber and toward the supports. In any case, the concrete must resist not only longitudinal compressions in the upper zone of the beam, but also inclined compressions inherent in shearing forces, which determine the minimum thickness of the web. For this purpose, wherever bent bars cannot be arranged for any particular reason, it will be necessary to provide stirrups, usually normal to the longitudinal bars (fig. 9:4b). Theoretically, these stirrups do not enter into action unless the concrete homogeneously deforms or cracks under excessive shear stress. Until this occurs there is no tendency for the stirrups to stretch in their own direction. They prevent shear cracks under 45° or failure caused by tensile stress components of the shear. In this case, the beam can be considered as triangular latticework or truss in which the stirrups work as verticals in tension, and the mass of concrete between them as diagonals in compression. The resultants of these forces produce increments of tension in the longitudinal reinforcement which, integrated from the supports to the mid-span of the beam, result in total tension balanced by the compression in the head of the beam.

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Those conditions make it possible to provide in every section of a structural member in reinforced concrete the reinforcement required by both longitudinal and transverse forces, and consequently, the most economical distribution of the reinforcing steel for both purposes. Since, furthermore, forms shaped for any desirable depth and width of the structural element can easily be provided, reinforced concrete as structural material can in each particular instance be adapted to any state of stress—a property almost impossible to obtain in other materials. The adequacy and suitability of the reinforcement for particular stress exigencies in the case of reinforced concrete beams is therefore more technical and organic than is apparent and visible on the exterior of the structural member. So in a technical sense reinforced concrete is not visually self-explanatory. This can no more be called a disadvantage than can the human skeleton, which cannot be seen in the body. Reinforced concrete is not, as is sometimes claimed, a commixture or a superposition of two independent structural materials; it is rather a unique harmonic aggregate or compound material. Reinforced concrete should be looked upon as a new structural material resisting both compression and tension, and it is far different from ordinary concrete, which is known as a brittle material, weak in tension. Although accustomed to brittle stone, we must now accept a stony material that is just as resistant to tension as steel. It has taken many years for the public to become accustomed to lintels, beams, and columns in the much more slender proportions of reinforced concrete than was usual in stone masonry. Prestressed concrete now offers further special advantages in some structural elements. Use of high-tensile steel wires having an ultimate strength ten and more times that of typical steel considerably reduces the sectional area of the reinforcement, and, consequently, also the thickness and other dimensions of concrete members, thus lowering the dead load and deflection, factors of importance for beams of long span. Furthermore, the permanent longitudinal compression induced by prestressing prevents fissures in concrete, and indirectly diminishes the tension that causes shear, all of these being important improvements over ordinary reinforced concrete. If the reinforcement is curved and made to follow approximately the lines of maximum tension, the result would be far more effective, but so far many difficulties in this project have been encountered owing to the friction between steel and concrete in the curved sections of the bars, when these are stressed. This causes considerable loss of stress. Finally, if the longitudinal prestressing (no matter whether pre- or posttensioning) is combined with transverse prestressing along the stirrups, any tension produced by axial forces or shear can easily be compensated or

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rescinded, which is equivalent to a considerable increase of the tensile strength of concrete. Application of these new prestressing techniques, combined with the techniques of precasting, has created new types of extremely light and resistant elements exposed to bending, such as joints, rafters, roof and floor slabs, etc., which, although not equal, are approximately as light and strong as similar members in steel. They possess greater strength and elasticity than do members in ordinary concrete and can be subjected to elastic deformation under which ordinary concrete would necessarily crack or sag permanently. It can be demonstrated that a beam of rectangular section, subjected to a certain bending moment, can still have uniform compressive stress over the whole cross section, as shown in figure 9:5b. It is only necessary that the external bending moment be balanced by the internal bending moment caused by prestressing, or by the product of the total load in the steel and approximately half of the effective depth of the section, instead of approximately eight-ninths of the depth of section—as is the case in typical reinforced concrete. But as the compressed zone covers the whole depth, instead of only one-third, and the distribution of stress instead of triangular is rectangular, the mean compression stress in the concrete will be about one-third. Thus it is apparent that in equal beam sections of the same quality of concrete, the reinforcement can be stressed until the resisting moment is at least trebled without any overstressing or rupture, and this will require that the prestressed reinforcement be able to resist a tension six times as great. However, this advantage is only theoretical, for a beam so designed would fail if the load were either increased or decreased beyond its precise magnitude. Consequently this type of prestressing could be useful only in

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structures of such great span and dead load that any live load on them would affect the total loading only very slightly. This is a delicate and very important point to be considered by the designer. It is derived from the state of stress, shown in figure 9:5c, caused by prestresSed steel embedded in concrete, but without consideration of stresses produced by external loading. It is not enough, therefore, that the concrete, after full loading (under dead and live load) remain in good condition (fig. 9:5e). Eccentric compression produced by prestressing of concrete can result, at least theoretically, in tensile stresses in the upper zone of the beam causing cracking, unless it is prevented either by top reinforcement or by changing the concrete section in such a way that the compressive force is exerted close to its center of gravity. This explains why sections of prestressed concrete beams very often have the shape of an inverted T (fig. 9:5d) or box, so that there is a sufficient amount of concrete around the prestressed or post-tensioned reinforcement. It is just as likely that the beam will fail after as before loading. Better quality of concrete allows greater slenderness and lightness in prestressed concrete beams made by precasting. These elements can thus be more economical and safe owing to the higher stresses permitted than in those of ordinary reinforced concrete. Finally, it should not be forgotten that with the slenderness increasing far beyond usual proportions of beams there is an increase of cost and also of deflection, and especially the latter imposes certain limitations of lightness. Development and broader use of continuous beams was made possible by advanced studies of hyperstatic structures and their analysis. There are very few ancient examples of structural continuity such as the prehistoric cave of La Menga (fig. 9:6), where monolithic stone slabs rest on walls and are also supported at the mid-span by stone columns. Here it was made sure that the three supports were equally effective in taking their share of the load by resting them, not on the rigid rock foundation, but on a layer of hard earth, which, owing to its greater plasticity, could in a certain measure readjust the reactions and heights of the three pilasters. But did the ancient designers realize this?

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Fig. 9:6. Megalithic dolmen of La Menga, Spain (from Summa Artis, by M. B. Cossio and J. Pijoan; Madrid, Espasa Calpe).

The advantage of a restrained beam is that the bending moment at the fixed supports is two-thirds and at the mid-span only one-third of the maximum bending moment in a simple beam uniformly loaded (fig. 9 : 7 a ) . It must be kept in mind, however, that the center moment in this continuous beam is more sensible to the weakening of the restraint and to the changes of continuity over the supports, and it increases sensibly with the decreasing rigidity over the supports. A continuous beam with equal spans ( b ) has analogous advantages if all spans are uniformly loaded, but if there is a concentrated or rolling load, which can act on one span and not on the others, these advantages are lessened, and there is a smaller reduction of the bending moments ( c ) . A continuous beam with constant section shows a better distribution of deflections and greater bearing if the end spans are smaller than the intermediate spans (d); besides it makes a better appearance. In the old Greek arrangement of columns, the favorable optical effect was enhanced not only by following a similar principle (reducing the end spans), but also by arranging the end columns so that they leaned slightly toward the center of the row. If a section of equal resistance for a continuous beam is studied, it can be observed that the intensity of the bending moments sensibly increases in the proximity of the supports within relatively short stretches, and simple haunches are sufficient to take care of this change. There is no great difference in distribution of shear in continuous and simply supported beams, although the latter can show greater values over the intermediate supports in accordance with the distribution of loading. Like any other hyperstatic structure, a continuous beam requires for better function a perfect restraint at the supports. If yielding supports are provided in a simple, isostatic beam, the interior forces are not affected by settlement of the supports. However, in a continuous, hyperstatic beam, settlement of any support causes considerable changes of stresses; but the continuity has advantages that partly compensate for this unfavorable phenomenon, as will be discussed later. In an isostatic beam one weak section can cause failure, whereas a continuous beam in a similar case can continue to bear as long as the sections at the supports are not overstressed, since the crack will act as an artificial hinge. Following the laws of flexure, a continuous beam has certain points where the bending moments vanish for any given loading. These points may be very close for different types of loading. If hinges in these points are inserted, the beam is converted into a statically determinate structure. Arrangement of hinges with free dilatability in alternate spans and approximately one-fifth of the span distant from the supports leads to the classical Gerber beam, which is nothing

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Fig. 9:7. Moment diagrams of continuous beams.

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The use of hinges and sliding supports permits longitudinal expansion and contraction of the beam, which under certain circumstances is of great importance, since without them practically irresistible forces can be produced. Smaller spans usually do not require this arrangement because there is always some possibility of yielding at the supports. A beam rigidly connected at both ends with elastic columns gives a monolithic frame. If a continuous beam is used in a similar way, a multispan frame is formed. Multiple-span and multiple-story structures will be discussed later. The cantilever beam might be said to be something like half such a beam turned upside down. To keep itself in position it needs one rigid attachment (fig. 9:80). It is therefore not so easy to place and keep it in position as the simply supported beam of double span, although both beams will resist the same maximum moment under their own weight. If the cantilever does not extend far into the wall it will lever up the stone above it, and it will bear heavily on the soffit at the edge of the wall with a heavy concentration of stress at these points. This places a limit on

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Fig. 9:8. (a) Example of an ancient bridge in Gaul as drawn by Viollet Le Due; (b) ceiling structure of Santa Maria de la Huerta, Soria, Spain, twelfth century (from Ars Hispaniae, Vol. IV, by L. Torres Balbas; Madrid, Editorial Plus-Ultra); (c) eave of the Portico del Partal, Granada, Spain, thirteenth century (from Ars Hispaniae, Vol. IV, by L. Torres Balbas; Madrid, Editorial Plus-Ultra).

its free span, and this type of structural element has not been any the less frequent or welcomed because of this. Individual projecting stones permit only small spans of relatively great thickness, as emphasized by the gargoyles typical of Gothic architecture. Timber cantilevers can project farther out (fig. 9:8b), but the free span soon becomes too great, and the beam must be supported by a bracket (fig. 9:8c), thus becoming a structure in which the bracket, pressing against the wall takes compression; the horizontal beam, above the bracket working as a tension chord, tends to pull out of its attachment in the wall. That is why it is frequent for this type of structure to be built in symmetrical pairs, so that they balance each other about a central support. But in this arrangement if the resultant of the load is offset from the center line, the resulting bending moment acting on the support has to be resisted by proper restraint in the support or by shifting the double cantilever for better balance. Before reaching the state of building stone vaults, the Roman builders managed to span open spaces by erecting successive cantilevers (fig. 9:8c), one upon the other, well anchored by counterweights, and each advancing farther out into the open, toward the opposite side. When sufficiently close, the two half-structures were connected by short beams. Naturally, steel and reinforced concrete permit much greater projection of cantilever structures with better anchorage and counterweight. For these cantilevers, the classical T-section is inverted, since the compression zone is at the bottom and wider seat is necessary. For longer spans the tension zone can be separated in form of an independent cable (fig. 9:9), in a triangular assembly with the beam compressed over the pier (Tempul Aqueduct, designed by the author). The cantilever is a structural member resistant to bending in a way similar to that of a simple beam supported at the mid-point. Even as cylindrical shells supported at the ends have a certain similarity to simple beams, the cantilever principle has been adapted also to shells in reinforced concrete, not only in cylindrical shape but also to shells with double curvature, as based upon conoid and hyperboloid. These structural types are used in modern design for large canopies and marquees, as shown in figure 12:19. Another structural element subjected to bending in one or two directions should not be overlooked—the slab, in which the state of stress is of great importance and merits more attention in our discussion. First of all, quite generally, if the width of a beam having a rectangular section is several times greater than its depth, the structural member is called a "slab," not a "beam." The slab constitutes a more elementary form with which to cover a surface or space between two parallel lines or walls, and was relatively little developed until reinforced concrete offered so many advantages due to its monolithic character.

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Fig. 9:9. Tempul Aqueduct at Jerez, Spain. Engineer, E. Torroja. Photograph, M. Garcia Moya. If a slab is subjected to a concentrated load or to an unequally distributed load from one parallel strip to the other (fig. 9:10a), deflections can be observed in both directions, between the supports and across, which gradually decrease and in some distant places become reversed in the form of upward bulging. In a rectangular slab of reinforced concrete supported along four edges, reinforcing steel in both directions will be necessary, as shown in figure 9:10fc, the main reinforcement between closer supports, and the transverse reinforcement across. Additional top reinforcement in the corners will take care of possible torsional stresses. Regulations for design contain all necessary requirements and specifications to be considered for calculation not only of the adequate reinforcement in both longitudinal and transverse sense, but also of the dimensions, thickness, and the width of the equivalent beam to be analyzed, since it is clear that these dimensions depend on the shape and plane dimensions of the slab. But here we are interested only in the concepts of the structural phenomenon and its qualitative characteristics.

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Fig. 9:10. Types of two-way slabs. If the monolithic slab is supported along the whole periphery instead of along two parallel edges (fig. 9:10c), all aforementioned phenomena will have more essential importance in the state of stress, and will create a new structural element—the two-way or four-way slab. This type of slab is characterized by flexure in two orthogonal directions, or simply by space deflection. Deflection in one direction is not possible without producing simultaneously deflection in the other direction, no matter in which way the loads are applied and distributed. The periphery of such a slab can have different shapes, rectangular or polygonal, circular, elliptic, etc. Each type of support and loading will require a different thickness, especially different amount and distribution of reinforcement in various directions, and will show different deflections. In

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each point will be two-way deflections and intensities of bending moments, determining, in combination with shear, the fundamental state of stress. In reinforced concrete the bending in any direction requires proper steel reinforcement, and its determination is the main problem in such slabs. On the other hand, no special consideration is necessary for the concrete in the compression zones, since this can withstand, simultaneously, compression in two perpendicular directions: the stresses in one direction not decreasing the strength in the direction normal to it so that the thickness of the slab is to be calculated for the bending moment in both directions. If two perpendicular bending moments are equal, as in the case of square or circular slabs, the strength of concrete, so to speak, doubles. And this, from a mechanical standpoint, is the main advantage of this type of structural form. This advantage is even greater if steel or any other material, resistant equally in compression and tension, is used, and the allowable bending stresses can be considerable. But the possibilities in application are limited by economical reasons and by methods of fabrication. Rectangular slabs supported along the periphery are necessarily subjected to torsions. This can be understood by realizing that deflection of a strip in one direction is affected by deflection of crossing strips (fig. 9:10c). This torsional phenomenon increases the resistibility and appreciably relieves flexural forces. But there is, however, the danger of cracking, especially near the corners, where torsional stresses reach their maximum. This phenomenon may be visualized by realizing that near the corners the spans, parallel to the diagonal, are very short (fig. 9:10c). Consequently, when the slab deflects, the line aa becomes strongly curved downward. This severe bending requires reinforcement of the lower face, in the direction aa. Similarly, the slab must be reinforced on the top face, in a direction normal to aa, since the slab, on deflecting, tends to lift up the corners, separating them from the supports. Shear in this type of slab is relatively insignificant because it is distributed in two directions along large widths. With increasing spans, the thickness, and consequently the weight of the slab, must be increased, and so the limit of the economy is soon reached. In order to reduce the amount of concrete and therefore the dead load and necessary reinforcement without reducing the effective depth, cross ribs are provided in which the stress conditions are similar to those in a full slab of the same thickness. The most convenient and economical type is the square slab. Analogous conditions occur in rectangular slabs of various ratio of spans in both directions; but when the ratio approaches approximately 2:1, there is but slight difference in bending moments and other stress conditions as compared with one-way slabs.

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We should mention the special case where the slab is supported only along three lengths of its periphery, thus making the state of stress very complicated and difficult to calculate. A special type is the so-called "mushroom slab" (fig. 9:10d), originated in the United States and applied in reinforced concrete for continouus floor slabs supported by columns arranged at considerable distances in both directions. This type is different in two essential aspects as compared with a slab supported all along its periphery: (1) Deformation and failure are possible, under certain circumstances, in one direction only, as it is shown in figure 9:10/. Normally, especially, if the columns are arranged at equal distances in both directions, the elastic deformation of the slab must be identical in both directions (fig. 9:lOe). If for any reason greater curvature and bending deflection is produced in one direction than in the other, this deformation will continue and will reach the plastic stage or the slab will fail without any contribution to its strength by the resistance of the slab in the other direction. (2) The fact that the support all along the periphery is restricted to points is of great importance in respect to shear concentration. In these points of supports the total load of a square is concentrated and distributed along a relatively small perimeter, and excessive shear stress could cause failure. For this reason column capitals, with additional drop panels for longer spans, are provided, which considerably extend the supporting periphery and thus reduce the shear stress. The size and the shape of the heavy capital is not always satisfactory from the aesthetic point of view. Therefore, drop panels of various shapes (circular or polygonal) are preferred, although the forming becomes more complicated and expensive. The system of slab, capital, and column has some distant resemblance to a ribbed vault, but it would be such a squat and far-fetched interpretation that it could not stand comparison on an artistic or mechanical level. Thus the flat slab, being an entirely different structural form, must seek artistic realization along entirely different, and to some extent, opposed directions. Mushroom slabs do not impose bending on their supports, when these are uniformly spaced and loaded; but when they are not, the frame formed by the slab and supports has the disadvantage that it imposes bending on these, and there is also a complex concentration of stresses at the joints of columns and slab. Hence its profitable use must be limited to instances where spans between supports are approximately equal in both directions. It is even better if the end spans are shorter than the central ones, or if the slab is cantilevered round the periphery; also, the live load should be fairly constant over all spans. Arrangement of columns supporting flat slabs can be rectangular, trapezoidal, quicuncial (four square supports with the fifth in the middle),

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Fig. 9:11. Tioga Building, Berkeley, California. Architect, Edwin J. Schruers. Engineers, Hollis Black and J. J. Polivka. Builder, Bay Shore Construction Co. staggered in various forms, or of equilateral triangles, which proved to be very practical. Forming all these shapes is simpler and more economical, and the thickness of the slab is less than that of a conventional flooring supported on joists, but a larger volume of concrete and a greater weight of steel are required than for the typical floor structure. Recently developed flat slabs of uniform thickness throughout poured one upon the other and—after sufficient hardening—lifted vertically along steel columns (so-called "lift-slabs") are much more economical. Also prestressing is being used for longer spans. The lift-slabs were used for multiplestory buildings by various methods, especially in the United States (Tioga Building, Berkeley, California, fig. 9:11). Their design and structural analysis considerably differs from that of typical flat slabs with standard specifications, and special attention should be given to the method of erection.

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chapter 10 trusses

During the first years of the eighteenth century, applied mechanics found a broad field for application of vector analysis in the development of new structural types: latticework or triangulated trusses. This structural type made it possible to use timber—a material that has always been under a certain handicap in joining members in tension—in long-span structures. The roof of the Riding School at Saint Petersburg, spanning 156 feet, is an interesting example of this application (fig. 10:1). Later, when steel with its rolled sections became predominant in the structural field, the development of triangulated trusses became rapid, exhuberant, and audacious, as demonstrated by the Eiffel Tower (fig. 1:2) and the Firth of Forth Bridge (fig. 14:3), among many other examples. Triangulated latticework as a structural element has many applications of interest and importance, such as columns, beams, arches, or frames in which trusses are connected with triangulated posts. Today, it can even be used in grid forms as a substitution for continuous surfaces of laminated structures. A triangulated type of structure is especially suitable for elements subjected to bending. Even in structural elements that are basically or exclusively subjected to compression, such as columns, trussed members provide greater stability against lateral forces and buckling. Consequently, the following discussion of their application to beams should refer also to problems and characteristics of their use for other structural elements, such as portal frames, arches, etc. There is a certain limitation to the depth of full-web girders. For long spans the weight of trusses is much lower and permits ernormous spans of metallic bridges, such as that shown in figure 10:2, a cantilever bridge with

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Fig. 10:1. Riding School in Saint Petersburg, nineteenth century. Engineer, A. Betancourt. a center span of 1,500 feet, whereas the maximum span of a full-web bridge girder is only 600 feet, built with less economy. Basically lattice trusses have a shape similar to that of full-web girders; the only difference is that the web is subdivided into a group of elements, some of them being in compression, and the others in tension, forces that result as components of the shear in the web.

Fig. 10:2. Bridge at Moissac, France.

In full-web girders of greater depths the material is not rationally utilized, since there is a limitation to the thickness of the web because of possible bulging. Trusses do not have this disadvantage because the internal stresses are concentrated along individual members of relatively short length and wide cross section and therefore of greater rigidity. They require less material and are simpler to fabricate. The first applications of triangulated structures were made, of course, in timber, of simplified scheme, the triangulated shape resulting from the pitched roof commonly used for better drainage. The two sloping members or chords must be held together by a tie member to prevent swaying of the supporting walls or sliding of the chords over the tops of the walls (fig. 10:3). In this way the first triangular truss was formed ( a ) . However, since the tie member for longer spans deflected, it had to be supported by a vertical member at the center which evidently is in tension ( b ) . And, finally, in order to reduce again the span of the two sloping members, two inclined braces were installed, which, as supports of a continuous beam, are in compression ( c ) . Figure 10:3d shows the vertical suspender in b split into two inclined hangers, which characterize, with additional supports as in c, the typical Polonceau truss. However, in order to lighten the web in steel and to make it more rigid, the first step was not so simple as it was in timber; the path of progressive improvement from simple to more complicated solutions was not followed. Here progress was more intuitive than mechanically rational, and the continuous web of uniform thickness was first replaced by multiple systems of web bracing (trelliswork, as shown in fig. 10:4a). It was soon realized that in this way the rigidity of this grid type of web was not essentially affected; but another problem arose: how to cope with the complicated structural analysis of a hyperstatic structure. Finally it was concluded that it is preferable to restrict the interior forces to action in the least possible number of truss members, not only for simplicity of design and calculations but also for greater economy as a result of lower weight of material. Some trusses were designed with a number of diagonal pairs supporting the truss at various equidistant points against the ends of the truss (fig. 10:4fo), but this arrangement has two instructive disadvantages. First, each pair of diagonals is stressed only by the loads acting on the vertical member passing through the common point of thé diagonals. It is intended to distribute the loads among the various diagonals, so that each, so to speak, will take its own share. But in this design no advantage is taken of all the members to contribute their strength when the live load is placed on some other part of the structure. The principle of economy requires the utilization of all parts of the structure simultaneously. In general, a single stronger

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Fig. 14:16. Font Pedrouse Viaduct, France. Engineer, P. Séjourné.

chapter 15

static-resisting

functionalism

W e have reviewed the panorama of structures in such a cursory way that only the fundamental classifications of structural elements, their assembly to greater units, and the intrinsic reasons for their existence and evolution could be appreciated. Now we should consider subjects of a different order, such as construction method and aesthetic expression. We shall try to discover what is imprinted on the structure by the static and resisting function of each material. In principle, this is the primary criterion by which the designer must make his selection of structural type and must plan the completed structure. In the course of the preceding chapters, several ideas, observations, and tendencies have appeared which may have seemed unrelated and collected by chance, but are in reality subject to general laws and orders and await the genius capable of organizing and synthesizing them into a general theory. Their complexity is so great and their heterogeneity so diverse that we may well wait a long time before such a genius appears. We can offer only vague and diversified opinions that, although of no apodictic value and with no pretense to unity, may still point out and establish certain fundamental ideas that are not by any means new but are sometimes neglected. This neglect makes reconsideration of them interesting, if only because by emphasizing them we can feel the satisfaction with which the spirit and the technical bent of the designer naturally radiate toward those ideas, even though he rarely stops to reflect over the causes and general laws that guide and mold his native capacity. To begin with, we must remember that basically the structure must find its support in the ground, forcing it to react in such a way as to keep in equilibrium the group of forces and weights which act on it. When hori-

230

zontal forces occur, a weight and a balancing arm are required over the support to put them in equilibrium, as the anchorage of the structure to the ground is, in general, a more difficult and costly step. In regard to resistance, the problem consists in transmitting the acting forces in such a way as to put them in balance with the supporting reactions, by internal forces in various structural elements. This must be accomplished with the utmost economy, without interfering with the functional exigencies but rather assisting them as much as possible. Without regard to constructional, aesthetic, and functional reasons, it seems that in principle the structure must achieve this transmission of forces with a minimum of material, matching the intensities of the stresses; or it must, with the same material, reduce the stresses, thus increasing the safety and sensation of unforced stability of the structure. In attaining these ends, all constructional elements should contribute whenever possible to the strength of the structure, instead of just being added as fill superimposed on the structure, which might be more economical in its own unadorned simplicity than with added elements. Structures can be classified as linear, plane, or three-dimensional bodies relatively comparable to one another. Among the first to which we must pay special attention are the triangulated structures or trusses, in which the fundamental structural element is the straight bar working in a simple tension or compression. Although flexures appear in these structural elements due to the rigidity of the joints, this phenomenon is secondary and should be reduced to a minimum because, even if this increases the bearing capacity of the structure, the gain is relatively small compared with the resistance to axial forces. In a separate group are structures consisting of posts and lintels assembled in simple, multiple, or superimposed porticos, schemes of typical building structures, in which the basic functional aim is to obtain maximum open space between horizontal floors and between vertical exterior walls and interior partitions. In these structures, also called trapezoidal, the transmission of forces required to put them into equilibrium is not accomplished only by axial components; there is also flexure, which makes the structures hyperstatic, due to their multiple rigid connections. Arched and polygonal spans similar to arches are structures of intermediate type in which flexure supersedes compression in proportion to the eccentricity of their directrixes with respect to the funiculars of the weights and other loads (compression lines) acting upon them and combined with resisting forces produced at the supports (reactions). The more these funiculars approach the directrix, the smaller will be the bending and the more justified will be the use of the arch.

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In so-called surface structures, as slabs, shells, or plates, it is necessary to distinguish between the two basic types of stress, flexural and plane stresses. Greatest advantages can be achieved with these structural types when modern monolithic materials are used, such as reinforced concrete. Plane slabs are subjected fundamentally to bending under loads normal to their plane. A slab can have circular, rectangular, triangular, or any other shape, and be supported on the periphery. Usually, a slab is supported along two or more parallel edges, determining the definition of a freely supported or continuous slab. The thickness of such slabs, as of a beam, is determined by the intensity of bending. Beside this, there are membranes. Among these the most simple and comprehensible is the thin plate or shell, for example, a vertical wall, whose height is close to the distance between vertical restraining supports, which carries its own weight and any additional loading. In such membranes, the stresses follow a quite irregular network (stress trajectories) in their plane, quite different of stress distribution in a simple beam. Such shells (if adequately thick and restrained) are very rigid, can withstand heavy loads with low stresses and light reinforcement. Similar stress conditions are encountered in certain curved membranes capable of establishing isostatic equilibrium with forces acting upon the surface of the membranes. These membranes offer the same advantage of rigidity and are very economical. The intermediate case occurs in the shell or lamina, in which the principal stresses acting in planes tangent to the surface of the membrane are to be combined with the flexural forces such as occur in a typical plate in order to establish an endohyperstatic equilibrium. It is evident that equilibrium is achieved more economically (with greater saving of material) in membranes than in laminas, but it is rarely possible to design a structure in such a way as to attain the adequate kind of equilibrium. Furthermore, just as in triangulated structures whose joints are not always entirely free, there are also some deformations in membranes which produce secondary bending. Shells (lamina) are not used as skeletons faced with various filling materials as required by utilitarian -or aesthetic functions; unless these materials, with their continuous surfaces, have the same properties as the bearing skeletons. Nevertheless, there are cases in which it is preferable to substitute for the continuous plate, generally of reinforced concrete, an open triangulated web (figs. 8:8 and 8:16). Such grid structures are still lighter and more economical in steel covered with waterproofing and sufficiently resistant material as a superimposed element. This principle was used for the Butterfly Bridge proposed for the San Francisco Bay in which the designers obtained all the benefits by using steel gridwork with double curvature and united continuous surfaces (fig. 8:8).

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The three-dimensional bodies, which always carry great volume, find their economical utilization when it is desired to obtain stability by weight if the stability cannot be attained by other, more economical means. Gravity dams are typical examples. It is easy to understand that the stressing of one element can always be resolved into various components as tension, compression, bending, or torsion. Only in the case of a tie or of a short piece in compression can all elements that form its mass be subjected to the same type of stressing of equal intensity. If a compression member is very slender, its bearing capacity is limited by dangerous bulging, which is proportional to the square of its slenderness. It is therefore advised to keep the bearing capacity of members in compression within safe limits by reducing their free length or by increasing the moment of inertia of the section. This can be done without increasing the cross-sectional area and consequently the weight by use of hollow, rigid sections or composite ones. Two very different phenomena occur very often simultaneously in bending: (1) the phenomenon of bending alone, which causes maximum longitudinal stresses in the extreme fibers and zero stresses in the neutral axis; and (2) the phenomenon of shear, which causes maximum stresses in the neutral axis and none in the extreme fibers. The shear stresses are usually much lower than the bending stresses. Both types of stresses vary also along the length of the member and vanish in certain points of the directrix (axis). Therefore, it is not possible to utilize the bearing capacity of a rectangular section to its maximum; and even with the advantages offered in this sense by an I-shaped section, the utilization is not complete along the whole length of the member. Furthermore, longitudinal stresses become greater when they must balance with exterior forces having a much larger lever arm than interior forces; and these stresses increase as the structural member becomes more slender. For this reason, there is a tendency to separate the structure's resisting parts in a latticework, if considerable loads are to be carried. Finally, we must not forget that zones in compression are also exposed to bulging of a deflected member. It could then be said, rather generally, that stressing a member in bending is more expensive than stressing it in compression, and the latter is in turn more expensive than stressing in tension. Furthermore, the thinner the member, the more expensive it becomes. This is the reason why the prejudicial influence of bulging is detrimental in materials having high strength, such as steel. In reality, in modern structures of materials which resist all kinds of stresses (tension, compression, or shear), members in tension are not more

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used than members in compression because, in general, the problem is to balance forces and weights, such as gravity forces, which are acting in vertical sense from the highest points toward the ground below; under such conditions, supports in compression or their equivalents become essential. But, if in a truss some diagonal members are stressed in tension, and others in compression, it is preferable to let the shorter diagonal struts take the compression, and the longer diagonal bars take the tension. For the same reasons, compressed steel gives w a y to suspension cables in large spans such as those in suspension bridges. However, this disposition requires two high piers in compression, although these piers or towers are much shorter Fig. 15:1. Walsall Power Station cooling towers, Staffordshire, England. Photograph, Craftsmen, Ltd.

than cables in tension. The solution of the problem is found not only in the lightness resulting from the change of the structural type or material (cable replacing rolled steel) but also in consideration of bulging and other unfavorable factors. In principle, the bearing conditions and the structural type are determined not only by the strength consideration, but also by concomitant deformations. Those of us who, being neither seamen nor fliers, prefer to walk on firm ground, will require not only resistance but also stability and fixity of a structure. But aside from the fact that this condition limits, for example, the slenderness of the floor beams with respect to the maximum acceptable deflection, it must always be considered that the resisting function tends to be absorbed by the more rigid members rather than by those that offer the most resistance, because stresses are always accompanied by corresponding deformations. It can thus be seen that if two crossed beams must support a load, the larger part of the load is absorbed by the more rigid of the beams, because the other beam would have much greater deflection if separated. In the same manner, in a two-span beam, the load, concentrated at the central support, is absorbed by it totally because the contraction of the support under the axial load has practically no displacement. Consequently no bending of the beam will occur as it would when the central support were removed, or made very elastic and deformable. Thin shells or lamina require similar considerations, although their distribution of stresses is more complex. Compressive stresses within the plane of the shell can cause lateral buckling that can be avoided by stiffening ribs. This is a case similar to that of a large plate girder, whose web must be stiffened to avoid the buckling caused by compressive stresses. Shell, on which forces in its plane and bending moments normal to it act simultaneously, will frequently require greater thickness or additional stiffening. A triangulated structure or truss with few lattice members resisting tension or compression is more economical and more rigid than a girder consisting of columns and beams (post and chords) such as the open-web girder (Vierendeel type) in which the bending in its members has an important function. For this reason, these structural types should be used only in cases where functional requirements make latticework unacceptable. Greater rigidity of a structure is generally more acceptable, not only for reasons already discussed but also because their deformation and oscillation under accidental or moving loads (live loads) are reduced, and because their periods of resonance causing prejudicial dynamic effects are lowered. Besides, deformations and vibrations frequently produce important secondary stresses.

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Secondary effects (e.g., that which is produced in a truss due to longitudinal elastic contraction and expansion of individual members of the latticework assembled without special hinged connections) always tend to induce a certain resisting function that partly relieves the structure from primary forces. However, reduction of primary stresses is much smaller than the increase of the stresses introduced by the secondary internal forces. Therefore, it is best to design types of trusses with proportions and arrangement in which these secondary stresses are as small as possible. Triangulations with extremely acute or obtuse angles and with small deformable members connected with other large and less deformable members always cause greater stress concentrations.. Any local concentration of stresses is dangerous and indicates a faulty conception of the design, since elastic work (produced by elastic deformation) is a function of the square of the force times the length along which it acts, and thus an excessive stress in a short zone which represents a very small part of the whole resistant work of the structural assembly can cause the failure of the whole structure, no matter how strong the other members are. An old English proverb, which is as old as chains—and these must indeed be very old—says that a chain is as strong as its weakest link. And, therefore, we must avoid in our design of a structure any points that are frankly weak in comparison with others. It is precisely for this reason that the study of joints is of fundamental importance, because in joints, where the stress state is strongly altered, are found dangerous concentrations of stresses. We should avoid many of the dispositions frequently adopted for economic reasons or for simplification of construction which may allow loosening or moving of certain members in respect to other members under some loads, even if such loads are abnormal. What is most important is not only the rigidity or the deformability of the whole homogeneous assembly, but also the tying together of some rigid elements with other, very deformable elements. Rigidity, however, presents disadvantages with respect to thermal deformations such as retraction, etc. The stresses increase with rigidity, as the thermal changes tend to produce a given deformation. When a structure is unable to deform in a particular direction, owing to the rigidity of the external supports, thermal changes impose very heavy stresses on the structure. A long-span arch between two firm abutments can easily deform under temperature changes, but it is practically impossible to build a straight beam under such conditions, because its great rigidity produces high stresses and thrusts due to thermal expansion or contraction. For this reason it is necessary to provide a larger number of expansion and contraction joints. Reinforced concrete arches without joints can span hundreds of

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yards, whereas a masonry wall, fixed at its ends, demands joints every few yards. Associated with this problem of deformation, joints, and articulations is that of isostasy versus hyperstasy. Much has been said about the advantages and disadvantages of isostasy as compared with hyperstasy in structures. That most present-day structures are really hyperstatic does not mean that structures of past ages were not also hyperstatic. In principle, the advantage of a hyperstatic structure over an isostatic structure is that a loaded member is assisted more in its bearing by neighboring members. Each member can therefore be more slender than if it would be separated. For example, the beams of a bridge, with several independent spans, are isostatic. Each of the beams, as the load passes over them, takes and resists the whole bending effort; but if the beams are continuous, the adjacent and following beams also are subjected to bending, which reduces the maximum stresses on the loaded span. Therefore, since hyperstasy permits the most integral advantage of the necessarily existing elements, it leads to a greater rigidity and to a greater economy of the structure. Yet, a hyperstatic structure, with all these advantages, requires a more thorough investigation of supports and connections than does an isostatic structure. The false movement of a support in an isostatic structure does not appreciably change conditions of equilibrium, but it may cause failure of a hyperstatic structure. However, the latter is more adaptable to abnormal loads and effects for which it has not been designed. If an element in an isostatic structure fails, two results are possible: (1) the failure can pertain only to the affected element, thus not endangering the rest of the structure (beam A in fig. 15:2a); or (2) the failure can spread to other elements (one of the parts, B, of the arch). If the structure is continuous and therefore hyperstatic (fig. 15:2b), the failure A changes the bearing of the continuous beam, which will in other points show larger bending moments than before the rupture. But these increments of bending will in general remain within the safety margin, so that the hyperstatic structure is more effective. Similarly, the failure at B (fig. 15:2b) can cause additional stresses in the rest of the arch, stresses that can produce its complete collapse if the failure at B becomes complete. But this probably will not happen if the failure is incomplete, because the arch acts then as if it were articulated at that point. Fig. 15:2. Examples of failure in viaduct arches.

This gives time to repair the damage, since the structure continues to hold. In other words, collapse is not inevitable as it would be in an isostatic structure, such as the three-hinged arch. The difference is even more noticeable when the structure and the material accept plastic deformations, as will a continuous metallic beam with a full-web section (fig. 15:3fl). Excessive loading can produce a considerable deformation in the sections over supports where bending is greatest. Thus a readjustment of stresses will take place, and an acceptable increment of the bending moment occurs at the center of that span. Collapse will not ensue until the three sections as indicated in the figure have exhausted their resistance. On the contrary, if the beam is freely supported (fig. 15:3b), plastic deformation starts in the center since the rest of the beam is not able to stop the movement. The safety factor must therefore be totally and essentially different in each case. To make the choice between iso- and hyperstatics, it is necessary to analyze the possible dangers, the abnormalities to be accounted for, and the defects (bad or worse) to be expected in each individual case. Whether the structure is isostatic or hyperstatic, it must be well coordinated, that is, organized in such a form that it will be safe in any possible event. Structures in general, unless with a specific aim, are not built to be bombed or to withstand enormous earthquakes. Disregarding such drastic extremes, we must take into account other, more likely effects that the structure must be able to resist, in order to localize and restrict damage and risk as much as possible. A multi-story apartment house, consisting of long parallel walls that carry floor joists without strong anchorage, has little or no lateral stiffness. It lacks bracing and is not monolithic. It lacks the tenacity, let us say, that would make it capable of resisting the forces and

Fig. 15:3. Deformations of continuous beams within the elastic and the plastic range.

deformations to which it can be subjected under abnormal circumstances (e.g., unequal settlement of the walls with the tendency to overturn, wind actions on the façade, etc.). It can readily be imagined how easily such a structure can fall apart and collapse, like a house of cards, as compared with another structure in which the floor beams, joists, and walls are solidly anchored, as, for example, by light beams in reinforced concrete, well tied to the columns, and forming a portico interconnected with the walls. Still better bracing is achieved if a few transverse walls are arranged normal to the exterior walls. Also a stiffening and enclosing girder along the whole perimeter of the building in its top floor, typing together all the vertical elements, is always an important contribution to safety in the face of innumerable dangers that escape consideration in the calculations. Basically, it is essential that, functionally, the transmission of forces, which from the points of application of loads to the points of reaction, makes equilibrium possible, be simple, clear, without twists—we might say, without discomforts. All really satisfactory structures have this property, and so do all the classic types, which are rigorously functional. Structure a (fig. 15:4) will always be better than structure b, in terms of resistance. Structure a will show smaller swaying and thrusts, and the bending in individual members will be less significant than in structure b. The latter is always of great advantage because, as has been observed before, members subjected to axial compression or tension are generally more economical than those designed for pure bending. It can be understood then that structure a resists much more comfortably and will be more gratifying than structure b, which presents with respect to a the same contortion as a hunchback to a normal man. A clean and clear structure, composed of a small number of elements whose respective functions are well defined, shows clearly to the eyes of an experienced designer the stress phenomena of the whole structure and of

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each of the elements within it. This is not to say that an element cannot have two simultaneous resisting functions, as is true of slabs monolithically connected with supporting beams, where the slab on top of the beams is bearing the loads by flexion in one direction between the two beams and simultaneously by compression in the other direction as a flange of the supporting beams. Those structures in which elements accumulate, giving the impression that it was necessary to keep adding them because it was feared that the first ones were not sufficient to the task, always give to a spectator a sensation of uneasiness and a suspicion of defect in their structural conception which should always be prevented (fig. 15:5). A structure with few and strong elements always gives an impression of ease and security, the feeling of a work well done, much more so than does a structure that is composed of a multitude of members and little elements, which mutually interweave and splice to form the whole. Of course, more than anything else, the material dictates the scarcity or abundance of elements; but, within each structural type proper to a material, this principle holds and has the character of generality. In closing this discussion, let us again remember that the material is an essential part of the structure and one of the most apparent determinants of its morphology. Thus, stone or brick is particularly indicated for use in elements in compression (e.g., walls, heavy pilasters, buttresses, arches), all of which permit, or even require, great thickness. These materials lead then to solid structures with continuous supports or to the use of arches, vaults, and domes, with which so many great works have been accomplished in almost all architectural styles. The designer of today, particularly the engineer, has an excessive contempt for brick, which is so characteristic of certain regions and with which so much was achieved by craftsmen of past centuries. Certainly if nature offers gravel and sand or stone in economic quantities, it will prob-

Fig. 15:5. Ahwillgate Bridge, India.

ably be advantageous to substitute for the solid and continuous surfaces of stone or brick others made of concrete, except where the thickness of concrete would not be sufficient and the cost of the double forms would make the unit more expensive. But reinforced concrete and even small hollow bricks, are most suitable materials out of which to make shell, slab, and membrane structures. When spans are not too large, vaults built up of small hollow bricks (and gypsum mortar) are very economical, if no tensile forces are involved, since no formwork is required in their construction (fig. 13:5). If additional light reinforcement is provided, they can also function effectively under tensile stresses. Percentage of steel in concrete should preferably be low. A reinforced concrete structure with a high proportion of reinforcement suggests that, except in rare cases, it could have been more aptly designed in steel. The use of symmetrical reinforcement in concrete sections of low depth to withstand heavy bending, the placing of heavy reinforcement in members taking compression, hoops, and so on, are devices to which one should resort only in particular cases within the general design of concrete structures, when there is ample justification. But they are only exceptional methods, where frequent practice should be consistently avoided. Concrete, which is highly resistant in compression, is weak in tension. For this reason the steel takes the function of tension and makes the concrete suitable to resist bending and tension as does any other homogeneous structural material. Its main advantage is that reinforcing can be placed where tension occurs, and thus it can be embedded in required quantities exactly in accordance with the internal state of stresses and all structural requirements. If the structural element is expected to be mainly in tension, or if flexure alternates the sign due to unfavorable distribution of the live load and the structural element demands tensile reinforcing on both sides, the advantages of reinforced concrete over steel are indeed lessened and, in many cases, totally lost. Concrete is basically a mass material that requires a sturdy section of the individual structural members in order to be economical, as the cost of additional reinforcing is higher than the cost of greater volume of concrete; the cost of scaffold and forms remains practically the same. Its heterogeneity also makes concrete more delicate in small thicknesses. Reinforced concrete is not very suitable for structural types that need more complicated forms, such as triangulated trusses, especially if their mesh is small. Lack of slenderness of concrete elements causes secondary stresses that make greater thickness of individual lattice members necessary. Finally, concrete in tension always has a tendency to crack; and if it is subjected to alternate tensions and compressions, the danger increases.

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Steel, on the contrary, gives to any structure a character of lightness which might be called "tendinous" as compared with the "massiveness" of concrete structures. Even as it is impossible to build in steel a dam 300 feet high, it is not feasible, at least at present, even to think of a reinforced concrete bridge beam with a span of 4,000 feet, although Freyssinet, one of the leading concrete engineers, already, 25 years ago, designed a concrete arch bridge with a span of 3,280 feet, and Polivka proposed to the California Toll Bridge Authority, in 1947, a main span of 3,000 feet for the San Francisco Bay Bridge. Dead weight and live load determine the dimensions of a structure. If for assumed dimensions of a structural member the resulting stresses are higher than allowable limits, the dimensions must be increased; but then the proper weight is increased more and, in consequence, also the internal forces. These conditions continue to be more and more severe in a vicious circle. In a gravity dam, however, the weight is the actively stabilizing element, and therefore concrete of the highest possible specific weight should be used. In plate structures (as in enclosures, roofs, floors, etc.) where, for functional reasons, a continuous surface is required, concrete not only can form the structure supporting the continuous surface, but it can at the same time form the surface itself. Besides, the possibility of working in two directions with the same material as, for example, in two-way slabs with crossing reinforcement, and the possibility of adjusting resistance to all kinds of states of stress simply by varying the thickness and quantity of the material in accordance with the type and intensity of the stress at each point, make- reinforced concrete most suitable in order to attain all required properties with the greatest economy. Concrete tends to produce structural monolithism and to form structures of rigid connections, whereas metallic structures, especially if they are riveted, tend to have simple or less rigid connections of supports as, for example, beams resting on girders, Even riveted connections frequently show plastic deformations—considered by many designers to be advantageous—which, however, greatly reduce the rigidity of the joints, thus creating states intermediate between those characteristic of monolithic and articulated structures. This indeterminacy, whose doubtful advantages are hard to ascertain, can certainly reduce secondary and parasitic forces to. an unknown degree, but all these differences are to a certain extent secondary and should not be overestimated. To connect concrete members, hinges with small rotations are simple and economical devices. Préfabrication is also useful in assembling a concrete structure with free support or with simple bearing joints or anchorage similar to those used in metallic struc-

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tures. In general, welding of the reinforcing bars in a simple way assures rigidity of connections if desired. The essential differences between concrete and steel are the mass and the molding possibilities of the former as compared with the lightness and hightensile and compressive resistance of the latter, as well as the rolled type of its profiles. In basic characteristics, prestressed and precast concrete falls between the two materials, since in it both properties are rationally combined. Concrete is the typical medium of large structures, wherein weight contributes to stability (e.g., dams), whereas steel will remain a material most suitable for light triangulated structures with large spans, where lightness is a basic requirement. The pyramids of Egypt might have been made in concrete. They, like all funerary monuments, call for the ageless stability of great masses, for there is nothing more enduring and lasting in this life than the inevitability of death. Conversely, it was steel that was chosen for the Eiffel Tower in those gay, optimistic years of the nineteenth century when technology was beginning to flourish. For, what better medium than steel to express the youthful exuberance symbolized by that famous tower: erected for a moment of holiday and exhibition with the same swiftness that a stem grows overnight to culminate in the sudden miracle of a flower. When we design very large spans, we shall use high-quality steel for suspension cables and light metals for members of the bridge deck. In any concrete structure, it is better to use few elements, solid and lightly reinforced, whereas in a steel structure we can afford the luxury of designing the web with a number of smaller elements in order to obtain, along with economy, maximum lightness. However, it is not necessary to select only one material for a given structure. It may be convenient to combine several materials, in consideration of the aforementioned ideas, For example, in a bridge (fig. 15:6) consisting of beams supported with trusses under the roadway, the upper part is working in compression, the lower part in tension, and the stresses in the diagonals alternate between the two. The functional purpose, which is the principal determinant of the structure, requires a continuous floor over the upper elements. Since a concrete deck is used as slab combined with beams resting upon steel trusses—that combination being the most suitable type for the floor structure—this arrangement will have simultaneously the advantages that the structure of the roadbed serves as the compression chord of the steel structure and the lower part of the truss has the function of the lower chord. This arrangement, as any other innovation, should be carefully considered. But there is no doubt that this combination of metallic

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Fig. 15:6. Tordera Bridge, Barcelona. Engineers, G. Andreu and E. Torroja. Photograph, M. Garcia Moya.

structure with reinforced concrete elements is a sound and clear solution of a composite structure. It is being used frequently with continual, steady improvements. In a roof structure with a large overhang (fig. 15:7), it will be necessary to use light materials and elements such as corrugated sheathing in asbestos cement, over light purlins and triangulated metallic brackets, whereas the rest of the structure—including the stands, stairways, and lounging and circulation areas—can be built of concrete, thus achieving with the combination of materials an appreciable economy.

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Fig. 15:7. Las Corts Football Ground, Barcelona. Architect, J. Saigner. Engineer, E. Torroja. Photograph, M. Garcia Moya.

Fig. 15:76

chapter 16 construction methods

A building cannot be manufactured in a shop and then set in place like a piece of furniture in a house. But even so, the préfabrication of essential structural elements, later to be assembled and joined together on the construction site, has been developed in recent years with increasing success and economy and has become a construction method of great importance, especially for large buildings or for large numbers of identical small buildings. This method has been used and has become typical of steel and timber structures: their parts are fabricated and partly assembled in shops, numbered, and transported to the construction site for subsequent erection. In stone masonry the pieces are cut to exact sizes, and in brick masonry the préfabrication is reduced to mass production of small bricks manufactured in series. In the past, concrete structures have been poured only in situ in wooden or steel forms. In recent years préfabrication, involving precasting structural elements, has contributed to speedy construction and economy, especially if prestressed concrete is used. The greater strength of both materials of today—concrete and steel—permits considerable reduction of dimensions and therefore of the proper weight, thus simplifying transportation and erection. The elimination of expensive forms and scaffolds contributes to economy. Many industrial buildings are inspiring examples of structures in precast, prestressed concrete. Similarly, deck beams of bridges have been constructed by joining at the working site shorter beams prefabricated in the workshop (fig. 16:2). This sacrifices partly or totally the monolithic properties of the structure for the sake of easier préfabrication. Indeed, préfabrication is no recent notion, for it had been the chosen method to build Solomon's temple, nearly 3,000 years ago.

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Modem techniques foster new problems in this field. The size and the weight of the prefabricated parts will be determined by factors particular to each individual case (e.g., conditions of transportation, availability of mechanical equipment, bearing capacity of cranes, etc.). It is obvious that steel members, because of their relative lightness, can be erected faster and more easily. Préfabrication methods can appreciably lessen the time necessary for construction, since many elements can be manufactured simultaneously. In contrast to this, at the working site often only a few men can work at the same time. For example, a bridge cantilevered from both ends limits the men engaged in its construction to two teams, each working from one springer, and advancing toward the crown. Yet, we should keep in mind that the construction method is of interest to us in this discussion only so far as it influences the selection of the most suitable structural type. In such structures as buildings only the assembly of parts and details related to that assembly are affected by exigencies and modalities of the construction method. However, in large engineering structures, and particularly in those of large span and surface, the construction method can fundamentally influence the choice of the most suitable structural type. Since it is difficult to establish general rules for the selection of any particular construction method, and since, therefore, the method selected will depend on the knowledge and experience of the designer, our discussion will be limited to specific cases. In this chapter, we cannot start with a discussion of developments from prehistoric times: there is but little information about the accomplishments of antiquity. We do not know very much about how the huge monoliths of Easter Island were moved into place, and how the single stone slab for the Ravenna Dome—40 feet in diameter, 3]/2 f e e t thick, 330 tons in weight— was transported across the Adriatic Sea, or how Gothic artisans were able to design keystones between the ribs of their domes, or how Solomon's temple was erected without the use of a single hammer, chisel, or any iron tool—a marvellous example of well-organized préfabrication. Even if we can conceive the enormous numbers of slaves who labored for Mesopotamian kings, it still is difficult to understand how it was possible to erect gigantic structures with the meager technical equipment shown in extant bas reliefs of the period, for example, the Assyrian bas relief, which we covetously admire in the British Museum. One of the broad fields in which constructional method can influence most decisively the choice of structural type is that of large—and medium— span bridges. With steel, the problem is generally easier than with masonry,

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for if typical erection methods and designs are adopted, the project will not be much influenced by constructional methods. It is assumed that the contractor has all necessary equipment for erecting the structural parts. Contemporary methods and facilities of transportation, lifting, and assembly make possible determination at the design stage of the most convenient size of structural members in order to achieve the greatest economy. Simultaneously, the time schedule for each operation must fit in with that of the others so that all can proceed in accordance with the construction program. When a span is relatively long, however, consideration of the construction methods in relation to the required auxiliary means becomes fundamental. Concrete bridges have been built whose members are of such light weight that the complete structure was lifted with cranes of normal capacity, thus reducing the labor on the site—which is always more expensive than shop labor—to a minimum. The economy of reinforced concrete—as compared to conventional steel—may be well overcompensated by savings in the cost of erection. This saving will increase with the number of identical parts, since a sufficient number of these will permit mass production in the factory and use of the asssembly line on the construction site. It is now customary, in order to avoid the high cost of centering, to fabricate girders or trusses on the approach to the bridge, in the same axis, and to push them longitudinally on rollers to their final position. (On railroad bridges the girders are ordinarily shifted transversely, not so much for economy as to get the job done quickly without interrupting traffic.) But when girders are placed lengthwise on the bridge, the method of erection must be considered at the design stage, as well as how the structure is going to be stressed in each intermediate position, that is, when the beam placed as cantilever just reaches the next pier and is subject to great bending. These are, of course, standard methods, well known to the designer. However, some cases offer special conditions, as does the bridge illustrated in figure 16:1. Here the most suitable structure was considered to be a tied arch, whose deck was carried by vertical hangers, without diagonal bracing. The reason for this design is not pertinent. The rigidity necessary to resist bending, caused by uneven distribution of live load, can be provided either by the arch or the suspended deck structure which acts as tie member. Here, to avoid expensive centering during construction, welded I-girders of the deck structure were aligned on the river bank, welded together, and joined by a few cross ties to make the whole assembly more rigid. Later, these assembled girders were pushed out over the river, sliding over the intermediate piers. Once this continuous structure had been placed in position, it was strengthened and braced with additional cross beams to become

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Fig. 16:1. Stages in the erection of an arch bridge. sufficiently strong for the erection of the arches, consisting of welded plates (and hence, relatively light). The proper weight of the girders and arches caused certain deflection in continuous girders that had to be corrected. For this purpose the arches had been left with temporary flexible joints, and the hangers supporting the girders had been fitted with turnbuckles, to adjust their length. By tightening these, the deck girders were brought to their undeflected position, hanging from the arches. Then the arches and beams were welded, and the continuous girders were cut into three separate spans, so that each span should work independently. W e mention this particular case, because it shows how economy can be effected by awareness at the very beginning of a project, and throughout its design, of the methods by which the structure is going to be built, and how best the design will aid erection. In this case, if the arch had been made rigid, instead of the tie-girder, none of these advantages would have been gained. To facilitate the actual construction process, the correct choice of struc-

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tural type and attention to details are of fundamental importance to the engineer. Each day it becomes easier to handle large structural parts, and although this is often difficult and dangerous, it has the advantage of avoiding scaffolding and formwork. The modern trend in reinforced concrete is moving increasingly toward the manufacture of large prestressed parts, to be erected later by assembling these parts in situ, and post-tensioning them together to ensure a rigid assembly. This technique promises to develop greatly in the near future. Freyssinet, for his bridge at Luzancy, France, used ingenious and original methods to position a 177-foot prestressed concrete beam (fig. 16:2). This beam is made up of prefabricated parts held together in compression by post-tensioned cables inserted in longitudinal holes. In spite of the great weight of the beam, its positioning was accomplished with ease by means of the rig shown in figure 16:2 and by use of a special reel for the cable whose radius, varying with each turn, permitted a calculated progress so that as the beam advanced toward its ultimate position, its two points of

Fig. 16:2. Luzancy Bridge, France. Engineer, E. Freyssinet. Photograph, H. Baranger.

Fig. 16:3. Erection of the steel centering of the Martin Gil Viaduct over the Esla River, Zamora, Spain. suspension were held constantly in a horizontal plane. In this way not only was it possible to use mechanical equipment of low standard capacity but to achieve the erection of the bridge with an over-all economy. Steel arches are frequently erected without centering as cantilevers, but they can also be built by suspending sections from a cable and adjusting their final position. This method was used for the railroad bridge over the Esla River at Zamora, Spain (the longest concrete arch for a railroad crossing in the world), where the light steel arches as rigid framework acted simultaneously as stiff reinforcement of concrete box sections and as centering (this very economical system was originated by Joseph Melan and was used for the first time in his bridge in the Golden Gate Park in San Francisco, built in 1889). Each section of the reinforcing latticework was strong enough to be cantilevered from one anchoring hanger to the next, starting from the abutment (fig. 16:3). In all, the arch had nine joints that could be adjusted so as to bring the arch into its correct position. When the adjustments were made, the steel sections were welded together to form a three-hinged arch, which served to support the concrete sections of the final arch. In the past, long-span masonry arches have been built over wooden centerings, and this method is still frequently used today. This construction method becomes more complicated for very long spans of today's bridges and by the special properties of the concrete we use. Account must be taken of the fact that as the concrete of the arch is poured, the centering

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is bound to undergo deformations under the increasing weight of the concrete. If the concrete for the entire arch could be poured in a few minutes, and if it could all set simultaneously at the moment of maximum deformation of the centering, the only problem with which to contend would be shrinkage of concrete. This would be true also if the concrete could be applied in sections or blocks, cemented together after they were in place, but such an expedient cannot be used because the enormous weight resting upon the centering would make it very expensive. To avoid this, the concrete arch of the Esla bridge was built in longitudinal strips, or rings, so that when one strip had been concreted and hardened, it added its own strength to the supporting steel skeleton used simultaneously as centering, and was able to take the load of the next strip, until this one also had hardened. In this case, the steel centering and each successive concrete strip are subject to varying forces according to the order and speed in which the different strips are concreted and according to the deformability and strength of the material. The distribution of stresses depends basically on the concreting procedure and on the flexibility of the steel centering. The volume of concrete most recently poured is still increasing the load, whereas the concrete poured earlier may already be contributing substantially to the strength of the arch. It is obvious that such considerations for particular construction methods must be taken up not at the time that exact stress analysis is made, but even before that, while the design is in its initial stage, when the dimensions of sections are being determined. Furthermore, stresses caused by shrinkage and plastic flow must also be considered at the same early design stage. Changes in internal forces during construction become especially important wherever timber, which deforms easier than steel, is used for centering. Centerings were used with good success for the Plougastel and Traneberg bridges. Because it is difficult—considering all these effects—to give to timber centering the exact final form of the concrete arch, designers have turned to the use of steel centerings (Melan system). Improvement of the Melan system consists of the following procedure. The load of the first section of the concrete arch is supported by the centering only; the second longitudinal section is a combined system consisting of the steel structure strengthened in the lower part of the arch with reinforced concrete; the load of the succeeding sections is carried in the same way, in accordance with the principle of composite structures.

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A typical example of this use of integral centering (and one which is perhaps unique so far) is the 690-foot arch over the Esla River at Zamora, Spain (fig. 16:4). After construction of the abutments and erection of the steel arch, the upper chord of the steel centering ( o ) was embedded in concrete ( 1 ) . To prevent serious deformations in the arch due to thermal changes, which occur in different intensities in the upper chord already concreted and in the lower chord of the steel arch not yet encased in concrete, the centering was built as a three-hinged arch—one hinge at the crown, two in the abutments. The steel structure alone ( o ) could not carry any load greater than its proper weight including the weight of the concreted upper chord. Once this concrete had set, the arch was strong enough to support the additional weight of the concrete poured around the lower chord ( 2 ) . This concreted lower chord had initially no strength, and remained hanging, as it were, from the top chord ( 1 ) . After setting, the lower chord was compressed to a predetermined intensity by means of hydraulic jacks installed at the crown in the axis of the lower chord. This relieved the upper chord almost entirely from stress. From this moment on, all four chords of the steel centering underwent equal deformation due to temperature variations, and the hinges were fastened in such a way that the structure acted as a fixed arch. Concreting of the other parts of box sections proceeded in succession, as shown in figure 16:4b. The function of the diagonals is simply to hold the chords in place and to resist shear resulting from different weights of individual block sections (voussoirs) poured along the directrix-in such a succession that the resulting compression line would not deviate noticeably from the predetermined axis. This method is slow but it permits use of quite light centering simultaneously utilized as additional concrete reinforcement. The weight of the Esla bridge was only 335 pounds per linear foot of the arch. In addition to such factors as elastic deformation, long-range deformations were considered. These varied with each section and were of considerable importance since the concrete was subjected to large loads during its hardening process. Forces caused by partial shrinkage were relatively small, because the elements of which each section consisted were allowed to shrink freely before pouring of the adjoining section. Joints between these curved sections were not filled until that time. Construction of the arch was completed when hydraulic jacks were inserted in a 3% 6 -inch opening at the crown to produce 7,700 tons of pressure, which was unequally distributed between upper and lower chords of the box section in order to obtain as uniform a stress distribution as possible at the final stage.

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Fig. 16:4a

Fig. 16:4 b

fel

3

Fig. 16:4. Steel arch used as centering and stiff reinforcement of the Martin Gili Viaduct. Engineers, C. Villalba, A. Salazar, and E. Torroja. Photograph, Heptener.

iC^EMXilS

Fig. 16:4c

Obviously, in this type of structure, the construction method had a tremendous influence on the design. The dimensions of the arch and its reinforcing would have been very different and the construction much more costly had a typical full centering been used. In large industrial buildings with reinforced concrete roofs, however, the problem is different. If the forms with scaffold can be reused without being dismantled, much greater economy in over-all construction cost can be achieved; but whether this is possible will depend entirely on the structural type, and the possibility of detaching the forms and moving them to an adjoining position. The classic example of this type is the hangar in Orly, France (figs. 16:5 and 12:3). Here arched sections and forms were designed so that they might be built without waste of material and labor, and with easy detachment and movement of the centering. The shoring was lowered under the concrete shell and, being mounted on rollers, it was moved to a new position, where it was again raised to the required height.

Fig. 16:5. Hangar for dirigibles, Orly, France. Engineer, E. Freyssinet. Fig. 16:5a

A similar construction method is being widely used in industrial buildings with barrel shell roofs with interior stiffening ribs. Still greater simplicity and economy can be achieved with exterior stiffening ribs leaving a continuous inner surface of the shell (fig. 16:6). In this way the gravity axis of the inverted T-section is slightly above the funicular of the external forces so that the compression is concentrated in the lower part of the stiffening rib and consequently a smaller amount of reinforcing is required. The details of these procedures must be kept constantly in mind during the design. In the example described, the abutments had to have a small angle of inclination in order to make possible the moving of the undisturbed scaffold. In other instances some modifications will be necessary in order to detach the forms from the concrete, for example, by turning the haunches down. In roof structures with tie rods, interchangeable scaffold should be considered. In the illustration shown in figure 16:6 the top part of the scaffold rests upon two sets of wedges which in turn rest on the lower part of the scaffold. When the scaffold reaches a tie member in its movement along the axis of the roof, the front set of wedges is removed and the tie member is allowed to pass between the top and bottom parts of the scaffold. These wedges are then put back in original position, and the rear wedges are removed. The centering is then moved past the tie member to its new position. A simpler and more economical solution is possible when the inner surface of the roof is inclined or has double curvature so that it becomes selfsustaining and the centering can be moved without any obstacle. This is true of the hyperbolic shed shown in figure 12:5. The double curvature of the shell makes it sufficiently rigid after completion of each section to hold up without bending, while the centering is detached and moved to the

next section to be poured. The open space between the adjoining arched elements was used for inclined skylights of triangular gridwork formed by steel diagonals which, interconnected with the hyperbolic concrete sections, imported to the whole sufficient strength and rigidity to resist the accidental loads due to wind pressure, etc. This is another case, similar to the Orly hangar, already discussed, where the construction method clearly and definitely influenced the exterior form of the structure. It is important to determine whether or not a structure can support itself during construction, since this will determine whether or not centering must be used. A dome, for example, can be built by successively supporting stable rings one upon the other, provided that the temporary inner circular edges will not sag. The central dome skylight (Algeciras Market Hall, fig. 8:15) was built without centering, the triangles or rhomboids being placed in successive rings. The fact that a dome can be constructed in this simple way makes of it a valuable structural type, especially for roofing large spans. It very often happens that the structure, during early stages of construction, is exposed to forces and to changes in the loading which put some elements in worse condition than when the structure is completed. Here are several examples: a beam whose function is the transmission of vertical loads to columns will exert inclined forces during erection if the centering is cantilevered; an arch built on two brackets projecting from the abutments to the crown will cause in these brackets during erection serious vertical bending to be counteracted later by hydraulic jacks inserted at the crown and by artificial compression. A series of arches on piers exert thrusts that are in good equilibrium when all spans are completed; however, the piers are subjected to eccentric forces during the construction of individual arches. A thousand other forms can be also mentioned which are influenced by specific construction methods. The following cases are mentioned as examples. In one particular structure the concrete cannot be poured at once in continuity, and requires some joints that are always weak in tension. It will be necessary to provide for the joints in the design and to change the scheme of the structure so as to allow the concrete work between joints to be done as fast as can be anticipated on the job. Because of the advantage of reusing the same forms, it is often more convenient and economical to use the same dimensions of structural members throughout, even if more concrete is required. Designers of welded structures are quite aware that, in order to obtain a good position for welding, the shapes of their built-up sections must sometimes be changed, the sequence of welding operations must be carefully determined, and the use of some structural types must be foregone, al-

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Fig. 16:7a

Fig. 16:7 b