Philosophy of Structures

By Eduardo Torroja

This title is part of UC Press's Voices Revived program, which commemorates University of California Press’s mission to seek out and cultivate the brightest minds and give them voice, reach, and impact. Drawing on a backlist dating to 1893, Voices Revived makes high-quality, peer-reviewed scholarship accessible once again using print-on-demand technology. This title was originally published in 1958.
This title is part of UC Press's Voices Revived program, which commemorates University of California Press’s mission to seek out and cultivate the brightest minds and give them voice, reach, and impact. Drawing on a backlist dating to 1893, Voices Revived

• Language: English
• Publisher: University of California Press
• Release date: Dec 22, 2023
• ISBN: 9780520328457

Eduardo Torroja

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philosophy off structures

by Eduardo Torroja

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 MILOS POLIVKA

contents

contents

general approach to the problem

phenomena of stressing

classical materials

timber and steel

reinforced and prestressed concrete

supports, walls, and foundations

the arch

the vault and the dome

the beam and the slab

trusses

retaining structures

the roof and other external covering

the floor and the building

bridges and aqueducts

static-resisting functionalism

construction methods

the beauty of structures

line and surface

genesis of the structural scheme

the calculation

the designer and the organization

list of illustrations

index of names

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

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.

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.

We 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.

We 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 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

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 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.

chapter 2

phenomena of 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 are 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 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 considera-

Fig. 2:1. Small prehistoric figure from Crete, now in Candia Museum, Crete (from Summa Artis, by M. B. Cossio and J. Pijoán; 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

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 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. 2:3c), 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).

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.

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. 2:4b) 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

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 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

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 reëntrant 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