# THEORY OF STRUCTURES TIMOSHENKO PDF

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Theory of Structures by Timoshenko & D. H. Young Free Download PDF This 2nd edition of "Theory of Structures", like the 1st, is intended primarily as a textbook for under-graduate & first-graduate courses in structural. Download Theory of Structures (2nd Edition) - Timoshenko & D. H. Young. [Timoshenko,_Stephen_P.]_History_of_Strength_of_Ma(raudone.info).pdf . In the same way, in dealing with the development of theory of structures, the portions.

He was elected dean of the Division of Structural Engineering in In he was awarded the D. Zhuravski prize of the St. Petersburg Ways of Communication Institute that helped him survive after losing his job.

He went to St Petersburg where he worked as a lecturer and then a Professor in the Electrotechnical Institute and the St Petersburg Institute of the Railways — During that time he developed the theory of elasticity and the theory of beam deflection, and continued to study buckling. In he returned to Kiev and assisted Vladimir Vernadsky in establishing the Ukrainian Academy of Sciences — the oldest academy among the Soviet republics other than Russia.

In — Timoshenko headed the newly established Institute of Mechanics of the Ukrainian Academy of Sciences , which today carries his name. During the Renaissance there was a revival of interest in science, and art leaders appeared in the field of architecture and engineering.

Leonard0 da Vinci was a most outstanding man of that period. Leonardo da Vinci was greatly interested in mechanirs and in one of his notes he states: He applies the notion of the principle of virtual displacements to analyze various systems of pulleys and levers such as are used in hoisting devices.

It seems that FIG. The famous Pont du Gard.

Leonardo da Vinci had a correct idea of the thrust produced by an arch. I n one of his manuscripts there is a sketch Fig. What forces are needed at a and b t o have equilibrium? From the dotted-line par- allelogram, in the sketch, it can be concluded that Leonardo d a Vinci had the correct answer in this case. Leonardo d a Vinci studied the strength of structural materials exper- imentally.

From the latter book Fig. The erection of the Vatican obelisk. Introducl ion 5 it, then attach a basket or any similar container to the wire and feed into the besket some fine sand through a small hole placed a t the end of a hopper. A spring is fixed so that it will close the hole as soon as the wire breaks. The basket is not upset while falling, since it falls through a very short distance.

The weight of sand and the location of the fracture of the wire are to be recorded. The test is repeated several times to check the results. Then a wire of one-half the previous length is tested and the additional weight it carries is recorded; then a wire of one-fourth length is tested and so forth, noting each time the ultimate strength and the location of the fracture. Leonardo da Vinci.

His conclusion was that the strength of beams supported at both ends varies inversely as the length and directly as the width. He also made some investigation of beams having one end fixed and the other free and states: Apparently Leonardo d a Vinci made some investigations of the strength of columns.

He states that this varies inversely as their lengths, but directly as some ratio of their cross sections.

These briefly discussed accomplish- ments of d a Vinci represent perhaps the first attempt to apply statics in finding the forces acting in members of structures and also the first experi- ments for determining the strength FIG. Tensile test of wire by Leonardo da Vinci. The first attempts t o find the safe dimensions of structural elements analytically were made in the seventeenth century.

It represents the beginning of the science of strength of materials. Preface This book was written on the basis of lectures on the history of strength of materials which I have given during the last twenty-five years to students in engineering mechanics who already had knowledge of strength of materials and theory of structures. During the preparation of the book for publication, considerable material was added to the initial contents of the lectures, but the general character of the course remained unchanged.

I n writing the book, I had in mind principally those stu- dents who, after a required course in strength of materials, would like to go deeper into the subject and learn something about the history of the development of strength of materials.

## Theory of Elastic Stability

Having this in mind, I did not try to prepare a repertorium of elasticity and give a complete bibliography of the subject. Klein and C. I n doing this, I considered it desirable to include in the history brief biographies of the most prominent workers in this subject and also to discuss the relation of the progress in strength of material to the state of engineering education and to the industrial development in various countries.

There is no doubt, for example, that the develop- ment of railroad transportation and the introduction of steel as structural material brought many new problems, dealing with strength of structures, and had a great influence on the development of strength of materials. The development of combustion engines and light airplane structures has had a similar effect in recent times. Progress in strength of materials cannot be satisfactorily discussed without considering the development of the adjacent sciences such as theory of elasticity and theory of structures.

There exists a close interrelation in the development of those sciences, and it was necessary to include some of their history in the book. In the same way, in dealing with the development of theory of structures, the portions having only technical interest were not included in this book. In this writing, I have tried to follow the chronological form of presenta- tion and divided the history of the subject into several periods.

For each of those periods I have discussed the progress made in strength of mate- rials and in adjacent sciences. This order was not always strictly followed, and in some discussions of the works of a particular author I have found it more expedient to put in the same place the review of all his publications, although some of them did not belong to the period under discussion.

In the preparation of the book, the existing publications on the history of sciences were very helpful. Among the biographies I found the following very useful: For the review of newer publications it was necessary to go through many periodicals in various tongues.

This took a considerable amount of time, but the writer will feel completely rewarded if his work will save some labor for other workers in history of strength of materials. I am thankful to my colleagues at Stanford University-to Prof.

Alfred S. Niles for his comments on the portions of the manuscript dealing with the early history of trusses and the Maxwell-Mohr method of analyzing statically indeterminate trusses; and to Prof. Donovan H. Young who gave much constructive advice a t the time of preparation of the manu- script. I am also very grateful to Dr.

Bishop for his reading of the entire manuscript and his numerous important comments, and t o our graduate student, James Gere, who checked the proofs.

## Engineering Mechanics

Stephen P. Timoshenko Stanford, Calif. V Introduction. Organization of the national academies of science. Robert Hooke. The mathematicians Bernoulli.

Engineering applications of strength of materials. Experimental study of the mechanical properties of structural materials in the eighteenth century. Theory of retaining walls in the eighteenth century. Theory of arches in the eighteenth century. The experimental work of French engineers between and The theories of arches and suspension bridges between and Thomas Young. Strength of mat,erials in England between and 98 Other notable European contributions to strength of materials.

Equations of equilibrium in the theory of elasticity.

Lam6 and B. The theory of plates.

## History of Strength of Materials - Timoshenko

Fairbairn and Hodgkinson. The growth of German engineering schools. Continuous beams. Tubular bridges. Early investigations on fatigue of metals. The work of Wohler. Moving loads. The early stages in the theory of trusses. Macquorn Rankine. Problems of elastic stability. Column formulas. Theory of retaining walls and arches between and Early work in elasticity at Cambridge University. Barre de Saint-Venant.

The semi-inverse method. The later work of Saint-Venant. Duhamel and Phillips. Franz Neumann. Lord Kelvin. James Clerk Maxwell. Mechanical Testing Laboratories. The work of 0 Mohr. Elastic stability problems. August Foppl. Statically determinate trusses.

Deflection of trusses. Statically indeterminate trusses. Arches and retaining walls. Lord Rayleigh. Theory of elasticity in England between and Theory of elasticity in Germany between and 71a. Solutions of two-dimensional problems between and Properties of materials within the elastic limit.

Fracture of brittle materials. Testing of ductile materials. Strength theories. Creep of metals a t elevated temperatures. Fatigue of metals. Experimental stress analysis. Felix Klein. Ludwig Prandtl. Approximate methods of solving elasticity problems.

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Three-dimensional problems of elasticity. Two-dimensional problems of elasticity Bending of plates and shells Elastic stability. Vibrations and impact. New methods of solving statically indeterminate systems Arches and suspension bridges. Stresses in railway tracks Theory of ship structures Name Index. Galileo Galileo was born in Pisa2 and was a descendant of a noble Florentine house. Galileo received his preliminary education in Latin, Greek, and logic in the monastery of Vallom- brosa, near Florence.

In , he was placed in Pisa University, where he was to study medicine. But very soon the lectures on mathematics began to attract his attention, and he threw all his energy into study- ing the work of Euclid and Archi- medes.

In , Galileo had to withdraw from the University, due to lack of means, without taking the degree and he returned to his home in Florence. There, Galileo gave private lessons FIG. Gall eo. Girard, Paris, Tabor, New York, Cardan discusses mechanics in some of his mathematical publications.

This work made him known, and in the middle of he was given the math- ematical professorship a t Pisa when he was twenty-five and a half years old.

During his time in Pisa , Galileo continued his work in mathematics and mechanics and made his famous experiments on falling bodies. The principal conclusions of this work were 1 all bodies fall from the same height in equal times; 2 in falling, the final velocities are proportional to the times; 3 the spaces fallen through are proportional to the squares of the times. These conclusions were in complete disagreement with those of Aristotelian mechanics, but Galileo did not hesitate t o use them in his disputes with the represent- atives of the Aristotelian school.

This produced feelings of animosity against the young Galileo, and finally he had to leave Pisa and return t o Florence. At this difficult time some friends helped him to get the profes- sorship in the University of Padua. Now Domino Galileo Galilei has been found, who lectured a t Pisa with very great honour and success, and who may be styled the first in his profes- sion, and who, being ready t o come at once to our said university, and there to give the said lectures, it is proper to accept him.

His lectures became so well known that students from other European countries came to Padua. A room capable of containing 2, students had t o be used eventually for these lectures.

I n this treatise various prob- lems of statics were treated by using the principle of virtual displace- ments.

The treatise attained a wide circulation in the form of manu- script copies. At about the same time, in connection with some problems in shipbuilding, Galileo became interested also in the strength of mate- rials. It is known that during his first years in Padua, Galileo taught the Ptolemaic system, as was the custom in those days. But in a letter to Kepler, as early as 1 See the book by J. Fahie, p. The Strength of Materials in the Seventeenth Century 9 , he states: With this instrument, he made FIG.

This last discovery had a great effect on the further development of astronomy, for the visible motion of this system became a very power- ful argument in favor of the Copernican theory.

All these discoveries made Galileo famous. In his new position Galileo had no duties other than to continue his scientific work and he put all his energy into astronomy. He discovered the peculiar shape of Saturn, observed the phases of Venus, and described the spots upon the sun.

In , the great work of Copernicus was con- demned by the Church, and, during the seven succeeding years, Galileo stopped publishing his controversial work in astronomy. Since the book definitely favored the Copernican theory, its sale was pro- hibited by the Church and Galileo was called to Rome by the Inquisition. There he was condemned and had to read his recantation. After his return to Florence, he had to live in his villa at Arcerti in strict seclusion, which he did during the remaining eight years of his life.

The book was printed by the Elzevirs at Leiden in Fig. A portion of the book, dealing with the mechanical properties of structural materials and with the strength of beams, constitutes the first publication in the field of strength of materials, and from that date the history of mechanics of elastic bodies begins. He states that if we make structures geometrically similar, then, with increase of the dimensions, they become weaker and weaker.

I n illustration he states: Galilee's of a bar, Galileo investigates the resistance to fracture lllustration Of ten- sile test. He states: This resistance opposes the separation of the part BD, lying outside the wall, from that portion lying inside.

From the preced- ing, it follows that the magnitude of the force applied at C bears to the magnitude of the resistance, found in the thickness of the prism, i.

The resisting couple corresponding to this stress distribution is equal only to one-third of the moment assumed by Galileo. On the basis of his theory Galileo draws several important conclusions. Considering a rectangular beam, he puts the question: Keeping the length of a circular cylinder constant and varying its radius, Galileo finds that the resisting moment increases as the cube of the radius.

Considering geometrically similar cantilever beams under the action of their weights Galileo concludes that, while the bending moment a t the built-in end increases as the fourth power of the length, the resisting moment is proportional to the cube of the linear dimensions.

This indi- cates that the geometrically similar beams are not equally strong. The beams become weaker with increase in dimensions and finally, when they are large, they may fail under the action of their weight only. He also observes that, to keep the strength constant, the cross-sectional dimen- sions must be increased at a greater rate than the length.

With these considerations in mind, Galileo makes the following impor- tant general remark: If the size of a body be diminished, the strength of that body is not diminished in the same proportion; indeed the FIG. Thus a small dog could prob- ably carry on his back two or three dogs of his own size; but I believe that a horse could not carry even one of his own size.

He observes that the possibility exists of economizing material by reducing the size of the cross section near the supports. Galileo gives a complete derivation of the form of a cantilever beam of equal strength, the cross section of which is rectangular. He shows also that if we remove half of the material and take the beam in the form of the wedge ABC, the strength at any cross section EF will be insufficient since, while the ratio of the bending moment at EF to that at A B is in the ratio E C: A C , the resisting moments, proportional to the square of the depth, will be in the ratio EC 2: AC 2at these cross sections.

To have the resisting moment varying in the same proportion as the bending moment, we must take the parabolic curve BFC Fig. The Strength of Materials in the Seventeenth Century 15 thousand operations for the purpose of greatly increasing strength without adding to weight; examples of these are seen in the bones of birds and in many kinds of reeds which are light and highly resistant both to bending and breaking.

For if a stem of straw which carries a head of wheat heavier than the entire stalk were made up of the same amount of mate- rial in solid form, it would offer less resistance to bending and breaking. This is an experience which has been verified and confirmed in practice where it is found that a hollow lance or a tube of wood or metal is much stronger than would be a solid one of the same length and weight. Comparing a hollow cylinder with a solid one of the same cross-sectional area, Galileo observes that their absolute strengths are the same and, since the resisting moments are equal to their absolute strength multiplied by the outer radius, the bending strength of the tube exceeds that of the solid cylinder in the same proportion aa that by which the diameter of the tube exceeds the diameter of the cylinder.

Organization of the National Academies of Science During the seventeenth century there was a rapid development in mathematics, astronomy, and in the natural sciences. Many learned men became interested in the sciences and experimental work in particular received much attention.

Many of the universities were controlled by the Church, and since this was not favorable for scientific progress, learned societies were organized in several European countries. The purpose of these was to bring men with scientific interests together and to facilitate experimental work.

This movement started in Italy where, in , the Accademia Secretorum Naturae was organized in Naples. The famous Accademia dei Lincei was founded in Rome in , and Galileo was one of its members. In the volume of the academy publications, considerable space is devoted to such prob- lems as those of the thermometer, barometer, and pendulum and also to various experiments relating to vacua. I In England at about the same time, scientific interest drew a group of men together, and they met whenever suitable opportunities presented themselves.

The mathematician Wallis describes these informal meet- ings as follows: Our business was preclud- ing matters of theology and state affairs to discourse and consider of Philosophical Enquiries, and such as related thereunto; as Physick, Anatomy, Geometry, Astronomy, Navigation, Staticks, Magnetics, Chymicks, Mechanicks, and Natural Experiments; with the state of these studies, as then cultivated a t home and abroad. Those meetings in London continued. In the list of those invited to become members of the Society we find the names of Robert Boyle, physicist and chemist; Christopher Wren, architect and mathematician, and John Wallis, mathematician.

The French Academy of Sciences also had its origin in the informal meetings of scientists. Father Mersenne established, and fostered until his death, a series of conferences which were attended by such men as Gassendi, Descartes, and Pascal.

Later on these private gatherings of scientists were continued in the house of Habert de Mont- mor. The Strength of Materials in the Seventeenth Century 17 Cassini, the Danish physicist Romer who measured the velocity of light , and the French physicist Mariotte appear in the first membership list of the Academy.

All these academies published their transactions and these had a great influence on the development of science in the eighteenth and nineteenth centuries. Robert Hooke 2 Robert Hooke was born in , the son of a parish minister who lived on the Isle of Wight. As a child he was weak and infirm, but very early he showed great interest in making mechanical toys and in drawing.

When he was thirteen years old, he entered Westminster School and lived in the house of Dr. Busby, master. There he learned Latin, Greek, and some Hebrew and became acquainted with the elements of Euclid and with other mathematical topics.

I n , Hooke was sent t o Christ Church, Oxford, where he was a chorister, and this gave him an opportu- nity to continue his study so that, in , he took the degree of Master of Arts. I n Oxford he came into contact with several scientists and, being a skilled mechanic, he helped them in their research work.

About he worked with Boyle and perfected an air pump. He writes: I made some trials for this end, which I found to succeed to my wish. The success of these made me farther think of improving it for finding the Longitude and the Method I had made for myself for Mechanick Inventions, quickly led me to the use of Springs, instead of Gravity, for the making a body vibrate in any Posture.

I n , on the recommendation of Robert Boyle, Hooke became curator of the experiments of the Royal Society and his knowledge of mechanics and inventive ability were put t o good use by the Society. He was always ready t o devise apparatus to demonstrate his own ideas or to illustrate and clarify any point arising in the discussions of the Fellows.

See also the paper by E. Andrade, PTOC.

London ,vol. The quotations given in this article are taken from these sources. In , Hooke became professor of geometry in Gresham College, but he continued to present his experiments, inventions and descriptions of new instruments to the Royal Society and to read his Cutlerian Lectures.

That all the heavenly bodies have not only a gravitation of their parts to their own proper centre, but that they also mutually attract each other within their spheres of action. That all bodies having a simple motion, will continue to move in a straight line, unless continually deflected from it by some extraneous force, causing them to describe a circle, an ellipse, or some other curve.

That this attraction is so much the greater as the bodies are nearer. As to the proportion in which those forces diminish by an increase of distance, I own [says he] I have not discovered it although I have made some experiments t o this purpose.

I leave this to others, who have time and knowledge sufficient for the task. After the Great Fire of London in September, , Hooke made a model incorporating his proposals for the rebuilding and the magistrates of the City made him a surveyor. He was very active in this reconstruc- tion work and designed several buildings. This is the first published paper in which the elastic properties of materials are discussed.

Regarding the experiments, he says: Then compare the several stretch- ings of the said string, and you will find that they will always bear the same proportions one to the other that the weights do that made them. From all these experiments Hooke draws the following conclusion: Nor is it observable in these bodies only, but in all other springy bodies whatsoever, whether metal, wood, stones, baked earth, hair, horns, silk, bones, sinews, glass, and the like. Respect being had to the particular figures of the bodies bended, and the advan- tagious or disadvantagious ways of bending them.

From this principle it will be easy to calculate the several strength of Bows. It will be easy to calculate the proportionate strength of the spring of a watch. From the same also it will be easy to give the reason of the Isochrone motion of a Spring or extended string, and of the uniform sound produced by those whose vibrations are quick enough to produce an audible sound.

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From this appears the reason why a spring applied to the balance of a watch doth make the vibrations thereof equal, whether they be greater or smaller. From this it will be easy to make a Philosophical Scale to examine the weight of any body without putting in weights. This Scale I contrived in order to examine the gravitation of bodies towards the Center of the Earth, viz. The Strength of Materials in the Seventeenth Century 21 5. Mariotte Mariotte spent most of his life in Dijon where he was the prior of St.

He became one of the first members of the French Academy of Sciences, in , and was largely responsible for the introduction of experimental methods into French science.

His experiments with air resulted in the well-known Boyle-Mariotte law which states that, at constant temperature, the pressure of a fixed mass of gas multiplied by its volume remains constant. In the mechanics of solid bodies, Mariotte established the laws of impact, for, using balls suspended by threads, he was able to demonstrate the conservation of momentum.

He invented the ballistic pendulum. He starts with simple tensile tests. Figure 2 0 d shows the arrange- ment used in his tensile tests of wood. I n Fig. Mariotte was not only interested in the absolute strength of materials but also in their elastic properties and found that, in all the materials tested, the elongations were proportional t o the applied forces.

He states that fracture occurs when the elongation exceeds a certain limit. In his discussion of the bending of a cantilever see Fig. If now the load F is increased somewhat, the lever begins to rotate about point C. The displacements of points A, D, and E are in proportion to their distances from C , but the forces applied at those points continue to be equal to 12 lb. Mariotte uses similar reasoning in considering the bending of a canti- lever beam Fig.

Assuming that, a t the instance of fracture, the right-hand portion of the beam rotates with respect to point D,he con- cludes that the forces in its longitudinal fibers will be in the same propor- tion as their distances from D. From this it follows that in the case of I FIG. Tensile and bending experiments made by Mariotte. Now, Mariotte goes further with his analysis and, referring again t o the rectangular beam Fig.

To calculate the load L, which is required to overcome the resistance of the fibers in tension, he uses Eq. Hence the contribution to the strength of the beam made by the compressed fibers will also be equal to L1, and is given by Eq.

The total strength will be given by the previously established equation a. We see that in his analysis Mariotte used a theory of stress distribution in elastic beams which is satisfactory. To verify his theory, Mariotte experimented with wooden cylindrical bars of 4 in. Mariotte tries to explain the discrepancy between his experimental results and those predicted by Eq. When he repeated the experi- ments with rods of glass, Mariotte again found that his formula [Eq. This French physicist also conducted experiments with beams sup- ported a t both ends, and he found that a beam with built-in ends can carry, at its center, twice the ultimate load for a simply supported beam of the same dimensions.

For this purpose he used a cylindrical drum A B Fig. In this way he deduced that the required thick- ness of pipes must be proportional to their internal pres- sure and to the diameter of the pipe. Dealing with the bending of uniformly loaded square F I G. We see that Mariotte considerably enhanced the theory of mechanics of elastic bodies. By introducing considerations of elastic deformation, he improved the theory of bending of beams and then used experiments to check his hypothesis.

He investigated the effects on the strength of a beam brought about by clamping its ends and gave a formula for the bursting strength of pipes.

The Mathematicians Bernoulli1 The Bernoulli family originally lived in Antwerp, but, because of the religious persecution of the Duke of Alba, they left Holland and, toward the end of the sixteenth century, settled in Basel. Starting near the end of the seventeenth century this family produced outstanding mathemati- cians for more than a hundred years. I n , the French Academy of Sciences elected the two brothers Jacob and John Bernoulli as foreign members, and up t o there were always representatives of the Ber- noulli family in that institution.

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During the last quarter of the seventeenth and the beginning of the eighteenth centuries a rapid develop- ment of the infinitesimal calculus took place. Started on the Con- tinent by Leibnitz ,2 it progressed principally by the work I of Jacob and John Bernoulli.

I n FIG. Jacob Bernoulli. Whereas Galileo and Mariotte investigated the strength of beams, Jacob Bernoulli made calculations of their deflection; he did not contribute to our knowledge of the physical properties of materials. Considering a rectangular beam built-in at one end and loaded a t the other by a force P , he takes the deflection curve as shown in Fig. Let ABFD represent an element of the beam the axial length of which is ds.

If, during bending, the cross FIG. However, the general form of Eq. John Bernoulli , the younger brother of Jacob, was con- sidered to be the greatest mathematician of his time. The original lectures of John Bernoulli on differential calculus were published by Naturforschende Gesellschaft of Basel in , on the occasion of the three hundredth anniversary of the Bernoullis having attained citizenship of Basel.

It was John Bernoulli who formu- lated the principle of virtual displacements in his letter toVarignon. Petemburg, The integration of this equation was done by Euler and it will be discussed later see page 35 , but Daniel Bernoulli made a series of verifying experiments and, about his results, he writes to Euler: Some of his experiments fur- nished new mathematical problems for Euler.

Euler Leonard Euler2 was born in the vicinity of Basel. The global dynamic stiffness matrix is expressed by: with denoting the matrix of the nodal concentrated springs and denoting the matrix of the nodal concentrated masses. Once the global equilibrium equation 80 has been assembled, the solution is obtained by: with the backslash operator denoting an appropriate solution method, based, for instance, on the Gauss elimination.

This code is part of the VIANDI computational system for the exact static and dynamic analysis of skeleton-like structures Dias, and freely available at www. In order to explore the above exact solution, some examples were judiciously chosen and compared with other analytical or numerical approaches from the literature, as now described. Therefore, the aim of this section is to verify the present method for the particular case when by comparing it with results from other authors.

The examples given in Tables 2 and 3 , where no damping is presented, show a very good agreement with the exact solution given in Abu-Hilal, , which uses Green functions obtained by the Laplace transform.

In both tables, additional results obtained by the approximate mode superposition method for different number of modes are also listed. Clearly, it can be seen that the mode superposition series solution improves with increasing number of modes and that the convergence is quite good for the deflection but not so good for the shear force.

This confirms the well-known fact that the mode superposition method lacks accuracy for predicting variables involving derivatives wrt position. For the damped cases in Table 4 , Abu-Hilal does not supply written tabular results. For this reason, the results were obtained by using the formulae presented there.In his confinement, without scientific books or any other contact with Western European culture, but with FIG. In Romny his schoolmate and friend was future famous semiconductor physicist Abram Ioffe.

There he learned Latin, Greek, and some Hebrew and became acquainted with the elements of Euclid and with other mathematical topics. He makes the usual assumption that the curvature is propor- tional to the bending moment and discusses several instances which might be of some interest in studying flat springs such as are used in watches.

Assuming, with Mariotte, that the cross section m rotates about the tangent nn Fig. All these academies published their transactions and these had a great influence on the development of science in the eighteenth and nineteenth centuries.

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