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Bible Encyclopedias
Elasticity
1911 Encyclopedia Britannica
eyz are connected by tha two sets of equations a2eyya2ezz a 2 eyz _ az 2 + ay 2 - ayaz a 2 ezz a 2 exx a 2 ezx _ ax 2 + az 2 - azax a 2 exx 8 2 e yy _ a2exy ay 2 ax 2 axay and a 2 a aeyz aezx aexy ` ayaz - ax - ax + ay + az J a 2 eyy _ a aeyz aezx aexy 2 azax ay ax ay az) a 2 e zz _ a aeyz aezx aexy 2 axay - az (ax + ay dz) These equations are known as the conditions of compatibility of strain-components. The components of strain which specify any possible strain satisfy them. Quantities arrived at in any way, and intended to be components of strain, if they fail to satisfy these equations, are not the components of any possible strain; and the theory or speculation by which they are reached must be modified or abandoned.
When the components of strain have been found in accordance with these and other necessary equations, the displacement is to be found by solving the equations (I) of § II, considered as differential equations to determine u, v, w. The most general possible solution will differ from any other solution by terms which contain arbitrary constants, and these terms represent a possible displacement. This " complementary displacement " involves no strain, and would be a possible displacement of an ideal perfectly rigid body.
17. The relations which connect the strains with each other and with the displacement are geometrical relations resulting from the definitions of the quantities and not requiring any experimental verification. They do not admit of such verification, because the strain within a body cannot be measured. The quantities (belonging to the same category) which can be measured are displacements of points on the surface of a body. For example, on the surface of a bar subjected to tension we may make two fine transverse scratches, and measure the distance between them before and after the bar is stretched. For such measurements very refined instruments are required. Instruments for this purpose are called barbarously " extensometers," and many different kinds have been devised. From measurements of displacement by an extensometer we may deduce the average extension of a filament of the bar terminated by the two scratches. In general, when we attempt to measure a strain, we really measure some displacements, and deduce the values, not of the strain at a point, but of the average extensions of some particular linear filaments of a body containing the point; and these filaments are, from the nature of the case, nearly always superficial filaments.
18. In the case of transparent materials such as glass there is available a method of studying experimentally the state of strain within a body. This method is founded upon the result that a piece of glass when strained becomes doubly refracting, with its optical principal axes at any point in the directions of the principal axes of strain (§ 15) at the point. When the piece has two parallel plane faces, and two of the principal axes of strain at any point are parallel to these faces, polarized light transmitted through the piece in a direction normal to the faces can be used to determine the directions of the principal axes of the strain at any point. If the directions of these axes are known theoretically the comparison of the experimental and theoretical results yields a test of the theory.
19. Relations between Stresses and Strains
The problem of the extension of a bar subjected to tension is the one which has been most studied experimentally, and as a result of this study it is found that for most materials, including all metals except cast metals, the measurable extension is proportional When this quantity is negative (2) to the applied tension, provided that this tension is not too great. In interpreting this result it is assumed that the tension is uniform over the cross-section of the bar, and that the extension of longitudinal filaments is uniform throughout the bar; and then the result takes the form of a law of proportionality connecting stress and strain: The tension is proportional to the extension. Similar results are found for the same materials when other methods of experimenting are adopted, for example, when a bar is supported at the ends and bent by an attached load and the deflexion is measured, or when a bar is twisted by an axial couple and the relative angular displacement of two sections is measured. We have thus very numerous experimental verifications of the famous law first enunciated by Robert Hooke in 1678 in the words " Lit Tensio sic vis "; that is, " the Power of any spring is in the same proportion as the Tension (- stretching) thereof." The most general statement of Hooke's Law in modern language would be: - Each of the six components of stress at any point of a body is a linear function of the six components of strain at the point. It is evident from what has been said above as to the nature of the measurement of stresses and strains that this law in all its generality does not admit of complete experimental verification, and that the evidence for it consists largely in the agreement of the results which are deduced from it in a theoretical fashion with the results of experiments. Of such results one of a general character may be noted here. If the law is assumed to be true, and the equations of motion of the body (§ 5) are transformed by means of it into differential equations for determining the components of displacement, these differential equations admit of solutions which represent periodic vibratory displacements '(see' § 85 below). The fact that solid bodies can be thrown into states of isochronous vibration has been emphasized by G. G. Stokes as a peremptory proof of the truth of Hooke's Law.
20. According to the statement of the generalized Hooke's Law the stress-components vanish when the strain-components vanish. The strain-components contemplated in experiments upon which the law is founded are measured from a zero of reckoning which corresponds to the state of the body subjected to experiment before the experiment is made, and the stresscomponents referred to in the statement of the law are those which are called into action by the forces applied to the body in the course of the experiment. No account is taken of the stress which must already exist in the body owing to the force of gravity and the forces by which the body is supported. When it is desired to take account of this stress it is usual to suppose that the strains which would be produced in the body if it could be freed from the action of gravity and from the pressures of supports are so small that the strains produced by the forces which are applied in the course of the experiment can be compounded with them by simple superposition. This supposition comes to the same thing as measuring the strain in the body, not from the state in which it was before the experiment, but from an ideal state (the " unstressed " state) in which it would be entirely free from internal stress, and allowing for the strain which would be produced by gravity and the supporting forces if these forces were applied to the body when free from stress. In most practical cases the initial strain to be allowed for is unimportant (see §§ 91-93 below).
21. Hooke's law of proportionality of stress and strain leads to the introduction of important physical constants: the moduluses of elasticity of a body. Let a bar of uniform section (of area w) be stretched with tension T, which is distributed uniformly over the section, so that the stretching force is Tw, and let the bar be unsupported at the sides. The bar will undergo a longitudinal extension of magnitude TÆ, where E is a constant quantity depending upon the material. This constant is called Young's modulus after Thomas Young, who introduced it into the science in 1807. The quantity E is of the same nature as a traction, that is to say, it is measured as a force estimated per unit of area. For steel it is about 204X10 12 dynes per square centimetre, or about 13,000 tons per sq. in.
22. The longitudinal extension of the bar under tension is not the only strain in the bar. It is accompanied by a lateral contraction by which all the transverse filaments of the bar are shortened. The amount of this contraction is vTÆ, where a is a certain number called Poisson's ratio, because its importance was at first noted by S. D. Poisson in 1828. Poisson arrived at the existence of this contraction, and the corresponding number a, from theoretical considerations, and his theory led him to assign to a the value 4. Many experiments have been made with the view of determining a, with the result that it has been found to be different for different materials, although for very many it does not differ much from 4. For steel the best value (Amagat's) is o. 268. Poisson's theory admits of being modified so as to agree with the results of experiment.
23. The behaviour of an elastic solid body, strained within the limits of its elasticity, is entirely determined by the constants E and a if the body is isotropic, that is to say, if it has the same quality in all dir. ctions around any point. Nevertheless it is convenient to introduce other constants which are related to the action of particular sorts of forces. The most important of these are the " modulus of compression " (or " bulk modulus ") and the " rigidity " (or "modulus of shear"). To define the modulus of compression, we suppose that a solid body of any form is subjected to uniform hydrostatic pressure of amount p. The state of stress within it will be one of uniform pressure, the same at all points, and the same in all directions round any point. There will be compression, the same at all points, and proportional to the pressure; and the amount of the compression can be expressed as p/k. The quantity k is the modulus of compression. In this case the linear contraction in any direction is p/3k; but in general the linear extension (or contraction) is not one-third of the cubical dilatation (or compression).
24. To define the rigidity, we suppose that a solid body is subjected to forces in such a way that there is shearing stress within it. For example, a cubical block may be subjected to opposing tractions on opposite faces acting in directions which are parallel to an edge of the cube and to both the faces. Let S be the amount of the traction, and let it be uniformly distributed over the faces. As we have seen (§ 7), equal tractions must act upon two other faces in suitable directions in order to maintain equilibrium (see fig. 2 of § 7). The two directions involved may be chosen as axes of x, y as in that figure. Then the state of stress will be one in which the stress-component denoted by X y is equal to S, and the remaining stress-components vanish; and the strain produced in the body is shearing strain of the type denoted by e,,,. The amount of the shearing strain is S/µ, and the quantity ,u is the " rigidity." 25. The modulus of compression and the rigidity are quantities of the same kind as Young's modulus. The modulus of compression of steel is about 1.43 X 10 12 dynes per square centimetre, the rigidity is about 8.19 X to ll dynes per square centimetre. It must be understood that the values for different specimens of nominally the same material may differ considerably.
The modulus of compression k and the rigidity of an isotropic material are connected with the Young's modulus E and Poisson's ratio a of the material by the equations k=E/3(1-2r), = E/2(1-?-o).
26. Whatever the forces acting upon an isotropic solid body may be, provided that the body is strained within its limits of elasticity, the strain-components are expressed in terms of the stress-components by the equations exx = (X x - 6Y y - o-Z Z) / E, eyz = YZ /,u, e y ,,=(Y 5 - oZz - o - X x)/ E, eZx = Zxl (I) e ZZ = (ZZ - o-X x - aY y)/ E, exy =Xy/A. If we introduce a quantity X, of the same nature as E or ,u, by the equation X=EO (2) we may express the stress-components in terms of the strain-components by the equations Xx = a(exx +eyy +eZ) - f -2yexx, =lA(exx+eyy+eZZ)+2keZZ, and then the behaviour of the body under the action of any forces YZ = = Xy = (3) depends upon the two constants X and µ. These two constants were introduced by G. Lame in his treatise of 1852. The importance of the quantity ,u had been previously emphasized by L. J. Vicat and G. G. Stokes.
27. The potential energy per unit of volume (often called the " resilience ") stored up in the body by the strain is equal to 2(X+ 2 /.)(exx+eyy+ez) 2 +2 (e2yz+ezx+exy- 4eyyezz-4ezzexx-4exxeyy) or the equivalent expression z [ (X x+ - 2 Q (Y y Z z -{- Z z X x+XxYy) + 2 (I + (i) (Y z+ Z x+ X x) l Æ. The former of these expressions is called the " strain-energy-function " 28. The Young's modulus E of a material is often determined experimentally by the direct method of the extensometer (§ 17), but more frequently it is determined indirectly by means of a result obtained in the theory of the flexure of a bar (see §§ 47, 53 below). The rigidity p. is usually determined indirectly by means of results obtained in the theory of the torsion of a bar (see §§ 41, 42 below). The modulus of compression k may be determined directly by means of the piezometer, as was done by E. H. Amagat, or it may be determined indirectly by means of a result obtained in the theory of a tube under pressure, as was done by A. Mallock (see § 78 below). The value of Poisson's ratio o is generally inferred from the relation connecting it with E and µ or with E and k, but it may also be determined indirectly by means of a result obtained in the theory of the flexure of a bar (§ 47 below), as was done by M. A. Cornu and A. Mallock, or directly by a modification of the extensometer method, as has been done recently by J. Morrow.
29. The elasticity of a fluid is always expressed by means of a single quantity of the same kind as the modulus of compression of a solid body. To any increment of pressure, which is not too great, there corresponds a proportional cubical compression, and the amount of this compression for an increment bp of pressure can be expressed as Sp/k. The quantity that is usually tabulated is the reciprocal of k, and it is called the coefficient of compressibility. It is the amount of compression per unit increase of pressure. As a physical quantity it is of the same dimensions as the reciprocal of a pressure (or of a force per unit of area). The pressures concerned are usually measured in atmospheres (1 atmosphere = 1.014 X Io 0 dynes per sq. cm.). For water the coefficient of compressibility, or the compression per atmosphere, is about 4.5 X 105. This gives for k the value 2.22 X 10 10 dynes per sq. cm. The Young's modulus and the rigidity of a fluid are always zero.
30. The relations between stress and strain in a material which is not isotropic are much more complicated. In such a material the Young's modulus depends upon the direction of the tension, and its variations about a point are expressed by means of a surface of the fourth degree. The Poisson's ratio depends upon the direction of the contracted lateral filaments as well as upon that of the longitudinal extended ones. The rigidity depends upon both the directions involved in the specification of the shearing stress. In general there is no simple relation between the Young's moduluses and Poisson's ratios and rigidities for assigned directions and the modulus of compression. Many materials in common use, all fibrous woods for example, are actually aeolotropic (that is to say, are not isotropic), but the materials which are aeolotropic in the most regular fashion are natural crystals. The elastic behaviour of crystals has been studied exhaustively by many physicists, and in particular by W. Voigt. The strain-energy-function is a homogeneous quadratic function of the six strain-components, and this function may have as many as 21 independent coefficients, taking the place in the general case of the 2 coefficients X, µ which occur when the material is isotropic - a result first obtained by George Green in 1837. The best experimental determinations of the coefficients have been made indirectly by Voigt by means of results obtained in the theories of the torsion and flexure of aeolotropic bars.
31. Limits of Elasticity
A solid body which has been strained by considerable forces does not in general recover its original size and shape completely after the forces cease to act. The strain that is left is called set. If set occurs the elasticity is said to be " imperfect," and the greatest strain (or the greatest load) of any specified type, for which no set occurs, defines the " limit of perfect elasticity " corresponding to the specified type of strain, or of stress. All fluids and many solid bodies, such as glasses and crystals, as well as some metals (copper, lead, silver) appear to be perfectly elastic as regards change of volume within wide limits; but malleable metals and alloys can have their densities permanently increased by considerable pressures. The limits of perfect elasticity as regards change of shape, on the other hand, are very low, if they exist at all, for glasses and other hard, brittle solids; but a class of metals including copper, brass, steel, platinum are very perfectly elastic as regards distortion, provided that the distortion is not too great. The question can be tested by observation of the torsional elasticity of thin fibres or wires. The limits of perfect elasticity are somewhat ill-defined, because an experiment cannot warrant us in asserting that there is no set, but only that, if there is any set, it is too small to be observed.
32. A different meaning may be, and often is, attached to the phrase " limits of elasticity " in consequence of the following experimental result: - Let a bar be held stretched under a moderate tension, and let the extension be measured; let the tension be slightly increased and the extension again measured; let this process be continued, the tension being increased by equal increments. It is found that when the tension is not too great the extension increases by equal increments (as nearly as experiment can decide), but that, as the tension increases, a stage is reached in which the extension increases faster than it would do if it continued to be proportional to the tension. The beginning of this stage is tolerably well marked. Some time before this stage is reached the limit of perfect elasticity is passed; that is to say, if the load is removed it is found that there is some permanent set. The limiting tension beyond which the above law of proportionality fails is often called the " limit of linear elasticity." It is higher than the limit of perfect elasticity. For steel bars of various qualities J. Bauschinger found for this limit values varying from io to 17 tons per square inch. The result indicates that, when forces which produce any kind of strain are applied to a solid body and are gradually increased, the strain at any instant increases proportionally to the forces up to a stage beyond that at which, if the forces were removed, the body would completely recover its original size and shape, but that the increase of strain ceases to be proportional to the increase of load when the load surpasses a certain limit. There would thus be, for any type of strain, a limit of linear elasticity, which exceeds the limit of perfect elasticity.
33A body which has been strained beyond the limit of linear elasticity is often said to have suffered an " over-strain. " When the load is removed, the set which can be observed is not entirely permanent; but it gradually diminishes with lapse of time. This phenomenon is named " elastic after-working." If, on the other hand, the load is maintained constant, the strain is gradually increased. This effect indicates a gradual flowing of solid bodies under great stress; and a similar effect was observed in the experiments of H. Tresca on the punching and crushing of metals. It appears that all solid bodies under sufficiently great loads become " plastic," that is to say, they take a set which gradually increases with the lapse of time. No plasticity is observed when the limit of linear elasticity is not exceeded.
34. The values of the elastic limits are affected by overstrain. If the load is maintained for some time, and then removed, the limit of linear elasticity is found to be higher than before. If the load is not maintained, but is removed and then reapplied, the limit is found to be lower than before. During a period of rest a test piece recovers its elasticity after overstrain.
35. The effects of repeated loading have been studied by A. Wbhler, J. Bauschinger, 0. Reynolds and others. It has been found that, after many repetitions of rather rapidly alternating stress, pieces are fractured by loads which they have many times withstood. It is not certain whether the fracture is in every case caused by the gradual growth of minute flaws from the beginning of the series of tests, or whether the elastic quality of the material suffers deterioration apart from such flaws. It appears, however, to be an ascertained result that, so long as the limit of linear elasticity is not exceeded, repeated loads and rapidly alternating loads do not produce failure of the material.
36. The question of the conditions of safety, or of the conditions in which rupture is produced, is one upon which there has been much speculation, but no completely satisfactory result has been obtained. It has been variously held that rupture occurs when the numerically greatest principal stress exceeds a certain limit, or when this stress is tension and exceeds a certain limit, or when the greatest difference of two principal stresses (called the " stress-difference ") exceeds a certain limit, or when the greatest extension or the greatest shearing strain or the greatest strain of any type exceeds a certain limit. Some of these hypotheses appear to have been disproved. It was held by G. F. Fitzgerald (Nature, Nov. 5, 1896) that rupture is not produced by pressure symmetrically applied all round a body, and this opinion has been confirmed by the recent experiments of A. Fdppl. This result disposes of the greatest stress hypothesis and also of the greatest strain hypothesis. The fact that short pillars can be crushed by longitudinal pressure disposes of the greatest tension hypothesis, for there is no tension in the pillar. The greatest extension hypothesis failed to satisfy some tests imposed by H. Wehage, who experimented with blocks of wrought iron subjected to equal pressures in two directions at right angles to each other. The greatest stressdifference hypothesis and the greatest shearing strain hypothesis would lead to practically identical results, and these results have been held by J. J. Guest to accord well with his experiments on metal tubes subjected to various systems of combined stress; but these experiments and Guest's conclusion have been criticized adversely by 0. Mohr, and the question cannot be regarded as settled. The fact seems to be that the conditions of rupture depend largely upon the nature of the test (tensional, torsional, flexural, or whatever it may be) that is applied to a specimen, and that no general formula holds for all kinds of tests. The best modern technical writings emphasize the importance of the limits of linear elasticity and of tests of dynamical resistance (§ 87 below) as well as of statical resistance.
37. The question of the conditions of rupture belongs rather to the science of the strength of materials than to the science of elasticity (§ i); but it has been necessary to refer to it briefly here, because there is no method except the methods of the theory of elasticity for determining the state of stress or strain in a body subjected to forces. Whatever view may ultimately be adopted as to the relation between the conditions of safety of a structure and the state of stress or strain in it, the calculation of this state by means of the theory or by experimental means (as in § 18) cannot be dispensed with.
38. Methods of determining the Stress in a Body subjected to given Forces
To determine the state of stress, or the state of strain, in an isotropic solid body strained within its limits of elasticity by given forces, we have to use (i.) the equations of equilibrium, (ii.) the conditions which hold at the bounding surface, (iii.) the relations between stress-components and strain-components, (iv.) the relations between strain-components and displacement. The equations of equilibrium are (with notation already used) three partial differential equations of the type aX x aX aZx ax +ay?, +az+PX.(I) The conditions which hold at the bounding surface are three equations of the type X x cos (x, v) +X 5 cos (y, v) -}-Z x cos (z, v) = xv, (2) where v denotes the direction of the outward-drawn normal to the bounding surface, and Xv denotes the x-component of the applied surface traction. The relations between stress-components and strain-components are expressed by either of the sets of equations (I) or (3) of § 26. The relations between strain-components and displacement are the equations (I) of § 11, or the equivalent conditions of compatibility expressed in equations (1) and (2) of § 16.
39. We may proceed by either of two ,nthods_ In one method we eliminate the stress-components and the strain-components and retain only the components of displacement. This method leads (with notation already used) to three partial differential equations of the type a v aw a2u 32u (32,(A+I-?)ax (a a+d+a +11 (axe+aye+azz) +PX =o, (3) and three boundary conditions of the type (au ay aw (au (gvx au) cos(x,v) + ay + az +,“ 2 cos(x,v) a x +cos(y,v) +ay +cos (z,v)(au + aw) } =Xv. az ax In the alternative method we eliminate the strain-components and the displacements. This method leads to a system of partial differential equations to be satisfied by the stress-components. In this system there are three equations of the type aX x ax ay +a w +P X = o, (I bis) three of the type a 2 X x a 2 X x a 2 X, t a 2 r) ax`2 ay- + az 2 + +0 ax2 (Xx+Yv+?z = ? (aX a Y aZl aX - v P ax + ay + az) --21)-67'I and three of the type a 2 Yz a 2 Y z 'a' 2 Yz I a 2 - / (Z 3Y ax 2 ay' + az 2 + 1-a ayaz (X z+Yy+Zz) = - az (6) ay the equations of the two latter types being necessitated by the conditions of compatibility of strain-components. The solutions of these equations have to be adjusted so that the boundary conditions of the type (2) may be satisfied.
40. It is evident that whichever method is adopted the mathematical problem is in general very complicated. It is also evident that, if we attempt to proceed by help of some intuition as to the nature of the stress or strain, our intuition ought to satisfy the tests provided by the above systems of equations. Neglect of this precaution has led to many errors. Another source of frequent error lies in the neglect of the conditions in which the above systems of equations are correct. They are obtained by help of the supposition that the relative displacements of the parts of the strained body are small. The solutions of them must therefore satisfy the test of smallness of the relative displacements.
Torsion.
As a first example of the application of the theory we take the problem of the torsion of prisms. This problem, considered first by C. A. Coulomb in 1784, was finally solved by B. de Saint-Venant in 1855. The problem is this: - A cylindrical or prismatic bar is held twisted by terminal couples; it is required to determine the state of stress and strain in the interior. When the bar is a circular cylinder the problem is easy. Any section is displaced by rotation about the central-line through a small angle, which is proportional to the distance z of the section from a fixed plane at right angles to this line. This plane is a terminal section if one of the two terminal sections is not displaced. The angle through which the section z rotates is Tz, where r is a constant, called the amount of the twist; and this constant T is equal to G/µI, where G is the twisting couple, and I is the moment of inertia of the cross-section about the central-line. This result is often called " Coulomb's law." The stress within the bar is shearing stress, consisting, as it must, of two sets of equal tangential tractions on two sets of planes which are at right angles to each other. These planes are the cross-sections and the axial planes of the bar. The tangential traction at any point of the crosssection is directed at right angles to the axial plane through the point, and the tangential traction on the axial plane is directed parallel to the length of the bar. The amount of either at a distance r from the axis is µTr or Gr/I. The result that G = µTI can be used to determine µ experimentally, for T may be measured and G and I are known.
42. When the cross-section of the bar is not circular it is clear that this solution fails; for the existence of tangential traction, near the prismatic bounding surface, on any plane which does not cut this surface at right angles, implies the existence of traction applied to this surface. We may attempt to modify the theory by retaining the supposition that the stress consists of shearing stress, involving tangential traction distributed in some way over the cross-sections. Such traction is obviously a necessary constituent of any stress-system which could be produced by terminal couples around the axis.
We should then know that there must be equal tangential traction directed along the length of the bar, and exerted across some planes or other which are parallel to this direction. We should also know that, at the bounding surface, these planes must cut this surface at right angles. The corresponding strain would be shearing strain which could involve (i.) a sliding of elements of one cross-section relative to another, (ii.) a relative sliding of elements of the above mentioned planes in the direction of the length of the bar. We could conclude that there may be a longitudinal displacement of the elements of the crosssections. We should then attempt to satisfy the conditions of the problem by supposing that this is the character of the strain, and that the corresponding displacement consists of (i.) a rotation of the cross-sections in their planes such as we found in the case of the circle, (ii.) a distortion of the crosssections into curved surfaces by a displacement (w) which is directed normally to their planes and varies in some manner from point to point of these planes. We could show that all the conditions of the problem are satisfied by this assumption, provided that the longitudinal displacement (w), considered as a function of the position of a point (x,y) in the cross-section, satisfies the equation axe+ay2 = o, and the boundary condition (aw =o, . (2) where T denotes the amount of the twist, and v the direction of the normal to the boundary. The solution is known for a great many forms of section. (In the particular case of a circular section w vanishes.) The tangential traction at any point of the cross-section is directed along the tangent to that curve of the family ik = const. which passes through the point, being the function determined by the equations aw/4 aw a,? ax - T a,, _ - T The amount of the twist T produced by terminal couples of magnitude G is G/C, where C is a constant, called the " torsional rigidity " of the prism, and expressed by the formula C =,u j ()2+ }dxdy, the integration being taken over the cross-section. When the coefficient of ,u in the expression for C is known for any section, can be determined by experiment with a bar of that form of section.
43. The distortion of the cross-sections into curved surfaces is shown graphically by drawing the contour lines (w = const.). In general the section is divided into a number of compartments, and the portions that lie within two adjacent compartments are respectively concave and convex. This result is illustrated in the accompanying figures (fig. 4 for the ellipse, given by x2/ fig. 5 for the equilateral triangle,given by (x+3a) (x2 - 3y2 - S ax+ 9 a 2) = o; fig. 6 for the square).
44. The distribution of the shearing stress over the cross-section is determined by the function >L already introduced. If we draw the curves ik= const., corresponding to any form of section, for equidifferent values of the constant, the tangential traction at any point on the cross-section is directed along the tangent to that curve of the family which passes through the point, and the magnitude of it is inversely proportional to the distance between consecutive curves of the family. Fig. 7 illustrates the result in the case of the equilateral triangle. The boundary is, of course, one of the lines. The " lines of shearing stress " which can thus be drawn are in every case identical with the lines of flow of frictionless liquid filling a cylindrical vessel of the same cross-section as the bar, when the liquid circulates in the plane of the section with uniform spin. They are also the same as the contour lines of a flexible and slightly extensible membrane, of which the edge has the same form as the bounding curve of the cross-section of the bar, when the membrane is fixed at the edge and slightly deformed by uniform pressure.
45.Saint-Venant's theory shows that the true torsional rigidity is in general less than that which would be obtained by extending Coulomb's law (G =µTI) to sections which are not circular. For an elliptic cylinder of sectional area co and moment of inertia I about its central-line the torsional rigidity is µco 4 /47r 2 I, and this formula is not far from being correct for a very large number of sections. For a bar of square section of side a centimetres, the torsional rigidity in C.G.S. units is (o1406)µa4 approximately, being expressed in dynes per square centimetre. How great the defect of the true value from that Y FIG. 6.
given by extending Coulomb's law may be in the case of sections with projecting corners is shown by the diagrams (fig. 8 especially no. 4). In these diagrams the upper of the two numbers under each figure indicates the fraction which the true torsional rigidity corresponding to the section is of that value which would be obtained by extending Coulomb's law; and the lower of the two numbers indicates the ratio which the torsional rigidity for a bar of the corresponding section bears to that of a bar of circular section of the same material and of equal sectional area. These results have an important practical application, inasmuch as they show that strengthening ribs and projections, such as are introduced in engineering to give stiffness to beams, have the reverse of a good effect when torsional stiffness is an object, although they are of great value in increasing the resistance to bending. The theory shows further that the resistance to torsion is very seriously diminished when there is in the surface any dent approaching to a re-entrant angle. At such a place the shearing strain tends to become infinite, and some (I) FIG. 4.
b2 +y2/c2 FIG. 5.
FIG. 7.
permanent set is produced by torsion. In the case of a section of any form, the strain and stress are greatest at points on the contour, and these points are in many cases the points of the contour which are nearest to the centroid of the section. The theory has also been applied to show that a longitudinal flaw 84346. '8186. -7783. '5374. 60000.
88326.8666. '8276. '6745. '72552.
FIG. 8. - Diagrams showing Torsional Rigidities.
near the axis of a shaft transmitting a torsional couple has little influence on the strength of the shaft, but that in the neighbourhood of a similar flaw which is much nearer to the surface than to the axis the shearing strain may be nearly doubled, and thus the possibility of such flaws is a source of weakness against which special provision ought to be made.
46. Bending of Beams
As a second example of the application of the general theory we take the problem of the flexure of a beam. In this case also we begin by forming a simple intuition as to the nature of the strain and the stress. On the side of the beam towards the centre of curvature the longitudinal filaments must be contracted, and on the other side they must be extended. If we assume that the cross-sections remain plane, and that the central-line is unaltered in length, we see (at once from fig. 9) that the extensions (or contractions) are given by the formula y/R, where y denotes the distance of a longitudinal filament from the plane drawn through the unstrained central-line at rightangles to the plane of bending, and R is the radius of curvature of the curve into which this line is bent (shown by the dotted line in the figure). Corresponding to this strain there must be traction acting across the crosssections. If we assume that there is no other stress, then the magnitude of the traction in question is Ey/R, where E is Young's modulus, and it is tension on the side where the filaments are extended and pressure on the side where they are contracted. If the plane of bending contains a set of principal axes of the cross-sections at their centroids, these tractions for the whole cross-section are equivalent to a couple of moment EI/R, where I now denotes the moment of inertia of the cross-section about an axis through its centroid at right angles to the plane of bending, and the plane of the couple is the plane of bending. Thus a beam of any form of section can be held bent in a "principal plane" by terminal couples of moment M, that is to say by a "bending moment" M; the central-line will take a curvature MÆI, so that it becomes an arc of a circle of radius EI/M; and the stress at any point will be tension of amount My/I, where y denotes distance (reckoned positive towards the side remote from the centre of curvature) from that plane which initially contains the central-line and is at right angles to the plane of the couple. This plane is called the " neutral plane." The restriction that the beam is bent in a principal plane means that the plane of bending contains one set of principal axes of the cross-sections at their centroids; in the case of a beam of rectangular section the plane would bisect two opposite edges at right angles. In order that the theory may hold good the radius of curvature must be very large.
47. In this problem of the bending of a beam by terminal couples the stress is tension, determined as above, and the corresponding strain consists therefore of longitudinal extension of amount MyÆI or y/R (contraction if y is negative), accompanied by lateral contraction of amount aMyÆI or ay/R(exten sion if y is negative), a being Poisson's ratio for the material. Our intuition of the nature of the strain was imperfect, inasmuch as it took no account of these lateral strains. The necessity for introducing them was pointed out by Saint-Venant. The effect of them is a change of shape of the crosssections in their own planes. This is shown in an exaggerated way in fig.
Io, where the rectangle AB CD represents the cross-section of the unstrained beam, or a rectangular portion of this cross-section, and the curvilinear figure A'B'C'D' represents in an exaggerated fashion the cross-section (or the corresponding portion of the cross-section) of the same beam, when bent so that the centre of curvature of the central-line (which is at right angles to the plane of the figure) is on the line EF produced beyond F. The lines A'B' and C'D' are approximately circles of radii R/a, when the central-line is a circle of radius R, and their centres are on the line FE produced beyond E. Thus the neutral plane, and each of the faces that is parallel to it, becomes strained into an anticlastic surface, whose principal curvatures are in the ratio a-: I. The general appearance of the bent beam is shown in an exaggerated fashion in fig. II, where the traces of the surface into which the neutral plane is bent are dotted. The result that the ratio of the principal curvatures of the anticlastic surfaces, into which the top and bottom planes of the beam (of rectangular section) are bent, is Poisson's ratio a-, has been used for the experimental determina tion of a. The result that the radius of curvature of the bent central-line is EI/M is used in the experimental determination of E. The quantity EI is often called the " flexural rigidity " of the beam. There are two principal flexural rigidities corresponding to bending in the two principal planes (cf. § 62 below).
48. That this theory requires modification, when the load does not consist simply of terminal couples, can be seen most easily by considering the problem of a beam loaded at one end with a weight W, and supported in a horizontal position at its other end. The forces that are exerted at any section p, to balance the weight W, must reduce statically to a vertical force W and a couple, and these forces arise from the action of the part Ap on the part Bp (see fig. 12), i.e. from the stresses across the section at p. The couple is equal to the moment of the applied load W about an axis drawn through the centroid of the section p at right angles to the plane of bending. This moment is called the " bending moment " at the section, it is the product of the load W and the distance of the section from the loaded end, so that it varies uniformly along the length of the beam. The stress that suffices in the simpler problem gives rise to no vertical force, and it is clear that in addition to longitudinal tensions and pressures there must be tangential tractions on the cross-sections. The resultant of these tangential tractions must be a force equal to W, and directed vertically; FIG. I I.
FIG. 9.
FIG. 10.