ficial width, 100 feet; depth, 112 feet. It was hampered by six sluices, but was used annually by some 4000 vessels. This was in 1887-95 converted into the Emperor William Canal, for which see CANALS.

Eiffel Tower.-Erected for the Exposition of 1889, the Eiffel Tower, in the Champ de Mars, Paris, is by far the highest artificial structure in the world, and its height of 300 metres (1092 feet) surpasses that of the obelisk at Washington by 537 feet, and that of St Paul's Cathedral by 688 feet. Its framework is composed essentially of four uprights, which rise from the corners of a square measuring 100 metres on the side; thus the area it covers at its base is nearly 2 acres. These uprights are supported on huge piers of masonry and concrete, the foundations for which were carried down, by the aid of iron caissons and compressed air, to a depth of about 15 metres on the side next the Seine, and about 9 metres on the other side. At first they curve upwards at an angle of 54°; then they gradually become straighter, until they unite in a single shaft rather more than half-way up. The first platform, at a height of 57 metres, has an area of 5860 square yards, and is reached either by staircases or lifts. The next, accessible by lifts only, is 115 metres up, and has an area of 32 square yards; while the third, at 276, supports a pavilion capable of holding 800 persons. Nearly 25 metres higher up still is the lantern, with a gallery 5 metres in diameter. The work of building this structure, which is mainly composed of iron latticework, was begun on 28th January 1887, and the full height was reached on 13th March 1889. Besides being one of the sights of Paris, to which visitors resort in order to enjoy the extensive view that can be had from its higher galleries on a clear day, the tower is used to some extent for scientific and semi-scientific purposes; thus meteorological observations are carried on. formed the objective in the flying-machine trials of October 1901, when M. Santos-Dumont succeeded in winning the Deutsch prize with an air-ship designed and constructed. by himself. The engineer under whose direction the tower was constructed was M. ALEXANDRE GUSTAVE EIFFEL (born at Dijon on 15th December 1832), who had already had a wide experience in the construction of large metal bridges, and who designed the huge sluices for the Panama Canal, in connexion with which there were the famous scandals in 1893.

It

Einbeck, a town of Prussia, province of Hanover, 50 miles by rail south from Hanover. The municipal antiquarian museum is preserved in the chapel (restored) of Zum Heiligen Geist. Here are an engineering school and a textile school. Population (1885), 7091; (1900), 7974. Eisenach, a town of Germany, second capital of the grand duchy of Saxe-Weimar, at the north-west foot of the Thuringian Forest, 32 miles by rail west. from Erfurt. The Nicolai Kirche was restored in 1887, and Georg Kirche in 1899. There are monuments to Sebastian Bach (1884), to Luther (1895), and to the war of 1870-71. The house (now museum) in which Fritz Reuter lived (1863-74), his grave in the new churchyard, and the Richard Wagner Museum should be mentioned. At Eisenach are a school of forestry, a school of design and the industrial arts, teachers' seminaries, an infirmary, and a prison. Population (1885), 19,743; (1901), 31,580.

Eisenerz ("Iron ore"), a market-place and old mining town in the government district of Leoben, Upper Styria, Austria, the chief centre of the Styrian iron industry. It is situated in a deep valley, dominated on the east by the Pfaffenstein (6140 feet) and on the west by the Kaiserschild (6830 feet). It has an interesting example of a mediæval fortified church, a Gothic edifice founded by Rudolph of Hapsburg in the 13th century and rebuilt in the 16th. The Erzberg, or Ore Mountain, which closes the valley on the south, furnishes such rich ore that it is quarried in the open air like stone, in the summer months. There is documentary evidence of the mines having been worked as far back as the 12th century. They afford employment to two to three thousand hands in summer and about half as many in winter, and yield some 150,000 tons of iron per annum. Eisenerz possesses, in addition, twenty-five furnaces, which produce iron, and particularly steel, of exceptional excellence. It is connected with the neighbouring mining and smelting centre of Vordernberg, at the other side of the Erzberg (3000 inhabitants), by a mountain railway on the cogged-wheel system, with an average gradient of 68: 1000. Population (1890), 5740; (1900), 6494.

Eisleben, a town of Prussia, province of Saxony, 24 miles by rail west by north from Halle. It was the birthplace and deathplace of Luther, to whom a monument by Siemering was unveiled in 1883. Eisleben has a mining school, and is the seat of copper, silver, and iron mines, and of works for smelting the ore, and also produces flower and vegetable seeds. Population (1885), 23,175; (1900), 23,898.

Ekaterinburg, a district town of Russia, 311 miles by rail south-east of Perm, on the Iset river. It is the most important and most rapidly developing town of the Urals. Population (1860), 19,830; (1897), 55,488. It is the seat of the central mining administration, and has a mining chemical laboratory for the assay of gold extracted both in the Urals and Siberia, a mining school, an Imperial stone-cutting factory, the Ural Society of Naturalists, a first-class magnetic and meteorological observatory, and many banks. Altogether, it is one of the best provincial towns of Russia. There are, besides, one large steam flour mill, candle works, several machinery works, woollen mills, paper works, soap works, and tanneries, and many stonecutting factories and workshops, the produce of which is widely exported. Altogether, over 2000 persons work in the factories, and nearly as many in the workshops. Many of the residents are engaged in gold mining. The trade in goods exported from the Urals and imported from Russia is very large, and two important fairs are held. Nearly 40 gold and platinum mines, 32 iron works, and numbers of other works and factories are scattered in the district, while wheels, travelling boxes, all sorts of hardware, boots, and so on are fabricated on a great scale in the villages.

Ekaterinodar, a town of Russia, North Caucasia, capital of the province of Kubañ, situated on the right bank of the Kubañ, 531 miles north-west of Tiflis, on the railway from Rostov to Novorossiysk, 85 miles by rail from this seaport. Founded in 1794 as a small fort, its population has grown from 9620 in 1860 to 65,700 in 1897. It has gymnasia for boys and girls, various professional schools, an experimental fruit-farm, and a natural history museum. A considerable trade, especially in grain, is carried on by its merchants.

Ekaterino-Nikolskaya, a Cossack village of Russia, province of the Amur, 340 miles below Blagovyeschensk, at the spot where the Amur enters a gorge through which it pierces the Little Khingan. The road which runs along the middle Amur ends at this spot, and when the

Ekaterinoslav, a province of South Russia, having Poltava and Kharkoff on the N., the Don province on the E., the Sea of Azov and Taurida on the S., and Kherson on the W. Area, 24,478 square miles. Its surface is undulating prairie gently sloping south and north, with a few hills reaching 1320 ft. in the north-east, where a gentle swelling (the Don Hills) compels the Don to make a great curve eastwards. Another row of hills, to which the eastward bend of the Dnieper is due, rises in the west. These hills have a crystalline core (granites, syenites, and diorites), while the surface strata are Carboniferous, Permian, Cretaceous, and Tertiary. The province is rich in minerals, especially in anthracite and coal, iron ores, and rock-salt. Granite, limestone, grindstone, slate, as also graphite, manganese, and mercury, are found. It is watered by the Dnieper, the Don, and their tributaries,

and several affluents of the Sea of Azov. The soil is a very fertile black earth, but the crops have occasionally to suffer from droughts-the average yearly rainfall being only 15 inches. Forests are scarce. Population (1860), 1,138,750; (1897), 2,112,651-chiefly Little Russians, partly Great Russians, and some Greeks. The land is owned as follows: the peasants hold in communal ownership 37 per cent., and in private ownership 4 per cent.; nobility, 31 per cent.; German colonists, 9.5 cent.; Greeks, 7 per cent. ; the remainder being owned by various persons and companies.

per

Wheat is grown extensively with the aid of modern machinery, and the average crops are 7,580,000 quarters of various cereals, 577,000 quarters of potatoes, and about 1000 cwt. of tobacco. Vines have been grown lately. There were in 1895, 295,670 horses, 653,770 horned cattle, and 2,174,350 sheep-nearly 6700 tons of raw wool being obtained every year. Mining attains every year more and more importance, the returns in South Russia (Ekaterinoslav and Don) being, in 1896, coal from 83 mines, 2,926,000 tons (12,000 workers); iron ore from 19 mines, 1,208,000 tons; manganese ore (4 mines), 45,550 tons; iron (all in Ekaterinoslav), 608,600 tons; steel, 51,100 tons; rails, 246,000 tons; iron goods, 50,000 tons. Besides, mercury, ore, and fireclay are extracted. Nearly 40,000 persons find occupation in factories, of which the iron works and the 34 agricultural machinery works are the most important, while in the district of Mariupol the making of agricultural implements and machinery has undergone a very great extension as a domestic industry in the villages. Grain is exported to a very large extent (about 950,000 tons, and 520,000 tons in transit), via the Dnieper, the Sebastopol railway, and the port of Mariupol. Nearly 4000 boats and 1000 rafts are loaded and unloaded every year, carrying over 2,000,000 tons of goods (value 13,000,000 roubles), and giving occupation to 27,000 men. The port of Mariupol is visited every year by about 80 steamers and 40 sailing vessels engaged in foreign trade, and 1600 vessels engaged in coasting. The government is better provided with schools, gymnasia, professional and primary, than many other governments of Russia. The chief towns of its 8 districts are: Ekaterinoslav (121,216 inhabitants), Alexandrovsk (16,393), Bakhmut (19,416), Mariupol (31,772), Novomoskovsk (12,862), Pavlograd (17,188), Slavyanoserbsk (3120), and Verkhnednyeprovsk (11,607).

(P. A. K.)

Ekaterinoslav, capital of the above province, on the right bank of the Dnieper, 607 miles by rail southsouth-west of Moscow. It is a rapidly growing town, well provided with schools, including a number of professional ones, and is an important depot for timber shipped down the Dnieper, as also for grain. Its iron works, steam flourmills, and agricultural machinery works give occupation to 5000 workers. Considerable trade is carried on in cattle, grain, and raw wool. Population (1861), 18,881, without suburbs; (1897), 121,216.

Elabuga, a district town of Russia, government and 279 miles south-east of Vyatka, on Kama river, near its

The famous Ananiinskiy Moghilnik (burial-place) is on the right bank of the Kama, 3 miles from above the town. It was discovered in 1858, was excavated by Alabin, Lerch, and Nevostruyeff, in charge of the Archæological Commission, and has since supplied extremely valuable collections of archæological objects It represented belonging to the Stone, Bronze, and Iron Ages. first a mound, about 500 feet in circumference, adorned with decorated stones (which have disappeared), and contained an inner wall, 65 feet in circumference, made of uncemented stone flags. Nearly 50 skeletons were discovered in it, mostly laid upon burned great variety of bronze decorations and glazed clay pearls were logs, surrounded with pots filled with partially burned bones. A strewn round the skeletons. The knives, daggers, and arrowpoints are of slate, bronze, and iron, the two last being very rough imitations of stone implements. On one of the funeral flags the image of a man, without moustaches or beard, dressed in a costume and helmet recalling those of the Circassians, was discovered. The exact period of these funeral mounds is not known.

Elastic Systems. In the article ELASTICITY (Ency. Brit. vol. vii.) there is an outline of the general mathematical theory of Elasticity, and an account of the results obtained by applying it to the solution of the very important problem of the torsion of prisms. The present article is meant to supplement the former one by giving an account of some of the results that have been arrived at in the application of the theory to other problems.

1. Flexure of a Beam.-The simple theory of the bending of a beam by applied couples has been explained in the original article, $$ 57-62. If M is the couple, the central line of the beam is bent into a circle of radius

EI/M, where E is the Young's modulus of the material,

and I is the moment of inertia of the cross-section about the axis through its centroid at right angles to the plane of flexure. This plane is supposed to contain one principal axis of inertia of each cross-section,1 and the locus of the perpendicular principal axes after flexure is a surface cutting the plane

of flexure at right angles and known as the "neutral surface"; the

traces of this surface are dotted in Fig. 1. The quantity EI is known

"flexural as the rigidity." The

Fig. 1.

applied couple M is balanced at any section by a couple arising from the stress exerted across the section. Supposing, to fix ideas, that the curve into which the central line is bent is concave downwards, the stress at any point above (below) the neutral surface is simple longitudinal tension (pressure) of amount per unit area equal to the product of M/I and the distance of the point from the neutral surface. The corresponding strain consists of extension (contraction) of the longitudinal fibres above (below) the neutral surface accompanied by lateral contraction (extension) of the perpendicular fibres, and the effect of the lateral strain is seen in the change of shape of the section producing the anticlastic curvature of the beam (cf. ELASTICITY, Ency. Brit. vol. vii. p. 809). The extension (contraction) of a longitudinal fibre distant y above (below) the neutral surface is My/EI, and the lateral contraction (extension) is oMy/EI, where σ is the "Poisson's ratio" of the material (a fraction nearly equal to for most hard solids). The displacements produced are—(1) a deflexion of the central line by which the centroid of each cross1 When this is the case the beam is said to be bent in a "" principal plane."

section comes to its proper place on the curved central | line; (2) a rotation of the plane of each cross-section about the axis through its centroid at right angles to the plane of flexure, of such an amount as to place it at right angles to the curved central line; (3) a distortion of the shape of each cross-section in its own plane producing the anticlastic curvature.

2. 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. 2), i.e., from the stresses across the section at p. 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 shearing

stresses at the cross-sections. The determination of the character of these, and of the corresponding strains and displacements, was effected by Saint-Venant and Clebsch for a number of forms of sections by means of an analysis of the same kind as that employed in the solution of the torsion problem.

3. Let 7 be the length of the beam, x the distance of the section p from the fixed end A, y the distance of any point below the horizontal plane through the centroid of the section at A, then the bending moment at p is W(1-x), and the longitudinal tension P at any point on the cross-section is W(1-x)y/I, and this is related to the bending moment exactly as in the simpler problem.

b

P

0

y Fig. 3.

YU

T

4. The expressions for the shearing stresses depend on the shape of the cross-section. Taking the beam to be of isotropic material and the cross-section to be an ellipse of semiaxes a and b (Fig. 3), the a axis being vertical in the unstrained state, and drawing the axis z at right angles to the plane of flexure, the vertical shearing stress U at any point (y, z) on any cross-section is 2W[(a2-y2){2a2(1+0)+b2} - 22a2(1-20)]. Ta3b(1+0) (3a2+b2)

The resultant of these stresses is W, but the amount at the centroid, which is the maximum amount, exceeds the average amount, W/rab, in the ratio

{4a2(1+0)+262)/(3a2+b2)(1+0).

The greatest value of U is in this case approximately twice its average value, but it is possible that these results for the bending of very thin tubes may be seriously at fault if the tube is not plugged, and if the load is not applied in the manner contemplated in the theory (cf. § 9). In such cases the extensions and contractions of the longitudinal fibres may be practically confined to a small part of the material near the ends of the tube, while the rest of the tube is deformed without stretching.

5. The shearing stresses U, T on the cross-sections are necessarily accompanied by shearing stresses on the longitudinal sections, and on each such section the shearing stress is parallel to the central line; on a vertical section z = const. its amount at any point is T, and on a horizontal section y = const. its amount at any point is U.

The internal stress at any point is completely determined by the components P, U, T, but these are not principal stresses. Clebsch has given an elegant geometrical construction for determining the principal stresses at any point when the values of P, U, T are known.

From the point O (Fig. 4) draw lines OP, OU, OT, to represent the stresses P, U, T at O, on the cross-section through O, in magnitude, direction and sense, and compound U and T into a resultant represented by OE; the

plane EOP is a principal plane of stress at O, and the principal stress at right angles to this plane vanishes. Take M the middle point of OP, and with centre M and radius ME describe a circle cutting the line OP in A and B; then OA and OB represent the magnitudes of the two remaining principal stresses. On AB describe a rectangle ABDC so that DC passes through E; then OC is the direction of the principal stress represented in magnitude by OA, and OD is the direction of the principal stress represented in magnitude by OB.

Fig. 4.

6. As regards the strain in the beam, the longitudinal and lateral extensions and contractions depend on the bending moment in the same way as in the simpler problem; but, the bending moment being variable, the anticlastic curvature produced is also variable. In addition to these extensions and contractions there are shearing strains corresponding to the shearing stresses T, U. The shearing strain corresponding to T consists of a relative sliding parallel to the central line of different longitudinal linear elements combined with a relative sliding in a transverse horizontal direction of elements of different cross-sections; the latter of these is concerned in the production of those displacements by which the variable anticlastic curvature is brought about; to see the effect of the former we may most suitably consider for the case of an elliptic crosssection the distortion of the shape of a rectangular portion

H

H

of a plane of the material which in the natural state was horizontal; all the boundaries of such a portion become parabolas of small curvature, which is variable along the length of the beam, and the particular effect under consideration is the change of the transverse horizontal linear elements from straight lines such as HK to parabolas such as H'K' (Fig. 5); the lines HL and KM are parallel to the central line, and the figure is drawn for a plane above the neutral plane. When the cross-section is not an ellipse the character of the strain is the same, but the curves are only approximately parabolic.

K'

K

Fig. 5.

M

and makes an angle s。 with the horizontal (see Fig. 8); it is, however, improbable that this condition is exactly realized in practice. In the application of the theory to the experimental determination of Young's modulus, the small angle which the central line at the support makes with the horizontal is an unknown quantity, to be eliminated by observation of the deflexion at two or more points.

8. We may suppose the displacement in a bent beam to be produced by the following operations: (1) the central line is deflected into its curved form, (2) the cross-sections are rotated about axes through their centroids at right angles to the plane of flexure so as to make angles equal tos with the central line, (3) each cross-section is distorted in its own plane in such a way that the appropriate variable anticlastic curvature is produced, (4) the crosssections are further distorted into curved surfaces. The contour lines of Fig. 7 show the disturbance from the central tangent plane, not from the original vertical plane.

9. Practical Application of Saint-Venant's Theory.— The theory above described is exact provided the forces applied to the loaded end, which have W for resultant, are distributed over the ter

minal section in a particular way, not likely to be realized in practice; and the application to practical problems depends on a principle due to Saint-Venant, to the effect that, except for comparatively small portions of the beam near to the loaded and fixed ends, the resultant only is effective, and its

not seriously affect the internal strain and stress. In

Fig. 8.

fact, the actual stress is that due to forces with the required

4W 2a2(1+0)+b2

Επαό 3a2+b2

for a circle with σ =, this becomes 7W/2Eπa2. The vertical filament through the centroid of any cross-section becomes a cubical parabola, as shown in Fig. 6, and the contour lines of the curved surface into which any crosssection is distorted are shown in Fig. 7 for a circular section. 7. The deflexion of the beam is determined from the equation

curvature of central line-bending moment÷flexural rigidity,

and the special conditions at the supported end; there 18 no alteration of this statement on account of the shears. As regards the special condition at an end which is encastrée, or built-in, SaintVenant proposed to assume that the central tangent plane of the crosssection at the end is vertical; with this assumption the tangent to the central line at the end is inclined downwards

Fig. 7.

theory, superposed upon that due to a certain distribution of forces on each terminal section which, if applied to a rigid body, would keep it in equilibrium; according to Saint-Venant's principle, the stresses and strains due to such distributions of force are unimportant except near the ends. For this principle to be exactly applicable it is necessary that the length of the beam should be very great compared with any linear dimension of its crosssection; for the practical application it is sufficient that the length should be about ten times the greatest diameter.

10. The theoretical determination of the stress in a bent beam under conditions as to load and support other than those considered in §§ 2-8 is attended by difficulties which have not yet been surmounted, but the equation for

the deflexion

curvature of central line-bending moment÷flexural rigidity

is sufficiently exact whenever the length is a considerable multiple of the greatest diameter of the crosssection. This result is indicated by the theories of indefinitely thin wires developed by Kirchhoff and Boussinesq, and has been confirmed by special researches made by Pochhammer and Pearson. The equation for the deflexion above written is the basis of the treatment of continuous beams resting on three or more supports and carrying distributed loads. The calculation of the bending moment can be replaced by a method of graphical construction, due to Mohr, and depending on the two following theorems :

(I.) The curve of the central line of each span of a

beam, when the bending moment M is given,1 is identical with the catenary or funicular curve passing through the ends of the span under a (fictitious) load per unit length of the span equal to M/EI, the horizontal tension in the funicular being unity.

(II.) The directions of the tangents to this funicular curve at the ends of the span are the same for all statically equivalent systems of (fictitious) load.

When M is known, the magnitude of the resultant shearing stress at any section is dM/dx, where x is measured along the beam.

11. Let be the length of a span of a loaded beam (Fig. 9), M1 and M, the bending moments at the ends, M the bending moment at a section distant x from the end (M1), M′ the bending moment at

3

12. When there is more than one span the funiculars in question may be drawn for each of the spans, and, if the bending moments at the ends of the extreme spans are known, the intermediate ones can be determined. This determination depends on two considerations: (1) the fictitious loads corresponding to the bending moment at any support are proportional to the lengths of the spans which abut on that support; (2) the sides of two funiculars that end at for the case of a uniform beam on three supports A, B, C, the ends any support coincide in direction. Fig. 11 illustrates the method A and C being freely supported. There will be an unknown bending moment M, at B, and the system of fictitious loads is wAB/EI at G the middle point of AB, BC/EI at G' the middle point of BC, MAB/EI at g and MBC/EI at g', where g and g' are the points of trisection nearer to B of the spans AB, BC. The centre of gravity of the two latter is a fixed point independent of Mo, and the line VK of the figure is the vertical through this point. We draw AD and CE to represent the loads at G and G' in magnitude; then D and E are fixed points. We construct any triangle UVW whose sides UV, UW pass through D, B, and whose vertices lie on the verticals gU, VK, g'W; the point F where VW meets DB is a fixed point,

M=M'+M+M2,

and thus a fictitious load statically equivalent to M/EI can be easily found when M' has been found. If we draw a curve (Fig. 10) to pass through the ends of the span, so that its ordinate represents the value of M'/EI, the corresponding fictitious loads are statically equivalent to a single load, of amount represented by the area of the

M1 A

N

Fig. 10.

M2

2

curve, placed at the

point of the span verB tically above the centre of gravity of this area. If PN is the ordinate of this curve, and if at the ends of the span we erect ordinates in the proper sense to represent M1/EI and M2/EI, the bending moment at any point is represented by the length PQ.2 For a uniformly distributed load the curve of M' is a parabola M'=wx(1-x), where w is the load per unit of length; and the statically equivalent fictitious load is w/EI placed at the middle point G of the span; also the loads statically equivalent to the fictitious loads M(1-x)/EI and Mx/EI are M1/EI and M2/EI placed at the points g, g' of trisection of the span. The funicular polygon for

two sides (2, 4) of the required funiculars which do not pass through A, B, or C. The remaining sides (1, 3, 5) can then be drawn, and the side 3 necessarily passes through B; for

M

A

1

Fig. 12.

5

E.

the triangle UVW and the triangle whose sides are 2, 3, 4 are in perspective.

The bending moment Mo is represented in the figure by the vertical line BH where H is on the continuation of the side 4, the scale being given by

BHM,BC CEBC

this appears from the diagrams of forces, Fig. 12, in which the oblique lines are marked to correspond to the sides of the funiculars to which they are parallel.

In the application of the method to more complicated cases there are two systems of fixed points corresponding to F, by means of which the sides of the funiculars are drawn.

13. Finite Bending of Thin Rod.-The equation curvature = bending moment÷flexural rigidity may also be applied to the problem of the flexure in a principal plane of a very thin rod or wire, for which the When the forces that procurvature need not be small.

duce the flexure are applied at the ends only, the curve into which the central line is bent is one of a definite family of curves, to which the name elastica has been given, and there is a division of the family into two species according as the external forces are applied directly to the ends or are applied to rigid arms attached to the

W

W

a

the fictitious loads can thus be drawn, and the direction of the central line at the supports is determined when the bending moments at the supports are known.

The sign of M is shown by the arrow-heads in Fig. 9, for which, with y downwards,

EI dx2

+M=0.

2 The figure is drawn for a case where the bending moment has the same sign throughout.

The length L of th curve between two inflexions corresponds to the time of oscillation of the pendulum from rest to rest, and we thus have L(W/EI)=2K,

where K is the real quarter period of elliptic functions of modulus

3 Mo is taken to have, as it obviously has, the opposite sense to that shown in Fig. 9.