It has been shown in Article 179, that if there be an inward radiating pressure upon a ring, of a given intensity per unit of arc, there is a thrust exerted all round that ring, whose amount is the product of that intensity into the radius of the ring. The same proposition is true, substituting an outward for an inward radiating pressure, and a tension all round the ring for a thrust. If, therefore, the horizontal radiating pressure of the dome at the joint CC be resisted by the tenacity of a hoop, the tension at each point of that hoop, being denoted by P,, is given by the equation

Now conceive the hoop to be removed to the circular joint D D, distant by the arc ds from CC, and let its tension in this new position be

P, -d Pr

The difference, d P,, when the tension of the hoop at CC is the greater, represents a thrust which must be exerted all round the ring of brickwork C C D D, and whose intensity per unit of length of the arc CD is

Every ring of brickwork for which p, is either nothing, or positive, is stable, independently of the tenacity of cement; for in each such ring there is no tension in any direction.

When p, becomes negative, that is, when P, has passed its maximum, and begins to diminish, there is tension horizontally round each ring of brickwork, which, in order to secure the stability of the dome, must be resisted by the tenacity of cement, or of external hoops, or by the resistance of abutments.

The inclination to

Such is the condition of stability of a dome. the horizon of the surface of the dome at the joint where p1 = 0, and below which that quantity becomes negative, is the angle of rupture of the dome; and the horizontal component of its thrust at that joint, is its total horizontal thrust against the abutment, hoop, or hoops, by which it is prevented from spreading.

A dome may have a circular opening in its crown. Oval arched openings may also be made at lower points, provided at such points there is no tension; and the ratio of the horizontal to the inclined axis of any such opening should be fixed by the equation

DOMES, SPHERICAL AND CONICAL.

267

Example I. Spherical Dome.-Uniform thickness, t; weight of material per unit of volume, w; radius, r.

x = r (1 − cos i); y = r sini; ds = rdv

and from this angle we obtain, for the horizontal thrust of the dome, per unit of periphery at the joint of rupture,

P1 = 0·382 w tr;

and for the tension on a hoop to resist that thrust,

P, = 0·3 w tr2.

(7.)

Example II. Truncated Conical Dome (fig. 113).—Apex, O. Depth of top of dome below apex, x; of base of dome, a; i, uniform inclination; t, uniform thickness; y = x cotan i.

P. being everywhere positive, there is in this dome no joint of rupture.

Example III. Truncated Conical Dome, supporting on its summit • turret or "lantern," of the weight L

235. Strength of Abutments and Vaults.—The dimensions required in an abutment, arch, or dome, to insure stability, are in most cases sufficient to insure strength also; but instances occur, in which the condition of sufficient strength requires to be independently considered, and it may be convenient here so far to anticipate the subject of strength as to state that condition, viz., that the intensity of the thrust in the materials shall at no point exceed a certain limit, found by dividing the resistance of the material to crushing by a number called the factor of safety. The factor of safety in existing bridges ranges from 3 or 4 to 50 and upwards. In tunnels it is about 4. Tredgold considers, that in bridges the best value for the factor of safety is about 8 (Treatise on Masonry). The resistance of some of the most important materials of masonry to crushing is stated in a table at the end of this volume; but a prudent engineer, who contemplates a great work in masonry, will not trust to tables alone, but will ascertain the strength of the materials at his command by direct experiment.

235 A. Transformation of Structures in Masonry. The principle already stated in Article 126, that to determine the intensity of a force in a transformed structure, the projected line representing the amount of the force must be divided by the projected area over which it is distributed, requires special attention in considering the strength of transformed structures of masonry.

To exemplify the application of that principle, conceive a rectangular prism whose dimensions are x, y, z, x being vertical: its volume is V = xyz. Let w be the weight of unity of volume of the material of which it is composed; and let the weight of the prism be represented by a line parallel to x, of the length W; then

W = wxyz.

The amount of an upward vertical pressure on the base of this prism, which balances W, will be represented by a line equal and opposite to W: that is

Now let there be a parallel projection of this prism, whose dimensions, xa x, y =by, z = cz, are oblique to each other. The weight of the new prism will be represented by a line parallel to x', of the length

Let

C=1

Λ

W' = a W.......

.(4.)

cos2 y' z'
x

- cos2x^y

Λ

(5.)

+ 2 cos y z⋅ cos zx cos x'y'.

Then the volume of the new prism is

V' = x' y' z √ C = V · a b c √ C ;......................(6.)

consequently the intensity of its weight is

W'
V

a b c √ C. V bc √ C

The area of the lower surface of the new prism is

.(7.)

being represented by a line P', which is the projection of P, and parallel to a'.

and if we consider the relation between stress and weight,

that is,

we find

(10.)

p' y z' sin y`x' — — w x y z √..............(11.).

270

CHAPTER III.

1

STRENGTH AND STIFFNESS.

SECTION 1.-Summary of the Theory of Elasticity as applied to
Strength and Stiffness.

236. The Theory of Elasticity relates to the laws which connect the stresses, or pressures and tensions, which act at the surface and in the interior of a body, with the alterations of dimensions and figure which the body and its parts simultaneously undergo. That theory, therefore, is the foundation of the principles of the strength and stiffness of materials of construction. The theory of elasticity has many other applications, to crystallography, to light, to sound, to heat, and to other branches of physics. Its full discussion would of itself require a voluminous work; in the present section, its principles are to be briefly summed in so far as they are applicable to the strength and stiffness of structures.

237. Elasticity is the property which bodies possess of occupying, and tending to occupy, portions of space of determinate volume and figure, at given pressures and temperatures, and which, in a homogeneous body, manifests itself equally in every part of appreciable magnitude.

238. An Elastic Force is a force exerted between two bodies at their surface of contact, or between two parts into which a body either is divided or is capable of being divided at the surface of actual or ideal separation between those parts. The intensity of an elastic force is stated in units of weight per unit of area of the surface at which it acts. That kind of force is in fact identical with stress, the statical laws of which have already been explained in Part I., Chapter V., Sections 2, 3, and 4, Articles 86 to 126.

239. Fluid Elasticity. The elasticity of a perfect fluid is such that its parts resist change of volume only, and not change of figure; whence it follows, that the pressure exerted by a perfectly fluid mass is wholly perpendicular to its surface at every point: principles which form the basis of hydrostatics and hydrodynamics. Fluids are either gaseous or liquid. A gaseous fluid is one whose parts (so far as is known by experiment) exert a pressure against