immersed plane. Let x = BE be the depth to which the lower edge of this plane is immersed below OY. From B draw BD = BE, and BF; produce the plane BF till it cuts the horizontal plane of no pressure, OY, in the line represented in section by O; through O and D draw a plane OH D, and conceive the prism BDHF to stand normally upon the base B F and to be bounded above by the plane D H. The pressure on the plane BF will be normal; its amount will be equal to the weight of fluid contained in the volume BDHF; that is to say, let a denote the depth of the centre of gravity of the plane BF below OY, and w the weight of unity of the volume of liquid; then the mean intensity of the pressure on BF is

(1.)

Let C be the centre of gravity of the volume BDHF; then the centre of pressure of the surface B F is the point where it is cut by the perpendicular CP let fall on it from C.

As the intensity of the pressure on any point of BF is proportional to its depth below OY, and consequently to its distance from O, this is a case of uniformly varying stress, and the formulæ of Article 94 are applicable to it. In the application of those formulæ it is to be observed, that the ordinates y are to be measured horizontally in the plane BF, whose centre of gravity is to be taken as the origin; that the co-ordinates x are to be measured in the same plane, along the direction of steepest declivity, and reckoned positive downwards; and that the value of the constant a in the equations of Article 94 is given by the formula

where is the angle of inclination of the plane B F to a horizontal plane.

Y

с

125. Pressure in an Indefinite Uniformly Sloping Solid.—Conceive a mass of homogeneous solid material to be indefinitely extended laterally and downwards, and to B be bounded above by a plane surface, making a given angle of declivity with a horizontal plane. In fig. 63, let YOY represent a vertical section of that upper sloping surface along its direction of greatest declivity, and OX a vertical plane perpendicular to the plane of vertical

B

Fig. 63.

PARALLEL PROJECTION OF STRESS AND WEIGHT.

127

Let w be the uniform

section which is represented by the paper. weight of unity of volume of the substance. Let BB be any plane parallel to, and at a vertical depth x below the plane Y Y. If the substance is exposed to no external force except its own weight, the only pressure which any portion of the plane B B can have to sustain is the weight of the material directly above it. Hence follows—

THEOREM I. In an indefinite homogeneous solid bounded above by a sloping plane, the pressure on any plane parallel to that sloping surface is vertical, and of an uniform intensity equal to the weight of the vertical prism which stands on unity of area of the given plane.

The area of the horizontal section of that prism is cos e, consequently, the intensity of the vertical pressure on the plane B B at the depth x is

P2 = wx cos .............

..(1.) From the above theorem, combined with the principle of conjugate stresses of Article 101, there follows

THEOREM II. The stress, if any, on any vertical plane is parallel to the sloping surface, and conjugate to the stress on a plane parallel to that surface.

Consider now the condition of a prismatic molecule A, bounded above and below by planes B B, C C, parallel to the sloping surface Y Y, and laterally by two pairs of parallel vertical planes. Let the common area of the upper and lower surfaces of this prism be unity, and its height Ax; then its volume is Ax cos e, and its weight w Ax cos e, which is equal and opposite to, and balanced by the excess of the vertical pressure on its lower face above the vertical pressure on its upper face. Therefore, the pressures parallel to the sloping surface, on the vertical faces of the prism, must balance each other independently; therefore they must be of equal mean intensity throughout the whole extent of the layer between the planes B B, CC; whence follows

THEOREM III. The state of stress, at a given uniform depth below the sloping surface, is uniform.

126. On the Parallel Projection of Stress and Weight. In applying the principles of parallel projection to distributed forces, it is to be borne in mind that those principles, as stated in Chapter IV., are applicable to lines representing the amounts or resultants of distributed forces, and not their intensities. The relations amongst the intensities of a system of distributed forces, whose resultants have been obtained by the method of projection, are to be arrived at by a subsequent process of dividing each projected resultant by the projected space over which it is distributed.

Examples of the application of processes of this kind to practical questions will appear in the Second Part.

128

CHAPTER VI.

ON STABLE AND UNSTABLE EQUILIBRIUM,

127. Stable and Unstable Equilibrium of a Free Body.—Sup pose a body, which is in equilibrio under a balanced system of forces, to be free to move, and to be caused to deviate to a small extent from its position of equilibrium. Then if the body tends to deviate further from its original position, its equilibrium is said to be unstable; and if it tends to return to its original position, its equilibrium is said to be stable.

Cases occur in which the equilibrium of the same body is stable for one kind or direction of deviation, and unstable for another. When the body neither tends to deviate further, nor to recover its original position, its equilibrium is said to be indifferent.

The solution of the question, whether the equilibrium of a given body under given forces is stable, unstable, or indifferent, for a given kind of deviation of position, is effected by supposing the deviation made, and finding the resultant of the forces which act on the body, altered as they may be by the deviation, in amount, in position, or in both. If this resultant acts towards the same direction with the deviation, the equilibrium is unstable—if towards the opposition direction, stable-and if the resultant is still nothing, the equilibrium is indifferent.

The disturbance of a free body from a position of stable equilibrium causes it to oscillate about that position.

128. Stability of a Fixed Body.-The term "stability," as applied to the condition of a body forming part of a structure, has, in most cases, a meaning different from that explained in the last Article, viz., the property of remaining in equilibrio, without sensible deviation of position, notwithstanding certain deviations of the load, or externally applied force, from its mean amount or position. Stability, in this sense, forms one of the principal subjects of the second part of this treatise.

PART II.

THEORY OF STRUCTURES.

CHAPTER I.

DEFINITIONS AND GENERAL PRINCIPLES.

129. Structures-— Pieces—Joints.-Structures have already, in Article 15, been distinguished from machines. A structure consists of two or more solid bodies, called its pieces, which touch each other, and are connected at portions of their surfaces called joints. 130. Supports-Foundations.-Although the pieces of a structure are fixed relatively to each other, the structure as a whole may be either fixed or moveable relatively to the earth.

A fixed structure is supported on a part of the solid material of the earth, called the foundation of the structure; the pressures by which the structure is supported, being the resistances of the various parts of the foundation, may be more or less oblique.

A moveable structure may be supported, as a ship, by floating in water, or as a carriage, by resting on the solid ground through wheels. When such a structure is actually in motion, it partakes to a certain extent of the properties of a machine; and the determination of the forces by which it is supported requires the consideration of dynamical as well as of statical principles; but when it is not in actual motion, though capable of being moved, the pressures which support it are determined by the principles of statics; and it is obvious that they must be wholly vertical, and have their resultant equal and directly opposed to the weight of the structure. 131. The Conditions of Equilibrium of a Structure are the three following:

1. That the forces exerted on the whole structure by external bodies shall balance each other. The forces to be considered under this head are-(1.) the Attraction of the Earth, that is, the weight of the structure; (2.) the External Load, arising from the pressures exerted against the structure by bodies not forming part of it nor of its foundation; (these two kinds of forces constitute the gross or total load; (3.) the Supporting Pressures, or resistance of the foundaThose three classes of forces will be spoken of together as the External Forces.

tion.

K

II. That the forces exerted on each piece of the structure shall balance each other. These consist of—(1.) the Weight of the piece, and (2.) the External Load on it, making together the Gross Load; and (3.) the Resistances, or stresses exerted at the joints, between the piece under consideration and the pieces in contact with it.

III. That the forces exerted on each of the parts into which the pieces of the structure can be conceived to be divided shall balance each other. Suppose an ideal surface to divide any part of any one of the pieces of the structure from the remainder of the piece; the forces which act on the part so considered are—(1.) its weight, and (2.) (if it is at the external surface of the piece) the external stress applied to it, if any, making together its gross load; (3.) the stress exerted at the ideal surface of division, between the part in question and the other parts of the piece.

132. Stability, Strength, and Stiffness.—It is necessary to the permanence of a structure, that the three foregoing conditions of equilibrium should be fulfilled, not only under one amount and one mode of distribution of load, but under all the variations of the load as to amount and mode of distribution which can occur in the use of the structure.

Stability consists in the fulfilment of the first and second conditions of equilibrium of a structure under all variations of load within given limits. A structure which is deficient in stability gives way by the displacement of its pieces from their proper positions.

Strength consists in the fulfilment of the third condition of equilibrium of a structure for all loads not exceeding prescribed limits; that is to say, the greatest internal stress produced in any part of any piece of the structure, by the prescribed greatest load, must be such as the material can bear, not merely without immediate breaking, but without such injury to its texture as might endanger its breaking in the course of time.

A piece of a structure may be rendered unfit for its purpose not merely by being broken, but by being stretched, compressed, bent, twisted, or otherwise strained out of its proper shape. It is necessary, therefore, that each piece of a structure should be of such dimensions that its alteration of figure under the greatest load applied to it shall not exceed given limits. This property is called stiffness, and is so connected with strength that it is necessary to consider them together.

From the foregoing considerations, it is evident that the theory of structures may be divided into two divisions, relating, the first to STABILITY, or the property of resisting displacement of the pieces, and the second to STRENGTH and STIFFNESS, or the power of each piece to resist fracture and disfigurement.