The following tests made by the author were on specimens 6 inches wide by 4 inches deep, on a span of 4 feet :

Lanza recommends that the modulus of rupture which may be safely assumed in practice should, for spruce and white-pine beams, be 3000 lbs. per square inch, but that for carefully selected timber we may take 4000 lbs. per square inch. For yellow pine he recommends 5000 lbs. per square inch, and that the factor of safety should be four, in building construction.

Lanza also found, by loading beams for a considerable time, that the long-continued application of a load may produce two or more times the deflection produced by the load when first applied, and that the modulus of elasticity that may be safely assumed for a beam that may have to carry its load for a considerable time should be from one-half to two-thirds the modulus obtained from quick tests: thus for spruce, he recommends that the modulus of elasticity should be assumed at from 666,300 to 888,300, and for yellow pine from 878,950 to 1,171,930 lbs. per square inch. He further considers that, although weakness in timber may be due to sap-woods, seasoncracks, and decay, by far the most frequent cause of weakness is the presence of knots, which in most cases determine the position of fracture; again, that it is not safe to rely on any extra strength due to seasoning.

The author recommends that for well-seasoned Australian timber the modulus of rupture used for 12" x 12" beams should

be 25 per cent. less than that recorded in the table-this recommendation is based on a few experiments on beams of this size of ironbark, red gum, and box-and that the factor of safety be taken as four for ordinary building construction, but that for bridges it should be eight for the live and four for the dead load. In view of Lanza's time-tests, the modulus of elasticity used in computing the deflections of beams should be taken at one-half that recorded in the tables. The strength and durability of Australian timber are greater for winter than for summer felled timber, and the strength is greater in seasoned than in unseasoned timber; but experiments prove that natural seasoning is better than either kiln-drying or steaming. The average life of timber bridges in Australia is about 25 years.

Generally the best timber is that which has grown slowly upon a soil rather dry than moist, and it is compact and heavy, the annual rings being narrow and uniform. Timber should show a hard, clear surface when cut, and should be free from clefts, radial cracks, cup-shakes, or cracks between the annual rings. The trees should be felled either in mid-summer or in mid-winter, when the sap is quiet; the latter is preferable. Timber beams should not be built into walls, or otherwise subjected to imperfect ventilation, and should be kept dry, although some timbers stand well when kept constantly wet; but most timbers decay rapidly when exposed so as to be wet and dry alternately.

Several different methods are in use for the preservation of timbers. Kyan's process consists of injecting corrosive sublimate (perchloride of mercury) into the pores. In Burnett's process chloride of zinc is used. In Boucherie's method sulphate of copper, while the Bethell process consists of saturating the timber with creosote. In these operations the air is exhausted from the tank in which the timber is placed, the sap drawn out from the pores, and the solution forced in under pressure.

A perfect antiseptic should be insoluble in water, and nonvolatile creosote appears to be the best, but enormous pressures would be necessary to force it into the hard woods of Australia. It is largely used for sleepers of soft timber.

CHAPTER III.

RESULTANT OF ANY NUMBER OF FORCES-GRAPHICAL REPRESENTATION OF MOMENTS OF FORCES-THE METHODS OF DETERMINING THE

STRESSES IN STRUCTURES.

THE determination of the stresses in structures such as trusses or girders is based upon the two following principles: (a) the principle of the resolution of forces; (b) the principle of moments. The former may be stated thus: If any number of forces in the same plane act at a point, or at different points, of a rigid body, and are in equilibrium, the algebraic sum of all their components in any direction is zero. That is to say, the tendency to move the body in one direction is exactly balanced by an equal tendency in the opposite. The moment of a force about a given point may be defined as the product of the force and the perpendicular distance from the point to the line of action of the force. The second principle may then be stated thus: If any number of forces in the same plane act at a point, or at different points, of a rigid body, and are in equilibrium, the algebraic sum of the moments of these forces, taken with reference to any point in their plane, is zero. That is to say, the tendency to produce rotation of the body in one direction is exactly balanced by an equal tendency to produce rotation in the opposite direction. These two fundamental principles give rise to two methods of calculation, and each may be applied analytically or graphically, but it will generally happen in any particular problem that one method offers advantages in simplicity over the others. In order to illustrate these principles, let P1, P2, P3, P4 represent forces acting in the same vertical plane, 01, 02, 03, 04 denote the angles they make with the horizon; let x1, x2, x3, x4 denote the perpendicular distances from any particular point in the plane

of the forces to the lines of action of the forces-that is to say, the moment of a force P, about a point O situated at a perpendicular distance x1, from it is P'x

1

Fig. 7 represents four forces in equilibrium, and the lines

are assumed to represent the forces in magnitude and direction, and the arrow-heads indicate the sense of each particular force. Now, the first principle may be stated thus

P1 sin 0, + P2 sin 02 + P3 sin 03 + P1 sin 0,= 0, or ΣY = 0

1

P1 cos 01+P2 cos 02 + P3 cos 03 + P, cos 0, 0, or ΣX = 0

2

The second principle may be stated thus—

=

P11 + P22+ P33+ P11 = 0, or ΣPx = 0

The first principle is applied graphically by drawing a polygon, termed the "force polygon," the sides of which are parallel to the forces and equal in length to the magnitude of the forces, as shown in Fig. 8.

This polygon will close, as the forces are in equilibrium. If the polygon does not close, the system of forces is not in equilibrium, and must have a resultant; thus Fig. 9 shows 7B

a system of forces not in equilibrium, and Fig. 10 the incomplete. polygon.

The closing line, shown dotted, represents the resultant force in magnitude, and is parallel to it in direction; but it does not show its line of action, and, in order to discover this, we proceed as follows:

Take any point O in Fig. 10 in the plane of the forces, which we will call the pole, and join it to the angular points of the polygon as shown in the dotted lines; draw lines in Fig. 9

D

parallel to the dotted lines radiating from O in Fig. 10, commencing at any point a on the line of action of the force P1; thus a b, Fig. 9, is parallel to the line from 0 to the join of P1 and P2, Fig. 10; be is parallel to the line from O to the join of P2 and P3, the points b and c lying on the forces P2 and P3 respectively. To find the point of application of the resultant R, we draw ad and cd parallel to the remaining lines in Fig. 10, viz. from O to the join of P, and R, and from O to the join of P, and R. The figure abed, Fig. 9, is termed the "funicular polygon." As the pole O, Fig. 1, may be selected anywhere in the plane, we have an infinite number of funicular polygons all passing through the point a, Fig. 9. Again, a may be taken anywhere in the line of action of the force P1, thus producing an infinite number of funicular polygons for the infinite number of positions of a, while the pole O, Fig. 10, remains unchanged. But whatever the position of the pole or the starting-point a, the intersection d, giving the position of the resultant, always falls on the line of action of the resultant. If the system of forces is in equilibrium, the funicular as well as the force polygon must close. The second principle is therefore briefly expressed by stating that for forces in equilibrium the funicular polygon must close.

Culman's Principle.-Suppose we have a single force P acting at A, Fig. 11. The force polygon in this case is represented by the straight line mHn, where mn represents the magnitude of

the force to scale, Fig. 12. Select a pole O, and join it with m and n, which is equivalent to resolving the force P in two directions. Draw through the point A, Fig. 11, two lines parallel to 81, and so, Fig. 12, and draw a line OH perpendicular to mn; the distance OH is called the polar distance. Draw xy perpendicular to the direction of P, Fig. 11, and draw lines ab, cd, ef, parallel to P. Then the moment of the force P with reference to any point is P x Ah. But in Fig. 12 we have by similar triangles