I now come to what may be called the process of substitution of forces: we have certain natural forces of which the directions are known, and for these we have to substitute equivalent forces going in some other directions, in which they may be met by the resistances of the materials used in the structure opposed to them.

Setting aside for the present lateral wind pressures and the like, the natural forces arising to be dealt with will proceed from gravitation, and hence will primarily be vertical in direction. The fact of these forces acting vertically affords great convenience for the description of the directions of parts of structures relatively to the original forces, as by the amount of elevation corresponding to a given horizontal extension: thus if two points be horizontally 3 feet apart, but one is 2 feet higher than the other, the inclination will be 2 to 3. These quantities being known, the distance between the points, and therefore the length of the element joining them, may be calculated from the property of a right-angled triangle, which is that in any rightangled triangle the sum of the squares of the sides enclosing the right angle is equal to the square of the remaining side.

a

W

b

It will readily be seen that if a structure be composed of bars jointed together so as to form a rigid combination, and then force be applied to it tending to distort it, such force will be split up into others, which will pass along and be resisted by the elastic strength of the component bars of the frame. If in Fig. 8, a b, a c represent two bars attached at b c to a solid mass, and a weight W is hung on at the joint a, the force due to the gravitation of this weight will be

Fig. 8.

с

split up into two others, one tending to pull a b away from and the other tending to thrust a c into the sustaining mass. The relations of the intensities of these strains to the weight W may be determined by the parallelogram of forces, and so the materials proportioned to sustain them. If the bars a b, a c, instead of being attached to a solid mass as shown, were connected to another framework of bars, there would be a further splitting up, or resolution of strains to be determined in a similar manner, and so we may go on, commencing with the primary load, and trace the resulting strains through all the bars of the most complicated structures up to the point where the strains pass away into the earth. By drawing the skeleton of centre lines of a framed structure, the diagram of strains will be formed, and the relations of the strains will be seen to vary with the relations of length of the various elements; but to this subject I shall return to deal particularly with the application of the methods to practical cases.

Having now acquired a clear conception of the manner in which forces internal and external come into action and produce definite results, the way is clear for the application of this knowledge to special cases, and for the elaboration of a series of definite formulæ or rules for the ordinary requirements of constructive art, and in the first place I shall treat of structures which depend upon the elastic strength of the materials composing them for their security and permanence.

CHAPTER III.

BENDING STRESS.

As the centres of gravity of surfaces and bodies will now constantly be referred to, a few words on the subject now may facilitate the subsequent operations.

The centre of gravity of a surface or solid is that point on which it balances exactly, and is the point about which the moments of weight of all the molecules in one direction will equal the moments of weight of all the molecules in the opposite direction. There can evidently be only one such point in any body or surface. A surface may, however, be balanced upon a knife-edge in more directions than one, but all the lines of direction thus found will intersect each other at the centre of gravity of the surface. The centre of gravity is the point at which we may consider the whole weight or effect of the surface or body as concentrated; and if any figure is freely suspended from a point, the centre of gravity will hang vertically under

1

44

2

such point. This fact furnishes a simple method by which the centre of gravity of any figure, however irregular, may be ascertained. Let it be required to find the centre of gravity of an irregular figure, abc, Fig. 9. Cut the figure out in paper or Bristol board; pierce holes at two corners, as at a and c. Suspend the figure on a pin stuck in a

Fig. 9.

vertical board, and on the same pin hang a plumb line de, and when it has become steady mark its direction and draw it on the figure, then the centre of gravity must be somewhere in the line thus drawn. Now suspend (as at 2) the figure by the hole at c, again apply the plumb line d e, and where it intersects the line already drawn will be the centre of gravity g.

b d

a

There are many, most in fact, of the forms with which we shall have to deal, of which the centre of gravity may readily be found without having recourse to this tentative method. Take a triangle, a b c, Fig. 10, for instance; this triangle may be regarded as being made up of indefinitely narrow strips parallel to bc, then each strip would balance on its centre; so if the side b c be bisected in e and a e joined, a e is a line passing through the centres of gravity of all the parts, and therefore through that of the whole system. Similarly, by bisecting a b in d and

f

h

m

0

n

j

K

Fig. 10.

76

joining e d, another line is found passing through the centre of gravity of the system; hence the intersection of these two lines determines the centre of gravity g of the triangle.

If a surface, then, is of symmetrical form about a rectilinear axis, its centre of gravity is somewhere in such axis, and if there be two axes of symmetry the centre of gravity will be at their intersection.

The centre of gravity of an area such as fg kj may be determined by finding the sum of the moments of the separate constituent areas about any convenient axis, and dividing it by the sum of the area, the quotient being the distance of the centre of gravity of the system from the axis chosen.

Let the dimensions of the figure be fg=15 inches, 7 m =9 inches, j k=2.5 inches, the other dimensions being as figured, and the axis selected about which to determine the moments be the boundary fg. The centre of gravity of each symmetrical element being its physical centre. Then the moment of the part

fgik areax distance from axis=(15X2)x1 =30.00 l m q p = " X nojk,, X

= (9×1)×2=22.50 =(2·5×3)×4 × 33.75

86.25

86.25

86.25

=

1.855 inches-distance

(15X2)+(9x1)+(2·5x3) 46-5

of centre of gravity of the whole figure from the axis fg. This term, centre of gravity, of course applies strictly to the centre of action of the force of gravitation, but custom has caused it to be used in speaking of the centres of forces other than that of gravity, to which its position applies, such as the centre of gravity of a resisting area of material; thus we say the centre of gravity of the flange of a girder, when that point is referred to in relation to the tensile or compressive stresses on the area of which it is centre of gravity.

The weights or loads producing strain are generally of geometrical form, having a centre of figure which, if the bodies be of uniform or homogeneous constitution, is also the centre of gravity; and if the load be not of such a character, it must be, for purposes of calculation, divided up into smaller parts, of which the centres of gravity may be observed.

I should advise the student to take for practice at haphazard a number of figures, and having determined the centres of gravity by calculation, to check them by the tentative method, in order to give him facility in working and confidence in his results.