other members. The stress on a particular member can always be found if the section taken does not cut more than three bars. If three or more bars are cut, the stress on any bar can be found if the stress on one or more of the bars cut is known; thus in Fig. 18 the section 00 cuts four bars, and we wish, for example, to find the stress CZ1. The stress in the bar AB has been found

.. 6·16CZ1 + 2436 × 20 + (– 1504 × 10·8) = 0

The method of sections would be applied in the same manner to obtain the stresses in more complicated roof and bridge. trusses; thus, if Fig. 19 represents a portion of a bridge truss, the stresses in the three bars cut may obviously be found by taking moments about suitable points; but if the points 3 4 be joined the section cuts four bars, and the method fails. The structure in this case is said to be redundant, as it is indeformable without the line 3 4. If both the bars 3 4 and 2 5 are omitted the structure is deformable, and may alter its shape without altering the length of the members 23, 35, 54, and 43. Generally, if n denote the number of sides of a closed figure, such as 2 3 4 5, and v the number of vertices or corners, then it may be proved that

FIG. 19.

For a deformable figure, n < 2v - 3

For an indeformable non-redundant figure, n = 2v

For a redundant figure, n>2v - 3

Every bridge or roof-truss should be divided up into triangular spaces, so that for every enclosed space bounded by lines, n = 2v - 3.

The stresses in braced structures, such as the roof-truss illustrated in Fig. 125, may be found graphically by means of a reciprocal figure. Two figures are said to be reciprocal if for every side of the one there is a corresponding side in the other. Corresponding sides are parallel, perpendicular, or inclined at some constant angle to each other. To every system of lines meeting at a point in the one figure there is a corresponding closed polygon in the other. A reciprocal figure can always be drawn if the original figure is indeformable and non-redundant.

Take, for example, the roof-truss illustrated in Fig. 125. Draw the force polygon to scale; in this case it becomes a straight line. The convenience of Bow's notation will now be more apparent. The reciprocal figure, Fig. 126, is drawn by commencing at K in the force polygon, and drawing the line KE parallel to the line KE in the roof-truss, Fig. 125, and drawing ZE parallel to ZE in Fig. 125 to meet it in E. Then, since KZ, Fig. 125, represents the reaction at the support 7, which is in equilibrium with the stresses KE, and EZ1, the triangle KEZ, Fig. 126, is the triangle of forces, and the length of the line KE, measured to the same scale as the force polygon, represents the stress KE,; also the length of the line ZE, Fig. 126, represents the stress ZE1, and the stress in every other member of the truss is found by completing the reciprocal figure in the manner shown, and measuring the length of each line in Fig. 126 to find the stress in the corresponding member of the roof-truss, Fig. 125. Every line in Fig. 126 is parallel to the corresponding line in Fig. 125, and, completing the figure from D by drawing DE parallel to DE, Fig. 125, and so on, we prove the accuracy of the work by the closing line AH, Fig. 126, being parallel to AH, Fig. 125. It will be noticed that for every space in Fig. 125 there are three lines diverging from a point in Fig. 126 having the same letter against it as the space; thus the space C, Fig. 125, bounded by the lines CB, CD, CZ is represented in Fig. 126 by the three diverging lines radiating from C. The determination of the character of the stresses is second only in importance to the determination of the stresses themselves. In Fig. 126 the thick lines denote the compressive stresses, and the thin lines the tensile, which are determined as follows::

The forces 2436, KE, and EZ, Fig. 125, are in equilibrium, and are represented by the closed polygon KEZ, Fig. 126. Follow round the polygon, Fig. 126, starting with the known upward reaction ZK = 2436, and indicate the direction of the forces by arrows as shown; put arrows on the corresponding stresses in Fig. 125, then it is seen that KE acts towards the point 7, and is therefore compressive; EZ acts away from it, and is tensile. If EK is produced to the right, it is seen that the stress must be tensile. Hence the following rule: Take any apex of the frame as a system of forces in equilibrium. Follow round the polygon formed by these forces in the direction

indicated by those forces whose direction is already known, and transfer the directions thus obtained from the forces to the apex under consideration. If the stress acts towards the apex, the piece is in compression; if away from the apex, the piece is in tension. The results found by measuring the lines to the proper scale in Fig. 126 may now be compared with those obtained by the method of sections. The principle of the resolution of forces may be applied algebraically by using the general equations of equilibrium, and the stresses determined by means of the method of sections and also by means of the reciprocal figure might have been determined by means of these equations. The objection to the algebraic method in such cases is the necessity of measuring all the angles which the bars make with their horizontal and vertical components. The method is too tedious when applied to a single roof or other truss, but may be useful when applied to a number of such trusses of similar angles, but with bars of different length.

CHAPTER IV.

BENDING MOMENTS AND SHEARING STRESSES.

THE action of transverse loads in producing stresses in beams may be conveniently studied with reference to a beam fixed at one end and loaded at the other.

Fig. 20 shows such a beam fixed by building it into a wall, and loaded at the extremity with a load W. The deflection or

bending produced by

the load is exaggerated for the sake of clearness, and it may be observed that the fibres on the convex side of the beam are lengthened, while those on the concave side are short

FIG. 20.

Tension

Neutral Layer
Compression

ened. The upper part of the beam is consequently in tension, and the lower part is in compression, while, somewhere between, tension must change to compression. The layer of fibres at which the change from tension to compression occurs is called the neutral layer, as it is neither lengthened nor shortened by the bending of the beam, and is therefore not subjected to either tension or compressive stresses.

If the beam is supported at both ends and loaded in any manner by vertical loads between the points of support, the tensile and compressive stresses developed in the beam will be as indicated in Fig. 21.

Besides the direct tensile and compressive stresses which are developed in the beam in consequence of the bending, there are other stresses which may be realized by clamping together a

number of planks, such as would be obtained by making horizontal sections of the beam shown in Fig. 21, and loading in a

Compression
Neutral Layer

Tension

FIG. 21.

similar manner. It

will then be observed that each section slides upon the other, and overlaps in the manner indicated in Fig. 22;

hence we conclude that there are horizontal stresses developed in a beam when loaded which tend to make the various

FIG. 22.

layers slide upon each other. Again, if, in the beam shown in Fig. 20, we imagine two forces applied at any section ab, each

equal to W, and acting in opposite directions so as to neutralize each other, it is clear that the three forces are equivalent to a couple tending to produce rotation of the beam to the right of the section ab, and an unbalanced force W.

Hence we conclude that there are vertical stresses developed in a loaded beam which tend to make the layers slide upon each

other in vertical planes, as indicated in Figs. 23 and 24. These horizontal and vertical stresses are termed shearing stresses.

The stresses produced in beams subjected to transverse loads may be summarized as follows:

(a) Direct tensile and compressive stresses. (b) Horizontal and vertical shearing stresses.

The term "bending moment" has been defined in Chapter III. The bending moments and shearing stresses for the various cases which occur frequently in connection with the construction of ordinary beams and girders will now be considered.

Case I., Figs. 25, 26, and 27.-A beam fixed at one end, termed a cantilever, is loaded at the other with a load denoted by W. This and the following problems may be solved by applying Culman's principle, but it is proposed to treat them algebraically as being rather more convenient for this class of problem. Let x denote the distance from the extremity to the