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THEORY or STRUCTURES]
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MECHANICS


because the resistances to displacement are the effect of a strained state of the pieces, which strained state is the effect of the load, and when the load is applied the strained state and the resistances produced by it increase until the resistances acquire just those magnitudes which are sufficient to balance the load, after which they increase no further.

This principle of least resistance renders determinate many problems in the statics of structures which were formerly considered indeterminate.

§ 7. Relations between Polygons of Loads and of Resistartces.—In a structure in which each piece is supported at two joints only, the well-known laws of statics show that the directions of the gross load on each piece and of the two resistances by which it is supported must lie in one plane, must either be parallel or meet in one point, and must bear to each other, if not parallel, the proportions of the sides of a triangle respectively parallel to their directions, and, if parallel, such proportions that each of the three forces shall be proportional to the distance between the other two, -all the three distances being measured along one direction. Considering, in the first place, the case in which the load and the two resistances by which each piece is balanced meet in one point, which may be called the centre of load, there will be as many such points of intersection, or centres of load, as there are pieces in the structure; and the directions and positions of the resistances or mutual pressures exerted between the pieces will be represented by the sides of a polygon joining

those points, as in fig.

86 Wl'l€I'€ Pl, Pg, P3,

P4 represent the centres

of load in a struc—

ture of four pieces,

and the sides of the

polygon og resistances

P1 P2 P3 4 represent

respectively the directions

and positions

of the resistances exerted

at the joints.

y let PL represent the

magnitude and direction of the gross load, and Pa, Pb the two resistances by which the piece to which that load is applied is supported; then will those three lines be respectively the diagonal and sidesof a parallelogram; or, what is the same thing, they will be equal to the three sides of a triangle; and they must be in the same lane, although the sides of the polygon of resistances may be in diligerent lanes.

p According to a well-known principle of statics, because the loads P7 bf lla' P5

3 L1 as

/'l "' bs

¢1~

Pt

L4




1'4"'~~lb4

Fig. 86.

Further, at an one of the centres of load or external pressures P1L1, &c., balance each other, they must be proportional to the sides of a closed polygon drawn respectively parallel to their directions. 3 1 In fig. 87 construct such a polygon of loads by 4' drawing the lines Ll, &c., parallel and propor-I-2 titpnil to, anfl jgined 'ind to encgl the order o, t egross oa sont epieceso t structure. 0 Then from the proportionality and parallelism of the load and the two resistances applied 1 to each piece of the structure to the three

, sides of a triangle, there results the following

L' theorem (originally due to Rankine):- If from the angles o the polygon of loads there be drawn lines (R1, 2, &c.), each of whuh is FIG 87 parallel to the resistance (as PIPQ, &c.) exerted at the joint between the pieces to which the two loads represented by the contiguous sides of the polygon of loads (such as Li, Lg, &c.) are applied; then will all those lines meet in one point (O), and their lengths, measured from that point to the angles of the polygon, will represent the magnitudes of the resistances to which they are respectively parallel. When the load on one of the pieces is parallel to the resistances which balance it, the polygon of resistances ceases to be closed, two of the sides becoming parallel to each other and to the load in uestion, and extending indefinitely. In the polygon of loads the direction of a load sustained by parallel resistances traverses the point 0.1

1

an enormous development has taken place in the subject of Graphic Statics, the first comprehensive textbook on the subject being Die Gfaphisehe Statik by K. Culmann, published at Zürich in 18661 Many of the graphical methods therein given have now passed into the textbooks usually studied by engineers. One of the most beautiful graphical constructions regularly used by engineers and known as “ the method of reciprocal figures ” is that for finding the loads supported by the several members of a braced structure, having given a system of external loads. The method was discovered by Clerk Maxwell, and the complete theory is discussed and exemplified in a paper “ On Reciprocal Figures, Frames and Diagrams of Forces, " Trans. Roy. Soc. Ed., vol. xxvi. (1870). Professor M. W. Crofton read a paper on “ Stress-Diagrams in Warren and Lattice Girders" at the meeting of the Mathematical Society (April 13, Since the relation discussed in § 7 was enunciated by Rankinei, § 8. H ow the Earth's Resistance is to be treated . . .When the pressure exerted by a structure on the earth (to which the earth's resistance is equal and opposite) consists either of one pressure, which is necessarily the resultant of the weight of the structure and of all the other forces applied to it, or of two or more parallel vertical forces, whose amount can be determined at the outset of the investigation, the resistance of the earth can be treated as one or more upward loads applied to the structure. But in other cases the earth is to be treated as one of the pieces of the structure, loaded with a force equal and opposite in direction and position to the resultant of the weight of the structure and of the other pressures applied to it. § 9. Partial Polygons of Resistance.-In a structure in which there are pieces supported at more than two joints, let a polygon be constructed of lines connecting the centres of load of any continuous series of pieces. This may be called a partial polygon of resistances. In considering its properties, the load at each centre of load is to be held to include the resistances of those joints which are not comprehended in the partial polygon of resistances, to which the theorem of § 7 will then apply in every respect. By constructing several partial polygons, and computing the relations between the loads and resistances which are determined by the application of that theorem 'to each of them, with the aid, if necessary, of Moseley's principle of the least resistance, the whole of the relations amongst the loads and resistances may be found. § 10. Line of Pressures-Centres and Line of Resistance.—The line of pressures is a line to which the directions of all the resistances in one polygon are tangents. The centre of resistance at any joint is the point where the line representing the total resistance exerted at that joint intersects the joint. The line of resistance is a line traversing all the centres of resistance of a series of joints, ~its form, in the positions intermediate between the actual joints of the structure, being determined by supposing the pieces and their loads to be subdivided by the introduction of intermediate joints ad injiniturn, and finding the continuous line, curved or straight, in which the intermediate centres of resistance are all situated, however great their number. The difference between the line of resistance and the line of pressures was first pointed out by Moseley. § II . The principles of the two preceding sections may be illustrated by the consideration of a particular case of a buttress of blocks forming a continuous series

of pieces (fig. 88), where aa,

bb, cc, dd represent plane

joints. Let the centre of

pressure C at the first joint

aa be known, and also the

pressure P acting at C in

direction and magnitude.

Find R; the resultant of this

pressure, the weight of the

block aabb acting through its

centre of gravity, and any

other external force which

may be acting on the block,

and produce its line of action

to cut the joint bb in Cl. C1

is then the centre of ressure

for the joint bb, and R1 is the

total force acting there. Repeating

this process for each

block in succession there will

be found the centres of pressure

C¢, C3, &c., and also the

resultant pressures R2, R3,

&c., acting at these respective

centres. The centres of pressure at the joints are also called centres of resistance, and the curve passing through these points is called a line of resistance. Let all the resultants acting at the several centres of resistance be produced until they cut one another in a series of points so as to form an unclosed polygon. This polygon is the partial polygon of resistance. A curve tangential to all the sides of the polygon is the line of pressures. § 12. Stability of Position, and Stability of Friction.-The resistances at the several joints having been determined by the principles set forth in § § 6, 7, 8, 9 and IO, not only under the ordinary load of the structure, but under all the variations to which the load is subject as to amount and distribution, the joints are now to be placed and shaped so that the pieces shall not suffer relative displacement under any of those loads. The relative displacement of the two pieces which abut against each other at a joint may take place either

/ .

Frass.

1871), and Professor O. Henrici illustrated the subject by a simple and ingenious notation. The application of the method of reciprocal ligu res was facilitated by a system of notation published in Economics of Construction in relation to framed Structures, by Robert H. Bow (London, 1873). A notable work on the general subject is that of Luigi Cremona, translated from the Italian by Professor T. H. Beare (Oxford, 1890), and a discussion of the subject of reciprocal figures from the special point of view of the engineering student is given in Vectors and Rotors by Henrici and Turner (London, 1903) See also above under “ Theoretical Mechanics, " Part I. § 5.