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point K, GL represents S and the maximum stress occurs at B. But S= M and M = N.KG, therefore fo, the bending stress at B, is equal to
= N. EG Thus the total stress at B, which is the maximum stress at the section, is given by
f = À + N. KG
....(6) In order that this may be true, it is, of course, necessary that the scale of the S-polygon should be the same as the linear scale of the figure and that the origin be G and the initial line Gx. $3. The Construction of the S-Polygon.
The best method of constructing the S-polygon of an angle is by locating the apices. Each apex corresponds to a side. It may be shown by transforming equation (5) into rectilinear coordinates, substituting the coördinates of any two points (xa ya), (xo y:), and solving for the apex (ab) of the S-polygon, that
(xo – x)– (ya – yo ) Iy
........ (7) yab = (x8 — xv ) Ix – (ya – yo ) I
xayь - xbya If the side considered be parallel to Gx, so that ya = yb, the above equation gives
while, if the side be parallel to Gy,
Considering, for example, the single-angle specimen I, used in the experiments to be described later and for which I x = Iy= 1.37 (in.)*, J =-0.81, and the distance from the centroid to the back of the angle is 0.85", the apex of the S-polygon corresponding to the side AB has the coördinates
0.81 0.85 The other apices may be determined in a similar manner, and the complete polygon is shown in Fig. 2.
VoL. CLXXX, No. 1076—11
The S-polygon furnishes the simplest method of determining the ratio of maximum to mean stress when a number of load axes have to be considered. It may also be used to establish a number of interesting results. For instance, as the point K moves down AB, starting from A, the radius vector GL terminates first on the f line, then on the a line, and finally on the b line, showing that the point of maximum stress changes from F to A and then to B. The ratio also changes considerably. It will be shown later that the S-polygon is of material aid in analyzing the prob able effect of a lock angle or a change in the restraining couple. $4. The Effect of Lateral Deflection and of End Restraints.
The earlier experiments of the writer have shown that in the case of a single angle loaded in tension and not effectively restrained at the ends, the axis K is along the line of rivets and slightly within the loading plate. In a long member there will be a measurable lateral deflection as the load is applied, the centroid of each section trying to set itself in the load axis. This will cause a change in the position of K relative to the section and consequently affect the distribution of stress over the section. This effect is not usually great enough to be of much practical importance, but must be considered in reducing experimental results. Of much greater interest is the effect of the gusset plate and lock angle in affecting the position of the load axis. A stiff end connection will introduce constraining couples both in and at right angles to the plane of the end plate, and these may have important effects upon the distribution of stress. A lock angle is often supposed to so constrain the ends that the eccentricity of pull may be neglected, although how far this is from actually being the case will be shown later. The chief object of the present paper is the investigation of these different constraints in the case of single- and double-angle members. $5. The Working Loads upon Eccentrically Loaded Members.
In a compression member it is, of course, always unsafe to allow the stress at any point to exceed the elastic limit of the material, even though the load which would cause this stress be well within the theoretical buckling load for a long column. Experiment has shown, however, that, in the case of a tension member, a redistribution of stress usually occurs under such con
ditions, and that the elastic limit may be considerably exceeded at certain parts of the section without danger to the structure as a whole. This would lead to the supposition that, when the stresses arising from eccentricity are taken into account, higher working stresses than usual might be used. It would, however, be unsafe to proceed on this assumption without more experimental evidence as to whether, in such case, the new distribution of stress persists without alteration after many reloadings. At present it would seem to be unwise to allow the stress over any considerable part of the structure to rise above the ordinary working stress of the material at any point of the cross-section, especially considering that there are always higher stresses locally near to the end connections. Thus it would appear that singleangle members, and to some extent double-angle members, should always be designed for the maximum stress, allowing for eccentricity,—i.e., including what are often termed “secondary” stresses, although they are frequently quite as large as, or even larger than, the so-called "primary” or uniform stress, leaving what may be termed the “tertiary” or local stresses due to the end connections to be included in the factor of safety.
PART II.—THE EXPERIMENTS. $1. The Extensometers.'
The extensometers used were a simplified form of the Martens' type, designed and constructed in the McGill Testing Laboratory, where they have been in use since 1906, and have been proved capable of giving very accurate results.
The principle of the instrument is shown in Fig. 3. It consists essentially of a double knife-edge, K, which fits between the specimen under test and a V groove in one end of a steel strip S, which is in contact with the specimen at A, and is pressed against it by means of a clip C. A change in the length of the specimen between A and B causes the knife-edge to tilt, and the tilt is measured by means of a telescope and scale, the scale being reflected in a mirror M attached to the knife-edge. In the actual instrument the steel strip is 38 inch wide, 16 inch thick, the length A B being chosen to suit requirements. The end A is is turned at right angles and brought to a sharp edge so that it may not slip on the specimen. The knife-edge is of hardened steel
• This section is reprinted from the writer's earlier paper, loc. cit.
about 0.18 inch x 0.12 inch x 0.45 inch, and the mirror is attached by means of a piece of steel knitting needle. The mirror is held in a clip of thin sheet steel which is arranged so that it can slide and rotate on the needle, a thin copper strip protecting its back from injury. This clip permits of a small amount of lateral adjustment. The mirror is about 1/2 inch square and must be as truly plane as possible, as otherwise there will be an error
introduced when the image of the scale moves to a different part of its surface, as it must do if the specimen deflects at all during test. In the original form of Martens' extensometer there was a device for adjusting the mirror and also a balance weight at the opposite side of the knife-edge, but these refinements are not only unnecessary but cumbersome, and make the instrument less adapted to use in restricted positions.
The extensometer is calibrated in a Whitworth measuring machine, and a calibrating rod is prepared for each instrument, giving the distance from the scale to the mirror, so that a definite