capable of extension to the investigation of built-up members, as the actual end conditions, connections between the parts, etc., cannot be correctly imitated in glass or xylonițe models. The thermoelectric method would appear to be more promising, but has not yet been sufficiently investigated. For the present it would seem as if reliance must be placed upon measurement with some form of extensometer. Most extensometers, however, measure only the average stress over a considerable length, are bulky and inconvenient, and cannot be used in positions difficult of access, such as the inner portions of a built-up member.

Some time ago the writer read a paper before the Canadian Society of Civil Engineers 3 in which he described investigations of the distribution of stress in certain members made with a simplified form of Martens' mirror extensometer constructed in the laboratories of McGill University, Montreal. This instrument, extremely simple in construction and operation, was shown to be capable, when certain precautions are observed in its use, of measuring strains accurately to inch on a length as small as 2 inches, and of being used in the most confined positions, such, for example, as the space between two angles placed back to back and separated by as little as 3% inch. It was thus proved to be eminently suitable for the laboratory investigation of strain distribution in built-up members.

1

100.000

In the paper cited above, the distribution of stress in singleand double-angle tension members riveted to end plates was considered. It was shown that the assumption of planar distribution of stress over the cross-section was justified and that the actual distribution in a single-angle member was closely in accordance with that given by the theory of eccentric stresses developed by Bach, Müller-Breslau and others, and put into workable form by Professor L. J. Johnson, of Harvard. It was also shown that a double-angle member did not act as one piece, but that each angle bent about its own neutral axis, and thus the correct way of considering a built-up member was not to regard it as a single piece bending as a beam, but as several pieces, each trying to bend about its own neutral axis, but restrained by the subsidiary members, such as tie plates or latticing.

In these experiments 3 inch x 3 inch x 14 inch angles were

& Transactions Canadian Society of Civil Engineers, vol. xxvi, p. 224.

used, riveted to end plates of the same width, that being the greatest which would fit into the grips of the testing machine. In this case the effect of the end constraints was found to be very little in the case of the single angle, but it was thought advisable to repeat the experiments with end plates more like the gusset plates to which such members are usually attached in practice. To this end special grips were made so that plates of considerable size could be used, and the results of the experiments to be described later show that there must be considerable end restraint under practical conditions. In the present paper the effect of this constraint is considered, and also the effect of changing the axis of pull on the gusset plate, but the chief object is the investigation of the effect of lock angles at the ends of the members. In practice the ends of single- and double-angle members are usually secured to the gusset plates directly by a row of rivets between the plate and the angle and indirectly through the medium of a small angle riveted both to the gusset plate and to the member as shown in Fig. 5, which represents the experimental specimens. The function of this small angle, called a "lock" or lug" angle, is supposed to be two-fold. First, by transmitting a pull through both legs of the angle or angles forming the member, it is supposed to lessen the eccentricity of connection, so that often the stress is calculated as though uniformly distributed over the cross-section; secondly, by allowing more rivets to be used at the ends of the member, it is supposed to be equivalent to lengthening the end connections. The writer hopes to show that both of these claims are to a large extent unfounded; that lock angles really serve very little useful purpose, and that the slight effect which they have, helpful or otherwise, cannot be predetermined.

66

Before entering upon a discussion of the experiments a brief account of the theory upon which the analysis of the results is based will be given.

PART I. THEORETICAL.

§1. The Distribution of Stress in Eccentrically Loaded Members of Uniform Cross-section.

A single angle such as is shown in Fig. 5, loaded in tension or compression through rivets in one leg, is an example of an eccentrically loaded member in which the load axis does not lie

in an axis of symmetry of the cross-section. If the ends of the member are not appreciably restrained from bending by the end connections, the axis of loading may be said to lie along the line of rivets and at the common face of the end plate and the angle. This axis is represented by K in Fig. 1, which represents a cross-section of the member. It will readily be seen that the ordinary theory of eccentric loading, which is true only for loads applied in an axis of symmetry of the cross-section, cannot be applied to this case, and recourse must be had to the general theory already mentioned.5 For a full account of this theory the reader is referred to the paper by Professor L. J. Johnson referred to above or to Appendix I of the paper by the present writer in the Transactions of the Canadian Society of Civil Engineers, vol. xxvi, p. 224. A brief résumé of the

N

theory will be given here, as it is essential to the understanding of the experimental results. Referring to Fig. 1, let G represent the centroid of the cross-section and Gr and Gy be a pair of coordinate axes parallel to the legs of the angle. Then, if N be the normal force applied at K, the stress at any point of the crosssection will be equal to where A is the area of the section, together with a stress arising from the bending moment N.KG to which the angle is subjected. This moment will cause bending about a neutral axis nn, which will not, as in the case of loading in a plane of symmetry, be perpendicular to KG, but at an angle a to Gr, given by the equation

'L. J. Johnson, loc. cit.

A '

tan a =

Ix J tan 2

J-Iy tan λ

(1)

where I and I, are the moments of inertia of the section about Gr and Gy respectively and J is the product of inertia of the

section with respect to the axes Gr and Gy,-i.e., the ffrydxdy with respect to these axes. The above equation is deduced by equating the sum of the moments of the stresses over the section. about Gr and Gy respectively to the components of the bending moment N.KG about these axes, assuming that the distribution of stress over the section follows a linear law. The stress at any point (xy) of the section may be shown to be given by either of the equations

where (x, y) are the coördinates of the load axis K. In order to find the maximum stress, it is only necessary to substitute for (ry) the coördinates of the point most distant from the neutral axis. If ƒ be equated to zero in either of the equations (2) or (3), the equation of the neutral axis nn will be obtained.

§2. The S-Polygon.

8

The S-polygon, a modification of the W-Fläche of Land, was first introduced by L. J. Johnson. It is a figure which gives at a glance the point of maximum bending stress and the value of the latter for any given load axis, and, as its use will facilitate certain deductions to be made later, its construction will be considered briefly here.

to

M I

f

y

For the bending of ordinary I sections the steel handbooks give a quantity called the " modulus " of the section which is equal where M is the bending moment applied to the section, y the distance of the skin from the neutral axis, f the maximum stress and I the moment of inertia of the section about the neutral axis. It may be defined as that quantity by which the bending moment at any section must be divided in order to give the maximum

* Ix and I, are given in the steel handbooks for all ordinary sections, but J has to be calculated. For the method of calculation see either of the papers cited above.

'This was shown to be the case experimentally by the writer, loc. cit. 8 Loc. cit.

stress at the section. In the general case of flexure as considered above there will be a similar quantity which may be termed the “flexure modulus” (S) of the section. Thus S=. The bending moment on the section is N.KG. The stress due to this at any point of the section is, from equation (2), given by

Eliminating a between this equation and equation (1), and rearranging terms, the section modulus for (xy) is given by

If the point (xy) remains fixed while λ changes, the above is readily seen to be the polar equation of a straight line, having

radius vector S and angle λ. This line may be termed the S-line for the point (xy). If S-lines be drawn for all points at which the maximum stress may occur, a polygon, called the S-polygon, will be obtained. Thus, for example, in the case of the single angle Fig. 2, the maximum stress may occur at A, B, C, D, or F. If G be chosen as the pole and Gr as the initial line, the figure (ab), (bc), (cd), etc., is the S-polygon drawn with S to the same scale as the linear dimensions of the figure. Thus for the load