AMERICAN SOCIETY OF CIVIL ENGINEERS

INSTITUTED 1852

PAPERS AND DISCUSSIONS

This Society is not responsible for any statement made or opinion expressed in its publications.

ON A NEW PRINCIPLE

IN THE THEORY OF STRUCTURES*

BY GEORGE F. SWAIN, PAST-PRESIDENT, AM. Soc. C. E.

SYNOPSIS.

This paper explains what is believed to be a new principle in the theory of structures, which has interesting and important applications. It is known that there is little new under the sun, and it may be that this principle has been known, and published elsewhere, but the writer has not seen it, and has arrived at it independently, from his own reflections on the subject.†

1.-DEFLECTION OF A POINT.

The deflection of any point, A, of any structure may be found by a well-known formula, first published in America by the writer in 1883. This formula is

This paper will not be presented for discussion at any meeting of the Society, but written communications on the subject are invited for subsequent publication in Proceedings, and with the paper in Transactions.

† In Mohr's "Abhandlungen aus dem Gebiete der Technischen Mechanik," p. 425, this principle is hinted at, but it is not fully explained or its general applicability shown. The same thing is referred to in Grimm's book on Secondary Stresses, but incompletely elucidated.

In a paper, entitled "On the Application of the Principle of Virtual_Velocities to the Determination of the Deflection and Stresses of Frames," in the Journal of the Franklin Institute.

§ "If this expression causes confusion in the mind of the reader by reason of the fact that the left member is a length while the right member is apparently a force times a length, it is only necessary to remind him that in reality the left member is a length times a unit force; or, differently stated, since t is the stress due to a unit force, it may be considered as a stress due to any given force divided by that force, and, therefore, is a ratio. At any rate, the equation as it stands is perfectly correct."

in which t=stress in any member due solely to a load of unity acting at the point the deflection of which is desired, under

the given actual loads, in the direction in which said deflection is desired;

s=stress in the same member, due solely to the given actual

loads;

7= length of the same member;

A: area of the same member;

E=modulus of elasticity of the same member.

This formula, as is well known, may be applied, not only to finding the deflection of a joint of a framed structure, but also to finding the deflection of any point of a beam. In the latter case, the area, A, must be replaced by an infinitesimal area, da, and the stresses, s and t, must be the stresses on such area. If the case is one of pure flexure, with no resultant axial force on any section, and if Mt is the bending moment due solely to the force, unity, and M, is the bending moment due to the actual loads, at any given section, the formula becomes

8

the integral being extended over the entire length.

(2)

The principle now to be explained is a similar one which affords a means of finding the rotation of the axis of any piece at any point.

2.-ANGULAR ROTATION OF THE AXIS OF A PIECE AT A POINT.
(THE NEW PRINCIPLE).

In addition to finding the deflection of any point of a structure in any given direction, it is often desired to find the angular rotation at a point, i. e., the change of direction of the axis of a piece. This may be obtained by a general method analogous to that used for finding the deflection.

Let it be desired to find the angular rotation, or change of direction of the axis of a piece, at a given point, A, due to any given loads. At the given point apply only a couple the moment, M, of which equals unity, and let the corresponding angular rotation, a, at A be first determined by the method of work. (It may be thought a contradiction, or an impossibility, to apply a couple at a point, but this obviously is the same as applying two very large forces a very short distance apart,

1

at a distance, d x, apart.

d x

This is

or two forces each equal to perfectly conceivable.) The work done by the couple the moment of which is M1, if gradually applied, will obviously be since the 2'

α

couple may be considered to consist of forces of unity a unit distance

apart, and its average value is

1

.

2

The only outer forces being this

α

couple, the work of these outer forces is and this must equal the

2

work of the inner forces. On any bar or fiber having a uniform stress intensity throughout its area and length, due to the applied moment, M=1, let r equal the total direct stress on the bar or fiber and ▲ l its change of length. Then the equation between the work of the outer forces and that of the inner forces is

α

2

= (change of length), or a = Σr (change of length).

2

Now, as in the case of deflection, it is clear that the deformation is purely geometrical phenomenon, and that any change of length of a bar or fiber, whatever its cause, will produce the same angular rotation at A. Hence, if for "change of length" in the above formula, which is the change of length due to M=1 at A, we substitute the change of length due to any given loading, the formula will give the angular rotation at A due to the given loading, or

r stress on the area, A, due to the couple, M:

point in a given sense;

=

1, at the given

s = stress on the area, A, due to the applied loads.

The applied couple, M, may be applied with either a right-handed or left-handed direction of rotation. If a comes positive, this means that the given loads produce a rotation in the same direction as that assumed for M.

In using this equation, tension and compression, of course, should be distinguished in sign, carefully using one of these terms consistently as positive and the other as negative.

If the stresses include bending stresses, the elementary fibers, da in area and dl in length, must be taken.

The formula shows the effect of direct stresses only, shearing effects being neglected.

In applying the couple, M=1, bending stresses may be produced in a bar, in addition to a direct stress in the same bar. The applied loads may produce bending or direct stresses, or both, in any bar. In such cases, for any fiber, s and r are the stresses on an area, da.

In general, if the applied loads cause in any bar at any section a direct force, Ts, and a flexural moment, M,, the value of s will be

and if the couple, M

=

1, produces in the same bar and section a direct

force, T,, and a moment, M,, the value of r will be

The value of the differential, da, therefore, will be

It is supposed that every bar is straight, with one principal axis of each cross-section in the plane of the applied loads, in which plane also the couple, M=1, is applied. I is the moment of inertia about the principal axis perpendicular to that plane, which, of course, passes through the center of gravity of the section, so that fyda = 0.

This differential refers to an infinitesimal length, dl, of any fiber of area, da, in any bar, and must be integrated over the cross-section, then over the length of the bar, and the sum of the results taken. The final result will easily be seen to be, if A is constant throughout the length of each bar,

The forces, T, and T ̧, are the direct forces acting in any bar, and are supposed to be constant throughout its length; and the moments, M, and M., are the bending moments at any section of any bar. Each of the forces, s and r, in Equation (3) may, in any bar, represent a direct stress only, or flexural stress only, or both. The following results clearly follow from Equation (4):

1.—If, in any bar, either the actual load system, or the applied moment, M: = 1, causes a direct stress only, and the other a flexural stress only, that bar may be neglected in the computation; for the value of a for that bar is zero.

2. The flexure in any bar may be neglected unless both the actual loads and the applied moment, M 1, cause flexural stress in it.

3. The direct stress in any bar may be neglected unless both the actual loads and the applied moment, M 1, cause a direct stress in it. 4.-If the case is one in which there is only flexural stress in the bars, without direct stress, the formula becomes

5.—If A as well as T varies along the length of any bar, the formula becomes

These principles apply in using Equation (1), to determine deflection, as in using Equation (3), to determine rotation.

3.-ILLUSTRATIONS OF ANGULAR ROTATION.

Although this method would seldom or never be used to find the slope of a beam exposed to flexure, this case may be illustrated to show the applicability of the method.

M=1

Case I.-Slope of a Beam Fixed at One End. Let it be desired to ascertain the slope of the free end of a beam of constant section (Fig. 1), fixed at one end and loaded at the other with a load, P. The load, P, produces on any section, distant

1, at the free end, if acting in the sense shown, likewise produces