CHAPTER XIII.

STRUTS.

In

General Statement.-The manner in which short compression pieces fail is shown in Chapter VIII.; but when their length is great in proportion to their diameter, they bend laterally, unless they are initially absolutely straight, exactly centrally loaded, and of perfectly uniform material-three conditions which are never fulfilled in practice. The nature of the stresses occurring in a strut is, therefore, that of a bar subjected to both bending and compressive stresses. Chapter XII. it was shown that if the load deviated but very slightly from the centre of gravity of the section, it very greatly increased the stress in the material; thus, in the case of a circular section, if the load only deviated by an amount equal to one-eighth diameter from the centre, the stress was doubled; hence a very slight initial bend in a compression member very seriously affects its strength.

Effects of Imperfect Loading.-Even if a strut be initially straight before loading, it does not follow that it will

remain so when loaded; either or both of the following causes may set up bending

(1) The one side of the strut may be harder and stiffer than the other; and consequently the soft side will yield most, and the strut will bend as shown in A, Fig. 453.

(2) The load may not be perfectly centrally applied, either through the ends not being true as shown in B, or through the load acting on one side, as in C.

Possible Discrepancies between Theory and Practice. We have shown that a very slight amount of bending makes a serious difference in the strength of struts ; hence such accidental circumstances as we have just mentioned may not only make a serious discrepancy between theory and experiment, but also between experiment and experiment. Then, again, the theoretical determination of the strength of struts does not rest on a very satisfactory basis, as in all the theories advanced somewhat questionable assumptions have to be made; but, in spite of it, the calculated buckling loads agree fairly well with experiments.

Bending of Long Struts.-The bending moment at the middle of the bent strut shown in Fig. 454 is evidently Wo.

W

Then W8 = ƒZ, using the same notation as in the preceding chapters.

If we increase the deflection we shall correspondingly increase the bending moment, and consequently the

stress.

From above we have

W

FIG. 454.

f

б

But as ƒ varies with 8,' { = a constant, say K ;

f

then WKZ

But Z for any given strut does not vary when the strut bends; hence there is only one value of W that will satisfy the equation.

When the strut is thus loaded, let an external bending moment M, indicated by the arrow (Fig. 455), be applied to it until the deflection is d1, and its stress f;

Then W8, + M = ƒZ
But Wo1 = f12

therefore M = 。

that is to say, that no external bending moment M is required to keep the strut in its bent position, or the strut, when thus loaded, is in a state of neutral equilibrium, and will remain

when left alone in any position in which it may be placed; this condition, of course, only holds so long as the strut is elastic, i.e. before the elastic limit is reached. This state of neutral equilibrium may be proved experimentally, if a long thin piece of elastic material be loaded as shown.

Now, place a load W, less than W on the strut, say W=W1+w, and let it again be bent by an external bending moment M till its deflection is d and the stress fi; then we have, as before—

W18, + M = ƒ¡Z = Wd1 = W18, + wò,
hence M = ιδι

Thus, in order to keep the strut in its bent position
with a deflection d1, we must subject it to a + bend-
ing moment M, i.e. one which tends to bend the
strut in the same direction as W1d,; hence, if we
remove the bending moment M, the deflection will
become zero, i.e. the strut will straighten itself.

2

W

FIG. 455.

Now, let a load W, greater than W be placed on the strut, say W = W1w, and let it again be bent until its deflection 6, and the stress f by an external bending moment M; then we have as before

Thus, in order to keep the strut in its bent position with a deflection &, we must subject it to a bending moment M, i.e. one which tends to bend the strut in the opposite direction to W; hence, if we remove the bending moment M, the deflection will go on increasing, and ere long the elastic limit will be reached when the strain will increase suddenly and much more rapidly than the stress, consequently the deflection will suddenly increase and the strut will buckle.

Thus, the strut may be in one of three conditions

Condition ii. is, of course, the only one in which a strut can exist for practical purposes; how much the working load must be less than W is determined by a suitable factor of safety.

Buckling Load of Long Thin Struts. Euler's Formula.--The results arrived at in the paragraph above

FIG. 456.

W

refer only to very long thin struts; we will now proceed to determine the value of W for such struts. If the deflection were entirely due to the eccentricity x of the load, then the bending moment at every section of the strut would be constant and equal to Wx, and the strut would then bend to the arc of a circle (see p. 434). For the present we will assume that struts do bend to an arc of a circle; we shall return to this point later on, and then give a more exact result.

=

Let the effective length of the strut (see Fig.

458);

E = Young's modulus of elasticity;

I

=

the least moment of inertia of a section of the strut (assumed to be of constant cross-section).

Then for a strut loaded thus

As the strut is very long and the deflection small, the length / remains practically constant, and the other quantities 8, E, I are also constant for any given strut; thus, W is equal to a constant, which we have previously shown must be the case.

FIG. 457.

Once the strut has begun to bend it cannot remain a circular arc, because the bending moment no longer remains constant at every section, but it will vary directly as the distance of any given section from the line of application of the load. Under these conditions assume as a second approximation that it bends to a parabolic arc, then the deflection

By Euler's theory, which we must not forget is only another approximation, since he neglects the direct stress on the section,

The I in this formula is the least moment of inertia of the section. Effect of End holding on the Buckling Load. In the case we have just considered the strut was supposed to be free or pivoted at the ends, but if the ends are not free the strut behaves in a different manner, as shown in the accompanying diagram.

Each strut is supposed to be of the same section, and loaded

with the same weight W.