with corresponding bars parallel, but without complete geometric
similarity. In the case of fig. 52 it may be shown that an equivalent
condition is that the six points A, B, C, D, E, F should lie on a conic
(M. W. Crofton). This is fulfilled when the opposite sides of the
hexagon are parallel, and (as a still more special case) when the
hexagon is regular.
When a frame has a critical form it may be in a state of stress
independently of the action of extraneous forces; moreover, the
stresses due to extraneous forces are
indeterminate, and may be infinite.
For suppose as before that one of the
bars is removed. If there are no extraneous
Fig. 53.
forces the equation of virtual work
reduces to S·δs = 0, where S is the stress
in the removed bar, and δs is the change
in the distance between the joints which
it connected. In a critical form we
have δs = 0, and the equation is satisfied
by an arbitrary value of S; a consistent
system of stresses in the remaining bars
can then be found by preceding rules. Again, when extraneous
forces P act on the joints, the equation is
where δp is the displacement of any joint in the direction of the corresponding force P. If Σ(P·δp) = 0, the stresses are merely indeterminate as before; but if Σ (P·δp) does not vanish, the equation cannot be satisfied by any finite value of S, since δs = 0. This means that, if the material of the frame were absolutely unyielding, no finite stresses in the bars would enable it to withstand the extraneous forces. With actual materials, the frame would yield elastically, until its configuration is no longer “critical.” The stresses in the bars would then be comparatively very great, although finite. The use of frames which approximate to a critical form is of course to be avoided in practice.
A brief reference must suffice to the theory of three dimensional frames. This is important from a technical point of view, since all structures are practically three-dimensional. We may note that a frame of n joints which is just rigid must have 3n − 6 bars; and that the stresses produced in such a frame by a given system of extraneous forces in equilibrium are statically determinate, subject to the exception of “critical forms.”
§ 10. Statics of Inextensible Chains.—The theory of bodies or structures which are deformable in their smallest parts belongs properly to elasticity (q.v.). The case of inextensible strings or chains is, however, so simple that it is generally included in expositions of pure statics.
It is assumed that the form can be sufficiently represented by a plane curve, that the stress (tension) at any point P of the curve, between the two portions which meet there, is in the direction of the tangent at P, and that the forces on any linear element δs must satisfy the conditions of equilibrium laid down in § 1. It follows that the forces on any finite portion will satisfy the conditions of equilibrium which apply to the case of a rigid body (§ 4).
 |
| Fig. 54. |
We will suppose in the first instance that the curve is plane. It is often convenient to resolve the forces on an element PQ (= δs) in the directions of the tangent and normal respectively. If T, T + δT be the tensions at P, Q, and δψ be the angle between the directions of the curve at these points, the components of the tensions along the tangent at P give (T + δT) cos ψ − T, or δT, ultimately; whilst for the component along the normal at P we have (T + δT) sin δψ, or Tδψ, or Tδs/ρ, where ρ is the radius of curvature.
Suppose, for example, that we have a light string stretched over a smooth curve; and let Rδs denote the normal pressure (outwards from the centre of curvature) on δs. The two resolutions give δT = 0, Tδψ = Rδs, or
The tension is constant, and the pressure per unit length varies as the curvature.
Next suppose that the curve is “rough”; and let Fδs be the tangential force of friction on δs. We have δT ± Fδs = 0, Tδψ = Rδs, where the upper or lower sign is to be taken according to the sense in which F acts. We assume that in limiting equilibrium we have F = μR, everywhere, where μ is the coefficient of friction. If the string be on the point of slipping in the direction in which ψ increases, the lower sign is to be taken; hence δT = Fδs = μTδψ, whence
if T0 be the tension corresponding to ψ = 0. This illustrates the resistance to dragging of a rope coiled round a post; e.g. if we put μ = .3, ψ = 2π, we find for the change of tension in one turn T/T0 = 6.5. In two turns this ratio is squared, and so on.
Again, take the case of a string under gravity, in contact with a smooth curve in a vertical plane. Let ψ denote the inclination to the horizontal, and wδs the weight of an element δs. The tangential and normal components of wδs are −s sinψ and −wδs cosψ. Hence
If we take rectangular axes Ox, Oy, of which Oy is drawn vertically upwards, we have δy = sin ψ δs, whence δT = wδy. If the string be uniform, w is constant, and
say; hence the tension varies as the height above some fixed level (y0). The pressure is then given by the formula
| R = T | dψ | −w cos ψ. |
| ds |
In the case of a chain hanging freely under gravity it is usually convenient to formulate the conditions of equilibrium of a finite portion PQ. The forces on this reduce to three, viz. the weight of PQ and the tensions at P, Q. Hence these three forces will be concurrent, and their ratios will be given by a triangle of forces. In particular, if we consider a length AP beginning at the lowest point A, then resolving horizontally and vertically we have
where T0 is the tension at A, and W is the weight of PA. The former equation expresses that the horizontal tension is constant.
 |
| Fig. 55. |
If the chain be uniform we have W = ws, where s is the arc AP: hence ws = T0 tan ψ. If we write T0 = wa, so that a is the length of a portion of the chain whose weight would equal the horizontal tension, this becomes
This is the “intrinsic” equation of the curve. If the axes of x and y be taken horizontal and vertical (upwards), we derive
Eliminating ψ we obtain the Cartesian equation
| y = a cosh xa | (9) |
of the common catenary, as it is called (fig. 56). The omission of the additive arbitrary constants of integration in (8) is equivalent to a special choice of the origin O of co-ordinates; viz. O is at a distance a vertically below the lowest point (ψ = 0) of the curve. The horizontal line through O is called the directrix. The relations
| s = a sinh xa, y2 = a2 + s 2, T = T0 sec ψ = wy, | (10) |
