398. A chain of given length being suspended between two given points in the same horizontal line: to determine the depth of the lowest point beneath the points of attachment; and, conversely, to determine the length of the chain whose lowest point shall hang at a given depth below its points of attachment.

The same notation being taken as before,

Integrating between the limits 0 and s, and observing that y=0 when s=0,

If H represent the depth of the lowest point, or the versed sine of the curve, then y=H when s=S.

399. The centre of gravity of the catenary.

If G represent the height of the centre of gravity above the lowest point, we have (Art. 32.)

ds

S. G=Syds=f yazdx.

ds

Substituting, therefore, for y and their values from equa

dx

tions (602) and (603), we have

400. THE SUSPENSION Bridge of GREATEST STRENGTH, THE WEIGHT OF THE SUSPENDING RODS BEING NEGLECTED.

Let ADB represent the chain, EF the road-way; and let

the weight of a bar of the material of the chain, one square inch in section and one foot long, be represented by μ,, the weight of each foot in the length of the road-way by μ, the aggregate section of the chains at any point P (in square inches) by K, the co-ordinates DM and MP of P by a and y, and the length of the portion DP of the chain by s. Then

will the weight of DP be represented by Kds, and the

μsKds,

weight of the portion CM of the road-way by μ; so that the whole load (u) borne by the portion DP of the chain will be represented (neglecting the weight of the suspending rods) by

μ1SKds+μ2t :. u=μ;
=μS Kds +μ2 . . . . . (614).

Let this load (u), supported by the portion DP of the chain, be represented by the line Da, and draw Dp in the direction of a tangent at D, representing on the same scale the tension e at that point; then will ap be parallel to a tangent to the chain at P (Art. 393).

Now let it be assumed that the aggregate section of the chains is made so to vary its dimensions, that their strength may at every point be equal to m times the strain which they have there to sustain. But this strain is represented in magnitude by the line ap (Art. 393.), or by (c2 + u2)*; if, therefore,

be taken to represent the tenacity of the material of the chain, per square inch of the section, then

ds

ds Kr

; therefore

dx

dx mc

=mc

T

mc

m

= mc (1+) (equation 615)

dx=

Also fKds=fKada

SK3dx="f{e2 + w2)dz (equation 616);

тс

dx

Substituting in this equation the value of u given by the preceding equation, and reducing,

T

y=

my, (log.

(mu,

log. sec.

which is the equation to the suspension chain of uniform strength, and therefore OF THE GREATEST STRENGTH WITH A

GIVEN QUANTITY OF MATERIAL.

401. To determine the variation of the section K of the chain of the suspension bridge of the greatest strength.

Let the value of u determined by equation (617) be substituted in equation (616); we shall thus obtain by reduction.

It is evident from this expression that the area of the section of the chains, of the suspension bridge of uniform strength, and therefore of the greatest economy of material, increases from the lowest point towards the points of suspension, where it is greatest.

*Hall's Diff. Cal. pp. 280, 283.

ds K t = dx mc

T

8=

SKdr. Now the function K (equation 619) may

be integrated in respect to x by known rules of the integral calculus; the value of s may therefore be determined in terms of x, and thence the length in terms of the span. The formula is omitted by reason of its length.

402. To determine the weight W of the chain of the suspension bridge of the greatest strength.

Let it be observed that W=
́=μ1ƒ'Kds=u—μ„r (equation

Εμι

614); substituting the value of u from equation (617), we have

W=c(1+ Tes)' tan. { (1+) = } - (620),

403. To determine the tension c upon the lowest point D of the chain of uniform strength.

Let H be taken to represent the depth of the lowest point D, beneath the points of suspension, and 2a the horizontal distance of those points: and let it be observed that H and a are corresponding values of y and x (equation 618);

.. H= log. sec. { "") (1+1) a}

ημι

T

Solving this equation in respect to c,

стр

Conceive the suspending rods to be replaced by a con

tinuous flexible lamina or plate connecting the roadway with the chain, and of such a uniform thickness that the material