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μα

с

)+2a}

But by equation (604) S=2_7) and by equation

8

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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

P

B

M P

B

the weight of a bar of the material of the chain, one square inch in section and one foot long, be represented by , the weight of each foot in the length of the road-way by ",, the aggregate section of the chains at any point P (in square inches) by K, the co-ordinates DM and MP of P by x and y, and the length of the portion DP of the chain by 8.

Then

will the weight of DP be represented by Kds, and the weight of the portion CM of the roadway by ; 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

Kds+μ2x, U=",
· ƒ Kd8+μx . ・ ・ ・ ・ (614).
SKd8+μ1æ

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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 c at that point; then will ap be parallel to a tangent to the chain at P (Art. 393).

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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+u3)*; if, therefore, be taken to represent the tenacity of the material of the chain, per square inch of the section, then

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

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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

mc ==

эти,

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K= {1+ (1+)tan. (1+). (619)+

T

T

τμ сти,

(619).†

It is evident from this expression that the area of the sec tion 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 suspen sion, where it is greatest.

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integrated in respect to x by known rules of the integral calculus, the value of 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. Church's Int. Cal. Art. 129, Case II.

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

Let it be observed that W=", SKds=u-, (equation

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

терия

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

ст

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

Let II 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);

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404. THE SUSPENSION BRIDGE OF GREATEST STRENGTH, THE WEIGHT OF THE SUSPENDING RODS BEING TAKEN INTO AOCOUNT.

Conceive the suspending rods to be replaced by a con

P

D

tinuous flexible lamina or plate connecting the roadway with the chain, and of such a uniform thickness that the material contained in it may be precisely equal in weight to the material of the suspending rods. It is evident that the conditions of the equilibrium will, on this hypothesis, be very nearly the same as in the actual case. Let, represent the

weight of each square foot of this plate, then will Syda

represent the weight of that portion of it which is suspended from the portion DP of the chain, and the whole load u upon that portion of the chain will be represented by

"JKds + M22
+μ2+μ
μ.
fydx........ (622).

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It may be shown, as before (Art. 400.), that

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A linear equation in u', the integration of which by a well known method gives

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Assuming the length of the shortest connecting rod DC to be represented by b, integrating between the limits b and y, and observing that when y=b, u=0,

* Church's Int. Cal. Art. 176.

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