The following statement shows in detail the cost of the entire scheme :

2. Excavating trenches, laying and jointing pipes, excavating and building manholes, &c.

3. Excavating and building five manholes

Total Expenditure.

R. A. P. 50,375 15 3

101,375 10 5

2,214 10 2

4. Excavating and building flushing siphons, and} 30,172 3 10

erecting ventilating pipes.

5. Excavating and erecting night-soil depôts

6. Building engine and boiler-house, engineer's bungalow, and works at farm-yard

7. Supplying, erecting, and fitting engines and boilers; supplying and erecting ejectors; supplying and laying sewage and air-mains

8. Establishment and contingencies

945 4 2

34,103 5 3

355,758 1 6

22,054 13 5

The Paper is accompanied by seven tracings, from which Plate 7

has been prepared.

(Paper No. 3085.)

"Stresses in Pipes bent at Right Angles caused by heating to the Temperature of Steam at various Pressures."

By THORNYCROFT DONALDSON, M.A., Assoc. M. Inst. C.E. NOTWITHSTANDING the recent great increase of boiler pressures, especially in marine engineering practice, data does not appear to be available to show whether the stresses in a steam-pipe, caused by expansion due to increase of temperature, are sufficiently

reduced by arranging a bend in the pipe. In the hope of contributing towards a fuller knowledge of this matter, the Author presents the following investigation, with an account of certain experiments bearing upon it.

Fig. 1 represents a pipe bent at right angles, of which the two arms, originally straight and bolted firmly by flanges to a rigid framework, have become deflected by increase of temperature. In the following investigation the "strut effect" of the reaction due to the short arm is regarded as having no effect upon the curvature and deflections of the long arm; that is, the long arm is considered to be affected only by a couple at its end and a force perpendicular to it, and not by a force parallel to it; and so for the short arm. It will be shown later that this assumption is approximately justified, except where the ratio of the long to the short arm is very large.

The nomenclature used is obvious from the Fig., observing

L 7

that = p.

Consider first the short arm OB ; at any point

distant a from B, the bending moment M

=

PB (σ1 - x); and at

O, where a becomes equal to 1, M = PB (σ1 - 1). The curvature M P(); and at point

y", at a point distant a from B, is

ΕΙ

=

ΕΙ

Hence it follows that the slope

and the deflection at O,

The value, y, is equal approximately to the expansion of OA, and may be written y = a L; where a is the coefficient of expansion for the range of temperature considered. Dealing similarly with the long arm O A, the curvature at O,

PA

y"

=

EI

(81-L); the

Since there is equilibrium and continuity at O, it follows that (1) the bending moments, and therefore the curvatures, in the two arms, are equal at O; (2) the slopes in the two arms are equal at O, but of opposite sign.

Hence the following equations may be formed:

(that is, the curvatures are equal at the point 0.)

(1)

. . (2)

(3)

1) I . . (4)

ΕΙ

(that is, the slopes are equal but of opposite sign at the point 0.)

There are thus four equations between the four unknown quan

tities P1, P, 8, and o1. Solving for 8, and σ,, to derive the points 81

Hence 8, may be evaluated in terms of L for various values of p. Also noting that when p < 1, 8, becomes σ1, σ may be expressed in terms of 7, though it is simpler to evaluate σ, from equation (7) 'and' from the values of 8, which have already been found. These values are given by the curves in Fig. 2, the dotted curve being for 81 and the full curve for σ1.

Writing σ, ol, where σ is the fraction given by the curve,

=

equation (1) gives

But at B, where the bending moment, and therefore the stress, is greatest,

where D is the external diameter of the pipe and

Now suppose p is constant, and, therefore also

f = aλ,

D

ī

= λ.

σ, then

and this equation, a being a constant, represents a series of straight lines giving the maximum stress at B for different values of λ,

along each of which p is constant. Such a series is given in Fig. 3, in which E has been taken as 15,000,000 and a the temperature range in degrees F. multiplied by 0.00000955, the pipe being assumed to be of copper. A horizontal line on this diagram through the maximum permissible stress (taken in this case to be 4,000 lbs.