Chap. XXVII.

CANALS AND RIVER LOCKS.

1541

is opposite, must form a resisting power, the energyof which should be equal to that of the wall, whichhas been supposed in the place of the gates. Eachof these masses is composed of a triangular prism,the base of which is ABE, and of a parallelepiped,the base of which is BCDE; the cube of the tri-angular prism is a 2 h) its arm of lever is %a~i~x,thus its energy will be \a 2 h + a 2 hx. , The cube ofthe parallelepiped is ahx , its arm of lever is £x,thusits energy is \ahx 2 , so that the energy of this re-sisting mass is %a 3 h + a 2 hx + % ahx 2 : the other massbeing the same, the total energy will be \a 3 h +

2 a s hx + ahx 2 , and as that energy should be equalto that of the wall, which we have supposed in theplace of the gates, which is \lh 3 , we shall have theequation, £a 3 A + 2a 2 kx + ahx 2 = \lh 3 , ora: 2 + 2ax =s

l/i 2 4a 8 Ik 2 a 2

---: whence we have x + a = V -•, and as

8a 3 8a 3

k 2 a 8

l=5 ‘2 metres, we shall have x + a— a /0 *65 —•-

a 3

If the thickness of the wall is equal to a third ofits height, or a = \h, we shall have # + a /1 '95h

+ 2 I 7 A a , or x + \h = \h + a/1 '95h + ^h 2 : if the thick-ness of the wall be made equal to half of its height,

we should have a=\h, and the equation would become x + \k = Vl'SOh + ^h 2 , or x + k= | k+ Vl' 30 h + -fch 2 . By means of these two equations, tables have been calculatedby Gauthey for the length of the shoulders.

.Fig. 2841.

mm

Fig. 2842.

Fig. 2843.

CHAMBER WALLS.

It may be remarked, that on giving to the thickness of walls half their height, theirlength need only be one-third, in order to have the requisite resistance, although their cubeis one-third greater. Supposing the rupture to take place from A in H, the lengthwould be less, but not considerably so; allowance must be made for the set-off of 12inches below the water, which increases the resistance of the wall, and puts it abovethe equilibrium. These shoulder walls are generally terminated by expanded wings, withan angle of 50°, on which it must be observed that, if the shoulders were shorter, the wing-walls would not be of much service in assisting the former to support the thrust of thewater, because the fracture would take place on the facing of the wing-walls, and therewould only be a small mass of masonry, A B C, to resist the thrust, whilst when the shoulderpieces A, D are long, the wing-w.alls may entirely resist the thrust, since the direction of therupture A E abuts at their extremity. The length of these walls should be equal to theradius of the gate, that they may open entirely and with facility, as the lockmen must walkon them to perform their operations.

The Wing-walls to the lower Gates should have a thickness equal to one-third of theirheight, on account of their supporting the earth; steps may be added to them when theysupport a talus, giving convenient access from the platform of the lock to the upper part:these walls generally terminate in the banks of the canal, and perpendicular return wallsmay be formed following their direction; but they are not so essential as in the upperpart, because the filtration of the*waters from the lower interval is not so dangerous as thatfrom the upper, or from the chamber of the lock.

Masses are generally constructed behind and below the hanging-posts, to receive theanchors which retain the collars, and to make them enter deeply into the masonry, in orderthat they may have a firm hold ; but they are also essential in preventing the water frominsinuating itself at the back of the side walls, and forming a disjunction between the earthand the masonry, where a void would allow the water from a chamber to percolate, andultimately cause the walls to give way : to prevent this alone a great thickness would nothe requisite ; but it must be increased on account of the anchors, two being attached to