OI=ON+NI=AV(AB+LM)+RN cos. RNI=ux{2a+x(tan. a, -tan. a,)}+Pcos.;

x-(λ+k) cot.px (2a+x(tan. a, - tan. a)}+Pcos.
a,- ¡°

Transposing and reducing,

y=

Aux (2a+x(tan. a, -tan. a)} + P(xsin.-k cos. )
μx{2a+x(tan. a,- tan. a)} +P cos.

;

but substituting x for c in equation (414), and multiplying
both sides of that equation by the denominator of the fraction
in the second member, and by the factor, we have

x{2a+x(tan. a,-tan. a,)}=1px3(tan. 2a-tan.a2)+x2a tan.a,+}μza2;

2

เ

4x3(tan.2 a, ―tan. a,)+μx2a tan. a,+μxa2+2P(x sin. —k cos.)
μx{2a+x(tan. a,―tan. a2)}+2P cos.

which is the equation to the line of resistance in a buttress.
If the intrados AD be vertical, tan. a, is to be assumed =0.
If AD be inclined on the opposite side of the vertical to that
shown in the figure, tan. a, is to be taken negatively. The
line of resistance being of three dimensions in x, it follows
that, for certain values of y, there are three possible values
of ; the curve has therefore a point of contrary flexure.
The conditions of the equilibrium of the buttress are deter-
mined from its line of resistance precisely as those of the wall.

Thus the thickness a of the buttress at its summit being
given, and its height c, and it being observed that the dis-
tance CE is represented by a+c tan. «,, the inclination a, of
its extrados to the vertical may be determined, so that its
line of resistance may intersect its foundation at a given
distance m from its extrados, by solving equation (415) in
respect to tan. a,, having first substituted c for z and
a+c tan. am for y; and any other of the elements deter-
mining the conditions of the stability of the buttress may in
like manner be determined by solving the equation (the same
substitutions being made in it) in respect to that element.

313. A WALL OF UNIFORM

THICKNESS

SUSTAINING THE

PRESSURE OF A FLUID.

If E be taken to represent the surface of the fluid, IK any

B

E

section of the wall, and EP two thirds the depth EK; then will P be the centre of pressure* of EK, the tendency of the fluid to overturn the portion AKIB of the wall being the same as would be produced by a single pressure applied perpendicular to its surface at P, and being equal in amount to the weight of a mass of water whose base is equal to EK, and its height to the depth of the centre of gravity of EK, or to EK. Let AK=x, AE=e, weight

of each cubic foot of the fluid=μ1;

...P=(x− e). (x—e)μ,=}(x—e)3⁄4μ ̧•

Let the direction of P intersect the axis of the wall in O; let it be represented in magnitude by OS; take ON to represent the weight of the portion AKIB of the wall; complete the parallelogram SN, and produce its diagonal to meet IK in Q; then will Q be a point in the line of resistance. Let QM=y, AB=a, weight of each cubic foot of material of QM RN

wall. By similar triangles, MO NO Now

QM=y, MO=PK=}EK=}(x-e), RN=OS=P
=}μ,(x—e)2, NO=weight of ABIK=μax ;

у

=

Dividing numerator and denominator of this equation

by 1, and observing that the fraction

represents the ratio

of the specific gravities of the material of the wall and the fluid, we have

Hydrostatics and Hydrodynamics," by the author of

* Treatise on 66 this work, Art. 38. p. 26.

which is the equation to the line of resistance in a wall of uniform thickness, sustaining the pressure of a fluid.

314. To determine the thickness, a, of the wall, so that its height, h, being given, the line of resistance may intersect its foundation at a given distance, m, within the extrados.

Substituting, in equation (416), h for x, and ¦a—m for y, and solving the resulting equation in respect to a, we obtain

whence it is apparent that y increases continually with x; so that the nearest approach is made by the line of resistance, to the extrados of the pier, at its lowest section. m therefore represents, in the above expression, the modulus of stability (Art. 286.).

315. The conditions necessary that the wall should not be overthrown by the slipping of the courses of stones on one another.

The angle SRO represents the inclination of the resultant pressure upon the section IK to the perpendicular; the proposed condition is therefore satisfied, so long as SRO is less than the limiting angle of resistance .

316. THE STABILITY OF A WALL OF VARIABLE THICKNESS SUSTAINING THE PRESSURE OF A FLUID.

Let us first suppose the internal face AB of the wall to be vertical; let XY be any section of it, P the centre of pressure of EX, and SM a vertical through the centre of gravity of the portion AXYD of the wall. Produce the horizontal direction of the pressure P of the fluid, supposed to be collected in its centre of pressure, to meet MS in S, and let SK be taken to represent it in magnitude, and ST to represent the weight of the portion AXYD of the wall, and complete the parallelogram STRK; then will its diagonal SR represent the direction and amount of the resultant pressure upon the mass AXYD, and if it be produced to intersect XY in Q, Q will be a point in the line of resistance.

=

Let AX=x, XQ=y, MX=λ, AE=e, AD-a, inclination of DC to verticala, μ weight of cubical foot of wall, weight of cubical foot of fluid. By similar triangles, QM RT

Now

QM=QX-MX=y-λ, SM=PX=}EX*=}(x−e); RT=pressure of fluid on EX=EX.μ,EX=1μ,(x−e)2+: ST=weight of mass AY= {2a+x tan. a} xu.

* The centre of pressure of a rectangular plane surface sustaining the pressure of a fluid is situated at two thirds the depth of its immersion.Hydrostatics, p. 26.

The pressure of a heavy fluid on any plane surface is equal to the weight of a prism of the fluid whose base is equal in area to the surface pressed, and its height to the depth of the centre of gravity of the surface pressed.-Hydrostatics, Art. 31.

Let M

μ

=σ; then, if the fluid be water,

represents the

specific gravity of the material of the wall; and if not, it represents the ratio of the specific gravities of the fluid and wall.

for

Now making a=0 in equation (414), and substituting

and x for c,

a2 tan.2a+ax tan. a+a23 tan. 2a + ax2 tan. « +a2x

x tan. a + 2a

Adding this equation to the preceding,

1

y=

2ax+x2 tan.<

(x-e)3 +3 tan.2 a + ax2 tan. a+ a2x

2ax + x2 tan. a

(419);

which is the equation to the line of resistance to the wall, the conditions of whose stability may be determined from it as before (see Arts. 291. 293.)

317. The conditions necessary that no course of stones composing the wall may slip upon the subjacent course.

This condition is satisfied when the inclination of SQ to the perpendicular to the surface of contact at Q is less than the limiting angle of resistance ; that is, when QSM<¢, or when

No course of stones will be made by the pressure of the fluid to slip upon the subjacent course so long as this condition is satisfied.

It is easily shown that the expression forming the second