EXAMPLES TO THE THIRD SERIES. 
XXXlll 
OF THE PRESSURE UPON BREST - SUMMERS, PLATES, AND 
PURLINS, SUSTAINING LOADED BEAMS AND RAFTERS, 
INCLINED AT GIVEN ANGLES TO THE HORIZONTAL 
PLANE. 
In Fig. 4, in the annexed plate, let the beam AC, fixed in position, 
be inclined to the horizontal and vertical planes, AB, BC at the angles 
CAB, ACB ; through G the centre of gravity of AC draw the vertical 
line FGH; through C draw any line ICE cutting FH in E; join EA 
and produce EA to K; then AC will be retained in its position by 
the ropes EC, EA ; or by the props CI, AK ; or by the rope EC and 
prop AK ; or by the rope EA and prop CT ; or by the brest-summers, 
plates or purhns P and Q ; the plane surfaces of which, viz. Qe, Ad, are 
at right angles to EC, EA respectively. 
Let the weight of the beam AC and the load thereon = W. 
Let Ec = 1-00 represent W ; draw ca, cb, parallel to EI, EK, cutting 
EK, EI in a and b ; then 
Pressure upon P in the direction EC = E6 XW (1); 
Pressure upon Q in the direction EA = EaXW (2). 
The beam AC, the centre of gravity G, and the vertical line FH 
through G, being fixed in position, let ICE, cutting FH in E, and the 
plate or purlin P, revolve round C as a centre, then E'AK' being drawn, 
the relative positions of P and Q are given which retain AC in equili- 
brium ; E'c'= 1-00 represents, as before, the weight AC, and E'6', E'a', 
the relative pressures upon P and Q respectively. 
When ICE coincides with BC, then KE is parallel to IE, and the 
surfaces Ce, Ad, perpendicular thereto, are horizontal, as in Figs. 7 and 
8 ; in which case there is evidently no horizontal thrust. 
In Fig. 7, let W = weight of AC ; then 
AD 
Vertical pressure p' upon P = ^^'^^^ 
DB 
Vertical pressure p upon Q = ;^ (4). 
If AD = DB, or the centre of gravity G be in the middle of AC, the 
vertical pressures upon P and Q — i weight of AC ; and 
In Fig. 8, the vertical pressure upon Q = weight of GG. 
Fig. 10.— When ICE, Fig. 4, is at right angles to AC, the relative 
positions of P and Q are as in Fig. 10. 
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