48 APPLIED MECHANICS 
A general construction for determining the centre of pressure or 
centre of stress of any plane figure when the pressure or stress varies ~ 
0 E: Zz 
Pp a H b B 
4 
R M 
S a N 
Ag LLLLi Lhe 
Q A, M, D B, N, Cc 
Fig. 46. 
uniformly in one direction is illustrated by Fig. 46. ABNCDMA is a 
plane figure supposed to be vertical, and AB and CD are horizontal, AA, 
is the altitude of the figure, and the pressure or stress is supposed to 
vary uniformly from an amount represented by AP at the level AB to an 
amount represented by A,Q at the level CD. AP and A,Q are horizontal. 
Join QP and produce it to meet A,A produced at O. Draw any 
horizontal SRMN to cut the given figure. Draw the horizontal OF, and 
the verticals MM, and NN). Through K, the middle point of MN, draw 
the vertical KF. Join FM, and FN, cutting MN at mand xn. If this 
construction be repeated at a sufficient number of levels, and all points 
corresponding to m be joined, also all points corresponding to m, a figure 
abnCDma is obtained, and the centroid of this figure will be the centre 
of pressure or centre of stress of the original figure. 
The proof is as follows. Suppose that the line MN is the centre line 
of a very narrow horizontal strip of the original figure, and let the width 
. of this strip be denoted by w. The magnitude of the resultant pressure 
or stress on this strip is equal to MN x wx RS, and it will act at K, the 
middle point of MN. 
Since SRMN is parallel to QA,DC, 
M,N, :mn::O0A,: OR, 
and, MN : mn::A,Q: RS, 
therefore MN x RS=mn x A,Q, 
and MN x wx RS=mn x w x A,Q, 
that is, the resultant of the pressure or stress on the strip of length mn 
when subjected to a pressure or stress A,Q will have the same magnitude 
as the resultant of the pressure or stress on the strip of length MN when 
subjected to a pressure or stress RS, and it will act at the same point K, 
which is also the middle point of mun. 
It follows that the resultant of the pressure or stress on the figure 
