or if the
stress varied rarely 5.6 tons per sq. in.; for members subjected to
alternations of tension and thrust frequently 3.3 tons per sq. in. or 5
tons per sq. in. if the alternations were infrequent. The shearing area of
rivets in tension members was made 11/2 times the useful section of plate in
tension. For compression members the shearing area of rivets in butt-joints
was made half the useful section of plate in compression.

[Illustration: FIG. 37.]

20. _Determination of Stresses in the Members of Bridges._--It is
convenient to consider beam girder or truss bridges, and it is the stresses
in the main girders which primarily require to be determined. A main girder
consists of an upper and lower flange, boom or chord and a vertical web.
The loading forces to be considered are vertical, the horizontal forces due
to wind pressure are treated separately and provided for by a horizontal
system of bracing. For practical purposes it is accurate enough to consider
the booms or chords as carrying exclusively the horizontal tension and
compression and the web as resisting the whole of the vertical and, in a
plate web, the equal horizontal shearing forces. Let fig. 37 represent a
beam with any system of loads W_1, W_2, ... W_n.

The reaction at the right abutment is

R_2 = W_1x_1/l+W_2x_2/l+...

That at the left abutment is

R_1 = W_1+W_2+...-R_2.

Consider any section a b. The total shear at a b is

S = R-[Sigma](W_1+W_2 ...)

where the summation extends to all the loads to the left of the section.
Let p_1, p_2 ... be the distances of the loads from a b, and p the distance
of R_1 from a b; then the bending moment at a b is

M = R_1p-[Sigma](W_1p_1+W_2p_2 ...)

where the summation extends to all the loads to the left of a b. If the
loads on the right of the section are considered the expressions are
similar and give the same results.

If A_t A_c are the cross sections of the tension and compression flanges or
chords, and h the distance between their mass centres, then on the
assumption that they resist all the direct horizontal forces the total
stress on each flange is

H_t = H_c = M/h

and the intensity of stress of tension or compression is

f_t = M/A_th,
f_c = M/A_ch.

If A is the area of the plate web in a vertical section, the intensity of
shearing stress is

f_x = S/A

and the intensity on horizontal sections is the same. If the web is a
braced web, then the vertical component