CHAPTER VII 
BEAMS AND BENDING 
99. Bending Moments and Shearing Forces on Beams.—In 
general the lines of action of the external forces acting on a beam are 
ndicular to the length of the beam, and they may be considered as 
in one plane, and this plane, in what follows, will be taken as the 
pene of the paper. The external forces acting on a beam must of course 
nee one another if the beam is at rest. 
Consider the case of a horizontal beam ABC (Fig. 106) acted on by 
any number of vertical forces P,, P,, etc. Take any cross section YY 
dividing the beam into two parts AB and BC. Consider the equilibrium 
of the part BC. Let the distances of P,, P,, etc., from YY be 2, 2g, ete. 
Let R (Fig. 107) be the resultant of the external forces P,, P,, ete., 
acting on BC. The magnitude of R will be the algebraical sum of the 
forees P,, P,, etc., acting on BC. (For the forces shown in Fig. 106, 
os Ly p fs Y i c 
TA Be xs 1 F ¢| a ee 
| 4 P, ae : R ‘s 
Fr@. 106. Fira. 107. Fria. 108. Fra. 109. 
“~<~ 
R=P,-P,-P,). The distance a of R from YY must be such that the 
‘moment of R: about a horizontal axis in YY is equal to the algebraical 
sum of the moments of P,, P,, etc., about the same axis. (For the 
forces shown in Fig. 106, Ra = P,x, — Patty — P,v,). Expressed in another 
way, R==P and Ra= (Px). 
3 The force R tends to turn BC about a horizontal axis in YY ; in other 
words, R tends to bend the beam at YY (Fig. 108), and the bending 
_ moment is Ra==Pz. R also tends to make BC slide on AB at YY; in 
other words, R tends to shear the beam at YY (Fig. 109), and the shearing 
» force is. R= =P. 
he bending and shearing effects of R on BC may perhaps be made 
_ more apparent by the artifice of applying at YY two opposite forces 
: a and Re» each equal and parallel to R, as shown in Fig. 110. The 
dition of the forces R, and R, will evidently not affect the equilibrium 
= of ae The forces R and R, being equal and acting in opposite 
directions, form a couple which can only produce a turning or 
action on BC, while the remaining force R, can only make or 
ee eens iS a 
87 
