Page 298 - Aircraft Stuctures for Engineering Student

9.1 Bending of open and closed section beams 279

9.1.3 Direct stress distribution due to bending
--a-----.-.--.--pp- I--Y-I_I--YIyYII_yI--~~,m *..IICILI..."~.CIII
Consider a beam having the arbitrary cross-section shown in Fig. 9.4(a). The beam
supports bending moments M, and My and bends about some axis in its cross-section
which is therefore an axis of zero stress or a neutral axis (NA). Let us suppose that the
origin of axes coincides with the centroid C of the cross-section and that the neutral
axis is a distancep from C. The direct stress a: on an element of area SA at a point
(x, )I) and a distance E from the neutral axis is, from the third of Eqs (1.42)
a= = E&? (9.1)
If the beam is bent to a radius of curvature p about the neutral axis at this particular
section then, since plane sections are assumed to remain plane after bending, and by a
comparison with symmetrical bending theory
E
& =-
P
Substituting for E= in Eq. (9.1) we have
EE
a, = -
P
The beam supports pure bending moments so that the resultant normal load on any
section must be zero. Hence
P

Therefore, replacing az in this equation from Eq. (9.2) and cancelling the constant
E/p gives
JdA = 0

Y Y
4

N

(a) (b)
Fig. 9.4 Determination of neutral axis position and direct stress due to bending.