3.17 BEAM
A rod or a bar of circular or rectangular cross-section, with its length very much greater than its thickness (so that there are no shearing stresses over any section of it) is called a beam.
If the beam be fixed only at one end and loaded at the other, it is called a cantilever.
3.18 BENDING OF A BEAM
Suppose we have a beam, of a rectangular cross-section, say, fixed at one end and loaded at the other (within the elastic limit) so as to be bent a little, as shown in figure with its upper surface becoming
slightly convex and the lower one concave. All the longitudinal filaments in the upper half of the beam thus get extended or lengthened, and therefore under tension, and all those in the lower half get compressed or shortened and therefore under pressure.
These extensions and compressions increase progressively as we proceed away from the axis on either side so that they are the maximum in the uppermost and the lowermost layers of the beam respectively. There must be a layer between the uppermost and the lowermost layers where the extensions in the upper half change sign to become compressions in the lower half. In this layer or plane, which is perpendicular to the section of the beam containing the axis, the filaments neither get extended nor compressed, i.e. retain their original lengths. This layer is therefore called the neutral surface of the beam.
3.19 THEORY OF SIMPLE BENDING
There are the following assumptions:
(i) That Hooke’s law is valid for both tensile and compressive stresses and that the value of Young‘s modulus (Y) for the material of the beam remains the same in either case. (ii) That there are no shearing stresses over any section of the beam when it is bent. This is
more or less ensured if the length of the beam is sufficiently large compared with its thickness.
(iii) That there is no change in cross-section of the beam on bending. The change in the shape of cross-section may result in a change in its area and hence also in its geometrical
moment of inertia Ig. Any such change is however always much too small and is, in
general, ignored.
(iv) That the radius of curvature of the neutral axis of the bent beam is very much greater than its thickness.
(v) That the minimum deflection of the beam is small compared with its length.
3.20 BENDING MOMENT
When two equal and opposite couples are applied at the ends of the rod it gets bent. The plane of bending is the same as the plane of the couple. Due to elongation and compression of the filaments above and below the neutral surface, internal restoring forces are developed which constitute a restoring couple. In the position of equilibrium the internal restoring couple is equal and opposite to the external couple producing bending of the beam. Both these couples lie in the plane of bending. The moment of this internal restoring couple is known as bending moment.
Expression for bending moment: Consider the forces acting on a cross-section through CD Fig. 3.7 of a bent beam. The external couple is acting on end B in the clockwise direction. The filaments above the neutral surface are elongated such that change in length is proportional to their distance from neutral axis. Therefore, the filaments to the right of CD and above neutral axis exert a pulling force towards left due to elastic reaction. Similarly, since the filaments below neutral axis are contracted, with change in length proportional to their distance from neutral axis, due to elastic reaction they exert a pushing force towards right as shown in figure.
Thus on CD, the forces are towards left above neutral axis while they are towards right below it. These forces form a system of anticlockwise couples whose resultant is the internal restoring couple. This couple is equal and opposite to the external couple producing bending in the beam, and keeps the position of beam to the right of CD in equilibrium. The moment of this internal restoring couple acting on the cross-section at CD is termed as bending moment. Its value is given by
G = YI R
g
Where Ig = Sda. z2 is called the geometrical moment of inertia of the cross-section of the
beam about the neutral filament (This quantity is analogous to the moment of inertia with the difference that mass is replaced by area). For a beam of rectangular cross-section of width b and thickness d
I = 1
12
3
bd
and bending moment G = Ybd
R
3
12
For a beam of circular cross-section of radius r its value is
I = 1
4
4
pr
and Bending moment G = Y r
R
p 4
4
In the position of equilibrium, this bending moment balances the enternal bending couple t, thus
t = C = YI
R
or R = YI
t
showing that the beam of uniform cross-section is bend into an arc of circle, since R is constant for given t.
3.21 THE CANTILEVER
When a beam of uniform cross-section is clamped hori- zontally at one end and could be bent by application of a load at or near the free end, the system is called a cantile- ver.
When the free end of the cantilever is loaded by a weight W (= Mg), the beam bends with curvature chang- ing along its length. The curvature is zero at the fixed end and increases with distance from this end becoming maxi- mum at the free end. This is because of the fact that at distance x from fixed end, for equilibrium of portion CB
of the beam the moment of external couple is W(l – x), where l is the length of the cantilever. Thus for equilibrium of portion CB of cantilever, we have
G = YI
R = W(l – x)
Here it is assumed that the weight of the beam is negligible.
Now the radius of curvature R of the neutral axis at P distant x from fixed end, and having depression y is given by 1 R = d y dx dy dx 2 2 2 3 2 1+
b
g
where (dy/dx) is the slope of the tangent at print P(x, y). If the depression of the beam is small
dy dx