The stresses produced in the beams due to pure bending moment are known as Bending stresses or Flexural stresses
Pure Bending Moment
The constant Bending Moment without Shear Force (zero Shear force) is called Pure bending moment.
If the beam is subjected pure bending moment, the beam will bend as arc of a circle with a radius of R as shown in figure.
In the process of bending there is a layer that do not extend or contract. The layer that do not extend or contract due bending is called the neutral layer and the axial line passing through this layer is called Neutral axis.
Due to bending of beam one side of the neutral axis will be extended and on the other is in contraction. These deformations are caused by the stresses due to bending moment are known as flexural or bending stresses.
The elongation in one side fibers are due to tensile bending stresses and contraction in the other side fibers are due to compressive bending stresses.
Therefore, the axial stress is zero at the neutral axis and increases linearly to the top and bottom fibers form neutral axis.
Theory of Bending: Flexure Formula
In deriving the relations between the bending moments and flexure (bending) stresses the following assumptions are made.
Assumptions in Theory of Bending:
1.Transverse sections of the beam that were plane before bending remain plane even after bending.
2.The material of the beam is isotropic and homogeneous and follows Hooke's law and has the same value of Young's Modulus in tension and compression.
3.The beam is subjected to Pure bending and therefore bends in an arc of a circle.
4.The radius of curvature is large compared to the dimension of the cross-section.
5.Each layer is independent to enlarge or contract.
6.The stresses are purely longitudinal and local effects of point loads are neglected.
The Bending Equation:
Take a small element ABCD of length dx of a beam which is subjected pure bending as shown in figure. Consider
M = Pure Bending Moment
I = Moment of inertia about Neutral axis (N.A.)
f = Bending stress in the fiber PQ
y = Distance of the fiber PQ from N.A.
R = Radius of Curvature
E = Young's Modulus
Consider a fiber PQ of depth dy at distance of y form Neutral axis NN'
The length of NN' = dx
Form the arc ONN', NN' = Rf
Before bending The length of fibers
AC = BD = PQ = NN' = dx = Rf -- (1) After bending the fibers AC, BD, and PQ are deformed to A'C', B'D' and P'Q' as shown in fig.
Let us consider the strain in the fiber PQ,
strain e = Change in length/ original length
= (PQ- P'Q')/PQ
From arc OPQ, P'Q' = (R-y) f
and PQ = Rf,
strain e = [Rf - (R-y) f]/Rf = y/R Let us apply the Hooke's law
strain = stress/ Young's modulus
e = f/E = y/R
The bending stress equation is,
f/y = E/R --- (2)
f = y (E/R)
Here E and R are constants, therefore within the elastic limit, the bending stress (f) is directly proportional to the distance between the neutral axis and the fibre (y).
Moment of resistance
As we have discussed that when a beam will be subjected with a sagging pure bending, layers above the neutral axis will be subjected with compressive stresses and layers below the neutral axis will be subjected with tensile stresses.
Therefore, there will be force acting on the layers of the beams due to these stresses. Therefore the total compressive force will be equal to total tensile force. Due to this couple there will be moment about the neutral axis.
Total moment of these forces about the neutral axis for a section will be termed as moment of resistance of that section.
Let us consider the cross-section of the beam as displayed here in following figure.
Let us assume one strip PQ of thickness dy and area dA at a distance y from the neutral axis as displayed in above figure.
Let us determine the force acting on the layer due to bending stress and we will have following equation
dF = f x dA
Let us determine the moment of this layer about the neutral axis, dM as mentioned here
dM = dF x y
dM = f x dA x y
Substitute f value form equation (2)
dM = (E/R) x y x dA x y
dM = (E/R) x y2 dA --- (3)
Total moment of the forces on the section of the beam around the neutral axis, also termed as moment of resistance, could be secured by integrating the equation(3)
dM = (E/R) x y2 dA and we will have
Where I is the area moment of Inertia of the section about neutral axis.
Note:
1.The the moment of resistance equation
M/I = E/R ---(4)
2.The bending stress equation is,
f/y = E/R --- (2)
we can write these two equations in a single equation form,
M/I = f/y = E/R
This equation may be remember as
" May I flow you Every Rule "
Oh
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