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BENDING STRESSES IN BEAMS - UINT 3

The stresses produced due to constant Bending Moment (with zero Shear Force or pure bending) are know as Bending stresses.

Assumptions in Theory of Bending :
In deriving the relations between the bending moments and flexure (bending)stresses and between the Shear forces and Sharing stresses the following assumptions are made.
1.Transverse sections of the beam that were plane before bending remain so 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.

Flexure Formula :

Stresses caused by the bending moment are known as flexural or bending stresses. Consider a beam to be loaded as shown.

THE BENDING EQUATION: 

where R is the radius of curvature of the beam in mm, M is the bending moment in N·mm, fb is the flexural stress in MPa., I is the centroidal moment of inertia in mm4, and y is the distance from the neutral axis to the  fiber in mm.

THE BENDING EQUATION IS GIVEN BY, 

M/I = f/y = E/R 
Where 
M = Bending Moment
I = Moment of inertia about Neutral axis (N.A.)
f = Bending stress
y = Distance of the fiber from N.A.
R = Radius of Curvature
E = Young's Modulus
This equation may be remember as 

" May I flow you Every Rule "

M/I = f/y = E/R

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