Bending stresses on Beams:
The term beam refers to a component that is designed to support transverse loads, that is, loads that act perpendicular to the longitudinal axis of the beam as shown in figure.
Fig: A supported beam loaded by a force and a distribution of pressure
·When the beam is bent by the action of downward transverse loads, the fibres near the top of the beam contract in length whereas the fibres near the bottom of the beam extend.
·Somewhere in between, there will be a plane where the fibres do not change length. This is called the neutral surface. Such a deformation of beam is called the bending.
Fig: Positive and Negative bending and draw for negative bending
Bending Moment Equation:
Following assumptions are made while deriving the bending moment equation:
The constraints put on the geometry would form the assumptions:
- Beam is initially straight and has a constant cross-section.
- Beam is made of homogeneous material and the beam has a longitudinal plane of symmetry.
- Resultant of the applied loads lies in the plane of symmetry.
- The geometry of the overall member is such that bending not buckling is the primary cause of failure.
- Elastic limit is nowhere exceeded, and ‘E’ is same in tension and compression.
- Plane cross - sections remains plane before and after bending.
The bending equation is given as:
where M is the Moment of resistance,
y is the distance of the considered strip from the Neutral Axis,
I is the moment of inertial of the section about the N.A,
E is the Young's modulus
σ is the bending stress
R is the radius of curvature of the beam,
From the equation, we also get,
The term 1/R (= ρ) is known as the curvature of the section and is inversely proportional to the flexural rigidity (EI) of the section.
From simple bending theory equation, the maximum stress obtained in any cross-section is given as:
For any given allowable stress, the maximum moment which can be accepted by a particular shape of cross-section is therefore
For ready comparison of the strength of various beam cross-section this relationship is sometimes written in the form
is termed as section modulus.
STRAIN ENERGY DUE TO BENDING:
As we know that strain energy per unit volume =
SHEAR STRESSES IN BEAMS:
·Consider a beam of rectangular cross section of width (b) and depth (d), subjected to a vertical force (F).
· The shear stress (τ) act parallel to the S.F. and the distribution of the shear stress is uniform across the width b of the beam.
Assumptions in finding out the expression for transverse shear stress:
- For all values of y, τ is uniform across the width of the cross-section, irrespective of its shape.
- is derived from the assumption that bending stress varies linearly across the section and is zero at the centroid.
- The material is homogeneous and isotropic, and the value of E is the same for tension as well as compression.
Expression for transverse shear stress:
where τ is the shear stress, F is the shear Force, I is the moment of Inertia. Z is the section modulus of the beam, A is the area of cross section and is the centroidal distance.
SHEAR STRESS FORMULA FOR DIFFRENT SECTIONS:
SHEAR STRESS DISTRIBUTION OVER OTHER SECTIONS:
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