Transverse Shear● In general, a beam will support both shear and bending moment.
● The shear force V is the result of shear stress distribution at the cross-section
● Due to complementary property of shear stress longitudinal shear stress will also
develop
Transverse Shear● Because of shear stresses shear strains will develop and the beam will deform
● The deformations cause complicated distortion of the cross section
● Cross section will warp
However we will assume that warping is small and cross sections remain plane.
The Shear Formula
Consider the horizontal force equilibrium of an element with thickness dx extracted from the beam
We exclude V, V + dV and w(x) from the free body diagram because we only consider horizontal force equilibrium
The Shear Formula
Now consider the shaded top portion of the element.
This segment has width t and both cross sectional
faces have an area A'. Since the moment acting on
both face differ by dM, equilibrium will not be satisfied
unless a longitudinal stress acts on the bottom face.
The Shear FormulaEquations for horizontal force equilibrium
Solving for
Noting that and define
Where
is the geometric center of the shaded area
The Shear Formula
= the shear stress in the member at the point located a distance y' from the neutral axis. Assumed to be constant and therefore averaged across the width t of the member
V = Internal shear force
I = Moment of inertia of entire cross sectional area
t = Width of the member's cross section area measured at the point y'
Q = , where A' is the area of the top (or bottom) portion of the member's cross-sectional area, above (or below) the section plane where t is measured, and is the distance from the neutral axis to the centroid of A'
The Shear Formula: Limitations● The shear formula was derived using the flexure formula therefore the
material has to be linear elastic and the same modulus of elasticity E in tension and compression
● It is assumed that the shear stress is uniform over the width t i.e. the average stress is computed. In reality the stress is not uniform.
● The maximum stress changes as function of the width and height ratio b/h of the cross section
● For b/h = 0.5, max
is 3% greater than the average stress ● For b/h = 2,
max is about 40 % greater than the average stress
● Therefore, the shear formula is not suitable for wide flange beams stress calculations
The Shear Formula: Limitations
Shear formula is only applicable across sections which are perpendicular to the outer boundary.
Irregular nonrectangular cross sections:
