Shear Stresses in Beams

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Shear Stresses in Beams
Previously, pure bending was assumed. ? V 0
In most beams, shear forces exist.
? We need to take this into consideration.
The cross section actually warps however, the
shear strain which accompanies the warping has
little effect in the normal strain thus, the
flexure formula remains approximately valid.
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The existence of horizontal shear in beams
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Recall




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Let ? y dA Q ? first moment of
the area wrt the C.A.
?

Recall that
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?
(1)
?
From Equilibrium of the FBD above,
?Fx 0 ? R (x?x) ? R (x) q ?x 0
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Dividing by ?x ?
?
?
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? From eqs. (1) (2) above
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OR
OR
This is the well-known shear Stress Formula
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Very Very Important (!!!)
How to calculate Q ??!!
Something is strange in this derivation! What
is it?
Only equilibrium consideration is utilized in
this derivation. Kinematic (Geometric
Compatibility) assumptions are not used!
This is one of the very few formulas in
structural mechanics in which compatibility is
not utilized in deriving the formula!
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