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6.1 Introduction
-- In a long beam, the dominating design factor:
m
McI
( )xy
Vaverage
A
-- Primary design factor
-- Minor design factor
-- In a short beam, the dominating design factor:
32max
VA
[due to transverse loading]
y component: xydA V 0z component: xzdA
Equation of equilibrium:
-- Shear stress xy is induced by transverse loading.
-- In pure bending -- no shear stress
6.2 Shear on the Horizontal Face of a Beam Element
0 0: ( )x D CF H dA A
MyI
C DI I I Knowing and
D CM MH ydA
I
AWe have (6.3)
SincedM
Vdx
( / )D CM M M dM dx x V x
VQH x
I
H VQq
x I
Therefore,
and
(6.4)
(6.5)
Defining Q ydA
= shear flow
= horizontal shear/length
here Q = the first moment w.t.to the neutral axis
Q = max at y = 0
The same result can be obtained for the lower element C'D' D''C''
6.3 Determination of the Shearing Stresses in a Beam
ave
H VQ x VQA I t x It
ave
VQIt
VQH x
I
(6.6)
= ave. shear stress
A = t x
At the N.A. Q = max, but ave may not be max, because of t
6.4 Shearing Stresses xy in Common Types of Beams
-- for narrow rectangular beams
xy
VQIt
2 21 12 2
( ) ( ) ( )Q Ay b c y c y b c y
1( )
2 y c y
t = b (6.7)
Q Ay
Also,3
3212 3bh
I bc
Hence,2 2
3
34xy
VQ c yV
Ib bc
Knowing A = 2bc, it follows
2
2
31
2( )xy
V yA c
maxxy
0xy
(6.9)
This is a parabolic equation with
@ y = c
@ y = 0 -- i.e. the neutral axis
At y = 0, max
3
2
V
A (6.10)
This is only true of rectangular cross-section beams.
ave
VQIt
maxweb
VA
Special cases:
American Standard beam (S-beam)
or a wide-flange beam (W-beam)
-- over section aa’ or bb’
-- Q = about cc’
(6.6)
(6.11)
For the web:
Assuming the entire V is carried by the web, since the flanges carry little shear force:
6.5 Further Discussion of the Distribution of Stresses in a Narrow Rectangular Beam
2
2
31
2( )xy
P yA c
x
Pxy MyI I
(6.12)
(6.13)
Plane sections do NOT remain plane – warping takes place, when a beam is subjected to a transverse shear loading
H VQq
x I
VQH x
I
0 0: ( )x D CF H dA A
Using similar procedures in Sec. 6.2, we have
= shear flow (6.5)
(6.4)
6.7 Shearing Stresses n Thin-Walled Members
VQH x
I
These two equations are valid for thin-walled members:
(6.4)
(6.6)ave
H VQ x VQA I t x It