Lecture15

1
• A. J. Clark School of Engineering •Department of Civil and Environmental Engineering• A. J. Clark School of Engineering •Department of Civil and Environmental Engineering• A. J. Clark School of Engineering •Department of Civil and Environmental Engineering• A. J. Clark School of Engineering •Department of Civil and Environmental Engineering• A. J. Clark School of Engineering •Department of Civil and Environmental Engineering
Third Edition
LECTURE
15
6.6 – 6.7
Chapter
BEAMS: SHEAR FLOW, THIN
WALLED MEMBERS
by
Dr. Ibrahim A. Assakkaf
SPRING 2003
ENES 220 – Mechanics of Materials
Department of Civil and Environmental Engineering
University of Maryland, College Park
LECTURE 15. BEAMS: SHEAR FLOW, THIN-WALLED MEMBERS (6.6 – 6.7) Slide No. 1
ENES 220 ©Assakkaf
Shear on the Horizontal Face of a Beam Element
• Consider prismatic beam
• For equilibrium of beam element
( )
∫
−
=∆
∑ ∫ −+∆==
A
CD
A
DDx
dAy
I
MM
H
dAHF σσ0
xVx
dx
dM
MM
dAyQ
CD
A
∆=∆=−
∫=
• Note,
flowshear
I
VQ
x
H
q
x
I
VQ
H
==
∆
∆
=
∆=∆
• Substituting,

2
ENES 220 ©Assakkaf
Shear on the Horizontal Face of a Beam Element
flowshear
I
VQ
x
H
q ==
∆
∆
=
• Shear flow,
• where
sectioncrossfullofmomentsecond
aboveareaofmomentfirst
'
2
1
=
∫=
=
∫=
+AA
A
dAyI
y
dAyQ
• Same result found for lower area
HH
QQ
q
I
QV
x
H
q
∆−=′∆
=
=′+
′−=
′
=
∆
′∆
=′
axisneutralto
respecthmoment witfirst
0
ENES 220 ©Assakkaf
Shearing Stress in Beams
Example 16
The transverse shear V at a certain
section of a timber beam is 600 lb. If
the beam has the cross section
shown in the figure, determine (a) the
vertical shearing stress 3 in. below
the top of the beam, and (b) the
maximum vertical stress on the cross
section.

3
ENES 220 ©Assakkaf
Example 16 (cont’d)
12 in.
8 in.
4 in.
8 in.
ENES 220 ©Assakkaf
From symmetry, the neutral axis is located
6 in. from either the top or bottom edge.
( ) ( )
( )( ) ( )( )[ ]
( )( ) ( )( )[ ] 3
3
3
4
33
in0.1124222528
in0.945.3212528
in3.981
12
84
12
128
=+=
=+=
=−=
′′
NAQ
Q
I
8 in.
4 in.
8 in.
12 in.
8 in.
4 in.
8 in.
N.A.
3 in.
( )
( )
( )
( )
psi2.171
43.981
1126000
)b(
psi7.143
43.981
946000
)a(
max
max
3
3
===
=== ′′
′′
It
VQ
It
VQ
Q
τ
τ
· ·· ·
··

4
ENES 220 ©Assakkaf
Longitudinal Shear on a Beam
Element of Arbitrary Shape
Consider a box beam obtained by
nailing together four planks as shown in
Fig. 1.
The shear per unit length (Shear flow) q
on a horizontal surfaces along which the
planks are joined is given by
flowshear
I
VQ
q == (1)
ENES 220 ©Assakkaf
Figure 1

5
ENES 220 ©Assakkaf
But couldBut could qq be determined if the planksbe determined if the planks
had been joined alonghad been joined along vertical surfacesvertical surfaces,,
as shown in Fig. 1b?as shown in Fig. 1b?
Previously, we had examined the
distribution of the vertical components
τxy of the stresses on a transverse
section of a W-beam or an S-beam as
shown in the following viewgraph.
ENES 220 ©Assakkaf
Shearing Stresses τxy in Common
Types of Beams
• For a narrow rectangular beam,
A
V
c
y
A
V
Ib
VQ
xy
2
3
1
2
3
max
2
2
=








−==
τ
τ
• For American Standard (S-beam)
and wide-flange (W-beam) beams
web
ave
A
V
It
VQ
=
=
maxτ
τ

6
ENES 220 ©Assakkaf
Longitudinal Shear on A Beam
But what about theBut what about the horizontalhorizontal
componentcomponent ττxzxz of the stresses in theof the stresses in the
flanges?flanges?
To answer these questions, the
procedure developed earlier must be
extended for the determination of the
shear per unit length q so that it will
apply to the cases just described.
ENES 220 ©Assakkaf
Longitudinal Shear on a Beam Element
of Arbitrary Shape
• We have examined the distribution of
the vertical components τxy on a
transverse section of a beam. We
now wish to consider the horizontal
components τxz of the stresses.
• Consider prismatic beam with an
element defined by the curved surface
CDD’C’.
( )∑ ∫ −+∆==
a
dAHF CDx σσ0
• Except for the differences in
integration areas, this is the same
result obtained before which led to
I
VQ
x
H
qx
I
VQ
H =
∆
∆
=∆=∆

7
ENES 220 ©Assakkaf
Example 17
A square box beam is constructed from
four planks as shown. Knowing that the
spacing between nails is 1.75 in. and the
beam is subjected to a vertical shear of
magnitude V = 600 lb, determine the
shearing force in each nail.
SOLUTION:
• Determine the shear force per unit
length along each edge of the upper
plank.
• Based on the spacing between nails,
determine the shear force in each
nail.
ENES 220 ©Assakkaf
For the upper plank,
( )( )( )
3
in22.4
.in875.1.in3in.75.0
=
=′= yAQ
For the overall beam cross-section,
( ) ( )
4
3
12
13
12
1
in42.27
in3in5.4
=
−=I
SOLUTION:
• Determine the shear force per unit
length along each edge of the upper
plank.
( )( )
lengthunitperforceedge
in
lb
15.46
2
in
lb
3.92
in27.42
in22.4lb600
4
3
=
==
===
q
f
I
VQ
q
• Based on the spacing between nails,
determine the shear force in each
nail.
( )in75.1
in
lb
15.46 





== lfF
lb8.80=F

8
ENES 220 ©Assakkaf
Shearing Stress in Thin-Walled
Members
It was noted earlier that Eq. 1 can be
used to determine the shear flow in an
arbitrary shape of a beam cross section.
This equation will be used in this section
to calculate both the shear flow and the
average shearing stress in thin-walled
members such as flanges of wide-
flange beams (Fig. 2) and box beams or
the walls of structural tubes.
ENES 220 ©Assakkaf
Members
Figure 2
Wide-Flange Beams

9
ENES 220 ©Assakkaf
Members
Consider a segment of a wide-flange
beam subjected to the vertical shear V.
The longitudinal shear force on the
element is
x
I
VQ
H ∆=∆ (2)
ENES 220 ©Assakkaf
Members
Figure 3

10
ENES 220 ©Assakkaf
Members
The corresponding shear stress is
Previously found a similar expression
for the shearing stress in the web
It
VQ
xt
H
xzzx =
∆
∆
≈=ττ (3)
It
VQ
xy =τ (4)
ENES 220 ©Assakkaf
Members
• The variation of shear flow across the
section depends only on the variation of
the first moment.
I
VQ
tq ==τ
• For a box beam, q grows smoothly from
zero at A to a maximum at C and C’ and
then decreases back to zero at E.
• The sense of q in the horizontal portions
of the section may be deduced from the
sense in the vertical portions or the
sense of the shear V.

11
ENES 220 ©Assakkaf
Members
• For a wide-flange beam, the shear flow
increases symmetrically from zero at A
and A’, reaches a maximum at C and the
decreases to zero at E and E’.
• The continuity of the variation in q and
the merging of q from section branches
suggests an analogy to fluid flow.
ENES 220 ©Assakkaf
Members
Example 18
Knowing that the
vertical shear is 50
kips in a W10 × 68
rolled-steel beam,
determine the
horizontal shearing
stress in the top
flange at the point a.

12
ENES 220 ©Assakkaf
Members
SOLUTION:
• For the shaded area,
( )( )( )
3
in98.15
in815.4in770.0in31.4
=
=Q
• The shear stress at a,
( )( )
( )( )in770.0in394
in98.15kips50
4
3
==
It
VQ
τ
ksi63.2=τ

Lecture15

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