Transcript Document

Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Example 6.04
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.
A square box beam is constructed from
four planks as shown. Knowing that the
spacing between nails is 44 mm. and the
beam is subjected to a vertical shear of
magnitude V = 2.7 kN, determine the
shearing force in each nail.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
6-1
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Example 6.04
For the upper plank,
Q  Ay  18mm76mm47mm
 64296mm3
For the overall beam cross-section,
I  121 112m m  121 76 m m
4
4
 10332m m4
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
6-2
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Example 6.04
SOLUTION:
• Determine the shear force per unit
length along each edge of the upper
plank.
VQ 2.7kN 64296m m
q

 16.8 N m m
4
I
10332m m
q
f   8.4 N m m
2
 edge forceper unit length
3
• Based on the spacing between nails,
determine the shear force in each nail.
N 

F  f    8.4
44mm
 mm 
F  369.6 N
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
6-3
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Shearing Stresses in Thin-Walled Members
• Consider a segment of a wide-flange
beam subjected to the vertical shear V.
• The longitudinal shear force on the
element is
H 
VQ
x
I
• The corresponding shear stress is
 zx   xz 
H VQ

t x It
• Previously found a similar expression
for the shearing stress in the web
 xy 
VQ
It
• NOTE:  xy  0
 xz  0
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
in the flanges
in the web
6-4
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Shearing Stresses in Thin-Walled Members
• The variation of shear flow across the
section depends only on the variation of
the first moment.
q t 
VQ
I
• 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.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
6-5
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Shearing Stresses in Thin-Walled Members
• For a wide-flange beam, the shear flow
increases symmetrically from zero at A
and A’, reaches a maximum at C and
then 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.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
6-6
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Plastic Deformations
I
M

 Y  maximum elastic moment
• Recall: Y
c
• For M = PL < MY , the normal stress does
not exceed the yield stress anywhere along
the beam.
• For PL > MY , yield is initiated at B and B’.
For an elastoplastic material, the half-thickness
of the elastic core is found from
 1 yY2 
3
Px  M Y 1  2 
 3c 
2


• The section becomes fully plastic (yY = 0) at
the wall when
3
PL  M Y  M p
2
• Maximum load which the beam can support is
Pmax 
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
Mp
L
6-7
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Plastic Deformations
• Preceding discussion was based on
normal stresses only
• Consider horizontal shear force on an
element within the plastic zone,
H   C   D dA   Y   Y dA  0
Therefore, the shear stress is zero in the
plastic zone.
• Shear load is carried by the elastic core,
3 P 
 xy 
1
2 A 
 max 
y 2 
where A  2byY
2
yY 
3P
2 A
• As A’ decreases, max increases and
may exceed Y
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
6-8
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Sample Problem 6.3
SOLUTION:
• For the shaded area,
Q  10819.6122.2
 259.3E 03mm3
• The shear stress at a,
Knowing that the vertical shear is 220
kN in a W250x101 rolled-steel beam,
determine the horizontal shearing
stress in the top flange at the point a
located 108 mm from the edge of the
beam.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.


VQ 220N 259.3E03 mm3


It
164E 06 mm4 19.6 mm


  17.7MPa
6-9
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Unsymmetric Loading of Thin-Walled Members
• If the shear load is applied such that the beam
does not twist, then the shear stress distribution
satisfies
D
B
E
VQ
 ave 
V   q ds F   q ds    q ds   F 
It
B
A
D
• F and F’ indicate a couple Fh and the need for
the application of a torque as well as the shear
load.
F h  Ve
• When the force P is applied at a distance e to the
left of the web centerline, the member bends in a
vertical plane without twisting.
• The point O is referred to as the shear center of
the beam section.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
6 - 10
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Unsymmetric Loading of Thin-Walled Members
• Beam loaded in a vertical plane
of symmetry deforms in the
symmetry plane without
twisting.
x  
My
I
 ave 
VQ
It
• Beam without a vertical plane
of symmetry bends and twists
under loading.
x  
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
My
I
 ave 
VQ
It
6 - 11
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Example 6.05
• Determine the location for the shear center of the
channel section with b = 100 mm., h = 150 ., and t = 4
mm.
e
• where
b
Fh
I
b VQ
Vb h
F   q ds  
ds   st ds
I0 2
0
0 I
Vthb2

4I
2
1 3
1 3
h 
I  I web  2 I flange  th  2  bt  bt   
12
 2  
12
1 th2 6b  h 
 12
• Combining,
e
b
2
h
3b

100m m
150m m
2
3100 mm.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
e  40 mm
6 - 12
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Example 6.06
• Determine the shear stress distribution for
V = 11 kN.

q VQ

t
It
• Shearing stresses in the flanges,
VQ V
h Vh
 st  
s
It It
2 2I
Vhb
6Vb
B  1 2

2 12 th 6b  h  th6b  h 




611kN100mm
 14.6MPa
4mm150mm6 100m m 150m m
• Shearing stress in the web,
VQ V  18 ht4b  h  3V 4b  h 
 max 
 1 2



It
th
6
b

h
t
2th6b  h 
12
311kN 4 100m m 150m m

 20.16MPa
24m m150m m6 100m m 150m m
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
6 - 13