Barber J.R. Intermediate Mechanics of Materials
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6.8 Summary 335
Further reading
Sections 6.1–6.3
W.B. Bickford (1998), Advanced Mechanics of Materials, Addison Wesley, Menlo
Park, CA, §§4.1–4.4.
A.P. Boresi, R.J. Schmidt, and O.M. Sidebottom (1993), Advanced Mechanics of
Materials, John Wiley, New York, 5th edn., §§8.1–8.3.
R.D. Cook and W.C. Young (1985), Advanced Mechanics of Materials, Macmillan,
New York, §§9.5–9.8.
Section 6.4, 6. 6
W.B. Bickford (1998), loc. cit., §§4.5–4.7.
R.D. Cook and W.C. Young (1985), loc. cit., §9.9.
Section 6.5
W.B. Bickford (1998), loc. cit., §§3.4, 3.5.
R.D. Cook and W.C. Young (1985), loc. cit., §8.7.
Section 6.7
J.R. Barber (2010), Elasticity, Springer, Dordrecht, 3rd edn., Chapter 16.
S.P. Timoshenko and J.N. Goodier (1970), Theory of Elasticity, McGraw-Hill, New
York, 3rd edn., Chapter 10.
336 6 Shear and Torsion of Thin-walled Beams
Problems
Section 6.1
In Problems 6.1–6.10, assume that the shear force V
y
acts through the shear centre
of the section, as defined in §6.2.
A
B
C
D
3
3
3
40
30
30
60
all dimensions in mm
Figure P6.1
6.1. The second moment of area for the symmetric beam section of Figure P6.1 is
I
x
=0.288 ×10
6
mm
4
. The beam is loaded by a vertical shear force of 4kN. Find the
maximum shear stress and compare it with the average value V
y
/A
web
, where A
web
is
the area of the vertical part AD of the section only.
6.2. Figure P6.2 shows the dimensions of a W36×230 I-beam (I
x
=15×10
3
in
4
) that
is loaded by a vertical shear force of 75,000 lbs. Find the maximum shear stress.
6.3. The wide flange I-beam of Figure P6.3 has second moment of area I
x
=7a
3
t/12
and is loaded by a vertical shear force V
y
. Find (i) the shear stress at A, (ii) the shear
stress at O and (iii) the average shear stress in the web.
Problems 337
1.26
0.76
35.7
16.5
t
t
a
a
O
A
.
all dimensions in inches
Figure P6.2 Figure P6.3
6.4. Figure P6.4 shows the cross section of a thin-walled beam subjected to a vertical
shear force V
y
. Find the distribution of shear stress in the section.
t
t
a
2a
30
o
Figure P6.4
338 6 Shear and Torsion of Thin-walled Beams
6.5. The slit cylindrical tube of Figure P6.5 has mean radius a and wall thickness
t. Find the distribution of shear stress in the section, when the beam is loaded by a
vertical shear force V
y
.
t
O
a
V
y
Figure P6.5
6.6. The crescent beam section of Figure P6.6 has mean radius a and thickness t =
t
0
cos
θ
. Find the distribution of shear stress in the section, when the beam is loaded
by a vertical shear force V
y
.
O
t
θ
a
Figure P6.6
6.7. Figure P6.7 shows an I-beam fabricated by welding together three rectangular
plates. Find the shear stress in the welds when the vertical shear force is 800 kN.
Problems 339
25
25
300
800
25
25
10
weld
detail
all dimensions in mm
Figure P6.7
6.8. An I-beam is fabricated by bolting four angle irons to a rectangular plate, using 1
inch diameter bolts, as shown in Figure P6.8. Determine the number of bolts needed
per unit length if the average shear stress in the bolts is to be equal to the maximum
shear stress at the neutral plane when the beam transmits a vertical shear force.
0.6
0.6
2.5
10
3
all dimensions in inches
Figure P6.8
340 6 Shear and Torsion of Thin-walled Beams
6.9. The beam of Figure P6.1 is fabricated from a channel and two bars by welding,
details of the weld being shown in Figure P6.9. If the minimum weld thickness is 1
mm, find the shear stresses in the weld.
3
1
A
B
3
all dimensions in mm
Figure P6.9
6.10. A team of industrial engineers wishes to attach a small bracket to the I-beam
of Figure P6.2 in order to mount a video camera to record factory operations. Details
of the bracket and its attachment are shown in Figure P6.10 It is proposed to use 1/8
inch diameter screws with an axial spacing of 6 inches. Estimate the average shear
stress in the screws. You may neglect the effect of the bracket on the second moment
of area of the beam.
Would you advise the engineers to relocate the bracket and if so, to where?
1.5
1.5
6
4
1
8
thick
all dimensions in inches
Figure P6.10
Section 6.2
6.11. Find the location of the shear centre for the beam of Figure P6.1.
6.12. Find the location of the shear centre for the slit box section shown in Figure
P6.12. All segments have thickness t.
Problems 341
a
a
Figure P6.12
6.13. Find the location of the shear centre for the slit cylindrical tube of Figure P6.5.
6.14. Find the location of the shear centre for the crescent section of Figure P6.6.
6.15. Find the location of the shear centre for the beam section of Figure P6.4.
6.16. Find the location of the shear centre for the beam section shown in Figure
P6.16.
t
a
o
45
2a
4a
t
t
Figure P6.16
342 6 Shear and Torsion of Thin-walled Beams
6.17. Find the distance c defining the location of the shear centre C for the beam
section shown in Figure P6.17.
t
O
a
.
C
3a
2
c
Figure P6.17
6.18. Find the distance c defining the location of the shear centre C for the beam
section shown in Figure P6.18.
t
a
.
C
c
a
2
Figure P6.18
Section 6.4
6.19. The closed thin-walled section of Figure P6.19 has an equilateral triangular
cross section of side a, but the vertical side is of thickness 2t, compared with t for
the two inclined walls. The section transmits a vertical shear force V
y
, whose line of
action passes through the point O. Find the shear stress distribution in the section.
In particular, find the maximum shear stress and identify the points where the shear
stress is zero.
Problems 343
t
30
o
30
o
a
a
2t
O
V
y
t
Figure P6.19
6.20. The outer and inner circular walls of a closed thin-walled circular tube have
slightly different centres, as shown in Figure P6.20, so that the thickness of the sec-
tion varies according to
t = t
0
(1 −
ε
cos
θ
) ,
where
ε
< 1. The tube is loaded by a vertical shear force V
y
acting through the point
O. Find the maximum shear stress in the section.
t
V
y
O
a
θ
Figure P6.20
344 6 Shear and Torsion of Thin-walled Beams
6.21. The closed thin-walled section of Figure P6.21 transmits a vertical shear force
V
y
, whose line of action passes through the point O
1
. Find the shear stresses at the
points A and B. Note: By symmetry, the shear centre for this section must be at C.
A
B
V
y
a
a
C
1
O
O
2
2a
t
Figure P6.21
6.22. The closed circular section of Figure P6.22 is loaded by a vertical shear force
V
y
whose line of action is such that the shear stress at B is zero. Find the shear stress
at A and the distance d defining the line of action. Does this mean that the same
distance d defines the shear centre for the open section of Figure P6.5?
t
O
a
A B
d
V
y
Figure P6.22