Third Edition
CHAPTER
8
MECHANICS OF
MATERIALS
Ferdinand P. Beer
E. Russell Johnston, Jr.
John T. DeWolf
Lecture Notes:
J. Walt Oler
Texas Tech University
Principle Stresses
Under a Given
Loading
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
Third
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MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Principle Stresses Under a Given Loading
Introduction
Principle Stresses in a Beam
Sample Problem 8.1
Sample Problem 8.2
Design of a Transmission Shaft
Sample Problem 8.3
Stresses Under Combined Loadings
Sample Problem 8.5
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Introduction
• In Chaps. 1 and 2, you learned how to determine the normal stress due
to centric loads
In Chap. 3, you analyzed the distribution of shearing stresses in a
circular member due to a twisting couple
In Chap. 4, you determined the normal stresses caused by bending
couples
In Chaps. 5 and 6, you evaluated the shearing stresses due to transverse
loads
In Chap. 7, you learned how the components of stress are transformed
by a rotation of the coordinate axes and how to determine the
principal planes, principal stresses, and maximum shearing stress
at a point.
• In Chapter 8, you will learn how to determine the stress in a structural
member or machine element due to a combination of loads and
how to find the corresponding principal stresses and maximum
shearing stress
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MECHANICS OF MATERIALS
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Principle Stresses in a Beam
• Prismatic beam subjected to transverse
loading
My
Mc
m 
I
I
VQ
VQ
 xy  
m 
It
It
x  
• Principal stresses determined from methods
of Chapter 7
• Can the maximum normal stress within
the cross-section be larger than
m 
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Mc
I
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MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Principle Stresses in a Beam
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MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Principle Stresses in a Beam
• Cross-section shape results in large values of t
xy
near the surface where sx is also large.
• smax may be greater than sm
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Sample Problem 8.1
SOLUTION:
• Determine shear and bending
moment in Section A-A’
• Calculate the normal stress at top
surface and at flange-web junction.
A 160-kN force is applied at the end
of a W200x52 rolled-steel beam.
• Evaluate the shear stress at flangeweb junction.
Neglecting the effects of fillets and
of stress concentrations, determine
whether the normal stresses satisfy a
design specification that they be
equal to or less than 150 MPa at
section A-A’.
• Calculate the principal stress at
flange-web junction
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Beer • Johnston • DeWolf
Sample Problem 8.1
SOLUTION:
• Determine shear and bending moment in
Section A-A’
M A  160 kN 0.375 m   60 kN - m
V A  160 kN
• Calculate the normal stress at top surface
and at flange-web junction.
MA
60 kN  m

S
512  106 m3
 117.2 MPa
y
90.4 mm
σb   a b  117.2 MPa 
c
103 mm
 102.9 MPa
a 
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MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Sample Problem 8.1
• Evaluate shear stress at flange-web junction.
Q  204  12.6 96.7  248.6  103 mm3
 248.6  106 m3


V AQ 160 kN  248.6  106 m3
b 

It
52.7  106 m 4 0.0079 m 


 95.5 MPa
• Calculate the principal stress at
flange-web junction
 max  12  b 
12  b 2   b2
2
102.9
 102.9 
2

 
  95.5
2
 2 
 169.9 MPa  150 MPa 
Design specification is not satisfied.
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Sample Problem 8.2
SOLUTION:
• Determine reactions at A and D.
• Determine maximum shear and
bending moment from shear and
bending moment diagrams.
The overhanging beam supports a
uniformly distributed load and a
concentrated load. Knowing that for
the grade of steel to used sall = 24 ksi
and t
all = 14.5 ksi, select the wideflange beam which should be used.
• Calculate required section modulus
and select appropriate beam section.
• Find maximum normal stress.
• Find maximum shearing stress.
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Sample Problem 8.2
SOLUTION:
• Determine reactions at A and D.
 M A  0  RD  59 kips
 M D  0  R A  41kips
• Determine maximum shear and bending
moment from shear and bending moment
diagrams.
M max  239.4 kip  in
V max  43 kips
with V  12.2 kips
• Calculate required section modulus
and select appropriate beam section.
M max 24 kip  in
S min 

 119.7 in 3
 all
24 ksi
select W21 62 beam section
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Sample Problem 8.2
• Find maximum shearing stress.
Assuming uniform shearing stress in web,
V
43 kips
 max  max 
 5.12 ksi  14.5 ksi
Aweb 8.40 in 2
• Find maximum normal stress.
a 
M max
60 kip  in
 2873
 22.6 ksi
3
S
127in
y
9.88
σb   a b  22.6 ksi 
 21.3 ksi
c
10.5
b 
V
12.2 kips

 1.45 ksii
Aweb 8.40 in 2
2
21.3 ksi
 21.3 ksi 
2
 max 
 
  1.45 ksi 
2
 2 
 21.4 ksi  24 ksi
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Design of a Transmission Shaft
• If power is transferred to and from the
shaft by gears or sprocket wheels, the
shaft is subjected to transverse loading
as well as shear loading.
• Normal stresses due to transverse loads
may be large and should be included in
determination of maximum shearing
stress.
• Shearing stresses due to transverse
loads are usually small and
contribution to maximum shear stress
may be neglected.
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Design of a Transmission Shaft
• At any section,
Mc
I
Tc
m 
J
m 
where M 2  M y2  M z2
• Maximum shearing stress,
2
2
 
 Mc   Tc 
 max   m    m 2  
  
 2 
 2I   J 
2
for a circular or annular cross - section, 2 I  J
 max 
c
M2 T2
J
• Shaft section requirement,
 M 2  T 2 

 max
J

 
 all
 c  min
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Sample Problem 8.3
SOLUTION:
• Determine the gear torques and
corresponding tangential forces.
• Find reactions at A and B.
• Identify critical shaft section from
torque and bending moment diagrams.
Solid shaft rotates at 480 rpm and
transmits 30 kW from the motor to
gears G and H; 20 kW is taken off at
gear G and 10 kW at gear H. Knowing
that sall = 50 MPa, determine the
smallest permissible diameter for the
shaft.
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• Calculate minimum allowable shaft
diameter.
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Sample Problem 8.3
SOLUTION:
• Determine the gear torques and corresponding
tangential forces.
TE 
P
30 kW

 597 N  m
2f 2 80 Hz 
T
597 N  m
FE  E 
 3.73 kN
rE
0.16 m
TC 
20 kW
 398 N  m
2 80 Hz 
FC  6.63 kN
TD 
10 kW
 199 N  m
2 80 Hz 
FD  2.49 kN
• Find reactions at A and B.
Ay  0.932 kN
Az  6.22 kN
B y  2.80 kN
Bz  2.90 kN
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Sample Problem 8.3
• Identify critical shaft section from torque and
bending moment diagrams.
 M 2  T 2 


 max
11602  3732  5972
 1357 N  m
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Sample Problem 8.3
• Calculate minimum allowable shaft diameter.
J
M 2  T 2 1357 N  m


 27.14  106 m3
c
 all
50 MPa
For a solid circular shaft,
J  3
 c  27.14  106 m3
c 2
c  0.02585 m  25.85 m
d  2c  51.7 mm
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Stresses Under Combined Loadings
• Wish to determine stresses in slender
structural members subjected to
arbitrary loadings.
• Pass section through points of interest.
Determine force-couple system at
centroid of section required to maintain
equilibrium.
• System of internal forces consist of
three force components and three
couple vectors.
• Determine stress distribution by
applying the superposition principle.
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Stresses Under Combined Loadings
• Axial force and in-plane couple vectors
contribute to normal stress distribution
in the section.
• Shear force components and twisting
couple contribute to shearing stress
distribution in the section.
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Stresses Under Combined Loadings
• Normal and shearing stresses are used to
determine principal stresses, maximum
shearing stress and orientation of principal
planes.
• Analysis is valid only to extent that
conditions of applicability of superposition
principle and Saint-Venant’s principle are
met.
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Sample Problem 8.5
SOLUTION:
• Determine internal forces in Section
EFG.
• Evaluate normal stress at H.
• Evaluate shearing stress at H.
Three forces are applied to a short
steel post as shown. Determine the
principle stresses, principal planes and
maximum shearing stress at point H.
• Calculate principal stresses and
maximum shearing stress.
Determine principal planes.
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Sample Problem 8.5
SOLUTION:
• Determine internal forces in Section EFG.
Vx  30 kN P  50 kN Vz  75 kN
M x  50 kN 0.130 m   75 kN 0.200 m 
 8.5 kN  m
M y  0 M z  30 kN 0.100 m   3 kN  m
Note: Section properties,
A  0.040 m 0.140 m   5.6  103 m 2
1 0.040 m 0.140 m 3  9.15  10 6 m 4
I x  12
1 0.140 m 0.040 m 3  0.747  10 6 m 4
I z  12
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Sample Problem 8.5
• Evaluate normal stress at H.
y 

P Mz a Mx b


A
Iz
Ix
3 kN  m 0.020 m 

5.6  10-3 m 2
0.747  106 m 4
50 kN

8.5 kN  m 0.025 m 
9.15  106 m 4
 8.93  80.3  23.2  MPa  66.0 MPa
• Evaluate shearing stress at H.
Q  A1 y1  0.040 m 0.045 m 0.0475 m 
 85.5  106 m3



Vz Q
75 kN  85.5  106 m3
 yz 

I xt
9.15  106 m 4 0.040 m 


 17.52 MPa
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Sample Problem 8.5
• Calculate principal stresses and maximum
shearing stress.
Determine principal planes.
 max  R  33.02  17.522  37.4 MPa
 max  OC  R  33.0  37.4  70.4 MPa
 min  OC  R  33.0  37.4  7.4 MPa
tan 2 p 
CY 17.52

2 p  27.96
CD 33.0
 p  13.98
 max  37.4 MPa
 max  70.4 MPa
 min  7.4 MPa
 p  13.98
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